Stability Results for a Coupled System of Impulsive Fractional Differential Equations

Abstract

In this paper, we establish sufficient conditions for the existence, uniqueness and Ulam–Hyers stability of the solutions of a coupled system of nonlinear fractional impulsive differential equations. The existence and uniqueness results are carried out via Banach contraction principle and Schauder’s fixed point theorem. The main theoretical results are well illustrated with the help of an example.

1. Introduction

Fractional differential equations (FDEs) provide an excellent tool for the description of memory and hereditary properties of different processes and materials. Thus, contrary to the classical derivative, the fractional derivative is nonlocal. Fractional calculus has played a very important role in enhancing the mathematical modeling of several phenomena occurring in engineering and scientific disciplines, such as blood flow systems, control theory, aerodynamics, the nonlinear oscillation of earthquake, the fluid-dynamic traffic model, polymer rheology, regular variation in thermodynamics, etc. FDEs are more accurate than the integer-order derivatives. Therefore, in the last few decades, fractional calculus has received great attention from researchers [1,2,3,4,5,6,7,8,9,10,11,12,13]. On the other hand, it is impossible to describe the complicated systems and processes with a single differential equation. Therefore, the coupled systems involving FDEs have also received incredible attention; consequently, many results are devoted to them [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31].

It is well known that the effects of a pulse cannot be ignored in many processes and phenomena. For example, in biological systems such as heart beats, blood flows, mechanical systems with impact, population dynamical systems and so on. Thus, researchers used differential equations with impulses to describe the aforesaid kinds of phenomena. Therefore, many mathematicians studied impulsive FDEs with different boundary conditions; see [32,33,34,35,36,37,38,39,40] and references cited therein.

In fields such as numerical analysis, optimization theory, and nonlinear analysis, we mostly deal with the approximate solutions and hence we need to check how close these solutions are to the actual solutions of the related system. For this purpose, many approaches can be used, but the approach of Ulam–Hyers stability is a simple and easy one. The aforesaid stability was first initiated by Ulam in 1940 and then was confirmed by Hyers in 1941 [41,42]. That’s why this stability is known as Ulam–Hyers stability. In 1978 [43], Rassias generalized the Ulam–Hyers stability by considering variables. Thereafter, mathematicians extended the work mentioned above to functional, differential, integrals and FDEs; for more information about the topic, the reader is recommended to [44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59].

Inspired from the above discussion, in this article, we study the existence, uniqueness and stability analysis of a coupled system of nonlinear FDEs with impulses of the form:

$$\left\{\begin{array}{c}{}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\alpha }}\mathrm{x}\left(\mathrm{t}\right)+\mathrm{h}\left(\mathrm{t}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{t}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{t}\right)\right)=0,\mathrm{t}\ne {\mathrm{t}}_{\mathrm{m}},\mathrm{m}=1,2\dots ,\mathrm{n},\hfill \\ {}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\beta }}\mathrm{y}\left(\mathrm{t}\right)+\mathrm{w}\left(\mathrm{t}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{t}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{t}\right)\right)=0,\mathrm{t}\ne {\mathrm{t}}_{\mathrm{m}},\mathrm{m}=1,2\dots ,\mathrm{n},\hfill \\ \Delta \mathrm{x}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{M}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right),\Delta {\mathrm{x}}^{\prime }{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{N}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right),\Delta {\mathrm{x}}^{″}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{O}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right),\hfill \\ \Delta \mathrm{y}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{M}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right),\Delta {\mathrm{y}}^{\prime }{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{N}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right),\Delta {\mathrm{y}}^{″}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{O}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right),\hfill \\ \mathrm{x}\left(0\right)={\mathrm{x}}^{\prime }\left(0\right)=0,{}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\epsilon }}\mathrm{x}\left(\mathsf{\Omega }\right)={\mathrm{x}}^{″}\left(1\right),\hfill \\ \mathrm{y}\left(0\right)={\mathrm{y}}^{\prime }\left(0\right)=0,{}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\rho }}\mathrm{y}\left(\mathsf{\Phi }\right)={\mathrm{y}}^{″}\left(1\right),\hfill \end{array}\right.$$

(1)

where $\mathrm{t}\in \mathrm{J}=[0,1],2<\mathsf{\alpha },\mathsf{\beta }\le 3,0<\mathrm{a},\mathrm{b},\mathsf{\epsilon },\mathsf{\Omega },\mathsf{\rho },\mathsf{\Phi }<1.$ ${}^{\mathrm{c}}\mathbf{D}$ stands for Caputo fractional derivative and $\mathrm{h},\mathrm{w}:\mathrm{J}\times {\mathbb{R}}^3\to \mathbb{R}$ are continuous functions. ${\mathrm{M}}_{1\mathrm{m}},{\mathrm{M}}_{2\mathrm{m}},{\mathrm{N}}_{1\mathrm{m}},{\mathrm{N}}_{2\mathrm{m}},{\mathrm{O}}_{1\mathrm{m}},{\mathrm{O}}_{2\mathrm{m}}\in \mathrm{C}(\mathbb{R},\mathbb{R})$ and ${\mathrm{t}}_{\mathrm{m}}$ satisfied $0={\mathrm{t}}_0<{\mathrm{t}}_1<\cdots <{\mathrm{t}}_{\mathrm{n}}<{\mathrm{t}}_{\mathrm{n}+1}=1$, $\Delta \mathrm{x}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}=\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}^{+}\right)-\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}^{-}\right)$, $\Delta {\mathrm{x}}^{\prime }{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{x}}^{\prime }\left({\mathrm{t}}_{\mathrm{m}}^{+}\right)-{\mathrm{x}}^{\prime }\left({\mathrm{t}}_{\mathrm{m}}^{-}\right)$, $\Delta {\mathrm{x}}^{″}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{x}}^{″}\left({\mathrm{t}}_{\mathrm{m}}^{+}\right)-{\mathrm{x}}^{″}\left({\mathrm{t}}_{\mathrm{m}}^{-}\right)$, $\Delta \mathrm{y}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}=\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}^{+}\right)-\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}^{-}\right)$, $\Delta {\mathrm{y}}^{\prime }{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{y}}^{\prime }\left({\mathrm{t}}_{\mathrm{m}}^{+}\right)-{\mathrm{y}}^{\prime }\left({\mathrm{t}}_{\mathrm{m}}^{-}\right)$, ${\mathrm{y}}^{″}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{y}}^{″}\left({\mathrm{t}}_{\mathrm{m}}^{+}\right)-{\mathrm{y}}^{″}\left({\mathrm{t}}_{\mathrm{m}}^{-}\right)$, $\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}^{+}\right),\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}^{+}\right)$, and $\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}^{-}\right),\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}^{-}\right)$ represent the right and left limits of $\mathrm{x}\left(\mathrm{t}\right),\mathrm{y}\left(\mathrm{t}\right)$, respectively, at $\mathrm{t}={\mathrm{t}}_{\mathrm{m}}$.

The remaining article is organized as follows: In Section 2, we give some definitions and lemmas related to fractional calculus. In Section 3, we establish our main results about the existence and uniqueness of solutions for the proposed system (1). In Section 4, we study the Ulam–Hyers stability. In Section 5, we provide an example to support our main results.

2. Background Materials

In this section, we give some basic definitions of fractional calculus that will be used throughout the article.

Definition 1.

(see [60]) If $\mathrm{x}:(0,\infty )\to \mathbb{R}$ and $\mathsf{\alpha }>0,$ then the Caputo fractional derivative of order α is defined as

$${}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\alpha }}\mathrm{x}\left(\mathrm{t}\right)=\frac{1}{\mathsf{\Gamma }(\mathrm{n}-\mathsf{\alpha })}{\int }_0^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathrm{n}-\mathsf{\alpha }-1}{\mathrm{x}}^{\left(\mathrm{n}\right)}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s},\mathrm{n}-1<\mathsf{\alpha }<\mathrm{n},\mathrm{n}=\left[\mathsf{\alpha }\right]+1,$$

where $\left[\mathsf{\alpha }\right]$ denotes the integer part of real number $\mathsf{\alpha },$ provided that the right side is pointwise defined on $(0,\infty ).$

Definition 2.

(see [60]) The Riemann–Liouville fractional integral of order $\mathsf{\alpha }>0$ for a function $\mathrm{x}:(0,\infty )\to \mathbb{R}$ is defined as

$${\mathbf{I}}^{\mathsf{\alpha }}\mathrm{x}\left(\mathrm{t}\right)=\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_0^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\alpha }-1}\mathrm{x}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s},\mathrm{t}>0,$$

provided that the right side is pointwise defined on $(0,\infty )$, where Γ is the Euler Gamma function.

Lemma 1.

(see [60]) The solution of the differential equations involving Caputo derivative ${}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\alpha }}\mathrm{x}\left(\mathrm{t}\right)=\mathrm{f}\left(\mathrm{t}\right),$ $\mathrm{t}\in \mathrm{J},$ has the form:

$${\mathbf{I}}^{\mathsf{\alpha }}{}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\alpha }}\mathrm{x}\left(\mathrm{t}\right)={\mathbf{I}}^{\mathsf{\alpha }}\mathrm{f}\left(\mathrm{t}\right)+{\mathrm{e}}_0+{\mathrm{e}}_1\mathrm{t}+\cdots +{\mathrm{e}}_{\mathrm{n}-1}{\mathrm{t}}^{\mathrm{n}-1},$$

for some ${\mathrm{e}}_{\mathrm{i}}\in \mathbb{R},\mathrm{i}=0,1,\dots ,\mathrm{n}-1,\mathrm{n}=\left[\mathsf{\alpha }\right]+1$.

Lemma 2.

(see [60]) If $\mathsf{\alpha },\mathsf{\beta }>0,\mathrm{t}\in \mathrm{J},$ then, for $\mathrm{x}\left(\mathrm{t}\right)$, we have

$$\begin{array}{c}\hfill {}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\alpha }}{\mathbf{I}}^{\mathsf{\alpha }}\mathrm{x}\left(\mathrm{t}\right)=\mathrm{x}\left(\mathrm{t}\right),{\mathbf{I}}^{\mathsf{\alpha }}{\mathbf{I}}^{\mathsf{\beta }}\mathrm{x}\left(\mathrm{t}\right)={\mathbf{I}}^{\mathsf{\alpha }+\mathsf{\beta }}\mathrm{x}\left(\mathrm{t}\right).\end{array}$$

Lemma 3.

(Banach contraction principle, see [59]) If X is real Banach space and $\mathbf{W}:\mathrm{X}⟶\mathrm{X}$ is a contraction mapping, then $\mathbf{W}$ has a unique fixed point in X.

Theorem 1.

(Schauder fixed point theorem, see [59]) If ω is a closed bounded convex subset of a Banach space X and $\mathbf{W}:\mathsf{\omega }⟶\mathsf{\omega }$ is completely continuous, then $\mathbf{W}$ has at least one fixed point in ω.

For the sake of convenience, we introduce the Banach space as follows:

Let $\mathrm{J}=[0,1],{\mathrm{J}}^{\prime }=\mathrm{J}/\{{\mathrm{t}}_1,{\mathrm{t}}_2,\dots ,{\mathrm{t}}_{\mathrm{n}}\}$. Define the set by

$$\begin{array}{cc}\hfill \mathrm{X}=\mathrm{PC}\left(\mathrm{J}\right)=& \{\mathrm{x}\left(\mathrm{t}\right):\mathrm{x}\left(\mathrm{t}\right),{\mathrm{x}}^{\prime }\left(\mathrm{t}\right),{\mathrm{x}}^{″}\left(\mathrm{t}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{t}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{x}\left(\mathrm{t}\right)\in \mathrm{C}\left({\mathrm{J}}^{\prime }\right),\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}^{+}\right)\mathrm{and}\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}^{-}\right)\hfill \\ & \mathrm{exists}\mathrm{and}\mathrm{satisfying}\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}^{-}\right)=\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right),1\le \mathrm{m}\le \mathrm{n}\}.\hfill \end{array}$$

It is easy to verify that X is a Banach space equipped with the norm:

$${\parallel \mathrm{x}\parallel }_0=\max \left\{\underset{\mathrm{t}\in \mathrm{J}}{\sup }|\mathrm{x}\left(\mathrm{t}\right)|,\underset{\mathrm{t}\in \mathrm{J}}{\sup }|{\mathrm{x}}^{\prime }\left(\mathrm{t}\right)|,\underset{\mathrm{t}\in \mathrm{J}}{\sup }|{\mathrm{x}}^{″}\left(\mathrm{t}\right)|,\underset{\mathrm{t}\in \mathrm{J}}{\sup }{|}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{t}\right)|,\underset{\mathrm{t}\in \mathrm{J}}{\sup }{|}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{x}\left(\mathrm{t}\right)|\right\},\forall \mathrm{x}\left(\mathrm{t}\right)\in \mathrm{P}\mathrm{C}\left(\mathrm{J}\right).$$

Similarly, we can define a set $\mathrm{Y}=\mathrm{P}\mathrm{C}\left(\mathrm{J}\right)$, which is a Banach space endowed with the defined norm:

$${\parallel \mathrm{y}\parallel }_0=\max \left\{\underset{\mathrm{t}\in \mathrm{J}}{\sup }|\mathrm{y}\left(\mathrm{t}\right)|,\underset{\mathrm{t}\in \mathrm{J}}{\sup }|{\mathrm{y}}^{\prime }\left(\mathrm{t}\right)|,\underset{\mathrm{t}\in \mathrm{J}}{\sup }|{\mathrm{y}}^{″}\left(\mathrm{t}\right)|,\underset{\mathrm{t}\in \mathrm{J}}{\sup }{|}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{y}\left(\mathrm{t}\right)|,\underset{\mathrm{t}\in \mathrm{J}}{\sup }{|}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{t}\right)|\right\},\forall \mathrm{y}\left(\mathrm{t}\right)\in \mathrm{P}\mathrm{C}\left(\mathrm{J}\right).$$

Furthermore, we define the Banach space ${\mathrm{Y}}^{\prime }=\mathrm{X}\times \mathrm{Y}$ with the norms $\parallel (\mathrm{x},\mathrm{y})\parallel ={\parallel \mathrm{x}\parallel }_0+{\parallel \mathrm{y}\parallel }_0$ and $\parallel (\mathrm{x},\mathrm{y})\parallel =\max \left\{{\parallel \mathrm{x}\parallel }_0,{\parallel \mathrm{y}\parallel }_0\right\}.$

Definition 3.

A pair of functions $(\mathrm{x}\left(\mathrm{t}\right),\mathrm{y}\left(\mathrm{t}\right))\in {\mathrm{Y}}^{\prime }$ is called a solution of (1) if $\left(\mathrm{x}\right(\mathrm{t}),\mathrm{y}(\mathrm{t}\left)\right)$ satisfy all the equations and boundary value conditions of the system (1).

Lemma 4.

Assume that $\mathrm{f}\in \mathrm{C}(\mathrm{J},\mathbb{R})$. A function $\mathrm{x}\in \mathrm{P}\mathrm{C}\left(\mathrm{J}\right)$ is a solution of the boundary value system

$$\left\{\begin{array}{c}{}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\alpha }}\mathrm{x}\left(\mathrm{t}\right)+\mathrm{f}\left(\mathrm{t}\right)=0,2<\mathsf{\alpha }\le 3,\hfill \\ \Delta \mathrm{x}{\mid }_{\mathrm{t}=\mathrm{t}\mathrm{m}}={\mathrm{M}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right),\mathrm{m}=1,2,\dots ,\mathrm{n},\hfill \\ \Delta {\mathrm{x}}^{\prime }{\mid }_{\mathrm{t}=\mathrm{t}\mathrm{m}}={\mathrm{N}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right),\mathrm{m}=1,2,\dots ,\mathrm{n},\hfill \\ \Delta {\mathrm{x}}^{″}{\mid }_{\mathrm{t}=\mathrm{t}\mathrm{m}}={\mathrm{O}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right),\mathrm{m}=1,2,\dots ,\mathrm{n},\hfill \\ \mathrm{x}\left(0\right)={\mathrm{x}}^{\prime }\left(0\right)=0,{}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\epsilon }}\mathrm{x}\left(\mathsf{\Omega }\right)={\mathrm{x}}^{″}\left(1\right),0<\mathsf{\epsilon },\mathsf{\Omega }<1,\hfill \end{array}\right.$$

(2)

if and only if $\mathrm{x}\in \mathrm{P}\mathrm{C}\left(\mathrm{J}\right)$ is the solution of integral equation

$$\mathrm{x}\left(\mathrm{t}\right)=\left\{\begin{array}{c}-\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_0^{\mathrm{t}}(\mathrm{t}-\mathrm{s})\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+{\mathrm{e}}^{\ast }{\mathrm{t}}^2,\mathrm{t}\in [0,{\mathrm{t}}_1],\hfill \\ -\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\alpha }-1}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-1}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ -\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\sum _{\mathrm{j}=1}^{\mathrm{m}}(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}}){\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-2}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\frac{1}{2\mathsf{\Gamma }(\mathsf{\alpha }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})}^2\hfill \\ \times {\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{M}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+\sum _{\mathrm{j}=1}^{\mathrm{m}}(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}}){\mathrm{N}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)\hfill \\ +\sum _{\mathrm{j}=1}^{\mathrm{m}}\frac{{(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})}^2}{2}{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\mathrm{e}}^{\ast }{\mathrm{t}}^2,\mathrm{t}\in ({\mathrm{t}}_{\mathrm{m}},{\mathrm{t}}_{\mathrm{m}+1}],1\le \mathrm{m}\le \mathrm{n},\hfill \end{array}\right.$$

(3)

where $\mathrm{t}\in ({\mathrm{t}}_{\mathrm{m}},{\mathrm{t}}_{\mathrm{m}+1}],1\le \mathrm{m}\le \mathrm{n}$,

$$\begin{array}{cc}\hfill {\mathrm{e}}^{\ast }=& \frac{\mathsf{\Gamma }(3-\mathsf{\epsilon })}{2(\mathsf{\Gamma }(3-\mathsf{\epsilon })-{\mathsf{\Omega }}^{2-\mathsf{\epsilon }})}[-\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon })}{\int }_{{\mathrm{t}}_{\mathrm{j}}}^{\mathsf{\Omega }}{(\mathsf{\Omega }-\mathrm{s})}^{\mathsf{\alpha }-\mathsf{\epsilon }-1}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\hfill \\ \hfill & \times \sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-2}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\frac{{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{N}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)\hfill \\ \hfill & +\frac{{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^1{(1-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)].\hfill \end{array}$$

Proof. 

Applying Lemma 1, for some constants ${\mathrm{e}}_0,{\mathrm{e}}_1,{\mathrm{e}}_2\in \mathbb{R}$, we have

$$\begin{array}{cc}\hfill \mathrm{x}\left(\mathrm{t}\right)& =-{\mathbf{I}}^{\mathsf{\alpha }}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+{\mathrm{e}}_0+{\mathrm{e}}_1\mathrm{t}+{\mathrm{e}}_2{\mathrm{t}}^2\hfill \\ \hfill & =-\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_0^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\alpha }-1}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+{\mathrm{e}}_0+{\mathrm{e}}_1\mathrm{t}+{\mathrm{e}}_2{\mathrm{t}}^2,\mathrm{t}\in [0,{\mathrm{t}}_1].\hfill \end{array}$$

(4)

Then, we obtain

$$\left\{\begin{array}{c}{\mathrm{x}}^{\prime }\left(\mathrm{t}\right)=-\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}{\int }_0^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\alpha }-2}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+{\mathrm{e}}_1+2{\mathrm{e}}_2\mathrm{t},\hfill \\ {\mathrm{x}}^{″}\left(\mathrm{t}\right)=-\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}{\int }_0^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+2{\mathrm{e}}_2.\hfill \end{array}\right.$$

(5)

When $\mathrm{t}\in ({\mathrm{t}}_1,{\mathrm{t}}_2)$, we have

$$\left\{\begin{array}{c}\mathrm{x}\left(\mathrm{t}\right)=-\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_{{\mathrm{t}}_1}^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\alpha }-1}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+{\mathrm{e}}_3+{\mathrm{e}}_4(\mathrm{t}-{\mathrm{t}}_1)+{\mathrm{e}}_5{(\mathrm{t}-{\mathrm{t}}_1)}^2,\hfill \\ {\mathrm{x}}^{\prime }\left(\mathrm{t}\right)=-\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}{\int }_{{\mathrm{t}}_1}^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\alpha }-2}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+{\mathrm{e}}_4+2{\mathrm{e}}_5(\mathrm{t}-{\mathrm{t}}_1),\hfill \\ {\mathrm{x}}^{″}\left(\mathrm{t}\right)=-\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}{\int }_{{\mathrm{t}}_1}^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+2{\mathrm{e}}_5,\hfill \end{array}\right.$$

(6)

where ${\mathrm{e}}_3,{\mathrm{e}}_4,{\mathrm{e}}_5$ are arbitrary constants, from (4)–(6), we can find

$$\begin{array}{cc}\hfill \mathrm{x}\left({\mathrm{t}}_1^{-}\right)& =-\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_0^{{\mathrm{t}}_1}{({\mathrm{t}}_1-\mathrm{s})}^{\mathsf{\alpha }-1}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+{\mathrm{e}}_0+{\mathrm{e}}_1{\mathrm{t}}_1+{\mathrm{e}}_2{\mathrm{t}}_1^2,\mathrm{x}\left({\mathrm{t}}_1^{+}\right)={\mathrm{e}}_3,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\mathrm{x}}^{\prime }\left({\mathrm{t}}_1^{-}\right)& =-\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}{\int }_0^{{\mathrm{t}}_1}{({\mathrm{t}}_1-\mathrm{s})}^{\mathsf{\alpha }-2}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+{\mathrm{e}}_1+2{\mathrm{e}}_2{\mathrm{t}}_1,{\mathrm{x}}^{\prime }\left({\mathrm{t}}_1^{+}\right)={\mathrm{e}}_4,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill {\mathrm{x}}^{″}\left({\mathrm{t}}_1^{-}\right)& =-\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}{\int }_0^{{\mathrm{t}}_1}{({\mathrm{t}}_1-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+2{\mathrm{e}}_2,{\mathrm{x}}^{″}\left({\mathrm{t}}_1^{+}\right)=2{\mathrm{e}}_5.\hfill \end{array}$$

(9)

Furthermore, $\Delta \mathrm{x}{\mid }_{\mathrm{t}={\mathrm{t}}_1}={\mathrm{M}}_{11}\left(\mathrm{x}\left({\mathrm{t}}_1\right)\right),\Delta {\mathrm{x}}^{\prime }{\mid }_{\mathrm{t}={\mathrm{t}}_1}={\mathrm{N}}_{11}\left(\mathrm{x}\left({\mathrm{t}}_1\right)\right),\Delta {\mathrm{x}}^{″}{\mid }_{\mathrm{t}={\mathrm{t}}_1}={\mathrm{O}}_{11}\left(\mathrm{x}\left({\mathrm{t}}_1\right)\right)$, and (7)–(9) give us:

$$\left\{\begin{array}{c}{\mathrm{e}}_3=-\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_0^{{\mathrm{t}}_1}{({\mathrm{t}}_1-\mathrm{s})}^{\mathsf{\alpha }-1}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+{\mathrm{e}}_0+{\mathrm{e}}_1{\mathrm{t}}_1+{\mathrm{e}}_2{\mathrm{t}}_1^2+{\mathrm{M}}_{11}\left(\mathrm{x}\left({\mathrm{t}}_1\right)\right),\hfill \\ {\mathrm{e}}_4=-\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}{\int }_0^{{\mathrm{t}}_1}{({\mathrm{t}}_1-\mathrm{s})}^{\mathsf{\alpha }-2}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+{\mathrm{e}}_1+2{\mathrm{e}}_2{\mathrm{t}}_1+{\mathrm{N}}_{11}\left(\mathrm{x}\left({\mathrm{t}}_1\right)\right),\hfill \\ {\mathrm{e}}_5=-\frac{1}{2\mathsf{\Gamma }(\mathsf{\alpha }-2)}{\int }_0^{{\mathrm{t}}_1}{({\mathrm{t}}_1-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+{\mathrm{e}}_2+\frac{1}{2}{\mathrm{O}}_{11}\left(\mathrm{x}\left({\mathrm{t}}_1\right)\right).\hfill \end{array}\right.$$

Plugging ${\mathrm{e}}_3,{\mathrm{e}}_4,\mathrm{a}\mathrm{n}\mathrm{d}{\mathrm{e}}_5$ into the first equation of (6) for $\mathrm{t}\in ({\mathrm{t}}_1,{\mathrm{t}}_2]$, we have

$$\begin{array}{cc}\hfill \mathrm{x}\left(\mathrm{t}\right)=& -\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_{{\mathrm{t}}_1}^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\alpha }-1}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_0^{{\mathrm{t}}_1}({\mathrm{t}}_1-\mathrm{s})\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\frac{(\mathrm{t}-{\mathrm{t}}_1)}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}{\int }_0^{{\mathrm{t}}_1}{({\mathrm{t}}_1-\mathrm{s})}^{\mathsf{\alpha }-2}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ & -\frac{{(\mathrm{t}-{\mathrm{t}}_1)}^2}{2\mathsf{\Gamma }(\mathsf{\alpha }-2)}{\int }_0^{{\mathrm{t}}_1}{({\mathrm{t}}_1-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+{\mathrm{M}}_{11}\left(\mathrm{x}\left({\mathrm{t}}_1\right)\right)+(\mathrm{t}-{\mathrm{t}}_1){\mathrm{N}}_{11}\left(\mathrm{x}\left({\mathrm{t}}_1\right)\right)+\frac{{(\mathrm{t}-{\mathrm{t}}_1)}^2}{2}{\mathrm{O}}_{11}\left(\mathrm{x}\left({\mathrm{t}}_1\right)\right)\hfill \\ \hfill & +{\mathrm{e}}_0+{\mathrm{e}}_1\mathrm{t}+{\mathrm{e}}_2{\mathrm{t}}^2.\hfill \end{array}$$

Repeating the same process for $\mathrm{t}\in ({\mathrm{t}}_{\mathrm{m}},{\mathrm{t}}_{\mathrm{m}+1}]$ such that $(\mathrm{m}=1,2,\dots ,\mathrm{n})$, then we can write

$$\begin{array}{cc}\hfill \mathrm{x}\left(\mathrm{t}\right)=& -\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\alpha }-1}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-1}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\sum _{\mathrm{j}=1}^{\mathrm{m}}(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})\hfill \\ \hfill & \times {\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-2}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\frac{1}{2\mathsf{\Gamma }(\mathsf{\alpha }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})}^2{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{M}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)\hfill \\ \hfill & +\sum _{\mathrm{j}=1}^{\mathrm{m}}(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}}){\mathrm{N}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+\sum _{\mathrm{j}=1}^{\mathrm{m}}\frac{{(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})}^2}{2}{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\mathrm{e}}_0+{\mathrm{e}}_1\mathrm{t}+{\mathrm{e}}_2{\mathrm{t}}^2.\hfill \end{array}$$

(10)

Furthermore, we have

$$\begin{array}{cc}{\mathrm{x}}^{″}\left(\mathrm{t}\right)=\hfill & -{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)\hfill \\ \hfill & +2{\mathrm{e}}_2.\hfill \end{array}$$

(11)

By utilizing conditions $\mathrm{x}\left(0\right)={\mathrm{x}}^{\prime }\left(0\right)=0$ in (4), we get ${\mathrm{e}}_0={\mathrm{e}}_1=0.$ In addition, it follows from (11) that

$$\begin{array}{cc}{\mathrm{x}}^{″}\left(1\right)=\hfill & -{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^1{(1-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)\hfill \\ \hfill & +2{\mathrm{e}}_2.\hfill \end{array}$$

(12)

In view of $\mathrm{j}\in \{0,1,\dots ,\mathrm{n}\}$ such that $\mathsf{\Omega }\in ({\mathrm{t}}_{\mathrm{j}},{\mathrm{t}}_{\mathrm{j}+1}]$, we have

$$\begin{array}{cc}\hfill \mathrm{x}\left(\mathsf{\Omega }\right)=& -\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_{{\mathrm{t}}_{\mathrm{j}}}^{\mathsf{\Omega }}{(\mathsf{\Omega }-\mathrm{s})}^{\mathsf{\alpha }-1}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-1}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\sum _{\mathrm{j}=1}^{\mathrm{m}}(\mathsf{\Omega }-{\mathrm{t}}_{\mathrm{j}})\hfill \\ \hfill & \times {\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-2}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\frac{1}{2\mathsf{\Gamma }(\mathsf{\alpha }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{(\mathsf{\Omega }-{\mathrm{t}}_{\mathrm{j}})}^2{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{M}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+\sum _{\mathrm{j}=1}^{\mathrm{m}}(\mathsf{\Omega }-{\mathrm{t}}_{\mathrm{j}}){\mathrm{N}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+\frac{{(\mathsf{\Omega }-{\mathrm{t}}_{\mathrm{j}})}^2}{2}{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\mathrm{e}}_2{\mathsf{\Omega }}^2.\hfill \end{array}$$

(13)

By applying result (5), we get

$$\begin{array}{cc}{}^{\mathrm{c}}{\mathrm{D}}_{0^{+}}^{\mathsf{\epsilon }}\mathrm{x}\left(\mathsf{\Omega }\right)=\hfill & -{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon })}}{\int }_{{\mathrm{t}}_{\mathrm{j}}}^{\mathsf{\Omega }}{(\mathsf{\Omega }-\mathrm{s})}^{\mathsf{\alpha }-\mathsf{\epsilon }-1}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-{\displaystyle \frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-2}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & -{\displaystyle \frac{{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(3-\mathsf{\epsilon })}}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+{\displaystyle \frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(2-\mathsf{\epsilon })}}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}\hfill \\ \hfill & \times {\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+{\displaystyle \frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{N}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+\frac{{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{O}}_{1\mathrm{j}}\hfill \\ \hfill & -{\displaystyle \frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}{\mathrm{O}}_{1\mathrm{j}}+{\displaystyle \frac{2{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}}{\mathrm{e}}_2.\hfill \end{array}$$

(14)

Since ${}^{\mathrm{c}}{\mathrm{D}}_{0^{+}}^{\mathsf{\epsilon }}\mathrm{x}\left(\mathsf{\Omega }\right)={\mathrm{x}}^{″}\left(1\right),$ thus (13) and (14) gives

$$\begin{array}{cc}\hfill {\mathrm{e}}_2=& \frac{\mathsf{\Gamma }(3-\mathsf{\epsilon })}{2(\mathsf{\Gamma }(3-\mathsf{\epsilon })-{\mathsf{\Omega }}^{2-\mathsf{\epsilon }})}[-\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon })}{\int }_{{\mathrm{t}}_{\mathrm{j}}}^{\mathsf{\Omega }}{(\mathsf{\Omega }-\mathrm{s})}^{\mathsf{\alpha }-\mathsf{\epsilon }-1}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\hfill \\ \hfill & \times \sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-2}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\frac{{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{N}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)\hfill \\ \hfill & +\frac{{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{O}}_{1\mathrm{j}}-\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}{\mathrm{O}}_{1\mathrm{j}}+\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^1{(1-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{f}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)].\hfill \end{array}$$

Plugging the values of ${\mathrm{e}}_0,{\mathrm{e}}_1$ and ${\mathrm{e}}_2$ into (4) and (10), (3) can thus be obtained. Conversely, we consider that $\mathrm{x}\left(\mathrm{t}\right)$ is a solution of (3). Then, it is obvious that (3) satisfies (2). □

Similarly as in Lemma 4, we can prove the following:

Lemma 5.

Let $\mathsf{\vartheta }\in \mathrm{C}(\mathrm{J},\mathbb{R})$. A function $\mathrm{y}\in \mathrm{P}\mathrm{C}\left(\mathrm{J}\right)$ is the solution of

$$\left\{\begin{array}{c}{}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\beta }}\mathrm{y}\left(\mathrm{t}\right)+\mathsf{\vartheta }\left(\mathrm{t}\right)=0,2<\mathsf{\beta }\le 3,\hfill \\ \Delta \mathrm{y}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{M}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right),\mathrm{m}=1,2,\dots ,\mathrm{n},\hfill \\ \Delta {\mathrm{y}}^{\prime }{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{N}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right),\mathrm{m}=1,2,\dots ,\mathrm{n},\hfill \\ \Delta {\mathrm{y}}^{″}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{O}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right),\mathrm{m}=1,2,\dots ,\mathrm{n},\hfill \\ \mathrm{y}\left(0\right)={\mathrm{y}}^{\prime }\left(0\right)=0,{}^{\mathrm{c}}{\mathbf{D}}_{0^{+}}^{\mathsf{\rho }}\mathrm{y}\left(\mathsf{\Phi }\right)={\mathrm{y}}^{″}\left(1\right),0<\mathsf{\rho },\mathsf{\Phi }<1,\hfill \end{array}\right.$$

(15)

if and only if $\mathrm{y}\in \mathrm{P}\mathrm{C}\left(\mathrm{J}\right)$ is the solution of the integral equation

$$\mathrm{y}\left(\mathrm{t}\right)=\left\{\begin{array}{c}-\frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}{\int }_0^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\beta }-1}\mathsf{\vartheta }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+{\mathrm{c}}^{\ast }{\mathrm{t}}^2,\mathrm{t}\in [0,{\mathrm{t}}_1],\hfill \\ -\frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\beta }-1}\mathsf{\vartheta }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\beta }-1}\mathsf{\vartheta }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ -\frac{1}{\mathsf{\Gamma }(\mathsf{\beta }-1)}\sum _{\mathrm{j}=1}^{\mathrm{m}}(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}}){\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\beta }-2}\mathsf{\vartheta }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\frac{1}{2\mathsf{\Gamma }(\mathsf{\beta }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})}^2\hfill \\ \times {\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\beta }-3}\mathsf{\vartheta }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{M}}_{2\mathrm{j}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+\sum _{\mathrm{j}=1}^{\mathrm{m}}(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}}){\mathrm{N}}_{2\mathrm{j}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)\hfill \\ +\sum _{\mathrm{j}=1}^{\mathrm{m}}\frac{{(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})}^2}{2}{\mathrm{O}}_{2\mathrm{j}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\mathrm{c}}^{\ast }{\mathrm{t}}^2,\mathrm{t}\in ({\mathrm{t}}_{\mathrm{m}},{\mathrm{t}}_{\mathrm{m}+1}],1\le \mathrm{m}\le \mathrm{n},\hfill \end{array}\right.$$

(16)

where $\mathrm{c}\in \{0,1,\dots ,\mathrm{n}\}$ such that $\mathsf{\Phi }\in ({\mathrm{t}}_{\mathrm{c}},{\mathrm{t}}_{\mathrm{c}+1}]$, and

$$\begin{array}{cc}\hfill {\mathrm{c}}^{\ast }=& \frac{\mathsf{\Gamma }(3-\mathsf{\rho })}{2(\mathsf{\Gamma }(3-\mathsf{\rho })-{\mathsf{\Phi }}^{2-\mathsf{\rho }})}[-\frac{1}{\mathsf{\Gamma }(\mathsf{\beta }-\mathsf{\rho })}{\int }_{{\mathrm{t}}_{\mathrm{j}}}^{\mathsf{\Phi }}{(\mathsf{\Phi }-\mathrm{s})}^{\mathsf{\beta }-\mathsf{\rho }-1}\mathsf{\vartheta }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\frac{{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(2-\mathsf{\rho })}\hfill \\ \hfill & \times \sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\beta }-2}\mathsf{\vartheta }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\frac{{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-2)\mathsf{\Gamma }(3-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\beta }-3}\mathsf{\vartheta }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-2)\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\beta }-3}\mathsf{\vartheta }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+\frac{{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{N}}_{2\mathrm{j}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)\hfill \\ \hfill & +\frac{{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(3-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{O}}_{2\mathrm{j}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-\frac{{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}{\mathrm{O}}_{2\mathrm{j}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+\frac{1}{\mathsf{\Gamma }(\mathsf{\beta }-2)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^1{(1-\mathrm{s})}^{\mathsf{\beta }-3}\mathsf{\vartheta }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{1}{\mathsf{\Gamma }(\mathsf{\beta }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\beta }-3}\mathsf{\vartheta }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{O}}_{2\mathrm{j}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)].\hfill \end{array}$$

3. Main Results

In this section, we use fixed point theorems to prove the existence of solutions to problem (1). According to Lemmas 4 and 5, we define operator $\mathbf{W}:{\mathrm{Y}}^{\prime }⟶{\mathrm{Y}}^{\prime }$ by

$$\mathbf{W}(\mathrm{x},\mathrm{y})\left(\mathrm{t}\right)={\left({\mathbf{W}}_1(\mathrm{x},\mathrm{y})\left(\mathrm{t}\right),{\mathbf{W}}_2(\mathrm{x},\mathrm{y})\left(\mathrm{t}\right)\right)}^{\mathrm{T}},\forall (\mathrm{x},\mathrm{y})\in {\mathrm{Y}}^{\prime },\mathrm{t}\in [0,1],$$

(17)

where

$${\mathbf{W}}_1(\mathrm{x},\mathrm{y})\left(\mathrm{t}\right)=\left\{\begin{array}{c}-\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_0^{\mathrm{t}}(\mathrm{t}-\mathrm{s})\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}+{\mathrm{e}}^{\ast }{\mathrm{t}}^2,\mathrm{t}\in [0,{\mathrm{t}}_1],\hfill \\ -\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\alpha }-1}\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}\hfill \\ -\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-1}\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}\hfill \\ -\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\sum _{\mathrm{j}=1}^{\mathrm{m}}(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}}){\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-2}\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}\hfill \\ -\frac{1}{2\mathsf{\Gamma }(\mathsf{\alpha }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})}^2{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}\hfill \\ +\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{M}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+\sum _{\mathrm{j}=1}^{\mathrm{m}}(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}}){\mathrm{N}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+\sum _{\mathrm{j}=1}^{\mathrm{m}}\frac{{(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})}^2}{2}{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\mathrm{e}}^{\ast }{\mathrm{t}}^2,\hfill \\ \mathrm{t}\in ({\mathrm{t}}_{\mathrm{m}},{\mathrm{t}}_{\mathrm{m}+1}],1\le \mathrm{m}\le \mathrm{n},\hfill \end{array}\right.$$

and

$${\mathbf{W}}_2(\mathrm{x},\mathrm{y})\left(\mathrm{t}\right)=\left\{\begin{array}{c}-\frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}{\int }_0^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\beta }-1}\mathrm{w}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}+{\mathrm{c}}^{\ast }{\mathrm{t}}^2,\mathrm{t}\in [0,{\mathrm{t}}_1],\hfill \\ -\frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\beta }-1}\mathrm{w}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}\hfill \\ -\frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\beta }-1}\mathrm{w}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}\hfill \\ -\frac{1}{\mathsf{\Gamma }(\mathsf{\beta }-1)}\sum _{\mathrm{j}=1}^{\mathrm{m}}(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}}){\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\beta }-2}\mathrm{w}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}\hfill \\ -\frac{1}{2\mathsf{\Gamma }(\mathsf{\beta }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})}^2{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\beta }-3}\mathrm{w}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}\hfill \\ +\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{M}}_{2\mathrm{j}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+\sum _{\mathrm{j}=1}^{\mathrm{m}}(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}}){\mathrm{N}}_{2\mathrm{j}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+\sum _{\mathrm{j}=1}^{\mathrm{m}}\frac{{(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})}^2}{2}{\mathrm{O}}_{2\mathrm{j}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\mathrm{c}}^{\ast }{\mathrm{t}}^2,\hfill \\ \mathrm{t}\in ({\mathrm{t}}_{\mathrm{m}},{\mathrm{t}}_{\mathrm{m}+1}],1\le \mathrm{m}\le \mathrm{n},\hfill \end{array}\right.$$

with

$$\begin{array}{cc}\hfill {\mathrm{e}}^{\ast }=& {\mathcal{Z}}_{\mathsf{\alpha }}[-\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon })}{\int }_{{\mathrm{t}}_{\mathrm{j}}}^{\mathsf{\Omega }}{(\mathsf{\Omega }-\mathrm{s})}^{\mathsf{\alpha }-\mathsf{\epsilon }-1}\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & -\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-2}\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & -\frac{{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{N}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+\frac{{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)\hfill \\ \hfill & +\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^1{(1-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}-\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)],\hfill \\ \hfill {\mathrm{c}}^{\ast }=& {\mathcal{Z}}_{\mathsf{\beta }}[-\frac{1}{\mathsf{\Gamma }(\mathsf{\beta }-\mathsf{\rho })}{\int }_{{\mathrm{t}}_{\mathrm{j}}}^{\mathsf{\Phi }}{(\mathsf{\Phi }-\mathrm{s})}^{\mathsf{\beta }-\mathsf{\rho }-1}\mathrm{w}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & -\frac{{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\beta }-2}\mathrm{w}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & -\frac{{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-2)\mathsf{\Gamma }(3-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\beta }-3}\mathrm{w}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-2)\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\beta }-3}\mathrm{w}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s})\hfill \\ \hfill & +\frac{{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{N}}_{2\mathrm{j}}(\mathrm{y}\left({\mathrm{t}}_{\mathrm{j}}\right)+\frac{{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(3-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{O}}_{2\mathrm{j}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-\frac{{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}{\mathrm{O}}_{2\mathrm{j}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)\hfill \\ \hfill & +\frac{1}{\mathsf{\Gamma }(\mathsf{\beta }-2)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^1{(1-\mathrm{s})}^{\mathsf{\beta }-3}\mathrm{w}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{1}{\mathsf{\Gamma }(\mathsf{\beta }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\beta }-3}\mathrm{w}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}-\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{O}}_{2\mathrm{j}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)],\hfill \end{array}$$

where ${\mathcal{Z}}_{\mathsf{\alpha }}=\frac{\mathsf{\Gamma }(3-\mathsf{\epsilon })}{2(\mathsf{\Gamma }(3-\mathsf{\epsilon })-{\mathsf{\Omega }}^{2-\mathsf{\epsilon }})}$ and ${\mathcal{Z}}_{\mathsf{\beta }}=\frac{\mathsf{\Gamma }(3-\mathsf{\rho })}{2(\mathsf{\Gamma }(3-\mathsf{\rho })-{\mathsf{\Phi }}^{2-\mathsf{\rho }})}.$ Thus, solving problem (1) is equivalent to obtain a fixed point of the operator $\mathbf{W}$. Next, we have to prove the uniqueness of solutions of problem (1).

Theorem 2.

Let the following conditions $\left({\mathbf{M}}_1\right)-\left({\mathbf{M}}_3\right)$ hold, and then the boundary value problem (1) has a unique solution.

$\left({\mathbf{M}}_1\right):$ For all $\mathrm{t}\in \mathrm{J}$ and ${\mathrm{x}}_{\mathrm{j}},{\mathrm{y}}_{\mathrm{j}}\in \mathbb{R}(\mathrm{j}=1,2)$ there exists some positive constants ${\mathsf{\mu }}_{\mathrm{j}},{\mathsf{\mu }}_{\mathrm{j}}^{\prime }(\mathrm{j}=1,2)$ such that

$$\begin{array}{cc}\hfill |\mathrm{h}(\mathrm{t},{\mathrm{x}}_1,{\mathrm{y}}_1)-\mathrm{h}(\mathrm{t},{\mathrm{x}}_2,{\mathrm{y}}_2)|& \le {\mathsf{\mu }}_1|{\mathrm{x}}_1-{\mathrm{x}}_2|+{\mathsf{\mu }}_2|{\mathrm{y}}_1-{\mathrm{y}}_2|,\hfill \\ \hfill |\mathrm{w}(\mathrm{t},{\mathrm{x}}_1,{\mathrm{y}}_1)-\mathrm{w}(\mathrm{t},{\mathrm{x}}_2,{\mathrm{y}}_2)|& \le {\mathsf{\mu }}_1^{\prime }|{\mathrm{x}}_1-{\mathrm{x}}_2|+{\mathsf{\mu }}_2^{\prime }|{\mathrm{y}}_1-{\mathrm{y}}_2|.\hfill \end{array}$$

$\left({\mathbf{M}}_2\right):$ For all $\mathrm{x},\mathrm{y}\in \mathbb{R}$, there exist some positive constants ${\mathrm{I}}_{\mathrm{j}\mathrm{k}},{\widehat{\mathrm{I}}}_{\mathrm{j}\mathrm{k}},{\check{\mathrm{I}}}_{\mathrm{j}\mathrm{k}}(\mathrm{j}=1,2;\mathrm{m}=1,2,\dots ,\mathrm{n})$ such that

$$\begin{array}{cc}\hfill |{\mathrm{M}}_{\mathrm{j}\mathrm{m}}\left(\mathrm{x}\right)-{\mathrm{M}}_{\mathrm{j}\mathrm{m}}\left(\mathrm{y}\right)|& \le {\mathrm{I}}_{\mathrm{j}\mathrm{m}}|\mathrm{x}-\mathrm{y}|,\hfill \\ \hfill |{\mathrm{N}}_{\mathrm{j}\mathrm{m}}\left(\mathrm{x}\right)-{\mathrm{N}}_{\mathrm{j}\mathrm{m}}\left(\mathrm{y}\right)|& \le {\widehat{\mathrm{I}}}_{\mathrm{j}\mathrm{m}}|\mathrm{x}-\mathrm{y}|,\hfill \\ \hfill |{\mathrm{O}}_{\mathrm{j}\mathrm{m}}\left(\mathrm{x}\right)-{\mathrm{O}}_{\mathrm{j}\mathrm{m}}\left(\mathrm{y}\right)|& \le {\check{\mathrm{I}}}_{\mathrm{j}\mathrm{m}}|\mathrm{x}-\mathrm{y}|.\hfill \end{array}$$

$\left({\mathbf{M}}_3\right):$

$$\begin{array}{c}\left[\left({\mathsf{\mu }}_1+{\mathsf{\mu }}_2\right)\right(\frac{2}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}+\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}+\frac{1}{2\mathsf{\Gamma }(\mathsf{\alpha }-1)}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon }+1)}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\hfill \\ +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{2{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)})+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{I}}_{1\mathrm{j}}+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{1\mathrm{j}}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{1\mathrm{j}}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}\hfill \\ +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}+{\mathcal{Z}}_{\mathsf{\alpha }}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}+\frac{1}{2}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}]<1\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \left[\left({\mathsf{\mu }}_1^{\prime }+{\mathsf{\mu }}_2^{\prime }\right)\right(\frac{2}{\mathsf{\Gamma }(\mathsf{\beta }+1)}+\frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}+\frac{1}{2\mathsf{\Gamma }(\mathsf{\beta }-1)}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}}{\mathsf{\Gamma }(\mathsf{\beta }-\mathsf{\rho }+1)}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)\mathsf{\Gamma }(2-\mathsf{\rho })}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(3-\mathsf{\rho })}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(2-\mathsf{\rho })}+\frac{2{\mathcal{Z}}_{\mathsf{\beta }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)})+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{I}}_{2\mathrm{j}}+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{2\mathrm{j}}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{2\mathrm{j}}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(3-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}+{\mathcal{Z}}_{\mathsf{\beta }}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}+\frac{1}{2}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}]<1.\hfill \end{array}$$

Proof. 

By using the Banach contraction principle, we can prove that $\mathbf{W}$, defined by (17), has a fixed point. Before proving the main result first, we will prove the contraction. When $\mathrm{t}\in \mathrm{J}$, from (17) and conditions $\left({\mathbf{M}}_1\right)-\left({\mathbf{M}}_2\right)$, for all $({\mathrm{x}}_1,{\mathrm{y}}_1),({\mathrm{x}}_2,{\mathrm{y}}_2)\in {\mathrm{Y}}^{\prime }$, we have

$$\begin{array}{cc}\hfill & |{\mathbf{W}}_1({\mathrm{x}}_1,{\mathrm{y}}_1)\left(\mathrm{t}\right)-{\mathbf{W}}_1({\mathrm{x}}_2,{\mathrm{y}}_2)\left(\mathrm{t}\right)|\hfill \\ \hfill \le & \frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_0^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\alpha }-1}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_2\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon })}{\int }_{{\mathrm{t}}_{\mathrm{j}}}^{\mathsf{\Omega }}{(\mathsf{\Omega }-\mathrm{s})}^{\mathsf{\alpha }-\mathsf{\epsilon }-1}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_2\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-2}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_2\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_2\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_2\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{N}}_{1\mathrm{j}}\left({\mathrm{x}}_1\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{N}}_{1\mathrm{j}}\left({\mathrm{x}}_2\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{O}}_{1\mathrm{j}}\left({\mathrm{x}}_1\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{O}}_{1\mathrm{j}}\left({\mathrm{x}}_2\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^1{(1-\mathrm{s})}^{\mathsf{\alpha }-3}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_2\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_2\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{t}}_{\mathrm{j}}||{\mathrm{O}}_{1\mathrm{j}}\left({\mathrm{x}}_1\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{O}}_{1\mathrm{j}}\left({\mathrm{x}}_2\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|+{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{O}}_{1\mathrm{j}}\left({\mathrm{x}}_1\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{O}}_{1\mathrm{j}}\left({\mathrm{x}}_2\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|,\hfill \\ \hfill \le & \frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}\left[{\mathsf{\mu }}_1{|}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){-}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right)|+{\mathsf{\mu }}_2{|}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right){-}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_2\left(\mathrm{s}\right)|\right]\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon }+1)}\left[{\mathsf{\mu }}_1{|}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){-}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right)|+{\mathsf{\mu }}_2{|}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right){-}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_2\left(\mathrm{s}\right)|\right]\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\left[{\mathsf{\mu }}_1{|}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){-}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right)|+{\mathsf{\mu }}_2{|}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right){-}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_2\left(\mathrm{s}\right)|\right]\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\left[{\mathsf{\mu }}_1{|}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){-}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right)|+{\mathsf{\mu }}_2{|}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{y}}_1\left(\mathrm{s}\right){-}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{y}}_2\left(\mathrm{s}\right)|\right]\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\left[{\mathsf{\mu }}_1{|}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){-}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right)|+{\mathsf{\mu }}_2{|}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right){-}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_2\left(\mathrm{s}\right)|\right]\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{N}}_{1\mathrm{j}}\left({\mathrm{x}}_1\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{N}}_{1\mathrm{j}}\left({\mathrm{x}}_2\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{O}}_{1\mathrm{j}}\left({\mathrm{x}}_1\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{O}}_{1\mathrm{j}}\left({\mathrm{x}}_2\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\left[{\mathsf{\mu }}_1{|}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){-}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right)|+{\mathsf{\mu }}_2{|}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right){-}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_2\left(\mathrm{s}\right)|\right]\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\left[{\mathsf{\mu }}_1{|}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){-}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right)|+{\mathsf{\mu }}_2{|}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right){-}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_2\left(\mathrm{s}\right)|\right]\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}|{\mathrm{O}}_{1\mathrm{j}}\left({\mathrm{x}}_1\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{O}}_{1\mathrm{j}}\left({\mathrm{x}}_2\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|+{\mathcal{Z}}_{\mathsf{\alpha }}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{O}}_{1\mathrm{j}}\left({\mathrm{x}}_1\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{O}}_{1\mathrm{j}}\left({\mathrm{x}}_2\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|,\hfill \\ \hfill \le & \frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}\left[{\mathsf{\mu }}_1\parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_0+{\mathsf{\mu }}_2{\parallel {\mathrm{y}}_1-{\mathrm{y}}_2\parallel }_0\right]+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon }+1)}\left[{\mathsf{\mu }}_1\parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_0+{\mathsf{\mu }}_2{\parallel {\mathrm{y}}_1-{\mathrm{y}}_2\parallel }_0\right]\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\left[{\mathsf{\mu }}_1\parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_0+{\mathsf{\mu }}_2{\parallel {\mathrm{y}}_1-{\mathrm{y}}_2\parallel }_0\right]+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\left[{\mathsf{\mu }}_1\parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_0+{\mathsf{\mu }}_2{\parallel {\mathrm{y}}_1-{\mathrm{y}}_2\parallel }_0\right]\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\left[{\mathsf{\mu }}_1\parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_0+{\mathsf{\mu }}_2{\parallel {\mathrm{y}}_1-{\mathrm{y}}_2\parallel }_0\right]+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{1\mathrm{j}}{\parallel {\mathrm{x}}_1-{\mathrm{x}}_2\parallel }_0+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}\hfill \\ \hfill & \times \parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_0+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\left[{\mathsf{\mu }}_1\parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_0+{\mathsf{\mu }}_2{\parallel {\mathrm{y}}_1-{\mathrm{y}}_2\parallel }_0\right]+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\left[{\mathsf{\mu }}_1\parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_0+{\mathsf{\mu }}_2{\parallel {\mathrm{y}}_1-{\mathrm{y}}_2\parallel }_0\right]\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}\parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_0+{\mathcal{Z}}_{\mathsf{\alpha }}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}{\parallel {\mathrm{x}}_1-{\mathrm{x}}_2\parallel }_0,\hfill \\ \hfill \le & \frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}\left[{\mathsf{\mu }}_1+{\mathsf{\mu }}_2\right]\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon }+1)}\left[{\mathsf{\mu }}_1+{\mathsf{\mu }}_2\right]\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel \hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\left[{\mathsf{\mu }}_1+{\mathsf{\mu }}_2\right]\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\left[{\mathsf{\mu }}_1+{\mathsf{\mu }}_2\right]\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel \hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\left[{\mathsf{\mu }}_1+{\mathsf{\mu }}_2\right]\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{1\mathrm{j}}\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel \hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\left[{\mathsf{\mu }}_1+{\mathsf{\mu }}_2\right]\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel \hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\left[{\mathsf{\mu }}_1+{\mathsf{\mu }}_2\right]\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel \hfill \\ \hfill & +{\mathcal{Z}}_{\mathsf{\alpha }}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel ,\hfill \\ \hfill \le & {\mathcal{Z}}_{\mathsf{\alpha }}\left[({\mathsf{\mu }}_1+{\mathsf{\mu }}_2)\right(\frac{1}{{\mathcal{Z}}_{\mathsf{\alpha }}\mathsf{\Gamma }(\mathsf{\alpha }+1)}+\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon }+1)}+\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(3-\mathsf{\epsilon })}+\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\hfill \\ \hfill & +\frac{2}{\mathsf{\Gamma }(\mathsf{\alpha }-1)})+\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{1\mathrm{j}}+\frac{{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}+\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}]\hfill \\ \hfill & \times \parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel ,\mathrm{t}\in [0,{\mathrm{t}}_1].\hfill \end{array}$$

(18)

When $\mathrm{t}\in \left({\mathrm{t}}_{\mathrm{m}},{\mathrm{t}}_{\mathrm{m}+1}\right]$, then

$$\begin{array}{cc}\hfill & |{\mathbf{W}}_1({\mathrm{x}}_1,{\mathrm{y}}_1)\left(\mathrm{t}\right)-{\mathbf{W}}_1({\mathrm{x}}_2,{\mathrm{y}}_2)\left(\mathrm{t}\right)|\hfill \\ \hfill \le & \frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\alpha }-1}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_2\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-1}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_2\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\sum _{\mathrm{j}=1}^{\mathrm{m}}|(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})|{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-2}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_2\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{1}{2\mathsf{\Gamma }(\mathsf{\alpha }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})}^2|{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_2\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{M}}_{1\mathrm{j}}\left({\mathrm{x}}_1\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{M}}_{1\mathrm{j}}\left({\mathrm{x}}_2\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|+\sum _{\mathrm{j}=1}^{\mathrm{m}}|(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})||{\mathrm{N}}_{1\mathrm{j}}\left({\mathrm{x}}_1\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{N}}_{1\mathrm{j}}\left({\mathrm{x}}_2\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon })}{\int }_{{\mathrm{t}}_{\mathrm{j}}}^{\mathsf{\Omega }}{(\mathsf{\Omega }-\mathrm{s})}^{\mathsf{\alpha }-\mathsf{\epsilon }-1}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_2\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-2}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_2\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +{\displaystyle \frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(3-\mathsf{\epsilon })}}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_2\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +{\displaystyle \frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(2-\mathsf{\epsilon })}}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{t}}_{\mathrm{j}}|{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_2\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{N}}_{1\mathrm{j}}\left({\mathrm{x}}_1\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{N}}_{1\mathrm{j}}\left({\mathrm{x}}_2\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{O}}_{1\mathrm{j}}\left({\mathrm{x}}_1\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{O}}_{1\mathrm{j}}\left({\mathrm{x}}_2\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^1{(1-\mathrm{s})}^{\mathsf{\alpha }-3}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_2\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_2\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_2\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{t}}_{\mathrm{j}}||{\mathrm{O}}_{1\mathrm{j}}\left({\mathrm{x}}_1\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{O}}_{1\mathrm{j}}\left({\mathrm{x}}_2\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|+{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{O}}_{1\mathrm{j}}\left({\mathrm{x}}_1\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{O}}_{1\mathrm{j}}\left({\mathrm{x}}_2\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|\hfill \\ \hfill & +\sum _{\mathrm{j}=1}^{\mathrm{m}}\frac{|{(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})}^2|}{2}|{\mathrm{O}}_{1\mathrm{j}}\left({\mathrm{x}}_1\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{O}}_{1\mathrm{j}}\left({\mathrm{x}}_2\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|.\hfill \end{array}$$

(19)

Utilizing $\left({\mathbf{M}}_1\right)$ and $\left({\mathbf{M}}_2\right)$ in (19) and taking the maximum, we get

$$\begin{array}{cc}\hfill \le & \frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}\left[{\mathsf{\mu }}_1\parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_0+{\mathsf{\mu }}_2{\parallel {\mathrm{y}}_1-{\mathrm{y}}_2\parallel }_0\right]+\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}\left[{\mathsf{\mu }}_1\parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_0+{\mathsf{\mu }}_2{\parallel {\mathrm{y}}_1-{\mathrm{y}}_2\parallel }_0\right]\hfill \\ \hfill & +\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}\left[{\mathsf{\mu }}_1\parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_0+{\mathsf{\mu }}_2{\parallel {\mathrm{y}}_1-{\mathrm{y}}_2\parallel }_0\right]+\frac{1}{2\mathsf{\Gamma }(\mathsf{\alpha }-1)}\left[{\mathsf{\mu }}_1\parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_0+{\mathsf{\mu }}_2{\parallel {\mathrm{y}}_1-{\mathrm{y}}_2\parallel }_0\right]\hfill \\ \hfill & +\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{I}}_{1\mathrm{j}}\parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_0+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{1\mathrm{j}}{\parallel {\mathrm{x}}_1-{\mathrm{x}}_2\parallel }_0+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon }+1)}\left[{\mathsf{\mu }}_1\parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_0+{\mathsf{\mu }}_2{\parallel {\mathrm{y}}_1-{\mathrm{y}}_2\parallel }_0\right]\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\left[{\mathsf{\mu }}_1\parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_0+{\mathsf{\mu }}_2{\parallel {\mathrm{y}}_1-{\mathrm{y}}_2\parallel }_0\right]+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\left[{\mathsf{\mu }}_1\parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_0+{\mathsf{\mu }}_2{\parallel {\mathrm{y}}_1-{\mathrm{y}}_2\parallel }_0\right]\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\left[{\mathsf{\mu }}_1\parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_0+{\mathsf{\mu }}_2{\parallel {\mathrm{y}}_1-{\mathrm{y}}_2\parallel }_0\right]+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{1\mathrm{j}}{\parallel {\mathrm{x}}_1-{\mathrm{x}}_2\parallel }_0\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}{\parallel {\mathrm{x}}_1-{\mathrm{x}}_2\parallel }_0+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\left[{\mathsf{\mu }}_1\parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_0+{\mathsf{\mu }}_2{\parallel {\mathrm{y}}_1-{\mathrm{y}}_2\parallel }_0\right]\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\left[{\mathsf{\mu }}_1\parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_0+{\mathsf{\mu }}_2{\parallel {\mathrm{y}}_1-{\mathrm{y}}_2\parallel }_0\right]+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}\parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_0+{\mathcal{Z}}_{\mathsf{\alpha }}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}{\parallel {\mathrm{x}}_1-{\mathrm{x}}_2\parallel }_0\hfill \\ \hfill & +\frac{1}{2}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}{\parallel {\mathrm{x}}_1-{\mathrm{x}}_2\parallel }_0,\hfill \\ \hfill \le & \frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}\left[{\mathsf{\mu }}_1+{\mathsf{\mu }}_2\right]\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel +\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}\left[{\mathsf{\mu }}_1+{\mathsf{\mu }}_2\right]\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel \hfill \\ \hfill & +\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}\left[{\mathsf{\mu }}_1+{\mathsf{\mu }}_2\right]\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel +\frac{1}{2\mathsf{\Gamma }(\mathsf{\alpha }-1)}\left[{\mathsf{\mu }}_1+{\mathsf{\mu }}_2\right]\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel \hfill \\ \hfill & +\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{I}}_{1\mathrm{j}}\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel +\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{1\mathrm{j}}\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon }+1)}\left[{\mathsf{\mu }}_1+{\mathsf{\mu }}_2\right]\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel \hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\left[{\mathsf{\mu }}_1+{\mathsf{\mu }}_2\right]\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\left[{\mathsf{\mu }}_1+{\mathsf{\mu }}_2\right]\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel \hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\left[{\mathsf{\mu }}_1+{\mathsf{\mu }}_2\right]\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{1\mathrm{j}}\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel \hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\left[{\mathsf{\mu }}_1+{\mathsf{\mu }}_2\right]\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel \hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\left[{\mathsf{\mu }}_1+{\mathsf{\mu }}_2\right]\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel \hfill \\ \hfill & +{\mathcal{Z}}_{\mathsf{\alpha }}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel +\frac{1}{2}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel \hfill \\ \hfill \le & \left[\left({\mathsf{\mu }}_1+{\mathsf{\mu }}_2\right)\right(\frac{2}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}+\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}+\frac{1}{2\mathsf{\Gamma }(\mathsf{\alpha }-1)}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon }+1)}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{2{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)})+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{I}}_{1\mathrm{j}}+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{1\mathrm{j}}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{1\mathrm{j}}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}+{\mathcal{Z}}_{\mathsf{\alpha }}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}+\frac{1}{2}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}]\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel .\hfill \end{array}$$

(20)

In the same fashion, we can obtain

$$\begin{array}{cc}\hfill & |{\mathbf{W}}_2({\mathrm{x}}_1,{\mathrm{y}}_1)\left(\mathrm{t}\right)-{\mathbf{W}}_2({\mathrm{x}}_2,{\mathrm{y}}_2)\left(\mathrm{t}\right)|\hfill \\ \le \hfill & {\mathcal{Z}}_{\mathsf{\beta }}\left[\left({\mathsf{\mu }}_1^{\prime }+{\mathsf{\mu }}_2^{\prime }\right)\right(\frac{1}{{\mathcal{Z}}_{\mathsf{\beta }}\mathsf{\Gamma }(\mathsf{\beta }+1)}+\frac{1}{\mathsf{\Gamma }(\mathsf{\beta }-\mathsf{\rho }+1)}+\frac{{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)\mathsf{\Gamma }(2-\mathsf{\rho })}+\frac{{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(3-\mathsf{\rho })}+\frac{{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(2-\mathsf{\rho })}\hfill \\ \hfill & +\frac{2}{\mathsf{\Gamma }(\mathsf{\beta }-1)})+\frac{{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{2\mathrm{j}}+\frac{{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(3-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}+\frac{{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}]\hfill \\ \hfill & \times \parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel ,\mathrm{t}\in [0,{\mathrm{t}}_1],\hfill \end{array}$$

(21)

and

$$\begin{array}{cc}\hfill & |{\mathbf{W}}_2({\mathrm{x}}_1,{\mathrm{y}}_1)\left(\mathrm{t}\right)-{\mathbf{W}}_2({\mathrm{x}}_2,{\mathrm{y}}_2)\left(\mathrm{t}\right)|\hfill \\ \le \hfill & \left[\left({\mathsf{\mu }}_1^{\prime }+{\mathsf{\mu }}_2^{\prime }\right)\right(\frac{2}{\mathsf{\Gamma }(\mathsf{\beta }+1)}+\frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}+\frac{1}{2\mathsf{\Gamma }(\mathsf{\beta }-1)}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}}{\mathsf{\Gamma }(\mathsf{\beta }-\mathsf{\rho }+1)}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)\mathsf{\Gamma }(2-\mathsf{\rho })}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(3-\mathsf{\rho })}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(2-\mathsf{\rho })}+\frac{2{\mathcal{Z}}_{\mathsf{\beta }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)})+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{I}}_{2\mathrm{j}}+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{2\mathrm{j}}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{2\mathrm{j}}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(3-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}+{\mathcal{Z}}_{\mathsf{\beta }}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}+\frac{1}{2}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}]\parallel ({\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{y}}_1-{\mathrm{y}}_2)\parallel ,\mathrm{t}\in ({\mathrm{t}}_{\mathrm{m}},{\mathrm{t}}_{\mathrm{m}+1}].\hfill \end{array}$$

(22)

Thus, from (18)–(22) and $\left({\mathbf{M}}_3\right),$ we infer that $\mathbf{W}$ is a contraction mapping. According to Lemma 3, $\mathbf{W}$ has a fixed point $({\mathrm{x}}^{\ast }\left(\mathrm{t}\right),{\mathrm{y}}^{\ast }\left(\mathrm{t}\right))\in {\mathrm{Y}}^{\prime }$, which is unique. Therefore, problem (1) has a unique solution $({\mathrm{x}}^{\ast }\left(\mathrm{t}\right),{\mathrm{y}}^{\ast }\left(\mathrm{t}\right)).$ □

Theorem 3.

Let $\left({\mathbf{M}}_1\right)-\left({\mathbf{M}}_2\right)$, $\left({\mathbf{M}}_4\right)$ and for all $\mathrm{t}\in \mathrm{J}$ such that $\mathrm{h}(\mathrm{t},0,0)=\mathrm{w}(\mathrm{t},0,0)=0,{\mathrm{M}}_{\mathrm{i}\mathrm{k}}={\mathrm{N}}_{\mathrm{i}\mathrm{k}}={\mathrm{O}}_{\mathrm{i}\mathrm{k}}=0,(\mathrm{i}=1,2;\mathrm{k}=1,2,\dots ,\mathrm{n})$ hold. Then, (1) has at least one solution $({\mathrm{x}}^{\ast }\left(\mathrm{t}\right),{\mathrm{y}}^{\ast }\left(\mathrm{t}\right)).$

Proof. 

For the sake of simplicity, let us denote

$$\begin{array}{cc}\hfill \mathsf{\varpi }=& \left[\left({\mathsf{\mu }}_1+{\mathsf{\mu }}_2\right)\right(\frac{3}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}+\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}+\frac{1}{2\mathsf{\Gamma }(\mathsf{\alpha }-1)}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon }+1)}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{2{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)})+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{I}}_{1\mathrm{j}}+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{1\mathrm{j}}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{1\mathrm{j}}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}+{\mathcal{Z}}_{\mathsf{\alpha }}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}+\frac{1}{2}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}],\hfill \\ \hfill \mathsf{\xi }=& \left[\left({\mathsf{\mu }}_1^{\prime }+{\mathsf{\mu }}_2^{\prime }\right)\right(\frac{3}{\mathsf{\Gamma }(\mathsf{\beta }+1)}+\frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}+\frac{1}{2\mathsf{\Gamma }(\mathsf{\beta }-1)}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}}{\mathsf{\Gamma }(\mathsf{\beta }-\mathsf{\rho }+1)}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)\mathsf{\Gamma }(2-\mathsf{\rho })}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(3-\mathsf{\rho })}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(2-\mathsf{\rho })}+\frac{2{\mathcal{Z}}_{\mathsf{\beta }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)})+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{I}}_{2\mathrm{j}}+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{2\mathrm{j}}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{2\mathrm{j}}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(3-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}+{\mathcal{Z}}_{\mathsf{\beta }}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}+\frac{1}{2}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}],\hfill \end{array}$$

and ${\mathcal{R}}_{\mathsf{\upsilon }}=\max \{(\frac{1}{\mathsf{\varpi }}+1,\frac{1}{\mathsf{\xi }}+1)\}$. Define the operator $\mathbf{W}$, as in (17), and a closed ball of Banach space ${\mathrm{Y}}^{\prime }$ as follows:

$$\begin{array}{c}\hfill \mathsf{\upsilon }=\left\{(\mathrm{x},\mathrm{y})\in {\mathrm{Y}}^{\prime }:\parallel (\mathrm{x},\mathrm{y})\parallel \le {\mathcal{R}}_{\mathsf{\upsilon }}\right\}.\end{array}$$

(23)

Similar to (18)–(22), we easily show that $\mathbf{W}\left(\mathsf{\upsilon }\right)\subset \mathsf{\upsilon }$ by applying $\left({\mathbf{M}}_4\right)$. $\mathbf{W}\left(\mathsf{\upsilon }\right)\subset \mathsf{\upsilon }$ indicates that $\mathbf{W}\left(\mathsf{\upsilon }\right)$ is uniformly bounded in ${\mathrm{Y}}^{\prime }$. The continuity of the operator $\mathbf{W}$ is follows from the continuity of $\mathrm{h},\mathrm{w},{\mathrm{M}}_{\mathrm{i}\mathrm{m}},{\mathrm{N}}_{\mathrm{i}\mathrm{m}}$ and ${\mathrm{O}}_{\mathrm{i}\mathrm{m}}$. Now, we need to prove that $\mathbf{W}:\mathsf{\upsilon }\to \mathsf{\upsilon }$ is equicontinuous. Let $(\mathrm{x},\mathrm{y})\in \mathsf{\upsilon }$ and ${\mathsf{\ell }}_1,{\mathsf{\ell }}_2\in [0,1]$ with ${\mathsf{\ell }}_1<{\mathsf{\ell }}_2$. When $0\le {\mathsf{\ell }}_1<{\mathsf{\ell }}_2\le {\mathrm{t}}_1$, similar to Equation (18), we have

$$\begin{array}{cc}\hfill & |{\mathbf{W}}_1(\mathrm{x},\mathrm{y})\left({\mathsf{\ell }}_2\right)-{\mathbf{W}}_1(\mathrm{x},\mathrm{y})\left({\mathsf{\ell }}_1\right)|\hfill \\ \hfill =& |\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_0^{{\mathsf{\ell }}_1}\left[{({\mathsf{\ell }}_2-\mathrm{s})}^{\mathsf{\alpha }-1}-{({\mathsf{\ell }}_1-\mathrm{s})}^{\mathsf{\alpha }-1}\right]\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}+\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_{{\mathsf{\ell }}_1}^{{\mathsf{\ell }}_2}{({\mathsf{\ell }}_2-\mathrm{s})}^{\mathsf{\alpha }-1}\hfill \\ \hfill & \times \mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}-{\mathrm{e}}^{\ast }{({\mathsf{\ell }}_2-{\mathsf{\ell }}_1)}^2|,\hfill \\ \hfill \le & \frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_0^{{\mathsf{\ell }}_1}\left[{({\mathsf{\ell }}_2-\mathrm{s})}^{\mathsf{\alpha }-1}-{({\mathsf{\ell }}_1-\mathrm{s})}^{\mathsf{\alpha }-1}\right]|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}+\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_{{\mathsf{\ell }}_1}^{{\mathsf{\ell }}_2}{({\mathsf{\ell }}_2-\mathrm{s})}^{\mathsf{\alpha }-1}\hfill \\ \hfill & \times |\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}+|{\mathrm{e}}^{\ast }|{({\mathsf{\ell }}_2-{\mathsf{\ell }}_1)}^2,\hfill \\ \hfill \le & \frac{{\mathsf{\mu }}_1+{\mathsf{\mu }}_2}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}\parallel (\mathrm{x},\mathrm{y})\parallel {\int }_0^{{\mathsf{\ell }}_1}\left[{({\mathsf{\ell }}_2-\mathrm{s})}^{\mathsf{\alpha }-1}-{({\mathsf{\ell }}_1-\mathrm{s})}^{\mathsf{\alpha }-1}\right]\mathrm{d}\mathrm{s}+\frac{{\mathsf{\mu }}_1+{\mathsf{\mu }}_2}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}\parallel (\mathrm{x},\mathrm{y})\parallel {({\mathsf{\ell }}_2-{\mathsf{\ell }}_1)}^{\mathsf{\alpha }}+|{\mathrm{e}}^{\ast }|{({\mathsf{\ell }}_2-{\mathsf{\ell }}_1)}^2,\hfill \\ \hfill \le & \frac{{\mathsf{\mu }}_1+{\mathsf{\mu }}_2}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}\parallel (\mathrm{x},\mathrm{y})\parallel \left[{({\mathsf{\ell }}_2-{\mathsf{\ell }}_1)}^{\mathsf{\alpha }}-({\mathsf{\ell }}_2^{\mathsf{\alpha }}-{\mathsf{\ell }}_1^{\mathsf{\alpha }})\right]+\frac{{\mathsf{\mu }}_1+{\mathsf{\mu }}_2}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}\parallel (\mathrm{x},\mathrm{y})\parallel {({\mathsf{\ell }}_2-{\mathsf{\ell }}_1)}^{\mathsf{\alpha }}+|{\mathrm{e}}^{\ast }|{({\mathsf{\ell }}_2-{\mathsf{\ell }}_1)}^2,\hfill \\ \hfill \le & {\mathcal{Z}}_{\mathsf{\alpha }}\left[({\mathsf{\mu }}_1+{\mathsf{\mu }}_2)\right(\frac{2}{{\mathcal{Z}}_{\mathsf{\alpha }}\mathsf{\Gamma }(\mathsf{\alpha }+1)}+\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon }+1)}+\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(3-\mathsf{\epsilon })}+\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\hfill \\ \hfill & +\frac{2}{\mathsf{\Gamma }(\mathsf{\alpha }-1)})+\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{1\mathrm{j}}+\frac{{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}+\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}]\parallel \mathrm{x},\mathrm{y}\parallel {({\mathsf{\ell }}_2-{\mathsf{\ell }}_1)}^2,\hfill \\ \hfill \le & \mathsf{\varpi }{\mathcal{R}}_{\mathsf{\upsilon }}{({\mathsf{\ell }}_2-{\mathsf{\ell }}_1)}^2.\hfill \end{array}$$

(24)

In the same fashion, we obtain

$$|{\mathbf{W}}_2(\mathrm{x},\mathrm{y})\left({\mathsf{\ell }}_2\right)-{\mathbf{W}}_2(\mathrm{x},\mathrm{y})\left({\mathsf{\ell }}_1\right)|\le \mathsf{\xi }{\mathcal{R}}_{\mathsf{\upsilon }}{({\mathsf{\ell }}_2-{\mathsf{\ell }}_1)}^2.$$

(25)

In addition, we obtain the same result when ${\mathrm{t}}_{\mathrm{m}}<{\mathsf{\ell }}_1<{\mathsf{\ell }}_2\le {\mathrm{t}}_{\mathrm{m}+1},1\le \mathrm{m}\le \mathrm{n},$ similar to (20)

$$|{\mathbf{W}}_1(\mathrm{x},\mathrm{y})\left({\mathsf{\ell }}_2\right)-{\mathbf{W}}_1(\mathrm{x},\mathrm{y})\left({\mathsf{\ell }}_1\right)|\le \mathsf{\varpi }{\mathcal{R}}_{\mathsf{\upsilon }}{({\mathsf{\ell }}_2-{\mathsf{\ell }}_1)}^2$$

(26)

and

$$|{\mathbf{W}}_2(\mathrm{x},\mathrm{y})\left({\mathsf{\ell }}_2\right)-{\mathbf{W}}_2(\mathrm{x},\mathrm{y})\left({\mathsf{\ell }}_1\right)|\le \mathsf{\xi }{\mathcal{R}}_{\mathsf{\upsilon }}{({\mathsf{\ell }}_2-{\mathsf{\ell }}_1)}^2.$$

(27)

Thus, it follows from (24)–(27) that, for any $\mathsf{ϵ}>0,$ there exists a positive constant $\mathsf{\sigma }=\frac{\mathsf{ϵ}}{{\mathcal{R}}_{\mathsf{\upsilon }}}\min \{\frac{1}{\mathsf{\varpi }},\frac{1}{\mathsf{\xi }}\}$ independent of ${\mathsf{\ell }}_1,{\mathsf{\ell }}_2$ and $(\mathrm{x},\mathrm{y})$ such that $\parallel \mathbf{W}(\mathrm{x},\mathrm{y})\left({\mathsf{\ell }}_2\right)-\mathbf{W}(\mathrm{x},\mathrm{y})\left({\mathsf{\ell }}_1\right)\parallel <\mathsf{ϵ}$, whenever $|{\mathsf{\ell }}_2-{\mathsf{\ell }}_1|\le \mathsf{\sigma }$. Thereby, $\mathbf{W}:{\mathrm{Y}}^{\prime }\to {\mathrm{Y}}^{\prime }$ is equicontinuous. By the Arzela–Ascoli theorem, we know that $\mathbf{W}:{\mathrm{Y}}^{\prime }\to {\mathrm{Y}}^{\prime }$ is completely continuous. In view of Theorem 1, $\mathbf{W}$ has a unique fixed point $({\mathrm{x}}^{\ast }\left(\mathrm{t}\right),{\mathrm{y}}^{\ast }\left(\mathrm{t}\right))\in \overline{\mathsf{\upsilon }},$ which is a solution of system (1). □

4. Ulam–Hyers Stability

In this section, we are interested in Ulam–Hyers stability and its types for the solution of (1).

Definition 4.

[61] Problem (1) is Ulam–Hyers stable if there exists a constant ${\mathbf{K}}_{\mathsf{\alpha },\mathsf{\beta }}=({\mathbf{K}}_{\mathsf{\alpha }},{\mathbf{K}}_{\mathsf{\beta }})>0$ such that, for any $\mathsf{ϵ}=({\mathsf{ϵ}}_{\mathsf{\alpha }},{\mathsf{ϵ}}_{\mathsf{\beta }})>0,$ and $\mathrm{m}=1,2,\dots ,\mathrm{n},$ there exists a solution $(\mathrm{x},\mathrm{y})\in {\mathrm{Y}}^{\prime }$ of:

$$\left\{\begin{array}{c}{|}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\alpha }}\mathrm{x}\left(\mathrm{t}\right)-\mathrm{h}\left(\mathrm{t}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{t}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{t}\right)\right)|\le {\mathsf{ϵ}}_{\mathsf{\alpha }},\hfill \\ |\Delta \mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)-{\mathrm{M}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)|\le {\mathsf{ϵ}}_{\mathsf{\alpha }},\hfill \\ |\Delta {\mathrm{x}}^{\prime }\left({\mathrm{t}}_{\mathrm{m}}\right)-{\mathrm{N}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)|\le {\mathsf{ϵ}}_{\mathsf{\alpha }},\hfill \\ |\Delta {\mathrm{x}}^{″}\left({\mathrm{t}}_{\mathrm{m}}\right)-{\mathrm{O}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)|\le {\mathsf{ϵ}}_{\mathsf{\alpha }},\hfill \\ {|}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\beta }}\mathrm{y}\left(\mathrm{t}\right)-\mathrm{w}\left(\mathrm{t}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{t}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{t}\right)\right)|\le {\mathsf{ϵ}}_{\mathsf{\beta }},\hfill \\ |\Delta \mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)-{\mathrm{M}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)|\le {\mathsf{ϵ}}_{\mathsf{\beta }},\hfill \\ |\Delta {\mathrm{y}}^{\prime }\left({\mathrm{t}}_{\mathrm{m}}\right)-{\mathrm{N}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)|\le {\mathsf{ϵ}}_{\mathsf{\beta }},\hfill \\ |\Delta {\mathrm{y}}^{″}\left({\mathrm{t}}_{\mathrm{m}}\right)-{\mathrm{O}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)|\le {\mathsf{ϵ}}_{\mathsf{\beta }},\hfill \end{array}\right.$$

(28)

corresponding to a solution $(\mathsf{\zeta },\mathsf{\chi })\in {\mathrm{Y}}^{\prime }$ of (1) such that

$$\begin{array}{c}\hfill |(\mathrm{x},\mathrm{y})\left(\mathrm{t}\right)-(\mathsf{\zeta },\mathsf{\chi })\left(\mathrm{t}\right)|\le {\mathbf{K}}_{\mathsf{\alpha },\mathsf{\beta }}\mathsf{ϵ}.\end{array}$$

Definition 5.

[61] Problem (1) is generalized Ulam–Hyers stable if there exists a function ${\mathrm{\Theta }}_{\mathsf{\alpha },\mathsf{\beta }}\in \mathrm{C}({\mathbb{R}}_{+},{\mathbb{R}}_{+})$, ${\mathrm{\Theta }}_{\mathsf{\alpha },\mathsf{\beta }}\left(0\right)=0$ for each $\mathsf{ϵ}>0,$ such that for every solution $(\mathrm{x},\mathrm{y})\in {\mathrm{Y}}^{\prime }$ of the inequality (28). there is a solution $(\mathsf{\zeta },\mathsf{\chi })\in {\mathrm{Y}}^{\prime }$ of (1) such that

$$|(\mathrm{x},\mathrm{y})\left(\mathrm{t}\right)-(\mathsf{\zeta },\mathsf{\chi })\left(\mathrm{t}\right)|\le {\mathrm{\Theta }}_{\mathsf{\alpha },\mathsf{\beta }}\left(\mathsf{ϵ}\right).$$

Definition 6.

[61] Problem (1) is Ulam–Hyers–Rassias stable with respect to $\left({\mathsf{\Psi }}_{\mathsf{\alpha },\mathsf{\beta }},{\mathsf{\phi }}_{\mathsf{\alpha },\mathsf{\beta }}\right)$, where ${\mathsf{\Psi }}_{\mathsf{\alpha },\mathsf{\beta }}=\left({\mathsf{\Psi }}_{\mathsf{\alpha }},{\mathsf{\Psi }}_{\mathsf{\beta }}\right)\in \mathrm{C}(\mathrm{J},\mathbb{R})$ and ${\mathsf{\phi }}_{\mathsf{\alpha },\mathsf{\beta }}=\left({\mathsf{\phi }}_{\mathsf{\alpha }},{\mathsf{\phi }}_{\mathsf{\beta }}\right)\in \mathrm{C}(\mathrm{J},\mathbb{R}),$ if, for every $\mathsf{ϵ}=({\mathsf{ϵ}}_{\mathsf{\alpha }},{\mathsf{ϵ}}_{\mathsf{\beta }})>0,$ there exists a real number ${\mathbf{K}}_{\mathsf{\Psi },\mathsf{\phi }}>0,$ such that for $\mathrm{m}=1,2,\dots ,\mathrm{n}$ and for a solution $(\mathrm{x},\mathrm{y})\in {\mathrm{Y}}^{\prime }$ of:

$$\left\{\begin{array}{c}{|}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\alpha }}\mathrm{x}\left(\mathrm{t}\right)-\mathrm{h}\left(\mathrm{t}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{t}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{t}\right)\right)|\le {\mathsf{\Psi }}_{\mathsf{\alpha }}\left(\mathrm{t}\right),\hfill \\ |\Delta \mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)-{\mathrm{M}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)|\le {\mathsf{\phi }}_{\mathsf{\alpha }},\hfill \\ |\Delta {\mathrm{x}}^{\prime }\left({\mathrm{t}}_{\mathrm{m}}\right)-{\mathrm{N}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)|\le {\mathsf{\phi }}_{\mathsf{\alpha }},\hfill \\ |\Delta {\mathrm{x}}^{″}\left({\mathrm{t}}_{\mathrm{m}}\right)-{\mathrm{O}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)|\le {\mathsf{\phi }}_{\mathsf{\alpha }},\hfill \\ {|}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\beta }}\mathrm{y}\left(\mathrm{t}\right)-\mathrm{w}\left(\mathrm{t}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{t}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{t}\right)\right)|\le {\mathsf{\Psi }}_{\mathsf{\beta }}\left(\mathrm{t}\right),\hfill \\ |\Delta \mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)-{\mathrm{M}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)|\le {\mathsf{\phi }}_{\mathsf{\beta }},\hfill \\ |\Delta {\mathrm{y}}^{\prime }\left({\mathrm{t}}_{\mathrm{m}}\right)-{\mathrm{N}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)|\le {\mathsf{\phi }}_{\mathsf{\beta }},\hfill \\ |\Delta {\mathrm{y}}^{″}\left({\mathrm{t}}_{\mathrm{m}}\right)-{\mathrm{O}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)|\le {\mathsf{\phi }}_{\mathsf{\beta }},\hfill \end{array}\right.$$

(29)

there is a solution $(\mathsf{\zeta },\mathsf{\chi })\in {\mathrm{Y}}^{\prime }$ of (1) such that

$$\begin{array}{c}\hfill |(\mathrm{x},\mathrm{y})\left(\mathrm{t}\right)-(\mathsf{\zeta },\mathsf{\chi })\left(\mathrm{t}\right)|\le {\mathbf{K}}_{\mathsf{\Psi },\mathsf{\phi }}\left({\mathsf{\Psi }}_{\mathsf{\alpha },\mathsf{\beta }}\left(\mathrm{t}\right)+{\mathsf{\phi }}_{\mathsf{\alpha },\mathsf{\beta }}\right)\mathsf{ϵ}.\end{array}$$

Definition 7.

[61] Problem (1) is generalized Ulam–Hyers–Rassias stable with respect to $\left({\mathsf{\Psi }}_{\mathsf{\alpha },\mathsf{\beta }},{\mathsf{\phi }}_{\mathsf{\alpha },\mathsf{\beta }}\right)\in \mathrm{C}(\mathrm{J},\mathbb{R}),$ if there exists a real number ${\mathbf{K}}_{\mathsf{\Psi },\mathsf{\phi }}>0,$ such that for $\mathrm{m}=1,2,\dots ,\mathrm{n}$ and for every solution $(\mathrm{x},\mathrm{y})\in {\mathrm{Y}}^{\prime }$ of the following:

$$\left\{\begin{array}{c}{|}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\alpha }}\mathrm{x}\left(\mathrm{t}\right)-\mathrm{h}\left(\mathrm{t}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{t}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{t}\right)\right)|\le {\mathsf{\Psi }}_{\mathsf{\alpha }}\left(\mathrm{t}\right){\mathsf{ϵ}}_{\mathsf{\alpha }},\hfill \\ |\Delta \mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)-{\mathrm{M}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)|\le {\mathsf{\phi }}_{\mathsf{\alpha }}{\mathsf{ϵ}}_{\mathsf{\alpha }},\hfill \\ |\Delta {\mathrm{x}}^{\prime }\left({\mathrm{t}}_{\mathrm{m}}\right)-{\mathrm{N}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)|\le {\mathsf{\phi }}_{\mathsf{\alpha }}{\mathsf{ϵ}}_{\mathsf{\alpha }},\hfill \\ |\Delta {\mathrm{x}}^{″}\left({\mathrm{t}}_{\mathrm{m}}\right)-{\mathrm{O}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)|\le {\mathsf{\phi }}_{\mathsf{\alpha }}{\mathsf{ϵ}}_{\mathsf{\alpha }},\hfill \\ {|}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\beta }}\mathrm{y}\left(\mathrm{t}\right)-\mathrm{w}\left(\mathrm{t}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{t}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{t}\right)\right)|\le {\mathsf{\Psi }}_{\mathsf{\beta }}\left(\mathrm{t}\right){\mathsf{ϵ}}_{\mathsf{\beta }},\hfill \\ |\Delta \mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)-{\mathrm{M}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)|\le {\mathsf{\phi }}_{\mathsf{\beta }}{\mathsf{ϵ}}_{\mathsf{\beta }},\hfill \\ |\Delta {\mathrm{y}}^{\prime }\left({\mathrm{t}}_{\mathrm{m}}\right)-{\mathrm{N}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)|\le {\mathsf{\phi }}_{\mathsf{\beta }}{\mathsf{ϵ}}_{\mathsf{\beta }},\hfill \\ |\Delta {\mathrm{y}}^{″}\left({\mathrm{t}}_{\mathrm{m}}\right)-{\mathrm{O}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)|\le {\mathsf{\phi }}_{\mathsf{\beta }}{\mathsf{ϵ}}_{\mathsf{\beta }},\hfill \end{array}\right.$$

(30)

there is a solution $(\mathsf{\zeta },\mathsf{\chi })\in {\mathrm{Y}}^{\prime }$ of (1) such that

$$\begin{array}{c}\hfill |(\mathrm{x},\mathrm{y})\left(\mathrm{t}\right)-(\mathsf{\zeta },\mathsf{\chi })\left(\mathrm{t}\right)|\le {\mathbf{K}}_{\mathsf{\Psi },\mathsf{\phi }}\left({\mathsf{\Psi }}_{\mathsf{\alpha },\mathsf{\beta }}\left(\mathrm{t}\right)+{\mathsf{\phi }}_{\mathsf{\alpha },\mathsf{\beta }}\right).\end{array}$$

Remark 1.

A function $(\mathrm{x},\mathrm{y})\in {\mathrm{Y}}^{\prime }$ is a solution of the inequality (28), if and only if there exist functions ${\mathrm{ϝ}}_{\mathrm{h}},{\widehat{\mathrm{ϝ}}}_{\mathrm{w}}\in {\mathrm{Y}}^{\prime }$ and a sequence ${\mathrm{ϝ}}_{\mathrm{m}},{\widehat{\mathrm{ϝ}}}_{\mathrm{m}},\mathrm{m}=1,2,\dots ,\mathrm{n}$ depending on $(\mathrm{x},\mathrm{y})$, such that

• $|{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{t}\right)|\le {\mathsf{ϵ}}_{\mathsf{\alpha }},|{\widehat{\mathrm{ϝ}}}_{\mathrm{w}}\left(\mathrm{t}\right)|\le {\mathsf{ϵ}}_{\mathsf{\beta }},|{\mathrm{ϝ}}_{\mathrm{m}}|\le {\mathsf{ϵ}}_{\mathsf{\alpha }},|{\widehat{\mathrm{ϝ}}}_{\mathrm{m}}|\le {\mathsf{ϵ}}_{\mathsf{\beta }},\mathrm{t}\in {\mathrm{J}}_{\mathrm{m}},\mathrm{m}=1,\dots ,\mathrm{n};$
• ${}^{\mathrm{c}}{\mathrm{D}}^{\mathsf{\alpha }}\mathrm{x}\left(\mathrm{t}\right)=-\mathrm{h}\left(\mathrm{t}{,}^{\mathrm{c}}{\mathrm{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{t}\right){,}^{\mathrm{c}}{\mathrm{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{t}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{t}\right);$
• $\Delta \mathrm{x}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{M}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}};$
• $\Delta \mathrm{x}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{N}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}};$
• $\Delta \mathrm{x}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{O}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}};$
• ${}^{\mathrm{c}}{\mathrm{D}}^{\mathsf{\beta }}\mathrm{x}\left(\mathrm{t}\right)=-\mathrm{w}\left(\mathrm{t}{,}^{\mathrm{c}}{\mathrm{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{t}\right){,}^{\mathrm{c}}{\mathrm{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{t}\right)\right)+{\widehat{\mathrm{ϝ}}}_{\mathrm{w}}\left(\mathrm{t}\right);$
• $\Delta \mathrm{y}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{M}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)+{\widehat{\mathrm{ϝ}}}_{\mathrm{m}};$
• $\Delta \mathrm{y}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{N}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)+{\widehat{\mathrm{ϝ}}}_{\mathrm{m}};$
• $\Delta \mathrm{y}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{O}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)+{\widehat{\mathrm{ϝ}}}_{\mathrm{m}}.$

Remark 2.

A function $(\mathrm{x},\mathrm{y})\in {\mathrm{Y}}^{\prime }$ is a solution of the inequality (29), if and only if there exist functions ${\mathrm{ϝ}}_{\mathrm{h}},{\widehat{\mathrm{ϝ}}}_{\mathrm{w}}\in {\mathrm{Y}}^{\prime }$ and a sequence ${\mathrm{ϝ}}_{\mathrm{m}},{\widehat{\mathrm{ϝ}}}_{\mathrm{m}},\mathrm{m}=1,2,\dots ,\mathrm{n}$ depending on $(\mathrm{x},\mathrm{y})$, such that

• $|{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{t}\right)|\le {\mathsf{\Psi }}_{\mathsf{\alpha }},|{\widehat{\mathrm{ϝ}}}_{\mathrm{w}}\left(\mathrm{t}\right)|\le {\mathsf{\Psi }}_{\mathsf{\beta }},|{\mathrm{ϝ}}_{\mathrm{m}}|\le {\mathsf{\phi }}_{\mathsf{\alpha }},|{\mathrm{ϝ}}_{\mathrm{m}}|\le {\mathsf{\phi }}_{\mathsf{\beta }},\mathrm{t}\in {\mathrm{J}}_{\mathrm{m}},\mathrm{m}=1,\dots ,\mathrm{n};$
• ${}^{\mathrm{c}}{\mathrm{D}}^{\mathsf{\alpha }}\mathrm{x}\left(\mathrm{t}\right)=-\mathrm{h}\left(\mathrm{t}{,}^{\mathrm{c}}{\mathrm{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{t}\right){,}^{\mathrm{c}}{\mathrm{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{t}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{t}\right);$
• $\Delta \mathrm{x}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{M}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}};$
• $\Delta \mathrm{x}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{N}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}};$
• $\Delta \mathrm{x}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{O}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}};$
• ${}^{\mathrm{c}}{\mathrm{D}}^{\mathsf{\beta }}\mathrm{x}\left(\mathrm{t}\right)=-\mathrm{w}\left(\mathrm{t}{,}^{\mathrm{c}}{\mathrm{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{t}\right){,}^{\mathrm{c}}{\mathrm{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{t}\right)\right)+{\widehat{\mathrm{ϝ}}}_{\mathrm{w}}\left(\mathrm{t}\right);$
• $\Delta \mathrm{y}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{M}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)+{\widehat{\mathrm{ϝ}}}_{\mathrm{m}};$
• $\Delta \mathrm{y}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{N}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)+{\widehat{\mathrm{ϝ}}}_{\mathrm{m}};$
• $\Delta \mathrm{y}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{O}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)+{\widehat{\mathrm{ϝ}}}_{\mathrm{m}}.$

Similarly, one can easily state such a remark for the inequality (30).

Theorem 4.

If the assumptions $\left({\mathbf{M}}_1\right)-\left({\mathbf{M}}_2\right)$ hold with

$${\mathsf{\Lambda }}_0=1-\frac{{\mathsf{\Lambda }}_2^{\ast }{\mathsf{\Lambda }}_4^{\ast }}{(1-{\mathsf{\Lambda }}_1^{\ast })(1-{\mathsf{\Lambda }}_3^{\ast })}>0,$$

(31)

then (1) is Ulam–Hyers and generalized Ulam–Hyers stable.

Proof. 

Let $(\mathrm{x},\mathrm{y})\in {\mathrm{Y}}^{\prime }$ be any solution of the inequality (28) and let $(\mathsf{\zeta },\mathsf{\chi })\in {\mathrm{Y}}^{\prime }$ be the unique solution of the following:

$$\left\{\begin{array}{c}{}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\alpha }}\mathsf{\zeta }\left(\mathrm{t}\right)+\mathrm{h}\left(\mathrm{t}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathsf{\zeta }\left(\mathrm{t}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathsf{\chi }\left(\mathrm{t}\right)\right)=0,\mathrm{t}\ne {\mathrm{t}}_{\mathrm{m}},\mathrm{m}=1,2,\dots ,\mathrm{n},\hfill \\ {}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\beta }}\mathsf{\chi }\left(\mathrm{t}\right)+\mathrm{w}\left(\mathrm{t}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathsf{\zeta }\left(\mathrm{t}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathsf{\chi }\left(\mathrm{t}\right)\right)=0,\mathrm{t}\ne {\mathrm{t}}_{\mathrm{m}},\mathrm{m}=1,2,\dots ,\mathrm{n},\hfill \\ \Delta \mathsf{\zeta }{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{M}}_{1\mathrm{m}}\left(\mathsf{\zeta }\left({\mathrm{t}}_{\mathrm{m}}\right)\right),\Delta {\mathsf{\zeta }}^{\prime }{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{N}}_{1\mathrm{m}}\left(\mathsf{\zeta }\left({\mathrm{t}}_{\mathrm{m}}\right)\right),\Delta {\mathsf{\zeta }}^{″}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{O}}_{1\mathrm{m}}\left(\mathsf{\zeta }\left({\mathrm{t}}_{\mathrm{m}}\right)\right),\hfill \\ \Delta \mathsf{\chi }{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{M}}_{2\mathrm{m}}\left(\mathsf{\chi }\left({\mathrm{t}}_{\mathrm{m}}\right)\right),\Delta {\mathsf{\chi }}^{\prime }{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{N}}_{2\mathrm{m}}\left(\mathsf{\chi }\left({\mathrm{t}}_{\mathrm{m}}\right)\right),\Delta {\mathsf{\chi }}^{″}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{O}}_{2\mathrm{m}}\left(\mathsf{\chi }\left({\mathrm{t}}_{\mathrm{m}}\right)\right),\hfill \\ \mathsf{\zeta }\left(0\right)={\mathsf{\zeta }}^{\prime }\left(0\right)=0,{}^{\mathrm{c}}{\mathrm{D}}^{\mathsf{\epsilon }}\mathsf{\zeta }\left(\mathsf{\Omega }\right)={\mathsf{\zeta }}^{″}\left(1\right),\hfill \\ \mathsf{\chi }\left(0\right)={\mathsf{\chi }}^{\prime }\left(0\right)=0,{}^{\mathrm{c}}{\mathrm{D}}^{\mathsf{\rho }}\mathsf{\chi }\left(\mathsf{\Phi }\right)={\mathsf{\chi }}^{″}\left(1\right).\hfill \end{array}\right.$$

(32)

By Lemma 2.4, we have

$$\left\{\begin{array}{c}{}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\alpha }}\mathrm{x}\left(\mathrm{t}\right)+\mathrm{h}\left(\mathrm{t}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{t}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{t}\right)\right)={\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{t}\right),\mathrm{t}\ne {\mathrm{t}}_{\mathrm{m}},\mathrm{m}=1,2\dots ,\mathrm{n},\hfill \\ {}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\beta }}\mathrm{y}\left(\mathrm{t}\right)+\mathrm{w}\left(\mathrm{t}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{t}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{t}\right)\right)={\mathrm{ϝ}}_{\mathrm{w}}\left(\mathrm{t}\right),\mathrm{t}\ne {\mathrm{t}}_{\mathrm{m}},\mathrm{m}=1,2\dots ,\mathrm{n},\hfill \\ \Delta \mathrm{x}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{M}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}},\Delta {\mathrm{x}}^{\prime }{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{N}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}},\Delta {\mathrm{x}}^{″}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{O}}_{1\mathrm{m}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}},\hfill \\ \Delta \mathrm{y}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{M}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)+{\widehat{\mathrm{ϝ}}}_{\mathrm{m}},\Delta {\mathrm{y}}^{\prime }{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{N}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)+{\widehat{\mathrm{ϝ}}}_{\mathrm{m}},\Delta {\mathrm{y}}^{″}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{O}}_{2\mathrm{m}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right)\right)+{\widehat{\mathrm{ϝ}}}_{\mathrm{m}},\hfill \\ \mathrm{x}\left(0\right)={\mathrm{x}}^{\prime }\left(0\right)=0,{}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\epsilon }}\mathrm{x}\left(\mathsf{\Omega }\right)={\mathrm{x}}^{″}\left(1\right),\hfill \\ \mathrm{y}\left(0\right)={\mathrm{y}}^{\prime }\left(0\right)=0,{}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\rho }}\mathrm{y}\left(\mathsf{\Phi }\right)={\mathrm{y}}^{″}\left(1\right).\hfill \end{array}\right.$$

(33)

Since $(\mathrm{x},\mathrm{y})$ is a solution of the inequality (28) and $\mathrm{t}\in \mathrm{J}$; hence, by Remark 1, we obtain

$$\mathrm{x}\left(\mathrm{t}\right)=\left\{\begin{array}{c}-\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_0^{\mathrm{t}}(\mathrm{t}-\mathrm{s})\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}+{\mathrm{e}}^{\ast }{\mathrm{t}}^2,\mathrm{t}\in [0,{\mathrm{t}}_1],\hfill \\ -\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\alpha }-1}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ -\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-1}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ -\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\sum _{\mathrm{j}=1}^{\mathrm{m}}(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}}){\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-2}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ -\frac{1}{2\mathsf{\Gamma }(\mathsf{\alpha }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})}^2{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ +\sum _{\mathrm{j}=1}^{\mathrm{m}}\left[{\mathrm{M}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}}\right]+\sum _{\mathrm{j}=1}^{\mathrm{m}}(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})\left[{\mathrm{N}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}}\right]+\sum _{\mathrm{j}=1}^{\mathrm{m}}\frac{{(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})}^2}{2}\hfill \\ \times \left[{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}}\right]+{\mathrm{e}}^{\ast }{\mathrm{t}}^2,\mathrm{t}\in \left({\mathrm{t}}_{\mathrm{m}},{\mathrm{t}}_{\mathrm{m}+1}\right],1\le \mathrm{m}\le \mathrm{n}.\hfill \end{array}\right.$$

$$\mathrm{y}\left(\mathrm{t}\right)=\left\{\begin{array}{c}-\frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}{\int }_0^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\beta }-1}\left[\mathrm{w}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{w}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}+{\mathrm{c}}^{\ast }{\mathrm{t}}^2,\mathrm{t}\in [0,{\mathrm{t}}_1],\hfill \\ -\frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\beta }-1}\left[\mathrm{w}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{w}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ -\frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\beta }-1}\mathrm{w}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}\hfill \\ -\frac{1}{\mathsf{\Gamma }(\mathsf{\beta }-1)}\sum _{\mathrm{j}=1}^{\mathrm{m}}(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}}){\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\beta }-2}\mathrm{w}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)\mathrm{d}\mathrm{s}\hfill \\ -\frac{1}{2\mathsf{\Gamma }(\mathsf{\beta }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})}^2{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\beta }-3}\left[\mathrm{w}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{w}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ +\sum _{\mathrm{j}=1}^{\mathrm{m}}\left[{\mathrm{M}}_{2\mathrm{j}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\widehat{\mathrm{ϝ}}}_{\mathrm{m}}\right]+\sum _{\mathrm{j}=1}^{\mathrm{m}}(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})\left[{\mathrm{M}}_{2\mathrm{j}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\widehat{\mathrm{ϝ}}}_{\mathrm{m}}\right]+\sum _{\mathrm{j}=1}^{\mathrm{m}}\frac{{(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})}^2}{2}\hfill \\ \times \left[{\mathrm{O}}_{2\mathrm{j}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\widehat{\mathrm{ϝ}}}_{\mathrm{m}}\right]+{\mathrm{c}}^{\ast }{\mathrm{t}}^2,\mathrm{t}\in \left({\mathrm{t}}_{\mathrm{m}},{\mathrm{t}}_{\mathrm{m}+1}\right],1\le \mathrm{m}\le \mathrm{n},\hfill \end{array}\right.$$

where

$$\begin{array}{cc}\hfill {\mathrm{e}}^{\ast }=& {\mathcal{Z}}_{\mathsf{\alpha }}[-\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon })}{\int }_{{\mathrm{t}}_{\mathrm{j}}}^{\mathsf{\Omega }}{(\mathsf{\Omega }-\mathrm{s})}^{\mathsf{\alpha }-\mathsf{\epsilon }-1}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & -\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-2}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & -\frac{{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}\left[{\mathrm{N}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}}\right]+\frac{{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}\left[{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}}\right]-\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}\hfill \\ \hfill & \times \left[{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}}\right]+\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^1{(1-\mathrm{s})}^{\mathsf{\alpha }-3}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}-\sum _{\mathrm{j}=1}^{\mathrm{m}}\left[{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}}\right]],\hfill \\ \hfill {\mathrm{c}}^{\ast }=& {\mathcal{Z}}_{\mathsf{\beta }}[-\frac{1}{\mathsf{\Gamma }(\mathsf{\beta }-\mathsf{\rho })}{\int }_{{\mathrm{t}}_{\mathrm{j}}}^{\mathsf{\Phi }}{(\mathsf{\Phi }-\mathrm{s})}^{\mathsf{\beta }-\mathsf{\rho }-1}\left[\mathrm{w}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{w}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & -\frac{{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\beta }-2}\left[\mathrm{w}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{w}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & -\frac{{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-2)\mathsf{\Gamma }(3-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\beta }-3}\left[\mathrm{w}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{w}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-2)\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\beta }-3}\left[\mathrm{w}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{w}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}\left[{\mathrm{N}}_{2\mathrm{j}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\widehat{\mathrm{ϝ}}}_{\mathrm{m}}\right]+\frac{{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(3-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}\left[{\mathrm{O}}_{2\mathrm{j}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\widehat{\mathrm{ϝ}}}_{\mathrm{m}}\right]-\frac{{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}\hfill \\ \hfill & \times \left[{\mathrm{O}}_{2\mathrm{j}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\widehat{\mathrm{ϝ}}}_{\mathrm{m}}\right]+\frac{1}{\mathsf{\Gamma }(\mathsf{\beta }-2)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^1{(1-\mathrm{s})}^{\mathsf{\beta }-3}\left[\mathrm{w}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{w}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{1}{\mathsf{\Gamma }(\mathsf{\beta }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\beta }-3}\left[\mathrm{w}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{w}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}-\sum _{\mathrm{j}=1}^{\mathrm{m}}\left[{\mathrm{O}}_{2\mathrm{j}}\left(\mathrm{y}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\widehat{\mathrm{ϝ}}}_{\mathrm{m}}\right]].\hfill \end{array}$$

For $\mathrm{t}\in [0,{\mathrm{t}}_1],$ we have

$$\begin{array}{cc}\hfill \mathrm{x}\left(\mathrm{t}\right)=& \frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_0^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\alpha }-1}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon })}{\int }_{{\mathrm{t}}_{\mathrm{j}}}^{\mathsf{\Omega }}{(\mathsf{\Omega }-\mathrm{s})}^{\mathsf{\alpha }-\mathsf{\epsilon }-1}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-2}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}\left[{\mathrm{N}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}}\right]+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}\left[{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}}\right]\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^1{(1-\mathrm{s})}^{\mathsf{\alpha }-3}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}{\mathrm{x}}_1\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}{\mathrm{y}}_1\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}\left[{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}}\right]+{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2\sum _{\mathrm{j}=1}^{\mathrm{m}}\left[{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}}\right].\hfill \end{array}$$

(34)

For computational convenience, we use $\mathrm{s}\left(\mathrm{t}\right)$ for the sum of terms which are free of $\mathrm{ϝ}$; then, (34) becomes

$$\begin{array}{cc}\hfill & |\mathrm{x}\left(\mathrm{t}\right)-{\mathrm{s}}_1\left(\mathrm{t}\right)|\hfill \\ \hfill \le & \frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_0^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\alpha }-1}|{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon })}{\int }_{{\mathrm{t}}_{\mathrm{j}}}^{\mathsf{\Omega }}{(\mathsf{\Omega }-\mathrm{s})}^{\mathsf{\alpha }-\mathsf{\epsilon }-1}|{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-2}|{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}|{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{t}}_{\mathrm{j}}|{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}|{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{ϝ}}_{\mathrm{m}}|+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{ϝ}}_{\mathrm{m}}|\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^1{(1-\mathrm{s})}^{\mathsf{\alpha }-3}|{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}|{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{t}}_{\mathrm{j}}||{\mathrm{ϝ}}_{\mathrm{m}}|+{\mathcal{Z}}_{\mathsf{\alpha }}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{ϝ}}_{\mathrm{m}}|.\hfill \end{array}$$

By utilizing Remark 1, we get

$$\begin{array}{cc}\hfill & |\mathrm{x}\left(\mathrm{t}\right)-{\mathrm{s}}_1\left(\mathrm{t}\right)|\hfill \\ \hfill \le & [\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon }+1)}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(3-\mathsf{\epsilon })}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{2{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}+\frac{2{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}+{\mathcal{Z}}_{\mathsf{\alpha }}]{\mathsf{ϵ}}_{\mathsf{\alpha }}.\hfill \end{array}$$

(35)

Let

$$\begin{array}{cc}\hfill {\mathcal{Q}}_1& =\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon }+1)}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(3-\mathsf{\epsilon })}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{2{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}+\frac{2{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}+{\mathcal{Z}}_{\mathsf{\alpha }}.\hfill \end{array}$$

Thus, (35) becomes

$$\begin{array}{cc}\hfill |\mathrm{x}\left(\mathrm{t}\right)-{\mathrm{s}}_1\left(\mathrm{t}\right)|& \le {\mathcal{Q}}_1{\mathsf{ϵ}}_{\mathsf{\alpha }}.\hfill \end{array}$$

(36)

Let

$$\begin{array}{c}\hfill |\mathrm{x}\left(\mathrm{t}\right)-\mathsf{\zeta }\left(\mathrm{t}\right)|=|\mathrm{x}\left(\mathrm{t}\right)-{\mathrm{s}}_1\left(\mathrm{t}\right)+{\mathrm{s}}_1\left(\mathrm{t}\right)-\mathsf{\zeta }\left(\mathrm{t}\right)|\le |\mathrm{x}\left(\mathrm{t}\right)-{\mathrm{s}}_1\left(\mathrm{t}\right)|+|{\mathrm{s}}_1\left(\mathrm{t}\right)-\mathsf{\zeta }\left(\mathrm{t}\right)|.\end{array}$$

(37)

Using (36) in (37), we have

$$\begin{array}{cc}\hfill & |\mathrm{x}(\mathrm{t})-\mathsf{\zeta }(\mathrm{t})|\hfill \\ \hfill \le & {\mathcal{Q}}_1{\mathsf{ϵ}}_{\mathsf{\alpha }}+\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_0^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\alpha }-1}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathsf{\zeta }\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathsf{\chi }\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon })}{\int }_{{\mathrm{t}}_{\mathrm{j}}}^{\mathsf{\Omega }}{(\mathsf{\Omega }-\mathrm{s})}^{\mathsf{\alpha }-\mathsf{\epsilon }-1}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathsf{\zeta }\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathsf{\chi }\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-2}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathsf{\zeta }\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathsf{\chi }\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathsf{\zeta }\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathsf{\chi }\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathsf{\zeta }\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathsf{\chi }\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{N}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{N}}_{1\mathrm{j}}\left(\mathsf{\zeta }\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{O}}_{1\mathrm{j}}\left(\mathsf{\zeta }\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^1{(1-\mathrm{s})}^{\mathsf{\alpha }-3}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathsf{\zeta }\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathsf{\chi }\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathsf{\zeta }\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathsf{\chi }\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{t}}_{\mathrm{j}}||{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{O}}_{1\mathrm{j}}\left(\mathsf{\zeta }\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|+{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{O}}_{1\mathrm{j}}\left(\mathsf{\zeta }\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|.\hfill \end{array}$$

Utilizing $\left({\mathbf{M}}_1\right)$ and $\left({\mathbf{M}}_2\right)$, we get

$$\begin{array}{cc}\hfill \le & {\mathcal{Q}}_1{\mathsf{ϵ}}_{\mathsf{\alpha }}+\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}\left[{\mathsf{\mu }}_1{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+{\mathsf{\mu }}_2{\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0\right]+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon }+1)}\left[{\mathsf{\mu }}_1{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+{\mathsf{\mu }}_2{\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0\right]\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\left[{\mathsf{\mu }}_1{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+{\mathsf{\mu }}_2{\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0\right]+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\left[{\mathsf{\mu }}_1{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+{\mathsf{\mu }}_2{\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0\right]\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\left[{\mathsf{\mu }}_1{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+{\mathsf{\mu }}_2{\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0\right]+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{1\mathrm{j}}{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\left[{\mathsf{\mu }}_1{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+{\mathsf{\mu }}_2{\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0\right]+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\hfill \\ \hfill & \times \left[{\mathsf{\mu }}_1{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+{\mathsf{\mu }}_2{\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0\right]+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+{\mathcal{Z}}_{\mathsf{\alpha }}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0.\hfill \end{array}$$

(38)

After some calculation and rearrangement in (38), we get

$$\begin{array}{c}\hfill {\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0-\frac{{\mathsf{\Lambda }}_2}{(1-{\mathsf{\Lambda }}_1)}{\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0\le \frac{{\mathcal{Q}}_1{\mathsf{ϵ}}_{\mathsf{\alpha }}}{(1-{\mathsf{\Lambda }}_1)},\end{array}$$

(39)

where

$$\begin{array}{cc}\hfill {\mathsf{\Lambda }}_1=& {\mathcal{Z}}_{\mathsf{\alpha }}\left[{\mathsf{\mu }}_1\right(\frac{1}{{\mathcal{Z}}_{\mathsf{\alpha }}\mathsf{\Gamma }(\mathsf{\alpha }+1)}+\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon }+1)}+\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(3-\mathsf{\epsilon })}+\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\hfill \\ \hfill & +\frac{2}{\mathsf{\Gamma }(\mathsf{\alpha }-1)})+\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{1\mathrm{j}}+\frac{{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}+\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}],\hfill \\ \hfill {\mathsf{\Lambda }}_2=& {\mathcal{Z}}_{\mathsf{\alpha }}\left[{\mathsf{\mu }}_2\right(\frac{1}{{\mathcal{Z}}_{\mathsf{\alpha }}\mathsf{\Gamma }(\mathsf{\alpha }+1)}+\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon }+1)}+\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(3-\mathsf{\epsilon })}+\frac{{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\hfill \\ \hfill & +\frac{2}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\left)\right].\hfill \end{array}$$

In addition, for $\mathrm{t}\in \left({\mathrm{t}}_{\mathrm{m}},{\mathrm{t}}_{\mathrm{m}+1}\right],$ we have

$$\begin{array}{cc}\hfill \mathrm{x}\left(\mathrm{t}\right)=& \frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\alpha }-1}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-1}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\sum _{\mathrm{j}=1}^{\mathrm{m}}(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}}){\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-2}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{1}{2\mathsf{\Gamma }(\mathsf{\alpha }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})}^2{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\sum _{\mathrm{j}=1}^{\mathrm{m}}\left[{\mathrm{M}}_{1\mathrm{j}}\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)+{\mathrm{ϝ}}_{\mathrm{m}}\right]+\sum _{\mathrm{j}=1}^{\mathrm{m}}(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})\left[{\mathrm{N}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}}\right]\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon })}{\int }_{{\mathrm{t}}_{\mathrm{j}}}^{\mathsf{\Omega }}{(\mathsf{\Omega }-\mathrm{s})}^{\mathsf{\alpha }-\mathsf{\epsilon }-1}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-2}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}\left[{\mathrm{N}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}}\right]+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}\left[{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}}\right]\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^1{(1-\mathrm{s})}^{\mathsf{\alpha }-3}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\left[\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)+{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)\right]\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}\left[{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}}\right]+{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2\sum _{\mathrm{j}=1}^{\mathrm{m}}\left[{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}}\right]\hfill \\ \hfill & +\sum _{\mathrm{j}=1}^{\mathrm{m}}\frac{{(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})}^2}{2}\left[{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)+{\mathrm{ϝ}}_{\mathrm{m}}\right].\hfill \end{array}$$

For computational convenience, we use ${\mathrm{s}}_1^{\ast }\left(\mathrm{t}\right)$ for the sum of terms which are free of $\mathrm{ϝ}$, so we have

$$\begin{array}{cc}\hfill & |\mathrm{x}\left(\mathrm{t}\right)-{\mathrm{s}}_1^{\ast }\left(\mathrm{t}\right)|\hfill \\ \hfill \le & \frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\alpha }-1}|{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}+\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-1}|{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\sum _{\mathrm{j}=1}^{\mathrm{m}}|(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})|{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-2}|{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}+\frac{1}{2\mathsf{\Gamma }(\mathsf{\alpha }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})}^2|{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}\hfill \\ \hfill & \times |{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}+\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{ϝ}}_{\mathrm{m}}|+\sum _{\mathrm{j}=1}^{\mathrm{m}}|(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})||{\mathrm{ϝ}}_{\mathrm{m}}|+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon })}{\int }_{{\mathrm{t}}_{\mathrm{j}}}^{\mathsf{\Omega }}{(\mathsf{\Omega }-\mathrm{s})}^{\mathsf{\alpha }-\mathsf{\epsilon }-1}|{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-2}|{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}|{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{t}}_{\mathrm{j}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}|{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{ϝ}}_{\mathrm{m}}|+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{ϝ}}_{\mathrm{m}}|\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^1{(1-\mathrm{s})}^{\mathsf{\alpha }-3}|{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}|{\mathrm{ϝ}}_{\mathrm{h}}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{t}}_{\mathrm{j}}||{\mathrm{ϝ}}_{\mathrm{m}}|+{\mathcal{Z}}_{\mathsf{\alpha }}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{ϝ}}_{\mathrm{m}}|+\sum _{\mathrm{j}=1}^{\mathrm{m}}\frac{|{(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})}^2|}{2}|{\mathrm{ϝ}}_{\mathrm{m}}|.\hfill \end{array}$$

By utilizing Remark 1, we get

$$\begin{array}{cc}\hfill & |\mathrm{x}\left(\mathrm{t}\right)-{\mathrm{s}}_1^{\ast }\left(\mathrm{t}\right)|\hfill \\ \hfill \le & (\frac{2}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}+\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}+\frac{4{\mathcal{Z}}_{\mathsf{\alpha }}+1}{2\mathsf{\Gamma }(\mathsf{\alpha }-1)}+\frac{3{\mathcal{Z}}_{\mathsf{\alpha }}+4}{2}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon }+1)}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{2{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(3-\mathsf{\epsilon })}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}){\mathsf{ϵ}}_{\mathsf{\alpha }}.\hfill \end{array}$$

(40)

For computational convenience, let

$$\begin{array}{cc}\hfill {\mathcal{Q}}_{\mathsf{\alpha }}=& \frac{2}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}+\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}+\frac{4{\mathcal{Z}}_{\mathsf{\alpha }}+1}{2\mathsf{\Gamma }(\mathsf{\alpha }-1)}+\frac{3{\mathcal{Z}}_{\mathsf{\alpha }}+4}{2}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon }+1)}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{2{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(3-\mathsf{\epsilon })}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}.\hfill \end{array}$$

Thus, (40) becomes

$$\begin{array}{cc}\hfill |\mathrm{x}\left(\mathrm{t}\right)-{\mathrm{s}}_1^{\ast }\left(\mathrm{t}\right)|& \le {\mathcal{Q}}_{\mathsf{\alpha }}{\mathsf{ϵ}}_{\mathsf{\alpha }}.\hfill \end{array}$$

(41)

Let

$$\begin{array}{c}\hfill |\mathrm{x}\left(\mathrm{t}\right)-\mathsf{\zeta }\left(\mathrm{t}\right)|=|\mathrm{x}\left(\mathrm{t}\right)-{\mathrm{s}}_1^{\ast }\left(\mathrm{t}\right)+{\mathrm{s}}_1^{\ast }\left(\mathrm{t}\right)-\mathsf{\zeta }\left(\mathrm{t}\right)|\le |\mathrm{x}\left(\mathrm{t}\right)-{\mathrm{s}}_1^{\ast }\left(\mathrm{t}\right)|+|{\mathrm{s}}_1^{\ast }\left(\mathrm{t}\right)-\mathsf{\zeta }\left(\mathrm{t}\right)|.\end{array}$$

(42)

Using (41) in (42), we get

$$\begin{array}{cc}\hfill & |\mathrm{x}(\mathrm{t})-\mathsf{\zeta }(\mathrm{t})|\hfill \\ \hfill \le & {\mathcal{Q}}_{\mathsf{\alpha }}{\mathsf{ϵ}}_{\mathsf{\alpha }}+\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^{\mathrm{t}}{(\mathrm{t}-\mathrm{s})}^{\mathsf{\alpha }-1}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathsf{\zeta }\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathsf{\chi }\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-1}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathsf{\zeta }\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathsf{\chi }\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\sum _{\mathrm{j}=1}^{\mathrm{m}}|(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})|{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-2}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathsf{\zeta }\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathsf{\chi }\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{1}{2\mathsf{\Gamma }(\mathsf{\alpha }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})}^2|{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathsf{\zeta }\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathsf{\chi }\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{M}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{M}}_{1\mathrm{j}}\left(\mathsf{\zeta }\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|+\sum _{\mathrm{j}=1}^{\mathrm{m}}|(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})||{\mathrm{N}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{N}}_{1\mathrm{j}}\left(\mathsf{\zeta }\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon })}{\int }_{{\mathrm{t}}_{\mathrm{j}}}^{\mathsf{\Omega }}{(\mathsf{\Omega }-\mathrm{s})}^{\mathsf{\alpha }-\mathsf{\epsilon }-1}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathsf{\zeta }\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathsf{\chi }\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-2}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathsf{\zeta }\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathsf{\chi }\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathsf{\zeta }\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathsf{\chi }\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-2)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{t}}_{\mathrm{j}}|{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathsf{\zeta }\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathsf{\chi }\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{N}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{N}}_{1\mathrm{j}}\left(\mathsf{\zeta }\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{O}}_{1\mathrm{j}}\left(\mathsf{\zeta }\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}{\int }_{{\mathrm{t}}_{\mathrm{m}}}^1{(1-\mathrm{s})}^{\mathsf{\alpha }-3}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathsf{\zeta }\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathsf{\chi }\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2}{\mathsf{\Gamma }(\mathsf{\alpha }-2)}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\int }_{{\mathrm{t}}_{\mathrm{j}-1}}^{{\mathrm{t}}_{\mathrm{j}}}{({\mathrm{t}}_{\mathrm{j}}-\mathrm{s})}^{\mathsf{\alpha }-3}|\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{s}\right)\right)-\mathrm{h}\left(\mathrm{s}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathsf{\zeta }\left(\mathrm{s}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathsf{\chi }\left(\mathrm{s}\right)\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{t}}_{\mathrm{j}}||{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{O}}_{1\mathrm{j}}\left(\mathsf{\zeta }\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|+{\mathcal{Z}}_{\mathsf{\alpha }}{\mathrm{t}}^2\sum _{\mathrm{j}=1}^{\mathrm{m}}|{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{O}}_{1\mathrm{j}}\left(\mathsf{\zeta }\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|\hfill \\ \hfill & +\sum _{\mathrm{j}=1}^{\mathrm{m}}\frac{|{(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})}^2|}{2}|{\mathrm{O}}_{1\mathrm{j}}\left(\mathrm{x}\left({\mathrm{t}}_{\mathrm{j}}\right)\right)-{\mathrm{O}}_{1\mathrm{j}}\left(\mathsf{\zeta }\left({\mathrm{t}}_{\mathrm{j}}\right)\right)|.\hfill \end{array}$$

Using $\left({\mathbf{M}}_1\right)$ and $\left({\mathbf{M}}_2\right)$, we have

$$\begin{array}{cc}\hfill & \le {\mathcal{Q}}_{\mathsf{\alpha }}{\mathsf{ϵ}}_{\mathsf{\alpha }}+\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}\left[{\mathsf{\mu }}_1{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+{\mathsf{\mu }}_2{\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0\right]+\frac{1}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}\left[{\mathsf{\mu }}_1{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+{\mathsf{\mu }}_2{\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0\right]\hfill \\ \hfill & +\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}\left[{\mathsf{\mu }}_1{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+{\mathsf{\mu }}_2{\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0\right]+\frac{1}{2\mathsf{\Gamma }(\mathsf{\alpha }-1)}\left[{\mathsf{\mu }}_1{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+{\mathsf{\mu }}_2{\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0\right]\hfill \\ \hfill & +\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{I}}_{1\mathrm{j}}{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{1\mathrm{j}}{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon }+1)}\left[{\mathsf{\mu }}_1{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+{\mathsf{\mu }}_2{\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0\right]\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\left[{\mathsf{\mu }}_1{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+{\mathsf{\mu }}_2{\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0\right]+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\left[{\mathsf{\mu }}_1{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+{\mathsf{\mu }}_2{\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0\right]\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}\left[{\mathsf{\mu }}_1{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+{\mathsf{\mu }}_2{\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0\right]+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{1\mathrm{j}}{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\left[{\mathsf{\mu }}_1{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+{\mathsf{\mu }}_2{\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0\right]\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}\left[{\mathsf{\mu }}_1{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+{\mathsf{\mu }}_2{\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0\right]+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+{\mathcal{Z}}_{\mathsf{\alpha }}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0\hfill \\ \hfill & +\frac{1}{2}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0.\hfill \end{array}$$

(43)

After some calculation and rearrangement in (43), we get

$$\begin{array}{c}\hfill {\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0-\frac{{\mathsf{\Lambda }}_2^{\ast }}{(1-{\mathsf{\Lambda }}_1^{\ast })}{\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0\le \frac{{\mathcal{Q}}_{\mathsf{\alpha }}{\mathsf{ϵ}}_{\mathsf{\alpha }}}{(1-{\mathsf{\Lambda }}_1^{\ast })},\end{array}$$

(44)

where

$$\begin{array}{cc}\hfill {\mathsf{\Lambda }}_1^{\ast }=& \left[{\mathsf{\mu }}_1\right(\frac{2}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}+\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}+\frac{1}{2\mathsf{\Gamma }(\mathsf{\alpha }-1)}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon }+1)}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{2{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)})+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{I}}_{1\mathrm{j}}+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{1\mathrm{j}}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{1\mathrm{j}}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}+{\mathcal{Z}}_{\mathsf{\alpha }}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}+\frac{1}{2}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}],\hfill \\ \hfill {\mathsf{\Lambda }}_2^{\ast }=& {\mathsf{\mu }}_2(\frac{2}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}+\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}+\frac{1}{2\mathsf{\Gamma }(\mathsf{\alpha }-1)}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon }+1)}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{2{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)}).\hfill \end{array}$$

On the similar fashion, for $\mathrm{t}\in [0,{\mathrm{t}}_1],$ and utilizing $\left({\mathbf{M}}_1\right)-\left({\mathbf{M}}_2\right)$, we can find

$$\begin{array}{cc}\hfill |\mathrm{y}\left(\mathrm{t}\right)-{\mathrm{s}}_2\left(\mathrm{t}\right)|& \le {\mathcal{Q}}_2{\mathsf{ϵ}}_{\mathsf{\beta }},\hfill \end{array}$$

(45)

where ${\mathrm{s}}_2\left(\mathrm{t}\right)$ are those terms which are free of $\mathrm{ϝ}$ and

$$\begin{array}{cc}\hfill {\mathcal{Q}}_2& =\frac{1}{\mathsf{\Gamma }(\mathsf{\beta }+1)}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}}{\mathsf{\Gamma }(\mathsf{\beta }-\mathsf{\rho }+1)}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)\mathsf{\Gamma }(2-\mathsf{\rho })}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(3-\mathsf{\rho })}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(2-\mathsf{\rho })}+\frac{2{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(2-\mathsf{\rho })}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(3-\mathsf{\rho })}+\frac{2{\mathcal{Z}}_{\mathsf{\beta }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)}+{\mathcal{Z}}_{\mathsf{\beta }},\hfill \end{array}$$

and

$$\begin{array}{c}\hfill {\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0-\frac{{\mathsf{\Lambda }}_4}{(1-{\mathsf{\Lambda }}_3)}{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0\le \frac{{\mathcal{Q}}_2{\mathsf{ϵ}}_{\mathsf{\beta }}}{(1-{\mathsf{\Lambda }}_3)},\end{array}$$

(46)

where

$$\begin{array}{cc}\hfill {\mathsf{\Lambda }}_3=& {\mathcal{Z}}_{\mathsf{\beta }}\left[{\mathsf{\mu }}_1^{\prime }\right(\frac{1}{{\mathcal{Z}}_{\mathsf{\beta }}\mathsf{\Gamma }(\mathsf{\beta }+1)}+\frac{1}{\mathsf{\Gamma }(\mathsf{\beta }-\mathsf{\rho }+1)}+\frac{{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)\mathsf{\Gamma }(2-\mathsf{\rho })}+\frac{{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(3-\mathsf{\rho })}+\frac{{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(2-\mathsf{\rho })}\hfill \\ \hfill & +\frac{2}{\mathsf{\Gamma }(\mathsf{\beta }-1)})+\frac{{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{2\mathrm{j}}+\frac{{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(3-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}+\frac{{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}],\hfill \\ \hfill {\mathsf{\Lambda }}_4=& {\mathcal{Z}}_{\mathsf{\beta }}\left[{\mathsf{\mu }}_2^{\prime }\right(\frac{1}{{\mathcal{Z}}_{\mathsf{\beta }}\mathsf{\Gamma }(\mathsf{\beta }+1)}+\frac{1}{\mathsf{\Gamma }(\mathsf{\beta }-\mathsf{\rho }+1)}+\frac{{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)\mathsf{\Gamma }(2-\mathsf{\rho })}+\frac{{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(3-\mathsf{\rho })}+\frac{{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(2-\mathsf{\rho })}\hfill \\ \hfill & +\frac{2}{\mathsf{\Gamma }(\mathsf{\beta }-1)}\left)\right].\hfill \end{array}$$

In addition, for $\mathrm{t}\in \left({\mathrm{t}}_{\mathrm{m}},{\mathrm{t}}_{\mathrm{m}+1}\right],1\le \mathrm{m}\le \mathrm{n}$, we can get

$$\begin{array}{cc}\hfill |\mathrm{y}\left(\mathrm{t}\right)-{\mathrm{s}}_2^{\ast }\left(\mathrm{t}\right)|& \le {\mathcal{Q}}_{\mathsf{\beta }}{\mathsf{ϵ}}_{\mathsf{\beta }},\hfill \end{array}$$

(47)

where ${\mathrm{s}}_2^{\ast }\left(\mathrm{t}\right)$ are those terms which are free of $\mathrm{ϝ}$ and

$$\begin{array}{cc}\hfill {\mathcal{Q}}_{\mathsf{\beta }}=& \frac{2}{\mathsf{\Gamma }(\mathsf{\beta }+1)}+\frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}+\frac{4{\mathcal{Z}}_{\mathsf{\beta }}+1}{2\mathsf{\Gamma }(\mathsf{\beta }-1)}+\frac{3{\mathcal{Z}}_{\mathsf{\beta }}+4}{2}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}}{\mathsf{\Gamma }(\mathsf{\beta }-\mathsf{\rho }+1)}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)\mathsf{\Gamma }(2-\mathsf{\rho })}+\frac{2{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(2-\mathsf{\rho })}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(3-\mathsf{\rho })}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(3-\mathsf{\rho })}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(2-\mathsf{\rho })}.\hfill \end{array}$$

In addition,

$$\begin{array}{c}\hfill {\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0-\frac{{\mathsf{\Lambda }}_4^{\ast }}{(1-{\mathsf{\Lambda }}_3^{\ast })}{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0\le \frac{{\mathcal{Q}}_{\mathsf{\beta }}{\mathsf{ϵ}}_{\mathsf{\beta }}}{(1-{\mathsf{\Lambda }}_3^{\ast })},\end{array}$$

(48)

where

$$\begin{array}{cc}\hfill {\mathsf{\Lambda }}_3^{\ast }=& \left[{\mathsf{\mu }}_2^{\prime }\right(\frac{2}{\mathsf{\Gamma }(\mathsf{\beta }+1)}+\frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}+\frac{1}{2\mathsf{\Gamma }(\mathsf{\beta }-1)}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}}{\mathsf{\Gamma }(\mathsf{\beta }-\mathsf{\rho }+1)}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)\mathsf{\Gamma }(2-\mathsf{\rho })}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(3-\mathsf{\rho })}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(2-\mathsf{\rho })}+\frac{2{\mathcal{Z}}_{\mathsf{\beta }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)})+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{I}}_{2\mathrm{j}}+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{2\mathrm{j}}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{2\mathrm{j}}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(3-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}+{\mathcal{Z}}_{\mathsf{\beta }}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}+\frac{1}{2}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}],\hfill \\ \hfill {\mathsf{\Lambda }}_4^{\ast }=& {\mathsf{\mu }}_1^{\prime }(\frac{2}{\mathsf{\Gamma }(\mathsf{\beta }+1)}+\frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}+\frac{1}{2\mathsf{\Gamma }(\mathsf{\beta }-1)}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}}{\mathsf{\Gamma }(\mathsf{\beta }-\mathsf{\rho }+1)}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)\mathsf{\Gamma }(2-\mathsf{\rho })}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(3-\mathsf{\rho })}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(2-\mathsf{\rho })}+\frac{2{\mathcal{Z}}_{\mathsf{\beta }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)}).\hfill \end{array}$$

The equivalent matrix of Equations (44) and (48) is given as:

$$\begin{array}{c}\left[\begin{array}{cc}1& -\frac{{\mathsf{\Lambda }}_2^{\ast }}{(1-{\mathsf{\Lambda }}_1^{\ast })}\\ -\frac{{\mathsf{\Lambda }}_4^{\ast }}{(1-{\mathsf{\Lambda }}_3^{\ast })}& 1\end{array}\right]\left[\begin{array}{c}{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0\\ {\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0\end{array}\right]\le \left[\begin{array}{c}\frac{{\mathcal{Q}}_{\mathsf{\alpha }}{\mathsf{ϵ}}_{\mathsf{\alpha }}}{(1-{\mathsf{\Lambda }}_1^{\ast })}\\ \frac{{\mathcal{Q}}_{\mathsf{\beta }}{\mathsf{ϵ}}_{\mathsf{\beta }}}{(1-{\mathsf{\Lambda }}_3^{\ast })}\end{array}\right].\hfill \end{array}$$

Solving the above inequality, we get

$$\begin{array}{c}\left[\begin{array}{c}{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0\\ {\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0\end{array}\right]\le \left[\begin{array}{cc}\frac{1}{{\mathsf{\Lambda }}_0}& \frac{{\mathsf{\Lambda }}_2^{\ast }}{{\mathsf{\Lambda }}_0(1-{\mathsf{\Lambda }}_1^{\ast })}\\ \frac{{\mathsf{\Lambda }}_4^{\ast }}{{\mathsf{\Lambda }}_0(1-{\mathsf{\Lambda }}_3^{\ast })}& \frac{1}{{\mathsf{\Lambda }}_0}\end{array}\right]\left[\begin{array}{c}\frac{{\mathcal{Q}}_{\mathsf{\alpha }}{\mathsf{ϵ}}_{\mathsf{\alpha }}}{(1-{\mathsf{\Lambda }}_1^{\ast })}\\ \frac{{\mathcal{Q}}_{\mathsf{\beta }}{\mathsf{ϵ}}_{\mathsf{\beta }}}{(1-{\mathsf{\Lambda }}_3^{\ast })}\end{array}\right],\hfill \end{array}$$

where

$${\mathsf{\Lambda }}_0=1-\frac{{\mathsf{\Lambda }}_2^{\ast }{\mathsf{\Lambda }}_4^{\ast }}{(1-{\mathsf{\Lambda }}_1^{\ast })(1-{\mathsf{\Lambda }}_3^{\ast })}>0.$$

Further simplification of the above system gives

$$\begin{array}{c}{\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0\le \frac{{\mathcal{Q}}_{\mathsf{\alpha }}{\mathsf{ϵ}}_{\mathsf{\alpha }}}{{\mathsf{\Lambda }}_0(1-{\mathsf{\Lambda }}_1^{\ast })}+\frac{{\mathsf{\Lambda }}_2^{\ast }{\mathcal{Q}}_{\mathsf{\beta }}{\mathsf{ϵ}}_{\mathsf{\beta }}}{{\mathsf{\Lambda }}_0(1-{\mathsf{\Lambda }}_1^{\ast })(1-{\mathsf{\Lambda }}_3^{\ast })},\hfill \\ {\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0\le \frac{{\mathcal{Q}}_{\mathsf{\beta }}{\mathsf{ϵ}}_{\mathsf{\beta }}}{{\mathsf{\Lambda }}_0(1-{\mathsf{\Lambda }}_3^{\ast })}+\frac{{\mathsf{\Lambda }}_4^{\ast }{\mathcal{Q}}_{\mathsf{\alpha }}{\mathsf{ϵ}}_{\mathsf{\alpha }}}{{\mathsf{\Lambda }}_0(1-{\mathsf{\Lambda }}_1^{\ast })(1-{\mathsf{\Lambda }}_3^{\ast })},\hfill \end{array}$$

from which we have

$$\begin{array}{cc}\hfill {\parallel \mathrm{x}-\mathsf{\zeta }\parallel }_0+{\parallel \mathrm{y}-\mathsf{\chi }\parallel }_0& \le \frac{{\mathcal{Q}}_{\mathsf{\alpha }}{\mathsf{ϵ}}_{\mathsf{\alpha }}}{{\mathsf{\Lambda }}_0(1-{\mathsf{\Lambda }}_1^{\ast })}+\frac{{\mathcal{Q}}_{\mathsf{\beta }}{\mathsf{ϵ}}_{\mathsf{\beta }}}{{\mathsf{\Lambda }}_0(1-{\mathsf{\Lambda }}_3^{\ast })}+\frac{{\mathsf{\Lambda }}_4^{\ast }{\mathcal{Q}}_{\mathsf{\alpha }}{\mathsf{ϵ}}_{\mathsf{\alpha }}}{{\mathsf{\Lambda }}_0(1-{\mathsf{\Lambda }}_1^{\ast })(1-{\mathsf{\Lambda }}_3^{\ast })}+\frac{{\mathsf{\Lambda }}_2^{\ast }{\mathcal{Q}}_{\mathsf{\beta }}{\mathsf{ϵ}}_{\mathsf{\beta }}}{{\mathsf{\Lambda }}_0(1-{\mathsf{\Lambda }}_1^{\ast })(1-{\mathsf{\Lambda }}_3^{\ast })}.\hfill \end{array}$$

(49)

Let $\max \left\{{\mathsf{ϵ}}_{\mathsf{\alpha }},{\mathsf{ϵ}}_{\mathsf{\beta }}\right\}=\mathsf{ϵ};$ then, from (49), we get

$$\parallel (\mathrm{x}-\mathsf{\zeta },\mathrm{y}-\mathsf{\chi })\parallel \le {\mathbf{K}}_{\mathsf{\alpha },\mathsf{\beta }}\mathsf{ϵ},$$

which implies that

$$\parallel (\mathrm{x},\mathrm{y})-(\mathsf{\zeta },\mathsf{\chi })\parallel \le {\mathbf{K}}_{\mathsf{\alpha },\mathsf{\beta }}\mathsf{ϵ},$$

(50)

where

$${\mathbf{K}}_{\mathsf{\alpha },\mathsf{\beta }}=\left[\frac{{\mathcal{Q}}_{\mathsf{\alpha }}}{{\mathsf{\Lambda }}_0(1-{\mathsf{\Lambda }}_1^{\ast })}+\frac{{\mathcal{Q}}_{\mathsf{\beta }}}{{\mathsf{\Lambda }}_0(1-{\mathsf{\Lambda }}_3^{\ast })}+\frac{{\mathsf{\Lambda }}_4^{\ast }{\mathcal{Q}}_{\mathsf{\alpha }}}{{\mathsf{\Lambda }}_0(1-{\mathsf{\Lambda }}_1^{\ast })(1-{\mathsf{\Lambda }}_3^{\ast })}+\frac{{\mathsf{\Lambda }}_2^{\ast }{\mathcal{Q}}_{\mathsf{\beta }}}{{\mathsf{\Lambda }}_0(1-{\mathsf{\Lambda }}_1^{\ast })(1-{\mathsf{\Lambda }}_3^{\ast })}\right].$$

Hence, problem (1) is Ulam–Hyers stable. Moreover, if we set $\mathrm{\Theta }\left(\mathsf{ϵ}\right)={\mathbf{K}}_{\mathrm{x},\mathrm{y}}\mathsf{ϵ}$; $\mathrm{\Theta }\left(0\right)=0$ in (50), then problem (1) is generalized Ulam–Hyers stable. □

$\left({\mathbf{M}}_5\right)$: Let ${\mathsf{\Psi }}_{\mathsf{\alpha }},{\mathsf{\Psi }}_{\mathsf{\beta }}\in \mathrm{P}\mathrm{C}(\mathrm{J},{\mathbb{R}}^{+})$ be nondecreasing functions; then, for $\mathrm{t}\in \mathrm{J}$, there are ${\mathrm{\lambda }}_{\mathsf{\alpha }},{\mathrm{\lambda }}_{\mathsf{\beta }}>0$ such that

$$\begin{array}{cc}\hfill {\mathbf{I}}^{\mathsf{\alpha }}{\mathsf{\Psi }}_{\mathsf{\alpha }}\left(\mathrm{t}\right)& \le {\mathrm{\lambda }}_{\mathsf{\alpha }}{\mathsf{\Psi }}_{\mathsf{\alpha }}\left(\mathrm{t}\right),{\mathbf{I}}^{\mathsf{\alpha }-1}{\mathsf{\Psi }}_{\mathsf{\alpha }}\left(\mathrm{t}\right)\le {\mathrm{\lambda }}_{\mathsf{\alpha }}{\mathsf{\Psi }}_{\mathsf{\alpha }}\left(\mathrm{t}\right),\hfill \\ \hfill {\mathbf{I}}^{\mathsf{\alpha }-2}{\mathsf{\Psi }}_{\mathsf{\alpha }}\left(\mathrm{t}\right)& \le {\mathrm{\lambda }}_{\mathsf{\alpha }}{\mathsf{\Psi }}_{\mathsf{\alpha }}\left(\mathrm{t}\right),{\mathbf{I}}^{\mathsf{\alpha }-\mathsf{\epsilon }}{\mathsf{\Psi }}_{\mathsf{\alpha }}\left(\mathrm{t}\right)\le {\mathrm{\lambda }}_{\mathsf{\alpha }}{\mathsf{\Psi }}_{\mathsf{\alpha }}\left(\mathrm{t}\right).\hfill \end{array}$$

Similarly,

$$\begin{array}{cc}\hfill {\mathbf{I}}^{\mathsf{\beta }}{\mathsf{\Psi }}_{\mathsf{\beta }}\left(\mathrm{t}\right)& \le {\mathrm{\lambda }}_{\mathsf{\beta }}{\mathsf{\Psi }}_{\mathsf{\beta }}\left(\mathrm{t}\right),{\mathbf{I}}^{\mathsf{\beta }-1}{\mathsf{\Psi }}_{\mathsf{\beta }}\left(\mathrm{t}\right)\le {\mathrm{\lambda }}_{\mathsf{\beta }}{\mathsf{\Psi }}_{\mathsf{\beta }}\left(\mathrm{t}\right),\hfill \\ \hfill {\mathbf{I}}^{\mathsf{\beta }-2}{\mathsf{\Psi }}_{\mathsf{\beta }}\left(\mathrm{t}\right)& \le {\mathrm{\lambda }}_{\mathsf{\beta }}{\mathsf{\Psi }}_{\mathsf{\beta }}\left(\mathrm{t}\right),{\mathbf{I}}^{\mathsf{\beta }-\mathsf{\rho }}{\mathsf{\Psi }}_{\mathsf{\beta }}\left(\mathrm{t}\right)\le {\mathrm{\lambda }}_{\mathsf{\beta }}{\mathsf{\Psi }}_{\mathsf{\beta }}\left(\mathrm{t}\right).\hfill \end{array}$$

Theorem 5.

Assume that $\left({\mathbf{M}}_1\right)-\left({\mathbf{M}}_2\right)$ and $\left({\mathbf{M}}_5\right)$ are satisfied; then, by Definition 6 and Definition 7, Problem (1) is Ulam–Hyers–Rassias stable with respect to $({\mathsf{\Psi }}_{\mathsf{\alpha },\mathsf{\beta }},{\mathsf{\phi }}_{\mathsf{\alpha },\mathsf{\beta }})$, as well as generalized Ulam–Hyers– Rassias stable.

5. Example

To substantiate the aforemention demonstrated theory, we supply the following problem:

$$\left\{\begin{array}{c}{}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\alpha }}\mathrm{x}\left(\mathrm{t}\right)+\mathrm{h}\left(\mathrm{t}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{t}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{t}\right)\right)=0,\mathrm{t}\ne {\mathrm{t}}_{\mathrm{m}},\hfill \\ {}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\beta }}\mathrm{y}\left(\mathrm{t}\right)+\mathrm{w}\left(\mathrm{t}{,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{a}}\mathrm{x}\left(\mathrm{t}\right){,}^{\mathrm{c}}{\mathbf{D}}^{\mathrm{b}}\mathrm{y}\left(\mathrm{t}\right)\right)=0,\hfill \\ \Delta \mathrm{x}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{M}}_{1\mathrm{m}}\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right),\Delta {\mathrm{x}}^{\prime }{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{N}}_{1\mathrm{m}}\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right),\Delta {\mathrm{x}}^{″}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{O}}_{1\mathrm{m}}\mathrm{x}\left({\mathrm{t}}_{\mathrm{m}}\right),\hfill \\ \Delta \mathrm{y}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{M}}_{2\mathrm{m}}\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right),\Delta {\mathrm{y}}^{\prime }{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{N}}_{2\mathrm{m}}\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right),\Delta {\mathrm{y}}^{″}{\mid }_{\mathrm{t}={\mathrm{t}}_{\mathrm{m}}}={\mathrm{O}}_{2\mathrm{m}}\mathrm{y}\left({\mathrm{t}}_{\mathrm{m}}\right),\hfill \\ \mathrm{x}\left(\mathrm{t}\right){=}^{\prime }\mathrm{x}\left(\mathrm{t}\right)=0,{}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\epsilon }}\mathrm{x}\left(\mathsf{\Omega }\right)={\mathrm{x}}^{″}\left(1\right),\mathrm{y}\left(0\right)={\mathrm{y}}^{\prime }\left(0\right)=0,{}^{\mathrm{c}}{\mathbf{D}}^{\mathsf{\rho }}\mathrm{y}\left(\mathsf{\Phi }\right)={\mathrm{y}}^{″}\left(1\right),\hfill \\ 0<\mathsf{\epsilon },\mathsf{\Omega }<1,0<\mathsf{\rho },\mathsf{\Phi }<1.\hfill \end{array}\right.$$

(51)

Take $\mathrm{J}=[0,1]$, $\mathsf{\alpha }=\frac{5}{2},\mathsf{\beta }=\sqrt{5},\mathsf{\epsilon }=\frac{1}{2},\mathsf{\Omega }=\frac{1}{3},\mathrm{a}=\frac{1}{2},\mathrm{b}=\frac{1}{8},{\mathrm{t}}_1=\frac{1}{2}$, $\mathsf{\rho }=\frac{1}{3},\mathsf{\Phi }=\frac{1}{5}$, $\mathrm{h}(\mathrm{t},\mathrm{x},\mathrm{y})={\mathrm{e}}^{\mathrm{t}}+\frac{|\mathrm{x}|+|\mathrm{y}|}{40\left(1+|\mathrm{x}|+|\mathrm{y}|\right)},\mathrm{w}(\mathrm{t},\mathrm{x},\mathrm{y})=\frac{\cos \mathrm{t}+|\mathrm{x}|+|\mathrm{y}|}{40\left(1+|\mathrm{x}|+|\mathrm{y}|\right)},$ ${\mathrm{M}}_{11}\left(\mathrm{x}\right)={\mathrm{M}}_{21}\left(\mathrm{x}\right)=\frac{|\mathrm{x}|}{5+|\mathrm{x}|},$ ${\mathrm{N}}_{11}\left(\mathrm{x}\right)={\mathrm{N}}_{21}\left(\mathrm{x}\right)=\frac{|\mathrm{x}|}{10+|\mathrm{x}|}$, ${\mathrm{O}}_{11}\left(\mathrm{x}\right)={\mathrm{O}}_{21}\left(\mathrm{x}\right)=\frac{|\mathrm{x}|}{15+|\mathrm{x}|}$. By direct computation, we have ${\mathsf{\mu }}_1={\mathsf{\mu }}_2={\mathsf{\mu }}_1^{\prime }={\mathsf{\mu }}_2^{\prime }=\frac{1}{40}$, ${\mathrm{I}}_{11}={\mathrm{I}}_{21}=\frac{1}{5}$, ${\widehat{\mathrm{I}}}_{11}={\widehat{\mathrm{I}}}_{21}=\frac{1}{10},$ ${\check{\mathrm{I}}}_{11}={\check{\mathrm{I}}}_{21}=\frac{1}{15},$

$$\begin{array}{cc}\hfill & \left[\left({\mathsf{\mu }}_1+{\mathsf{\mu }}_2\right)\right(\frac{2}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}+\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}+\frac{1}{2\mathsf{\Gamma }(\mathsf{\alpha }-1)}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-\mathsf{\epsilon }+1)}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(3-\mathsf{\epsilon })}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)\mathsf{\Gamma }(2-\mathsf{\epsilon })}+\frac{2{\mathcal{Z}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }(\mathsf{\alpha }-1)})+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{I}}_{1\mathrm{j}}+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{1\mathrm{j}}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{1\mathrm{j}}+\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{2-\mathsf{\epsilon }}}{\mathsf{\Gamma }(3-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\alpha }}{\mathsf{\Omega }}^{1-\mathsf{\epsilon }}}{\mathsf{\Gamma }(2-\mathsf{\epsilon })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}+{\mathcal{Z}}_{\mathsf{\alpha }}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}+\frac{1}{2}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{1\mathrm{j}}]\approx 0.680277<1.\hfill \end{array}$$

Similarly,

$$\begin{array}{cc}\hfill & \left[\left({\mathsf{\mu }}_1^{\prime }+{\mathsf{\mu }}_2^{\prime }\right)\right(\frac{2}{\mathsf{\Gamma }(\mathsf{\beta }+1)}+\frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}+\frac{1}{2\mathsf{\Gamma }(\mathsf{\beta }-1)}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}}{\mathsf{\Gamma }(\mathsf{\beta }-\mathsf{\rho }+1)}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)\mathsf{\Gamma }(2-\mathsf{\rho })}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(3-\mathsf{\rho })}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)\mathsf{\Gamma }(2-\mathsf{\rho })}+\frac{2{\mathcal{Z}}_{\mathsf{\beta }}}{\mathsf{\Gamma }(\mathsf{\beta }-1)})+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{I}}_{2\mathrm{j}}+\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{2\mathrm{j}}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\widehat{\mathrm{I}}}_{2\mathrm{j}}+\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{2-\mathsf{\rho }}}{\mathsf{\Gamma }(3-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}\hfill \\ \hfill & +\frac{{\mathcal{Z}}_{\mathsf{\beta }}{\mathsf{\Phi }}^{1-\mathsf{\rho }}}{\mathsf{\Gamma }(2-\mathsf{\rho })}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}+{\mathcal{Z}}_{\mathsf{\beta }}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}+\frac{1}{2}\sum _{\mathrm{j}=1}^{\mathrm{m}}{\check{\mathrm{I}}}_{2\mathrm{j}}]\approx 0.427420<1.\hfill \end{array}$$

Thus, by Theorem 2, Problem (51) has a unique solution. Furthermore, all of the assumptions are satisfied, so the Problem (51) is Ulam–Hyers, generalized Ulam–Hyers, Ulam–Hyers–Rassias and generalized Ulam–Hyers–Rassias stable.

6. Conclusions

In the above study, we have successfully built up existence theory for the solutions of system (1). The required analysis has been developed with the help of the Banach contraction principle and Schauder fixed point theorem. We found that the fractional order coupled system is additionally complicated and challenging as compared to the single FDEs. We also concluded that, if we increase the order or boundary conditions, then the end result turns into extra accurate. Our results are new and fascinating. Our methods can be used to study the existence of solutions for the high order or multiple-point boundary value systems of a nonlinear coupled system of FDEs. Furthermore, we have presented different kinds of Ulam–Hyers stability results for the solution of the considered system (1). In addition, we have presented our main theoretical results with the help of an example. In the future, this concept can be extended to more applied and complicated problems of applied nature. The obtained results can be used in fields like numerical analysis and managerial sciences including business mathematics and economics, etc.