Stability and Controllability Study for Mixed Integral Fractional Delay Dynamic Systems Endowed with Impulsive Effects on Time Scales

Abstract

In this article, we investigate a novel class of mixed integral fractional delay dynamic systems with impulsive effects on time scales. Also, fixed-point techniques are applied to study the existence and uniqueness of a solution to the considered systems. Furthermore, sufficient conditions for Ulam–Hyers stability and controllability of the considered systems are established. It turns out that controllability is a very relevant property in dynamic systems and also in differential equations since, if controllability holds, then the solution of a system of differential equations also holds. Finally, an illustrative example of the obtained results is provided.

1. Background and Building Systems

Fractional differential equations (FDEs) and partial differential equations have been used to model some natural phenomena and physical problems as initial and boundary value problems. Numerous studies on the concept of FDEs have been conducted during the last few decades. Due to a chance meeting between Leibniz and L’Hospital in 1695, the concept of FDEs was initially put forth. The development of mathematical models for a wide range of issues emerging in mathematical networks, control theory, aerodynamics, bioengineering, physics, blood flows, signal processing, engineering, etc., is now significantly facilitated by FDEs [1,2].

From several investigations, we can deduce that FDEs have a far higher prominence than integer-order derivatives. As a result, fractional calculus attracted a lot of interest and attention from experts and scholars. Additionally, it improved the sketch of the hereditary characteristics of diverse materials and processes. As a result, numerous monographs, and research articles have been published in this area [3,4,5,6,7,8,9,10].

For many types of functional equations, the theory of stability analysis, including Mittag-Leffler function (MLF), exponential, Lyapunov, and finite time stability, has recently been addressed. In 1940, Ulam and Hyers developed the most significant and intriguing type of stability, known as Ulam-Hyers (UH) stability [11]. The stability of homomorphisms across groups was a topic Ulam touched on during his lecture at Wisconsin University. Hyers [12] responded favorably to Ulam’s dilemma in 1941 on the basis of the idea that groups can be thought of as Banach spaces (BS), and this stability came to be known as UH stability. See [13,14,15,16] for more details.

There are many phenomena in the real world that are exposed to short-term external influences as they evolve. These external effects durations are incredibly short compared to the overall period of the phenomenon being witnessed. Therefore, it is reasonable to assume that these outside forces are genuine impulses. Now to investigate these abrupt changes, impulsive differential equations play a key role in modeling physical real-world problems. Such type of impulsive differential equations has argued concern to different applications, including biological systems such as heartbeat, blood flow, and impulse rate. Also, it has many applications in electrodynamics, population dynamics, viscoelastic, radio physics, metallurgy, mathematical economy, electric technology, theoretical physics, pharmacokinetics, control theory, and chemical engineering technology [17,18,19,20,21,22,23].

The core idea in mathematical control theory, controllability, was introduced by Kalman in 1960. In general, controllability refers to the ability to use a proper control function to steer the state of a control dynamical equation from an arbitrary initial state to the desired terminal state. The controllability results have been studied by many authors [24,25]. Additionally, the study of controllability findings on temporal scales is a recent field with limited evidence [26,27]. The existence, controllability, and Ulam type stability of the impulsive fractional dynamical system with respect to a mixed structure have specifically been investigated in a few studies.

Based on the above contributions, in this manuscript, we consider a new class of dynamic systems with mixed integral fractional delays and impulsive effects on time scales. Also, fixed-point approaches are used to demonstrate the existence and originality of solutions to the systems under discussion. Moreover, Ulam-Hyers stability is also proven for the aforementioned mixed impulsive systems, as well as a generalized version of it. Furthermore, the controllability of the aforementioned systems was then examined. Finally, an illustrative example is used to explore the results from the earlier parts.

Now, we investigate the following mixed integral fractional dynamic systems on the time scale $\tau$, which are generalizations of the results of [28]:

$$\left\{\begin{array}{l}{}^{c,\tau }{D}^{\rho }\nu \left(\vartheta \right)=B\left(\vartheta \right)\nu \left(\vartheta \right)+Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\\ +\Theta \left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s\right),\\ {}^{c,\tau }{D}^{\rho }\xi \left(\vartheta \right)=B\left(\vartheta \right)\xi \left(\vartheta \right)+Z\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)\\ +\Theta \left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s\right),\\ \vartheta \in {\tau }^{\prime }=\tau \backslash \{{\vartheta }_1,{\vartheta }_2,{\vartheta }_1\dots ,{\vartheta }_n\},\rho \in (0,1),\nu \left({\vartheta }_0\right)={\nu }_0,\xi \left({\vartheta }_0\right)={\xi }_0,\\ \nu \left({\vartheta }_r^{+}\right)-\nu \left({\vartheta }_r^{-}\right)=O_r\left(\nu \left({\vartheta }_r^{-}\right),\xi \left({\vartheta }_r^{-}\right)\right)+D_r\left({\vartheta }_r^{-},\nu \left({\vartheta }_r^{-}\right),\xi \left({\vartheta }_r^{-}\right)\right),r=1,2,\dots ,n,\\ \xi \left({\vartheta }_r^{+}\right)-\xi \left({\vartheta }_r^{-}\right)=O_r\left(\xi \left({\vartheta }_r^{-}\right),\nu \left({\vartheta }_r^{-}\right)\right)+D_r\left({\vartheta }_r^{-},\xi \left({\vartheta }_r^{-}\right),\nu \left({\vartheta }_r^{-}\right)\right),r=1,2,\dots ,n,\end{array}\right.$$

(1)

and

$$\left\{\begin{array}{c}{}^{c,\tau }{D}^{\rho }\nu \left(\vartheta \right)=B\left(\vartheta \right)\nu \left(\vartheta \right)+Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\\ +\Xi \left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(s,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s\right),\\ {}^{c,\tau }{D}^{\rho }\xi \left(\vartheta \right)=B\left(\vartheta \right)\xi \left(\vartheta \right)+Z\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)\\ +\Xi \left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s\right),\\ \vartheta \in (s_i,{\vartheta }_{i+1}]\cap \tau ,i=1,2,\dots ,n,\rho \in (0,1),\nu \left({\vartheta }_0\right)={\nu }_0,\xi \left({\vartheta }_0\right)={\xi }_0,\\ \nu \left(\vartheta \right)=\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\xi \left(s\right),\nu \left(s\right))\Delta s,\vartheta \in ({\vartheta }_i,s_i]\cap \tau ,i=1,2,\dots ,n,\\ \xi \left(\vartheta \right)=\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\nu \left(s\right),\xi \left(s\right))\Delta s,\vartheta \in ({\vartheta }_i,s_i]\cap \tau ,i=1,2,\dots ,n.\end{array}\right.$$

(2)

Furthermore, we study the controllability of the following systems:

$$\left\{\begin{array}{c}{}^{c,\tau }{D}^{\rho }\nu \left(\vartheta \right)=B\left(\vartheta \right)\nu \left(\vartheta \right)+Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\\ +\Theta \left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s\right)+H\ell \left(\vartheta \right),\\ {}^{c,\tau }{D}^{\rho }\xi \left(\vartheta \right)=B\left(\vartheta \right)\xi \left(\vartheta \right)+Z\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)\\ +\Theta \left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s\right)+H\ell \left(\vartheta \right),\\ \vartheta \in {\tau }^{\prime }=\tau \backslash \{{\vartheta }_1,{\vartheta }_2,{\vartheta }_1\dots ,{\vartheta }_n\},\rho \in (0,1),\nu \left({\vartheta }_0\right)={\nu }_0,\xi \left({\vartheta }_0\right)={\xi }_0,\\ \nu \left({\vartheta }_r^{+}\right)-\nu \left({\vartheta }_r^{-}\right)=O_r\left(\nu \left({\vartheta }_r^{-}\right),\xi \left({\vartheta }_r^{-}\right)\right)+D_r\left({\vartheta }_r^{-},\nu \left({\vartheta }_r^{-}\right),\xi \left({\vartheta }_r^{-}\right)\right),r=1,2,\dots ,n,\\ \xi \left({\vartheta }_r^{+}\right)-\xi \left({\vartheta }_r^{-}\right)=O_r\left(\xi \left({\vartheta }_r^{-}\right),\nu \left({\vartheta }_r^{-}\right)\right)+D_r\left({\vartheta }_r^{-},\xi \left({\vartheta }_r^{-}\right),\nu \left({\vartheta }_r^{-}\right)\right),r=1,2,\dots ,n,\end{array}\right.$$

(3)

and

$$\left\{\begin{array}{c}{}^{c,\tau }{D}^{\rho }\nu \left(\vartheta \right)=B\left(\vartheta \right)\nu \left(\vartheta \right)+Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\\ +\Xi \left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(s,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s\right)+H\ell \left(\vartheta \right),\\ {}^{c,\tau }{D}^{\rho }\xi \left(\vartheta \right)=B\left(\vartheta \right)\xi \left(\vartheta \right)+Z\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)\\ +\Xi \left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s\right)+H\ell \left(\vartheta \right),\\ \vartheta \in (s_i,{\vartheta }_{i+1}]\cap \tau ,i=1,2,\dots ,n,\rho \in (0,1),\nu \left({\vartheta }_0\right)={\nu }_0,\xi \left({\vartheta }_0\right)={\xi }_0,\\ \nu \left(\vartheta \right)=\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\nu \left(s\right),\xi \left(s\right))\Delta s,\vartheta \in ({\vartheta }_i,s_i]\cap \tau ,i=1,2,\dots ,n,\\ \xi \left(\vartheta \right)=\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\xi \left(s\right),\nu \left(s\right))\Delta s,\vartheta \in ({\vartheta }_i,s_i]\cap \tau ,i=1,2,\dots ,n,\end{array}\right.$$

(4)

where ${}^{c,\tau }{D}^{\rho }$ denotes the Caputo derivative (CD) [1] of fractional order $\rho$ on time scales $\tau$, $B\left(\vartheta \right)$ represents a regressive square matrix, which is piecewise continuous, $H:\tau \to \tau$ is a bounded linear operator. By assuming $\mathbb{R}$ as the real number, $\nu ,\xi \in L^2(I,\mathbb{R})$ are control maps, ${\tau }^0={[{\vartheta }_0,{\vartheta }_l]}_{\tau },$ the pre-fixed numbers are ${\vartheta }_0=s_0<{\vartheta }_1<s_1<{\vartheta }_2<\cdots <{\vartheta }_n<s_n<{\vartheta }_{n+1}={\vartheta }_l$ and the mappings $Z:{\tau }^0\times {\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n,$$Z_{1,2}:{\tau }^0\times {\tau }^0\times {\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n,$$\Theta :{\tau }^0\times {\left({\mathbb{R}}^n\right)}^4\to {\mathbb{R}}^n,$ (where ${\left({\mathbb{R}}^n\right)}^4={\mathbb{R}}^n\times {\mathbb{R}}^n\times {\mathbb{R}}^n\times {\mathbb{R}}^n),$${\phi }_i:({\vartheta }_i,s_i]\cap \tau \times {\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n,$$i=1,2,\dots ,n,$$\Xi :(s_i,{\vartheta }_{i+1}]\cap \tau \times {\left({\mathbb{R}}^n\right)}^4\to {\mathbb{R}}^n,$$i=1,2,\dots ,n,$$O_r:{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n,$$D_r:{\tau }^0\times {\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ are continuous. We also provide the right limit and the left limit of the $\nu \left(\vartheta \right)$ and $\xi \left(\vartheta \right)$ at ${\vartheta }_r$ as $\nu \left({\vartheta }_r^{+}\right)={lim}_{ϵ\to 0^{+}}\nu ({\vartheta }_r+ϵ),$$\nu \left({\vartheta }_r^{-}\right)={lim}_{ϵ\to 0^{-}}\nu ({\vartheta }_r-ϵ),$$\xi \left({\vartheta }_r^{+}\right)={lim}_{ϵ\to 0^{+}}\xi ({\vartheta }_r+ϵ),$$\xi \left({\vartheta }_r^{-}\right)={lim}_{ϵ\to 0^{-}}\xi ({\vartheta }_r-ϵ),$ respectively.

2. Definitions and Auxiliary Lemmas

In this part, we give some definitions, fundamental ideas, lemmas and hypotheses for this study.

Assume that $C(I,\mathbb{R})$ is the space of all continuous mapping from the interval I onto $\mathbb{R}$. Clearly, $C(I,\mathbb{R})$ is a BS equipped with the norm ${∥\nu ∥}_C={sup}_{\vartheta \in \tau }\left|\nu \left(\vartheta \right)\right|.$ The product space (PS) $Q=C_4(I,\mathbb{R})$ (where $C_4(I,\mathbb{R})=C(I,\mathbb{R})\times C(I,\mathbb{R})\times C(I,\mathbb{R})\times C(I,\mathbb{R}))$ is also BS under the norm

$${∥\left({\nu }_1,{\nu }_2,{\nu }_3,{\nu }_4\right)∥}_C={∥{\nu }_1∥}_C+{∥{\nu }_2∥}_C+{∥{\nu }_3∥}_C+{∥{\nu }_4∥}_C.$$

Also, we define a BS $C^1(I,\mathbb{R})=\left\{\nu \in C(I,\mathbb{R}):{\nu }^{\Delta }\in C(I,\mathbb{R})\right\}$ with the norm ${∥\nu ∥}_{C^1}=max\left\{{∥\nu ∥}_C,{∥{\nu }^{\Delta }∥}_{C^1}\right\}.$ Moreover, $Q^1=C_4^1(I,\mathbb{R})$ is a PS with

$${∥\left({\nu }_1,{\nu }_2,{\nu }_3,{\nu }_4\right)∥}_{C^1}={∥{\nu }_1∥}_{C^1}+{∥{\nu }_2∥}_{C^1}+{∥{\nu }_3∥}_{C^1}+{∥{\nu }_4∥}_{C^1}.$$

A time scale $\tau$ is a closed, nonempty subset of $\mathbb{R}$. An interval on a time scale is defined as ${[a,b]}_{\tau }=\left\{\vartheta \in \tau :a\le \vartheta \le b\in C(I,\mathbb{R})\right\}.$ The same is true for ${(a,b)}_{\tau }$ and ${[a,b)}_{\tau }.$

The operators for forward and backward jumps $\rho :\tau \to \tau$ and $\upsilon :\tau \to \tau$ are described as

$$\rho \left(\vartheta \right)=inf\{s\in \tau :s>\vartheta \}\mathrm{and}\upsilon \left(\vartheta \right)=sup\{s\in \tau :s<\vartheta \},$$

respectively. To determine the existing distance between two consecutive points, the operator $\Omega :\tau \to [0,\infty )$, which is defined as $\Omega \left(\vartheta \right)=\rho \left(\vartheta \right)-\vartheta$ is applied. Along these lines, the derived version ${\tau }^r$ of $\tau$ is

$${\tau }^r=\left\{\begin{array}{cc}\tau \backslash (\upsilon ,sup\tau ],& \mathrm{if}sup\tau <\infty ,\\ \tau ,& \mathrm{if}sup\tau =\infty .\end{array}\right.$$

For all $\vartheta \in {\tau }^r,$ the regressive (respectively positively regressive) function $\beth :\tau \to \mathbb{R}$ is described as $1+\Omega \left(\vartheta \right)\beth \left(\vartheta \right)\ne 0$ (respectively $1+\Omega \left(\vartheta \right)\beth \left(\vartheta \right)>0).$

Definition 1

([29]). The delta derivative $f^{\Delta }\left(\vartheta \right)$ of a mapping $f:\tau \to \mathbb{R}$ at a point $\vartheta \in {\tau }^r$ is a number (assuming it exists) if for $\epsilon >0,$ a neighborhood V of ϑ exists if and only if

$$\left|\left(f\left(\rho \left(\vartheta \right)\right)-f\left(\varrho \right)\right)-f^{\Delta }\left(\vartheta \right)\left(\rho \left(\vartheta \right)-\varrho \right)\right|\le \epsilon \left|\rho \left(\vartheta \right)-\varrho \right|,\forall \varrho \in V.$$

The following results are very important in the sequel.

Theorem 1

([29]). Assume that $a,b\in \tau$ and $h\in C_{rd}\left(\tau ,\mathbb{R}\right).$ Then

(i)

$\tau =\mathbb{R}$ leads to

$${\int }_a^{b}f\left(\vartheta \right)\Delta \vartheta ={\int }_a^{b}f\left(\vartheta \right)d\vartheta .$$

(ii)

If only isolated points make up $[a,b)$, then

$${\int }_a^{b}f\left(\vartheta \right)\Delta \vartheta =\left\{\begin{array}{cc}\sum _{\vartheta \in [a,b)}\varkappa \left(\vartheta \right)f\left(\vartheta \right),& \mathit{if}a<b,\\ 0,& \mathit{if}a=b,\\ -\sum _{\vartheta \in [a,b)}\varkappa \left(\vartheta \right)f\left(\vartheta \right),& \mathit{if}a>b.\end{array}\right.$$

(iii)

If $\tau =y\mathbb{Z}=\{yr:r\in \mathbb{Z}\},$$y>0,$ we obtain

$${\int }_a^{b}f\left(\vartheta \right)\Delta \vartheta =\left\{\begin{array}{cc}{\displaystyle \sum _{r=\frac{a}{y}}^{\frac{b}{y}-1}}yf\left(yr\right),& \mathit{if}a<b,\\ 0,& \mathit{if}a=b,\\ -{\displaystyle \sum _{r=\frac{a}{y}}^{\frac{b}{y}-1}}yf\left(yr\right),& \mathit{if}a>b.\end{array}\right.$$

Theorem 2

([30]). Let the function $f:\mathbb{R}\to \mathbb{R}$ be a continuous and nondecreasing, and $a,b\in \tau ,$ then

$${\int }_a^{b}f\left(\vartheta \right)\Delta \vartheta ={\int }_a^{b}f\left(\vartheta \right)d\vartheta .$$

Definition 2

([30]). The delta fractional integral for the integrable mapping $\psi :{[a,b]}_{\tau }\to \mathbb{R}$ is described as

$${}^{\Delta }{I}_{c^{+}}^{\rho }\psi \left(\vartheta \right)=\frac{1}{\Gamma \left(\rho \right)}\underset{c}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}\psi \left(s\right)\Delta s.$$

Definition 3

([30]). For the mapping $f:\tau \to \mathbb{R},$ the fractional CD on the time scale is

$${}^{c,\tau }{D}_{c^{+}}^{\rho }f\left(\vartheta \right)=\frac{1}{\Gamma (n-\rho )}\underset{c}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{n-\rho -1}{f}^{{\Delta }^n}\left(s\right)\Delta s,$$

where $n=\left[\rho \right]+1$ and $f^{{\Delta }_n}$ is the n-th derivative of $f.$

• If $\tau ={\cup }_{i=0}^{\infty }[2i,2i+1],$ we have

  $$\begin{array}{ccc}\hfill {}^{c,\tau }{D}_{c^{+}}^{\rho }f\left(\vartheta \right)& =& \frac{1}{\Gamma (n-\rho )}\underset{c}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{n-\rho -1}{f}^{{\Delta }^n}\left(s\right)\Delta s\hfill \\ & =& \frac{1}{\Gamma (n-\rho )}\left(\sum _{r=0}^{i-1}\underset{2r}{\overset{2r+1}{\int }}{\left(\vartheta -s\right)}^{n-\rho -1}{f}^{{\Delta }^n}\left(s\right)\Delta s+\underset{2i}{\overset{2i+1}{\int }}{\left(\vartheta -s\right)}^{n-\rho -1}{f}^{{\Delta }^n}\left(s\right)\Delta s\right),\hfill \end{array}$$

  for $\vartheta \in [2i,2i+1],$$i=0,1,\dots$.
• If $\tau =y\mathbb{Z},$$y>0,$ we get

  $$\begin{array}{ccc}\hfill {}^{c,\tau }{D}_{c^{+}}^{\rho }f\left(\vartheta \right)& =& \frac{1}{\Gamma (n-\rho )}\underset{c}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{n-\rho -1}{f}^{{\Delta }^n}\left(s\right)\Delta s\hfill \\ & =& \frac{1}{\Gamma (n-\rho )}\sum _{r=0}^{\frac{\vartheta }{y}-1}y{\left(\vartheta -ry\right)}^{n-\rho -1}{f}^{{\Delta }^n}\left(y\right),\vartheta \in \tau .\hfill \end{array}$$
• If $\tau =\left\{q^n:q>1,n\in \mathbb{Z}\right\}\cup \mathbb{Z},$ then

  $$\begin{array}{ccc}\hfill {}^{c,\tau }{D}_{c^{+}}^{\rho }f\left(\vartheta \right)& =& \frac{1}{\Gamma (n-\rho )}\underset{c}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{n-\rho -1}{f}^{{\Delta }^n}\left(s\right)\Delta s\hfill \\ & =& \frac{1}{\Gamma (n-\rho )}\sum _{\vartheta \in \tau }\varkappa \left(s\right){\left(\vartheta -s\right)}^{n-\rho -1}{f}^{{\Delta }^n}\left(s\right)\Delta s.\hfill \end{array}$$

Take into account the MLF as

$$O_{\rho ,\alpha }\left(\vartheta \right)=\sum _{r=0}^{\infty }\frac{{\vartheta }^r}{\Gamma \left(\rho r+\alpha \right)}\mathrm{for}\rho ,\alpha >0.$$

If $\alpha =1,$ we get

$$O_{\rho ,1}\left(\varsigma {\vartheta }^{\rho }\right)=O_{\rho }\left(\varsigma {\vartheta }^{\rho }\right)=\sum _{r=0}^{\infty }\frac{{\varsigma }^r{\vartheta }^{\rho r}}{\Gamma \left(\rho r+1\right)},\varsigma ,\vartheta \in \mathbb{C},$$

has the fascinating property ${}^c{D}_{0^{+}}^{\rho }{O}_{\rho }\left(\varsigma {\vartheta }^{\rho }\right)=\varsigma O_{\rho }\left({\vartheta }^{\rho }\right).$

Remark 1.

Motivated by the results of [31], the solution of the problem (1) takes the form

$$\nu \left(\vartheta \right)=\left\{\begin{array}{c}{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right),\vartheta \in ({\vartheta }_0,{\vartheta }_1],\\ {\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right),\\ +\sum _{t=1}^i{O}_t\left(\nu \left({\vartheta }_t^{-}\right),\xi \left({\vartheta }_t^{-}\right)\right)+D_t\left({\vartheta }_t^{-},\nu \left({\vartheta }_t^{-}\right),\xi \left({\vartheta }_t^{-}\right)\right),\vartheta \in ({\vartheta }_i,{\vartheta }_{i+1}],i=1,2,\dots ,n,\end{array}\right.$$

and

$$\xi \left(\vartheta \right)=\left\{\begin{array}{c}{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\xi }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Theta \left(s,\xi \left(s\right),\nu \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p\right),\vartheta \in ({\vartheta }_0,{\vartheta }_1],\\ {\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\xi }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Theta \left(s,\xi \left(s\right),\nu \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p\right),\\ +\sum _{t=0}^i{O}_t\left(\xi \left({\vartheta }_t^{-}\right),\nu \left({\vartheta }_t^{-}\right)\right)+D_t\left({\vartheta }_t^{-},\xi \left({\vartheta }_t^{-}\right),\nu \left({\vartheta }_t^{-}\right)\right),\vartheta \in ({\vartheta }_i,{\vartheta }_{i+1}],i=1,2,\dots ,n,\end{array}\right.$$

where ${\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right)$ represents the matrix for the aforementioned MLF, which is given by

$${\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right)=\sum _{t=0}^i\frac{B^t{\vartheta }^{\rho t}}{\Gamma \left(\rho t+1\right)}.$$

Now, in order to reach our results, we take into account the following assertions:

(1)

For the mappings $\Theta ,\Xi :{\tau }^0\times {\left({\mathbb{R}}^n\right)}^4\to {\mathbb{R}}^n$, there are positive constants $T_{{\Theta }_i},T_{{\Xi }_i},$$i=1,2,3,4$ so that

$$\begin{array}{ccc}\hfill \left|\Theta \left(\vartheta ,{\sigma }_1,{\sigma }_2,{\sigma }_3,{\sigma }_4\right)-\Theta \left(\vartheta ,{\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4\right)\right|& \le & \sum _{i=1}^4{T}_{{\Theta }_i}\left|{\sigma }_i-{\omega }_i\right|,\mathrm{for}\mathrm{each}\vartheta \in I,{\sigma }_i,{\omega }_i\in \mathbb{R},\hfill \\ \hfill \left|\Xi \left(\vartheta ,{\sigma }_1,{\sigma }_2,{\sigma }_3,{\sigma }_4\right)-\Xi \left(\vartheta ,{\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4\right)\right|& \le & \sum _{i=1}^4{T}_{{\Xi }_i}\left|{\sigma }_i-{\omega }_i\right|,\mathrm{for}\mathrm{each}\vartheta \in I,{\sigma }_i,{\omega }_i\in \mathbb{R}.\hfill \end{array}$$

(2)

The mappings $\Theta ,\Xi :{\tau }^0\times {\left({\mathbb{R}}^n\right)}^4\to {\mathbb{R}}^n$ are continuous and there are positive constants $c_i,h_i,$$i=1,2,3,4$ so that

$$\begin{array}{ccc}\hfill \left|\Theta \left(\vartheta ,{\sigma }_1,{\sigma }_2,{\sigma }_3,{\sigma }_4\right)\right|& \le & c_0+c_1\left|{\sigma }_1\right|+c_2\left|{\sigma }_2\right|+c_3\left|{\sigma }_3\right|+c_4\left|{\sigma }_4\right|,\mathrm{for}\mathrm{each}\vartheta \in I,{\sigma }_i\in \mathbb{R},\hfill \\ \hfill \left|\Xi \left(\vartheta ,{\sigma }_1,{\sigma }_2,{\sigma }_3,{\sigma }_4\right)\right|& \le & h_0+h_1\left|{\sigma }_1\right|+h_2\left|{\sigma }_2\right|+h_3\left|{\sigma }_3\right|+h_4\left|{\sigma }_4\right|,\mathrm{for}\mathrm{each}\vartheta \in I,{\sigma }_i\in \mathbb{R}.\hfill \end{array}$$

(3)

There is a bounded invertible operator ${{(}^{\rho }{\Re }_{{\vartheta }_0}^A)}^{-1}$ for the linear operator $\left({}^{\rho }{\Re }_{{\vartheta }_0}^A\right):L^2(I,\mathbb{R})\to \mathbb{R},$ which is described as

$${}^{\rho }{\Re }_{{\vartheta }_0}^{A}L=\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(A-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)HL\left(s\right)\Delta s,$$

(5)

these operators admit values in $L^2(I,\mathbb{R})\backslash ker{(}^{\rho }{\Re }_{{\vartheta }_0}^A).$

(4)

There exists a positive constant $Y_{\Re }^{\rho }$ so that $∥{{(}^{\rho }{\Re }_{{\vartheta }_0}^A)}^{-1}∥\le Y_{\Re }^{\rho }.$

(5)

The operator $H:\tau \to \tau$ is continuous and there is $Y_{\Re }$ so that $∥H∥\le Y_{\Re }.$

Equation (5) can be computed for various $\tau$ using Theorem 1 as follows:

• If $\tau =y\mathbb{Z},$$y>0,$ we have

  $${}^{\rho }{\Re }_{{\vartheta }_0}^{A}L=\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(A-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)HL\left(s\right)\Delta s=\sum _{r=0}^{\frac{\vartheta }{y}-1}y{\left(A-rsy\right)}^{\rho -1}HL\left(sy\right).$$
• If $\tau ={\cup }_{i=0}^{\infty }[2i,2i+1]$ and $\vartheta \in [4,5],$ we obtain

  $$\begin{array}{ccc}& & {}^{\rho }{\Re }_{{\vartheta }_0}^{A}L\hfill \\ & =& \underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(A-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)HL\left(s\right)\Delta s\hfill \\ & =& \underset{0}{\overset{1}{\int }}{\left(A-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)HL\left(s\right)\Delta s+\underset{2}{\overset{3}{\int }}{\left(A-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)HL\left(s\right)\Delta s\hfill \\ & & +\underset{4}{\overset{A}{\int }}{\left(A-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)HL\left(s\right)\Delta s.\hfill \end{array}$$
• When $\tau =\left\{q^m:q>1,m\in \mathbb{Z}\right\}\cup \mathbb{Z}.$ Then

  $${}^{\rho }{\Re }_{{\vartheta }_0}^{A}L=\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(A-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)HL\left(s\right)\Delta s=\sum _{\vartheta \in [0,A]}\varkappa {(A-\vartheta )}^{\rho -1}HL\left(\vartheta \right).$$

For simplicity, we consider the following:

$$\begin{array}{ccc}& & \begin{array}{ccc}{u}_1=\underset{\vartheta \in \tau }{sup}∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0∥,& u_2=\underset{\vartheta \in \tau }{sup}∥{\Lambda }_{\rho ,\rho }\left(B{\left(\vartheta -s\right)}^{\rho }\right)∥,& u_3=\underset{\vartheta \in \tau }{sup}∥{\left(\vartheta -s\right)}^{\rho -1}∥,\\ u_4=\underset{\vartheta \in \tau }{sup}∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\xi }_0∥,& E_1=\sum _{j=1}^i\left(T_O+T_D\right)\delta +u_1,& E_2=\sum _{j=1}^i\left(T_O+T_D\right),\\ E_3=\frac{1}{\Gamma \left(\rho \right)}{u}_3{T}_{\phi }\delta \left(s_i-{\vartheta }_i\right)+u_1,& E_4=\frac{1}{\Gamma \left(\rho \right)}{u}_3{T}_{\phi }\delta \left(s_i-{\vartheta }_i\right),& E_5=\sum _{j=1}^i\left(T_O+T_D\right)\delta +u_4,\end{array}\hfill \\ & & \begin{array}{c}{E}_6=\frac{1}{\Gamma \left(\rho \right)}{u}_3{T}_{\phi }\delta \left(s_i-{\vartheta }_i\right)+u_4,\end{array}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill J_1& =& u_3\left(u_2{T}_Z+u_2{T}_{{\Theta }_1}+u_2{T}_{{\Theta }_2}+u_2\left(T_{{\Theta }_3}{T}_{Z_1}+T_{{\Theta }_4}{T}_{Z_2}\right)(s_l-s_0)+Y_{\Theta }^{*}\right)({\vartheta }_l-{\vartheta }_0)\hfill \\ \hfill J_2& =& u_3\left(u_2{T}_Z+u_2{T}_{{\Theta }_1}+u_2{T}_{{\Theta }_2}+u_2\left(T_{{\Theta }_3}{T}_{Z_1}+T_{{\Theta }_4}{T}_{Z_2}\right)(s_l-s_0)\right)({\vartheta }_l-{\vartheta }_0)\hfill \\ \hfill J_1^{*}& =& u_3\left(u_2{T}_Z+u_2{T}_{{\Xi }_1}+u_2{T}_{{\Xi }_2}+u_2\left(T_{{\Theta }_3}{T}_{Z_1}+T_{{\Theta }_4}{T}_{Z_2}\right)(s_l-s_0)+Y_{\Xi }^{*}\right)({\vartheta }_l-{\vartheta }_0),\hfill \\ \hfill J_2^{*}& =& u_3\left(u_2{T}_Z+u_2{T}_{{\Xi }_1}+u_2{T}_{{\Xi }_2}+u_2\left(T_{{\Theta }_3}{T}_{Z_1}+T_{{\Theta }_4}{T}_{Z_2}\right)(s_l-s_0)\right)({\vartheta }_l-{\vartheta }_0)\hfill \\ \hfill J_3& =& E_2+J_2,J_3^{*}=E_4+J_2^{*}.\hfill \end{array}$$

3. The Existence and Uniqueness Study

In this part, existence and uniqueness criteria are studied.

Theorem 3.

Under the assertion $\left(1\right)$, the mixed impulsive system (MIS) (1) possesses a unique solution (US) provided that

$$\underset{1\le i\le 3}{max}\left\{J_i\right\}<0.5.$$

(6)

Proof. 

Assume that $\mho \subseteq Q$ and $\mho =\left\{\left(K_1,K_2,K_3,K_4\right)\in Q:∥K_1,K_2,K_3,K_4∥\le {\delta }_2\right\},$ where ${\delta }_2=max\{\delta ,{\delta }_1\}$ and $\delta ,{\delta }_1\in (0,1)$ so that

$$\delta >max\{E_1,E_2,E_3\}\mathrm{and}\delta >max\{E_5,E_2,E_3\},$$

and the rest constants are presented in the sequel. Define an operator ${\Im }_{\rho }:\mho \times \mho \to \mho$ by

$${\Im }_{\rho }(\nu ,\xi )\left(\vartheta \right)=\left\{\begin{array}{c}{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right),\vartheta \in ({\vartheta }_0,{\vartheta }_1],\\ {\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right),\\ +\sum _{t=0}^i{O}_t\left(\nu \left({\vartheta }_t^{-}\right),\xi \left({\vartheta }_t^{-}\right)\right)+D_t\left({\vartheta }_t^{-},\nu \left({\vartheta }_t^{-}\right),\xi \left({\vartheta }_t^{-}\right)\right),\vartheta \in ({\vartheta }_i,{\vartheta }_{i+1}],i=1,2,\dots ,n,\end{array}\right.$$

(7)

Suppose that

$$\begin{array}{ccc}\hfill ∥Z\left(\vartheta ,\nu ,\xi \right)∥& \le & ∥Z\left(\vartheta ,\nu ,\xi \right)-Z\left(\vartheta ,0,0\right)∥+∥Z\left(\vartheta ,0,0\right)∥\le T_Z\left(∥\nu ∥+∥\xi ∥\right)+Y_Z,\hfill \\ \hfill ∥O\left(\nu \left({\vartheta }^{-}\right),\xi \left({\vartheta }^{-}\right)\right)∥& \le & T_O(∥\nu ∥+∥\xi ∥),\hfill \\ \hfill D_t\left({\vartheta }^{-},\nu \left({\vartheta }^{-}\right),\xi \left({\vartheta }^{-}\right)\right)& \le & T_D(∥\nu ∥+∥\xi ∥),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill ∥\Theta \left(\vartheta ,K_1,K_2,K_3,K_4\right)∥& \le & ∥\Theta \left(\vartheta ,K_1,K_2,K_3,K_4\right)-\Theta \left(\vartheta ,0,0,0,0\right)∥-∥\Theta \left(\vartheta ,0,0,0,0\right)∥\hfill \\ & \le & T_{{\Theta }_1}∥K_1∥+T_{{\Theta }_2}∥K_2∥+T_{{\Theta }_3}∥K_3∥+T_{{\Theta }_4}∥K_4∥+Y_{\Theta },\hfill \end{array}$$

where $Y_Z={sup}_{\vartheta \in \tau }∥Z\left(\vartheta ,0,0\right)∥,$$Y_{\Theta }={sup}_{\vartheta \in \tau }∥\Theta \left(\vartheta ,0,0,0,0\right)∥,$ and $Y_{\Theta }^{*}=Y_Z+Y_{\Theta }.$ Additionally, $K_1=\nu \left(s\right),$$K_2=\xi \left(s\right),$

$$K_3=\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\mathrm{and}{K}_4=\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p.$$

Now, we show that ${\Im }_{\rho }:\mho \times \mho \to \mho$ is a self-mapping. For $\vartheta \in ({\vartheta }_i,{\vartheta }_{i+1}],$$i=1,2,\dots ,n,$ we can obtain

$$\begin{array}{ccc}\hfill ∥{\Im }_{\rho }(\nu ,\xi )\left(\vartheta \right)∥& \le & \sum _{j=1}^i∥O_j\left(\nu \left({\vartheta }_j^{-}\right),\xi \left({\vartheta }_j^{-}\right)\right)∥+\sum _{j=1}^i∥D_j\left({\vartheta }_j^{-},\nu \left({\vartheta }_j^{-}\right),\xi \left({\vartheta }_j^{-}\right)\right)∥+∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0∥\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥∥\left(Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\right.\hfill \\ & & +\left.\Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right)\right)∥\Delta s\hfill \end{array}$$

$$\begin{array}{ccc}\hfill & \le & \sum _{j=1}^i{T}_O∥\nu \left({\vartheta }_j^{-}\right)+\xi \left({\vartheta }_j^{-}\right)∥+\sum _{j=1}^i{T}_D∥\nu \left({\vartheta }_j^{-}\right)+\xi \left({\vartheta }_j^{-}\right)∥+∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0∥\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥\left(∥Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)∥\right.\hfill \\ & & +∥\Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right)∥\Delta s,\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill ∥{\Im }_{\rho }(\nu ,\xi )\left(\vartheta \right)∥& \le & \sum _{j=1}^i{T}_O∥\nu \left({\vartheta }_j^{-}\right)+\xi \left({\vartheta }_j^{-}\right)∥+\sum _{j=1}^i{T}_D∥\nu \left({\vartheta }_j^{-}\right)+\xi \left({\vartheta }_j^{-}\right)∥+∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0∥\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥\left(T_Z\left(∥\nu ∥+∥\xi ∥\right)+T_{{\Theta }_1}∥\nu \left(s\right)∥+T_{{\Theta }_2}∥\xi \left(s\right)∥\right.\hfill \\ & & +\left.T_{{\Theta }_3}∥\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p∥+T_{{\Theta }_4}∥\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p∥+Y_{\Theta }^{*}\right)\Delta s\hfill \end{array}$$

$$\begin{array}{ccc}\hfill & \le & \sum _{j=1}^i{T}_O∥\nu \left({\vartheta }_j^{-}\right)+\xi \left({\vartheta }_j^{-}\right)∥+\sum _{j=1}^i{T}_D∥\nu \left({\vartheta }_j^{-}\right)+\xi \left({\vartheta }_j^{-}\right)∥+∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0∥\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥\left(T_Z\left(∥\nu ∥+∥\xi ∥\right)+T_{{\Theta }_1}∥\nu \left(s\right)∥+T_{{\Theta }_2}∥\xi \left(s\right)∥\right.\hfill \\ & & +\left.T_{{\Theta }_3}{T}_{Z_1}\underset{s_0}{\overset{s_l}{\int }}∥\nu \left(p\right)+\xi \left(p\right)∥\Delta p+T_{{\Theta }_4}{T}_{Z_2}\underset{s_0}{\overset{s_l}{\int }}∥\nu \left(p\right)+\xi \left(p\right)∥\Delta p+Y_{\Theta }^{*}\right)\Delta s,\hfill \end{array}$$

hence

$$\begin{array}{ccc}\hfill ∥{\Im }_{\rho }(\nu ,\xi )\left(\vartheta \right)∥& \le & \left(T_O+T_D\right)\sum _{j=1}^i∥\nu \left({\vartheta }_j^{-}\right)+\xi \left({\vartheta }_j^{-}\right)∥+∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0∥\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥\left(T_Z\left(∥\nu ∥+∥\xi ∥\right)+T_{{\Theta }_1}∥\nu \left(s\right)∥+\right.\hfill \\ & & \left.+T_{{\Theta }_2}∥\xi \left(s\right)∥+\left(T_{{\Theta }_3}{T}_{Z_1}+T_{{\Theta }_4}{T}_{Z_2}\right)∥\nu \left(p\right)+\xi \left(p\right)∥(s_l-s_0)+Y_{\Theta }^{*}\right)\Delta s\hfill \\ & \le & \left(T_O+T_D\right)\sum _{j=1}^i\underset{\vartheta \in \tau }{sup}∥\nu \left({\vartheta }_j^{-}\right)+\xi \left({\vartheta }_j^{-}\right)∥+\underset{\vartheta \in \tau }{sup}∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0∥\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}\underset{\vartheta \in \tau }{sup}∥{\left(\vartheta -s\right)}^{\rho -1}∥\underset{\vartheta \in \tau }{sup}∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥\left(T_Z\left(\underset{\vartheta \in \tau }{sup}∥\nu ∥+\underset{\vartheta \in \tau }{sup}∥\xi ∥\right)+T_{{\Theta }_1}\underset{\vartheta \in \tau }{sup}∥\nu \left(s\right)∥\right.\hfill \\ & & \left.+T_{{\Theta }_2}\underset{\vartheta \in \tau }{sup}∥\xi \left(s\right)∥+\left(T_{{\Theta }_3}{T}_{Z_1}+T_{{\Theta }_4}{T}_{Z_2}\right)\underset{\vartheta \in \tau }{sup}∥\nu \left(p\right)+\xi \left(p\right)∥(s_l-s_0)+Y_{\Theta }^{*}\right)\Delta s,\hfill \end{array}$$

which leads to

$$\begin{array}{ccc}\hfill ∥{\Im }_{\rho }(\nu ,\xi )\left(\vartheta \right)∥& \le & \left(T_O+T_D\right)\sum _{j=1}^i{∥\nu +\xi ∥}_{\infty }+u_1+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{u}_2{u}_3\left(T_Z{∥\nu +\xi ∥}_{\infty }+T_{{\Theta }_1}{∥\nu ∥}_{\infty }\right.\hfill \\ & & +\left.T_{{\Theta }_2}{∥\xi ∥}_{\infty }+\left(T_{{\Theta }_3}{T}_{Z_1}+T_{{\Theta }_4}{T}_{Z_2}\right){∥\nu \left(p\right)+\xi \left(p\right)∥}_{\infty }(s_l-s_0)+Y_{\Theta }^{*}\right)\Delta s\hfill \\ & \le & \sum _{j=1}^i\left(T_O+T_D\right)\delta +u_1+u_3\left(\delta u_2{T}_Z+\delta u_2{T}_{{\Theta }_1}+\delta u_2{T}_{{\Theta }_2}\right.\hfill \\ & & +\left.\delta u_2\left(T_{{\Theta }_3}{T}_{Z_1}+T_{{\Theta }_4}{T}_{Z_2}\right)(s_l-s_0)+\delta Y_{\Theta }^{*}\right)\times \underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}\Delta s\hfill \\ & \le & \sum _{j=1}^i\left(T_O+T_D\right)\delta +u_1+\delta u_3\left(u_2{T}_Z+u_2{T}_{{\Theta }_1}+u_2{T}_{{\Theta }_2}\right.\hfill \\ & & +\left.u_2\left(T_{{\Theta }_3}{T}_{Z_1}+T_{{\Theta }_4}{T}_{Z_2}\right)(s_l-s_0)+Y_{\Theta }^{*}\right)({\vartheta }_l-{\vartheta }_0).\hfill \end{array}$$

So

$$∥{\Im }_{\rho }(\nu ,\xi )\left(\vartheta \right)∥\le E_1+\delta J_1\le \delta +\delta J_1={\delta }_1.$$

Similarly, one can prove that

$$∥{\Im }_{\rho }(\xi ,\nu )\left(\vartheta \right)∥\le E_5+\delta J_1\le \delta +\delta J_1={\delta }_1,$$

where ${\delta }_1=\delta +\delta J_1.$ This implies that

$$∥{\Im }_{\rho }(\nu ,\xi )\left(\vartheta \right)∥\le {\delta }_2\mathrm{and}∥{\Im }_{\rho }(\xi ,\nu )\left(\vartheta \right)∥\le {\delta }_2.$$

(8)

It follows from (8) that $\Im \left(\mho \times \mho \right)\subseteq \mho .$ Again for $\vartheta \in ({\vartheta }_i,{\vartheta }_{i+1}],$$i=1,2,\dots ,n,$ with ${\nu }_0={\tilde{\nu }}_0$ and ${\xi }_0={\tilde{\xi }}_0,$ one gets

$$\begin{array}{ccc}& & ∥{\Im }_{\rho }(\nu ,\xi )\left(\vartheta \right)-{\Im }_{\rho }(\tilde{\nu },\tilde{\xi })\left(\vartheta \right)∥\hfill \\ & \le & \sum _{j=1}^i∥O_j\left(\nu \left({\vartheta }_j^{-}\right),\xi \left({\vartheta }_j^{-}\right)\right)-O_j\left(\tilde{\nu }\left({\vartheta }_j^{-}\right),\tilde{\xi }\left({\vartheta }_j^{-}\right)\right)∥+\sum _{j=1}^i∥D_j\left({\vartheta }_j^{-},\nu \left({\vartheta }_j^{-}\right),\xi \left({\vartheta }_j^{-}\right)\right)-D_j\left({\vartheta }_j^{-},\tilde{\nu }\left({\vartheta }_j^{-}\right),\tilde{\xi }\left({\vartheta }_j^{-}\right)\right)∥\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥∥\left(Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\right.\hfill \\ & & +\left.\Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right)\right)\hfill \\ & & -\left(Z\left(\vartheta ,\tilde{\nu }\left(\vartheta \right),\tilde{\xi }\left(\vartheta \right)\right)+\Theta \left(s,\tilde{\nu }\left(s\right),\tilde{\xi }\left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\tilde{\nu }\left(p\right),\tilde{\xi }\left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\tilde{\nu }\left(p\right),\tilde{\xi }\left(p\right)\right)\Delta p\right)\right)∥\Delta s\hfill \end{array}$$

which yields that

$$\begin{array}{ccc}& & ∥{\Im }_{\rho }(\nu ,\xi )\left(\vartheta \right)-{\Im }_{\rho }(\tilde{\nu },\tilde{\xi })\left(\vartheta \right)∥\hfill \\ & \le & \sum _{j=1}^i{T}_O\left(∥\nu \left({\vartheta }_j^{-}\right)-\tilde{\nu }\left({\vartheta }_j^{-}\right)∥+∥\xi \left({\vartheta }_j^{-}\right)-\tilde{\xi }\left({\vartheta }_j^{-}\right)∥\right)+\sum _{j=1}^i{T}_D\left(∥\nu \left({\vartheta }_j^{-}\right)-\tilde{\nu }\left({\vartheta }_j^{-}\right)∥+∥\xi \left({\vartheta }_j^{-}\right)-\tilde{\xi }\left({\vartheta }_j^{-}\right)∥\right)\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥\left[T_Z\left(∥\nu \left(\vartheta \right)-\tilde{\nu }\left(\vartheta \right)∥+∥\xi \left(\vartheta \right)-\tilde{\xi }\left(\vartheta \right)∥\right)\right.\hfill \\ & & +T_{{\Theta }_1}∥\nu \left(s\right)-\tilde{\nu }\left(s\right)∥+T_{{\Theta }_2}∥\xi \left(s\right)-\tilde{\xi }\left(s\right)∥+T_{{\Theta }_3}\underset{s_0}{\overset{s_l}{\int }}∥Z_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)-Z_1\left(s,p,\tilde{\nu }\left(p\right),\tilde{\xi }\left(p\right)\right)∥\Delta p\hfill \\ & & \left.+T_{{\Theta }_4}\underset{s_0}{\overset{s_l}{\int }}∥Z_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)-Z_2\left(s,p,\tilde{\nu }\left(p\right),\tilde{\xi }\left(p\right)\right)∥\Delta p\right]\Delta s.\hfill \end{array}$$

Hence

$$\begin{array}{ccc}& & ∥{\Im }_{\rho }(\nu ,\xi )\left(\vartheta \right)-{\Im }_{\rho }(\tilde{\nu },\tilde{\xi })\left(\vartheta \right)∥\hfill \\ & \le & \sum _{j=1}^i\left(T_O+T_D\right)\left(∥\nu \left({\vartheta }_j^{-}\right)-\tilde{\nu }\left({\vartheta }_j^{-}\right)∥+∥\xi \left({\vartheta }_j^{-}\right)-\tilde{\xi }\left({\vartheta }_j^{-}\right)∥\right)\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥\left[T_Z\left(∥\nu \left(\vartheta \right)-\tilde{\nu }\left(\vartheta \right)∥+∥\xi \left(\vartheta \right)-\tilde{\xi }\left(\vartheta \right)∥\right)+T_{{\Theta }_1}∥\nu \left(s\right)-\tilde{\nu }\left(s\right)∥\right.\hfill \\ & & +T_{{\Theta }_2}∥\xi \left(s\right)-\tilde{\xi }\left(s\right)∥+T_{{\Theta }_3}{T}_{Z_1}\underset{s_0}{\overset{s_l}{\int }}\left(∥\nu \left(p\right)-\tilde{\nu }\left(p\right)∥+∥\xi \left(p\right)-\tilde{\xi }\left(p\right)∥\right)\Delta p\hfill \\ & & \left.+T_{{\Theta }_3}{T}_{Z_2}\underset{s_0}{\overset{s_l}{\int }}\left(∥\nu \left(p\right)-\tilde{\nu }\left(p\right)∥+∥\xi \left(p\right)-\tilde{\xi }\left(p\right)∥\right)\Delta p\right]\Delta s\hfill \end{array}$$

$$\begin{array}{ccc}& \le & \sum _{j=1}^i\left(T_O+T_D\right)\left(\underset{\vartheta \in \tau }{sup}∥\nu \left({\vartheta }_j^{-}\right)-\tilde{\nu }\left({\vartheta }_j^{-}\right)∥+\underset{\vartheta \in \tau }{sup}∥\xi \left({\vartheta }_j^{-}\right)-\tilde{\xi }\left({\vartheta }_j^{-}\right)∥\right)\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}\underset{\vartheta \in \tau }{sup}∥{\left(\vartheta -s\right)}^{\rho -1}∥\underset{\vartheta \in \tau }{sup}∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥\hfill \\ & & \times \left[\begin{array}{c}{T}_Z\left(\underset{\vartheta \in \tau }{sup}∥\nu \left(\vartheta \right)-\tilde{\nu }\left(\vartheta \right)∥+\underset{\vartheta \in \tau }{sup}∥\xi \left(\vartheta \right)-\tilde{\xi }\left(\vartheta \right)∥\right)\\ +T_{{\Theta }_1}\underset{\vartheta \in \tau }{sup}∥\nu \left(s\right)-\tilde{\nu }\left(s\right)∥\end{array}\right.\hfill \\ & & +T_{{\Theta }_2}\underset{\vartheta \in \tau }{sup}∥\xi \left(s\right)-\tilde{\xi }\left(s\right)∥+T_{{\Theta }_3}{T}_{Z_1}\left(\underset{\vartheta \in \tau }{sup}∥\nu \left(p\right)-\tilde{\nu }\left(p\right)∥+\underset{\vartheta \in \tau }{sup}∥\xi \left(p\right)-\tilde{\xi }\left(p\right)∥\right)(s_l-s_0)\hfill \\ & & \left.+T_{{\Theta }_3}{T}_{Z_2}\left(\underset{\vartheta \in \tau }{sup}∥\nu \left(p\right)-\tilde{\nu }\left(p\right)∥+\underset{\vartheta \in \tau }{sup}∥\xi \left(p\right)-\tilde{\xi }\left(p\right)∥\right)(s_l-s_0)\right]\Delta s.\hfill \end{array}$$

Using the fact ${∥\nu -\tilde{\nu }∥}_{\infty }\le {∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }$ and ${∥\xi -\tilde{\xi }∥}_{\infty }\le {∥\xi -\tilde{\xi }∥}_{\infty }+{∥\nu -\tilde{\nu }∥}_{\infty },$ one can write

$$\begin{array}{ccc}& & ∥{\Im }_{\rho }(\nu ,\xi )\left(\vartheta \right)-{\Im }_{\rho }(\tilde{\nu },\tilde{\xi })\left(\vartheta \right)∥\hfill \\ & \le & \sum _{j=1}^i\left(T_O+T_D\right)\left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right)+u_3{u}_2\left[T_Z\left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right)+T_{{\Theta }_1}{∥\nu -\tilde{\nu }∥}_{\infty }\right.\hfill \\ & & +T_{{\Theta }_2}{∥\xi -\tilde{\xi }∥}_{\infty }+T_{{\Theta }_3}{T}_{Z_1}\left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right)(s_l-s_0)\hfill \\ & & +\left.T_{{\Theta }_3}{T}_{Z_2}\left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right)(s_l-s_0)\right]\times \underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}\Delta s,\hfill \\ & \le & \sum _{j=1}^i\left(T_O+T_D\right)+u_3\left[u_2{T}_Z+u_2{T}_{{\Theta }_1}+u_2{T}_{{\Theta }_2}+u_2{T}_{{\Theta }_3}{T}_{Z_1}(s_l-s_0)+u_2{T}_{{\Theta }_3}{T}_{Z_2}(s_l-s_0)\right]({\vartheta }_l-{\vartheta }_0)\hfill \\ & & \times \left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right)\hfill \\ & =& \left(E_2+J_2\right)\left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right)=J_3\left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right).\hfill \end{array}$$

(9)

Analogously, we obtain that

$$∥{\Im }_{\rho }(\nu ,\xi )\left(\vartheta \right)-{\Im }_{\rho }(\tilde{\nu },\tilde{\xi })\left(\vartheta \right)∥\le J_3\left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right).$$

(10)

From (6), (9) and (10), we conclude that the operator ${\Im }_{\rho }$ is strictly contractive. As a result, using the Banach principle, the MIS (1) has a US. □

We now have the following conclusion for the MIS (2):

Theorem 4.

Via the assertion $\left(1\right)$, the MIS (2) admits a US provided that

$$\underset{1\le i\le 3}{max}\left\{J_i^{*}\right\}<0.5.$$

(11)

Proof. 

Assume that $\mho \subseteq Q$ and $\mho =\left\{\left(K_1,K_2,K_3,K_4\right)\in Q:∥K_1,K_2,K_3,K_4∥\le {\delta }_2\right\},$ where ${\delta }_2=max\{\delta ,{\delta }_1\}$ and $\delta ,{\delta }_1\in (0,1)$ so that

$$\delta >max\{E_1,E_2,E_3\}\mathrm{or}\delta >max\{E_5,E_2,E_3\}.$$

Describe ${\wp }_{\rho }:\mho \times \mho \to \mho$ as

$${\wp }_{\rho }(\nu ,\xi )\left(\vartheta \right)=\left\{\begin{array}{c}{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Xi \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right),\vartheta \in ({\vartheta }_i,s_i]\cap \tau ,\\ {\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Xi \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right),\\ \frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{s_i}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\nu \left(s\right),\xi \left(s\right))\Delta s,\vartheta \in (s_i,{\vartheta }_{i+1}]\cap \tau ,i=1,2,\dots ,n.\end{array}\right.$$

Also, let

$$\begin{array}{ccc}\hfill ∥Z\left(\vartheta ,\nu ,\xi \right)∥& \le & ∥Z\left(\vartheta ,\nu ,\xi \right)-Z\left(\vartheta ,0,0\right)∥+∥Z\left(\vartheta ,0,0\right)∥\le T_Z\left(∥\nu ∥+∥\xi ∥\right)+Y_Z,\hfill \\ \hfill ∥\phi (s,\nu (s),\xi (s\left)\right)∥& \le & T_{\phi }\left(∥\nu ∥+∥\xi ∥\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill ∥\Xi \left(\vartheta ,K_1,K_2,K_3,K_4\right)∥& \le & ∥\Xi \left(\vartheta ,K_1,K_2,K_3,K_4\right)-\Xi \left(\vartheta ,0,0,0,0\right)∥-∥\Xi \left(\vartheta ,0,0,0,0\right)∥\hfill \\ & \le & T_{{\Xi }_1}∥K_1∥+T_{{\Xi }_2}∥K_2∥+T_{{\Xi }_3}∥K_3∥+T_{{\Xi }_4}∥K_4∥+Y_{\Xi },\hfill \end{array}$$

where $Y_Z={sup}_{\vartheta \in \tau }∥Z\left(\vartheta ,0,0\right)∥,$$Y_{\Xi }={sup}_{\vartheta \in \tau }∥\Xi \left(\vartheta ,0,0,0,0\right)∥,$ and $Y_{\Xi }^{*}=Y_Z+Y_{\Xi }.$ Additionally, $K_1=\nu \left(s\right),$$K_2=\xi \left(s\right),$

$$K_3=\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\mathrm{and}{K}_4=\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p.$$

Now, we claim that ${\wp }_{\rho }:\mho \times \mho \to \mho$ is a self-mapping. For $\vartheta \in ({\vartheta }_i,{\vartheta }_{i+1}]\cap \tau ,$$i=1,2,\dots ,n,$ we have

$$\begin{array}{ccc}\hfill ∥{\wp }_{\rho }(\nu ,\xi )\left(\vartheta \right)∥& \le & ∥\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{s_i}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\nu \left(s\right),\xi \left(s\right))\Delta s∥+∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0∥\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥∥\left(Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\right.\hfill \\ & & +\left.\Xi \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right)\right)∥\Delta s\hfill \end{array}$$

$$\begin{array}{ccc}\hfill & \le & \frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{s_i}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\phi }_i(s,\nu \left(s\right),\xi \left(s\right))∥\Delta s+∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0∥\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥\left(∥Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)∥\right.\hfill \\ & & +\left.∥\Xi \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right)∥\right)\Delta s,\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill ∥{\wp }_{\rho }(\nu ,\xi )\left(\vartheta \right)∥& \le & \frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{s_i}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥T_{\phi }\left(∥\nu \left(s\right)∥+∥\xi \left(s\right)∥\right)\Delta s+∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0∥\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥\left(T_Z\left(∥\nu ∥+∥\xi ∥\right)+T_{{\Xi }_1}∥\nu \left(s\right)∥+T_{{\Xi }_2}∥\xi \left(s\right)∥\right.\hfill \\ & & +\left.T_{{\Xi }_3}∥\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p∥+T_{{\Xi }_4}∥\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p∥+Y_{\Xi }^{*}\right)\Delta s\hfill \\ & \le & \frac{1}{\Gamma \left(\rho \right)}∥{\left(\vartheta -s\right)}^{\rho -1}∥T_{\phi }∥\nu +\xi ∥\left(s_i-{\vartheta }_i\right)+∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0∥\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥\left(T_Z∥\nu +\xi ∥+T_{{\Xi }_1}∥\nu \left(s\right)∥+T_{{\Xi }_2}∥\xi \left(s\right)∥\right.\hfill \\ & & +\left.T_{{\Theta }_3}{T}_{Z_1}\underset{s_0}{\overset{s_l}{\int }}∥\nu \left(p\right)+\xi \left(p\right)∥\Delta p+T_{{\Theta }_4}{T}_{Z_2}\underset{s_0}{\overset{s_l}{\int }}∥\nu \left(p\right)+\xi \left(p\right)∥\Delta p+Y_{\Xi }^{*}\right)\Delta s,\hfill \end{array}$$

hence

$$\begin{array}{ccc}& & ∥{\wp }_{\rho }(\nu ,\xi )\left(\vartheta \right)∥\hfill \\ & \le & \frac{1}{\Gamma \left(\rho \right)}∥{\left(\vartheta -s\right)}^{\rho -1}∥T_{\phi }∥\nu \left(s\right)+\xi \left(s\right)∥\left(s_i-{\vartheta }_i\right)+∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0∥\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥\left(T_Z∥\nu +\xi ∥+T_{{\Xi }_1}∥\nu \left(s\right)∥+T_{{\Xi }_2}∥\xi \left(s\right)∥\right.\hfill \\ & & \left.T_{{\Theta }_3}{T}_{Z_1}∥\nu \left(p\right)+\xi \left(p\right)∥(s_l-s_0)+T_{{\Theta }_4}{T}_{Z_2}∥\nu \left(p\right)+\xi \left(p\right)∥(s_l-s_0)+Y_{\Xi }^{*}\right)\Delta s\hfill \\ & \le & \frac{1}{\Gamma \left(\rho \right)}\underset{\vartheta \in \tau }{sup}∥{\left(\vartheta -s\right)}^{\rho -1}∥T_{\phi }\underset{\vartheta \in \tau }{sup}∥\nu \left(s\right)+\xi \left(s\right)∥\left(s_i-{\vartheta }_i\right)+\underset{\vartheta \in \tau }{sup}∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0∥\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}\underset{\vartheta \in \tau }{sup}∥{\left(\vartheta -s\right)}^{\rho -1}∥\underset{\vartheta \in \tau }{sup}∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥\left(T_Z\underset{\vartheta \in \tau }{sup}∥\nu +\xi ∥+T_{{\Xi }_1}\underset{\vartheta \in \tau }{sup}∥\nu \left(s\right)∥+T_{{\Xi }_2}\underset{\vartheta \in \tau }{sup}∥\xi \left(s\right)∥\right.\hfill \\ & & \left.T_{{\Theta }_3}{T}_{Z_1}\underset{\vartheta \in \tau }{sup}∥\nu \left(p\right)+\xi \left(p\right)∥(s_l-s_0)+T_{{\Theta }_4}{T}_{Z_2}\underset{\vartheta \in \tau }{sup}∥\nu \left(p\right)+\xi \left(p\right)∥(s_l-s_0)+Y_{\Xi }^{*}\right)\Delta s,\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}& & ∥{\wp }_{\rho }(\nu ,\xi )\left(\vartheta \right)∥\hfill \\ & \le & \frac{1}{\Gamma \left(\rho \right)}{u}_3{T}_{\phi }{∥\nu +\xi ∥}_{\infty }\left(s_i-{\vartheta }_i\right)+u_1+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{u}_2{u}_3\left(T_Z{∥\nu +\xi ∥}_{\infty }+T_{{\Xi }_1}{∥\nu ∥}_{\infty }+T_{{\Xi }_2}{∥\xi ∥}_{\infty }\right.\hfill \\ & & \left.T_{{\Theta }_3}{T}_{Z_1}{∥\nu +\xi ∥}_{\infty }(s_l-s_0)+T_{{\Theta }_4}{T}_{Z_2}{∥\nu +\xi ∥}_{\infty }(s_l-s_0)+Y_{\Xi }^{*}\right)\Delta s\hfill \\ & \le & \frac{1}{\Gamma \left(\rho \right)}{u}_3{T}_{\phi }\delta \left(s_i-{\vartheta }_i\right)+u_1+u_3\left(u_2{T}_Z\delta +u_2{T}_{{\Xi }_1}\delta +u_2{T}_{{\Xi }_2}\delta \right.\hfill \\ & & +\left.u_2{T}_{{\Theta }_3}{T}_{Z_1}(s_l-s_0)+T_{{\Theta }_4}{T}_{Z_2}\delta (s_l-s_0)+Y_{\Xi }^{*}\right)\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}\Delta s\hfill \\ & \le & \frac{1}{\Gamma \left(\rho \right)}{u}_3{T}_{\phi }\delta \left(s_i-{\vartheta }_i\right)+u_1+\delta u_3\left(u_2{T}_Z+u_2{T}_{{\Xi }_1}+u_2{T}_{{\Xi }_2}\right.\hfill \\ & & +\left.u_2\left(T_{{\Theta }_3}{T}_{Z_1}+T_{{\Theta }_4}{T}_{Z_2}\right)(s_l-s_0)+Y_{\Xi }^{*}\right)\left({\vartheta }_l-{\vartheta }_0\right).\hfill \end{array}$$

Thus

$$∥{\wp }_{\rho }(\nu ,\xi )\left(\vartheta \right)∥\le E_3+\delta J_1^{*}={\delta }_1.$$

Similarly, one can write

$$∥{\wp }_{\rho }(\xi ,\nu )\left(\vartheta \right)∥\le E_6+\delta J_1^{*}={\delta }_1.$$

where ${\delta }_1=\delta +\delta J_1^{*}.$ Hence

$$∥{\wp }_{\rho }(\nu ,\xi )\left(\vartheta \right)∥\le {\delta }_2\mathrm{and}∥{\wp }_{\rho }(\xi ,\nu )\left(\vartheta \right)∥\le {\delta }_2.$$

(12)

Therefore, from (12), we get $\wp \left(\mho \times \mho \right)\subseteq \mho .$ Also, for $\vartheta \in (s_i,{\vartheta }_{i+1}]\cap \tau ,$$i=1,2,\dots ,n,$ with ${\nu }_0={\tilde{\nu }}_0$ and ${\xi }_0={\tilde{\xi }}_0,$ we get

$$\begin{array}{ccc}& & ∥{\wp }_{\rho }(\nu ,\xi )\left(\vartheta \right)-{\wp }_{\rho }(\tilde{\nu },\tilde{\xi })\left(\vartheta \right)∥\hfill \\ & \le & ∥\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{s_i}{\int }}{\left(\vartheta -s\right)}^{\rho -1}\left[{\phi }_i(s,\nu \left(s\right),\xi \left(s\right))-{\phi }_i(s,\nu \left(s\right),\xi \left(s\right))\right]\Delta s∥\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥∥\left(Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\right.\hfill \\ & & +\left.\Xi \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right)\right)\hfill \\ & & -\left(Z\left(\vartheta ,\tilde{\nu }\left(\vartheta \right),\tilde{\xi }\left(\vartheta \right)\right)+\Xi \left(s,\tilde{\nu }\left(s\right),\tilde{\xi }\left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\tilde{\nu }\left(p\right),\tilde{\xi }\left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\tilde{\nu }\left(p\right),\tilde{\xi }\left(p\right)\right)\Delta p\right)\right)∥\Delta s\hfill \end{array}$$

$$\begin{array}{ccc}& \le & \frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{s_i}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\phi }_i(s,\nu \left(s\right),\xi \left(s\right))-{\phi }_i(s,\tilde{\nu }\left(s\right),\tilde{\xi }\left(s\right))∥\Delta s\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥\left[T_Z\left(∥\nu \left(\vartheta \right)-\tilde{\nu }\left(\vartheta \right)∥+∥\xi \left(\vartheta \right)-\tilde{\xi }\left(\vartheta \right)∥\right)\right.\hfill \\ & & +T_{{\Xi }_1}∥\nu \left(s\right)-\tilde{\nu }\left(s\right)∥+T_{{\Xi }_2}∥\xi \left(s\right)-\tilde{\xi }\left(s\right)∥+T_{{\Xi }_3}\underset{s_0}{\overset{s_l}{\int }}∥Z_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)-Z_1\left(s,p,\tilde{\nu }\left(p\right),\tilde{\xi }\left(p\right)\right)∥\Delta p\hfill \\ & & \left.+T_{{\Xi }_4}\underset{s_0}{\overset{s_l}{\int }}∥Z_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)-Z_2\left(s,p,\tilde{\nu }\left(p\right),\tilde{\xi }\left(p\right)\right)∥\Delta p\right]\Delta s,\hfill \end{array}$$

hence

$$\begin{array}{ccc}& & ∥{\wp }_{\rho }(\nu ,\xi )\left(\vartheta \right)-{\wp }_{\rho }(\tilde{\nu },\tilde{\xi })\left(\vartheta \right)∥\hfill \\ & \le & \frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{s_i}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥L_{\phi }\left[∥\nu \left(s\right)-\tilde{\nu }\left(s\right)∥+∥\xi \left(s\right)-\tilde{\xi }\left(s\right)∥\right]\Delta s\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥\left[T_Z\left(∥\nu \left(\vartheta \right)-\tilde{\nu }\left(\vartheta \right)∥+∥\xi \left(\vartheta \right)-\tilde{\xi }\left(\vartheta \right)∥\right)+T_{{\Xi }_1}∥\nu \left(s\right)-\tilde{\nu }\left(s\right)∥\right.\hfill \\ & & +T_{{\Xi }_2}∥\xi \left(s\right)-\tilde{\xi }\left(s\right)∥+T_{{\Xi }_3}{T}_{Z_1}\underset{s_0}{\overset{s_l}{\int }}\left(∥\nu \left(p\right)-\tilde{\nu }\left(p\right)∥+∥\xi \left(p\right)-\tilde{\xi }\left(p\right)∥\right)\Delta p\hfill \\ & & \left.+T_{{\Xi }_3}{T}_{Z_2}\underset{s_0}{\overset{s_l}{\int }}\left(∥\nu \left(p\right)-\tilde{\nu }\left(p\right)∥+∥\xi \left(p\right)-\tilde{\xi }\left(p\right)∥\right)\Delta p\right]\Delta s\hfill \\ & \le & \frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{s_i}{\int }}\underset{\vartheta \in \tau }{sup}∥{\left(\vartheta -s\right)}^{\rho -1}∥L_{\phi }\underset{\vartheta \in \tau }{sup}\left[∥\nu \left(s\right)-\tilde{\nu }\left(s\right)∥+∥\xi \left(s\right)-\tilde{\xi }\left(s\right)∥\right]\Delta s\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}\underset{\vartheta \in \tau }{sup}∥{\left(\vartheta -s\right)}^{\rho -1}∥\underset{\vartheta \in \tau }{sup}∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥\left[\begin{array}{c}{T}_Z\left(\underset{\vartheta \in \tau }{sup}∥\nu \left(\vartheta \right)-\tilde{\nu }\left(\vartheta \right)∥+\underset{\vartheta \in \tau }{sup}∥\xi \left(\vartheta \right)-\tilde{\xi }\left(\vartheta \right)∥\right)\\ +T_{{\Xi }_1}\underset{\vartheta \in \tau }{sup}∥\nu \left(s\right)-\tilde{\nu }\left(s\right)∥\end{array}\right.\hfill \\ & & +T_{{\Xi }_2}\underset{\vartheta \in \tau }{sup}∥\xi \left(s\right)-\tilde{\xi }\left(s\right)∥+T_{{\Xi }_3}{T}_{Z_1}\left(\underset{\vartheta \in \tau }{sup}∥\nu \left(p\right)-\tilde{\nu }\left(p\right)∥+\underset{\vartheta \in \tau }{sup}∥\xi \left(p\right)-\tilde{\xi }\left(p\right)∥\right)(s_l-s_0)\hfill \\ & & \left.+T_{{\Xi }_3}{T}_{Z_2}\left(\underset{\vartheta \in \tau }{sup}∥\nu \left(p\right)-\tilde{\nu }\left(p\right)∥+\underset{\vartheta \in \tau }{sup}∥\xi \left(p\right)-\tilde{\xi }\left(p\right)∥\right)(s_l-s_0)\right]\Delta s,\hfill \end{array}$$

it follows that

$$\begin{array}{ccc}& & ∥{\wp }_{\rho }(\nu ,\xi )\left(\vartheta \right)-{\wp }_{\rho }(\tilde{\nu },\tilde{\xi })\left(\vartheta \right)∥\hfill \\ & \le & \frac{1}{\Gamma \left(\rho \right)}{u}_3{L}_{\phi }\left[{∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right]\left(s_i-{\vartheta }_i\right)+u_3{u}_2\left[\begin{array}{c}{T}_Z\left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right)\\ +T_{{\Xi }_1}{∥\nu -\tilde{\nu }∥}_{\infty }\end{array}\right.\hfill \\ & & +T_{{\Xi }_2}{∥\xi -\tilde{\xi }∥}_{\infty }+T_{{\Xi }_3}{T}_{Z_1}\left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right)(s_l-s_0)\hfill \\ & & +\left.T_{{\Xi }_3}{T}_{Z_2}\left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right)(s_l-s_0)\right]\times \underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}\Delta s,\hfill \end{array}$$

$$\begin{array}{ccc}& \le & \frac{1}{\Gamma \left(\rho \right)}{u}_3{L}_{\phi }\left(s_i-{\vartheta }_i\right)+u_3\left[\begin{array}{c}{u}_2{T}_Z+u_2{T}_{{\Theta }_1}+u_2{T}_{{\Theta }_2}\\ +u_2{T}_{{\Theta }_3}{T}_{Z_1}(s_l-s_0)+u_2{T}_{{\Theta }_3}{T}_{Z_2}(s_l-s_0)\end{array}\right]({\vartheta }_l-{\vartheta }_0)\hfill \\ & & \times \left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right)\hfill \\ & =& \left(E_4+J_2^{*}\right)\left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right)=J_3^{*}\left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right).\hfill \end{array}$$

(13)

Analogously, we can write

$$∥{\Im }_{\rho }(\nu ,\xi )\left(\vartheta \right)-{\Im }_{\rho }(\tilde{\nu },\tilde{\xi })\left(\vartheta \right)∥\le J_3^{*}\left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right).$$

(14)

It follows from (11), (13) and (14) that the impulsive problem (2) has a US based on the Banach principle. □

The existence of at least one solution is next examined for both the MISs (1) and (2) using the Leray-Schauder alternative fixed point (FP) method and the weaker stipulation $\left(2\right)$.

Theorem 5.

Assume that the assertion $\left(2\right)$ holds. If there is $\gimel >0$ so that

$$u_1+u_4+2J_3\gimel <\gimel ,$$

(15)

then the MIS (1) has at least on solution.

Proof. 

In the beginning, we demonstrate that the operator ${\Im }_{\rho }$ defined by (7) is completely continuous (CC). We can see that it is a continuous operator due to the continuity of the mappings O, D, Z, and $\Theta .$ Moreover, suppose that ${\mho }_1\subseteq Q$ and by using the fact O, D, Z, and $\Theta$ are bounded operators. Then there are positive constants $T_1,$$T_2,$$Y_1$ and $Y_2$ so that

$$\begin{array}{ccc}\hfill \sum _{j=1}^i{O}_j\left(\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right),\sum _{j=1}^i{O}_j\left(\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)& \le & T_1,\sum _{j=1}^i{D}_j\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right),\sum _{j=1}^i{D}_j\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)\le T_2,\hfill \\ \hfill Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right),Z\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)& \le & Y_1\mathrm{and}∥\Theta \left(\vartheta ,K_1,K_2,K_3,K_4\right)∥\le Y_2,\hfill \end{array}$$

where $K_1=\nu \left(s\right),$$K_2=\xi \left(s\right),$$K_3=\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p$ and $K_4=\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p.$ Consider $T_1+T_2+u_1=\daleth ,$$T_1+T_2+u_4={\daleth }^{*},$$Y_1+Y_2=Y,$$∥{(\vartheta -s)}^{\rho -1}∥\le {\daleth }_1,$$\daleth +{\daleth }_1{u}_{2}Y({\vartheta }_l-{\vartheta }_0)\le \aleph ,$ and ${\daleth }^{*}+{\daleth }_1{u}_{2}Y({\vartheta }_l-{\vartheta }_0)\le \aleph .$

For any $\nu ,\xi \in {\mho }_1$ and $\vartheta \in ({\vartheta }_i,{\vartheta }_{i+1}],$$i=1,2,\dots ,n,$ we get

$$\begin{array}{ccc}\hfill ∥{\Im }_{\rho }(\nu ,\xi )\left(\vartheta \right)∥& \le & \sum _{j=1}^i∥O_j\left(\nu \left({\vartheta }_j\right),\xi \left({\vartheta }_j\right)\right)∥+\sum _{j=1}^i∥D_j\left({\vartheta }_j,\nu \left({\vartheta }_j\right),\xi \left({\vartheta }_j\right)\right)∥+∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0∥\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥∥\left(Z\left(s,\nu \left(s\right),\xi \left(s\right)\right)\right.\hfill \\ & & \left.\Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right)\right)∥\Delta s\hfill \\ & \le & T_1+T_2+u_1+{\daleth }_1{u}_2\left(Y_1+Y_2\right)\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}\Delta s\hfill \\ & =& \daleth +{\daleth }_1{u}_{2}Y\left({\vartheta }_l-{\vartheta }_0\right).\hfill \end{array}$$

(16)

Similarly

$$∥{\Im }_{\rho }(\xi ,\nu )\left(\vartheta \right)∥\le T_1+T_2+u_4+{\daleth }_1{u}_2\left(Y_1+Y_2\right)\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}\Delta s={\daleth }^{*}+{\daleth }_1{u}_{2}Y\left({\vartheta }_l-{\vartheta }_0\right).$$

(17)

From (16) and (17), we have

$$∥{\Im }_{\rho }(\nu ,\xi )\left(\vartheta \right)∥\le \aleph \mathrm{and}∥{\Im }_{\rho }(\xi ,\nu )\left(\vartheta \right)∥\le \aleph .$$

Therefore, ℑ is uniformly bounded.

Now, we establish that ℑ is CC. For this regard, we discuss the following options:

Option 1.

Consider that time scales $\tau$ are made up of distinct points, with each point on $\tau$ being isolated. Based on Theorem 1, ${\Im }_{\rho }$ takes the form

$${\Im }_{\rho }(\nu ,\xi )\left(\vartheta \right)=\left\{\begin{array}{c}{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\sum _{\vartheta \in \tau }{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\\ +\sum _{\vartheta \in \tau }{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right),\vartheta \in ({\vartheta }_0,{\vartheta }_1],\\ {\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\sum _{\vartheta \in \tau }{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\\ +\sum _{\vartheta \in \tau }{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right),\\ +\sum _{t=1}^i{O}_t\left(\nu \left({\vartheta }_t^{-}\right),\xi \left({\vartheta }_t^{-}\right)\right)+D_t\left({\vartheta }_t^{-},\nu \left({\vartheta }_t^{-}\right),\xi \left({\vartheta }_t^{-}\right)\right),\vartheta \in ({\vartheta }_i,{\vartheta }_{i+1}],i=1,2,\dots ,n.\end{array}\right.$$

(18)

It is obvious that (18) is a collection of summing operators on a discrete finite set. Furthermore, it is implied from the continuity of oj, $O_t,$$D_t,$Z and $\Theta$ that ${\Lambda }_{\rho }$ is CC.

Option 2.

Suppose that $\tau$ is continuous and ${\vartheta }_{l_1},{\vartheta }_{l_2}\in ({\vartheta }_i,{\vartheta }_{i+1}],$$i=1,2,\dots ,n$ so that ${\vartheta }_{l_1}<{\vartheta }_{l_2},$ then

$$\begin{array}{ccc}& & ∥{\Im }_{\rho }(\nu ,\xi )\left({\vartheta }_{l_2}\right)-{\Im }_{\rho }(\nu ,\xi )\left({\vartheta }_{l_1}\right)∥\hfill \\ & \le & ∥\sum _{j=1}^i{O}_j\left(\nu \left({\vartheta }_{l_{2j}}^{-}\right),\xi \left({\vartheta }_{l_{2j}}^{-}\right)\right)-O_j\left(\nu \left({\vartheta }_{l_{1j}}^{-}\right),\xi \left({\vartheta }_{l_{1j}}^{-}\right)\right)\hfill \\ & & +\sum _{j=1}^i{D}_j\left({\vartheta }_{l_{2j}}^{-},\nu \left({\vartheta }_{l_{2j}}^{-}\right),\xi \left({\vartheta }_{l_{2j}}^{-}\right)\right)-D_j\left({\vartheta }_{l_{1j}}^{-},\nu \left({\vartheta }_{l_{1j}}^{-}\right),\xi \left({\vartheta }_{l_{1j}}^{-}\right)\right)\hfill \\ & & +\left[\left(\underset{{\vartheta }_0}{\overset{{\vartheta }_{l_2}}{\int }}{\left({\vartheta }_{l_2}-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left({\vartheta }_{l_2}-s\right)}^{\rho }\right)\right)\left(Z\left(s,\nu \left(s\right),\xi \left(s\right)\right)\right.\right.\right.\hfill \\ & & +\left.\left.\Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right)\right)\Delta s\right)\hfill \\ & & -\left(\underset{{\vartheta }_0}{\overset{{\vartheta }_{l_1}}{\int }}{\left({\vartheta }_{l_1}-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left({\vartheta }_{l_1}-s\right)}^{\rho }\right)\right)\left(Z\left(s,\nu \left(s\right),\xi \left(s\right)\right)\right.\right.\hfill \\ & & +\left.\left.\Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right)\right)\Delta s\right]∥,\hfill \end{array}$$

it follows that

$$\begin{array}{ccc}& & ∥{\Im }_{\rho }(\nu ,\xi )\left({\vartheta }_{l_2}\right)-{\Im }_{\rho }(\nu ,\xi )\left({\vartheta }_{l_1}\right)∥\hfill \\ & \le & \sum _{j=1}^i∥O_j\left(\nu \left({\vartheta }_{l_{2j}}^{-}\right),\xi \left({\vartheta }_{l_{2j}}^{-}\right)\right)-O_j\left(\nu \left({\vartheta }_{l_{1j}}^{-}\right),\xi \left({\vartheta }_{l_{1j}}^{-}\right)\right)∥\hfill \\ & & +\sum _{j=1}^i∥D_j\left({\vartheta }_{l_{2j}}^{-},\nu \left({\vartheta }_{l_{2j}}^{-}\right),\xi \left({\vartheta }_{l_{2j}}^{-}\right)\right)-D_j\left({\vartheta }_{l_{1j}}^{-},\nu \left({\vartheta }_{l_{1j}}^{-}\right),\xi \left({\vartheta }_{l_{1j}}^{-}\right)\right)∥\hfill \\ & & +∥\underset{{\vartheta }_0}{\overset{{\vartheta }_{l_2}}{\int }}\left[{\left({\vartheta }_{l_1}-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left({\vartheta }_{l_2}-s\right)}^{\rho }\right)\right)-{\left({\vartheta }_{l_1}-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left({\vartheta }_{l_1}-s\right)}^{\rho }\right)\right)\right]\times Z\left(s,\nu \left(s\right),\xi \left(s\right)\right)∥\hfill \\ & & +∥\underset{{\vartheta }_0}{\overset{{\vartheta }_{l_2}}{\int }}\left[{\left({\vartheta }_{l_1}-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left({\vartheta }_{l_2}-s\right)}^{\rho }\right)\right)-{\left({\vartheta }_{l_1}-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left({\vartheta }_{l_1}-s\right)}^{\rho }\right)\right)\right]\hfill \end{array}$$

$$\begin{array}{ccc}& & \times \Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right)\Delta s∥\hfill \\ & & +∥\underset{{\vartheta }_{l_1}}{\overset{{\vartheta }_{l_2}}{\int }}\left[{\left({\vartheta }_{l_2}-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left({\vartheta }_{l_2}-s\right)}^{\rho }\right)\right)\right]Z\left(s,\nu \left(s\right),\xi \left(s\right)\right)∥\hfill \\ & & +∥\underset{{\vartheta }_{l_1}}{\overset{{\vartheta }_{l_2}}{\int }}\left[{\left({\vartheta }_{l_2}-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left({\vartheta }_{l_2}-s\right)}^{\rho }\right)\right)\right]Z\left(s,\nu \left(s\right),\xi \left(s\right)\right)\hfill \\ & & \times \Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right)\Delta s∥.\hfill \end{array}$$

It is easy to see that the above inequality tends to 0 as ${\vartheta }_{l_1}\to {\vartheta }_{l_2}$. The operator ${\Im }_{\rho }$ is hence equicontinuous. Finally, we come to the conclusion that ${\Im }_{\rho }$ is CC using the Arzela-Ascoli theorem.

Option 3.

Suppose that $\tau$ contains isolated points along with dense ones, that is $\tau$ is continuous and discrete. In light of Theorem 1, we can represent ${\Im }_{\rho }$ as the summation operator, which is CC (analyzed in option 1). Also, one may establish that ${\Im }_{\rho }$ is a CC operator for the dense points (analyzed in option 2). Therefore, for isolated and dense points, ${\Im }_{\rho }$ can be expressed as the sum of two operators. Since the two operators are CC, we can infer that their sum is also CC. As a result, the operator ${\Im }_{\rho }$ is CC. Thus, by adding together the three aforementioned scenarios, we determine that ${\Im }_{\rho }$ is a CC operator.

Ultimately, let $\alpha \in [0,1]$ and there are $\nu$ and $\xi$ so that $\nu \left(\vartheta \right)={\Im }_{\rho }(\nu ,\xi )\left(\vartheta \right)$ and $\xi \left(\vartheta \right)={\Im }_{\rho }(\xi ,\nu )\left(\vartheta \right).$ Hence for $\vartheta \in ({\vartheta }_i,{\vartheta }_{i+1}],$$i=1,2,\dots ,n,$ one can write

$$\begin{array}{ccc}& & ∥\xi \left(\vartheta \right)∥\hfill \\ & =& ∥\alpha \left({\Im }_{\rho }(\xi ,\nu )\left({\vartheta }_l\right)\right)∥\hfill \\ & & ∥\alpha \left[\sum _{j=1}^i{O}_j\left(\nu \left({\vartheta }_{l_j}^{-}\right),\xi \left({\vartheta }_{l_j}^{-}\right)\right)+\sum _{j=1}^i{D}_j\left({\vartheta }_{l_j}^{-},\nu \left({\vartheta }_{l_j}^{-}\right),\xi \left({\vartheta }_{l_j}^{-}\right)\right)+{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0\right.\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left({\vartheta }_l-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left({\vartheta }_l-s\right)}^{\rho }\right)\right)\left(Z\left(s,\nu \left(s\right),\xi \left(s\right)\right)\right.\hfill \\ & & +\left.\left.\Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right)\right)\Delta s\right]∥\hfill \end{array}$$

$$\begin{array}{ccc}\hfill & \le & \sum _{j=1}^i{L}_O{∥\nu +\xi ∥}_{\infty }+\sum _{j=1}^i{L}_D{∥\nu +\xi ∥}_{\infty }+u_1+∥{\left({\vartheta }_l-s\right)}^{\rho -1}∥u_2\left(L_Z{∥\nu +\xi ∥}_{\infty }+L_{{\Theta }_1}{∥\nu ∥}_{\infty }\right.\hfill \\ & & \left.+L_{{\Theta }_2}{∥\xi ∥}_{\infty }+L_{{\Theta }_3}{L}_{Z_1}{∥\nu +\xi ∥}_{\infty }(s_l-s_0)+L_{{\Theta }_4}{L}_{Z_2}{∥\nu +\xi ∥}_{\infty }(s_l-s_0)\right)({\vartheta }_l-{\vartheta }_0)\hfill \\ & =& u_1+\left[\begin{array}{c}\sum _{j=1}^i{L}_O+\sum _{j=1}^i{L}_D\\ +u_3\left(u_2{L}_Z+u_2{L}_{{\Theta }_1}+u_2{L}_{{\Theta }_2}+u_2\left(L_{{\Theta }_3}{L}_{Z_1}+L_{{\Theta }_4}{L}_{Z_2}\right)(s_l-s_0)\right)({\vartheta }_l-{\vartheta }_0)\end{array}\right]\hfill \\ & & \times {∥\nu +\xi ∥}_{\infty },\hfill \end{array}$$

which yields that

$${∥\xi ∥}_{\infty }\le u_1+\left[E_2+J_2\right]{∥\nu +\xi ∥}_{\infty }=u_1+J_3{∥\nu +\xi ∥}_{\infty }.$$

(19)

Similarly, we have

$${∥\nu ∥}_{\infty }\le u_4+\left[E_2+J_2\right]{∥\nu +\xi ∥}_{\infty }=u_4+J_3{∥\nu +\xi ∥}_{\infty }.$$

(20)

Combining (19) and (20), we obtain that

$${∥\xi +\nu ∥}_{\infty }\le {∥\xi ∥}_{\infty }+{∥\nu ∥}_{\infty }=u_1+u_4+2J_3{∥\nu +\xi ∥}_{\infty }.$$

Hence

$$\frac{{∥\xi +\nu ∥}_{\infty }}{u_1+u_4+2J_3{∥\nu +\xi ∥}_{\infty }}\le 1.$$

From (15), we have $\gimel >0$ so that ${∥\xi +\nu ∥}_{\infty }\ne \gimel .$ Let us consider the set

$$\Phi =\{\xi ,\nu \in \tau ,{∥\xi +\nu ∥}_{\infty }<\gimel \}.$$

Consequently, the operator ${\Im }_{\rho }:\Phi \times \Phi \to \tau$ is both continuously and CC. Therefore from the choice of $\Phi$ there are no $\xi ,\nu \in {\mho }_1(\Phi )$ provided that $\xi =\alpha \left({\Im }_{\rho }(\xi ,\nu )\left(\vartheta \right)\right)$ and $\nu =\alpha \left({\Im }_{\rho }(\nu ,\xi )\left(\vartheta \right)\right),$$\alpha \in [0,1].$ Due to the nonlinear Leray-Schauder alternative, ℑ has a coupled FP, which is a solution to the MIS (1). □

Theorem 6.

Assume that the assertion $\left(2\right)$ is true. If there is ${\gimel }^{*}>0$ so that

$$u_1+u_4+2J_3^{*}{\gimel }^{*}<{\gimel }^{*},$$

then the MIS (2) has at least on solution.

Proof. 

It is comparable to the earlier defense of ${\wp }_{\rho }$ in Theorem 5. □

4. Stability Study

We begin this part with the inequalities below:

$$\left\{\begin{array}{c}∥{}^{c,\tau }{D}^{\rho }\nu \left(\vartheta \right)-B\left(\vartheta \right)\nu \left(\vartheta \right)+Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\\ -\Theta \left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s\right)∥\le \epsilon ,\vartheta \in {\tau }^{\prime }\\ ∥{}^{c,\tau }{D}^{\rho }\xi \left(\vartheta \right)-B\left(\vartheta \right)\xi \left(\vartheta \right)-Z\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)\\ -\Theta \left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s\right)∥\le \epsilon ,\vartheta \in {\tau }^{\prime }\\ ∥\nu \left({\vartheta }_r^{+}\right)-\nu \left({\vartheta }_r^{-}\right)-O_r\left(\nu \left({\vartheta }_r^{-}\right),\xi \left({\vartheta }_r^{-}\right)\right)-D_r\left({\vartheta }_r^{-},\nu \left({\vartheta }_r^{-}\right),\xi \left({\vartheta }_r^{-}\right)\right)∥\le \epsilon ,r=1,2,\dots ,n,\\ ∥\xi \left({\vartheta }_r^{+}\right)-\xi \left({\vartheta }_r^{-}\right)-O_r\left(\xi \left({\vartheta }_r^{-}\right),\nu \left({\vartheta }_r^{-}\right)\right)-D_r\left({\vartheta }_r^{-},\xi \left({\vartheta }_r^{-}\right),\nu \left({\vartheta }_r^{-}\right)\right)∥\le \epsilon ,r=1,2,\dots ,n,\end{array}\right.$$

(21)

and

$$\left\{\begin{array}{c}∥{}^{c,\tau }{D}^{\rho }\nu \left(\vartheta \right)-B\left(\vartheta \right)\nu \left(\vartheta \right)-Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\\ -\Xi \left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(s,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s\right)∥\le \epsilon ,\\ ∥{}^{c,\tau }{D}^{\rho }\xi \left(\vartheta \right)-B\left(\vartheta \right)\xi \left(\vartheta \right)-Z\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)\\ -\Xi \left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s\right)∥\le \epsilon ,\\ \vartheta \in (s_i,{\vartheta }_{i+1}]\cap \tau ,i=1,2,\dots n,\\ ∥\nu \left(\vartheta \right)-\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\xi \left(s\right),\nu \left(s\right))\Delta s∥\le \epsilon ,\\ ∥\xi \left(\vartheta \right)-\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\nu \left(s\right),\xi \left(s\right))\Delta s∥\le \epsilon ,\vartheta \in ({\vartheta }_i,s_i]\cap \tau ,i=1,2,\dots ,n.\end{array}\right.$$

(22)

for $\epsilon >0.$

Definition 4.

We say that the MIS (1) is UH stable on $\tau ,$ if for every $\xi ,\nu \in {\mho }_1(\tau ,{\mathbb{R}}^n)$ satisfying (21), there are $\widehat{\xi },\widehat{\nu }\in Q^1(\tau ,{\mathbb{R}}^n)$ as a solution of (1) so that

$$∥\xi \left(s\right)-\widehat{\xi }\left(s\right)∥+∥\nu \left(s\right)-\widehat{\nu }\left(s\right)∥\le F\epsilon ,\mathrm{for}F>0,s\in \tau .$$

Definition 5.

We say that the MIS (2) is UH stable on $\tau ,$ if for every $\xi ,\nu \in Q^1(\tau ,{\mathbb{R}}^n)$ fulfilling (22), there are $\widehat{\xi },\widehat{\nu }\in Q^1(\tau ,{\mathbb{R}}^n)$ as a solution of (2) so that

$$∥\xi \left(\vartheta \right)-\widehat{\xi }\left(\vartheta \right)∥+∥\nu \left(\vartheta \right)-\widehat{\nu }\left(\vartheta \right)∥\le F^{*}\epsilon ,\mathrm{for}{F}^{*}>0,s\in \tau .$$

Remark 2.

The solutions $\xi ,\nu \in Q^1(\tau ,{\mathbb{R}}^n)$ fulfill (21), if there is $l\in Q(\tau ,{\mathbb{R}}^n)$ with the sequence $l_r$ so that $∥l_r∥\le \epsilon$ and

$$\left\{\begin{array}{c}{}^{c,\tau }{D}^{\rho }\nu \left(\vartheta \right)=B\left(\vartheta \right)\nu \left(\vartheta \right)+Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\\ +\Theta \left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s\right)+l\left(\vartheta \right),\\ {}^{c,\tau }{D}^{\rho }\xi \left(\vartheta \right)=B\left(\vartheta \right)\xi \left(\vartheta \right)+Z\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)\\ +\Theta \left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s\right)+l\left(\vartheta \right),\\ \vartheta \in {\tau }^{\prime },\nu \left({\vartheta }_0\right)={\nu }_0,\xi \left({\vartheta }_0\right)={\xi }_0,\\ \nu \left({\vartheta }_r^{+}\right)-\nu \left({\vartheta }_r^{-}\right)=O_r\left(\nu \left({\vartheta }_r^{-}\right),\xi \left({\vartheta }_r^{-}\right)\right)+D_r\left({\vartheta }_r^{-},\nu \left({\vartheta }_r^{-}\right),\xi \left({\vartheta }_r^{-}\right)\right)+l_r,r=1,2,\dots ,n,\\ \xi \left({\vartheta }_r^{+}\right)-\xi \left({\vartheta }_r^{-}\right)=O_r\left(\xi \left({\vartheta }_r^{-}\right),\nu \left({\vartheta }_r^{-}\right)\right)+D_r\left({\vartheta }_r^{-},\xi \left({\vartheta }_r^{-}\right),\nu \left({\vartheta }_r^{-}\right)\right)+l_r,r=1,2,\dots ,n,\end{array}\right.$$

Lemma 1.

Every functions $\xi ,\nu \in Q^1(\tau ,{\mathbb{R}}^n)$ that fulfill (21) also satisfy the inequalities shown below:

$$\left\{\begin{array}{c}∥\nu \left(\vartheta \right)-{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0-{\displaystyle \sum _{r=1}^n}\left(O_r\left(\nu \left({\vartheta }_r^{-}\right),\xi \left({\vartheta }_r^{-}\right)\right)+D_r\left({\vartheta }_r^{-},\nu \left({\vartheta }_r^{-}\right),\xi \left({\vartheta }_r^{-}\right)\right)\right)\\ -\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s-\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\\ \times \Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\right)\Delta s∥\le \delta \epsilon ,\\ ∥\xi \left(\vartheta \right)-{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\xi }_0-\sum _{r=1}^n\left(O_r\left(\xi \left({\vartheta }_r^{-}\right),\nu \left({\vartheta }_r^{-}\right)\right)+D_r\left({\vartheta }_r^{-},\xi \left({\vartheta }_r^{-}\right),\nu \left({\vartheta }_r^{-}\right)\right)\right)\\ -\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s-\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\\ \times \Theta \left(s,\xi \left(s\right),\nu \left(s\right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\right)\Delta s∥\le \delta \epsilon ,\end{array}\right.$$

for $\vartheta \in ({\vartheta }_r,{\vartheta }_{r+1}]\subset \tau ,$ where $∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥\le \frac{1}{2}{u}_2$ and $\delta =(n+\frac{1}{2}{u}_3{u}_2({\vartheta }_l-{\vartheta }_0)).$

Proof. 

When $\xi ,\nu \in Q^1(\tau ,{\mathbb{R}}^n)$ justify (21), then by Remark 2, we have

$$\left\{\begin{array}{c}{}^{c,\tau }{D}^{\rho }\nu \left(\vartheta \right)=B\left(\vartheta \right)\nu \left(\vartheta \right)+Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\\ +\Theta \left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s\right)+l\left(\vartheta \right),\\ {}^{c,\tau }{D}^{\rho }\xi \left(\vartheta \right)=B\left(\vartheta \right)\xi \left(\vartheta \right)+Z\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)\\ +\Theta \left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s\right)+l\left(\vartheta \right),\\ \vartheta \in {\tau }^{\prime },\nu \left({\vartheta }_0\right)={\nu }_0,\xi \left({\vartheta }_0\right)={\xi }_0,\\ \nu \left({\vartheta }_r^{+}\right)-\nu \left({\vartheta }_r^{-}\right)=O_r\left(\nu \left({\vartheta }_r^{-}\right),\xi \left({\vartheta }_r^{-}\right)\right)+D_r\left({\vartheta }_r^{-},\nu \left({\vartheta }_r^{-}\right),\xi \left({\vartheta }_r^{-}\right)\right)+l_r,r=1,2,\dots ,n,\\ \xi \left({\vartheta }_r^{+}\right)-\xi \left({\vartheta }_r^{-}\right)=O_r\left(\xi \left({\vartheta }_r^{-}\right),\nu \left({\vartheta }_r^{-}\right)\right)+D_r\left({\vartheta }_r^{-},\xi \left({\vartheta }_r^{-}\right),\nu \left({\vartheta }_r^{-}\right)\right)+l_r,r=1,2,\dots ,n,\end{array}\right.$$

yields

$$\begin{array}{ccc}\hfill \nu \left(\vartheta \right)& =& {\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\sum _{r=1}^n\left(O_r\left(\nu \left({\vartheta }_r^{-}\right),\xi \left({\vartheta }_r^{-}\right)\right)+D_r\left({\vartheta }_r^{-},\nu \left({\vartheta }_r^{-}\right),\xi \left({\vartheta }_r^{-}\right)\right)\right)+\sum _{t=1}^n{l}_t\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\hfill \\ & & \times \Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\right)\Delta s\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)f\left(s\right)\Delta s\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \xi \left(\vartheta \right)& =& {\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\xi }_0+\sum _{r=1}^n\left(O_r\left(\xi \left({\vartheta }_r^{-}\right),\nu \left({\vartheta }_r^{-}\right)\right)+D_r\left({\vartheta }_r^{-},\xi \left({\vartheta }_r^{-}\right),\nu \left({\vartheta }_r^{-}\right)\right)\right)+\sum _{t=1}^n{l}_t\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\hfill \\ & & \times \Theta \left(s,\xi \left(s\right),\nu \left(s\right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\right)\Delta s\hfill \\ & & \underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)f\left(s\right)\Delta s.\hfill \end{array}$$

So

$$\begin{array}{ccc}& & ∥\nu \left(\vartheta \right)-{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0-\sum _{r=1}^n\left(O_r\left(\nu \left({\vartheta }_r^{-}\right),\xi \left({\vartheta }_r^{-}\right)\right)+D_r\left({\vartheta }_r^{-},\nu \left({\vartheta }_r^{-}\right),\xi \left({\vartheta }_r^{-}\right)\right)\right)\hfill \\ & & -\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s-\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\hfill \\ & & \times \Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\right)\Delta s∥\hfill \\ & \le & \underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥∥f\left(s\right)∥\Delta s+\sum _{t=1}^n∥l_t∥\le \delta \epsilon .\hfill \end{array}$$

Similarly, we can show that

$$\begin{array}{ccc}& & ∥\xi \left(\vartheta \right)-{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\xi }_0-\sum _{r=1}^n\left(O_r\left(\xi \left({\vartheta }_r^{-}\right),\nu \left({\vartheta }_r^{-}\right)\right)+D_r\left({\vartheta }_r^{-},\xi \left({\vartheta }_r^{-}\right),\nu \left({\vartheta }_r^{-}\right)\right)\right)\hfill \\ & & -\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s-\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\hfill \\ & & \times \Theta \left(s,\xi \left(s\right),\nu \left(s\right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\right)\Delta s∥\hfill \\ & \le & \underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥∥f\left(s\right)∥\Delta s+\sum _{t=1}^n{l}_t\le \delta \epsilon .\hfill \end{array}$$

Thus, we get the desired result. □

Remark 3.

The mappings $\xi ,\nu \in Q^1(\tau ,{\mathbb{R}}^n)$ verify (22), if there is a map $l\in Q(\tau ,{\mathbb{R}}^n)$ and bounded sequences $\{l_r,$$r=1,2,\dots ,n\}$ provided that $∥l_r\left(\vartheta \right)∥\le \epsilon ,$$\vartheta \in \tau ,$$r=1,2,\dots ,n$ so that

$$\left\{\begin{array}{c}{}^{c,\tau }{D}^{\rho }\nu \left(\vartheta \right)=B\left(\vartheta \right)\nu \left(\vartheta \right)+Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\\ +\Xi \left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s\right)+l\left(\vartheta \right),\\ {}^{c,\tau }{D}^{\rho }\xi \left(\vartheta \right)=B\left(\vartheta \right)\xi \left(\vartheta \right)+Z\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)\\ +\Xi \left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s\right)+l\left(\vartheta \right),\\ \vartheta \in (s_i,{\vartheta }_{i+1}]\cap \tau ,i=1,2,\dots n,\\ \nu \left(\vartheta \right)=\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\xi \left(s\right),\nu \left(s\right))\Delta s+l_i,\\ \xi \left(\vartheta \right)=\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\nu \left(s\right),\xi \left(s\right))\Delta s+l_i,\vartheta \in ({\vartheta }_i,s_i]\cap \tau ,i=1,2,\dots ,n.\end{array}\right.$$

Lemma 2.

Every mappings $\xi ,\nu \in Q^1(\tau ,{\mathbb{R}}^n)$ that fulfill (22) also fulfill the inequalities shown below:

$$\left\{\begin{array}{c}∥\nu \left(\vartheta \right)-{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0-\underset{s_i}{\overset{\vartheta }{\int }}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(s,\nu \left(s\right),\xi \left(s\right)\right)\\ -\underset{s_i}{\overset{\vartheta }{\int }}{\Lambda }_{\rho ,\rho }\left(B{\left(\vartheta -s\right)}^{\rho }\right)\Xi \left(s,\nu \left(s\right),\xi \left(s\right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\right)\Delta s\\ -\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\xi \left(s\right),\nu \left(s\right))\Delta s∥\le \left(u_3{u}_2\left(\vartheta -{\vartheta }_i\right)+n\right)\epsilon ,\\ ∥\xi \left(\vartheta \right)-{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\xi }_0-\underset{s_i}{\overset{\vartheta }{\int }}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(s,\xi \left(s\right),\nu \left(s\right)\right)\\ -\underset{s_i}{\overset{\vartheta }{\int }}{\Lambda }_{\rho ,\rho }\left(B{\left(\vartheta -s\right)}^{\rho }\right)\Xi \left(s,\xi \left(s\right),\nu \left(s\right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\right)\Delta s\\ -\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\xi \left(s\right),\nu \left(s\right))\Delta s∥\le \left(u_3{u}_2\left(\vartheta -{\vartheta }_i\right)+n\right)\epsilon ,\\ \vartheta \in (s_i,{\vartheta }_{i+1}]\cap \tau ,i=1,2,\dots n,\end{array}\right.$$

and

$$\left\{\begin{array}{c}∥\xi \left(s\right)-\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\xi \left(s\right),\nu \left(s\right))\Delta s∥\le n\epsilon ,\\ ∥\nu \left(s\right)-\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\nu \left(s\right),\xi \left(s\right))\Delta s∥\le n\epsilon ,\end{array}\right.\vartheta \in ({\vartheta }_i,s_i]\cap \tau ,i=1,2,\dots ,n,$$

in which $∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥\le \frac{1}{2}{u}_2.$

Proof. 

If $\xi ,\nu \in Q^1(\tau ,{\mathbb{R}}^n)$ justify (21), by virtue of Remark 3, we get

$$\left\{\begin{array}{c}{}^{c,\tau }{D}^{\rho }\nu \left(\vartheta \right)=B\left(\vartheta \right)\nu \left(\vartheta \right)+Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\\ +\Xi \left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s\right)+l\left(\vartheta \right),\\ {}^{c,\tau }{D}^{\rho }\xi \left(\vartheta \right)=B\left(\vartheta \right)\xi \left(\vartheta \right)+Z\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)\\ +\Xi \left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s\right)+l\left(\vartheta \right),\\ \nu \left({\vartheta }_0\right)={\nu }_0,\xi \left({\vartheta }_0\right)={\xi }_0,\vartheta \in (s_i,{\vartheta }_{i+1}]\cap \tau ,i=1,2,\dots ,n,\\ \nu \left(\vartheta \right)=\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\xi \left(s\right),\nu \left(s\right))\Delta s+l_i,\vartheta ,s\in ({\vartheta }_i,s_i]\cap \tau ,i=1,2,\dots ,n,\\ \xi \left(\vartheta \right)=\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\nu \left(s\right),\xi \left(s\right))\Delta s+l_i,\vartheta ,s\in ({\vartheta }_i,s_i]\cap \tau ,i=1,2,\dots ,n.\end{array}\right.$$

(23)

Obviously, Equation (23) leads to

$$\nu \left(\vartheta \right)=\left\{\begin{array}{c}{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s+\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\\ \times \Xi \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right)\Delta s\\ +\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\nu \left(s\right),\xi \left(s\right))\Delta s,\vartheta \in (s_i,{\vartheta }_{i+1}]\cap \tau ,i=1,2,\dots ,n,\\ \frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\nu \left(s\right),\xi \left(s\right))\Delta s+l_i,\vartheta \in ({\vartheta }_i,s_i]\cap \tau ,i=1,2,\dots ,n,\end{array}\right.$$

and

$$\xi \left(\vartheta \right)=\left\{\begin{array}{c}{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s+\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\\ \times \Xi \left(s,\xi \left(s\right),\nu \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p\right)\Delta s\\ +\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\xi \left(s\right),\nu \left(s\right))\Delta s,\vartheta \in (s_i,{\vartheta }_{i+1}]\cap \tau ,i=1,2,\dots ,n,\\ \frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\xi \left(s\right),\nu \left(s\right))\Delta s+l_i,\vartheta \in ({\vartheta }_i,s_i]\cap \tau ,i=1,2,\dots ,n.\end{array}\right.$$

For $\vartheta \in (s_i,{\vartheta }_{i+1}]\cap \tau ,$$i=1,2,\dots ,n,$ we have

$$\begin{array}{c}∥\nu \left(\vartheta \right)-{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0-\underset{s_i}{\overset{\vartheta }{\int }}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(s,\nu \left(s\right),\xi \left(s\right)\right)\\ -\underset{s_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B{\left(\vartheta -s\right)}^{\rho }\right)\Xi \left(s,\nu \left(s\right),\xi \left(s\right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\nu \left(s\right),\xi \left(s\right)\right)\right)\Delta s\\ -\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\xi \left(s\right),\nu \left(s\right))\Delta s∥\\ \le \underset{s_i}{\overset{\vartheta }{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥∥f\left(s\right)∥\Delta s+\sum _{t=1}^n∥l_t∥\le \left(\frac{1}{2}{u}_3{u}_2\left(\vartheta -{\vartheta }_i\right)+n\right)\epsilon ,\end{array}$$

and

$$\begin{array}{c}∥\xi \left(\vartheta \right)-{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\xi }_0-\underset{s_i}{\overset{\vartheta }{\int }}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(s,\xi \left(s\right),\nu \left(s\right)\right)\\ -\underset{s_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B{\left(\vartheta -s\right)}^{\rho }\right)\Xi \left(s,\nu \left(s\right),\xi \left(s\right),\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_1\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\Delta s,\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{Z}_2\left(\vartheta ,s,\xi \left(s\right),\nu \left(s\right)\right)\right)\Delta s\\ -\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\xi \left(s\right),\nu \left(s\right))\Delta s∥\\ \le \underset{s_i}{\overset{\vartheta }{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥∥f\left(s\right)∥\Delta s+{\displaystyle \sum _{t=1}^n}∥l_t∥\le \left(\frac{1}{2}{u}_3{u}_2\left(\vartheta -{\vartheta }_i\right)+n\right)\epsilon ,\end{array}$$

with a similar approach, we obtain

$$\left\{\begin{array}{c}∥\xi \left(s\right)-\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\xi \left(s\right),\nu \left(s\right))\Delta s∥\le n\epsilon ,\\ ∥\nu \left(s\right)-\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{\vartheta }{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\nu \left(s\right),\xi \left(s\right))\Delta s∥\le n\epsilon ,\end{array}\right.\vartheta \in ({\vartheta }_i,s_i]\cap \tau ,i=1,2,\dots ,n.$$

Consequently, the debate is over. □

Currently, we present a sufficient stipulation for the UH stability of MISs (1) and (2).

Theorem 7.

If both inequality (6) and assumption $\left(1\right)$ are true, the MIS (1) is UH stable.

Proof. 

Assume that $\xi$ and $\nu$ are a solution of the MIS (1) and $\tilde{\xi }$ and $\tilde{\nu }$ are a solution of (21). Using Theorem 3, one has

$$\nu \left(\vartheta \right)=\left\{\begin{array}{c}{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right),\vartheta \in ({\vartheta }_0,{\vartheta }_1],\\ {\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right),\\ +\sum _{t=0}^i{O}_t\left(\nu \left({\vartheta }_t^{-}\right),\xi \left({\vartheta }_t^{-}\right)\right)+D_t\left({\vartheta }_t^{-},\nu \left({\vartheta }_t^{-}\right),\xi \left({\vartheta }_t^{-}\right)\right),\vartheta \in ({\vartheta }_i,{\vartheta }_{i+1}],i=1,2,\dots ,n,\end{array}\right.$$

where ${\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right)$ is the matrix form of the MLF. In a manner similar to Theorem 3, we obtain

$$\begin{array}{ccc}& & ∥\nu \left(\vartheta \right)-\tilde{\nu }\left(\vartheta \right)∥\hfill \\ & \le & \sum _{j=1}^i∥O_j\left(\nu \left({\vartheta }_j^{-}\right),\xi \left({\vartheta }_j^{-}\right)\right)-O_j\left(\tilde{\nu }\left({\vartheta }_j^{-}\right),\tilde{\xi }\left({\vartheta }_j^{-}\right)\right)∥\hfill \\ & & +\sum _{j=1}^i∥D_j\left({\vartheta }_j^{-},\nu \left({\vartheta }_j^{-}\right),\xi \left({\vartheta }_j^{-}\right)\right)-D_j\left({\vartheta }_j^{-},\tilde{\nu }\left({\vartheta }_j^{-}\right),\tilde{\xi }\left({\vartheta }_j^{-}\right)\right)∥\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥∥\left(Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\right.\hfill \\ & & +\left.\Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right)\right)\hfill \\ & & -\left(Z\left(\vartheta ,\tilde{\nu }\left(\vartheta \right),\tilde{\xi }\left(\vartheta \right)\right)\right.\hfill \\ & & +\left.\Theta \left(s,\tilde{\nu }\left(s\right),\tilde{\xi }\left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\tilde{\nu }\left(p\right),\tilde{\xi }\left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\tilde{\nu }\left(p\right),\tilde{\xi }\left(p\right)\right)\Delta p\right)\right)∥\Delta s\hfill \\ & \le & \frac{1}{2}{u}_3{u}_2({\vartheta }_l-{\vartheta }_i)\epsilon +J_3\left(∥\nu \left(\vartheta \right)-\tilde{\nu }\left(\vartheta \right)∥+∥\xi \left(\vartheta \right)-\tilde{\xi }\left(\vartheta \right)∥\right).\hfill \end{array}$$

(24)

Similarly, we can obtain

$$∥\xi \left(\vartheta \right)-\tilde{\xi }\left(\vartheta \right)∥\le \frac{1}{2}{u}_3{u}_2({\vartheta }_l-{\vartheta }_i)\epsilon +J_3\left(∥\xi \left(\vartheta \right)-\tilde{\xi }\left(\vartheta \right)∥+∥\nu \left(\vartheta \right)-\tilde{\nu }\left(\vartheta \right)∥\right).$$

(25)

From (24) and (25), we get

$$∥\nu \left(\vartheta \right)-\tilde{\nu }\left(\vartheta \right)∥+∥\xi \left(\vartheta \right)-\tilde{\xi }\left(\vartheta \right)∥\le u_3{u}_2({\vartheta }_l-{\vartheta }_i)\frac{\epsilon }{1-2J_3}=\nabla \epsilon ,$$

where $\nabla =\frac{u_3{u}_2({\vartheta }_l-{\vartheta }_i)}{1-2J_3}.$ It follows that the MIS (1) is UH stable. In addition, if $\overline{\nabla }\left(\epsilon \right)=\overline{\nabla }\left(0\right)=0,$ once this happens, our considered system (1) becomes generalized UH stable. □

Theorem 8.

If both inequality (11) and assumption $\left(1\right)$ are true, the MIS (2) is UH stable.

Proof. 

Assume that $\xi$ and $\nu$ are a solution of the MIS (1) and $\tilde{\xi }$ and $\tilde{\nu }$ are a solution of (22). From Theorem 4, we have

$$\nu \left(\vartheta \right)=\left\{\begin{array}{c}{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Xi \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right),\vartheta \in ({\vartheta }_i,s_i]\cap \tau ,\\ {\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Xi \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right),\\ +\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{s_i}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\nu \left(s\right),\xi \left(s\right))\Delta s,\vartheta \in (s_i,{\vartheta }_{i+1}]\cap \tau ,i=1,2,\dots ,n,\end{array}\right.$$

and

$$\xi \left(\vartheta \right)=\left\{\begin{array}{c}{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Xi \left(s,\xi \left(s\right),\nu \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p\right),\vartheta \in ({\vartheta }_i,s_i]\cap \tau ,\\ {\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Xi \left(s,\xi \left(s\right),\nu \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p\right),\\ +\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{s_i}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\xi \left(s\right),\nu \left(s\right))\Delta s,\vartheta \in (s_i,{\vartheta }_{i+1}]\cap \tau ,i=1,2,\dots ,n,\end{array}\right.$$

where ${\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right)$ is the matrix form of the MLF. In a manner similar to Theorem 4, we get

$$\begin{array}{ccc}& & ∥\nu \left(\vartheta \right)-\tilde{\nu }\left(\vartheta \right)∥\hfill \\ & \le & \frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{s_i}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\phi }_i(s,\nu \left(s\right),\xi \left(s\right))-{\phi }_i(s,\tilde{\nu }\left(s\right),\tilde{\xi }\left(s\right))∥\Delta s\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥∥\left(Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\right.\hfill \\ & & +\left.\Xi \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right)\right)\hfill \\ & & -\left(Z\left(\vartheta ,\tilde{\nu }\left(\vartheta \right),\tilde{\xi }\left(\vartheta \right)\right)+\Xi \left(s,\tilde{\nu }\left(s\right),\tilde{\xi }\left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\tilde{\nu }\left(p\right),\tilde{\xi }\left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\tilde{\nu }\left(p\right),\tilde{\xi }\left(p\right)\right)\Delta p\right)\right)∥\Delta s\hfill \\ & \le & \frac{1}{2}{u}_3{u}_2({\vartheta }_l-{\vartheta }_i)\epsilon +J_3^{*}\left(∥\nu \left(\vartheta \right)-\tilde{\nu }\left(\vartheta \right)∥+∥\xi \left(\vartheta \right)-\tilde{\xi }\left(\vartheta \right)∥\right).\hfill \end{array}$$

(26)

Analogously, we can obtain that

$$∥\xi \left(\vartheta \right)-\tilde{\xi }\left(\vartheta \right)∥\le \frac{1}{2}{u}_3{u}_2({\vartheta }_l-{\vartheta }_i)\epsilon +J_3^{*}\left(∥\xi \left(\vartheta \right)-\tilde{\xi }\left(\vartheta \right)∥+∥\nu \left(\vartheta \right)-\tilde{\nu }\left(\vartheta \right)∥\right).$$

(27)

Adding (26) and (27), we find that

$$∥\nu \left(\vartheta \right)-\tilde{\nu }\left(\vartheta \right)∥+∥\xi \left(\vartheta \right)-\tilde{\xi }\left(\vartheta \right)∥\le \left(u_3{u}_2({\vartheta }_l-{\vartheta }_i)\right)\frac{\epsilon }{1-2J_3^{*}}=\Delta \epsilon ,$$

where $\Delta =\frac{u_3{u}_2({\vartheta }_l-{\vartheta }_i)}{1-2J_3^{*}}.$ Therefore, the MIS (2) is UH stable. Further, if ${\Delta }^{*}={\Delta }^{*}\left(0\right)=0,$ once this happens, our supposed system (2) becomes generalized UH stable. □

5. Controllability Study

The third part of the article analyzes the controllability of given impulsive systems. We start by providing some definitions in this line.

Definition 6.

The MIS (3) has a solution $\left(\nu ,\xi \right)\in \tau \times \tau$ if $\nu \left(0\right)={\nu }_0$, $\xi \left(0\right)={\xi }_0$ and the pair $\left(\xi ,\nu \right)$ is a solution to the integral equations below:

$$\nu \left(\vartheta \right)=\left\{\begin{array}{c}{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right),\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)H\ell \left(\vartheta \right)\Delta s,\vartheta \in ({\vartheta }_0,{\vartheta }_1],\\ {\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right),\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)H\ell \left(\vartheta \right)\Delta s\\ +\sum _{t=1}^i{O}_t\left(\nu \left({\vartheta }_t^{-}\right),\xi \left({\vartheta }_t^{-}\right)\right)+D_t\left({\vartheta }_t^{-},\nu \left({\vartheta }_t^{-}\right),\xi \left({\vartheta }_t^{-}\right)\right),\vartheta \in ({\vartheta }_i,{\vartheta }_{i+1}],i=1,2,\dots ,n,\end{array}\right.$$

(28)

and

$$\xi \left(\vartheta \right)=\left\{\begin{array}{c}{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\xi }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Theta \left(s,\xi \left(s\right),\nu \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p\right),\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)H\ell \left(\vartheta \right)\Delta s,\vartheta \in ({\vartheta }_0,{\vartheta }_1],\\ {\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\xi }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Theta \left(s,\xi \left(s\right),\nu \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p\right),\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)H\ell \left(\vartheta \right)\Delta s\\ +\sum _{t=1}^i{O}_t\left(\xi \left({\vartheta }_t^{-}\right),\nu \left({\vartheta }_t^{-}\right)\right)+D_t\left({\vartheta }_t^{-},\xi \left({\vartheta }_t^{-}\right),\nu \left({\vartheta }_t^{-}\right)\right),\vartheta \in ({\vartheta }_i,{\vartheta }_{i+1}],i=1,2,\dots ,n,\end{array}\right.$$

(29)

where ${\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right)$ is the matrix form of the MLF.

Definition 7.

We say that the MIS (3) is controllable on $\tau ,$ if for each ${\nu }_0,{\xi }_0,{\nu }_A,{\xi }_A\in \tau$ where ${\vartheta }_{i+1}=A$ is any arbitrary point, there is an rd-continuous function $\ell \in L^2(I,\mathbb{R})$ so that the relevant solution of (3) fulfills $\nu \left(0\right)={\nu }_0$, $\xi \left(0\right)={\xi }_0$ and $\ell \left(A\right)={\ell }_A.$

For convenience, we choose to use:

$$\begin{array}{ccc}\hfill J_4& =& u_3\left(u_2{T}_Z+u_2{T}_{{\Theta }_1}+u_2{T}_{{\Theta }_2}+u_2\left(T_{{\Theta }_3}{T}_{Z_1}+T_{{\Theta }_4}{T}_{Z_2}\right)(s_l-s_0)+Y_{\Theta }^{*}\right)\hfill \\ & & \times ({\vartheta }_l-{\vartheta }_0)\left(1+T_H{T}_{\Re }^{\rho }({\vartheta }_l-{\vartheta }_0)\right),\hfill \\ \hfill J_5& =& u_3\left(u_2{T}_Z+u_2{T}_{{\Xi }_1}+u_2{T}_{{\Xi }_2}+u_2\left(T_{{\Xi }_3}{T}_{Z_1}+T_{{\Xi }_4}{T}_{Z_2}\right)(s_l-s_0)\right)({\vartheta }_l-{\vartheta }_0),\hfill \\ \hfill J_6& =& \left(1+T_H{T}_{\Re }^{\rho }({\vartheta }_l-{\vartheta }_0)\right)\times \left(\sum _{j=1}^i\left(T_O+T_D\right)\right.\hfill \\ & & +\left.\left(u_2{T}_Z+u_2{T}_{{\Theta }_1}+u_2{T}_{{\Theta }_2}+u_2\left(T_{{\Theta }_3}{T}_{Z_1}+T_{{\Theta }_4}{T}_{Z_2}\right)(s_l-s_0)\right)({\vartheta }_l-{\vartheta }_0)\right),\hfill \\ \hfill J_7& =& \left(1+T_H{T}_{\Re }^{\rho }(\vartheta -{\vartheta }_l)\right)\times \left(\frac{1}{\Gamma \left(\rho \right)}{u}_3{T}_{\phi }\left(s_i-{\vartheta }_i\right)\right.\hfill \\ & & +\left.\left(u_2{T}_Z+u_2{T}_{{\Xi }_1}+u_2{T}_{{\Xi }_2}+u_2\left(T_{{\Xi }_3}{T}_{Z_1}+T_{{\Xi }_4}{T}_{Z_2}\right)(s_l-s_0)\right)({\vartheta }_l-{\vartheta }_0)\right),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill E_7& =& \left(1+T_H{T}_{\Re }^{\rho }\right)\left(\sum _{j=1}^i\left(T_O+T_D\right)\delta +u_1\right)+T_H{T}_H^{\rho }∥{\nu }_A∥,\hfill \\ \hfill E_8& =& \left(1+T_H{T}_{\Re }^{\rho }\right)\left(\sum _{j=1}^i\left(T_O+T_D\right)\delta +u_4\right)+T_H{T}_H^{\rho }∥{\xi }_A∥,\hfill \\ \hfill E_9& =& \left(1+T_H{T}_{\Re }^{\rho }\right)\left(\frac{1}{\Gamma \left(\rho \right)}{u}_3{T}_{\phi }{\delta }^{{}^{{}^{″}}}\left(s_i-{\vartheta }_i\right)+u_1+Y_{\Theta }^{*}\right)+T_H{T}_H^{\rho }∥{\nu }_A∥,\hfill \\ \hfill E_{10}& =& \left(1+T_H{T}_{\Re }^{\rho }\right)\left(\frac{1}{\Gamma \left(\rho \right)}{u}_3{T}_{\phi }{\delta }^{{}^{{}^{″}}}\left(s_i-{\vartheta }_i\right)+u_4+Y_{\Theta }^{*}\right)+T_H{T}_H^{\rho }∥{\xi }_A∥.\hfill \end{array}$$

Lemma 3.

Assume that the hypotheses $\left(1\right)$, $\left(3\right)-\left(5\right)$ are true and ${\nu }_A,{\xi }_A\in \tau$ where ${\vartheta }_{i+1}=A$ is any chosen point. Then the pair $\left(\nu ,\xi \right)\in \tau \times \tau ,$ which is defined in (28) and (29) is a solution to the MIS (3) with the control function

$$\ell \left(\vartheta \right)=\left\{\begin{array}{c}{{(}^{\rho }{\Re }_{{\vartheta }_0}^A)}^{-1}\left[{\nu }_T-{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0-\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\right.\\ -\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \left.\times \Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right)\right],\vartheta \in ({\vartheta }_0,{\vartheta }_1],\\ {{(}^{\rho }{\Re }_{{\vartheta }_0}^A)}^{-1}\left[{\nu }_T-{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0-\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\right.\\ -\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right),\\ -\left.\sum _{t=1}^i\left(O_t\left(\nu \left({\vartheta }_t^{-}\right),\xi \left({\vartheta }_t^{-}\right)\right)-D_t\left({\vartheta }_t^{-},\nu \left({\vartheta }_t^{-}\right),\xi \left({\vartheta }_t^{-}\right)\right)\right)\right],\vartheta \in ({\vartheta }_i,{\vartheta }_{i+1}],i=1,2,\dots ,n,\end{array}\right.$$

(30)

or

$$\ell \left(\vartheta \right)=\left\{\begin{array}{c}{{(}^{\rho }{\Re }_{{\vartheta }_0}^A)}^{-1}\left[{\xi }_T-{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\xi }_0-\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)\Delta s\right.\\ -\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \left.\times \Theta \left(s,\xi \left(s\right),\nu \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p\right)\right],\vartheta \in ({\vartheta }_0,{\vartheta }_1],\\ {{(}^{\rho }{\Re }_{{\vartheta }_0}^A)}^{-1}\left[{\xi }_T-{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\xi }_0-\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)\Delta s\right.\\ -\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Theta \left(s,\xi \left(s\right),\nu \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p\right),\\ -\left.\sum _{t=1}^i\left(O_t\left(\xi \left({\vartheta }_t^{-}\right),\nu \left({\vartheta }_t^{-}\right)\right)+D_t\left({\vartheta }_t^{-},\xi \left({\vartheta }_t^{-}\right),\nu \left({\vartheta }_t^{-}\right)\right)\right)\right],\vartheta \in ({\vartheta }_i,{\vartheta }_{i+1}],i=1,2,\dots ,n,\end{array}\right.$$

(31)

where ${\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right)$ is the matrix form of the MLF, $\nu \left(A\right)={\nu }_A$ and $\xi \left(A\right)={\xi }_A.$ Further, the control function $\ell \left(\vartheta \right)$ estimated by $∥\ell \left(\vartheta \right)∥\le {\Pi }_{\ell }$ or $∥\ell \left(\vartheta \right)∥\le {\widehat{\Pi }}_{\ell }$, where ${\Pi }_{\ell }$ and ${\widehat{\Pi }}_{\ell }$ are defined by

$$\begin{array}{ccc}\hfill {\Pi }_{\ell }& =& {\Pi }_{\Re }^{\rho }\left[∥{\nu }_A∥+∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0∥+\left(\sum _{j=1}^i\left(T_O+T_D\right)\right){∥\nu +\xi ∥}_{\infty }\right.\hfill \\ & & \left.+\left(u_3{u}_2\left(T_Z+T_{{\Theta }_1}+T_{{\Theta }_2}+\left(T_{{\Theta }_3}{T}_{Z_1}+T_{{\Theta }_4}{T}_{Z_2}\right)(s_l-s_0)\right){∥\nu +\xi ∥}_{\infty }+Y_{\Theta }^{*}\right)({\vartheta }_l-{\vartheta }_0)\right],\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\widehat{\Pi }}_{\ell }& =& {\Pi }_{\Re }^{\rho }\left[∥{\xi }_A∥+∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\xi }_0∥+\left(\sum _{j=1}^i\left(T_O+T_D\right)\right){∥\nu +\xi ∥}_{\infty }\right.\hfill \\ & & \left.+\left(u_3{u}_2\left(T_Z+T_{{\Theta }_1}+T_{{\Theta }_2}+\left(T_{{\Theta }_3}{T}_{Z_1}+T_{{\Theta }_4}{T}_{Z_2}\right)(s_l-s_0)\right){∥\nu +\xi ∥}_{\infty }+Y_{\Theta }^{*}\right)({\vartheta }_l-{\vartheta }_0)\right],\hfill \end{array}$$

for $\vartheta \in ({\vartheta }_i,{\vartheta }_{i+1}],$$i=1,2,\dots ,n.$

Proof. 

For $\vartheta \in ({\vartheta }_i,{\vartheta }_{i+1}],$$i=1,2,\dots ,n,$ let $\left(\nu ,\xi \right)\in \tau \times \tau$, which is defined in (28) and (29)) is a solution to the MIS (3), then for $\vartheta \in A,$ and using (30), we get

$$\begin{array}{ccc}\hfill \nu \left(A\right)& =& {\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(A-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(A-s\right)}^{\rho }\right)\right)Z\left(s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(A-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(A-s\right)}^{\rho }\right)\right)\hfill \\ & & \times \Theta \left(s,\xi \left(s\right),\nu \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p\right)\Delta s\hfill \\ & & +\sum _{t=1}^i\left(O_t\left(\xi \left({\vartheta }_t^{-}\right),\nu \left({\vartheta }_t^{-}\right)\right)+D_t\left({\vartheta }_t^{-},\xi \left({\vartheta }_t^{-}\right),\nu \left({\vartheta }_t^{-}\right)\right)\right)\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(A-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(A-s\right)}^{\rho }\right)\right)H\ell \left(A\right)\Delta s\hfill \end{array}$$

$$\begin{array}{ccc}& =& {\Lambda }_{\rho }\left(B{A}^{\rho }\right){\nu }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(A-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(A-s\right)}^{\rho }\right)\right)Z\left(s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(A-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(A-s\right)}^{\rho }\right)\right)\hfill \\ & & \times \Theta \left(s,\xi \left(s\right),\nu \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p\right)\Delta s\hfill \\ & & +\sum _{t=1}^i\left(O_t\left(\xi \left(A_t^{-}\right),\nu \left(A_t^{-}\right)\right)+D_t\left(A_t^{-},\xi \left(A_t^{-}\right),\nu \left(A_t^{-}\right)\right)\right)+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(A-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(A-s\right)}^{\rho }\right)\right)H\hfill \end{array}$$

$$\begin{array}{ccc}& & \times {{(}^{\rho }{\Re }_{{\vartheta }_0}^A)}^{-1}\left[{\nu }_T-{\Lambda }_{\rho }\left(B{A}^{\rho }\right){\nu }_0-\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(A-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(A-s\right)}^{\rho }\right)\right)Z\left(s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s\right.\hfill \\ & & -\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(A-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(A-s\right)}^{\rho }\right)\right)\hfill \\ & & \times \Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right)\Delta s\hfill \\ & & \left.-\sum _{t=1}^i\left(O_t\left(\nu \left(A_t^{-}\right),\xi \left(A_t^{-}\right)\right)+D_t\left(A_t^{-},\nu \left(A_t^{-}\right),\xi \left(A_t^{-}\right)\right)\right)\right]\Delta s,\hfill \end{array}$$

which yields that

$$\begin{array}{ccc}\hfill \nu \left(A\right)& =& {\Lambda }_{\rho }\left(B{A}^{\rho }\right){\nu }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(A-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(A-s\right)}^{\rho }\right)\right)Z\left(s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(A-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(A-s\right)}^{\rho }\right)\right)\hfill \\ & & \times \Theta \left(s,\xi \left(s\right),\nu \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p\right)\Delta s\hfill \\ & & +\sum _{t=1}^i\left(O_t\left(\xi \left(A_t^{-}\right),\nu \left(A_t^{-}\right)\right)-D_t\left(A_t^{-},\xi \left(A_t^{-}\right),\nu \left(A_t^{-}\right)\right)\right)+\hfill \\ & & \times {(}^{\rho }{\Re }_{{\vartheta }_0}^A){{(}^{\rho }{\Re }_{{\vartheta }_0}^A)}^{-1}\left[{\nu }_T-{\Lambda }_{\rho }\left(B{A}^{\rho }\right){\nu }_0-\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(A-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(A-s\right)}^{\rho }\right)\right)Z\left(s,\nu \left(s\right),\xi \left(s\right)\right)\Delta s\right.\hfill \\ & & -\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(A-s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(A-s\right)}^{\rho }\right)\right)\hfill \\ & & \times \Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right)\Delta s\hfill \\ & & \left.-\sum _{t=1}^i\left(O_t\left(\nu \left(A_t^{-}\right),\xi \left(A_t^{-}\right)\right)+D_t\left(A_t^{-},\nu \left(A_t^{-}\right),\xi \left(A_t^{-}\right)\right)\right)\right]\Delta s\hfill \\ & =& {\nu }_A.\hfill \end{array}$$

In the same way, by using (31), we can prove that $\xi \left(A\right)={\xi }_A.$ Also, for $\vartheta \in ({\vartheta }_i,{\vartheta }_{i+1}],$$i=1,2,\dots ,n,$ the estimation

$$\begin{array}{ccc}\hfill ∥\ell \left(\vartheta \right)∥& =& {\Pi }_{\Re }^{\rho }\left[∥{\nu }_A∥-∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0∥\right.\hfill \\ & & -\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥∥Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)∥\Delta s\hfill \\ & & -\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥\hfill \\ & & \times ∥\Theta \left(s,\xi \left(s\right),\nu \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p\right)∥\Delta s\hfill \\ & & -\left.\sum _{t=1}^i\left(∥O_t\left(\nu \left({\vartheta }_t^{-}\right),\xi \left({\vartheta }_t^{-}\right)\right)∥+∥D_t\left({\vartheta }_t^{-},\nu \left({\vartheta }_t^{-}\right),\xi \left({\vartheta }_t^{-}\right)\right)∥\right)\right],\hfill \end{array}$$

leads to

$$\begin{array}{ccc}\hfill ∥\ell \left(\vartheta \right)∥& =& {\Pi }_{\Re }^{\rho }\left[∥{\nu }_A∥+∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0∥+\left(\sum _{j=1}^i\left(T_O+T_D\right)\right){∥\nu +\xi ∥}_{\infty }\right.\hfill \\ & & \left.+\left(u_3{u}_2\left(T_Z+T_{{\Theta }_1}+T_{{\Theta }_2}+\left(T_{{\Theta }_3}{T}_{Z_1}+T_{{\Theta }_4}{T}_{Z_2}\right)(s_l-s_0)\right){∥\nu +\xi ∥}_{\infty }+Y_{\Theta }^{*}\right)({\vartheta }_l-{\vartheta }_0)\right]={\Pi }_{\ell }\hfill \end{array}$$

Similarly,

$$\begin{array}{ccc}\hfill ∥\ell \left(\vartheta \right)∥& =& {\Pi }_{\Re }^{\rho }\left[∥{\xi }_A∥+∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\xi }_0∥+\left(\sum _{j=1}^i\left(T_O+T_D\right)\right){∥\nu +\xi ∥}_{\infty }\right.\hfill \\ & & \left.+\left(u_3{u}_2\left(T_Z+T_{{\Theta }_1}+T_{{\Theta }_2}+\left(T_{{\Theta }_3}{T}_{Z_1}+T_{{\Theta }_4}{T}_{Z_2}\right)(s_l-s_0)\right){∥\nu +\xi ∥}_{\infty }+Y_{\Theta }^{*}\right)({\vartheta }_l-{\vartheta }_0)\right]={\widehat{\Pi }}_{\ell }.\hfill \end{array}$$

This finishes the proof. □

Definition 8.

The MIS (4) has a solution $\left(\nu ,\xi \right)\in \tau \times \tau$ if $\nu \left(0\right)={\nu }_0$, $\xi \left(0\right)={\xi }_0$ and the pair $\left(\xi ,\nu \right)$ is a solution to the following integral equations:

$$\nu \left(\vartheta \right)=\left\{\begin{array}{c}{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Xi \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right),\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)H\ell \left(\vartheta \right)\Delta s,\vartheta \in ({\vartheta }_i,s_{i+1}]\cap \tau ,i=1,2,\dots ,n,\\ {\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Xi \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right),\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)H\ell \left(\vartheta \right)\Delta s\\ +\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{s_i}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\nu \left(s\right),\xi \left(s\right))\Delta s,\vartheta \in (s_i,{\vartheta }_{i+1}]\cap \tau ,i=1,2,\dots ,n,\end{array}\right.$$

(32)

and

$$\xi \left(\vartheta \right)=\left\{\begin{array}{c}{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\xi }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Xi \left(s,\xi \left(s\right),\nu \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p\right),\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)H\ell \left(\vartheta \right)\Delta s,\vartheta \in ({\vartheta }_i,s_{i+1}]\cap \tau ,i=1,2,\dots ,n,\\ {\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\xi }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Xi \left(s,\xi \left(s\right),\nu \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p\right),\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)H\ell \left(\vartheta \right)\Delta s\\ +\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{s_i}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\xi \left(s\right),\nu \left(s\right))\Delta s,\vartheta \in (s_i,{\vartheta }_{i+1}]\cap \tau ,i=1,2,\dots ,n,\end{array}\right.$$

(33)

where ${\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right)$ is the matrix form of the MLF.

Definition 9.

We say that the MIS (4) is controllable on $\tau ,$ if for each ${\nu }_0,{\xi }_0,{\nu }_A,{\xi }_A\in \tau$ where ${\vartheta }_{i+1}=A,$ there is an rd-continuous function $\ell \in L^2(I,\mathbb{R})$ so that the corresponding solution of (4) justifies $\nu \left(0\right)={\nu }_0$, $\xi \left(0\right)={\xi }_0$ and $\ell \left(A\right)={\ell }_A.$

Lemma 4.

Suppose that the hypotheses $\left(1\right),\left(3\right),\left(4\right),\left(5\right)$ hold and ${\nu }_A,{\xi }_A\in \tau$ where ${\vartheta }_{i+1}=A$ is any chosen point. Then the pair $\left(\nu ,\xi \right)\in \tau \times \tau ,$ which is defined in (32) and (33) is a solution to the MIS (3) on $(s_i,{\vartheta }_{i+1}]\cap \tau ,$$i=1,2,\dots ,n,$ with the control function

$$\ell \left(\vartheta \right)=\left\{\begin{array}{c}{{(}^{\rho }{\Re }_{{\vartheta }_0}^A)}^{-1}\left[{\nu }_T-{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0-\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\right.\\ -\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \left.\times \Xi \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right)\right],\vartheta \in ({\vartheta }_i,s_i]\cap \tau ,\\ {{(}^{\rho }{\Re }_{{\vartheta }_0}^A)}^{-1}\left[{\nu }_T-{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0-\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\right.\\ -\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Xi \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right),\\ \left.-\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{s_i}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\nu \left(s\right),\xi \left(s\right))\Delta s\right],\vartheta \in (s_i,{\vartheta }_{i+1}],i=1,2,\dots ,n,\end{array}\right.$$

or

$$\ell \left(\vartheta \right)=\left\{\begin{array}{c}{{(}^{\rho }{\Re }_{{\vartheta }_0}^A)}^{-1}\left[{\xi }_T-{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\xi }_0-\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)\Delta s\right.\\ -\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \left.\times \Xi \left(s,\xi \left(s\right),\nu \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p\right)\right],\vartheta \in ({\vartheta }_i,s_i]\cap \tau ,\\ {{(}^{\rho }{\Re }_{{\vartheta }_0}^A)}^{-1}\left[{\xi }_T-{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\xi }_0-\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\xi \left(\vartheta \right),\nu \left(\vartheta \right)\right)\Delta s\right.\\ -\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Xi \left(s,\xi \left(s\right),\nu \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\xi \left(p\right),\nu \left(p\right)\right)\Delta p\right),\\ \left.-\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{s_i}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\xi \left(s\right),\nu \left(s\right))\Delta s\right],\vartheta \in (s_i,{\vartheta }_{i+1}]\cap \tau ,i=1,2,\dots ,n,,\end{array}\right.$$

where ${\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right)$ is the matrix form of the MLF, $\nu \left(A\right)={\nu }_A$, and $\xi \left(A\right)={\xi }_A.$ Further, the control function $\ell \left(\vartheta \right)$ estimated by $∥\ell \left(\vartheta \right)∥\le {\beth }_{\ell }$ or $∥\ell \left(\vartheta \right)∥\le {\tilde{\beth }}_{\ell },$ where ${\beth }_{\ell }$ and ${\tilde{\beth }}_{\ell }$ are defined by

$$\begin{array}{ccc}\hfill {\beth }_{\ell }& =& {\Pi }_{\Re }^{\rho }\left[∥{\nu }_A∥+∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0∥+\frac{1}{\Gamma \left(\rho \right)}(s_i-{\vartheta }_i)u_1{T}_{\phi }{∥\nu +\xi ∥}_{\infty }\right.\hfill \\ & & \left.+\left(u_3{u}_2\left(T_Z+T_{{\Theta }_1}+T_{{\Theta }_2}+\left(T_{{\Theta }_3}{T}_{Z_1}+T_{{\Theta }_4}{T}_{Z_2}\right)(s_l-s_0)\right){∥\nu +\xi ∥}_{\infty }+Y_{\Theta }^{*}\right)({\vartheta }_l-{\vartheta }_0)\right],\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\tilde{\beth }}_{\ell }& =& {\Pi }_{\Re }^{\rho }\left[∥{\xi }_A∥+∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\xi }_0∥+\frac{1}{\Gamma \left(\rho \right)}(s_i-{\vartheta }_i)u_4{T}_{\phi }{∥\nu +\xi ∥}_{\infty }\right.\hfill \\ & & \left.+\left(u_3{u}_2\left(T_Z+T_{{\Theta }_1}+T_{{\Theta }_2}+\left(T_{{\Theta }_3}{T}_{Z_1}+T_{{\Theta }_4}{T}_{Z_2}\right)(s_l-s_0)\right){∥\nu +\xi ∥}_{\infty }+Y_{\Theta }^{*}\right)({\vartheta }_l-{\vartheta }_0)\right],\hfill \end{array}$$

>for $\vartheta \in (s_i,{\vartheta }_{i+1}]\cap \tau ,$$i=1,2,\dots ,n.$

Proof. 

Similar to Lemma 3, the proof is given. □

Theorem 9.

If the assertions $\left(1\right),\left(3\right),\left(4\right),\left(5\right)$ are true, then the MIS (3) is controllable on τ provided that

$$max\left\{J_4,J_5,J_6\right\}<1.$$

(34)

Proof. 

Assume that ${\mho }^{″}\subseteq Q$ where ${\mho }^{″}=\left\{\left(K_1,K_2,K_3,K_4\right)\in Q:∥K_1,K_2,K_3,K_4∥\le {\delta }_2^{″}\right\},$ where ${\delta }_2^{″}=max\{{\delta }^{″},{\delta }_1^{″}\}$ and ${\delta }^{″},{\delta }_1^{″}\in (0,1),$ and also ${\delta }_2^{″}\ge E_7$. Define an operator ${\Im }_{\rho }^{″}:{\mho }^{″}\times {\mho }^{″}\to {\mho }^{″}$ by

$${\Im }_{\rho }^{″}(\nu ,\xi )\left(\vartheta \right)=\left\{\begin{array}{c}{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right),\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)H\ell \left(\vartheta \right)\Delta s,\vartheta \in ({\vartheta }_0,{\vartheta }_1],\\ {\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right),\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)H\ell \left(\vartheta \right)\Delta s\\ +\sum _{t=1}^i\left(O_t\left(\nu \left({\vartheta }_t^{-}\right),\xi \left({\vartheta }_t^{-}\right)\right)+D_t\left({\vartheta }_t^{-},\nu \left({\vartheta }_t^{-}\right),\xi \left({\vartheta }_t^{-}\right)\right)\right),\vartheta \in ({\vartheta }_i,{\vartheta }_{i+1}],i=1,2,\dots ,n,\end{array}\right.$$

where ${\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right)$ is the matrix form of the MLF. Now, we show that ${\Im }_{\rho }^{″}:{\mho }^{″}\times {\mho }^{″}\to {\mho }^{″}$ is a self-mapping. For $\vartheta \in ({\vartheta }_i,{\vartheta }_{i+1}],$$i=1,2,\dots ,n,$ we have

$$\begin{array}{ccc}\hfill ∥{\Im }_{\rho }^{″}(\nu ,\xi )\left(\vartheta \right)∥& \le & \sum _{j=1}^i∥O_j\left(\nu \left({\vartheta }_j^{-}\right),\xi \left({\vartheta }_j^{-}\right)\right)∥+\sum _{j=1}^i∥D_j\left({\vartheta }_j^{-},\nu \left({\vartheta }_j^{-}\right),\xi \left({\vartheta }_j^{-}\right)\right)∥+∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0∥\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥∥\left(Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\right.\hfill \\ & & +\left.\Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right)\right)∥\Delta s\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥∥H∥∥\ell \left(s\right)∥\Delta s\hfill \\ & \le & \sum _{j=1}^i\left(T_O+T_D\right){\delta }^{″}+u_1+{\delta }^{″}{u}_3\left(u_2{T}_Z+u_2{T}_{{\Theta }_1}+u_2{T}_{{\Theta }_2}\right.\hfill \\ & & +\left.u_2\left(T_{{\Theta }_3}{T}_{Z_1}+T_{{\Theta }_4}{T}_{Z_2}\right)(s_l-s_0)+Y_{\Theta }^{*}\right)({\vartheta }_l-{\vartheta }_0)\hfill \\ & & +T_H{T}_{\Re }^{\rho }({\vartheta }_l-{\vartheta }_0)\left[\sum _{j=1}^i\left(T_O+T_D\right){\delta }^{″}+{∥\nu ∥}_A+u_1+{\delta }^{″}{u}_3\left(u_2{T}_Z+u_2{T}_{{\Theta }_1}+u_2{T}_{{\Theta }_2}\right.\right.\hfill \\ & & \left.\left.+u_2\left(T_{{\Theta }_3}{T}_{Z_1}+T_{{\Theta }_4}{T}_{Z_2}\right)(s_l-s_0)+Y_{\Theta }^{*}\right)({\vartheta }_l-{\vartheta }_0)\right]\hfill \end{array}$$

which implies that

$$∥{\Im }_{\rho }^{″}(\nu ,\xi )\left(\vartheta \right)∥\le E_7+{\delta }^{″}{J}_6\le {\delta }^{″}+{\delta }^{″}{J}_6={\delta }_1^{″}$$

Similarly, if ${\delta }_2^{″}\ge E_8,$ one can show that

$$∥{\Im }_{\rho }(\xi ,\nu )\left(\vartheta \right)∥\le E_8+{\delta }^{″}{J}_1\le {\delta }^{″}+{\delta }^{″}{J}_1={\delta }_1^{″},$$

Hence

$$∥{\Im }_{\rho }^{″}(\nu ,\xi )\left(\vartheta \right)∥\le {\delta }_2^{″}\mathrm{and}∥{\Im }_{\rho }^{″}(\xi ,\nu )\left(\vartheta \right)∥\le {\delta }_2^{″}.$$

(35)

It follows from (35) that ${\Im }^{″}\left({\mho }^{″}\times {\mho }^{″}\right)\subseteq {\mho }^{″}.$ Also, for $\vartheta \in ({\vartheta }_i,{\vartheta }_{i+1}],$$i=1,2,\dots ,n,$ with ${\nu }_0={\tilde{\nu }}_0$ and ${\xi }_0={\tilde{\xi }}_0,$ one has

$$\begin{array}{ccc}& & ∥{\Im }_{\rho }^{″}(\nu ,\xi )\left(\vartheta \right)-{\Im }_{\rho }^{″}(\tilde{\nu },\tilde{\xi })\left(\vartheta \right)∥\hfill \\ & \le & \sum _{j=1}^i∥O_j\left(\nu \left({\vartheta }_j^{-}\right),\xi \left({\vartheta }_j^{-}\right)\right)-O_j\left(\tilde{\nu }\left({\vartheta }_j^{-}\right),\tilde{\xi }\left({\vartheta }_j^{-}\right)\right)∥+\sum _{j=1}^i∥D_j\left({\vartheta }_j^{-},\nu \left({\vartheta }_j^{-}\right),\xi \left({\vartheta }_j^{-}\right)\right)-D_j\left({\vartheta }_j^{-},\tilde{\nu }\left({\vartheta }_j^{-}\right),\tilde{\xi }\left({\vartheta }_j^{-}\right)\right)∥\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥∥\left(Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\right.\hfill \\ & & +\left.\Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right)\right)\hfill \\ & & -\left(Z\left(\vartheta ,\tilde{\nu }\left(\vartheta \right),\tilde{\xi }\left(\vartheta \right)\right)+\Theta \left(s,\tilde{\nu }\left(s\right),\tilde{\xi }\left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\tilde{\nu }\left(p\right),\tilde{\xi }\left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\tilde{\nu }\left(p\right),\tilde{\xi }\left(p\right)\right)\Delta p\right)\right)∥\Delta s\hfill \\ & & -\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥∥H∥∥{{(}^{\rho }{\Re }_{{\vartheta }_0}^A)}^{-1}∥\left(\vartheta -{\vartheta }_l\right)\hfill \\ & & \times \left[\sum _{j=1}^i∥O_j\left(\nu \left({\vartheta }_j^{-}\right),\xi \left({\vartheta }_j^{-}\right)\right)-O_j\left(\tilde{\nu }\left({\vartheta }_j^{-}\right),\tilde{\xi }\left({\vartheta }_j^{-}\right)\right)∥+\sum _{j=1}^i∥D_j\left({\vartheta }_j^{-},\nu \left({\vartheta }_j^{-}\right),\xi \left({\vartheta }_j^{-}\right)\right)-D_j\left({\vartheta }_j^{-},\tilde{\nu }\left({\vartheta }_j^{-}\right),\tilde{\xi }\left({\vartheta }_j^{-}\right)\right)∥\right.\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥\Delta s∥\left(Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\right.\hfill \\ & & +\left.\Theta \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right)\right)\hfill \\ & & -\left.\left(Z\left(\vartheta ,\tilde{\nu }\left(\vartheta \right),\tilde{\xi }\left(\vartheta \right)\right)+\Theta \left(s,\tilde{\nu }\left(s\right),\tilde{\xi }\left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\tilde{\nu }\left(p\right),\tilde{\xi }\left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\tilde{\nu }\left(p\right),\tilde{\xi }\left(p\right)\right)\Delta p\right)\right)∥\right]\Delta s,\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}& & ∥{\Im }_{\rho }^{″}(\nu ,\xi )\left(\vartheta \right)-{\Im }_{\rho }^{″}(\tilde{\nu },\tilde{\xi })\left(\vartheta \right)∥\hfill \\ & \le & \left(\sum _{j=1}^i\left(T_O+T_D\right)+\left[u_2{T}_Z+u_2{T}_{{\Theta }_1}+u_2{T}_{{\Theta }_2}+u_2\left(T_{{\Theta }_3}{T}_{Z_1}+T_{{\Theta }_3}{T}_{Z_2}\right)(s_l-s_0)\right]({\vartheta }_l-{\vartheta }_0)\right)\hfill \\ & & \times \left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right)\hfill \\ & & +T_H{T}_{\Re }^{\rho }({\vartheta }_l-{\vartheta }_0)\left[\sum _{j=1}^i\left(T_O+T_D\right)+\left[u_2{T}_Z+u_2{T}_{{\Theta }_1}+u_2{T}_{{\Theta }_2}+u_2\left(T_{{\Theta }_3}{T}_{Z_1}+T_{{\Theta }_3}{T}_{Z_2}\right)(s_l-s_0)\right]({\vartheta }_l-{\vartheta }_0)\right]\hfill \\ & & \times \left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right)\hfill \\ & \le & (1+T_H{T}_{\Re }^{\rho }({\vartheta }_l-{\vartheta }_0))\hfill \\ & & \times \left[\sum _{j=1}^i\left(T_O+T_D\right)+\left[u_2{T}_Z+u_2{T}_{{\Theta }_1}+u_2{T}_{{\Theta }_2}+u_2\left(T_{{\Theta }_3}{T}_{Z_1}+T_{{\Theta }_3}{T}_{Z_2}\right)(s_l-s_0)\right]({\vartheta }_l-{\vartheta }_0)\right]\hfill \\ & & \times \left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right).\hfill \end{array}$$

Hence

$$∥{\Im }_{\rho }^{″}(\nu ,\xi )\left(\vartheta \right)-{\Im }_{\rho }^{″}(\tilde{\nu },\tilde{\xi })\left(\vartheta \right)∥\le J_6\left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right).$$

Analogously, we can write

$$∥{\Im }_{\rho }^{″}(\nu ,\xi )\left(\vartheta \right)-{\Im }_{\rho }^{″}(\tilde{\nu },\tilde{\xi })\left(\vartheta \right)∥\le J_6\left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right).$$

Therefore, the operator ${\Im }_{\rho }^{″}$ is strictly contractive. As a result, using the Banach FP theorem, ${\Im }_{\rho }^{″}$ has only one FP, that is the MIS (3) has a US. Moreover, we deduce from Lemma 3 that $\nu \left(\vartheta \right)$ and $\xi \left(\vartheta \right)$ satisfy the conditions $\nu \left(A\right)={\nu }_A$ and $\xi \left(A\right)={\xi }_A$. As a result, the controllability exists for the MIS (3). □

Theorem 10.

If the assertions (1), (3), (4) and (5) hold, then the MIS (4) is controllable on τ provided that

$$max\left\{J_i\right\}<1,i=5,7.$$

Proof. 

Assume that ${\mho }^{″}\subseteq Q$ where ${\mho }^{″}=\left\{\left(K_1,K_2,K_3,K_4\right)\in Q:∥K_1,K_2,K_3,K_4∥\le {\delta }_2^{″}\right\},$ where ${\delta }_2^{″}=max\{{\delta }^{″},{\delta }_1^{″}\}$ and ${\delta }^{″},{\delta }_1^{″}\in (0,1),$ and also ${\delta }_2^{″}\ge E_9$. Define an operator ${\Im }_{\rho }^{**}:{\mho }^{″}\times {\mho }^{″}\to {\mho }^{″}$ by

$${\Im }_{\rho }^{**}(\nu ,\xi )\left(\vartheta \right)=\left\{\begin{array}{c}{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Xi \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right),\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)H\ell \left(\vartheta \right)\Delta s,\vartheta \in ({\vartheta }_i,s_i]\cap \tau ,i=1,2,\dots ,n,\\ {\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0+\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)\Delta s\\ \times \Xi \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right),\\ +\frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{s_i}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\phi }_i(s,\xi \left(s\right),\nu \left(s\right))\Delta s\\ +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}{\left(\vartheta -s\right)}^{\rho -1}{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)H\ell \left(s\right)\Delta s,\vartheta \in (s_i,{\vartheta }_{i+1}]\cap \tau ,i=1,2,\dots ,n,\end{array}\right.$$

where ${\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right)$ is the matrix form of the MLF. Now, we illustrate that ${\Im }_{\rho }^{**}:{\mho }^{″}\times {\mho }^{″}\to {\mho }^{″}$ is a self-mapping. For $\vartheta \in (s_i,{\vartheta }_{i+1}]\cap \tau ,$$i=1,2,\dots ,n,$ we get

$$\begin{array}{ccc}\hfill ∥{\Im }_{\rho }^{**}(\nu ,\xi )\left(\vartheta \right)∥& \le & \frac{1}{\Gamma \left(\rho \right)}\underset{{\vartheta }_i}{\overset{s_i}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\phi }_i(s,\nu \left(s\right),\xi \left(s\right))∥\Delta s+∥{\Lambda }_{\rho }\left(B{\vartheta }^{\rho }\right){\nu }_0∥\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥\left(∥Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right)\right)∥\right.\hfill \\ & & +\left.∥\Xi \left(s,\nu \left(s\right),\xi \left(s\right),\underset{s_0}{\overset{s_l}{\int }}{Z}_1\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p,\underset{s_0}{\overset{s_l}{\int }}{Z}_2\left(s,p,\nu \left(p\right),\xi \left(p\right)\right)\Delta p\right)∥\right)\Delta s\hfill \\ & & +\underset{{\vartheta }_0}{\overset{{\vartheta }_l}{\int }}∥{\left(\vartheta -s\right)}^{\rho -1}∥∥{\Lambda }_{\rho ,\rho }\left(B\left({\left(\vartheta -s\right)}^{\rho }\right)\right)∥∥H∥∥\ell \left(s\right)∥\Delta s\hfill \\ & \le & \frac{1}{\Gamma \left(\rho \right)}{u}_3{T}_{\phi }\delta \left(s_i-{\vartheta }_i\right)+u_1+u_3\left(u_2{T}_Z{\delta }^{″}+u_2{T}_{{\Xi }_1}{\delta }^{″}+u_2{T}_{{\Xi }_2}{\delta }^{″}\right.\hfill \\ & & \left.+u_2{\delta }^{″}{T}_{{\Theta }_3}{T}_{Z_1}(s_l-s_0)+T_{{\Theta }_4}{T}_{Z_2}{\delta }^{″}(s_l-s_0)+Y_{\Xi }^{*}\right)\left({\vartheta }_l-{\vartheta }_0\right)\hfill \\ & & +T_H{T}_{\Re }^{\rho }\left[\frac{1}{\Gamma \left(\rho \right)}{u}_3{T}_{\phi }\delta \left(s_i-{\vartheta }_i\right)+u_1+u_3\left(u_2{T}_Z{\delta }^{″}+u_2{T}_{{\Xi }_1}{\delta }^{″}+u_2{T}_{{\Xi }_2}{\delta }^{″}\right.\right.\hfill \\ & & \left.+\left.u_2{\delta }^{″}{T}_{{\Theta }_3}{T}_{Z_1}(s_l-s_0)+T_{{\Theta }_4}{T}_{Z_2}{\delta }^{″}(s_l-s_0)+Y_{\Xi }^{*}\right)\left({\vartheta }_l-{\vartheta }_0\right)\right]\hfill \\ & \le & E_9+{\delta }^{″}{J}_5\le {\delta }^{″}+{\delta }^{″}{J}_5={\delta }_1^{″},\hfill \end{array}$$

also, if $E_{10}\le {\delta }^{″},$ we have

$$∥{\Im }_{\rho }^{**}(\xi ,\nu ,)\left(\vartheta \right)∥\le E_{10}+{\delta }^{″}{J}_5\le {\delta }^{″}+{\delta }^{″}{J}_5={\delta }_1^{″}.$$

Hence

$$∥{\Im }_{\rho }^{**}(\nu ,\xi )\left(\vartheta \right)∥\le {\delta }_2^{″}\mathrm{and}∥{\Im }_{\rho }^{**}(\xi ,\nu ,)\left(\vartheta \right)∥\le {\delta }_2^{″}.$$

(36)

It follows from (36) that ${\Im }^{**}\left({\mho }^{″}\times {\mho }^{″}\right)\subseteq {\mho }^{″}.$ Also, for $\vartheta \in ({\vartheta }_i,{\vartheta }_{i+1}],$$i=1,2,\dots ,n,$ with ${\nu }_0={\tilde{\nu }}_0$ and ${\xi }_0={\tilde{\xi }}_0,$ one has

$$\begin{array}{ccc}& & ∥{\Im }_{\rho }^{**}(\nu ,\xi )\left(\vartheta \right)-{\Im }_{\rho }^{**}(\tilde{\nu },\tilde{\xi })\left(\vartheta \right)∥\hfill \\ & \le & \left(\frac{1}{\Gamma \left(\rho \right)}{u}_3{T}_{\phi }(s_i-{\vartheta }_i)+u_3\left[u_2{T}_Z+u_2{T}_{{\Xi }_1}+u_2{T}_{{\Xi }_2}+u_2\left(T_{{\Xi }_3}{T}_{Z_1}+T_{{\Xi }_3}{T}_{Z_2}\right)(s_l-s_0)\right]({\vartheta }_l-{\vartheta }_0)\right)\hfill \\ & & \times \left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right)\hfill \\ & & +T_H{T}_{\Re }^{\rho }(\vartheta -{\vartheta }_l)\left[\frac{1}{\Gamma \left(\rho \right)}{u}_3{T}_{\phi }(s_i-{\vartheta }_i)+u_3\left[\begin{array}{c}{u}_2{T}_Z+u_2{T}_{{\Xi }_1}+u_2{T}_{{\Xi }_2}\\ +u_2\left(T_{{\Xi }_3}{T}_{Z_1}+T_{{\Xi }_3}{T}_{Z_2}\right)(s_l-s_0)\end{array}\right]({\vartheta }_l-{\vartheta }_0)\right]\hfill \\ & & \times \left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right)\hfill \\ & \le & (1+T_H{T}_{\Re }^{\rho }(\vartheta -{\vartheta }_l))\hfill \\ & & \times \left[\frac{1}{\Gamma \left(\rho \right)}{u}_3{T}_{\phi }(s_i-{\vartheta }_i)+u_3\left[u_2{T}_Z+u_2{T}_{{\Xi }_1}+u_2{T}_{{\Xi }_2}+u_2\left(T_{{\Xi }_3}{T}_{Z_1}+T_{{\Xi }_3}{T}_{Z_2}\right)(s_l-s_0)\right]({\vartheta }_l-{\vartheta }_0)\right]\hfill \\ & & \times \left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right).\hfill \end{array}$$

Hence

$$∥{\Im }_{\rho }^{**}(\nu ,\xi )\left(\vartheta \right)-{\Im }_{\rho }^{**}(\tilde{\nu },\tilde{\xi })\left(\vartheta \right)∥\le J_7\left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right).$$

Similarly, we can obtain

$$∥{\Im }_{\rho }^{**}(\xi ,\nu )\left(\vartheta \right)-{\Im }_{\rho }^{**}(\tilde{\xi },\tilde{\nu })\left(\vartheta \right)∥\le J_7\left({∥\nu -\tilde{\nu }∥}_{\infty }+{∥\xi -\tilde{\xi }∥}_{\infty }\right).$$

Therefore, the operator ${\Im }_{\rho }^{**}$ is strictly contractive. Thus, from the Banach FP theorem, ${\Im }_{\rho }^{**}$ has a unique FP, which is a US to the MIS (4). In addition, we deduce from Lemma 4 that $\nu \left(\vartheta \right)$ and $\xi \left(\vartheta \right)$ satisfy the stipulations $\nu \left(A\right)={\nu }_A$ and $\xi \left(A\right)={\xi }_A$. This proves that, the MIS (4) is controllable. □

6. Supportive Example

This part is devoted to analyzing the findings from the earlier steps.

Example 1.

Consider the MIS below:

$$\left\{\begin{array}{c}{}^{c,\tau }{D}^{\rho }\nu \left(\vartheta \right)=\frac{3}{\vartheta -1}\nu \left(\vartheta \right)+\frac{3}{\vartheta -1.3}\varphi \left(\vartheta \right)+e_u\left(\vartheta ,\psi \left(\nu \right(\vartheta \left)\right),\psi \left(\xi \right(\vartheta \left)\right)\right)+{\int }_0^{\vartheta }{\Lambda }_{\rho ,\gamma }\left(\nu ,\xi \right)\Delta s+\varphi \left(\vartheta \right),\\ {}^{c,\tau }{D}^{\rho }\xi \left(\vartheta \right)=\frac{3}{\vartheta -1}\xi \left(\vartheta \right)+\frac{3}{\vartheta -1.3}\varphi \left(\vartheta \right)+e_u\left(\vartheta ,\psi \left(\xi \right(\vartheta \left)\right),\psi \left(\nu \right(\vartheta \left)\right)\right)+{\int }_0^{\vartheta }{\Lambda }_{\rho ,\gamma }\left(\xi ,\nu \right)\Delta s+\varphi \left(\vartheta \right),\\ \nu \left(0\right)=1,\xi \left(0\right)=1,\vartheta ={[0,4]}_{\tau }\backslash \{1,1.3\},\\ \nu \left({\vartheta }_r\right)=O_r\left(\nu \left({\vartheta }_r\right),\xi \left({\vartheta }_r\right)\right)+D_r\left({\vartheta }_r^{-},\nu \left({\vartheta }_r^{-}\right),\xi \left({\vartheta }_r^{-}\right),\varphi \left({\vartheta }_r^{-}\right)\right),r=1,2,\\ \xi \left({\vartheta }_r\right)=O_r\left(\xi \left({\vartheta }_r^{-}\right),\nu \left({\vartheta }_r^{-}\right)\right)+D_r\left({\vartheta }_r^{-},\xi \left({\vartheta }_r^{-}\right),\nu \left({\vartheta }_r^{-}\right),\varphi \left({\vartheta }_r^{-}\right)\right),r=1,2.\end{array}\right.$$

(37)

It is equivalent to

$$\left\{\begin{array}{c}\left|{}^{c,\tau }{D}^{\rho }\nu \left(\vartheta \right)-\frac{3}{\vartheta -1}\nu \left(\vartheta \right)-\frac{3}{\vartheta -1.3}\varphi \left(\vartheta \right)-e_u\left(\vartheta ,\psi \left(\nu \right(\vartheta \left)\right),\psi \left(\xi \right(\vartheta \left)\right)\right)-{\int }_0^{\vartheta }{\Lambda }_{\rho ,\gamma }\left(\nu ,\xi \right)\Delta s+\varphi \left(\vartheta \right)\right|\le 1,\\ \left|{}^{c,\tau }{D}^{\rho }\xi \left(\vartheta \right)-\frac{3}{\vartheta -1}\xi \left(\vartheta \right)+\frac{3}{\vartheta -1.3}\varphi \left(\vartheta \right)+e_u\left(\vartheta ,\psi \left(\xi \right(\vartheta \left)\right),\psi \left(\nu \right(\vartheta \left)\right)\right)+{\int }_0^{\vartheta }{\Lambda }_{\rho ,\gamma }\left(\xi ,\nu \right)\Delta s+\varphi \left(\vartheta \right)\right|\le 1,\\ \vartheta ={[0,4]}_{\tau }\backslash \{1,1.3\},\\ \left|\nu \left({\vartheta }_r\right)-O_r\left(\nu \left({\vartheta }_r\right),\xi \left({\vartheta }_r\right)\right)-D_r\left({\vartheta }_r^{-},\nu \left({\vartheta }_r^{-}\right),\xi \left({\vartheta }_r^{-}\right),\varphi \left({\vartheta }_r^{-}\right)\right)\right|\le 1,r=1,2,\\ \left|\xi \left({\vartheta }_r\right)-O_r\left(\xi \left({\vartheta }_r^{-}\right),\nu \left({\vartheta }_r^{-}\right)\right)-D_r\left({\vartheta }_r^{-},\xi \left({\vartheta }_r^{-}\right),\nu \left({\vartheta }_r^{-}\right),\varphi \left({\vartheta }_r^{-}\right)\right)\right|\le 1,r=1,2.\end{array}\right.$$

(38)

We put ${\tau }^r={[0,4]}_{\tau }\backslash \{1,1.3\},$${\vartheta }_1=1,$${\vartheta }_2=1.3,$$K_1\left(\vartheta \right)=\frac{3}{\vartheta -1},$$K_2\left(\vartheta \right)=\frac{3}{\vartheta -1.3},$${\Lambda }_{\rho ,\gamma }\left(\nu ,\xi \right)={\displaystyle \sum _{r=0}^{\infty }}\frac{{\nu }^r{\xi }^r}{\Gamma \left(r\rho +\gamma \right)},$ for $\rho ,\gamma >0.$ Also, we take

$$Z\left(\vartheta ,\nu \left(\vartheta \right),\xi \left(\vartheta \right),S_{\nu ,\xi }\left(\vartheta \right),\varphi \left(\vartheta \right)\right)=e_u\left(\vartheta ,\psi \left(\nu \right(\vartheta \left)\right),\psi \left(\xi \right(\vartheta \left)\right)\right)+{\int }_0^t{\Lambda }_{\rho ,\gamma }\left(\nu ,\xi \right)\Delta s+\varphi \left(\vartheta \right),$$

where $\varphi \left(\vartheta \right)$ is a control map for $\vartheta \in {\tau }^r$ and $S_{\nu ,\xi }\left(\vartheta \right)={\int }_0^t{\Lambda }_{\rho ,\gamma }\left(\nu ,\xi \right)\Delta s$ and put $\epsilon =1.$ Assume that $\widehat{\nu },\widehat{\xi }\in Q^1\left([0,3],\mathbb{R}\right)$ satisfy (38), then there is $\phi \in Q^1\left({[0,3]}_{{\tau }^r},\mathbb{R}\right)$ with ${\phi }_0\in \mathbb{R}$ so that $\left|\phi (\vartheta ,\nu ,\xi )\right|\le 1,$ for all $\vartheta \in {\tau }^r$ and $\left|{\phi }_0\right|\le 1.$ So, (38) can be written as

$$\left\{\begin{array}{c}{}^{c,\tau }{D}^{\rho }\widehat{\nu }\left(\vartheta \right)=\frac{3}{\vartheta -1}\widehat{\nu }\left(\vartheta \right)+\frac{3}{\vartheta -1.3}\varphi \left(\vartheta \right)+e_u\left(\vartheta ,\psi \left(\widehat{\nu }\left(\vartheta \right)\right),\psi \left(\widehat{\xi }\left(\vartheta \right)\right)\right)+{\int }_0^{\vartheta }{\Lambda }_{\rho ,\gamma }\left(\widehat{\nu },\widehat{\xi }\right)\Delta s+\varphi \left(\vartheta \right)+\phi (\vartheta ,\widehat{\nu },\widehat{\xi }),\vartheta \in {\tau }^r,\\ {}^{c,\tau }{D}^{\rho }\widehat{\xi }\left(\vartheta \right)=\frac{3}{\vartheta -1}\widehat{\xi }\left(\vartheta \right)+\frac{3}{\vartheta -1.3}\varphi \left(\vartheta \right)+e_u\left(\vartheta ,\psi \left(\widehat{\xi }\left(\vartheta \right)\right),\psi \left(\widehat{\nu }\left(\vartheta \right)\right)\right)+{\int }_0^{\vartheta }{\Lambda }_{\rho ,\gamma }\left(\widehat{\xi },\widehat{\nu }\right)\Delta s+\varphi \left(\vartheta \right)+\phi (\vartheta ,\widehat{\xi },\widehat{\nu }),\vartheta \in {\tau }^r,\\ \widehat{\nu }\left({\vartheta }_r\right)=O_r\left(\widehat{\nu }\left({\vartheta }_r\right),\widehat{\xi }\left({\vartheta }_r\right)\right)+D_r\left({\vartheta }_r^{-},\widehat{\nu }\left({\vartheta }_r^{-}\right),\widehat{\xi }\left({\vartheta }_r^{-}\right),\varphi \left({\vartheta }_r^{-}\right)\right)+{\phi }_0\left(\widehat{\vartheta },\widehat{\nu },\xi \right),r=1,2,\\ \widehat{\xi }\left({\vartheta }_r\right)=O_r\left(\widehat{\xi }\left({\vartheta }_r^{-}\right),\widehat{\nu }\left({\vartheta }_r^{-}\right)\right)+D_r\left({\vartheta }_r^{-},\widehat{\xi }\left({\vartheta }_r^{-}\right),\widehat{\nu }\left({\vartheta }_r^{-}\right),\varphi \left({\vartheta }_r^{-}\right)\right)+{\phi }_0\left(\vartheta ,\widehat{\xi },\widehat{\nu }\right),r=1,2.\end{array}\right.$$

So, (37) has the solution

$$\begin{array}{c}\nu \left(\vartheta \right)=O_1\left(\nu \left({\vartheta }_1\right),\xi \left({\vartheta }_1\right)\right)+O_2\left(\nu \left({\vartheta }_2\right),\xi \left({\vartheta }_2\right)\right)+D_1\left({\vartheta }_1^{-},\nu \left({\vartheta }_1^{-}\right),\xi \left({\vartheta }_1^{-}\right),\varphi \left({\vartheta }_1^{-}\right)\right)+D_2\left({\vartheta }_2^{-},\nu \left({\vartheta }_2^{-}\right),\xi \left({\vartheta }_2^{-}\right),\varphi \left({\vartheta }_2^{-}\right)\right)\\ +{\int }_0^t{e}_u\left(\vartheta ,\psi \left(\vartheta \right)\right)\left[e_u\left(\vartheta ,\psi \left(\nu \right(\vartheta \left)\right),\psi \left(\xi \right(\vartheta \left)\right)\right)+{\int }_0^s{\Lambda }_{\rho ,\gamma }\left(\nu ,\xi \right)\Delta p+\varphi \left(s\right)\right]\Delta s,\\ \xi \left(\vartheta \right)=O_1\left(\xi \left({\vartheta }_1\right),\nu \left({\vartheta }_1\right)\right)+O_2\left(\xi \left({\vartheta }_2\right),\nu \left({\vartheta }_2\right)\right)+D_1\left({\vartheta }_1^{-},\xi \left({\vartheta }_1^{-}\right),\nu \left({\vartheta }_1^{-}\right),\varphi \left({\vartheta }_1^{-}\right)\right)+D_2\left({\vartheta }_2^{-},\xi \left({\vartheta }_2^{-}\right),\nu \left({\vartheta }_2^{-}\right),\varphi \left({\vartheta }_2^{-}\right)\right)\\ +{\int }_0^t{e}_u\left(\vartheta ,\psi \left(\vartheta \right)\right)\left[e_u\left(\vartheta ,\psi \left(\xi \right(\vartheta \left)\right),\psi \left(\nu \right(\vartheta \left)\right)\right)+{\int }_0^s{\Lambda }_{\rho ,\gamma }\left(\xi ,\nu \right)\Delta p+\varphi \left(s\right)\right]\Delta s.\end{array}$$

Therefore, the MIS (37) has just one solution on $Q^1\left({[0,3]}_{\tau },\mathbb{R}\right)$, according to our obtained results, and the system (37) is UH stable on ${\tau }^r$.

7. Conclusions and Future Works

Over the past year, impulsive fractional differential equations have advanced quickly. Particularly for those issues that are subject to abrupt changes and discontinuous jumps and cannot be described by integer-order differential equations, it plays a significant role in providing a natural framework for modeling various real processes arising in phenomena discussed in various scientific fields. Numerous mathematicians have recently demonstrated an interest in the qualitative theory of impulsive differential equations. In a connected fractional dynamic system with initial boundary and impulsive conditions on time scales, we have successfully demonstrated its existence, uniqueness, UH stability and controllability criterion. For the existence of at least one solution, the Leray-Schauder alternative type FP theorem was applied. The unique solution was looked into while using the Banach contraction theorem. In addition, an example was given to demonstrate the results drawn from the analysis. For future projects, the main aim of the authors is that these qualitative specifications can be checked and established on some real-world impulsive systems arising in mathematical models of brain.