A Novel Implementation of Dhage’s Fixed Point Theorem to Nonlinear Sequential Hybrid Fractional Differential Equation

Abstract

In this work, the existence and uniqueness of solutions to a sequential fractional (Hybrid) differential equation with hybrid boundary conditions were investigated by the generalization of Dhage’s fixed point theorem and Banach contraction mapping, respectively. In addition, the U-H technique is employed to verify the stability of this solution. This study ends with two examples illustrating the theoretical findings.

1. Introduction

The theory of fractional calculus is an interesting field to explore in recent years. Additionally, this theory has many applications to describe many events in the real world and deal with a group of phenomena in several fields such as blood flow phenomena, mechanics, biophysics, automatic, aerodynamics, some branches of medicine, and electronics. For instance, the authors of [1] discussed the applicability of fractional differential equations in electric circuits, and in 2019, Saqib, M. et al. applied the fractional differential equation to heat transfer in a hybrid nanofluid, see [2]. For more details, one can refer to [3,4,5,6,7,8,9,10,11,12,13,14,15,16].

In addition to the great importance of studying the existence of solutions to fractional differential equations using the many theories of the fixed point, several studies have been conducted over the years to investigate how stability concepts such as the Mittag–Leffler function, exponential, and Lyapunov stability apply to various types of dynamic systems. Ulam and Hyers, on the other hand, identified previously unknown types of stability known as Ulam-stability [17]. This example is not exclusive, as many similar works can be found in [18,19,20,21,22,23,24,25,26,27].

To name a few studies on hybrid fractional equations, recently, the authors in [28] verified the existence and uniqueness of a solution for the following IVP:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}_a{\mathcal{D}}^{\widehat{\nu },\varrho ,v}\left(\frac{u\left({\chi }_1\right)}{\mathsf{\Psi }({\chi }_1,x\left({\chi }_1\right))}\right)=\mathsf{\Phi }({\chi }_1,u\left({\chi }_1\right)),{\chi }_1\in J:=[a,b],\hfill \\ {}_a{I}^{1-\widehat{\nu },\varrho ,v}{\left(\frac{u\left({\chi }_1\right)}{\mathsf{\Psi }({\chi }_1,x\left({\chi }_1\right))}\right)}_{{\chi }_1=a}=\lambda \in \mathbb{R},\hfill \end{array}\right.\end{array}$$

where $0<\widehat{\nu }\le 1,\varrho \in (0,1]{,}_a{\mathcal{D}}^{\widehat{\nu },\varrho ,v}$ is the proportional fractional derivative of order $\widehat{\nu },$ with respect to a certain continuously differentiable and increasing function v with $v^{\prime }\left(\theta \right)>0$ for all ${\chi }_1\in J{,}_a{I}^{1-\widehat{\nu },\varrho ,v}$ is the proportional fractional integral of order of $(1-\widehat{\nu })$ with respect to a continuously differentiable and increasing function $v,\mathsf{\Psi }:J\times \mathbb{R}⟶\mathbb{R}\backslash \left\{0\right\}$ and $\mathsf{\Phi }:J\times \mathbb{R}⟶\mathbb{R}\backslash$ are continuous functions.

In [29], the stability, existence, and uniqueness were found for the following hybrid FDE:

$$\begin{array}{ccc}\hfill {}^{CH}{\mathcal{D}}^p\left(\frac{u\left({\chi }_1\right)}{Z_1({\chi }_1,u\left({\chi }_1\right),v\left({\chi }_1\right))}\right)& =& {\widehat{\mathsf{\Psi }}}_1({\chi }_1,u\left({\chi }_1\right),v\left({\chi }_1\right)),{\chi }_1\in J:=[1,e],p\in (1,2]\hfill \\ \hfill {}^{CH}{\mathcal{D}}^q\left(\frac{u\left({\chi }_1\right)}{Z_2({\chi }_1,u\left({\chi }_1\right),v\left({\chi }_1\right))}\right)& =& {\widehat{\mathsf{\Psi }}}_2({\chi }_1,u\left({\chi }_1\right),v\left({\chi }_1\right)),q\in (1,2]\hfill \\ \hfill \end{array}$$

Subject to the following boundary conditions:

$$\begin{array}{ccc}\hfill {\left(\frac{u\left({\chi }_1\right)}{Z_1({\chi }_1,u\left({\chi }_1\right),v\left({\chi }_1\right))}\right)}_{{\chi }_1=1}& =& 0,\hfill \\ \hfill {}^{CH}D{\left(\frac{u\left({\chi }_1\right)}{Z_1({\chi }_1,u\left({\chi }_1\right),v\left({\chi }_1\right))}\right)}_{{\chi }_1=e}& =& {\mathsf{\Theta }}_2^{CH}D{\left(\frac{v\left({\chi }_1\right)}{Z_1({\chi }_1,u\left({\chi }_1\right),v\left({\chi }_1\right))}\right)}_{{\chi }_1={\vartheta }_1},\hfill \\ \hfill {\left(\frac{v\left({\chi }_1\right)}{Z_2({\chi }_1,u\left({\chi }_1\right),v\left({\chi }_1\right))}\right)}_{{\chi }_1=1}& =& 0,\hfill \\ \hfill {}^{CH}D{\left(\frac{v\left({\chi }_1\right)}{Z_2({\chi }_1,u\left({\chi }_1\right),v\left({\chi }_1\right))}\right)}_{{\chi }_1=e}& =& {\mathsf{\Theta }}_1^{CH}D{\left(\frac{v\left({\chi }_1\right)}{Z_1({\chi }_1,u\left({\chi }_1\right),v\left({\chi }_1\right))}\right)}_{{\chi }_1={\vartheta }_2},\hfill \end{array}$$

where ${}^{CH}{\mathcal{D}}^{\gamma },\gamma \in \{p,q\}$ is the Caputo–Hadamard fractional derivative of order $1<\gamma \le 2,{\mathsf{\Theta }}_1,{\mathsf{\Theta }}_2\in [0,1),$ and ${\vartheta }_1,{\vartheta }_2\in (1,e)$.

Recently [30], existence results were obtained via the well-known fixed point theories for the following system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^c{\mathcal{D}}^{\omega }\left(\frac{{\mathcal{D}}^{\widehat{\mathsf{\Psi }}}v\left({\chi }_1\right)-{\sum }_{i=1}^m{I}^{{\beta }_i}{h}_i({\chi }_1,x\left({\chi }_1\right))}{f_1({\chi }_1,x\left({\chi }_1\right))}\right)=g_1({\chi }_1,x\left({\chi }_1\right),I^{\gamma }x\left({\chi }_1\right))a.e.{\chi }_1\in J=[0,1],\hfill \\ {}^c{\mathcal{D}}^{\omega }\left(\frac{{\mathcal{D}}^{\widehat{\mathsf{\Psi }}}x\left({\chi }_1\right)-{\sum }_{i=1}^m{I}^{{\beta }_i}{h}_i({\chi }_1,v\left({\chi }_1\right))}{f_2({\chi }_1,v\left({\chi }_1\right))}\right)=g_2({\chi }_1,x\left({\chi }_1\right),I^{\gamma }x\left({\chi }_1\right))a.e.{\chi }_1\in J=[0,1],\hfill \\ {}^c{\mathcal{D}}^{\widehat{\mathsf{\Psi }}}v\left(0\right)=0{=}^c{\mathcal{D}}^{\widehat{\mathsf{\Psi }}}x\left(0\right),v^{\prime }\left(0\right)=0=x^{\prime }\left({\chi }_1\right),v\left(1\right)={\rho }_1\left(x\left(\vartheta \right)\right),x\left(1\right)={\rho }_2\left(v\left(\vartheta \right)\right).\hfill \end{array}\right.\end{array}$$

and the parameters satisfy the condition $0<\omega \le ,1<\widehat{\mathsf{\Psi }}\le 2,0<\vartheta <1,$ for $j=1,2,f_j:J\times \mathbb{R}⟶\mathbb{R}\backslash \left\{0\right\},g_j:J\times \mathbb{R}\times \mathbb{R}⟶\mathbb{R}$ and for $i=1,\dot{,}m$ the function $h_j:J\times \mathbb{R}⟶\mathbb{R}$ fulfill the Caratheodory property, the special boundary function ${\rho }_j:\mathbb{R}⟶\mathbb{R}$ my the nonlinear, and the notation $v^{\prime }$ is used for the first order ordinary derivative of v and ${}^{c}D$ stands for fractional derivative in the sense of Caputo. In the literature, there are other definitions of fractional derivatives. The present paper is a continuation of the work [9], in order to study the existence of a solution for a nonlinear sequential hybrid fractional differential equation:

$$\left\{\begin{array}{c}{}^c{\mathcal{D}}^{\mathsf{\xi }}{(}^c{\mathcal{D}}^{\widehat{\iota }}+\gamma )\left(\frac{\widehat{\nu }\left({\chi }_1\right)}{f({\chi }_1,\widehat{\nu }\left({\chi }_1\right))}\right)=\widehat{\mathsf{\Psi }}({\chi }_1,\widehat{\nu }\left({\chi }_1\right))\mathrm{a.e.}t\in J=[0,1]\hfill \\ \hfill \\ \frac{\widehat{\nu }\left(0\right)}{f(0,\widehat{\nu }\left(0\right))}+\vartheta \frac{\widehat{\nu }\left(1\right)}{f(1,\widehat{\nu }\left(1\right))}={\tau }_1{\int }_0^1{\widehat{\mathsf{\Psi }}}_1(\widehat{\varkappa },\widehat{\nu }\left(\varkappa \right))d\widehat{\varkappa },\hfill \\ {}^c{\mathcal{D}}^{\widehat{\iota }}\left(\frac{\widehat{\nu }\left(0\right)}{f(0,\widehat{\nu }\left(0\right))}\right)+\vartheta {}^c{\mathcal{D}}^{\widehat{\iota }}\left(\frac{\widehat{\nu }\left(1\right)}{f(1,\widehat{\nu }\left(1\right))}\right)={\tau }_2{\int }_0^1{\widehat{\mathsf{\Psi }}}_2(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa },\hfill \\ {}^c{\mathcal{D}}^{2\widehat{\iota }}\left(\frac{\widehat{\nu }\left(0\right)}{f(0,\widehat{\nu }\left(0\right))}\right)+\vartheta {}^c{\mathcal{D}}^{2\widehat{\iota }}\left(\frac{\widehat{\nu }\left(1\right)}{f(1,\widehat{\nu }\left(1\right))}\right)={\tau }_3{\int }_0^1{\widehat{\mathsf{\Psi }}}_3(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa },\hfill \end{array}\right.$$

(1)

where $0<\widehat{\iota }<1,1<\mathsf{\xi }\le 2,\gamma ,\vartheta ,{\tau }_i,\in {\mathbb{R}}^{*}$ for $i=1,2,3,$ with $\vartheta \ne -1$. ${}^c{\mathcal{D}}^{\mathsf{\xi }},$ ${}^c{\mathcal{D}}^{\widehat{\iota }}$ are the Caputo’s fractional derivatives, and $f:[0;1]\times \mathbb{R}\to \mathbb{R}\backslash \left\{0\right\},$ $\widehat{\mathsf{\Psi }}\in \mathcal{E}([0,1]\times \mathbb{R},\mathbb{R})$ and ${\widehat{\mathsf{\Psi }}}_1,{\widehat{\mathsf{\Psi }}}_2,{\widehat{\mathsf{\Psi }}}_3:[0,1]\times \mathbb{R}\to \mathbb{R}$ are the given continuous functions.

By a solution of the problem (1), we mean a function $\widehat{\nu }\in C(J,\mathbb{R})$ such that

(i)

the function ${\chi }_1\mapsto \frac{\widehat{\nu }}{f({\chi }_1,\widehat{\nu })}$ is continuous for each $\widehat{\nu }\in \mathbb{R},$ and

(ii)

x satisfies the equations in (1).

Despite popular belief, researchers have paid less attention to studies of hybrid fractional differential equations associated with hybrid boundary conditions. According to the authors’ observations, there is no analytical literature on the existence of mixed hybrid fractional differential equations of a sequential type involving hybrid BCs. Adding to this, Dhage’s fixed point theorem is employed in investigating the existence results for the proposed BVP, which gives this work novelty and originality.

The rest of the article is as follows: Section 2 presents the basic definitions, lemmas, and theorems that underpin our main conclusions. In Section 3, we find solutions to the given fractional differential Equation (1) using Dahg’s fixed point theorem. Section 4 looks at the Ulam–Hyers stability of the provided fractional differential Equation (1). In Section 5, an example is provided to further clarify the study’s findings. In Section 6, a conclusion and a future work are introduced.

2. Preliminaries

Let us introduce some preliminary results that will be useful for proving our results in the subsequent sections.

Let the Banach space denoted by $\mathcal{X}=\mathcal{E}(J,\mathbb{R})$ with the norm

$$\parallel \widehat{\nu }\parallel =sup\{|\widehat{\nu }\left({\chi }_1\right)|,t\in J\}.$$

Definition 1.

([31]). The R-L fractional integral of θ with order $\widehat{\iota }$ is given by

$$I_a^{\widehat{\iota }}g\left({\chi }_1\right)={\int }_a^{{\chi }_1}\frac{{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }-1}}{\mathsf{\Gamma }\left(\widehat{\iota }\right)}g\left(\widehat{\varkappa }\right)d\widehat{\varkappa }.$$

Definition 2.

([31]). The Caputo fractional integral of order $\widehat{\iota }$ for a function θ is given by

$${(}^c{D}_{a^{+}}^{\widehat{\iota }}g)\left({\chi }_1\right)=\frac{1}{\mathsf{\Gamma }(n-\widehat{\iota })}{\int }_a^{{\chi }_1}\frac{{({\chi }_1-\widehat{\varkappa })}^{n-\widehat{\iota }-1}}{\mathsf{\Gamma }\left(\widehat{\iota }\right)}{g}^{\left(n\right)}\left(\widehat{\varkappa }\right)d\widehat{\varkappa }$$

where $n=\left[\widehat{\iota }\right]+1$ and $\left[\widehat{\iota }\right]$ denotes the integer part of $\widehat{\iota }.$

Next, we state Dhage’s fixed point theorem followed by our main auxiliary lemma.

Theorem 1.

([32]). For any $\mathcal{S}$ nonempty, a closed convex and bounded subset of a Banach algebra $\mathcal{X}$ and for any operators $\mathcal{P},\mathcal{E}:\mathcal{X}⟶\mathcal{X}$ and $\mathcal{F}:\mathcal{S}⟶\mathcal{X}$ such that:

$\left(i\right)$

$\mathcal{P}$ and $\mathcal{E}$ are Lipschitzian with Lipschitz constants $\widehat{\nu }$ and $\rho ,$ respectively,

$\left(ii\right)$

$\mathcal{F}$ is compact and continuous,

$\left(iii\right)$

$\widehat{\nu }=\mathcal{P}\widehat{\nu }\mathcal{F}\rho +\mathcal{E}\widehat{\nu }⟹\widehat{\nu }\in \mathcal{S}$ for all $\rho \in ,\mathcal{S}$

$\left(iv\right)$

$\widehat{\nu }\mathcal{W}+\rho <1,$ where $\mathcal{W}=\parallel \mathcal{F}\left(\mathcal{S}\right)\parallel$.

Then, the equation $\widehat{\nu }=\mathcal{D}\widehat{\nu }\mathcal{F}\rho +\mathcal{E}\widehat{\nu }$ has a solution.

Lemma 1.

([31]). Let $n\in \mathbb{N}$ and $n-1<\widehat{\iota }<n$. If f is a continuous function, then we have

$$I^{\widehat{\iota }c}{\mathcal{D}}^{\widehat{\iota }}f\left({\chi }_1\right)=f\left({\chi }_1\right)+{\omega }_0+{\omega }_{1}t+{\omega }_2{t}^2+...+{\omega }_{n-1}{t}^{n-1}.$$

For the proof of our auxiliary Lemma, we state the following hypothesis:

$\left(H_0\right)$ The function $x⟼\frac{\widehat{\nu }}{f({\chi }_1,\widehat{\nu })}$ is increasing in $\mathbb{R}$ almost everywhere for $t\in J.$

Lemma 2.

Assume that hypothesis $\left(H_0\right)$ holds. Then, for any $\rho \in L^1(J;\mathbb{R})$. The function $\widehat{\nu }\in C(J,\mathbb{R})$ is a solution of the problem

$$\left\{\begin{array}{c}{}^c{\mathcal{D}}^{\mathsf{\xi }}{(}^c{\mathcal{D}}^{\widehat{\iota }}+\gamma )\left(\frac{\widehat{\nu }\left({\chi }_1\right)}{f({\chi }_1,\widehat{\nu }\left({\chi }_1\right))}\right)=\rho \left({\chi }_1\right)a.e.t\in J=[0,1]\hfill \\ \hfill \\ \frac{\widehat{\nu }\left(0\right)}{f(0,\widehat{\nu }\left(0\right))}+\vartheta \frac{\widehat{\nu }\left(1\right)}{f(1,\widehat{\nu }\left(1\right))}={\tau }_1{\int }_0^1{\widehat{\mathsf{\Psi }}}_1(\widehat{\varkappa },\widehat{\nu })\left(\widehat{\varkappa }\right)d\widehat{\varkappa },\hfill \\ {}^c{\mathcal{D}}^{\widehat{\iota }}\left(\frac{\widehat{\nu }\left(0\right)}{f(0,\widehat{\nu }\left(0\right))}\right)+\vartheta {}^c{\mathcal{D}}^{\widehat{\iota }}\left(\frac{\widehat{\nu }\left(1\right)}{f(1,\widehat{\nu }\left(1\right))}\right)={\tau }_2{\int }_0^1{\widehat{\mathsf{\Psi }}}_2(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa },\hfill \\ {}^c{\mathcal{D}}^{2\widehat{\iota }}\left(\frac{\widehat{\nu }\left(0\right)}{f(0,\widehat{\nu }\left(0\right))}\right)+\vartheta {}^c{\mathcal{D}}^{2\widehat{\iota }}\left(\frac{\widehat{\nu }\left(1\right)}{f(1,\widehat{\nu }\left(1\right))}\right)={\tau }_3{\int }_0^1{\widehat{\mathsf{\Psi }}}_3(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa },\hfill \end{array}\right.$$

(2)

is given by

$$\begin{array}{ccc}\widehat{\nu }\left({\chi }_1\right)\hfill & {\displaystyle =f({\chi }_1,\widehat{\nu }\left({\chi }_1\right))[{\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}\rho \left(\widehat{\varkappa }\right)d\widehat{\varkappa }}\hfill & \\ & {\displaystyle +{\mathsf{\Theta }}_1\left({\chi }_1\right){\int }_0^1{\widehat{\mathsf{\Psi }}}_2(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill & \\ & {\displaystyle +{\mathsf{\Theta }}_2\left({\chi }_1\right){\int }_0^1{\widehat{\mathsf{\Psi }}}_1(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }7}\hfill \\ & {\displaystyle +{\mathsf{\Theta }}_3\left({\chi }_1\right){\int }_0^1{\widehat{\mathsf{\Psi }}}_3(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }+{\displaystyle \frac{{\mathsf{\Theta }}_4\left({\chi }_1\right)}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-\widehat{\iota }-1}\rho \left(\widehat{\varkappa }\right)d\widehat{\varkappa }}\hfill \\ & +{\displaystyle \frac{{\mathsf{\Theta }}_5\left({\chi }_1\right)}{\mathsf{\Gamma }\left(\widehat{\iota }\right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-1}\rho \left(\widehat{\varkappa }\right)d\widehat{\varkappa }+{\displaystyle \frac{\vartheta \gamma }{(1+\vartheta )\mathsf{\Gamma }\left(\widehat{\iota }\right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }-1}\widehat{\nu }\left(\widehat{\varkappa }\right)d\widehat{\varkappa }\hfill \\ & -{\displaystyle \frac{\vartheta }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}\rho \left(\widehat{\varkappa }\right)d\widehat{\varkappa }]-\frac{\gamma }{\mathsf{\Gamma }\left(\widehat{\iota }\right)}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }-1}\widehat{\nu }\left(\widehat{\varkappa }\right)d\widehat{\varkappa },\hfill \end{array}$$

where

$$\begin{array}{ccc}{\mathsf{\Theta }}_1\left({\chi }_1\right)\hfill & =\hfill & {\displaystyle \frac{{{\chi }_1}^{\widehat{\iota }}{\tau }_2\left(1-\gamma \mathsf{\Gamma }(2-\widehat{\iota })\right)}{\mathsf{\Gamma }(\widehat{\iota }+1)(1+\vartheta )}}+{\displaystyle \frac{{{\chi }_1}^{\widehat{\iota }+1}\gamma {\tau }_2\mathsf{\Gamma }(2-\widehat{\iota })}{\mathsf{\Gamma }(2+\widehat{\iota })\vartheta }}\hfill \\ & & +\left({\displaystyle \frac{\vartheta }{{(1+\vartheta )}^2\mathsf{\Gamma }(\widehat{\iota }+1)}}-{\displaystyle \frac{1}{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+2)}}\right)\mathsf{\Gamma }(2-\widehat{\iota })\gamma {\tau }_2-{\displaystyle \frac{\vartheta {\tau }_2}{{(1+\vartheta )}^2\mathsf{\Gamma }(\widehat{\iota }+1)}},\hfill \\ {\mathsf{\Theta }}_2\left({\chi }_1\right)\hfill & =\hfill & {\displaystyle \frac{{{\chi }_1}^{\widehat{\iota }}\gamma {\tau }_1}{\mathsf{\Gamma }(\widehat{\iota }+1)(1+\vartheta )}}-{\displaystyle \frac{{\tau }_1}{1+\vartheta }}+{\displaystyle \frac{\vartheta \gamma {\tau }_1}{{(1+\vartheta )}^2\mathsf{\Gamma }(\widehat{\iota }+1)}},\hfill \\ {\mathsf{\Theta }}_3\left({\chi }_1\right)\hfill & =\hfill & -{\displaystyle \frac{{{\chi }_1}^{\widehat{\iota }}\mathsf{\Gamma }(2-\widehat{\iota }){\tau }_3}{\mathsf{\Gamma }(\widehat{\iota }+1)(1+\vartheta )}}+{\displaystyle \frac{{{\chi }_1}^{\widehat{\iota }+1}\mathsf{\Gamma }(2-\widehat{\iota }){\tau }_3}{\mathsf{\Gamma }(\widehat{\iota }+2)\vartheta }}+{\displaystyle \frac{\vartheta \mathsf{\Gamma }(2-\widehat{\iota }){\tau }_3}{{(1+\vartheta )}^2\mathsf{\Gamma }(\widehat{\iota }+1)}}-{\displaystyle \frac{\mathsf{\Gamma }(2-\widehat{\iota }){\tau }_3}{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+2)}},\hfill \\ {\mathsf{\Theta }}_4\left({\chi }_1\right)\hfill & =\hfill & {\displaystyle \frac{{{\chi }_1}^{\widehat{\iota }}\mathsf{\Gamma }(2-\widehat{\iota })\vartheta }{\mathsf{\Gamma }(\widehat{\iota }+1)(1+\vartheta )}}-{\displaystyle \frac{{{\chi }_1}^{\widehat{\iota }+1}\mathsf{\Gamma }(2-\widehat{\iota })}{\mathsf{\Gamma }(\widehat{\iota }+2)}}+\mathsf{\Gamma }(2-\widehat{\iota })\left({\displaystyle \frac{\vartheta }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+2)}}-{\displaystyle \frac{{\vartheta }^2}{{(1+\vartheta )}^2\mathsf{\Gamma }(\widehat{\iota }+1)}}\right),\hfill \\ {\mathsf{\Theta }}_5\left({\chi }_1\right)\hfill & =\hfill & -{\displaystyle \frac{{{\chi }_1}^{\widehat{\iota }}\vartheta }{\mathsf{\Gamma }(\widehat{\iota }+1)(1+\vartheta )}}-{\displaystyle \frac{{\vartheta }^2}{{(1+\vartheta )}^2\mathsf{\Gamma }(\widehat{\iota }+1)}},\hfill \end{array}$$

Proof. 

By Lemma 1, we obtain

$${(}^c{\mathcal{D}}^{\widehat{\iota }}+\gamma )\left(\frac{\widehat{\nu }\left({\chi }_1\right)}{f({\chi }_1,\widehat{\nu }\left({\chi }_1\right))}\right)=I^{\mathsf{\xi }}\rho \left({\chi }_1\right)+{\omega }_0+{\omega }_{1}t,$$

$${}^c{\mathcal{D}}^{\widehat{\iota }}\left(\frac{\widehat{\nu }\left({\chi }_1\right)}{f({\chi }_1,\widehat{\nu }\left({\chi }_1\right))}\right)=I^{\mathsf{\xi }}\rho \left({\chi }_1\right)+{\omega }_0+{\omega }_{1}t-\gamma \left(\frac{\widehat{\nu }\left({\chi }_1\right)}{f({\chi }_1,\widehat{\nu }\left({\chi }_1\right)}\right),$$

$$\frac{\widehat{\nu }\left({\chi }_1\right)}{f({\chi }_1,\widehat{\nu }\left({\chi }_1\right))}=I^{\widehat{\iota }+\mathsf{\xi }}\rho \left({\chi }_1\right)+I^{\widehat{\iota }}{\omega }_0+I^{\widehat{\iota }}{\omega }_{1}t-I^{\widehat{\iota }}\lambda \left(\frac{\widehat{\nu }\left({\chi }_1\right)}{f({\chi }_1,\widehat{\nu }\left({\chi }_1\right))}\right)+{\omega }_2,$$

where ${\omega }_0,{\omega }_1,{\omega }_2\in \mathbb{R}$. By using to the condition ${}^c{\mathcal{D}}^{2\widehat{\iota }}\left(\frac{\widehat{\nu }\left(0\right)}{f(0,\widehat{\nu }\left(0\right))}\right)+\vartheta {}^c{\mathcal{D}}^{2\widehat{\iota }}\left(\frac{\widehat{\nu }\left(1\right)}{f(1,\widehat{\nu }\left(1\right))}\right)={\tau }_3{\int }_0^1{\widehat{\mathsf{\Psi }}}_3(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa },$ we find that:

$$\begin{array}{c}\hfill {\displaystyle {\omega }_1=\mathsf{\Gamma }(2-\widehat{\iota })\left({\displaystyle \frac{{\tau }_3}{\vartheta }}{\int }_0^1{\widehat{\mathsf{\Psi }}}_3(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }+{\displaystyle \frac{\gamma {\tau }_2}{\vartheta }}{\int }_0^1{\widehat{\mathsf{\Psi }}}_2(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }-\frac{1}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota })}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-{\widehat{\iota }}_1}\rho \left(\widehat{\varkappa }\right)d\widehat{\varkappa }\right).}\end{array}$$

Using the ${\displaystyle {}^c{\mathcal{D}}^{\widehat{\iota }}\left(\frac{\widehat{\nu }\left(0\right)}{f(0,\widehat{\nu }\left(0\right))}\right)+\vartheta {}^c{\mathcal{D}}^{\widehat{\iota }}\left(\frac{\widehat{\nu }\left(1\right)}{f(1,\widehat{\nu }\left(1\right))}\right)={\tau }_2{\int }_0^1{\widehat{\mathsf{\Psi }}}_2(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}$ and ${\displaystyle \left(\frac{\widehat{\nu }\left(0\right)}{f(0,\widehat{\nu }\left(0\right))}\right)+\vartheta \left(\frac{\widehat{\nu }\left(1\right)}{f(1,\widehat{\nu }\left(1\right))}\right)={\tau }_1{\int }_0^1{\widehat{\mathsf{\Psi }}}_1(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa },}$ we have

$${\displaystyle \begin{array}{ccc}{\omega }_0\hfill & =\hfill & {\displaystyle {\displaystyle \frac{-\mathsf{\Gamma }(2-\widehat{\iota }){\tau }_3}{1+\vartheta }}{\int }_0^1{\widehat{\mathsf{\Psi }}}_3(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }+{\displaystyle \frac{(1-\gamma \mathsf{\Gamma }(2-\widehat{\iota })){\tau }_2}{1+\vartheta }}{\int }_0^1{\widehat{\mathsf{\Psi }}}_2(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle }\hfill & +\frac{\gamma {\tau }_1}{1+\vartheta }{\int }_0^1{\widehat{\mathsf{\Psi }}}_1(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }\hfill \\ & & {\displaystyle +{\displaystyle \frac{\mathsf{\Gamma }(2-\widehat{\iota })\vartheta }{(1+\vartheta )\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-\widehat{\iota }-1}\rho \left(\widehat{\varkappa }\right)d\widehat{\varkappa }-\frac{\vartheta }{(1+\vartheta )\mathsf{\Gamma }\left(\mathsf{\xi }\right)}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-1}\rho \left(\widehat{\varkappa }\right)d\widehat{\varkappa },}\hfill \\ {\omega }_2\hfill & =\hfill & {\displaystyle \frac{\vartheta \gamma }{(1+\vartheta )\mathsf{\Gamma }\left(\widehat{\iota }\right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }-1}\widehat{\nu }\left(\widehat{\varkappa }\right)d\widehat{\varkappa }+\left({\displaystyle \frac{{\tau }_1}{1+\vartheta }}-{\displaystyle \frac{\vartheta \gamma {\tau }_1}{\mathsf{\Gamma }(\widehat{\iota }+1){(1+\vartheta )}^2}}\right){\int }_0^1{\widehat{\mathsf{\Psi }}}_1(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }\hfill \\ & & -{\displaystyle \frac{\vartheta }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}\rho \left(\widehat{\varkappa }\right)d\widehat{\varkappa }+{\displaystyle \frac{{\vartheta }^2}{{(1+\vartheta )}^2\mathsf{\Gamma }\left(\mathsf{\xi }\right)\mathsf{\Gamma }(\widehat{\iota }+1)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-1}\rho \left(\widehat{\varkappa }\right)d\widehat{\varkappa }\hfill \\ & & +{\displaystyle \frac{\mathsf{\Gamma }(2-\widehat{\iota })}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota })}}\left({\displaystyle \frac{\vartheta }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+2)}}-{\displaystyle \frac{{\vartheta }^2}{{(1+\vartheta )}^2\mathsf{\Gamma }(\widehat{\iota }+1)}}\right){\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-\widehat{\iota }-1}\rho \left(\widehat{\varkappa }\right)d\widehat{\varkappa }\hfill \\ & & +\left[\left({\displaystyle \frac{{\vartheta }^2}{{(1+\vartheta )}^2\mathsf{\Gamma }(\widehat{\iota }+1)}}-{\displaystyle \frac{\vartheta }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+2)}}\right){\displaystyle \frac{\mathsf{\Gamma }(2-\widehat{\iota })\gamma {\tau }_2}{\vartheta }}-{\displaystyle \frac{\vartheta {\sigma }_2}{{(1+\vartheta )}^2\mathsf{\Gamma }(\widehat{\iota }+1)}}\right]\hfill \\ & & \times {\int }_0^1{\widehat{\mathsf{\Psi }}}_2(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }.\hfill \end{array}}$$

Substituting the value of ${\omega }_0,$ ${\omega }_1,$ and ${\omega }_2,$ we obtain

$$\begin{array}{cc}\widehat{\nu }\left({\chi }_1\right)\hfill & {\displaystyle =f({\chi }_1,\widehat{\nu }\left({\chi }_1\right))[{\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}\rho \left(\widehat{\varkappa }\right)d\widehat{\varkappa }+{\mathsf{\Theta }}_1\left({\chi }_1\right){\int }_0^1{\widehat{\mathsf{\Psi }}}_2(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\mathsf{\Theta }}_2\left({\chi }_1\right){\int }_0^1{\widehat{\mathsf{\Psi }}}_1(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }+{\mathsf{\Theta }}_3\left({\chi }_1\right){\int }_0^1{\widehat{\mathsf{\Psi }}}_3(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }+{\displaystyle \frac{{\mathsf{\Theta }}_4\left({\chi }_1\right)}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-\widehat{\iota }-1}\rho \left(\widehat{\varkappa }\right)d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\displaystyle \frac{{\mathsf{\Theta }}_5\left({\chi }_1\right)}{\mathsf{\Gamma }\left(\mathsf{\xi }\right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-1}\rho \left(\widehat{\varkappa }\right)d\widehat{\varkappa }+{\displaystyle \frac{\vartheta \gamma }{(1+\vartheta )\mathsf{\Gamma }\left(\widehat{\iota }\right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }-1}\widehat{\nu }\left(\widehat{\varkappa }\right)d\widehat{\varkappa }}\hfill \\ & {\displaystyle -{\displaystyle \frac{\vartheta }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}\rho \left(\widehat{\varkappa }\right)d\widehat{\varkappa }]-{\displaystyle \frac{\gamma }{\mathsf{\Gamma }\left(\widehat{\iota }\right)}}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }-1}\widehat{\nu }\left(\widehat{\varkappa }\right)d\widehat{\varkappa }.}\hfill \end{array}$$

□

3. Existence Result

Based on Dhage’s fixed point theorem, we will present in this section the verification of the existence of a solution to Equation (1). To simplify the calculations, we set ${\mathsf{\Theta }}_i={\displaystyle \underset{{\chi }_1\in [0,1]}{max}}|{\mathsf{\Theta }}_i\left({\chi }_1\right)|$ for $i=1,2,\dots ,5.$

The following assumptions are the necessary conditions to derive our main results:

$\left(H_1\right)$- $f_{:}[0,1]\times \mathbb{R}\to \mathbb{R}\backslash \left\{0\right\}$ and ${\widehat{\mathsf{\Psi }}}_1,{\widehat{\mathsf{\Psi }}}_2,{\widehat{\mathsf{\Psi }}}_3,:[0,1]\times \mathbb{R}\to \mathbb{R}$ are continuous functions.

$\left(H_2\right)$- There exists a constant $q_0,q_1$ such that

$|f({\chi }_1,{\widehat{\nu }}_1)-f({\chi }_1,{\widehat{\nu }}_2)|\le q_0|{\widehat{\nu }}_1-{\widehat{\nu }}_2|,$

$|w({\chi }_1,{\widehat{\nu }}_1)-w({\chi }_1,{\widehat{\nu }}_2)|\le q_1|{\widehat{\nu }}_1-{\widehat{\nu }}_2|,$ for all ${\chi }_1\in J$ and ${\widehat{\nu }}_1,{\widehat{\nu }}_2\in \mathbb{R}$.

$\left(H_3\right)$- There exist positive constants $q_2,q_3,q_4,$ such that:

$|{\widehat{\mathsf{\Psi }}}_1({\chi }_1,{\widehat{\nu }}_1)-{\widehat{\mathsf{\Psi }}}_1({\chi }_1,{\widehat{\nu }}_2)|\le q_2|{\widehat{\nu }}_1-{\widehat{\nu }}_2|,$

$|{\widehat{\mathsf{\Psi }}}_2({\chi }_1,{\widehat{\nu }}_1)-{\widehat{\mathsf{\Psi }}}_2({\chi }_1,{\widehat{\nu }}_2)|\le q_3|{\widehat{\nu }}_1-{\widehat{\nu }}_2|,$

$|{\widehat{\mathsf{\Psi }}}_3({\chi }_1,{\widehat{\nu }}_1)-{\widehat{\mathsf{\Psi }}}_3({\chi }_1,{\widehat{\nu }}_2)|\le q_4|{\widehat{\nu }}_1-{\widehat{\nu }}_2|.$

$\left(H_4\right)$- $|f({\chi }_1,\widehat{\nu })|\le \phi \left({\chi }_1\right),$$|w({\chi }_1,\widehat{\nu })|\le m\left({\chi }_1\right),$$|{\widehat{\mathsf{\Psi }}}_2({\chi }_1,\widehat{\nu })|\le \rho \left({\chi }_1\right),$$|{\widehat{\mathsf{\Psi }}}_3({\chi }_1,\widehat{\nu })|\le \psi \left({\chi }_1\right),$

$|{\widehat{\mathsf{\Psi }}}_1({\chi }_1,\widehat{\nu })|\le \varphi \left({\chi }_1\right),$$\forall ({\chi }_1,\widehat{\nu })\in [0,1]\times \mathbb{R}$ with $m,\phi ,\rho ,\varphi ,\psi \in \mathcal{E}([0,1];{\mathbb{R}}^{+})$.

Theorem 2.

Assume that hypothesis $\left(H_0\right)-\left(H_4\right)$ holds. Further, if

$$\begin{array}{cc}& q_0\left(\right[q_1({\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}+{\displaystyle \frac{{\mathsf{\Theta }}_4}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota }+1)}}+{\displaystyle \frac{{\mathsf{\Theta }}_5}{\mathsf{\Gamma }(\mathsf{\xi }+1)}}\hfill \\ & +|{\displaystyle \frac{\vartheta }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}|)+|{\displaystyle \frac{\vartheta \gamma }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+1)}}|+{\mathsf{\Theta }}_1{q}_3+{\mathsf{\Theta }}_2{q}_2\hfill \\ & +{\mathsf{\Theta }}_3{q}_4]+{\displaystyle \frac{{\mathsf{{\rm Y}}}_0}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}+{\displaystyle \frac{{\mathsf{{\rm Y}}}_0{\mathsf{\Theta }}_4}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota }+1)}}+{\mathsf{\Theta }}_1{\mathsf{{\rm Y}}}_2+A_2{\mathsf{{\rm Y}}}_1\hfill \\ & +{\mathsf{\Theta }}_3{\mathsf{{\rm Y}}}_3+{\displaystyle \frac{{\mathsf{\Theta }}_5{\mathsf{{\rm Y}}}_0}{\mathsf{\Gamma }(\mathsf{\xi }+1)}}+|{\displaystyle \frac{\vartheta {\mathsf{{\rm Y}}}_0}{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}})+{\displaystyle \frac{\gamma }{\mathsf{\Gamma }(\widehat{\iota }+1)}}<1.\hfill \end{array}$$

then the problem (1) has a solution defined on $[0,1]$.

Proof. 

Let the subset $\mathcal{S}$ of $\mathcal{X}$ be defined as

$$\begin{array}{c}\hfill \mathcal{S}=\{\widehat{\nu }\in \mathcal{X}/\parallel \widehat{\nu }\parallel \le r\}\end{array}$$

with

$$\begin{array}{c}r\ge {\displaystyle \frac{|\parallel \phi \parallel \left({\displaystyle \frac{\parallel m\parallel }{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}})+{\mathsf{\Theta }}_1\parallel \rho \parallel +{\mathsf{\Theta }}_2\parallel \varphi \parallel +{\mathsf{\Theta }}_3\parallel \psi \parallel +{\displaystyle \frac{{\mathsf{\Theta }}_4\parallel m\parallel }{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota }+1)}}+{\displaystyle \frac{{\mathsf{\Theta }}_5\parallel m\parallel }{\mathsf{\Gamma }(\mathsf{\xi }+1)}}+{\displaystyle \frac{|\vartheta |\parallel m\parallel }{(|1+\vartheta |)\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}\right)}{1-\left({\displaystyle \frac{\parallel \phi \parallel |\vartheta \lambda |}{(|1+\vartheta |)\mathsf{\Gamma }(\widehat{\iota }+1)}}+{\displaystyle \frac{|\gamma |}{\mathsf{\Gamma }(\widehat{\iota }+1)}}\right)}}\hfill \end{array}.$$

Now,

$$\begin{array}{cc}\widehat{\nu }\left({\chi }_1\right)\hfill & {\displaystyle =f({\chi }_1,\widehat{\nu }\left({\chi }_1\right))[{\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}w(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\mathsf{\Theta }}_1\left({\chi }_1\right){\int }_0^1{\widehat{\mathsf{\Psi }}}_2(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\mathsf{\Theta }}_2\left({\chi }_1\right){\int }_0^1{\widehat{\mathsf{\Psi }}}_1(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }+{\mathsf{\Theta }}_3\left({\chi }_1\right){\int }_0^1{\widehat{\mathsf{\Psi }}}_3(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\displaystyle \frac{{\mathsf{\Theta }}_4\left({\chi }_1\right)}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-\widehat{\iota }-1}w(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & +{\displaystyle \frac{{\mathsf{\Theta }}_5\left({\chi }_1\right)}{\mathsf{\Gamma }\left(\mathsf{\xi }\right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-1}w(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }+{\displaystyle \frac{\vartheta \gamma }{(1+\mu )\mathsf{\Gamma }\left(\widehat{\iota }\right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }-1}\widehat{\nu }\left(\widehat{\varkappa }\right)d\widehat{\varkappa }\hfill \\ & -{\displaystyle \frac{\mu }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}w(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }]-\frac{\gamma }{\mathsf{\Gamma }\left(\widehat{\iota }\right)}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }-1}\widehat{\nu }\left(\widehat{\varkappa }\right)d\widehat{\varkappa }.\hfill \end{array}$$

Define three operators $\mathcal{P}:\mathcal{X}⟶\mathcal{X},$ $\mathcal{F}:\mathcal{S}⟶\mathcal{X}$ and $\mathcal{E}:\mathcal{S}⟶\mathcal{X}$ by:

$$\mathcal{P}\widehat{\nu }\left({\chi }_1\right)=f({\chi }_1,\widehat{\nu }\left({\chi }_1\right)),{\chi }_1\in J$$

(3)

$\begin{array}{cc}\mathcal{F}\widehat{\nu }\left({\chi }_1\right)\hfill & {\displaystyle ={\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}w(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }-\frac{\gamma }{\mathsf{\Gamma }\left(\widehat{\iota }\right)}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }-1}\widehat{\nu }\left(\widehat{\varkappa }\right)d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\mathsf{\Theta }}_1\left({\chi }_1\right){\int }_0^1{\widehat{\mathsf{\Psi }}}_2(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\mathsf{\Theta }}_2\left({\chi }_1\right){\int }_0^1{\widehat{\mathsf{\Psi }}}_1(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }+{\mathsf{\Theta }}_3\left({\chi }_1\right){\int }_0^1{\widehat{\mathsf{\Psi }}}_3(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\displaystyle \frac{{\mathsf{\Theta }}_4\left({\chi }_1\right)}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-\widehat{\iota }-1}w(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & +{\displaystyle \frac{{\mathsf{\Theta }}_5\left({\chi }_1\right)}{\mathsf{\Gamma }\left(\mathsf{\xi }\right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-1}w(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }+{\displaystyle \frac{\vartheta \gamma }{(1+\vartheta )\mathsf{\Gamma }\left(\widehat{\iota }\right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }-1}\widehat{\nu }\left(\widehat{\varkappa }\right)d\widehat{\varkappa }\hfill \\ & -{\displaystyle \frac{\vartheta }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}w(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa },\hfill \end{array}$

and

$$\mathcal{E}\widehat{\nu }\left({\chi }_1\right)=\frac{-\gamma }{\mathsf{\Gamma }\left(\widehat{\iota }\right)}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }-1}\widehat{\nu }\left(\widehat{\varkappa }\right)d\widehat{\varkappa }.$$

Accordingly, Equation (3) can be transformed into the following operator equation:

$$\widehat{\nu }\left({\chi }_1\right)=\mathcal{P}\widehat{\nu }\left({\chi }_1\right)\mathcal{F}\widehat{\nu }\left({\chi }_1\right)+\mathcal{E}\widehat{\nu }\left({\chi }_1\right),{\chi }_1\in [0,1].$$

(4)

To achieve all the conditions of Theorem 1, we split our proof into four claims as follows.

Claim 1. Let $\widehat{\nu },\rho \in \mathcal{X}$ then by hypothesis $\left(H_1\right)$,

$$|\mathcal{P}\widehat{\nu }\left({\chi }_1\right)-\mathcal{P}\rho \left({\chi }_1\right)|=|f({\chi }_1,\widehat{\nu }\left({\chi }_1\right))-f({\chi }_1,\rho \left({\chi }_1\right))|\le q_0|\widehat{\nu }\left({\chi }_1\right)-\rho \left({\chi }_1\right)|\le q_0\parallel \widehat{\nu }-\rho \parallel$$

for all ${\chi }_1\in J$. Then,

$$\parallel \mathcal{P}\widehat{\nu }-\mathcal{P}\rho \parallel \le q_0\parallel \widehat{\nu }-\rho \parallel forall\widehat{\nu },\rho \in \mathcal{X}.$$

Additionally,

$$|\mathcal{E}\widehat{\nu }\left({\chi }_1\right)-\mathcal{E}\rho \left({\chi }_1\right)|=\frac{\gamma }{\mathsf{\Gamma }\left(\widehat{\iota }\right)}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }-1}|\widehat{\nu }\left(\widehat{\varkappa }\right)-\rho \left(\widehat{\varkappa }\right)|ds\le \frac{|\gamma |}{\mathsf{\Gamma }(\widehat{\iota }+1)}|\widehat{\nu }-\rho |$$

Claim 2. We show the continuity of the operator $\mathcal{F}$ in $\mathcal{S}$.

Let ${\left({\widehat{\nu }}_n\right)}_n$ be a sequence in $\mathcal{S}$ converging to a point $\widehat{\nu }\in \mathcal{S}$. Then, by Lebesgue dominated convergence theorem,

$${\displaystyle \underset{n\to \infty }{lim}{\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}w(\widehat{\varkappa },{\widehat{\nu }}_n\left(\widehat{\varkappa }\right))d\widehat{\varkappa }={\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}\underset{n\to \infty }{lim}w(\widehat{\varkappa },{\widehat{\nu }}_n\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}$$

Then,

$$\begin{array}{cc}{lim}_{n\to \infty }\mathcal{F}{\widehat{\nu }}_n\left({\chi }_1\right)\hfill & {\displaystyle =\underset{n\to \infty }{lim}[{\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}w(\widehat{\varkappa },{\widehat{\nu }}_n\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & +{\mathsf{\Theta }}_1\left({\chi }_1\right){\int }_0^1{\widehat{\mathsf{\Psi }}}_2(\widehat{\varkappa },{\widehat{\nu }}_n\left(\widehat{\varkappa }\right))d\widehat{\varkappa }\hfill \\ & {\displaystyle +{\mathsf{\Theta }}_2\left({\chi }_1\right){\int }_0^1{\widehat{\mathsf{\Psi }}}_1(\widehat{\varkappa },{\widehat{\nu }}_n\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & +{\mathsf{\Theta }}_3\left({\chi }_1\right){\int }_0^1{\widehat{\mathsf{\Psi }}}_2(\widehat{\varkappa },{\widehat{\nu }}_n\left(\widehat{\varkappa }\right))d\widehat{\varkappa }\hfill \\ & {\displaystyle +{\displaystyle \frac{{\mathsf{\Theta }}_4\left({\chi }_1\right)}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-\widehat{\iota }-1}w(\widehat{\varkappa },{\widehat{\nu }}_n\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & +{\displaystyle \frac{{\mathsf{\Theta }}_5\left({\chi }_1\right)}{\mathsf{\Gamma }\left(\mathsf{\xi }\right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-1}w(\widehat{\varkappa },x_n\left(\widehat{\varkappa }\right))d\widehat{\varkappa }\hfill \\ & {\displaystyle +{\displaystyle \frac{\vartheta \gamma }{(1+\vartheta )\mathsf{\Gamma }\left(\widehat{\iota }\right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }-1}{\widehat{\nu }}_n\left(\widehat{\varkappa }\right)d\widehat{\varkappa }}\hfill \\ & -{\displaystyle \frac{\vartheta }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}w(\widehat{\varkappa },{\widehat{\nu }}_n\left(\widehat{\varkappa }\right))d\widehat{\varkappa }]\hfill \\ & {\displaystyle -\frac{\gamma }{\mathsf{\Gamma }\left(\widehat{\iota }\right)}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }-1}{\widehat{\nu }}_n\left(\widehat{\varkappa }\right)d\widehat{\varkappa },}\hfill \\ & {\displaystyle ={\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}\underset{n\to \infty }{lim}w(\widehat{\varkappa },{\widehat{\nu }}_n\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\mathsf{\Theta }}_1\left({\chi }_1\right){\int }_0^1\underset{n\to \infty }{lim}{\widehat{\mathsf{\Psi }}}_2(\widehat{\varkappa },{\widehat{\nu }}_n\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\mathsf{\Theta }}_2\left({\chi }_1\right){\int }_0^1\underset{n\to \infty }{lim}{\widehat{\mathsf{\Psi }}}_1(\widehat{\varkappa },{\widehat{\nu }}_n\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\mathsf{\Theta }}_3\left({\chi }_1\right){\int }_0^1\underset{n\to \infty }{lim}{\widehat{\mathsf{\Psi }}}_3(\widehat{\varkappa },{\widehat{\nu }}_n\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\displaystyle \frac{{\mathsf{\Theta }}_4\left({\chi }_1\right)}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-\widehat{\iota }-1}\underset{n\to \infty }{lim}w(\widehat{\varkappa },{\widehat{\nu }}_n\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & +{\displaystyle \frac{{\mathsf{\Theta }}_5\left({\chi }_1\right)}{\mathsf{\Gamma }\left(\mathsf{\xi }\right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-1}{lim}_{n\to \infty }w(\widehat{\varkappa },{\widehat{\nu }}_n\left(\widehat{\varkappa }\right))d\widehat{\varkappa }\hfill \\ & {\displaystyle +{\displaystyle \frac{\vartheta \gamma }{(1+\vartheta )\mathsf{\Gamma }\left(\widehat{\iota }\right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }-1}\underset{n\to \infty }{lim}{\widehat{\nu }}_n\left(\widehat{\varkappa }\right)d\widehat{\varkappa }}\hfill \\ & -{\displaystyle \frac{\vartheta }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}{lim}_{n\to \infty }w(\widehat{\varkappa },{\widehat{\nu }}_n\left(\widehat{\varkappa }\right))d\widehat{\varkappa }\hfill \\ & -\frac{\gamma }{\mathsf{\Gamma }\left(\widehat{\iota }\right)}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }-1}{lim}_{n\to \infty }{\widehat{\nu }}_n\left(\widehat{\varkappa }\right)d\widehat{\varkappa },\hfill \\ & {\displaystyle ={\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}w(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\mathsf{\Theta }}_1\left({\chi }_1\right){\int }_0^1{\widehat{\mathsf{\Psi }}}_2(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }+{\mathsf{\Theta }}_2\left({\chi }_1\right){\int }_0^{1}g(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\mathsf{\Theta }}_3\left({\chi }_1\right){\int }_0^1{\widehat{\mathsf{\Psi }}}_3(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\displaystyle \frac{{\mathsf{\Theta }}_4\left({\chi }_1\right)}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-\widehat{\iota }-1}w(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\displaystyle \frac{{\mathsf{\Theta }}_5\left({\chi }_1\right)}{\mathsf{\Gamma }\left(\mathsf{\xi }\right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-1}w(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & +{\displaystyle \frac{\vartheta \gamma }{(1+\vartheta )\mathsf{\Gamma }\left(\omega \right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }-1}\widehat{\nu }\left(\widehat{\varkappa }\right)d\widehat{\varkappa }\hfill \\ & {\displaystyle -{\displaystyle \frac{\vartheta }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}w(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & -\frac{\gamma }{\mathsf{\Gamma }\left(\widehat{\iota }\right)}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }-1}\widehat{\nu }\left(\widehat{\varkappa }\right)d\widehat{\varkappa },\hfill \\ & =\mathcal{F}\widehat{\nu }\left({\chi }_1\right)\hfill \end{array}$$

for all ${\chi }_1\in [0,1].$ This shows the continuity of $\mathcal{F}$ in $\mathcal{S}.$

Claim 3. We show the compactness of $\mathcal{F}$ on $\mathcal{S}.$ Let $\widehat{\nu }\in \mathcal{S},$ for all ${\chi }_1\in J$ define $\underset{0\le {\chi }_1\le 1}{sup}|w({\chi }_1,0)|={\mathsf{{\rm Y}}}_0,$ $\underset{0\le t\le 1}{sup}|g({\chi }_1,0)|={\mathsf{{\rm Y}}}_1,$ $\underset{0\le t\le 1}{sup}|h({\chi }_1,0)|={\mathsf{{\rm Y}}}_2,$ $\underset{0\le t\le 1}{sup}|k({\chi }_1,0)|={\mathsf{{\rm Y}}}_3.$

with

$$\begin{array}{cc}r\hfill & ={\displaystyle \frac{{\mathsf{{\rm Y}}}_0}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}+{\displaystyle \frac{{\mathsf{{\rm Y}}}_0{\mathsf{\Theta }}_4}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota }+1)}}+{\mathsf{\Theta }}_1{\mathsf{{\rm Y}}}_2+{\mathsf{\Theta }}_2{\mathsf{{\rm Y}}}_1+{\mathsf{\Theta }}_3{\mathsf{{\rm Y}}}_3+{\displaystyle \frac{{\mathsf{\Theta }}_5{\mathsf{{\rm Y}}}_0}{\mathsf{\Gamma }(\mathsf{\xi }+1)}}\hfill \\ & +|{\displaystyle \frac{\vartheta {\mathsf{{\rm Y}}}_0}{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}|.\hfill \end{array}$$

First, we show that $\mathcal{F}\left(\mathcal{S}\right)$ is a uniformly bounded set in $\mathcal{S}$. For $\widehat{\nu }\in \mathcal{S}$, ${\chi }_1\in [0,1],$ we have:

$$\begin{array}{cc}|\mathcal{F}\widehat{\nu }\left({\chi }_1\right)|\hfill & {\displaystyle \le {\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}\left[|w(\widehat{\varkappa },\widehat{\nu })-w(\widehat{\varkappa },0)|+|w(\widehat{\varkappa },0)|\right]d\widehat{\varkappa }}\hfill \\ & +|{\mathsf{\Theta }}_1\left({\chi }_1\right)|{\int }_0^1|{\widehat{\mathsf{\Psi }}}_2(s;\widehat{\nu }\left(\widehat{\varkappa }\right))-{\widehat{\mathsf{\Psi }}}_2(\widehat{\varkappa };0)|+|{\widehat{\mathsf{\Psi }}}_2(\widehat{\varkappa };0)|d\widehat{\varkappa }\hfill \\ & {\displaystyle +|{\mathsf{\Theta }}_2\left({\chi }_1\right)|{\int }_0^1|{\widehat{\mathsf{\Psi }}}_1(s;\widehat{\nu }\left(\widehat{\varkappa }\right))-{\widehat{\mathsf{\Psi }}}_1(\widehat{\varkappa };0)|+|g(\widehat{\varkappa };0)|d\widehat{\varkappa }}\hfill \\ & {\displaystyle +|{\mathsf{\Theta }}_3\left({\chi }_1\right)|{\int }_0^1|{\widehat{\mathsf{\Psi }}}_3(s;\widehat{\nu }\left(\widehat{\varkappa }\right))-{\widehat{\mathsf{\Psi }}}_3(\widehat{\varkappa };0)|+|{\widehat{\mathsf{\Psi }}}_3(\widehat{\varkappa };0)|ds+{\displaystyle \frac{|{\mathsf{\Theta }}_4\left({\chi }_1\right)|}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota })}}}\hfill \\ & {\displaystyle \times {\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-\widehat{\iota }-1}\left[|w(\widehat{\varkappa },\widehat{\nu })-w(\widehat{\varkappa },0)|+|w(\widehat{\varkappa },0)|\right]ds}\hfill \\ & {\displaystyle +{\displaystyle \frac{|{\mathsf{\Theta }}_5\left({\chi }_1\right)|}{\mathsf{\Gamma }\left(\mathsf{\xi }\right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-1}\left[|f(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))-f(\widehat{\varkappa },0)|+|f(\widehat{\varkappa },0)|\right]d\widehat{\varkappa }}\hfill \\ & {\displaystyle +|{\displaystyle \frac{\vartheta \gamma }{(1+\vartheta )\mathsf{\Gamma }\left(\widehat{\iota }\right)}}|{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }}|\widehat{\nu }\left(\widehat{\varkappa }\right)|ds+|{\displaystyle \frac{\vartheta }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}|}\hfill \\ & {\displaystyle \times {\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}\left[|w(\widehat{\varkappa },\widehat{\nu })-w(\widehat{\varkappa },0)|+|w(\widehat{\varkappa },0)|\right]ds,}\hfill \end{array}$$

Consequently, $\begin{array}{cc}\parallel \mathcal{F}\widehat{\nu }\parallel \hfill & \le {\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}\left(q_1\parallel \widehat{\nu }\parallel +{\mathsf{{\rm Y}}}_0\right)+{\mathsf{\Theta }}_1[q_3\parallel \widehat{\nu }\parallel +{\mathsf{{\rm Y}}}_2]+{\mathsf{\Theta }}_2[q_2\parallel \widehat{\nu }\parallel +{\mathsf{{\rm Y}}}_1]+{\mathsf{\Theta }}_3[q_4\parallel \widehat{\nu }\parallel +{\mathsf{{\rm Y}}}_3]\hfill \\ & +{\displaystyle \frac{{\mathsf{\Theta }}_4}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota }+1)}}\left(q_1\parallel \widehat{\nu }\parallel +{\mathsf{{\rm Y}}}_0\right)+{\displaystyle \frac{{\mathsf{\Theta }}_5}{\mathsf{\Gamma }(\mathsf{\xi }+1)}}\left(q_1\parallel \widehat{\nu }\parallel +{\mathsf{{\rm Y}}}_0\right)+|{\displaystyle \frac{\vartheta \gamma }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+1)}}|\parallel \widehat{\nu }\parallel \hfill \\ & +|{\displaystyle \frac{\vartheta }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}|\left(q_1\parallel \widehat{\nu }\parallel +{\mathsf{{\rm Y}}}_0\right)\hfill \\ & \le [q_1\left({\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}+{\displaystyle \frac{{\mathsf{\Theta }}_4}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota }+1)}}+{\displaystyle \frac{{\mathsf{\Theta }}_5}{\mathsf{\Gamma }(\mathsf{\xi }+1)}}+|{\displaystyle \frac{\vartheta }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}|\right)\hfill \\ & +|{\displaystyle \frac{\vartheta \gamma }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+1)}}|+{\mathsf{\Theta }}_1{q}_3+{\mathsf{\Theta }}_2{q}_2+{\mathsf{\Theta }}_3{q}_4]\parallel \widehat{\nu }\parallel \hfill \\ & +{\displaystyle \frac{{\mathsf{{\rm Y}}}_0}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}+{\displaystyle \frac{{\mathsf{{\rm Y}}}_0{\mathsf{\Theta }}_4}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota }+1)}}+A_1{\mathsf{{\rm Y}}}_2+{\mathsf{\Theta }}_2{\mathsf{{\rm Y}}}_1\hfill \\ & +{\mathsf{\Theta }}_3{\mathsf{{\rm Y}}}_3+{\displaystyle \frac{{\mathsf{\Theta }}_5{\mathsf{{\rm Y}}}_0}{\mathsf{\Gamma }(\mathsf{\xi }+1)}}+|{\displaystyle \frac{\vartheta {\mathsf{{\rm Y}}}_0}{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}|,\hfill \end{array}$

thus,

$\begin{array}{cc}\parallel \mathcal{F}\widehat{\nu }\parallel \hfill & \le [q_1\left({\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}+{\displaystyle \frac{{\mathsf{\Theta }}_4}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota }+1)}}+{\displaystyle \frac{{\mathsf{\Theta }}_5}{\mathsf{\Gamma }(\mathsf{\xi }+1)}}+|{\displaystyle \frac{\vartheta }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}|\right)\hfill \\ & +|{\displaystyle \frac{\vartheta \gamma }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+1)}}|+{\mathsf{\Theta }}_1{q}_3+{\mathsf{\Theta }}_2{q}_2+{\mathsf{\Theta }}_3{q}_4]r\hfill \\ & +{\displaystyle \frac{{\mathsf{{\rm Y}}}_0}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}+{\displaystyle \frac{{\mathsf{{\rm Y}}}_0{\mathsf{\Theta }}_4}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota }+1)}}+{\mathsf{\Theta }}_1{\mathsf{{\rm Y}}}_2+{\mathsf{\Theta }}_2{\mathsf{{\rm Y}}}_1\hfill \\ & +{\mathsf{\Theta }}_3{\mathsf{{\rm Y}}}_3+{\displaystyle \frac{{\mathsf{\Theta }}_5{\mathsf{{\rm Y}}}_0}{\mathsf{\Gamma }(\mathsf{\xi }+1)}}+|{\displaystyle \frac{\vartheta {\mathsf{{\rm Y}}}_0}{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}|\hfill \end{array}$

for all $\widehat{\nu }\in \mathcal{S}$. This shows that $\mathcal{F}$ is uniformly bounded on $\mathcal{S}$. Next, we show the equicontinuity of $\mathcal{F}\left(\mathcal{S}\right)$ on $\mathcal{X}$. Let ${{\chi }_1}_1,{{\chi }_1}_2\in J,$ then for any $\widehat{\nu }\in \mathcal{S}$

$\begin{array}{cc}|\mathcal{F}\widehat{\nu }\left({{\chi }_1}_2\right)-\mathcal{F}\widehat{\nu }\left({{\chi }_1}_1\right)|\hfill & {\displaystyle \le |{\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}({\int }_0^{{{\chi }_1}_2}{({{\chi }_1}_2-s)}^{\widehat{\iota }+\mathsf{\xi }-1}w(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))}\hfill \\ & {\displaystyle -{\int }_0^{{{\chi }_1}_1}{({{\chi }_1}_1-s)}^{\widehat{\iota }+\mathsf{\xi }-1}w(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right)))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +({\mathsf{\Theta }}_1\left({{\chi }_1}_2\right)-{\mathsf{\Theta }}_1\left({{\chi }_1}_1\right)){\int }_0^1{\widehat{\mathsf{\Psi }}}_2(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +({\mathsf{\Theta }}_2\left({{\chi }_1}_2\right)-{\mathsf{\Theta }}_2\left({{\chi }_1}_1\right)){\int }_0^1{\widehat{\mathsf{\Psi }}}_1(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +({\mathsf{\Theta }}_3\left({{\chi }_1}_2\right)-{\mathsf{\Theta }}_3\left({{\chi }_1}_1\right)){\int }_0^1{\widehat{\mathsf{\Psi }}}_3(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\displaystyle \frac{({\mathsf{\Theta }}_4\left({{\chi }_1}_2\right)-{\mathsf{\Theta }}_4\left({{\chi }_1}_1\right))}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-\widehat{\iota }-1}w(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & +{\displaystyle \frac{({\mathsf{\Theta }}_5\left({{\chi }_1}_2\right)-{\mathsf{\Theta }}_5\left({{\chi }_1}_1\right))}{\mathsf{\Gamma }\left(\mathsf{\xi }\right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-1}w(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }|,\hfill \\ & {\displaystyle \le {\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_{{{\chi }_1}_1}^{{{\chi }_1}_2}|w(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))|ds}\hfill \\ & {\displaystyle +({\mathsf{\Theta }}_1\left({{\chi }_1}_2\right)-{\mathsf{\Theta }}_1\left({{\chi }_1}_1\right)){\int }_0^1{\widehat{\mathsf{\Psi }}}_2(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +({\mathsf{\Theta }}_2\left({{\chi }_1}_2\right)-{\mathsf{\Theta }}_2\left({{\chi }_1}_1\right)){\int }_0^1{\widehat{\mathsf{\Psi }}}_1(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +(A_3\left({{\chi }_1}_2\right)-A_3\left({{\chi }_1}_1\right)){\int }_0^1{\widehat{\mathsf{\Psi }}}_3(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\displaystyle \frac{({\mathsf{\Theta }}_4\left({{\chi }_1}_2\right)-{\mathsf{\Theta }}_4\left({{\chi }_1}_1\right))}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-\widehat{\iota }-1}w(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & +{\displaystyle \frac{({\mathsf{\Theta }}_5\left({{\chi }_1}_2\right)-{\mathsf{\Theta }}_5\left({{\chi }_1}_1\right))}{\mathsf{\Gamma }\left(\mathsf{\xi }\right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-1}w(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }|.\hfill \end{array}$ Hence,

$$\forall \epsilon >0,\exists \vartheta >0:|{{\chi }_1}_1-{{\chi }_1}_2|<\vartheta ⟹|\mathcal{F}\widehat{\nu }\left({{\chi }_1}_1\right)-\mathcal{F}\widehat{\nu }\left({{\chi }_1}_2\right)|<\epsilon$$

for all ${{\chi }_1}_1,{{\chi }_1}_2\in J$ and for all $\widehat{\nu }\in X$.

Therefore, the operator $\mathcal{F}\left(\mathcal{S}\right)$ is equicontinuously set in $\mathcal{X}$.

By applying the Arzelá–Ascoli theorem, we deduce that the operator $\mathcal{F}$ is compact and continuous.

Claim 4. We show Hypothesis 3 of Theorem 1.

Let $\widehat{\nu }\in \mathcal{X}$ and $\rho \in \mathcal{S}$ be arbitrary such that $\widehat{\nu }=\mathcal{P}\widehat{\nu }\mathcal{F}\rho +\mathcal{E}\widehat{\nu }$. Then,

$$\begin{array}{cc}|\widehat{\nu }\left({\chi }_1\right)|\hfill & =|\mathcal{P}\widehat{\nu }\left({\chi }_1\right)||\mathcal{F}\rho \left({\chi }_1\right)|+|\mathcal{E}\widehat{\nu }\left({\chi }_1\right)|\hfill \\ & {\displaystyle \le |f({\chi }_1,\widehat{\nu }\left({\chi }_1\right))|[{\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}|w(\widehat{\varkappa },\rho \left(\widehat{\varkappa }\right))|ds+|{\mathsf{\Theta }}_1\left({\chi }_1\right)|}\hfill \\ & {\displaystyle \times {\int }_0^1|{\widehat{\mathsf{\Psi }}}_2(s;\rho \left(\widehat{\varkappa }\right))|ds}\hfill \\ & {\displaystyle +|{\mathsf{\Theta }}_2\left({\chi }_1\right)|{\int }_0^1|{\widehat{\mathsf{\Psi }}}_1(s;\rho \left(\widehat{\varkappa }\right))|d\widehat{\varkappa }}\hfill \\ & {\displaystyle +|{\mathsf{\Theta }}_3\left({\chi }_1\right)|{\int }_0^1|{\widehat{\mathsf{\Psi }}}_3(s;\rho \left(\widehat{\varkappa }\right))|ds+{\displaystyle \frac{|{\mathsf{\Theta }}_4\left({\chi }_1\right)|}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}|w(\widehat{\varkappa },\rho \left(\widehat{\varkappa }\right))|d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\displaystyle \frac{|{\mathsf{\Theta }}_5\left({\chi }_1\right)|}{\mathsf{\Gamma }\left(\mathsf{\xi }\right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-1}|w(\widehat{\varkappa },\rho \left(\widehat{\varkappa }\right))|ds}\hfill \\ & {\displaystyle +{\displaystyle \frac{|\vartheta |}{|1+\vartheta |\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\omega +\beta -1}|w(\widehat{\varkappa },\rho \left(\widehat{\varkappa }\right))|d\widehat{\varkappa }}\hfill \\ & {\displaystyle +|{\displaystyle \frac{\vartheta \gamma }{(1+\vartheta )\mathsf{\Gamma }\left(\widehat{\iota }\right)}}|{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }-1}|\rho \left(\widehat{\varkappa }\right)|ds]+{\displaystyle \frac{|\gamma |}{\mathsf{\Gamma }\left(\widehat{\iota }\right)}}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }-1}|\widehat{\nu }\left(\widehat{\varkappa }\right)|d\widehat{\varkappa }}\hfill \\ & {\displaystyle \le |\phi \left({\chi }_1\right)|[{\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}m\left(\widehat{\varkappa }\right)d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\mathsf{\Theta }}_1{\int }_0^1\rho \left(\widehat{\varkappa }\right)d\widehat{\varkappa }+{\mathsf{\Theta }}_2{\int }_0^1\varphi \left(\widehat{\varkappa }\right)d\widehat{\varkappa }+{\mathsf{\Theta }}_3{\int }_0^1\psi \left(\widehat{\varkappa }\right)d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\displaystyle \frac{{\mathsf{\Theta }}_4}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-\widehat{\iota }-1}m\left(\widehat{\varkappa }\right)d\widehat{\varkappa }+{\displaystyle \frac{{\mathsf{\Theta }}_5}{\mathsf{\Gamma }\left(\mathsf{\xi }\right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-1}m\left(\widehat{\varkappa }\right)d\widehat{\varkappa }}\hfill \\ & {\displaystyle +\frac{|\vartheta |}{(|1+\vartheta |)\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}m\left(\widehat{\varkappa }\right)d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\displaystyle \frac{|\vartheta \gamma |}{(|1+\vartheta |)\mathsf{\Gamma }\left(\widehat{\iota }\right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }-1}|\rho \left(\widehat{\varkappa }\right)|ds]+{\displaystyle \frac{|\gamma |}{\mathsf{\Gamma }\left(\widehat{\iota }\right)}}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }-1}|\widehat{\nu }\left(\widehat{\varkappa }\right)|d\widehat{\varkappa }}\hfill \\ & \le ||\phi ||({\displaystyle \frac{\parallel m\parallel }{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}+{\mathsf{\Theta }}_1\parallel \rho \parallel +{\mathsf{\Theta }}_2\parallel \varphi \parallel +{\mathsf{\Theta }}_3\parallel \psi \parallel +{\displaystyle \frac{{\mathsf{\Theta }}_4\parallel m\parallel }{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota }+1)}}\hfill \\ & +{\displaystyle \frac{{\mathsf{\Theta }}_5\parallel m\parallel }{\mathsf{\Gamma }(\mathsf{\xi }+1)}}+{\displaystyle \frac{|\vartheta |\parallel m\parallel }{(|1+\vartheta |)\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}+{\displaystyle \frac{|\vartheta \lambda |}{(|1+\vartheta |)\mathsf{\Gamma }(\widehat{\iota }+1)}}\parallel \rho \parallel )\hfill \\ & +{\displaystyle \frac{|\gamma |}{\mathsf{\Gamma }|(\widehat{\iota }+1)}}\parallel \widehat{\nu }\parallel .\hfill \end{array}$$

Further, we obtain

$$\begin{array}{c}\parallel x\parallel \le r\hfill \end{array}$$

Then, $\widehat{\nu }\in \mathcal{S}$ and, hence, the 3 in Theorem 1 is verified.

Lastly, we obtain

$$\begin{array}{c}M=\parallel \mathcal{F}\left(\mathcal{S}\right)\parallel \hfill \\ & =sup\{\parallel \mathcal{F}\widehat{\nu }\parallel :\widehat{\nu }\in \mathcal{S}\}\hfill \\ & \le \left[q_1\right({\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}+{\displaystyle \frac{{\mathsf{\Theta }}_4}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota }+1)}}+{\displaystyle \frac{{\mathsf{\Theta }}_5}{\mathsf{\Gamma }(\mathsf{\xi }+1)}}\hfill \\ & +|{\displaystyle \frac{\vartheta }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}|)+|{\displaystyle \frac{\vartheta \gamma }{(1+\vartheta )\mathsf{\Gamma }(\gamma +1)}}|+{\mathsf{\Theta }}_1{q}_3+{\mathsf{\Theta }}_2{q}_2\hfill \\ & +{\mathsf{\Theta }}_3{q}_4]r+{\displaystyle \frac{{\mathsf{{\rm Y}}}_0}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}+{\displaystyle \frac{{\mathsf{{\rm Y}}}_0{\mathsf{\Theta }}_4}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota }+1)}}+{\mathsf{\Theta }}_1{\mathsf{{\rm Y}}}_2+{\mathsf{\Theta }}_2{\mathsf{{\rm Y}}}_1\hfill \\ & +{\mathsf{\Theta }}_3{\mathsf{{\rm Y}}}_3+{\displaystyle \frac{{\mathsf{\Theta }}_5{\mathsf{{\rm Y}}}_0}{\mathsf{\Gamma }(\mathsf{\xi }+1)}}+|{\displaystyle \frac{\mu {\mathsf{{\rm Y}}}_0}{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}.\hfill \end{array}$$

and so,

$$\begin{array}{cc}{q}_{0}M+{\displaystyle \frac{\lambda }{\mathsf{\Gamma }(\widehat{\iota }+1)}}\hfill & \le q_0\left(\right[q_1({\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}+{\displaystyle \frac{{\mathsf{\Theta }}_4}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota }+1)}}+{\displaystyle \frac{{\mathsf{\Theta }}_5}{\mathsf{\Gamma }(\mathsf{\xi }+1)}}\hfill \\ & +|{\displaystyle \frac{\vartheta }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}|)+|{\displaystyle \frac{\vartheta \lambda }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+1)}}|+{\mathsf{\Theta }}_1{q}_3+{\mathsf{\Theta }}_2{q}_2\hfill \\ & +{\mathsf{\Theta }}_3{q}_4]r+{\displaystyle \frac{{\mathsf{{\rm Y}}}_0}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}+{\displaystyle \frac{{\mathsf{{\rm Y}}}_0{\mathsf{\Theta }}_4}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota }+1)}}+{\mathsf{\Theta }}_1{\mathsf{{\rm Y}}}_2+{\mathsf{\Theta }}_2{\mathsf{{\rm Y}}}_1\hfill \\ & +{\mathsf{\Theta }}_3{\mathsf{{\rm Y}}}_3+{\displaystyle \frac{{\gamma }_5{\mathsf{{\rm Y}}}_0}{\mathsf{\Gamma }(\beta +1)}}+|{\displaystyle \frac{\vartheta {\mathsf{{\rm Y}}}_0}{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}})+{\displaystyle \frac{\gamma }{\mathsf{\Gamma }(\widehat{\iota }+1)}}<1.\hfill \end{array}$$

Hence, the operator $\mathcal{P}\widehat{\nu }\mathcal{F}\widehat{\nu }+\mathcal{E}\widehat{\nu }=\widehat{\nu }$ has at least one solution in $\mathcal{S}$. So, a solution can be defined for the problem (1) on J, and this finalizes the proof. □

4. Stability Results

Consider the nonlinear operator ${\mathcal{Z}}_1\in \mathcal{E}([0,1],\mathbb{R})\to \mathcal{E}([0,1],\mathbb{R})$, where $\widehat{\nu }$ is define by (4)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^c{\mathcal{D}}^{\mathsf{\xi }}{(}^c{\mathcal{D}}^{\widehat{\iota }}+\gamma )\left(\frac{\widehat{\nu }\left({\chi }_1\right)}{f({\chi }_1,\widehat{\nu }\left({\chi }_1\right))}\right)-\widehat{\mathsf{\Psi }}({\chi }_1,\widehat{\nu }\left({\chi }_1\right))={\mathcal{Z}}_1\left(\widehat{\nu }\right)\left({\chi }_1\right),{\chi }_1\in (0,1),\hfill \end{array}\right.\end{array}$$

for ${\chi }_1\in [0,1]$. For some ${\varsigma }_1>0$, the following inequality is valid.

$$\begin{array}{c}\hfill ||{\mathcal{Z}}_1,\left(\widehat{\nu }\right)||\le {\varsigma }_1,\end{array}$$

(5)

Definition 3.

The problem (1) is Hyers–Ulam stable if ${\mathcal{M}}_1>0$ exist with a unique solution $\left(\widehat{\nu }\right)\in \mathcal{E}([0,1],\mathbb{R})$ of problem (1) with

$$\begin{array}{c}\hfill ||\widehat{\nu }-\left(\widehat{\widehat{\nu }}\right)||\le {\mathcal{M}}_1{\varsigma }_1\end{array}$$

for each solution $\left(\widehat{\widehat{\nu }}\right)$ of the previous inequality in belongs to $\mathcal{E}\left(\right[0,1],\mathbb{R})$.

Theorem 3.

Let the assumptions of $\left(H_2\right)$ hold. Then, problem (1) is Ulam–Hyers stable.

Proof. 

Let $\left(\widehat{\nu }\right)\in \mathcal{E}([0,1],\mathbb{R})$ be the solution of the problem that satisfies (1). Let $\left(\widehat{x}\right)$ be any satisfying solution for (1).

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^c{\mathcal{D}}^{\mathsf{\xi }}{(}^c{\mathcal{D}}^{\widehat{\iota }}+\gamma )\left(\frac{\widehat{\nu }\left({\chi }_1\right)}{f({\chi }_1,\widehat{\nu }\left({\chi }_1\right))}\right)=\widehat{\mathsf{\Psi }}({\chi }_1,\widehat{\nu }\left({\chi }_1\right))+{\mathcal{Z}}_1\left(\widehat{\nu }\right)\left({\chi }_1\right),{\chi }_1\in (0,1),\hfill \end{array}\right.\end{array}$$

for ${\chi }_1\in [0,1]$. Therefore,

$$\begin{array}{cc}\hfill \widehat{\widehat{\nu }}\left({\chi }_1\right)=& {\displaystyle \widehat{\nu }\left(\widehat{x}\right)\left({\chi }_1\right)+f({\chi }_1,\widehat{\nu }\left({\chi }_1\right))[{\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}\rho \left(\widehat{\varkappa }\right)d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\mathsf{\Theta }}_1\left({\chi }_1\right){\int }_0^1{\widehat{\mathsf{\Psi }}}_2(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ \hfill & {\displaystyle +{\mathsf{\Theta }}_2\left({\chi }_1\right){\int }_0^1{\widehat{\mathsf{\Psi }}}_1(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\mathsf{\Theta }}_3\left({\chi }_1\right){\int }_0^1{\mathsf{\Theta }}_3(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\displaystyle \frac{{\mathsf{\Theta }}_4\left({\chi }_1\right)}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-\widehat{\iota }-1}\rho \left(\widehat{\varkappa }\right)d\widehat{\varkappa }}\hfill \\ \hfill & +{\displaystyle \frac{{\mathsf{\Theta }}_5\left({\chi }_1\right)}{\mathsf{\Gamma }\left(\mathsf{\xi }\right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-1}\rho \left(\widehat{\varkappa }\right)d\widehat{\varkappa }\hfill \\ & {\displaystyle +{\displaystyle \frac{\vartheta \gamma }{(1+\mu )\mathsf{\Gamma }\left(\omega \right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\omega -1}\widehat{\nu }\left(\widehat{\varkappa }\right)d\widehat{\varkappa }}\hfill \\ \hfill & -{\displaystyle \frac{\vartheta }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}\rho \left(\widehat{\varkappa }\right)d\widehat{\varkappa }]\hfill \\ & {\displaystyle -\frac{\gamma }{\mathsf{\Gamma }\left(\widehat{\iota }\right)}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }-1}\widehat{\nu }\left(\widehat{\varkappa }\right)d\widehat{\varkappa },}\hfill \end{array}$$

which implies

$$\begin{array}{cc}\hfill |\widehat{\nu }\left(\widehat{x}\right)\left({\chi }_1\right)-\widehat{\widehat{\nu }}\left({\chi }_1\right)|\le & {\displaystyle f({\chi }_1,\widehat{\nu }\left({\chi }_1\right))[{\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\omega +\beta -1}\rho \left(\widehat{\varkappa }\right)d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\mathsf{\Theta }}_1\left({\chi }_1\right){\int }_0^1{\widehat{\mathsf{\Psi }}}_2(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ \hfill & {\displaystyle +{\mathsf{\Theta }}_2\left({\chi }_1\right){\int }_0^1{\widehat{\mathsf{\Psi }}}_1(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\mathsf{\Theta }}_3\left({\chi }_1\right){\int }_0^1{\widehat{\mathsf{\Psi }}}_3(\widehat{\varkappa },\widehat{\nu }\left(\widehat{\varkappa }\right))d\widehat{\varkappa }}\hfill \\ & {\displaystyle +{\displaystyle \frac{A_4\left({\chi }_1\right)}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-\widehat{\iota }-1}\rho \left(\widehat{\varkappa }\right)d\widehat{\varkappa }}\hfill \\ \hfill & +{\displaystyle \frac{{\mathsf{\Theta }}_5\left({\chi }_1\right)}{\mathsf{\Gamma }\left(\mathsf{\xi }\right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\mathsf{\xi }-1}\rho \left(\widehat{\varkappa }\right)d\widehat{\varkappa }\hfill \\ & {\displaystyle +{\displaystyle \frac{\vartheta \gamma }{(1+\mu )\mathsf{\Gamma }\left(\widehat{\iota }\right)}}{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }-1}\widehat{\nu }\left(\widehat{\varkappa }\right)d\widehat{\varkappa }}\hfill \\ \hfill & -{\displaystyle \frac{\vartheta }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi })}}{\int }_0^1{(1-\widehat{\varkappa })}^{\widehat{\iota }+\mathsf{\xi }-1}\rho \left(\widehat{\varkappa }\right)d\widehat{\varkappa }]\hfill \\ & {\displaystyle -\frac{\gamma }{\mathsf{\Gamma }\left(\widehat{\iota }\right)}{\int }_0^{{\chi }_1}{({\chi }_1-\widehat{\varkappa })}^{\widehat{\iota }-1}\widehat{\nu }\left(\widehat{\varkappa }\right)d\widehat{\varkappa },}\hfill \end{array}$$

$$\begin{array}{c}\hfill |\left(\widehat{\nu }\right)-\left(\widehat{\widehat{\nu }}\right)|\le {\mathcal{M}}_1{\varsigma }_1.\end{array}$$

By this, the previously presented problem (1) is U-H stable. □

5. Examples

Example 1.

The following equation is of a fractional hybrid type:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^c{\mathcal{D}}^{\frac{4}{3}}\left({}^c{\mathcal{D}}^{\frac{1}{3}}+{\displaystyle \frac{1}{300}}\right)\frac{\widehat{\nu }\left({\chi }_1\right)}{f({\chi }_1,\widehat{\nu }\left({\chi }_1\right))}={\displaystyle \frac{1}{500+t^2}}\left(sin\left(\widehat{\nu }\left({\chi }_1\right)\right)+{\displaystyle \frac{|\widehat{\nu }\left({\chi }_1\right)|}{1+|\widehat{\nu }\left({\chi }_1\right)|}}+\widehat{\nu }\left({\chi }_1\right)\right),{\chi }_1\in [0,1],\hfill \\ {\displaystyle \frac{\widehat{\nu }\left(0\right)}{f(0,\widehat{\nu }\left(0\right))}+\frac{\widehat{\nu }\left(1\right)}{f(1,\widehat{\nu }\left(1\right))}={\displaystyle \frac{1}{200}}{\int }_0^1\frac{|\widehat{\nu }\left(\widehat{\varkappa }\right)|}{300+|\widehat{\nu }\left(\widehat{\varkappa }\right)|}ds,}\hfill \\ {\displaystyle {}^c{\mathcal{D}}^{\frac{1}{3}}\left(\frac{\widehat{\nu }\left(0\right)}{f(0,\widehat{\nu }\left(0\right))}\right)+{}^c{\mathcal{D}}^{\frac{1}{3}}\left(\frac{\widehat{\nu }\left(1\right)}{f(1,\widehat{\nu }\left(1\right))}\right)={\displaystyle \frac{1}{200}}{\int }_0^1{\left(\frac{1}{s+2}\right)}^3\frac{|\widehat{\nu }\left(\widehat{\varkappa }\right)|}{30+|\widehat{\nu }\left(\widehat{\varkappa }\right)|}ds,}\hfill \\ {\displaystyle {}^c{\mathcal{D}}^{\frac{2}{3}}\left(\frac{\widehat{\nu }\left(0\right)}{f(0,\widehat{\nu }\left(0\right))}\right)+{}^c{\mathcal{D}}^{\frac{2}{3}}\left(\frac{\widehat{\nu }\left(1\right)}{f(1,\widehat{\nu }\left(1\right))}\right)={\displaystyle \frac{1}{200}}{\int }_0^1{\left({\displaystyle \frac{1}{s+4}}\right)}^2{\displaystyle \frac{|\widehat{\nu }\left(\widehat{\varkappa }\right)|}{30+|\widehat{\nu }\left(\widehat{\varkappa }\right)|}}ds,}\hfill \end{array}\right.\end{array}$$

(6)

where $\mathsf{\xi }={\displaystyle \frac{4}{3}},$ $\widehat{\iota }={\displaystyle \frac{1}{3}},$ $\gamma ={\displaystyle \frac{1}{300}},$ $\vartheta =1,$ ${\tau }_1={\tau }_2={\tau }_3={\displaystyle \frac{1}{200}},$ and

$$\widehat{\mathsf{\Psi }}({\chi }_1,\widehat{\nu })={\displaystyle \frac{1}{500+t^2}}\left(sin\left(\widehat{\nu }\left({\chi }_1\right)\right)+{\displaystyle \frac{|\widehat{\nu }\left({\chi }_1\right)|}{1+|\widehat{\nu }\left({\chi }_1\right)|}}+\widehat{\nu }\left({\chi }_1\right)\right),$$

$$f({\chi }_1,\widehat{\nu })={\displaystyle \frac{{({\chi }_1+1)}^2}{100}}\left(sin\widehat{\nu }\left({\chi }_1\right)+{\displaystyle \frac{|\widehat{\nu }\left({\chi }_1\right)|}{1+|\widehat{\nu }\left({\chi }_1\right)|}}+3\right)$$

$${\widehat{\mathsf{\Psi }}}_2({\chi }_1,\widehat{\nu })={\displaystyle \frac{|\widehat{\nu }\left({\chi }_1\right)|}{300+|\widehat{\nu }\left({\chi }_1\right)|}},{\widehat{\mathsf{\Psi }}}_1({\chi }_1,\widehat{\nu })={\left(\frac{1}{{\chi }_1+2}\right)}^3{\displaystyle \frac{|\widehat{\nu }\left({\chi }_1\right)|}{30+|\widehat{\nu }\left({\chi }_1\right)|}},$$

$${\widehat{\mathsf{\Psi }}}_3({\chi }_1,\widehat{\nu })={\left(\frac{1}{{\chi }_1+4}\right)}^2{\displaystyle \frac{|\widehat{\nu }\left({\chi }_1\right)|}{30+|\widehat{\nu }\left({\chi }_1\right)|}}.$$

Clearly, $q_0=q_1=q_2=q_3={\displaystyle \frac{1}{500}},$$q_4={\displaystyle \frac{1}{300}},$

furthermore, we have

$$\begin{array}{cc}& q_0\left(\right[q_1({\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}+{\displaystyle \frac{A_4}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota }+1)}}+{\displaystyle \frac{{\mathsf{\Theta }}_5}{\mathsf{\Gamma }(\mathsf{\xi }+1)}}\hfill \\ & +|{\displaystyle \frac{\vartheta }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}|)+|{\displaystyle \frac{\vartheta \gamma }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+1)}}|+{\mathsf{\Theta }}_1{q}_3+A_2{q}_2\hfill \\ & +{\mathsf{\Theta }}_3{q}_4]+{\displaystyle \frac{{\mathsf{{\rm Y}}}_0}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}+{\displaystyle \frac{{\mathsf{{\rm Y}}}_0{\mathsf{\Theta }}_4}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota }+1)}}+{\mathsf{\Theta }}_1{\mathsf{{\rm Y}}}_2+{\mathsf{\Theta }}_2{\mathsf{{\rm Y}}}_1\hfill \\ & +{\mathsf{\Theta }}_3{\mathsf{{\rm Y}}}_3+{\displaystyle \frac{{\mathsf{\Theta }}_5{\mathsf{{\rm Y}}}_0}{\mathsf{\Gamma }(\mathsf{\xi }+1)}}+|{\displaystyle \frac{\vartheta {\mathsf{{\rm Y}}}_0}{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}})+{\displaystyle \frac{\gamma }{\mathsf{\Gamma }(\widehat{\iota }+1)}}\approx 0.026215478961<1.\hfill \end{array}$$

Hence, the problem 6 has a solution, by Theorem 2.

Example 2.

Consider a problem of the following form

$$\begin{array}{c}\hfill \left\{\begin{array}{c}c{\mathcal{D}}^{\frac{4}{3}}{(}^c{\mathcal{D}}^{\frac{1}{3}}+{\displaystyle \frac{1}{700}})\frac{x\left({\chi }_1\right)}{f({\chi }_1,\widehat{\nu }\left({\chi }_1\right))}={\displaystyle \frac{1}{1+t^4}}\left({\displaystyle \frac{{{\chi }_1}^4|\widehat{\nu }\left({\chi }_1\right)|}{\left(4|\widehat{\nu }\left({\chi }_1\right)|+10\right)}}+{\displaystyle \frac{|\widehat{\nu }\left({\chi }_1\right)|}{\left(4|\widehat{\nu }\left({\chi }_1\right)|+6\right)}}+{\displaystyle \frac{1}{1+{\widehat{\nu }}^2\left(\widehat{\varkappa }\right)}}\right),{\chi }_1\in [0,1],\hfill \\ {\displaystyle \frac{\widehat{\nu }\left(0\right)}{f(0,\widehat{\nu }\left(0\right))}+\frac{\widehat{\nu }\left(1\right)}{f(1,\widehat{\nu }\left(1\right))}={\displaystyle \frac{1}{300}}{\int }_0^1\frac{1}{s+100}\frac{|x(\widehat{\varkappa })|}{300+|x(\widehat{\varkappa })|}ds,}\hfill \\ {\displaystyle {}^c{\mathcal{D}}^{\frac{1}{3}}\left(\frac{\widehat{\nu }\left(0\right)}{f(0,\widehat{\nu }\left(0\right))}\right)+{}^c{\mathcal{D}}^{\frac{1}{3}}\left(\frac{\widehat{\nu }\left(1\right)}{f(1,\widehat{\nu }\left(1\right))}\right)={\displaystyle \frac{1}{300}}{\int }_0^1\left({\displaystyle \frac{1}{s+8000}}\right){\displaystyle \frac{|\widehat{\nu }\left(\widehat{\varkappa }\right)|}{30+|\widehat{\nu }\left(\widehat{\varkappa }\right)|}}ds,}\hfill \\ {\displaystyle {}^c{\mathcal{D}}^{\frac{2}{3}}\left(\frac{\widehat{\nu }\left(0\right)}{f(0,\widehat{\nu }\left(0\right))}\right)+{}^c{\mathcal{D}}^{\frac{2}{3}}\left(\frac{\widehat{\nu }\left(1\right)}{f(1,\widehat{\nu }\left(1\right))}\right)={\displaystyle \frac{1}{300}}{\int }_0^1\left(\frac{1}{s+1600}\right)\frac{|\widehat{\nu }\left(\widehat{\varkappa }\right)|}{30+|\widehat{\nu }\left(\widehat{\varkappa }\right)|}ds.}\hfill \end{array}\right.\end{array}$$

(7)

where $\mathsf{\xi }={\displaystyle \frac{3}{2}},$ $\widehat{\iota }={\displaystyle \frac{1}{2}},$ $\gamma ={\displaystyle \frac{1}{700}},$ $\vartheta =1,$ ${\tau }_1={\tau }_2={\tau }_3={\displaystyle \frac{1}{300}},$ and

$$\widehat{\mathsf{\Psi }}({\chi }_1,\widehat{\nu })={\displaystyle \frac{1}{{{\chi }_1}^2+4}}\left({\displaystyle \frac{{{\chi }_1}^2|x\left({\chi }_1\right)|}{\left(3|\widehat{\nu }\left({\chi }_1\right)|+1\right)}}+{\displaystyle \frac{|\widehat{\nu }\left({\chi }_1\right)|}{\left(6|\widehat{\nu }\left({\chi }_1\right)|+10\right)}}+{\displaystyle \frac{1}{1+{\widehat{\nu }}^2\left({\chi }_1\right)}}\right)$$

$$f({\chi }_1,\widehat{\nu })={\displaystyle \frac{1}{2(35+t)}}\left({\displaystyle \frac{{\widehat{\nu }}^2\left({\chi }_1\right)+|\widehat{\nu }\left({\chi }_1\right)|}{1+|\widehat{\nu }\left({\chi }_1\right)|}}\right)+t^{{\displaystyle \frac{1}{3}}}+1$$

$${\widehat{\mathsf{\Psi }}}_1({\chi }_1,\widehat{\nu })={\displaystyle \frac{1}{{\chi }_1+100}}{\displaystyle \frac{|\widehat{\nu }\left({\chi }_1\right)|}{300+|\widehat{\nu }\left({\chi }_1\right)|}},$$

$${\widehat{\mathsf{\Psi }}}_2({\chi }_1,\widehat{\nu })=\left(\frac{1}{{\chi }_1+8000}\right){\displaystyle \frac{|\widehat{\nu }\left({\chi }_1\right)|}{30+|\widehat{\nu }\left({\chi }_1\right)|}},{\widehat{\mathsf{\Psi }}}_3({\chi }_1,\widehat{\nu })=\left(\frac{1}{{\chi }_1+1600}\right){\displaystyle \frac{|\widehat{\nu }\left({\chi }_1\right)|}{30+|\widehat{\nu }\left({\chi }_1\right)|}}.$$

After calculating, we obtain

$$\begin{array}{cc}& q_0\left(\right[q_1({\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}+{\displaystyle \frac{A_4}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota }+1)}}+{\displaystyle \frac{{\mathsf{\Theta }}_5}{\mathsf{\Gamma }(\mathsf{\xi }+1)}}\hfill \\ & +|{\displaystyle \frac{\vartheta }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}|)+|{\displaystyle \frac{\vartheta \gamma }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+1)}}|+{\mathsf{\Theta }}_1{q}_3+A_2{q}_2\hfill \\ & +{\mathsf{\Theta }}_3{q}_4]+{\displaystyle \frac{{\mathsf{{\rm Y}}}_0}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}+{\displaystyle \frac{{\mathsf{{\rm Y}}}_0{\mathsf{\Theta }}_4}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota }+1)}}+{\mathsf{\Theta }}_1{\mathsf{{\rm Y}}}_2+{\mathsf{\Theta }}_2{\mathsf{{\rm Y}}}_1\hfill \\ & +{\mathsf{\Theta }}_3{\mathsf{{\rm Y}}}_3+{\displaystyle \frac{{\mathsf{\Theta }}_5{\mathsf{{\rm Y}}}_0}{\mathsf{\Gamma }(\mathsf{\xi }+1)}}+|{\displaystyle \frac{\vartheta {\mathsf{{\rm Y}}}_0}{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}})+{\displaystyle \frac{\gamma }{\mathsf{\Gamma }(\widehat{\iota }+1)}}\approx 0.145781245789<1.\hfill \end{array}$$

Hence, the problem 8 has a least one solution, by Theorem 2.

Example 3.

For this example, we consider the following problem of the form:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^c{\mathcal{D}}^{\frac{3}{2}}{(}^c{\mathcal{D}}^{\frac{1}{2}}+{\displaystyle \frac{1}{400}}){\displaystyle \frac{\widehat{\nu }\left({\chi }_1\right)}{f({\chi }_1,\widehat{\nu }\left({\chi }_1\right))}}={\displaystyle \frac{1}{\sqrt{{\chi }_1}+e}}\left({\displaystyle \frac{\left(3|\widehat{\nu }\left({\chi }_1\right)|+1\right)}{e^t|\widehat{\nu }\left({\chi }_1\right)|}}+{\displaystyle \frac{|\widehat{\nu }\left({\chi }_1\right)|}{\left(8|{\widehat{\nu }}^2\left({\chi }_1\right)|+10\right)}}+e^{-\widehat{\nu }\left({\chi }_1\right)}\right),{\chi }_1\in [0,1],\hfill \\ {\displaystyle \frac{\widehat{\nu }\left(0\right)}{f(0,\widehat{\nu }\left(0\right))}+\frac{\widehat{\nu }\left(1\right)}{f(1,\widehat{\nu }\left(1\right))}={\displaystyle \frac{1}{1000}}{\int }_0^1\frac{1}{s+20}\frac{|\widehat{\nu }\left(\widehat{\varkappa }\right)|}{300+|\widehat{\nu }\left(\widehat{\varkappa }\right)|}ds,}\hfill \\ {\displaystyle {}^c{\mathcal{D}}^{\frac{1}{3}}\left(\frac{\widehat{\nu }\left(0\right)}{f(0,\widehat{\nu }\left(0\right))}\right)+{}^c{\mathcal{D}}^{\frac{1}{3}}\left(\frac{\widehat{\nu }\left(1\right)}{f(1,\widehat{\nu }\left(1\right))}\right)={\displaystyle \frac{1}{1000}}{\int }_0^1\left({\displaystyle \frac{1}{s+8000}}\right){\displaystyle \frac{|\widehat{\nu }\left(\widehat{\varkappa }\right)|}{300+|\widehat{\nu }\left(\widehat{\varkappa }\right)|}}ds,}\hfill \\ {\displaystyle {}^c{\mathcal{D}}^{\frac{2}{3}}\left(\frac{\widehat{\nu }\left(0\right)}{f(0,\widehat{\nu }\left(0\right))}\right)+{}^c{\mathcal{D}}^{\frac{2}{3}}\left(\frac{\widehat{\nu }\left(1\right)}{f(1,\widehat{\nu }\left(1\right))}\right)={\displaystyle \frac{1}{1000}}{\int }_0^1\left(\frac{1}{s+1600}\right)\frac{|\widehat{\nu }\left(\widehat{\varkappa }\right)|}{300+|\widehat{\nu }\left(\widehat{\varkappa }\right)|}ds.}\hfill \end{array}\right.\end{array}$$

(8)

where $\mathsf{\xi }={\displaystyle \frac{3}{2}},$ $\widehat{\iota }={\displaystyle \frac{1}{2}},$ $\gamma ={\displaystyle \frac{1}{400}},$ $\vartheta =1,$ ${\tau }_1={\tau }_2={\tau }_3={\displaystyle \frac{1}{1000}},$ and

$$\widehat{\mathsf{\Psi }}({\chi }_1,\widehat{\nu })={\displaystyle \frac{1}{\sqrt{{\chi }_1}+e}}\left({\displaystyle \frac{\left(3|\widehat{\nu }\left({\chi }_1\right)|+1\right)}{e^t|\widehat{\nu }\left({\chi }_1\right)|}}+{\displaystyle \frac{|\widehat{\nu }\left({\chi }_1\right)|}{\left(8|{\widehat{\nu }}^2\left({\chi }_1\right)|+10\right)}}+e^{-\widehat{\nu }\left({\chi }_1\right)}\right)$$

$$f({\chi }_1,\widehat{\nu })={\displaystyle \frac{1}{{{\chi }_1}^2}}\left(cos\widehat{\nu }\left({\chi }_1\right)+{\displaystyle \frac{|\widehat{\nu }\left({\chi }_1\right)|}{1+|\widehat{\nu }\left({\chi }_1\right)|}}+3\right),$$

$${\widehat{\mathsf{\Psi }}}_1({\chi }_1,\widehat{\nu })={\displaystyle \frac{1}{{\chi }_1+20}}{\displaystyle \frac{|\widehat{\nu }\left({\chi }_1\right)|}{300+|\widehat{\nu }\left({\chi }_1\right)|}},$$

$${\widehat{\mathsf{\Psi }}}_2({\chi }_1,\widehat{\nu })=\left(\frac{1}{{\chi }_1+8000}\right){\displaystyle \frac{|\widehat{\nu }\left({\chi }_1\right)|}{300+|\widehat{\nu }\left({\chi }_1\right)|}},{\widehat{\mathsf{\Psi }}}_3({\chi }_1,\widehat{\nu })=\left(\frac{1}{{\chi }_1+1600}\right){\displaystyle \frac{|\widehat{\nu }\left({\chi }_1\right)|}{300+|\widehat{\nu }\left({\chi }_1\right)|}}.$$

After calculating, we obtain

$$\begin{array}{cc}& q_0\left(\right[q_1({\displaystyle \frac{1}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}+{\displaystyle \frac{A_4}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota }+1)}}+{\displaystyle \frac{{\mathsf{\Theta }}_5}{\mathsf{\Gamma }(\mathsf{\xi }+1)}}\hfill \\ & +|{\displaystyle \frac{\vartheta }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}|)+|{\displaystyle \frac{\vartheta \gamma }{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+1)}}|+{\mathsf{\Theta }}_1{q}_3+A_2{q}_2\hfill \\ & +{\mathsf{\Theta }}_3{q}_4]+{\displaystyle \frac{{\mathsf{{\rm Y}}}_0}{\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}}+{\displaystyle \frac{{\mathsf{{\rm Y}}}_0{\mathsf{\Theta }}_4}{\mathsf{\Gamma }(\mathsf{\xi }-\widehat{\iota }+1)}}+{\mathsf{\Theta }}_1{\mathsf{{\rm Y}}}_2+{\mathsf{\Theta }}_2{\mathsf{{\rm Y}}}_1\hfill \\ & +{\mathsf{\Theta }}_3{\mathsf{{\rm Y}}}_3+{\displaystyle \frac{{\mathsf{\Theta }}_5{\mathsf{{\rm Y}}}_0}{\mathsf{\Gamma }(\mathsf{\xi }+1)}}+|{\displaystyle \frac{\vartheta {\mathsf{{\rm Y}}}_0}{(1+\vartheta )\mathsf{\Gamma }(\widehat{\iota }+\mathsf{\xi }+1)}})+{\displaystyle \frac{\gamma }{\mathsf{\Gamma }(\widehat{\iota }+1)}}\approx 0.2569878945<1.\hfill \end{array}$$

Hence, the problem 8 has a least one solution, by Theorem 2.

6. Conclusions

Most natural phenomena are treated using different types of fractional differential equations. This diversity in this type of equation helps us to scrutinize the integration of many phenomena in various fields. This helps us in creating programs that enable us to consume rational materials. In this paper, based on the generalization of Dhage’s fixed point theorem, Banach contraction mapping, and U-H stability, we treated the existence, uniqueness, and stability of solutions to a sequential fractional (Hybrid) differential equation with hybrid boundary conditions, respectively. This equation plays an important role in the field of the control system. For future work, we suggest using other types of fractional derivative operators such as the generalized Hilfer fractional derivative, and the one who is interested in the subject can also investigate the existence and uniqueness of the solutions for the tripled systems via several fixed points theorems such as Leray–Schuader’s alternative, and Mönch’s fixed point theorem.