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Ulam–Hyers stability of impulsive integrodifferential equations with Riemann–Liouville boundary conditions

Abstract

This paper is concerned with a class of impulsive implicit fractional integrodifferential equations having the boundary value problem with mixed Riemann–Liouville fractional integral boundary conditions. We establish some existence and uniqueness results for the given problem by applying the tools of fixed point theory. Furthermore, we investigate different kinds of stability such as Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and generalized Ulam–Hyers–Rassias stability. Finally, we give two examples to demonstrate the validity of main results.

Introduction

During the last few decades, boundary value problems of fractional differential equations have been utilized in different problems of applied nature; for example, we can find it in analytical formulations of systems and processes. Due to a more accurate behavior of fractional differential equations, it got the interest of research community in various applied fields of sciences such as chemistry, engineering, mechanics, physics, and so on. For the readers’ convenience, we refer to the monographs [9, 11, 15, 23] and their references. Also, an experimental study was presented in [21].

For boundary value problems of fractional differential equations, the existence of solutions is an important and basic requirement. Furthermore, the uniqueness of solutions is the next important feature for more specific behavior of solutions. In the literature, many results are available about these two necessary properties of solutions; see, for example [2, 7, 8, 20, 22, 27]. Integral boundary conditions are very important in the solutions of many practical systems [1, 51].

The impulsive phenomena and their models are investigated and analyzed in different practical problems. The theory of impulsive mathematical models based on fractional differential equations has very significant applications in many applied problems in natural sciences and engineering. Many evolutionary processes that possess abrupt changes at certain moments can be described with the help of aforesaid models. The abrupt changes in evolutionary processes can be of two types. The first one, characterized by short-term perturbations with negligible duration in comparison with the duration of the whole processes, is called instantaneous impulses. The second one is characterized by abrupt changes that remain active for a finite interval of time is called noninstantaneous impulses. Many evolutionary processes can be modeled using noninstantaneous impulses such as the flow of drugs in blood streams (hemodynamic equilibrium of a person), decompensation, and many others. In this context, impulsive fractional differential equations are studied in different aspects; see, for example [13, 14, 17, 24, 26, 30, 32, 34, 41, 49].

Stability analysis, which has been solely studied for differential equations of arbitrary order and abundantly discussed by the researchers, is the theory related to the stability of differential equations. In stability theory, the Ulam stability was first established by Ulam [35] in 1940 and then was extended by Hyers and Rassias [12, 25]. More recent results on the so-called Hyers–Ulam stability have relaxed the stability conditions. Many mathematicians extended the Hyers results in different directions [4, 18, 19, 2831, 33, 36, 37, 39, 4145, 4749]. The monographs [5, 6, 16, 38] treated fractional differential equations with instantaneous impulses of the following form:

$$ \textstyle\begin{cases} ^{c}\mathcal{D}^{r}v(\tau )=u(\tau,v(\tau )),\quad \tau \in [0, \mathrm{T}], \mathrm{T}>0, \tau \neq \tau _{k}, k=1,2,\dots,m, \\ \Delta v(\tau )=\varUpsilon _{k}(v(\tau _{k}^{-})), \quad k=1,2,\dots,m, \end{cases} $$

where \({}^{c}\mathcal{D}^{r}\) is the Caputo fractional derivative of order \(r\in (n-1,n), n\) is any natural number with lower bound 0, \(u:[0,\mathrm{T}]\times \mathbb{R}\rightarrow \mathbb{R}\) is continuous, \(\varUpsilon _{k}:\mathbb{R}\rightarrow \mathbb{R}\) is instantaneous impulse, and \(\tau _{k}\) satisfies \(0=\tau _{0}<\tau _{1}<\cdots <\tau _{m}=\mathrm{T}\), \(v(\tau _{k}^{+})= \lim_{\epsilon \rightarrow 0}v( \tau _{k}+\epsilon )\) and \(v(\tau _{k}^{-})= \lim_{\epsilon \rightarrow 0}v(\tau _{k}+\epsilon )\) denotes the right and left limits of \(v(\tau )\) at \(\tau =\tau _{k}\), respectively.

Ahmad et al. [3] studied an implicit type of nonlinear impulsive fractional differential equations given by

$$ \textstyle\begin{cases} ^{c}\mathcal{D}^{r}y(\tau )=f(\tau,y(\tau ),{}^{c}\mathcal{D}^{r}y( \tau )), \quad \tau \in [0,1], \tau \neq \tau _{k}, k=1,2,\dots,m, \\ \Delta y(\tau )=\varUpsilon _{k}(y(\tau _{k})), \qquad \Delta y'(\tau )= \hat{\varUpsilon }_{k}(y(\tau _{k})), \quad k=1,2,\dots,m, \\ y(0)=g(y),\qquad y(1)=h(y), \end{cases} $$

where \({}^{c}\mathcal{D}^{r}\) is Caputo fractional derivative of order \(1< r\leq 2, f:[0,1]\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}\) and \(\varUpsilon _{k}, \hat{\varUpsilon }_{k}:\mathbb{R}\rightarrow \mathbb{R}\) are continuous functions, and

$$\begin{aligned} &\Delta y(\tau _{k}) = y\bigl(\tau _{k}^{+} \bigr)-y\bigl(\tau _{k}^{-}\bigr), \\ &\Delta y'(\tau _{k}) = y'\bigl(\tau _{k}^{+}\bigr)-y'\bigl(\tau _{k}^{-}\bigr), \end{aligned}$$

where \((\tau _{k}^{+}), y'(\tau _{k}^{+}), y(\tau _{k}^{-}), y'( \tau _{k}^{-})\) are the respective left and right limits of \(y(\tau _{k})\) at \(\tau =\tau _{k}\).

Recently, Wang et al. [39] studied the existence, uniqueness, and different kinds of stability in the sense of Ulam for the following nonlinear implicit fractional integrodifferential equation of the form

$$ \textstyle\begin{cases} ^{c}\mathcal{D}^{p}u(\tau )=\alpha (\tau,u(\tau ),{}^{c}\mathcal{D} ^{p}u(\tau )) +\frac{1}{\varGamma (\delta )}\int _{0}^{\tau }(\tau -s)^{ \sigma -1}g(s,u(s),{}^{c}\mathcal{D}^{p}u(s))\,ds,\\ \quad \tau \in \mathcal{J}, \\ u(\tau )|_{\tau =0}=-u(\tau )|_{\tau =T},\qquad {}^{c}\mathcal{D}^{r}u( \tau )|_{\tau =0}=-^{c}\mathcal{D}^{r}u(\tau )|_{\tau = \mathrm{T}}, \end{cases} $$
(1.1)

where \({}^{c}\mathcal{D}^{p}\) and \({}^{c}\mathcal{D}^{r}\) is the Caputo fractional derivatives of orders \(1< p\leq 2\) and \(0\leq r\leq 2\), \(\mathcal{J}=[0,\mathrm{T}]\) with \(\mathrm{T},\sigma \), \(\delta >0\), and the functions α, \(g:\mathcal{J}\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}\) are continuous. Also, they performed the same analysis for the proposed implicit coupled system:

$$ \textstyle\begin{cases} ^{c}\mathcal{D}^{p}u(\tau )-\alpha (\tau,y(\tau ),{}^{c}\mathcal{D} ^{p}u(\tau ))-\frac{1}{\varGamma (\delta )} \int _{0}^{\tau }(\tau -s)^{ \sigma -1}g(s,y(s),{}^{c}\mathcal{D}^{p}u(s))\,ds=0,\\\quad \tau \in \mathcal{J}, \\ ^{c}\mathcal{D}^{q}y(\tau )-\chi (\tau,u(\tau ),{}^{c}\mathcal{D}^{q}y( \tau ))-\frac{1}{\varGamma (\delta )} \int _{0}^{\tau }(\tau -s)^{\sigma -1}f(s,u(s),{}^{c}\mathcal{D}^{p}y(s))\,ds=0,\\ \quad \tau \in \mathcal{J}, \\ u(\tau )|_{\tau =0}=-u(\tau )|_{\tau =\mathrm{T}},\qquad {}^{c} \mathcal{D}^{r}u(\tau )|_{\tau =0}=-^{c}\mathcal{D}^{r}u(\tau ) |_{\tau =\mathrm{T}}, \\ y(\tau )|_{\tau =0}=-y(\tau )|_{\tau =\mathrm{T}},\qquad {}^{c} \mathcal{D}^{\omega }y(\tau )|_{\tau =0}=-^{c}\mathcal{D}^{\omega }y(\tau )|_{\tau =\mathrm{T}}, \end{cases} $$
(1.2)

where \({}^{c}\mathcal{D}^{p}\), \({}^{c}\mathcal{D}^{r}\), \({}^{c}\mathcal{D} ^{q}\), and \({}^{c}\mathcal{D}^{\omega }\) are the Caputo fractional derivatives of orders \(1< p\), \(q\leq 2\) and \(0\leq r\), \(\omega \leq 2\), σ, \(\delta >0\), \(\mathcal{J}=[0,\mathrm{T}]\), \(\mathrm{T}>0\), and the functions α, χ, \(g, f:\mathcal{J}\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}\) are continuous.

In the present study, we extend models (1.1) and (1.2) to impulsive systems with Riemann–Liouville boundary conditions instead of antiperiodic boundary condition. More precisely, we study the model

$$ \textstyle\begin{cases} ^{c}\mathcal{D}^{r}\omega (\tau )=\mathcal{A}(\tau,\omega (\tau ),{}^{c} \mathcal{D}^{r}\omega (\tau )) +\int _{0}^{\tau }\frac{(\tau -s)^{ \sigma -1}}{\varGamma (\delta )} \mathcal{B}(s,\omega (s),{}^{c} \mathcal{D}^{r}\omega (s))\,ds, \\ \quad\text{where } \tau \in \mathcal{J}, \tau \neq \tau _{i}, i=1,2, \dots,m, \\ \Delta \omega (\tau _{i})=\varUpsilon _{i}(\omega (\tau _{i})), \qquad \Delta \omega '(\tau _{i})=\hat{\varUpsilon _{i}}(\omega (\tau _{i})),\quad i=1,2,\dots,m, \\ \eta _{1}\omega (0)+\xi _{1}I^{r}\omega (0)=\nu _{1}, \qquad \eta _{2}\omega ( \mathrm{T})+\xi _{2}I^{r}\omega (\mathrm{T})=\nu _{2}, \end{cases} $$
(1.3)

where \({}^{c}\mathcal{D}^{r}\) is the Caputo fractional derivative with \(1< r\leq 2, \mathcal{J}=[0,\mathrm{T}]\text{ with }\mathrm{T}>0\), and \(\sigma, \delta >0\), the functions \(\mathcal{A}, \mathcal{B}: \mathcal{J}\times \mathbb{R}^{2}\rightarrow \mathbb{R}\) are continuous, and \(\eta _{1}, \eta _{2}, \xi _{1}, \xi _{2}\) are positive constants.

The first results of this paper establish the existence and uniqueness of solution for this problem. Also, we investigate the following implicit coupled system:

$$ \textstyle\begin{cases} ^{c}\mathcal{D}^{r}\omega (\tau )=\mathcal{A}(\tau,y(\tau ),{}^{c} \mathcal{D}^{r}\omega (\tau )) +\int _{0}^{\tau }\frac{(\tau -s)^{ \sigma -1}}{\varGamma (\delta )}\mathcal{B}(s,y(s),{}^{c}\mathcal{D}^{r} \omega (s))\,ds, \\ \quad \text{where } \tau \in \mathcal{J}, \tau \neq \tau _{i}, i=1,2, \dots,m, \\ ^{c}\mathcal{D}^{p}y(\tau )=\mathcal{A}'(\tau,\omega (\tau ),{}^{c} \mathcal{D}^{p}y(\tau )) +\int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{ \varGamma (\delta )}\mathcal{B}'(s,\omega (s),{}^{c}\mathcal{D}^{p}y(s))\,ds, \\ \quad \text{where } \tau \in \mathcal{J}, \tau \neq \tau _{j}, j=1,2, \dots,n, \\ \Delta \omega (\tau _{i})=\varUpsilon _{i}(\omega (\tau _{i})), \qquad \Delta \omega '(\tau _{i})=\hat{\varUpsilon _{i}}(\omega (\tau _{i})),\quad i=1,2,\dots,m, \\ \Delta y(\tau _{j})=\varUpsilon _{j}(y(\tau _{j})),\qquad \Delta y'(\tau _{j})= \hat{\varUpsilon _{j}}(y(\tau _{j})),\quad j=1,2,\dots,n, \\ \eta _{1}\omega (0)+\xi _{1}I^{r}\omega (0)=\nu _{1},\qquad \eta _{2}\omega ( \mathrm{T})+\xi _{2}I^{r}\omega (\mathrm{T})=\nu _{2}, \\ \eta _{3}y(0)+\xi _{3}I^{p}y(0)=\nu _{3}, \qquad \eta _{4}y(\mathrm{T})+\xi _{4}I^{p}y(\mathrm{T})=\nu _{4}, \end{cases} $$
(1.4)

where \({}^{c}\mathcal{D}^{r}\) and \({}^{c}\mathcal{D}^{p}\) are the Caputo fractional derivatives with \(1< r, p\leq 2, \mathcal{J}=[0, \mathrm{T}]\) with \(\mathrm{T}>0\), \(\sigma, \delta >0\), the functions \(\mathcal{A}, \mathcal{A}', \mathcal{B}, \mathcal{B}': \mathcal{J}\times \mathbb{R}^{2}\rightarrow \mathbb{R}\) are continuous, and \(\eta _{1}, \eta _{2}, \eta _{3}, \eta _{4}, \xi _{1}, \xi _{2}, \xi _{3}, \xi _{4}\) are positive constants. Coupled systems of fractional integrodifferential equations have also been extensively studied due to their applications. Some recent works dealing with coupled systems of Caputo fractional differential equations involving different kinds of integral boundary conditions can be found in [50].

The second main results are devoted to the study of stability results for both systems. There are two main classes of stability results considered here, Ulam–Hyers and Ulam–Hyers–Rassias stability, and their generalized equivalents. To be more specific, our aim is to build connections between stability results in both systems.

It is important to note that problem (1.3) and the coupled one (1.4) considered in this paper extend the study of fractional integrodifferential systems, and from this point of view, we believe that the obtained results will contribute to the existing literature on the topic.

The rest of the paper is organized as follows: In Sect. 2, we first establish an equivalent integral equation for the fractional integrodifferential equations with impulse, and we obtain existence results by using the Banach contraction principle, Schauder’s fixed point theorem, and Krasnoselskii’s fixed point theorem to the proposed problems (1.3) and (1.4), respectively. In Sect. 3, we consider four types of Ulam–Hyers stability concepts. Finally, in Sect. 4, we construct two examples to illustrate the obtained results. Fundamental definitions, essential lemmas, and the proofs of the main theorems are given in Appendices 1, 2, and 3.

Notation: We denote by \(\mathcal{M}\) the space of all piecewise continuous functions \(\mathrm{PC}(\mathcal{J},\mathbb{R})\); \(\mathcal{J}=\mathcal{J}_{0}\cup \mathcal{J}_{1}\cup \mathcal{J}_{2} \cup \cdots \cup \mathcal{J}_{i}\), where \(\mathcal{J}_{0}=[0,\tau _{1}], \mathcal{J}_{1}=(\tau _{1},\tau _{2}], \mathcal{J}_{2}=(\tau _{2}, \tau _{3}],\dots,\mathcal{J}_{i}=(\tau _{i},\tau _{i+1}], i=1,2, \dots,m\), and \(\mathcal{J'}=\mathcal{J}-\{\tau _{1},\tau _{2},\tau _{3}, \dots,\tau _{i}\}\).

We define \(\mathcal{M}=\{\omega:\mathcal{J}\rightarrow \mathbb{R}: \omega \in C(\mathcal{J}_{i},\mathbb{R})\text{ and }\omega (\tau _{i}^{+}), \omega (\tau _{i}^{-})\text{ exist such that }\Delta \omega (\tau _{i})=\omega (\tau _{i}^{+})-\omega (\tau _{i}^{-})\text{ for }i=1,2,\dots,m\}\).

Existence and uniqueness

The aim of this section is giving conditions under which the fractional integrodifferential equation (1.3) and coupled system (1.4) provide existence and uniqueness results.

Existence and uniqueness solution for system (1.3)

Our first result is stated as follows.

Theorem 2.1

Let\(1< r\leq 2\), and let\(\alpha \in \mathcal{M}\)be a continuous function. Then a function\(\omega \in \mathcal{M}\)is solution to the problem

$$ \textstyle\begin{cases} ^{c}\mathcal{D}^{r}\omega (\tau )=\alpha (\tau ),\quad \tau \in \mathcal{J}, \tau \neq \tau _{i}, i=1,2,\dots,m, \\ \Delta \omega (\tau _{i})=\varUpsilon _{i}(\omega (\tau _{i})), \qquad \Delta \omega '(\tau _{i})=\hat{\varUpsilon _{i}}(\omega (\tau _{i})),\quad i=1,2,\dots,m, \\ \eta _{1}\omega (0)+\xi _{1}I^{r}\omega (0)=\nu _{1}, \qquad \eta _{2}\omega ( \mathrm{T})+\xi _{2}I^{r}\omega (\mathrm{T})=\nu _{2}, \end{cases} $$
(2.1)

where

$$ \alpha (\tau )=\mathcal{A}\bigl(\tau,\omega (\tau ),{}^{c} \mathcal{D}^{r} \omega (\tau )\bigr) + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B}\bigl(s, \omega (s),{}^{c}\mathcal{D}^{r}\omega (s)\bigr)\,ds, $$

if and only ifωsatisfies

$$ \omega (\tau )=\textstyle\begin{cases} \frac{1}{\varGamma (r)}\int _{0}^{\tau }(\tau -s)^{r-1}\alpha (s)\,ds -\frac{ \tau }{\eta _{2}\mathrm{T}}\frac{\xi _{2}}{\varGamma (r)}\int _{0}^{ \mathrm{T}}(\mathrm{T}-s)^{r-1}\omega (s)\,ds \\\quad{}+\frac{\nu _{1}}{\eta _{1}}-\frac{ \tau \nu _{1}}{\mathrm{T}\eta _{1}}+\frac{\tau \nu _{2}}{\mathrm{T}\eta _{2}} \\ \quad{}-\frac{\tau }{\mathrm{T}} [\frac{1}{\varGamma (r)}\int _{\tau _{1}} ^{\mathrm{T}}(\mathrm{T}-s)^{r-1}\alpha (s)\,ds +\frac{1}{\varGamma (r)} \int _{0}^{\tau _{1}}(\tau _{1}-s)^{r-1}\alpha (s)\,ds \\ \quad{}+\frac{(\mathrm{T}-\tau _{1})}{\varGamma (r-1)}\int _{0}^{\tau _{1}}(\tau _{1}-s)^{r-2}\alpha (s)\,ds +(\mathrm{T}-\tau _{1})\hat{\varUpsilon _{1}}( \omega (\tau _{1}))+\varUpsilon _{1}(\omega (\tau _{1})) ],\\ \quad \tau \in \mathcal{J}_{0}, \\ \frac{1}{\varGamma (r)}\int _{0}^{\tau }(\tau -s)^{r-1}\alpha (s)\,ds -\frac{ \tau }{\eta _{2}\mathrm{T}}\frac{\xi _{2}}{\varGamma (r)}\int _{0}^{ \mathrm{T}}(\mathrm{T}-s)^{r-1}\omega (s)\,ds \\\quad{}+\frac{\nu _{1}}{\eta _{1}}-\frac{ \tau \nu _{1}}{\mathrm{T}\eta _{1}}+\frac{\tau \nu _{2}}{\mathrm{T}\eta _{2}} \\ \quad{}-\frac{\tau }{\mathrm{T}}\sum_{i=1}^{m} [\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}\alpha (s)\,ds +\frac{1}{ \varGamma (r)}\int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1}\alpha (s)\,ds \\ \quad{}+\frac{(\mathrm{T}-\tau _{i})}{\varGamma (r-1)}\int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-2}\alpha (s)\,ds +(\mathrm{T}-\tau _{i}) \hat{\varUpsilon _{i}}(\omega (\tau _{i}))+\varUpsilon _{i}(\omega (\tau _{i})) ], \\\quad \tau \in \mathcal{J}_{i}, i=1,2,\dots,m. \end{cases} $$
(2.2)

Proof

Applying Lemma A.3 (see Appendix 1) to (2.1) with \(a_{0}, a_{1}\in \mathbb{R}\), we have

$$\begin{aligned} \omega (\tau )=I^{r}\alpha (\tau )-a_{0}-a_{1} \tau = \frac{1}{\varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1}\alpha (s) \,ds-a_{0}-a _{1}\tau, \quad \tau \in [0,\tau _{1}]. \end{aligned}$$
(2.3)

Furthermore, we obtain

$$\begin{aligned} \omega '(\tau )=I^{r-1}\alpha (\tau )-a_{1}= \frac{1}{\varGamma (r-1)} \int _{0}^{\tau }(\tau -s)^{r-2}\alpha (s) \,ds-a_{1}, \quad \tau \in [0,\tau _{1}]. \end{aligned}$$

For \(\tau \in (\tau _{1},\tau _{2}]\), there are \(b_{0}, b_{1}\in \mathbb{R}\) such that

$$\begin{aligned} \textstyle\begin{cases} \omega (\tau )=\frac{1}{\varGamma (r)}\int _{\tau _{1}}^{\tau }(\tau -s)^{r-1} \alpha (s)\,ds-b_{0}-b_{1}(\tau -\tau _{1}), \\ \omega '(\tau )=\frac{1}{\varGamma (r-1)}\int _{\tau _{1}}^{\tau }( \tau -s)^{r-2}\alpha (s)\,ds-b_{1}. \end{cases}\displaystyle \end{aligned}$$

Hence it follows that

$$\begin{aligned} \textstyle\begin{cases} \omega (\tau _{1}^{-})=\frac{1}{\varGamma (r)}\int _{0}^{\tau _{1}}(\tau _{1}-s)^{r-1}\alpha (s)\,ds-a_{0}-a_{1}\tau _{1}, \\ \omega (\tau _{1}^{+})=-b_{0}, \\ \omega '(\tau _{1}^{-})=\frac{1}{\varGamma (r-1)}\int _{0}^{\tau _{1}}( \tau _{1}-s)^{r-2}\alpha (s)\,ds-a_{1}, \\ \omega '(\tau _{1}^{+})=-b_{1}. \end{cases}\displaystyle \end{aligned}$$

Using

$$\begin{aligned} \textstyle\begin{cases} \Delta \omega (\tau _{1})=\omega (\tau _{1}^{+})-\omega (\tau _{1}^{-})= \varUpsilon _{1}(\omega (\tau _{1})), \\ \Delta \omega '(\tau _{1})=\omega '(\tau _{1}^{+})-\omega '(\tau _{1} ^{-})=\hat{\varUpsilon _{1}}(\omega (\tau _{1})), \end{cases}\displaystyle \end{aligned}$$

we obtain

$$\begin{aligned} \textstyle\begin{cases} -b_{0}=\frac{1}{\varGamma (r)}\int _{0}^{\tau _{1}}(\tau _{1}-s)^{r-1} \alpha (s)\,ds-a_{0}-a_{1}\tau _{1} +\varUpsilon _{1}(\omega (\tau _{1})), \\ -b_{1}=\frac{1}{\varGamma (r-1)}\int _{0}^{\tau _{1}}(\tau _{1}-s)^{r-2} \alpha (s)\,ds-a_{1}+\hat{\varUpsilon _{1}}(\omega (\tau _{1})). \end{cases}\displaystyle \end{aligned}$$

Thus

$$\begin{aligned} \omega (\tau ) = {}&\frac{1}{\varGamma (r)} \int _{\tau _{1}}^{\tau }(\tau -s)^{r-1} \alpha (s)\,ds +\frac{1}{\varGamma (r)} \int _{0}^{\tau _{1}}(\tau _{1}-s)^{r-1} \alpha (s)\,ds\\ &{} +\frac{\tau -\tau _{1}}{\varGamma (r-1)} \int _{0}^{\tau _{1}}( \tau _{1}-s)^{r-2} \alpha (s)\,ds \\ &{}+(\tau -\tau _{1})\hat{\varUpsilon }_{1}\bigl(\omega (\tau _{1})\bigr)+\varUpsilon _{1}\bigl(\omega (\tau _{1})\bigr)-a_{0}-a_{1}\tau,\quad \tau \in (\tau _{1},\tau _{2}]. \end{aligned}$$

Similarly, we have

$$\begin{aligned} \omega (\tau ) ={}& \sum_{i=1}^{m} \biggl[\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\tau }(\tau -s)^{r-1}\alpha (s)\,ds +\frac{1}{\varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \alpha (s)\,ds \\ &{}+\frac{\tau -\tau _{i}}{\varGamma (r-1)} \int _{\tau _{i-1}}^{\tau _{i}}( \tau _{i}-s)^{r-2} \alpha (s)\,ds +(\tau -\tau _{i})\hat{\varUpsilon _{i}}\bigl( \omega (\tau _{i})\bigr)+\varUpsilon _{i}\bigl(\omega (\tau _{i})\bigr) \biggr] \\ &{}-a_{0}-a_{1} \tau, \\ &\tau \in (\tau _{i},\tau _{i+1}], i=1,2,\dots,m. \end{aligned}$$
(2.4)

Finally, after applying \(\eta _{1}\omega (0)+\xi _{1}I^{r}\omega (0)=\nu _{1}\) and \(\eta _{2}\omega (\mathrm{T})+\xi _{2}I^{r}\omega (\mathrm{T})= \nu _{2}\) to (2.4) and calculating the values of \(a_{0}\) and \(a_{1}\), we obtain equation (2.2).

Conversely, if \(\omega (\tau )\) is a solution of (2.2), then it is obvious that \({}^{c}\mathcal{D}^{r}\omega (\tau )=\alpha (\tau )\) and \(\eta _{1}\omega (0)+\xi _{1}I^{r}\omega (0)=\nu _{1}, \eta _{2}\omega (\mathrm{T})+\xi _{2}I^{r}\omega (\mathrm{T})=\nu _{2}, \Delta \omega (\tau _{i})=\varUpsilon _{i}(\omega (\tau _{i})), \Delta \omega '(\tau _{i})=\hat{\varUpsilon _{i}}(\omega (\tau _{i}))\), \(i=1,2,\dots,m\). □

Corollary 2.2

In light of Theorem 2.1, problem (1.3) has the solution

$$ \omega (\tau )=\textstyle\begin{cases} \frac{1}{\varGamma (r)}\int _{0}^{\tau }(\tau -s)^{r-1}\alpha (s)\,ds -\frac{ \tau }{\eta _{2}\mathrm{T}}\frac{\xi _{2}}{\varGamma (r)}\int _{0}^{ \mathrm{T}}(\mathrm{T}-s)^{r-1}\omega (s)\,ds +\frac{\nu _{1}}{\eta _{1}}-\frac{ \tau \nu _{1}}{\mathrm{T}\eta _{1}}+\frac{\tau \nu _{2}}{\mathrm{T}\eta _{2}} \\ \quad {}-\frac{\tau }{\mathrm{T}} [\frac{1}{\varGamma (r)}\int _{\tau _{1}} ^{\mathrm{T}}(\mathrm{T}-s)^{r-1}\alpha (s)\,ds +\frac{1}{\varGamma (r)} \int _{0}^{\tau _{1}}(\tau _{1}-s)^{r-1}\alpha (s)\,ds \\ \quad{}+\frac{(\mathrm{T}-\tau _{1})}{\varGamma (r-1)}\int _{0}^{\tau _{1}}(\tau _{1}-s)^{r-2}\alpha (s)\,ds +(\mathrm{T}-\tau _{1})\hat{\varUpsilon _{1}}( \omega (\tau _{1}))+\varUpsilon _{1}(\omega (\tau _{1})) ], \quad \tau \in \mathcal{J}_{0}, \\ \frac{1}{\varGamma (r)}\int _{0}^{\tau }(\tau -s)^{r-1}\alpha (s)\,ds -\frac{ \tau }{\eta _{2}\mathrm{T}}\frac{\xi _{2}}{\varGamma (r)}\int _{0}^{ \mathrm{T}}(\mathrm{T}-s)^{r-1}\omega (s)\,ds +\frac{\nu _{1}}{\eta _{1}}-\frac{ \tau \nu _{1}}{\mathrm{T}\eta _{1}}+\frac{\tau \nu _{2}}{\mathrm{T}\eta _{2}} \\ \quad{}-\frac{\tau }{\mathrm{T}}\sum_{i=1}^{m} [\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}\alpha (s)\,ds +\frac{1}{ \varGamma (r)}\int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1}\alpha (s)\,ds \\ \quad{}+\frac{(\mathrm{T}-\tau _{i})}{\varGamma (r-1)}\int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-2}\alpha (s)\,ds +(\mathrm{T}-\tau _{i}) \hat{\varUpsilon _{i}}(\omega (\tau _{i}))+\varUpsilon _{i}(\omega (\tau _{i})) ], \\\quad \tau \in \mathcal{J}_{i}, i=1,2,\dots,m, \end{cases} $$

where

$$ \alpha (\tau )=\mathcal{A}\bigl(\tau,\omega (\tau ),{}^{c} \mathcal{D}^{r} \omega (\tau )\bigr) + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )}\mathcal{B}\bigl(s, \omega (s),{}^{c}\mathcal{D}^{r}\omega (s)\bigr)\,ds. $$

Let

$$\begin{aligned} v(\tau ) &=\mathcal{A}\bigl(\tau,\omega (\tau ),{}^{c} \mathcal{D}^{r}\omega (\tau )\bigr) + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )}\mathcal{B}\bigl(s, \omega (s),{}^{c}\mathcal{D}^{r}\omega (s)\bigr)\,ds \\ &=\mathcal{A}\bigl(\tau,\omega (\tau ),v(\tau )\bigr) + \int _{0}^{\tau }\frac{( \tau -s)^{\sigma -1}}{\varGamma (\delta )}\mathcal{B}\bigl(s, \omega (s),v(s)\bigr)\,ds. \end{aligned}$$

Also, we consider \(\mathcal{M}=\mathrm{PC}(\mathcal{J},\mathbb{R})\) endowed with the norm

$$ \Vert \omega \Vert _{\mathcal{M}}=\max \bigl\{ \bigl\vert \omega (\tau ) \bigr\vert : \tau \in \mathcal{J}\bigr\} . $$

We can easily see that \({\mathcal{M}}\) is a Banach space. Further, if ω is a solution of problem (1.3), then

$$\begin{aligned} \omega (\tau ) = {}&\frac{1}{\varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1} \alpha (s)\,ds -\frac{\tau }{\eta _{2}\mathrm{T}} \frac{\xi _{2}}{\varGamma (r)} \int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \omega (s)\,ds +\frac{\nu _{1}}{\eta _{1}}-\frac{\tau \nu _{1}}{\mathrm{T} \eta _{1}}+\frac{\tau \nu _{2}}{\mathrm{T}\eta _{2}} \\ &{}-\frac{\tau }{\mathrm{T}}\sum_{i=1}^{m} \biggl[\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}\alpha (s)\,ds +\frac{1}{ \varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \alpha (s)\,ds \\ &{}+\frac{(\mathrm{T}-\tau _{i})}{\varGamma (r-1)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-2} \alpha (s)\,ds +(\mathrm{T}-\tau _{i}) \hat{\varUpsilon _{i}} \bigl(\omega (\tau _{i})\bigr)+\varUpsilon _{i}\bigl(\omega ( \tau _{i})\bigr) \biggr],\\ & \tau \in \mathcal{J}_{i}, i=1,2, \dots,m. \end{aligned}$$

Now, to study (1.3) by fixed point theory, let \(\mathcal{T}: \mathcal{M}\rightarrow \mathcal{M}\) be the operator defined as

$$ \bigl(\mathcal{T}\omega (\tau )\bigr)=\textstyle\begin{cases} \frac{1}{\varGamma (r)}\int _{0}^{\tau }(\tau -s)^{r-1}v(s)\,ds -\frac{ \tau }{\eta _{2}\mathrm{T}}\frac{\xi _{2}}{\varGamma (r)}\int _{0}^{ \mathrm{T}}(\mathrm{T}-s)^{r-1}\omega (s)\,ds \\ \quad{}+\frac{\nu _{1}}{\eta _{1}}-\frac{ \tau \nu _{1}}{\mathrm{T}\eta _{1}}+\frac{\tau \nu _{2}}{\mathrm{T}\eta _{2}} \\ \quad{}-\frac{\tau }{\mathrm{T}} [\frac{1}{\varGamma (r)}\int _{\tau _{1}} ^{\mathrm{T}}(\mathrm{T}-s)^{r-1}v(s)\,ds +\frac{1}{\varGamma (r)}\int _{0} ^{\tau _{1}}(\tau _{1}-s)^{r-1}v(s)\,ds \\ \quad{}+\frac{(\mathrm{T}-\tau _{1})}{\varGamma (r-1)}\int _{0}^{\tau _{1}}(\tau _{1}-s)^{r-2}v(s)\,ds +(\mathrm{T}-\tau _{1})\hat{\varUpsilon _{1}}(\omega (\tau _{1}))+\varUpsilon _{1}(\omega (\tau _{1})) ],\\ \quad \tau \in \mathcal{J}_{0}, \\ \frac{1}{\varGamma (r)}\int _{0}^{\tau }(\tau -s)^{r-1}v(s)\,ds -\frac{ \tau }{\eta _{2}\mathrm{T}}\frac{\xi _{2}}{\varGamma (r)}\int _{0}^{ \mathrm{T}}(\mathrm{T}-s)^{r-1}\omega (s)\,ds \\ \quad{}+\frac{\nu _{1}}{\eta _{1}}-\frac{ \tau \nu _{1}}{\mathrm{T}\eta _{1}}+\frac{\tau \nu _{2}}{\mathrm{T}\eta _{2}} \\ \quad{}-\frac{\tau }{\mathrm{T}}\sum_{i=1}^{m} [\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}v(s)\,ds +\frac{1}{ \varGamma (r)}\int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1}v(s)\,ds \\ \quad{}+\frac{(\mathrm{T}-\tau _{i})}{\varGamma (r-1)}\int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-2}v(s)\,ds +(\mathrm{T}-\tau _{i}) \hat{\varUpsilon _{i}}(\omega (\tau _{i}))+\varUpsilon _{i}(\omega (\tau _{i})) ], \\ \quad \tau \in \mathcal{J}_{i}, i=1,2,\dots,m, \end{cases} $$
(2.5)

where

$$ v(\tau )=\mathcal{A}\bigl(\tau,\omega (\tau ),v(\tau )\bigr) + \int _{0}^{\tau }\frac{( \tau -s)^{\sigma -1}}{\varGamma (\delta )}\mathcal{B}\bigl(s, \omega (s),v(s)\bigr)\,ds. $$

Let us assume the following hypotheses:

  • \([A_{1}]\) There exist constants \(\mathrm{M}_{1}>0\) and \(\mathrm{N} _{1}\in (0,1)\) such that, for all \(\tau \in \mathcal{J}\), \(u, \overline{u}\in \mathcal{M}\), and \(w, \overline{w}\in \mathbb{R}\),

    $$ \bigl\vert \mathcal{A}(\tau,u,w)-\mathcal{A}(\tau,\overline{u},\overline{w}) \bigr\vert \leq \mathrm{M}_{1} \vert u-\overline{u} \vert + \mathrm{N}_{1} \vert w-\overline{w} \vert . $$

    Similarly, there exist constants \(\mathrm{M}_{2}>0\) and \(\mathrm{N} _{2}\in (0,1)\) such that, for all \(\tau \in \mathcal{J}\), \(u, \overline{u}\in \mathcal{M}\), and \(w, \overline{w}\in \mathbb{R}\),

    $$ \bigl\vert \mathcal{B}(\tau,u,w)-\mathcal{B}(\tau,\overline{u},\overline{w}) \bigr\vert \leq \mathrm{M}_{2} \vert u-\overline{u} \vert + \mathrm{N}_{2} \vert w-\overline{w} \vert ; $$
  • \([A_{2}]\) For any \(u, \overline{u}\in \mathcal{M}\), there exist constants \(\mathbb{A}, \mathbb{B}>0\) such that

    $$\begin{aligned} & \bigl\vert \varUpsilon _{i}\bigl(u(\tau _{i})\bigr)- \varUpsilon _{i}\bigl(\overline{u}(\tau _{i})\bigr) \bigr\vert \leq \mathbb{A} \bigl\vert u(\tau _{i})-\overline{u}(\tau _{i}) \bigr\vert , \\ & \bigl\vert \hat{\varUpsilon _{i}}\bigl(u(\tau _{i}) \bigr)-\hat{\varUpsilon _{i}}\bigl(\overline{u}( \tau _{i}) \bigr) \bigr\vert \leq \mathbb{B} \bigl\vert u(\tau _{i})- \overline{u}(\tau _{i}) \bigr\vert ,\quad i=1,2, \dots,m; \end{aligned}$$
  • \([A_{3}]\) There exist bounded functions \(l_{1}, m_{1}, n_{1} \in \mathcal{M}\) such that

    $$ \bigl\vert \mathcal{A}\bigl(\tau,u(\tau ),w(\tau )\bigr) \bigr\vert \leq l_{1}(\tau )+m_{1}(\tau ) \bigl\vert u( \tau ) \bigr\vert +n_{1}(\tau ) \bigl\vert w(\tau ) \bigr\vert $$

    with \(l_{1}^{*}=\sup_{\tau \in \mathcal{J}}l_{1}(\tau ), m _{1}^{*}=\sup_{\tau \in \mathcal{J}}m_{1}(\tau )\) and \(n_{1}^{*}=\sup_{\tau \in \mathcal{J}}n_{1}(\tau )<1\).

    Similarly, there exist bounded functions \(l_{2}, m_{2}, n_{2} \in \mathcal{M}\) such that

    $$ \bigl\vert \mathcal{B}\bigl(\tau,u(\tau ),w(\tau )\bigr) \bigr\vert \leq l_{2}(\tau )+m_{2}(\tau ) \bigl\vert u( \tau ) \bigr\vert +n_{2}(\tau ) \bigl\vert w(\tau ) \bigr\vert $$

    with \(l_{2}^{*}=\sup_{\tau \in \mathcal{J}}l_{2}(\tau ), m _{2}^{*}=\sup_{\tau \in \mathcal{J}}m_{2}(\tau )\), and \(n_{2}^{*}=\sup_{\tau \in \mathcal{J}}n_{2}(\tau )<1 \text{ with }1-n_{1}^{*}-n_{2}^{*}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}>0\);

  • \([A_{4}]\) The functions \(\varUpsilon _{i}:\mathbb{R}\rightarrow \mathbb{R}, i=1,2,\dots,m\), are continuous for each \(u\in \mathbb{R}\). There exist constants \(\mathcal{K}_{\varUpsilon _{i}}, \mathcal{L}_{\varUpsilon _{i}}>0\) such that \(|\varUpsilon _{i}(u(\tau _{i}))| \leq \mathcal{K}_{\varUpsilon _{i}}|u(\tau )|+\mathcal{L}_{\varUpsilon _{i}}\).

    Similarly, for each \(u\in \mathbb{R}\), the functions \(\hat{\varUpsilon _{i}}:\mathbb{R}\rightarrow \mathbb{R}; i=1,2,\dots,m\), are continuous, and for constants \(\mathcal{K}'_{\hat{\varUpsilon _{i}}}, \mathcal{L}'_{\hat{\varUpsilon _{i}}}>0\), we have the inequality \(|\hat{\varUpsilon _{i}}(u(\tau _{i}))|\leq \mathcal{K}'_{ \hat{\varUpsilon _{i}}}|u(\tau )|+\mathcal{L}'_{\hat{\varUpsilon _{i}}}\).

The main results of this section are presented in the following theorems.

Theorem 2.3

If hypotheses\([A_{1}]\)\([A_{4}]\)are satisfied, then problem (1.3) has at least one solution.

Proof

See Appendix 2. □

Theorem 2.4

If hypotheses\([A_{1}]\)\([A_{2}]\)and the inequality

$$\begin{aligned} & \biggl[ \biggl(\frac{m\mathrm{T}^{r}}{\varGamma (r+1)} +\frac{m\mathrm{T} ^{r-1}}{\varGamma (r)} \biggr) \biggl(\frac{\mathrm{M}_{1}}{1-\mathrm{N} _{1}-\mathrm{N}_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{M}_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma ( \delta )}}{1-\mathrm{N}_{1} -\mathrm{N}_{2}\frac{\mathrm{T}^{\sigma }}{ \sigma \varGamma (\delta )}} \biggr) \\ &\quad {}+\frac{\xi _{2}\mathrm{T}^{r}}{\eta _{2}\varGamma (r+1)}+m(\mathbb{A}+ \mathbb{B}) \biggr]< 1 \quad\textit{with }1- \mathrm{N}_{1}-\mathrm{N}_{2}\frac{ \mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}>0 \end{aligned}$$
(2.6)

are satisfied, then problem (1.3) has a unique solution.

Proof

See Appendix 2. □

Our approach to prove the existence of the solution for problem (1.3) from Theorem 2.3 is based on Theorem A.5 (see Appendix 1). Also, the proof of the uniqueness for problem (1.3) treated in Theorem 2.4 is based on the arguments from Theorem A.6 (see Appendix 1).

In Sect. 4, we will provide an example demonstrating how (2.6) can be computed in a specific case.

Existence and uniqueness solution for system (1.4)

In this section, we consider the coupled system of nonlinear implicit fractional differential equation with impulsive conditions from (1.4). First, we have the following:

Theorem 2.5

The system

$$ \textstyle\begin{cases} {}^{c}D^{r}\omega (\tau )=\alpha (\tau ), \quad \tau \in \mathcal{J}, \\ {}^{c}D^{p}y(\tau )=\beta (\tau ), \quad\tau \in \mathcal{J}, \\ \Delta \omega (\tau _{i})=\varUpsilon _{i}(\omega (\tau _{i})), \qquad \Delta \omega '(\tau _{i})=\hat{\varUpsilon _{i}}(\omega (\tau _{i})),\quad i=1,2,\dots,m, \\ \Delta y(\tau _{j})=\varUpsilon _{j}(y(\tau _{j})), \qquad \Delta y'(\tau _{j})= \hat{\varUpsilon _{j}}(y(\tau _{j})),\quad j=1,2,\dots,n, \\ \eta _{1}\omega (0)+\xi _{1}I^{r}\omega (0)=\nu _{1},\qquad \eta _{2}\omega (\mathrm{T})+\xi _{2}I^{r}\omega (\mathrm{T})=\nu _{2}, \\ \eta _{3}y(0)+\xi _{3}I^{p}y(0)=\nu _{3},\qquad \eta _{4}y(\mathrm{T})+\xi _{4}I^{p}y(\mathrm{T})=\nu _{4} \end{cases} $$

has a solution \((\omega,y)\) if and only if

$$ \omega (\tau )=\textstyle\begin{cases} \frac{1}{\varGamma (r)}\int _{0}^{\tau }(\tau -s)^{r-1}\alpha (s)\,ds+\frac{ \nu _{1}}{\eta _{1}} -\frac{\tau }{\mathrm{T}} [\frac{\xi _{2}}{\eta _{2}\varGamma (r)}\int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}\omega (s)\,ds \\ \quad{}+\frac{1}{ \varGamma (r)}\int _{\tau _{1}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}\alpha (s)\,ds \\ \quad{}+\frac{1}{\varGamma (r)}\int _{0}^{\tau _{1}}(\tau _{1}-s)^{r-1}\alpha (s)\,ds +\frac{\mathrm{T}-\tau _{1}}{\varGamma (r-1)}\int _{0}^{\tau _{1}}(\tau _{1}-s)^{r-2} \alpha (s)\,ds \\ \quad{}+(\mathrm{T}-\tau _{1})\hat{\varUpsilon _{1}}(\omega (\tau _{1}))+\varUpsilon _{1}(\omega (\tau _{1})) +\frac{\nu _{1}}{\eta _{1}}-\frac{\nu _{2}}{\eta _{2}} ] , \quad \tau \in \mathcal{J}_{0}, \\ \frac{1}{\varGamma (r)}\int _{0}^{\tau }(\tau -s)^{r-1}\alpha (s)\,ds+\frac{ \nu _{1}}{\eta _{1}} -\frac{\tau }{\mathrm{T}} [\frac{\xi _{2}}{\eta _{2}\varGamma (r)}\int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}\omega (s)\,ds +\frac{ \nu _{1}}{\eta _{1}}-\frac{\nu _{2}}{\eta _{2}} ] \\ \quad{}-\frac{\tau }{\mathrm{T}}\sum_{i=1}^{m} [\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}\alpha (s)\,ds +\frac{1}{ \varGamma (r)}\int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1}\alpha (s)\,ds \\ \quad{}+\frac{\mathrm{T}-\tau _{i}}{\varGamma (r-1)}\int _{\tau _{i-1}}^{\tau _{i}}( \tau _{i}-s)^{r-2}\alpha (s)\,ds \\ \quad{}+(\mathrm{T}-\tau _{i})\hat{\varUpsilon _{i}}(\omega (\tau _{i}))+\varUpsilon _{i}(\omega (\tau _{i})) ], \quad \tau \in \mathcal{J}_{i}, \end{cases} $$

and

$$ y(\tau )=\textstyle\begin{cases} \frac{1}{\varGamma (p)}\int _{0}^{\tau }(\tau -s)^{p-1}\beta (s)\,ds+\frac{ \nu _{3}}{\eta _{3}} -\frac{\tau }{\mathrm{T}} [\frac{\xi _{4}}{\eta _{4}\varGamma (p)}\int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{p-1}y(s)\,ds \\ \quad{}+\frac{1}{ \varGamma (p)}\int _{\tau _{1}}^{\mathrm{T}}(\mathrm{T}-s)^{p-1}\beta (s)\,ds \\ \quad{}+\frac{1}{\varGamma (p)}\int _{0}^{\tau _{1}}(\tau _{1}-s)^{p-1}\beta (s)\,ds +\frac{\mathrm{T}-\tau _{1}}{\varGamma (p-1)}\int _{0}^{\tau _{1}}(\tau _{1}-s)^{p-2} \beta (s)\,ds \\ \quad{}+(\mathrm{T}-\tau _{1})\hat{\varUpsilon _{1}}(y(\tau _{1}))+\varUpsilon _{1}(y( \tau _{1})) +\frac{\nu _{3}}{\eta _{3}}-\frac{\nu _{4}}{\eta _{4}} ] , \quad \tau \in \mathcal{J}_{0}, \\ \frac{1}{\varGamma (p)}\int _{0}^{\tau }(\tau -s)^{p-1}\beta (s)\,ds+\frac{ \nu _{3}}{\eta _{3}} -\frac{\tau }{\mathrm{T}} [\frac{\xi _{4}}{\eta _{4}\varGamma (p)}\int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{p-1}y(s)\,ds +\frac{ \nu _{3}}{\eta _{3}}-\frac{\nu _{4}}{\eta _{4}} ] \\ \quad{}-\frac{\tau }{\mathrm{T}}\sum_{j=1}^{n} [\frac{1}{\varGamma (p)} \int _{\tau _{j}}^{\mathrm{T}}(\mathrm{T}-s)^{p-1}\beta (s)\,ds +\frac{1}{ \varGamma (p)}\int _{\tau _{j-1}}^{\tau _{j}}(\tau _{j}-s)^{p-1}\beta (s)\,ds \\ \quad{}+\frac{ \mathrm{T}-\tau _{j}}{\varGamma (p-1)}\int _{\tau _{j-1}}^{\tau _{j}}(\tau _{j}-s)^{p-2}\beta (s)\,ds \\ \quad{}+(\mathrm{T}-\tau _{j})\hat{\varUpsilon _{j}}(y(\tau _{j}))+\varUpsilon _{j}(y( \tau _{j})) ],\quad \tau \in \mathcal{J}_{j}, \end{cases} $$

where

$$ \alpha (\tau )=\mathcal{A}\bigl(\tau,y(\tau ),{}^{c}D^{r} \omega (\tau )\bigr) + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B} \bigl(s,y(s),{}^{c}D^{r}\omega (s)\bigr)\,ds $$

and

$$ \beta (\tau )=\mathcal{A}'\bigl(\tau,\omega (\tau ),{}^{c}D^{p}y(\tau )\bigr) + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B}'\bigl(s,\omega (s),{}^{c}D^{p}y(s) \bigr)\,ds. $$

Proof

The proof is similar to that given in Theorem 2.1 and hence is not included here. □

For \(\tau _{i}\in \mathcal{J}\) such that \(\tau _{1}<\tau _{2}<\cdots <\tau _{m}\) and \(\mathcal{J}'=\mathcal{J}-\{\tau _{1},\tau _{2},\dots,\tau _{m}\}\), we define the space \(\mathcal{X}=\{\omega:\mathcal{J}\rightarrow \mathbb{R} | \omega \in \mathcal{C}(\mathcal{J}'),\text{ right limit }\omega (\tau ^{+}_{i})\text{ and left limit }\omega (\tau ^{-}_{i})\text{ exist, and }\Delta \omega (\tau _{i})= \omega (\tau ^{-}_{i})-\omega (\tau ^{+}_{i}), 1< i\leq m\}\). Clearly, \((\mathcal{X},\|\cdot \|)\) is a Banach space endowed with the norm \(\|\omega \|=\max_{\tau \in \mathcal{J}}|\omega |\).

Similarly, for \(\tau _{j}\in \mathcal{J}\) such that \(\tau _{1}<\tau _{2}< \cdots <\tau _{n}\) and \(\mathcal{J}'=\mathcal{J}-\{\tau _{1},\tau _{2}, \dots,\tau _{n}\}\), we define the space \(\mathcal{Y}=\{y:\mathcal{J} \rightarrow \mathbb{R} | y\in \mathcal{C}(\mathcal{J}'), \text{ right limit } y(\tau ^{+}_{j}) \text{ and left limit } y( \tau ^{-}_{j}) \text{ exist, and } \Delta y(\tau _{i})=y(\tau ^{-}_{j})-y( \tau ^{+}_{j}), 1< j\leq n\}\), which is a Banach space endowed with the norm \(\|y\|=\max_{\tau \in \mathcal{J}}|y|\).

Consequently, the product space \(\mathcal{X}\times \mathcal{Y}\) is a Banach space with the norm \(\|(\omega,y)\|=\|\omega \|+\|y\|\) or \(\|(\omega,y)\|=\max \{\|\omega \|,\|y\|\}\).

Theorem 2.6

Let\(\mathcal{A}, \mathcal{B}, \mathcal{A}', \mathcal{B}'\)be continuous functions. Then\((\omega,y)\in \mathcal{X}\times \mathcal{Y}\)is a solution of problem (1.4) if and only if\((\omega,y)\)is a solution of

$$\begin{aligned} \omega (\tau ) ={}& \frac{1}{\varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1}\alpha (s)\,ds+ \frac{ \nu _{1}}{\eta _{1}} -\frac{\tau }{\mathrm{T}} \biggl[\frac{\xi _{2}}{\eta _{2}\varGamma (r)} \int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}\omega (s)\,ds \\ &{} +\frac{\nu _{1}}{\eta _{1}}-\frac{\nu _{2}}{\eta _{2}} \biggr] -\frac{ \tau }{\mathrm{T}} \sum_{i=1}^{m} \biggl[\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}\alpha (s)\,ds + \frac{1}{\varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \alpha (s)\,ds \\ &{} +\frac{\mathrm{T}-\tau _{i}}{\varGamma (r-1)} \int _{\tau _{i-1}}^{\tau _{i}}( \tau _{i}-s)^{r-2} \alpha (s)\,ds +(\mathrm{T}-\tau _{i}) \hat{\varUpsilon _{i}} \bigl(\omega (\tau _{i})\bigr)+\varUpsilon _{i}\bigl(\omega ( \tau _{i})\bigr) \biggr], \\ & \tau \in \mathcal{J}_{i}, \end{aligned}$$
(2.7)

and

$$\begin{aligned} y(\tau ) ={}& \frac{1}{\varGamma (p)} \int _{0}^{\tau }(\tau -s)^{p-1}\beta (s)\,ds+ \frac{ \nu _{3}}{\eta _{3}} -\frac{\tau }{\mathrm{T}} \biggl[\frac{\xi _{4}}{\eta _{4}\varGamma (p)} \int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{p-1}y(s)\,ds \\ &{} +\frac{\nu _{3}}{\eta _{3}}-\frac{\nu _{4}}{\eta _{4}} \biggr] -\frac{ \tau }{\mathrm{T}} \sum_{j=1}^{n} \biggl[\frac{1}{\varGamma (p)} \int _{\tau _{j}}^{\mathrm{T}}(\mathrm{T}-s)^{p-1}\beta (s) \,ds + \frac{1}{\varGamma (p)} \int _{\tau _{j-1}}^{\tau _{j}}(\tau _{j}-s)^{p-1} \beta (s)\,ds \\ &{} +\frac{\mathrm{T}-\tau _{j}}{\varGamma (p-1)} \int _{\tau _{j-1}}^{\tau _{j}}( \tau _{j}-s)^{p-2} \beta (s)\,ds +(\mathrm{T}-\tau _{j}) \hat{\varUpsilon _{j}} \bigl(y(\tau _{j})\bigr)+\varUpsilon _{j}\bigl(y(\tau _{j})\bigr) \biggr], \quad\tau \in \mathcal{J}_{j}. \end{aligned}$$

Proof

If \((\omega,y)\) is a solution of system (1.4), then it is a solution of (2.7). Conversely, if \((\omega,y)\) is a solution of (2.7), then

$$ \textstyle\begin{cases} {}^{c}\mathcal{D}^{r}\omega (\tau )=\mathcal{A}(\tau,y(\tau ),{}^{c} \mathcal{D}^{r}\omega (\tau )) +\int _{0}^{\tau }\frac{(\tau -s)^{ \sigma -1}}{\varGamma (\delta )}\mathcal{B}(s,y(s),{}^{c}\mathcal{D}^{r} \omega (s))\,ds \\ \quad\text{where } \tau \in \mathcal{J}, \tau \neq \tau _{i} \text{ for } i=1,2,\dots,m, \\ ^{c}\mathcal{D}^{p}y(\tau )=\mathcal{A}'(\tau,\omega (\tau ),{}^{c} \mathcal{D}^{p}y(\tau )) +\int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{ \varGamma (\delta )}\mathcal{B}'(s,\omega (s),{}^{c}\mathcal{D}^{p}y(s))\,ds \\ \quad\text{where } \tau \in \mathcal{J}, \tau \neq \tau _{j} \text{ for } j=1,2,\dots,n, \\ \Delta \omega (\tau _{i})=\varUpsilon _{i}(\omega (\tau _{i})), \qquad \Delta \omega '(\tau _{i})=\hat{\varUpsilon _{i}}(\omega (\tau _{i})),\quad i=1,2,\dots,m, \\ \Delta y(\tau _{j})=\varUpsilon _{j}(y(\tau _{j})), \qquad \Delta y'(\tau _{j})= \hat{\varUpsilon _{j}}(y(\tau _{j})),\quad j=1,2,\dots,n, \\ \eta _{1}\omega (0)+\xi _{1}I^{r}\omega (0)=\nu _{1}, \qquad \eta _{2}\omega ( \mathrm{T})+\xi _{2}I^{r}\omega (\mathrm{T})=\nu _{2}, \\ \eta _{3}y(0)+\xi _{3}I^{p}y(0)=\nu _{3}, \qquad \eta _{4}y(\mathrm{T})+\xi _{4}I^{p}y(\mathrm{T})=\nu _{4}. \end{cases} $$

Thus \((\omega,y)\) is a solution of (1.4). □

For convenience, we use the following notations:

$$\begin{aligned} v(\tau ) & = \mathcal{A}\bigl(\tau,y(\tau ),{}^{c}D^{r}\omega ( \tau )\bigr) + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B} \bigl(s,y(s),{}^{c}D^{r}\omega (s)\bigr)\,ds \\ & = \mathcal{A}\bigl(\tau,y(\tau ),v(\tau )\bigr) + \int _{0}^{\tau }\frac{( \tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B} \bigl(s,y(s),v(s)\bigr)\,ds, \\ z(\tau ) & = \mathcal{A}'\bigl(\tau,\omega (\tau ),{}^{c}D^{p}y(\tau )\bigr) + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B}'\bigl(s,\omega (s),{}^{c}D^{p}y(s) \bigr)\,ds \\ & = \mathcal{A}'\bigl(\tau,\omega (\tau ),z(\tau )\bigr) + \int _{0}^{\tau }\frac{( \tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B}'\bigl(s,\omega (s),z(s)\bigr)\,ds. \end{aligned}$$

System (1.4) can be transformed into a fixed point problem.

Define the operators \(\mathcal{T}_{r}, \mathcal{T}_{p}:\mathcal{X} \times \mathcal{Y}\rightarrow \mathcal{X}\times \mathcal{Y}\) by

$$ \mathcal{T}_{r}(\omega,y) (\tau )=\textstyle\begin{cases} \frac{1}{\varGamma (r)}\int _{0}^{\tau }(\tau -s)^{r-1}v(s)\,ds+\frac{\nu _{1}}{\eta _{1}} -\frac{\tau }{\mathrm{T}} [\frac{\xi _{2}}{\eta _{2}\varGamma (r)}\int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}\omega (s)\,ds \\ \quad{}+\frac{1}{ \varGamma (r)}\int _{\tau _{1}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}v(s)\,ds \\ \quad{}+\frac{1}{\varGamma (r)}\int _{0}^{\tau _{1}}(\tau _{1}-s)^{r-1}v(s)\,ds +\frac{ \mathrm{T}-\tau _{1}}{\varGamma (r-1)}\int _{0}^{\tau _{1}}(\tau _{1}-s)^{r-2}v(s)\,ds \\ \quad{}+(\mathrm{T}-\tau _{1})\hat{\varUpsilon _{1}}(\omega (\tau _{1}))+\varUpsilon _{1}(\omega (\tau _{1})) +\frac{\nu _{1}}{\eta _{1}}-\frac{\nu _{2}}{\eta _{2}} ] , \quad\tau \in \mathcal{J}_{0}, \\ \frac{1}{\varGamma (r)}\int _{0}^{\tau }(\tau -s)^{r-1}v(s)\,ds+\frac{\nu _{1}}{\eta _{1}} -\frac{\tau }{\mathrm{T}} [\frac{\xi _{2}}{\eta _{2}\varGamma (r)}\int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}\omega (s)\,ds +\frac{ \nu _{1}}{\eta _{1}}-\frac{\nu _{2}}{\eta _{2}} ] \\ \quad{}-\frac{\tau }{\mathrm{T}}\sum_{i=1}^{m} [\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}v(s)\,ds +\frac{1}{ \varGamma (r)}\int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1}v(s)\,ds \\\quad {}+\frac{ \mathrm{T}-\tau _{i}}{\varGamma (r-1)}\int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-2}v(s)\,ds \\ \quad{}+(\mathrm{T}-\tau _{i})\hat{\varUpsilon _{i}}(\omega (\tau _{i}))+\varUpsilon _{i}(\omega (\tau _{i})) ], \quad \tau \in \mathcal{J}_{i}, \end{cases} $$

and

$$ \mathcal{T}_{p}(\omega,y) (\tau )=\textstyle\begin{cases} \frac{1}{\varGamma (p)}\int _{0}^{\tau }(\tau -s)^{p-1}z(s)\,ds+\frac{\nu _{3}}{\eta _{3}} -\frac{\tau }{\mathrm{T}} [\frac{\xi _{4}}{\eta _{4}\varGamma (p)}\int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{p-1}y(s)\,ds \\ \quad{}+\frac{1}{ \varGamma (p)}\int _{\tau _{1}}^{\mathrm{T}}(\mathrm{T}-s)^{p-1}z(s)\,ds \\ \quad{}+\frac{1}{\varGamma (p)}\int _{0}^{\tau _{1}}(\tau _{1}-s)^{p-1}z(s)\,ds +\frac{ \mathrm{T}-\tau _{1}}{\varGamma (p-1)}\int _{0}^{\tau _{1}}(\tau _{1}-s)^{p-2}z(s)\,ds \\ \quad{}+(\mathrm{T}-\tau _{1})\hat{\varUpsilon _{1}}(y(\tau _{1}))+\varUpsilon _{1}(y( \tau _{1})) +\frac{\nu _{3}}{\eta _{3}}-\frac{\nu _{4}}{\eta _{4}} ] ,\quad \tau \in \mathcal{J}_{0}, \\ \frac{1}{\varGamma (p)}\int _{0}^{\tau }(\tau -s)^{p-1}z(s)\,ds+\frac{\nu _{3}}{\eta _{3}} -\frac{\tau }{\mathrm{T}} [\frac{\xi _{4}}{\eta _{4}\varGamma (p)}\int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{p-1}y(s)\,ds +\frac{ \nu _{3}}{\eta _{3}}-\frac{\nu _{4}}{\eta _{4}} ] \\ \quad{}-\frac{\tau }{\mathrm{T}}\sum_{j=1}^{n} [\frac{1}{\varGamma (p)} \int _{\tau _{j}}^{\mathrm{T}}(\mathrm{T}-s)^{p-1}z(s)\,ds +\frac{1}{ \varGamma (p)}\int _{\tau _{j-1}}^{\tau _{j}}(\tau _{j}-s)^{p-1}z(s)\,ds \\ \quad{}+\frac{ \mathrm{T}-\tau _{j}}{\varGamma (p-1)}\int _{\tau _{j-1}}^{\tau _{j}}(\tau _{j}-s)^{p-2}z(s)\,ds \\ \quad{}+(\mathrm{T}-\tau _{j})\hat{\varUpsilon _{j}}(y(\tau _{j}))+\varUpsilon _{j}(y( \tau _{j})) ], \quad \tau \in \mathcal{J}_{j}, \end{cases} $$

with \(\mathcal{T}(\omega,y)(\tau )=(\mathcal{T}_{r}(\omega,y)( \tau ),\mathcal{T}_{p}(\omega,y)(\tau ))\).

We further need the following hypotheses:

  • \([\tilde{A_{1}}]\) there exist constants \(\mathrm{M}_{1}>0\) and \(\mathrm{N}_{1}\in (0,1)\) such that, for all \(\tau \in \mathcal{J}\), \(u, \overline{u}\in \mathcal{X}\), and \(w, \overline{w}\in \mathbb{R}\), we have

    $$ \bigl\vert \mathcal{A}(\tau,u,w)-\mathcal{A}(\tau,\overline{u},\overline{w}) \bigr\vert \leq \mathrm{M}_{1} \vert u-\overline{u} \vert + \mathrm{N}_{1} \vert w-\overline{w} \vert . $$

    Similarly, there exist constants \(\mathrm{M}_{2}>0\) and \(\mathrm{N} _{2}\in (0,1)\) such that, for all \(\tau \in \mathcal{J}\), \(u, \overline{u}\in \mathcal{X}\), and \(w, \overline{w}\in \mathbb{R}\), we have

    $$ \bigl\vert \mathcal{B}(\tau,u,w)-\mathcal{B}(\tau,\overline{u},\overline{w}) \bigr\vert \leq \mathrm{M}_{2} \vert u-\overline{u} \vert + \mathrm{N}_{2} \vert w-\overline{w} \vert ; $$
  • \([\tilde{A_{2}}]\) there exist constants \(\mathrm{M}'_{1}>0\) and \(\mathrm{N}'_{1}\in (0,1)\) such that, for all \(\tau \in \mathcal{J}\), \(u, \overline{u}\in \mathcal{Y}\), and \(w, \overline{w}\in \mathbb{R}\), we have

    $$ \bigl\vert \mathcal{A}'(\tau,u,w)-\mathcal{A}'(\tau, \overline{u},\overline{w}) \bigr\vert \leq \mathrm{M}'_{1} \vert u-\overline{u} \vert +\mathrm{N}'_{1} \vert w-\overline{w} \vert . $$

    Similarly, there exist constants \(\mathrm{M}'_{2}>0\) and \(\mathrm{N}'_{2} \in (0,1)\) such that, for all \(\tau \in \mathcal{J}\), \(u, \overline{u}\in \mathcal{Y}\), and \(w, \overline{w}\in \mathbb{R}\), we have

    $$ \bigl\vert \mathcal{B}'(\tau,u,w)-\mathcal{B}'(\tau, \overline{u},\overline{w}) \bigr\vert \leq \mathrm{M}'_{2} \vert u-\overline{u} \vert +\mathrm{N}'_{2} \vert w-\overline{w} \vert ; $$
  • \([\tilde{A_{3}}]\) for any \(w, \overline{w}\in \mathcal{X}\times \mathcal{Y}\), there exist constants \(A_{\varUpsilon _{i}}, A_{ \hat{\varUpsilon _{i}}}>0\) such that

    $$\begin{aligned} & \bigl\vert \varUpsilon _{i}\bigl(w(\tau _{i})\bigr)- \varUpsilon _{i}\bigl(\overline{w}(\tau _{i})\bigr) \bigr\vert \leq A_{\varUpsilon _{i}} \bigl\vert w(\tau _{i})-\overline{w}( \tau _{i}) \bigr\vert ; \\ & \bigl\vert \hat{\varUpsilon _{i}}\bigl(w(\tau _{i}) \bigr)-\hat{\varUpsilon _{i}}\bigl(\overline{w}( \tau _{i}) \bigr) \bigr\vert \leq A_{\hat{\varUpsilon _{i}}} \bigl\vert w(\tau _{i})- \overline{w}(\tau _{i}) \bigr\vert ,\quad i=1,2,\dots,m. \end{aligned}$$

    Similarly, for any \(y, \overline{y}\in \mathcal{X}\times \mathcal{Y}\), there exist constants \(A_{\varUpsilon _{j}}, A_{ \hat{\varUpsilon _{j}}}>0\) such that

    $$\begin{aligned} & \bigl\vert \varUpsilon _{j}\bigl(w(\tau _{j})\bigr)- \varUpsilon _{j}\bigl(\overline{w}(\tau _{j})\bigr) \bigr\vert \leq A_{\varUpsilon _{j}} \bigl\vert w(\tau _{j})-\overline{w}( \tau _{j}) \bigr\vert ; \\ & \bigl\vert \hat{\varUpsilon _{j}}\bigl(w(\tau _{j}) \bigr)-\hat{\varUpsilon _{j}}\bigl(\overline{w}( \tau _{j}) \bigr) \bigr\vert \leq A_{\hat{\varUpsilon _{j}}} \bigl\vert w(\tau _{j})- \overline{w}(\tau _{j}) \bigr\vert ,\quad j=1,2,\dots,n; \end{aligned}$$
  • \([\tilde{A_{4}}]\) there exist \(a_{1}, b_{1}, c_{1}\in \mathcal{X}\) such that

    $$ \bigl\vert \mathcal{A}\bigl(\tau,u(\tau ),w(\tau )\bigr) \bigr\vert \leq a_{1}(\tau )+b_{1}(\tau ) \bigl\vert u( \tau ) \bigr\vert +c_{1}(\tau ) \bigl\vert w(\tau ) \bigr\vert $$

    with \(a_{1}^{*}=\sup_{\tau \in \mathcal{J}}a_{1}(\tau ), b _{1}^{*}=\sup_{\tau \in \mathcal{J}}b_{1}(\tau ), c_{1}^{*}= \sup_{\tau \in \mathcal{J}}c_{1}(\tau )<1\).

    Similarly, there exist \(a_{2}, b_{2}, c_{2}\in \mathcal{X}\) such that

    $$ \bigl\vert \mathcal{B}\bigl(\tau,u(\tau ),w(\tau )\bigr) \bigr\vert \leq a_{2}(\tau )+b_{2}(\tau ) \bigl\vert u( \tau ) \bigr\vert +c_{2}(\tau ) \bigl\vert w(\tau ) \bigr\vert $$

    with \(a_{2}^{*}=\sup_{\tau \in \mathcal{J}}a_{2}(\tau ), b _{2}^{*}=\sup_{\tau \in \mathcal{J}}b_{2}(\tau ), c_{2}^{*}= \sup_{\tau \in \mathcal{J}}c_{2}(\tau )<1 \text{ with } 1-c _{1}^{*}-c_{2}^{*} \frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}\);

  • \([\tilde{A_{5}}]\) there exist \(l_{1}, m_{1}, n_{1}\in \mathcal{Y}\) such that

    $$ \bigl\vert \mathcal{A}'\bigl(\tau,u(\tau ),w(\tau )\bigr) \bigr\vert \leq l_{1}(\tau )+m_{1}(\tau ) \bigl\vert u( \tau ) \bigr\vert +n_{1}(\tau ) \bigl\vert w(\tau ) \bigr\vert $$

    with \(l_{1}^{*}=\sup_{\tau \in \mathcal{J}}l_{1}(\tau ), m _{1}^{*}=\sup_{\tau \in \mathcal{J}}m_{1}(\tau ), n_{1}^{*}= \sup_{\tau \in \mathcal{J}}n_{1}(\tau )<1\).

    Similarly, there exist \(l_{2}, m_{2}, n_{2}\in \mathcal{Y}\) such that

    $$ \bigl\vert \mathcal{B}'\bigl(\tau,u(\tau ),w(\tau )\bigr) \bigr\vert \leq l_{2}(\tau )+m_{2}(\tau ) \bigl\vert u( \tau ) \bigr\vert +n_{2}(\tau ) \bigl\vert w(\tau ) \bigr\vert $$

    with \(l_{2}^{*}=\sup_{\tau \in \mathcal{J}}l_{2}(\tau ), m _{2}^{*}=\sup_{\tau \in \mathcal{J}}m_{2}(\tau ), n_{2}^{*}= \sup_{\tau \in \mathcal{J}}n_{2}(\tau )<1 \text{ with} 1-n _{1}^{*}-n_{2}^{*} \frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}>0\);

  • \([\tilde{A_{6}}]\) The functions \(\varUpsilon _{i}:\mathbb{R}\rightarrow \mathbb{R}; i=1,2,\dots,m\), are continuous for each \(u\in \mathbb{R}\). There exist constants \(\mathcal{K}_{\varUpsilon _{i}}, \mathcal{L}_{\varUpsilon _{i}}>0\) such that \(|\varUpsilon _{i}(u(\tau _{i}))| \leq \mathcal{K}_{\varUpsilon _{i}}|u(\tau )|+\mathcal{L}_{\varUpsilon _{i}}\).

    Similarly, the functions \(\hat{\varUpsilon _{i}}:\mathbb{R}\rightarrow \mathbb{R}; i=1,2,\dots,m\), are continuous for each \(u\in \mathbb{R}\). There exist constants constants \(\mathcal{K}'_{ \hat{\varUpsilon _{i}}}, \mathcal{L}'_{\hat{\varUpsilon _{i}}}>0\) such that \(|\hat{\varUpsilon _{i}}(u(\tau _{i}))|\leq \mathcal{K}'_{ \hat{\varUpsilon _{i}}}|u(\tau )|+\mathcal{L}'_{\hat{\varUpsilon _{i}}}\);

  • \([\tilde{A_{7}}]\) The functions \(\varUpsilon _{j}:\mathbb{R}\rightarrow \mathbb{R}; j=1,2,\dots,n\), are continuous for each \(u\in \mathbb{R}\). There exist constants \(\mathcal{K}_{\varUpsilon _{j}}, \mathcal{L}_{\varUpsilon _{j}}>0\) such that \(|\varUpsilon _{j}(u(\tau _{j}))| \leq \mathcal{K}_{\varUpsilon _{j}}|u(\tau )|+\mathcal{L}_{\varUpsilon _{j}}\).

    Similarly, the functions \(\hat{\varUpsilon _{j}}:\mathbb{R}\rightarrow \mathbb{R}; j=1,2,\dots,n\), are continuous for each \(u\in \mathbb{R}\). There exist constants \(\mathcal{K}'_{\hat{\varUpsilon _{i}}}, \mathcal{L}'_{\hat{\varUpsilon _{i}}}>0\) such that \(| \hat{\varUpsilon _{j}}(u(\tau _{j}))|\leq \mathcal{K}'_{ \hat{\varUpsilon _{i}}}|u(\tau )|+\mathcal{L}'_{\hat{\varUpsilon _{i}}}\);

  • \([\tilde{A_{8}}]\) Denote

    $$\begin{aligned} \Delta _{1} ={} & \biggl[ \biggl(\frac{m\mathrm{T}^{r}}{\varGamma (r+1)} +\frac{m \mathrm{T}^{r-1}}{\varGamma (r)} \biggr) \biggl(\frac{\mathrm{M}_{1}}{1- \mathrm{N}_{1}-\mathrm{N}_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{M}_{2}\frac{\mathrm{T}^{\sigma }}{ \sigma \varGamma (\delta )}}{1-\mathrm{N}_{1} -\mathrm{N}_{2}\frac{ \mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} \biggr) \\ &{} +\frac{\xi _{2}\mathrm{T}^{r}}{\eta _{2}\varGamma (r+1)} +m(\mathbb{A}_{ \hat{\varUpsilon _{i}}}+ \mathbb{A}_{\varUpsilon _{i}}) \biggr]< 1 \quad\text{with } 1-\mathrm{N}_{1}- \mathrm{N}_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}>0 \end{aligned}$$

    and

    $$\begin{aligned} \Delta _{2} ={}& \biggl[ \biggl(\frac{n\mathrm{T}^{p}}{\varGamma (p+1)} +\frac{n \mathrm{T}^{p-1}}{\varGamma (p)} \biggr) \biggl(\frac{\mathrm{M}'_{1}}{1- \mathrm{N}'_{1}-\mathrm{N}'_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{M}'_{2}\frac{T^{\sigma }}{\sigma \varGamma (\delta )}}{1 -\mathrm{N}'_{1}-\mathrm{N}'_{2}\frac{\mathrm{T} ^{\sigma }}{\sigma \varGamma (\delta )}} \biggr) \\ &{} +\frac{\xi _{4}\mathrm{T}^{p}}{\eta _{4}\varGamma (p+1)} +n(\mathbb{A}_{ \hat{\varUpsilon _{j}}}+ \mathbb{A}_{\varUpsilon _{j}}) \biggr]< 1 \quad\text{with } 1-\mathrm{N}'_{1}- \mathrm{N}'_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}>0. \end{aligned}$$

Now, we are in position to state the main results of this section.

Theorem 2.7

If hypotheses\([\tilde{A_{1}}]\)\([\tilde{A_{4}}]\)are satisfied, then problem (1.4) has at least one solution.

Proof

See Appendix 2. □

Theorem 2.8

If\(\Delta =\max (\Delta _{1},\Delta _{2})<1\), then under hypotheses\([\tilde{A_{1}}]\)\([\tilde{A_{7}}]\), system (1.4) has a unique solution.

Proof

See Appendix 2. □

Hyers–Ulam stability

In this section, we provide novel characterizations of the Hyers–Ulam stability for systems (1.3) and (1.4). We rely on stability notions from [21]; for various concepts of Hyers–Ulam stability, see, for example [37, 43, 46, 47].

Hyers–Ulam stability concepts for system (1.3)

For \(\omega \in \mathcal{M}, \epsilon _{r}>0, \phi _{r}\geq 0\), and a nondecreasing function \(\psi _{r}\in C(\mathcal{J},\mathbb{R}_{+})\), the following set of inequalities are satisfied:

$$\begin{aligned} & \textstyle\begin{cases} \vert ^{c}\mathcal{D}^{r}\omega (\tau )-\mathcal{A}(\tau,\omega (\tau ),{}^{c} \mathcal{D}^{r}\omega (\tau )) -\int _{0}^{\tau }\frac{(\tau -s)^{ \sigma -1}}{\varGamma (\delta )}\mathcal{B}(s,\omega (s),{}^{c}\mathcal{D} ^{r}\omega (s))\,ds \vert \leq \epsilon _{r}, \\ \quad \tau \in \mathcal{J}_{i}, i=1,2,\dots,m, \\ \vert \Delta \omega (\tau _{i})-\varUpsilon _{i}(\omega (\tau _{i})) \vert \leq \epsilon _{r}, \quad i=1,2,\dots,m, \end{cases}\displaystyle \end{aligned}$$
(3.1)
$$\begin{aligned} &\textstyle\begin{cases} \vert ^{c}\mathcal{D}^{r}\omega (\tau )-\mathcal{A}(\tau,\omega (\tau ),{}^{c} \mathcal{D}^{r}\omega (\tau )) -\int _{0}^{\tau }\frac{(\tau -s)^{ \sigma -1}}{\varGamma (\delta )}\mathcal{B}(s,\omega (s),{}^{c}\mathcal{D} ^{r}\omega (s))\,ds \vert \leq \psi _{r}(\tau ), \\ \quad \tau \in \mathcal{J}_{i}, i=1,2,\dots,m, \\ \vert \Delta \omega (\tau _{i})-\varUpsilon _{i}(\omega (\tau _{i})) \vert \leq \phi _{r}, \quad i=1,2,\dots,m, \end{cases}\displaystyle \end{aligned}$$
(3.2)

and

$$ \textstyle\begin{cases} \vert ^{c}\mathcal{D}^{r}\omega (\tau )-\mathcal{A}(\tau,\omega (\tau ),{}^{c} \mathcal{D}^{r}\omega (\tau )) -\int _{0}^{\tau }\frac{(\tau -s)^{ \sigma -1}}{\varGamma (\delta )}\mathcal{B}(s,\omega (s),{}^{c}\mathcal{D} ^{r}\omega (s))\,ds \vert \\ \quad\leq \epsilon _{r}\psi _{r}(\tau ),\quad \tau \in \mathcal{J}_{i}, i=1,2,\dots,m, \\ \vert \Delta \omega (\tau _{i})-\varUpsilon _{i}(\omega (\tau _{i})) \vert \leq \epsilon _{r}\phi _{r}, \quad i=1,2,\dots,m. \end{cases} $$
(3.3)

Recall the definitions of stability concepts from [21].

Definition 3.1

Problem (1.3) is said to be Hyers–Ulam stable if there exists \(\mathcal{C}_{\mathcal{A},\mathcal{B}}>0\) such that, for each \(\epsilon _{r}>0\) and any solution \(\omega \in \mathcal{M}\) of inequality (3.1), there exists a unique solution \(\omega ^{*}\in \mathcal{M}\) of problem (1.3) such that

$$ \bigl\Vert \omega (\tau )-\omega ^{*}(\tau ) \bigr\Vert _{\mathcal{M}}\leq \mathcal{C} _{\mathcal{A},\mathcal{B}}\epsilon _{r}\quad \text{for all } \tau \in \mathcal{J}. $$

Definition 3.2

Problem (1.3) is said to be generalized Hyers–Ulam stable if there exists a function \(\vartheta \in \mathcal{C}(\mathbb{R}_{+}, \mathbb{R}_{+})\) with \(\vartheta (0)=0\) such that, for each \(\epsilon _{r}>0\) and any solution \(\omega \in \mathcal{M}\) of inequality (3.1), there exists a unique solution \(\omega ^{*}\in \mathcal{M}\) of problem (1.3) such that

$$ \bigl\Vert \omega (\tau )-\omega ^{*}(\tau ) \bigr\Vert _{\mathcal{M}}\leq \mathcal{C} _{\mathcal{A},\mathcal{B}}\vartheta (\epsilon _{r})\quad \text{for all } \tau \in \mathcal{J}. $$

Definition 3.3

Problem (1.3) is said to be Hyers–Ulam–Rassias stable with respect to \((\phi _{r},\psi _{r})\) if there exists \(\mathcal{C}_{ \mathcal{A},\mathcal{B}}>0\) such that, for each \(\epsilon _{r}>0\) and any solution \(\omega \in \mathcal{M}\) of inequality (3.3), there exists a unique solution \(\omega ^{*}\in \mathcal{M}\) of problem (1.3) such that

$$ \bigl\Vert \omega (\tau )-\omega ^{*}(\tau ) \bigr\Vert _{\mathcal{M}}\leq \mathcal{C} _{\mathcal{A},\mathcal{B}}\epsilon _{r}\bigl(\phi _{r}+\psi _{r}(\tau )\bigr)\quad \text{for all } \tau \in \mathcal{J}. $$

Definition 3.4

Problem (1.3) is said to be generalized Hyers–Ulam–Rassias stable with respect to \((\phi _{r},\psi _{r})\) if there exists \(\mathcal{C}_{\mathcal{A},\mathcal{B}}>0\) such that, for each \(\epsilon _{r}>0\) and any solution \(\omega \in \mathcal{M}\) of inequality (3.2), there exists a unique solution \(\omega ^{*}\in \mathcal{M}\) of problem (1.3) such that

$$ \bigl\Vert \omega (\tau )-\omega ^{*}(\tau ) \bigr\Vert _{\mathcal{M}}\leq \mathcal{C} _{\mathcal{A},\mathcal{B}}\bigl(\phi _{r}+\psi _{r}(\tau )\bigr) \quad\text{for all } \tau \in \mathcal{J}. $$

Some remarks are in order.

Remark 3.5

Definition 3.1 implies Definition 3.2, and Definition 3.3 implies Definition 3.4.

Remark 3.6

A function \(\omega \in \mathcal{M}\) is a solution of inequality (3.1) if there exist a function \(\varPhi \in \mathcal{M}\) and a sequence \(\varPhi _{i}\) (which depends on ω) such that

  1. (i)

    \(|\varPhi (\tau )|\leq \epsilon _{r}\) and \(|\varPhi _{i}|\leq \epsilon _{r} \text{ for all }\tau \in \mathcal{J}, i=1,2,\dots,m\);

  2. (ii)

    \({}^{c}\mathcal{D}^{r}\omega (\tau )=\mathcal{A}(\tau,\omega ( \tau ),{}^{c}\mathcal{D}^{r}\omega (\tau )) +\int _{0}^{\tau }\frac{( \tau -s)^{\sigma -1}}{\varGamma (\delta )}\mathcal{B}(s,\omega (s),{}^{c} \mathcal{D}^{r}\omega (s))\,ds+\varPhi (\tau )\text{ for all}\tau \in \mathcal{J}\); and

  3. (iii)

    \(\Delta \omega (\tau _{i})=\varUpsilon _{i}(\omega (\tau _{i}))+\varPhi _{i} \text{ for all } \tau \in \mathcal{J}, i=1,2,\dots,m\).

Remark 3.7

A function \(\omega \in \mathcal{M}\) is a solution of inequality (3.2) if there exist a function \(\varPhi \in \mathcal{M}\) and a sequence \(\varPhi _{i}\) (which depends on ω) such that

  1. (i)

    \(|\varPhi (\tau )|\leq \psi _{r}(\tau )\) and \(|\varPhi _{i}|\leq \phi _{r} \text{ for all }\tau \in \mathcal{J}, i=1,2,\dots,m\);

  2. (ii)

    \({}^{c}\mathcal{D}^{r}\omega (\tau )=\mathcal{A}(\tau,\omega ( \tau ),{}^{c}\mathcal{D}^{r}\omega (\tau )) +\int _{0}^{\tau }\frac{( \tau -s)^{\sigma -1}}{\varGamma (\delta )}\mathcal{B}(s,\omega (s),{}^{c} \mathcal{D}^{r}\omega (s))\,ds+\varPhi (\tau )\) for all \(\tau \in \mathcal{J}\); and

  3. (iii)

    \(\Delta \omega (\tau _{i})=\varUpsilon _{i}(\omega (\tau _{i}))+\varPhi _{i}\text{ for all }\tau \in \mathcal{J}, i=1,2,\dots,m\).

Remark 3.8

A function \(\omega \in \mathcal{M}\) is a solution of inequality (3.3) if there exist a function \(\varPhi \in \mathcal{M}\) and a sequence \(\varPhi _{i}\) (which depends on ω) such that

  1. (i)

    \(|\varPhi (\tau )|\leq \psi _{r}(\tau )\) and \(|\varPhi _{i}|\leq \epsilon _{r}\phi _{r}\text{ for all }\tau \in \mathcal{J}, i=1,2, \dots,m\);

  2. (ii)

    \({}^{c}\mathcal{D}^{r}\omega (\tau )=\mathcal{A}(\tau,\omega ( \tau ),{}^{c}\mathcal{D}^{r}\omega (\tau )) +\int _{0}^{\tau }\frac{( \tau -s)^{\sigma -1}}{\varGamma (\delta )}\mathcal{B}(s,\omega (s),{}^{c} \mathcal{D}^{r}\omega (s))\,ds+\varPhi (\tau )\) for all \(\tau \in \mathcal{J}\); and

  3. (iii)

    \(\Delta \omega (\tau _{i})=\varUpsilon _{i}(\omega (\tau _{i}))+\varPhi _{i}\text{ for all }\tau \in \mathcal{J}, i=1,2,\dots,m\).

Definition 3.9

A function \(\omega \in \mathcal{M}\) that satisfies (1.3) and its conditions on \(\mathcal{J}\) is a solution of problem (1.3).

Theorem 3.10

If\(\omega \in \mathcal{M}\)is a solution of inequality (3.1), thenωis a solution of the inequality

$$ \bigl\vert \omega (\tau )-q(\tau ) \bigr\vert \leq \biggl( \frac{\tau ^{r}}{\varGamma (r+1)}-\frac{m \tau ^{r+1}}{\mathrm{T}\varGamma (r+1)}-\frac{\tau m}{\mathrm{T}} \biggr) \epsilon _{r}. $$

Proof

Let ω be a solution of inequality (3.1). Then by Remark 3.6ω is also a solution of

$$ \textstyle\begin{cases} {}^{c}\mathcal{D}^{r}\omega (\tau )=\mathcal{A}(\tau,\omega (\tau ),{}^{c} \mathcal{D}^{r}\omega (\tau )) \\ \phantom{{}^{c}\mathcal{D}^{r}\omega (\tau )=}{}+\int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B}(s,\omega (s),{}^{c}\mathcal{D}^{r}\omega (s))\,ds+\varPhi ( \tau ),\\ \quad \tau \in \mathcal{J}, \tau \neq \tau _{i}, i=1,2,\dots,m, \\ \Delta \omega (\tau _{i})=\varUpsilon _{i}(\omega (\tau _{i})),\qquad \Delta \omega '(\tau _{i})=\hat{\varUpsilon _{i}}(\omega (\tau _{i})),\quad i=1,2,\dots,m, \\ \eta _{1}\omega (0)+\xi _{1}I^{r}\omega (0)=\nu _{1}, \qquad \eta _{2}\omega ( \mathrm{T})+\xi _{2}I^{r}\omega (\mathrm{T})=\nu _{2}, \end{cases} $$
(3.4)

that is,

$$\begin{aligned} \omega (\tau ) ={}& \frac{1}{\varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1}v(s)\,ds \\ &{}+ \frac{1}{ \varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1}\varPhi (s)\,ds+ \frac{\nu _{1}}{ \eta _{1}} -\frac{\tau }{\mathrm{T}} \biggl[\frac{\nu _{1}}{\eta _{1}}- \frac{ \nu _{2}}{\eta _{2}} \\ & +\frac{\xi _{2}}{\eta _{2}\varGamma (r)} \int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \omega (s)\,ds \biggr] -\frac{\tau }{\mathrm{T}}\sum_{i=1}^{m} \biggl[ \frac{1}{ \varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}v(s)\,ds \\ &{} +\frac{1}{\varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \varPhi (s)\,ds +\frac{1}{\varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1}v(s) \,ds \\ &{}+\frac{\mathrm{T}-\tau _{i}}{\varGamma (r-1)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-2}v(s) \,ds +(\mathrm{T}-\tau _{i}) \hat{\varUpsilon _{i}}\bigl(\omega (\tau _{i})\bigr)+\varUpsilon _{i}\bigl(\omega (\tau _{i})\bigr)+ \varPhi _{i} \biggr]. \end{aligned}$$
(3.5)

For simplicity, let \(q(\tau )\) denote the terms of \(\omega (\tau )\) that are free from \(\varPhi (\tau )\), that is,

$$\begin{aligned} q(\tau ) ={}& \frac{1}{\varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1}v(s)\,ds+ \frac{\nu _{1}}{\eta _{1}} \\ &{}-\frac{\tau }{\mathrm{T}} \biggl[\frac{\nu _{1}}{\eta _{1}}- \frac{\nu _{2}}{\eta _{2}} +\frac{\xi _{2}}{\eta _{2}\varGamma (r)} \int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}\omega (s)\,ds \biggr] \\ &{} -\frac{\tau }{\mathrm{T}}\sum_{i=1}^{m} \biggl[\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}v(s)\,ds +\frac{1}{ \varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1}v(s) \,ds \\ &{}+\frac{\mathrm{T}-\tau _{i}}{\varGamma (r-1)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-2}v(s) \,ds +(\mathrm{T}-\tau _{i}) \hat{\varUpsilon _{i}}\bigl(\omega (\tau _{i})\bigr)+\varUpsilon _{i}\bigl(\omega (\tau _{i})\bigr) \biggr]. \end{aligned}$$

Thus (3.5) can be written as

$$\begin{aligned} &\bigl\vert \omega (\tau )-q(\tau ) \bigr\vert \\ &\quad\leq \frac{1}{\varGamma (r)} \int _{0}^{\tau }( \tau -s)^{r-1} \bigl\vert \varPhi (s) \bigr\vert \,ds -\frac{\tau }{\mathrm{T}}\sum_{i=1}^{m} \biggl[\frac{1}{\varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \bigl\vert \varPhi (s) \bigr\vert \,ds + \vert \varPhi _{i} \vert \biggr]. \end{aligned}$$

Using (i) from Remark 3.6, we get

$$\begin{aligned} \bigl\vert \omega (\tau )-q(\tau ) \bigr\vert &\leq \biggl( \frac{\tau ^{r}}{\varGamma (r+1)}-\frac{m \tau ^{r+1}}{T\varGamma (r+1)}-\frac{\tau m}{\mathrm{T}} \biggr)\epsilon _{r}. \end{aligned}$$

 □

Theorem 3.11

If hypothesis\([A_{1}]\)holds and

$$\begin{aligned} & \biggl[ \biggl(\frac{m\mathrm{T}^{r}}{\varGamma (r+1)}+\frac{m\mathrm{T} ^{r-1}}{\varGamma (r)} \biggr) \biggl(\frac{\mathrm{M}_{1}}{1-\mathrm{N} _{1}-\mathrm{N}_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{M}_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma ( \delta )}}{1-\mathrm{N}_{1} -\mathrm{N}_{2}\frac{\mathrm{T}^{\sigma }}{ \sigma \varGamma (\delta )}} \biggr) \\ &\quad{} +\frac{\xi _{2}\mathrm{T}^{r}}{\eta _{2}\varGamma (r+1)}+m(\mathbb{A}+ \mathbb{B}) \biggr]< 1, \end{aligned}$$
(3.6)

then problem (1.3) is Ulam–Hyers and generalized Ulam–Hyers stable.

Proof

See Appendix 3. □

Assume that

  • \([A_{5}]\) there exist a nondecreasing function \(\psi _{r}\in \mathcal{M}\) and a constant \(\varrho _{\psi _{r}}>0\) such that, for each \(\tau \in \mathcal{J}\), we have

    $$ I^{\varrho }\psi _{r}(\tau )\leq \varrho _{\psi _{r}}\psi _{r}(\tau ). $$

From Theorem 3.11 and \([A_{5}]\) we obtain the following theorem.

Theorem 3.12

Under hypotheses\([A_{1}]\)\([A_{5}]\)and condition (3.6), problem (1.3) is Ulam–Hyers–Rassias and generalized Ulam–Hyers–Rassias stable.

Hyers–Ulam stability concepts for system (1.4)

Let \(\epsilon _{r}, \epsilon _{p}>0, \mathcal{A}, \mathcal{B}, \mathcal{A}', \mathcal{B}'\) be continuous functions, and \(\psi _{r}, \psi _{p}:\mathcal{J}\rightarrow \mathbb{R}^{+}\) be nondecreasing functions. Consider the following inequalities:

$$\begin{aligned} &\textstyle\begin{cases} \vert ^{c}\mathcal{D}^{r}\omega (\tau )-\mathcal{A}(\tau,y(\tau ),{}^{c} \mathcal{D}^{r}\omega (\tau )) \\ \quad{}-\int _{0}^{\tau }\frac{(\tau -s)^{ \sigma -1}}{\varGamma (\delta )} \mathcal{B}(s,y(s),{}^{c}\mathcal{D}^{r} \omega (s))\,ds \vert \leq \epsilon _{r}, \quad \tau \in \mathcal{J}, \\ \vert ^{c}\mathcal{D}^{p}y(\tau )-\mathcal{A}'(\tau,\omega (\tau ),{}^{c} \mathcal{D}^{p}y(\tau )) \\ \quad{}-\int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{ \varGamma (\delta )} \mathcal{B}'(s,\omega (s),{}^{c}\mathcal{D}^{p}y(s))\,ds \vert \leq \epsilon _{p}, \quad \tau \in \mathcal{J}, \\ \vert \Delta \omega (\tau _{i})-\varUpsilon _{i}(\omega (\tau _{i})) \vert \leq \epsilon _{r},\quad i=1,2,\dots,m, \\ \vert \Delta y(\tau _{j})-\varUpsilon _{j}(y(\tau _{j})) \vert \leq \epsilon _{p},\quad j=1,2,\dots,n, \end{cases}\displaystyle \end{aligned}$$
(3.7)
$$\begin{aligned} &\textstyle\begin{cases} \vert ^{c}\mathcal{D}^{r}\omega (\tau )-\mathcal{A}(\tau,y(\tau ),{}^{c} \mathcal{D}^{r}\omega (\tau )) \\ \quad{}-\int _{0}^{\tau }\frac{(\tau -s)^{ \sigma -1}}{\varGamma (\delta )} \mathcal{B}(s,y(s),{}^{c}\mathcal{D}^{r} \omega (s))\,ds \vert \leq \psi _{r}, \quad \tau \in \mathcal{J}, \\ \vert ^{c}\mathcal{D}^{p}y(\tau )-\mathcal{A}'(\tau,\omega (\tau ),{}^{c} \mathcal{D}^{p}y(\tau )) \\ \quad{}-\int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{ \varGamma (\delta )} \mathcal{B}'(s,\omega (s),{}^{c}\mathcal{D}^{p}y(s))\,ds \vert \leq \psi _{p}, \quad \tau \in \mathcal{J}, \\ \vert \Delta \omega (\tau _{i})-\varUpsilon _{i}(\omega (\tau _{i})) \vert \leq \phi _{r},\quad i=1,2,\dots,m, \\ \vert \Delta y(\tau _{j})-\varUpsilon _{j}(y(\tau _{j})) \vert \leq \phi _{p}, \quad j=1,2,\dots,n, \end{cases}\displaystyle \end{aligned}$$
(3.8)

and

$$ \textstyle\begin{cases} \vert ^{c}\mathcal{D}^{r}\omega (\tau )-\mathcal{A}(\tau,y(\tau ),{}^{c} \mathcal{D}^{r}\omega (\tau )) \\ \quad{}-\int _{0}^{\tau }\frac{(\tau -s)^{ \sigma -1}}{\varGamma (\delta )} \mathcal{B}(s,y(s),{}^{c}\mathcal{D}^{r} \omega (s))\,ds \vert \leq \epsilon _{r}\psi _{r}, \quad \tau \in \mathcal{J}, \\ \vert ^{c}\mathcal{D}^{p}y(\tau )-\mathcal{A}'(\tau,\omega (\tau ),{}^{c} \mathcal{D}^{p}y(\tau )) \\ \quad{}-\int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{ \varGamma (\delta )}\mathcal{B}'(s,\omega (s),{}^{c}\mathcal{D}^{p}y(s))\,ds \vert \leq \epsilon _{p}\psi _{p}, \quad \tau \in \mathcal{J}, \\ \vert \Delta \omega (\tau _{i})-\varUpsilon _{i}(\omega (\tau _{i})) \vert \leq \epsilon _{r}\phi _{r},\\ \quad i=1,2,\dots,m, \\ \vert \Delta y(\tau _{j})-\varUpsilon _{j}(y(\tau _{j})) \vert \leq \epsilon _{p}\phi _{p},\quad j=1,2,\dots,n. \end{cases} $$
(3.9)

Recall the appropriate definitions of stability concepts from [21].

Definition 3.13

Problem (1.4) is said to be Hyers–Ulam stable if there exists \(\mathcal{C}_{r,p}=\max (\mathcal{C}_{r},\mathcal{C}_{p})>0\) for some \(\epsilon =(\epsilon _{r},\epsilon _{p})\) and for each solution \((\omega,y)\in \mathcal{X}\times \mathcal{Y}\) of (3.7), there exists a solution \((\omega ^{*},y^{*})\in \mathcal{X}\times \mathcal{Y}\) of (1.4) with

$$ \bigl\Vert (\omega,y) (\tau )-\bigl(\omega ^{*},y^{*} \bigr) (\tau ) \bigr\Vert _{\mathcal{X}\times \mathcal{Y}}\leq \mathcal{C}_{r,p}\epsilon \quad\text{for all } \tau \in \mathcal{J}. $$

Definition 3.14

Problem (1.4) is said to be generalized Hyers–Ulam stable if there exists a function \(\varTheta \in C(\mathcal{J},\mathbb{R})\) with \(\varTheta (0)=0\) such that for each solution \((\omega,y)\in \mathcal{X}\times \mathcal{Y}\) of (3.7), there exists a solution \((\omega ^{*},y^{*})\in \mathcal{X}\times \mathcal{Y}\) of (1.4) with

$$ \bigl\Vert (\omega,y) (\tau )-\bigl(\omega ^{*},y^{*} \bigr) (\tau ) \bigr\Vert _{\mathcal{X}\times \mathcal{Y}}\leq \varTheta (\epsilon ) \quad\text{for all } \tau \in \mathcal{J}. $$

Definition 3.15

Problem (1.4) is said to be Hyers–Ulam–Rassias stable with respect to \(\psi _{r,p}=(\psi _{r},\psi _{p})\in C^{1}(\mathcal{J}, \mathbb{R})\) if there exists a constant \(\mathcal{C}_{\psi _{r},\psi _{p}}=\max (\mathcal{C}_{\psi _{r}},\mathcal{C}_{\psi _{p}})\) such that, for some \(\epsilon =(\epsilon _{r},\epsilon _{p})>0\) and for each solution \((\omega,y)\in \mathcal{X}\times \mathcal{Y}\) of (3.8), there exists a solution \((\omega ^{*},y^{*})\in \mathcal{X}\times \mathcal{Y}\) of (1.4) with

$$ \bigl\Vert (\omega,y) (\tau )-\bigl(\omega ^{*},y^{*} \bigr) (\tau ) \bigr\Vert _{\mathcal{X}\times \mathcal{Y}}\leq \mathcal{C}_{\psi _{r},\psi _{p}}\epsilon \quad\text{for all } \tau \in \mathcal{J}. $$

Definition 3.16

Problem (1.4) is said to be generalized Hyers–Ulam–Rassias stable with respect to \(\psi _{r,p}=(\psi _{r},\psi _{p})\in C^{1}( \mathcal{J},\mathbb{R})\) if there exists a constant \(\mathcal{C}_{\psi _{r},\psi _{p}}=\max (\mathcal{C}_{\psi _{r}},\mathcal{C}_{\psi _{p}})>0\) such that, for each solution \((\omega,y)\in \mathcal{X}\times \mathcal{Y}\) of (3.9), there exists a solution \((\omega ^{*},y ^{*})\in \mathcal{X}\times \mathcal{Y}\) of (1.4) with

$$ \bigl\Vert (\omega,y) (\tau )-\bigl(\omega ^{*},y^{*} \bigr) (\tau ) \bigr\Vert _{\mathcal{X}\times \mathcal{Y}}\leq \mathcal{C}_{\psi _{r},\psi _{p}}\psi _{r,p} \quad\text{for all } \tau \in \mathcal{J}. $$

We have two remarks.

Remark 3.17

Definition 3.13 implies Definition 3.14, and Definition 3.15 implies Definition 3.16.

Remark 3.18

We say that \((\omega,y)\in \mathcal{X}\times \mathcal{Y}\) is a solution of (3.7) if there exist the functions \(\mu _{\mathcal{A}, \mathcal{B}}\), \(\varLambda _{\mathcal{A}',\mathcal{B}'}\in \mathcal{X} \times \mathcal{Y}\), depending upon \(\omega, y\), respectively, such that

  1. (i)

    \(|\mu _{\mathcal{A},\mathcal{B}}(\tau )|\leq \epsilon _{r}, | \varLambda _{\mathcal{A}',\mathcal{B}'}(\tau )|\leq \epsilon _{p} \text{ for all } \tau \in \mathcal{J}\);

  2. (ii)
    $$\begin{aligned} ^{c}\mathcal{D}^{r}\omega (\tau )={}&\mathcal{A}\bigl(\tau,y( \tau ),{}^{c} \mathcal{D}^{r}\omega (\tau )\bigr)\\ &{} + \int _{0}^{\tau }\frac{(\tau -s)^{ \sigma -1}}{\varGamma (\delta )} \mathcal{B} \bigl(s,y(s),{}^{c}\mathcal{D}^{r} \omega (s)\bigr)\,ds +\mu _{\mathcal{A},\mathcal{B}}(\tau ),\\ & \tau \in \mathcal{J}_{i}, \end{aligned}$$

    and

    $$\begin{aligned} ^{c}\mathcal{D}^{p}y(\tau )={}&\mathcal{A}\bigl(\tau,\omega ( \tau ),{}^{c} \mathcal{D}^{p}y(\tau )\bigr)\\ &{} + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{ \varGamma (\delta )} \mathcal{B}\bigl(s, \omega (s),{}^{c}\mathcal{D}^{r}y(s)\bigr)\,ds +\varLambda _{\mathcal{A}',\mathcal{B}'}(\tau ),\\ & \tau \in \mathcal{J} _{j}; \end{aligned}$$
  3. (iii)

    \(\Delta \omega (\tau _{i})=\varUpsilon _{i}(\omega (\tau _{i}))+\mu _{i}, \tau \in \mathcal{J}_{i}, i=1,2,\dots,m\), and \(\Delta y( \tau _{j})=\varUpsilon _{j}(y(\tau _{j}))+\varLambda _{j}\), \(\tau \in \mathcal{J}_{j}\), \(j=1,2,\dots,n\).

Theorem 3.19

Let\((\omega,y)\in \mathcal{X}\times \mathcal{Y}\)be a solution of inequality (3.7). Then we have

$$ \textstyle\begin{cases} \vert \omega (\tau )-q(\tau ) \vert \leq (\frac{\tau ^{r}}{\varGamma (r+1)}-\frac{m \tau ^{r+1}}{\mathrm{T}\varGamma (r+1)}-\frac{\tau m}{\mathrm{T}} ) \epsilon _{r},\quad \tau \in \mathcal{J}, \\ \vert y(\tau )-q'(\tau ) \vert \leq (\frac{\tau ^{p}}{\varGamma (p+1)}-\frac{n \tau ^{p+1}}{\mathrm{T}\varGamma (p+1)}-\frac{\tau n}{\mathrm{T}} ) \epsilon _{p}, \quad \tau \in \mathcal{J}. \end{cases} $$

Proof

Let \((\omega,y)\) be a solution of inequality (3.7). Then by Remark 3.18\((\omega,y)\) is also a solution of

$$ \textstyle\begin{cases} ^{c}\mathcal{D}^{r}\omega (\tau )=\mathcal{A}(\tau,y(\tau ),{}^{c} \mathcal{D}^{r}\omega (\tau )) +\int _{0}^{\tau }\frac{(\tau -s)^{ \sigma -1}}{\varGamma (\delta )} \mathcal{B}(s,y(s),{}^{c}\mathcal{D}^{r} \omega (s))\,ds+\mu _{\mathcal{A},\mathcal{B}} \\ \quad\text{where } \tau \in \mathcal{J}, \tau \neq \tau _{i} \text{ for } i=1,2,\dots,m, \\ ^{c}\mathcal{D}^{p}y(\tau )=\mathcal{A}'(\tau,\omega (\tau ),{}^{c} \mathcal{D}^{p}y(\tau )) +\int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{ \varGamma (\delta )} \mathcal{B}'(s,\omega (s),{}^{c}\mathcal{D}^{p}y(s))\,ds+ \varLambda _{\mathcal{A}',\mathcal{B}'} \\ \quad\text{where } \tau \in \mathcal{J}, \tau \neq \tau _{j}, j=1,2, \dots,n, \\ \Delta \omega (\tau _{i})=\varUpsilon _{i}(\omega (\tau _{i})),\qquad \Delta \omega '(\tau _{i})=\hat{\varUpsilon _{i}}(\omega (\tau _{i})),\quad i=1,2,\dots,m, \\ \Delta y(\tau _{j})=\varUpsilon _{j}(y(\tau _{j})), \qquad \Delta y'(\tau _{j})= \hat{\varUpsilon _{j}}(y(\tau _{j})),\quad j=1,2,\dots,n, \\ \eta _{1}\omega (0)+\xi _{1}I^{r}\omega (0)=\nu _{1}, \qquad \eta _{2}\omega ( \mathrm{T})+\xi _{2}I^{r}\omega (\mathrm{T})=\nu _{2}, \\ \eta _{3}y(0)+\xi _{3}I^{p}y(0)=\nu _{3},\qquad \eta _{4}y(\mathrm{T})+\xi _{4}I^{p}y(\mathrm{T})=\nu _{4}, \end{cases} $$
(3.10)

that is,

$$\begin{aligned} \omega (\tau )= {}& \frac{1}{\varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1}\alpha (s)\,ds +\frac{1}{ \varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1}\mu (s)\,ds+ \frac{\nu _{1}}{ \eta _{1}} -\frac{\tau }{\mathrm{T}} \biggl[\frac{\nu _{1}}{\eta _{1}}- \frac{ \nu _{2}}{\eta _{2}} \\ &{}+\frac{\xi _{2}}{\eta _{2}\varGamma (r)} \int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \omega (s)\,ds \biggr] -\frac{\tau }{\mathrm{T}}\sum_{i=1}^{m} \biggl[\frac{1}{ \varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}\alpha (s)\,ds \\ &{}+\frac{1}{\varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \mu (s)\,ds +\frac{1}{\varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \alpha (s)\,ds \\ &{}+\frac{\mathrm{T}-\tau _{i}}{\varGamma (r-1)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-2} \alpha (s)\,ds +(\mathrm{T}-\tau _{i}) \hat{\varUpsilon _{i}} \bigl(\omega (\tau _{i})\bigr)+\varUpsilon _{i}\bigl(\omega ( \tau _{i})\bigr)+ \mu _{i} \biggr] \end{aligned}$$
(3.11a)

and

$$\begin{aligned} y(\tau )={} & \frac{1}{\varGamma (p)} \int _{0}^{\tau }(\tau -s)^{p-1}\beta (s)\,ds + \frac{1}{ \varGamma (p)} \int _{0}^{\tau }(\tau -s)^{p-1}\lambda (s) \,ds+\frac{\nu _{3}}{ \eta _{3}} -\frac{\tau }{\mathrm{T}} \biggl[\frac{\nu _{3}}{\eta _{3}}- \frac{ \nu _{4}}{\eta _{4}} \\ &{}+\frac{\xi _{4}}{\eta _{4}\varGamma (p)} \int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{p-1}y(s)\,ds \biggr] -\frac{\tau }{\mathrm{T}}\sum_{j=1}^{n} \biggl[ \frac{1}{\varGamma (p)} \int _{\tau _{j}}^{\mathrm{T}}(\mathrm{T}-s)^{p-1} \beta (s)\,ds \\ &{}+\frac{1}{\varGamma (p)} \int _{\tau _{j-1}}^{\tau _{j}}(\tau _{j}-s)^{p-1} \lambda (s)\,ds +\frac{1}{\varGamma (p)} \int _{\tau _{j-1}}^{\tau _{j}}(\tau _{j}-s)^{p-1} \beta (s)\,ds \\ &{}+\frac{\mathrm{T}-\tau _{j}}{\varGamma (p-1)} \int _{\tau _{j-1}}^{\tau _{j}}(\tau _{j}-s)^{p-2} \beta (s)\,ds +(\mathrm{T}-\tau _{j}) \hat{\varUpsilon _{j}} \bigl(y(\tau _{j})\bigr)+\varUpsilon _{j}\bigl(y(\tau _{j})\bigr)+\lambda _{j} \biggr], \end{aligned}$$
(3.11b)

where

$$ \alpha (\tau )=\mathcal{A}\bigl(\tau,y(\tau ),{}^{c} \mathcal{D}^{r}\omega ( \tau )\bigr) + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B} \bigl(s,y(s),{}^{c}\mathcal{D}^{r}\omega (s)\bigr)\,ds $$

and

$$ \beta (\tau )=\mathcal{A}'\bigl(\tau,\omega (\tau ),{}^{c} \mathcal{D}^{p}y( \tau )\bigr) + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B}'\bigl(s,\omega (s),{}^{c}\mathcal{D}^{p}y(s) \bigr)\,ds. $$

From (3.11a) we have

$$\begin{aligned} \omega (\tau )= {}& \frac{1}{\varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1}\alpha (s)\,ds +\frac{1}{ \varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1}\mu (s)\,ds+ \frac{\nu _{1}}{ \eta _{1}} -\frac{\tau }{\mathrm{T}} \biggl[\frac{\nu _{1}}{\eta _{1}}- \frac{ \nu _{2}}{\eta _{2}} \\ &{}+\frac{\xi _{2}}{\eta _{2}\varGamma (r)} \int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \omega (s)\,ds \biggr] -\frac{\tau }{\mathrm{T}}\sum_{i=1}^{m} \biggl[\frac{1}{ \varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}\alpha (s)\,ds \\ &{}+\frac{1}{\varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \mu (s)\,ds +\frac{1}{\varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \alpha (s)\,ds \\ &{}+\frac{\mathrm{T}-\tau _{i}}{\varGamma (r-1)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-2} \alpha (s)\,ds +(\mathrm{T}-\tau _{i}) \hat{\varUpsilon _{i}} \bigl(\omega (\tau _{i})\bigr)+\varUpsilon _{i}\bigl(\omega ( \tau _{i})\bigr)+ \mu _{i} \biggr]. \end{aligned}$$
(3.12)

Thus (3.12) becomes

$$ \bigl\vert \omega (\tau )-q(\tau ) \bigr\vert \leq \frac{1}{\varGamma (r)} \int _{0}^{\tau }( \tau -s)^{r-1} \bigl\vert \mu (s) \bigr\vert \,ds -\frac{\tau }{\mathrm{T}}\sum_{i=1}^{m} \biggl[\frac{1}{\varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \bigl\vert \mu (s) \bigr\vert \,ds + \vert \mu _{i} \vert \biggr], $$

where

$$\begin{aligned} q(\tau )= {}& \frac{1}{\varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1}\alpha (s)\,ds+ \frac{ \nu _{1}}{\eta _{1}} -\frac{\tau }{\mathrm{T}} \biggl[\frac{\nu _{1}}{\eta _{1}}- \frac{\nu _{2}}{\eta _{2}} +\frac{\xi _{2}}{\eta _{2}\varGamma (r)} \int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}\omega (s)\,ds \biggr] \\ &{}-\frac{\tau }{\mathrm{T}}\sum_{i=1}^{m} \biggl[ \frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}\alpha (s)\,ds +\frac{1}{ \varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \alpha (s)\,ds \\ &{}+\frac{\mathrm{T}-\tau _{i}}{\varGamma (r-1)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-2} \alpha (s)\,ds +(\mathrm{T}-\tau _{i}) \hat{\varUpsilon _{i}} \bigl(\omega (\tau _{i})\bigr)+\varUpsilon _{i}\bigl(\omega ( \tau _{i})\bigr) \biggr]. \end{aligned}$$

Using (i) from Remark 3.18, we obtain

$$ \bigl\vert \omega (\tau )-q(\tau ) \bigr\vert \leq \biggl( \frac{\tau ^{r}}{\varGamma (r+1)}-\frac{m \tau ^{r+1}}{\mathrm{T}\varGamma (r+1)}-\frac{\tau m}{\mathrm{T}} \biggr) \epsilon _{r}. $$

Repeating a similar procedure for (3.11b) together with (i) from Remark 3.18, we have

$$ \bigl\vert y(\tau )-q'(\tau ) \bigr\vert \leq \biggl( \frac{\tau ^{p}}{\varGamma (p+1)}-\frac{n \tau ^{p+1}}{\mathrm{T}\varGamma (p+1)}-\frac{\tau n}{\mathrm{T}} \biggr) \epsilon _{p}, $$

where

$$\begin{aligned} q'(\tau )= {}& \frac{1}{\varGamma (p)} \int _{0}^{\tau }(\tau -s)^{p-1}\beta (s)\,ds+ \frac{ \nu _{3}}{\eta _{3}} -\frac{\tau }{\mathrm{T}} \biggl[\frac{\nu _{3}}{\eta _{3}}- \frac{\nu _{4}}{\eta _{4}} +\frac{\xi _{4}}{\eta _{4}\varGamma (p)} \int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{p-1}y(s)\,ds \biggr] \\ &{}-\frac{\tau }{\mathrm{T}}\sum_{j=1}^{n} \biggl[ \frac{1}{\varGamma (p)} \int _{\tau _{j}}^{\mathrm{T}}(\mathrm{T}-s)^{p-1}\beta (s) \,ds +\frac{1}{ \varGamma (p)} \int _{\tau _{j-1}}^{\tau _{j}}(\tau _{j}-s)^{p-1} \beta (s)\,ds \\ &{}+\frac{\mathrm{T}-\tau _{j}}{\varGamma (p-1)} \int _{\tau _{j-1}}^{\tau _{j}}(\tau _{j}-s)^{p-2} \beta (s)\,ds +(\mathrm{T}-\tau _{j}) \hat{\varUpsilon _{j}} \bigl(y(\tau _{j})\bigr)+\varUpsilon _{j}\bigl(y(\tau _{j})\bigr) \biggr]. \end{aligned}$$

Thus the proof is complete. □

Theorem 3.20

If hypotheses\([\tilde{A}_{1}]\)\([\tilde{A}_{3}]\)hold with

$$ \Delta =1-\mathcal{Q}_{r}\mathcal{Q}_{p}> 0, $$
(3.13)

then system (1.4) is stable, in the sense of Ulam–Hyers.

Proof

See Appendix 3. □

In the next section, we provide an example demonstrating how (3.13) can be computed in a specific case. We conclude this section with two remarks.

Remark 3.21

We set \(\varTheta (\epsilon )=C_{r,p}\epsilon \), \(\varTheta (0)=0\) in (C.10). By Definition 3.14 the proposed system (1.4) is generalized Ulam–Hyers stable.

To obtain the connections between the Ulam–Hyers–Rassias stability concepts, we introduce the following hypothesis.

  • \([\tilde{A_{9}}]\) Let \(\varOmega _{r}, \varOmega _{p}\in \mathcal{C}( \mathcal{J},\mathbb{R}^{+})\) be an increasing functions. Then there exist \(\varLambda _{\varOmega _{r}}, \varLambda _{\varOmega _{p}}>0\) such that, for each \(\tau \in \mathcal{J}\),

    $$ {I}^{{r}}\varOmega _{r}(\tau )\leq \varLambda _{\varOmega _{r}}\varOmega _{r}(\tau ) \quad\text{and} \quad {I}^{r-1}\varOmega _{r}(\tau ) \leq \varLambda _{\varOmega _{r}}\varOmega _{r}(\tau ) $$

    and

    $$ {I}^{p}\varOmega _{p(\tau )}\leq \varLambda _{\varOmega _{p}}\varOmega _{p}(\tau ) \quad\text{and} \quad {I}^{p-1}\varOmega _{p}(\tau ) \leq \varLambda _{\varOmega _{p}}\varOmega _{p}(\tau ). $$

Remark 3.22

Under hypotheses \([\tilde{A_{1}}]\)\([\tilde{A_{9}}]\), by (3.13) and Theorems 3.19 and 3.20 system (1.4) is Ulam–Hyers–Rassias and generalized Ulam–Hyers–Rassias stable.

Illustrative examples

We present two examples to demonstrate the existence and stability of our obtained results.

Example 4.1

Consider

$$\begin{aligned} \textstyle\begin{cases} {}^{c}\mathcal{D}^{\frac{3}{2}}\omega (\tau )=\frac{ \vert \omega (\tau ) \vert + \cos \vert ^{c}\mathcal{D}^{\frac{3}{2}}\omega (\tau ) \vert }{90e^{\tau +2}(1+ \vert \omega (\tau ) \vert + \vert ^{c}\mathcal{D}^{\frac{3}{2}}\omega (\tau ) \vert )} \\ \phantom{{}^{c}\mathcal{D}^{\frac{3}{2}}\omega (\tau )=}{}+\frac{1}{ \varGamma (\frac{5}{2})}\int _{0}^{1}(\tau -s)^{\frac{3}{2}}\frac{ \vert \omega (s) \vert +\sin \vert ^{c}\mathcal{D}^{\frac{3}{2}}\omega (s) \vert }{101e^{\tau +2}(1+ \vert \omega (s) \vert + \vert ^{c}\mathcal{D}^{\frac{3}{2}}\omega (s) \vert )}, \quad \tau \neq \frac{1}{3}, \\ \omega (0)+I^{\frac{3}{2}}\omega (0)=\frac{1}{2},\qquad \omega (1)+I^{ \frac{3}{2}}\omega (1)=\frac{1}{2}, \\ \Delta \omega (\frac{1}{3})=\varUpsilon (\omega (\frac{1}{3})),\qquad \Delta \omega '(\frac{1}{3})=\hat{\varUpsilon }(\omega (\frac{1}{3})), \end{cases}\displaystyle \end{aligned}$$
(4.1)

where \(r=\frac{3}{2}\), \(\mathcal{J}_{0}=[0,\frac{1}{3}]\), \(\mathcal{J}_{1}=(\frac{1}{3},1]\).

Set

$$\begin{aligned} &\mathcal{A}(\tau,\omega,y)=\frac{ \vert \omega (\tau ) \vert +\cos \vert ^{c} \mathcal{D}^{\frac{3}{2}}\omega (\tau ) \vert }{90e^{\tau +2}(1+ \vert \omega ( \tau ) \vert + \vert ^{c}\mathcal{D}^{\frac{3}{2}}\omega (\tau ) \vert )}, \\ &\mathcal{B}(\tau,\omega,y)=\frac{ \vert \omega (\tau ) \vert +\sin \vert ^{c} \mathcal{D}^{\frac{3}{2}}\omega (\tau ) \vert }{101e^{\tau +2}(1+ \vert \omega ( \tau ) \vert + \vert ^{c}\mathcal{D}^{\frac{3}{2}}\omega (\tau ) \vert )}. \end{aligned}$$

Obviously, \(\mathcal{A}\) and \(\mathcal{B}\) are jointly continuous functions. Now, for all \(\omega, \overline{\omega }\in \mathcal{M}\), \(y, \overline{y}\in \mathbb{R}\), and \(\tau \in [0,1]\), we have

$$\begin{aligned} \bigl\vert \mathcal{A}(\tau,\omega,y)-\mathcal{A}(t,\overline{\omega }, \overline{y}) \bigr\vert \leq \frac{1}{90e^{2}}\bigl( \vert \omega - \overline{\omega } \vert + \vert y- \overline{y} \vert \bigr) \end{aligned}$$

and

$$\begin{aligned} \bigl\vert \mathcal{B}(\tau,\omega,y)-\mathcal{B}(\tau,\overline{\omega }, \overline{y}) \bigr\vert \leq \frac{1}{101e^{2}}\bigl( \vert \omega - \overline{\omega } \vert + \vert y- \overline{y} \vert \bigr). \end{aligned}$$

These satisfy condition \([A_{1}]\) with \(\mathrm{M}_{1}=\mathrm{N}_{1}=\frac{1}{90e ^{2}}\) and \(\mathrm{M}_{2}=\mathrm{N}_{2}=\frac{1}{101e^{2}}\).

Set

$$ \varUpsilon _{1}\biggl(\omega \biggl(\frac{1}{3}\biggr)\biggr)= \frac{ \vert \omega (\frac{1}{3}) \vert }{40+ \vert \omega (\frac{1}{3}) \vert }\quad \text{for } \omega \in \mathcal{M} $$

and

$$ \hat{\varUpsilon _{1}}\biggl(\omega \biggl(\frac{1}{3}\biggr) \biggr)=\frac{ \vert \omega (\frac{1}{3}) \vert }{20+ \vert \omega (\frac{1}{3}) \vert }\quad \text{for } \omega \in \mathcal{M}. $$

Then we have

$$ \biggl\vert \varUpsilon _{1}\biggl(\omega \biggl(\frac{1}{3} \biggr)\biggr)-\varUpsilon _{1}\biggl(\overline{\omega }\biggl( \frac{1}{3}\biggr)\biggr) \biggr\vert \leq \frac{1}{35} \vert \omega -\overline{\omega } \vert $$

and

$$ \biggl\vert \hat{\varUpsilon _{1}}\biggl(\omega \biggl( \frac{1}{3}\biggr)\biggr)-\hat{\varUpsilon _{1}}\biggl(\overline{ \omega }\biggl(\frac{1}{3}\biggr)\biggr) \biggr\vert \leq \frac{1}{20} \vert \omega -\overline{\omega } \vert , $$

respectively. Hence \(\mathbb{A}=\frac{1}{35}\) and \(\mathbb{B}= \frac{1}{20}\). Thus condition \([A_{2}]\) is satisfied.

Also,

$$\begin{aligned} & \biggl[ \biggl(\frac{m\mathrm{T}^{r}}{\varGamma (r+1)} +\frac{m\mathrm{T} ^{r-1}}{\varGamma (r)} \biggr) \biggl(\frac{\mathrm{M}_{1}}{1-\mathrm{N} _{1}-\mathrm{N}_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{M}_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma ( \delta )}}{1-\mathrm{N}_{1} -\mathrm{N}_{2}\frac{\mathrm{T}^{\sigma }}{ \sigma \varGamma (\delta )}} \biggr) \\ &\quad{}+\frac{\xi _{2}\mathrm{T}^{r}}{\eta _{2}\varGamma (r+1)}+m(\mathbb{A}+ \mathbb{B}) \biggr]\thickapprox 0.83374< 1 \end{aligned}$$

with \(m=1, \mathrm{T}=1, \xi _{2}=\eta _{2}=1, \sigma =\delta = \frac{5}{2}, r=\frac{3}{2}, \mathrm{M}_{1}=\mathrm{N}_{1}=\frac{1}{90e ^{2}}, \mathrm{M}_{2}=\mathrm{N}_{2}=\frac{1}{101e^{2}}, \mathbb{A}=\frac{1}{35}, \mathbb{B}=\frac{1}{20}\). Therefore by Theorem 2.4 problem (4.1) has a unique solution. Also, letting \(\psi (\tau )=|\tau |, \tau \in [0,1]\), we have

$$ I^{\frac{1}{2}}\psi (\tau )=\frac{1}{\varGamma (\frac{1}{2})} \int _{0} ^{\tau }(\tau -s)^{(\frac{1}{2}-1)} \vert s \vert \,ds =\frac{4^{\frac{3}{2}}}{3\sqrt{ \pi }}\leq \frac{2\tau }{\sqrt{\pi }}. $$

Hence \([A_{5}]\) is satisfied with \(\mathcal{L}_{\psi }=\frac{2}{\sqrt{ \pi }}\). Therefore by Theorem 3.12 the given problem is Ulam–Hyers–Rassias stable and consequently generalized Ulam–Hyers–Rassias stable.

Example 4.2

Consider

$$ \textstyle\begin{cases} ^{c}\mathcal{D}^{\frac{1}{2}}\omega (\tau )= \frac{1+ \vert y(\tau ) \vert + \cos \vert ^{c}\mathcal{D}^{\frac{1}{2}}\omega (\tau ) \vert }{104e^{\tau +5}(1+ \vert y( \tau ) \vert + \vert ^{c}\mathcal{D}^{\frac{1}{2}}\omega (\tau ) \vert )} \\ \phantom{^{c}\mathcal{D}^{\frac{1}{2}}\omega (\tau )=}{}+\int _{0}^{1}\frac{(\tau -s)^{\frac{3}{2}}}{\varGamma (\frac{5}{2})} \frac{1+ \vert y(s) \vert + \sin \vert ^{c}\mathcal{D}^{\frac{1}{2}}\omega (s) \vert }{104e^{s+5}(1+ \vert y(s) \vert + \vert ^{c} \mathcal{D}^{\frac{1}{2}}\omega (s) \vert )}\,ds,\quad \tau \in [0,1], \tau \neq \frac{1}{3}, \\ ^{c}\mathcal{D}^{\frac{1}{2}}y(\tau )= \frac{2+ \vert \omega (\tau ) \vert + \cos \vert ^{c}\mathcal{D}^{\frac{1}{2}}y(\tau ) \vert }{70e^{\tau +2}(1+ \vert \omega (\tau ) \vert + \vert ^{c}\mathcal{D}^{\frac{1}{2}}y(\tau ) \vert )} \\ \phantom{^{c}\mathcal{D}^{\frac{1}{2}}y(\tau )=}{}+\int _{0}^{1}\frac{(\tau -s)^{\frac{3}{2}}}{\varGamma (\frac{5}{2})} \frac{ \vert \omega (s) \vert +\cos \vert ^{c}\mathcal{D}^{\frac{1}{2}}y(s) \vert }{70e^{s+2}(1+ \vert \omega (s) \vert + \vert ^{c}\mathcal{D}^{\frac{1}{2}}y(s) \vert )}\,ds, \quad \tau \in [0,1], \tau \neq \frac{1}{4}, \\ \omega (0)+I^{\frac{1}{2}}\omega (0)=\frac{3}{2}, \qquad \omega (1)+I ^{\frac{1}{2}}\omega (1)=\frac{3}{2}, \\ y(0)+I^{\frac{1}{2}}y(0)=\frac{3}{2},\qquad y(1)+I^{\frac{1}{2}}y(1)= \frac{3}{2}, \\ \Delta \omega (\frac{1}{3})=\varUpsilon (\omega (\frac{1}{3})), \qquad \Delta \omega '(\frac{1}{3})=\hat{\varUpsilon }(\omega (\frac{1}{3})), \\ \Delta y(\frac{1}{4})=\varUpsilon (y(\frac{1}{4})), \qquad \Delta y'( \frac{1}{4})=\hat{\varUpsilon }(y(\frac{1}{4})), \end{cases} $$
(4.2)

\(\tau _{i}=\frac{1}{3}\text{ for }i=1,2,3,\dots,60,\text{ and } \tau _{j}=\frac{1}{4}\text{ for } j=1,2,3,\dots,100\).

For any \(\omega, \overline{\omega }, y, \overline{y}\in \mathbb{R}\) and \(\tau \in [0,1]\), we obtain

$$ \bigl\vert \mathcal{A}(\tau,\omega,y)-\mathcal{A}(\tau,\overline{\omega }, \overline{y}) \bigr\vert \leq \frac{1}{104e^{5}}\bigl( \vert \omega - \overline{\omega } \vert + \vert y- \overline{y} \vert \bigr) $$

and

$$ \bigl\vert \mathcal{B}(\tau,\omega,y)-\mathcal{B}(\tau,\overline{\omega }, \overline{y}) \bigr\vert \leq \frac{1}{104e^{5}}\bigl( \vert \omega - \overline{\omega } \vert + \vert y- \overline{y} \vert \bigr). $$

Similarly, for any \(\omega, \overline{\omega }, y, \overline{y}\in \mathbb{R}\), and \(\tau \in [0,1]\), we obtain

$$ \bigl\vert \mathcal{A}'(\tau,\omega,y)-\mathcal{A}'( \tau,\overline{\omega }, \overline{y}) \bigr\vert \leq \frac{1}{70e^{2}}\bigl( \vert \omega -\overline{\omega } \vert + \vert y- \overline{y} \vert \bigr) $$

and

$$ \bigl\vert \mathcal{B}'(\tau,\omega,y)-\mathcal{B}'( \tau,\overline{\omega }, \overline{y}) \bigr\vert \leq \frac{1}{70e^{2}}\bigl( \vert \omega -\overline{\omega } \vert + \vert y- \overline{y} \vert \bigr). $$

These satisfy condition \([\tilde{A}_{1}]\) with \(\mathrm{M}_{1}= \mathrm{M}_{2}=\mathrm{N}_{1}=\mathrm{N}_{2}=\frac{1}{104e^{5}}, \mathrm{M}'_{1}=\mathrm{M}'_{2}=\mathrm{N}'_{1}=\mathrm{N}'_{2}=\frac{1}{70e ^{2}}\).

Set

$$ \varUpsilon _{i}\biggl(\omega \biggl(\frac{1}{3}\biggr)\biggr)= \frac{ \vert \omega (\frac{1}{3}) \vert }{40+ \vert \omega (\frac{1}{3}) \vert } \quad\text{for } \omega \in \mathcal{X} $$

and

$$ \hat{\varUpsilon _{1}}\biggl(\omega \biggl(\frac{1}{3}\biggr) \biggr)=\frac{ \vert \omega (\frac{1}{3}) \vert }{20+ \vert \omega (\frac{1}{3}) \vert } \quad\text{for } \omega \in \mathcal{X}. $$

Then for \(\omega, \overline{\omega }\in \mathcal{X}\), we have

$$\begin{aligned} \biggl\vert \varUpsilon _{i}\biggl(\omega \biggl(\frac{1}{3} \biggr)\biggr)-\varUpsilon _{i}\biggl(\overline{ \omega }\biggl( \frac{1}{3}\biggr)\biggr) \biggr\vert &= \biggl\vert \frac{ \vert \omega (\frac{1}{3}) \vert }{40+ \vert \omega (\frac{1}{3}) \vert }-\frac{ \vert \overline{\omega }(\frac{1}{3}) \vert }{40+ \vert \overline{ \omega }(\frac{1}{3}) \vert } \biggr\vert \\ &\leq \frac{1}{35} \vert \omega -\overline{\omega } \vert \end{aligned}$$

and

$$ \biggl\vert \hat{\varUpsilon _{1}}\biggl(\omega \biggl( \frac{1}{3}\biggr)\biggr)-\hat{\varUpsilon _{1}}\biggl(\overline{ \omega }\biggl(\frac{1}{3}\biggr)\biggr) \biggr\vert \leq \frac{1}{20} \vert \omega -\overline{\omega } \vert , $$

respectively. Hence \(\mathbb{A}_{\varUpsilon _{i}}=\frac{1}{35}\) and \(\mathbb{A}_{\hat{\varUpsilon }_{i}}=\frac{1}{20}\). Thus condition \([\tilde{A}_{2}]\) is satisfied. Similarly, if

$$ \varUpsilon _{j}\biggl(y\biggl(\frac{1}{4}\biggr)\biggr)= \frac{ \vert y(\frac{1}{4}) \vert }{50+ \vert y( \frac{1}{4}) \vert }\quad \text{for } y\in \mathcal{Y}, $$

then for \(y, \overline{y}\in \mathcal{Y}\), we have

$$\begin{aligned} \biggl\vert \varUpsilon _{j}\biggl(y\biggl(\frac{1}{4}\biggr) \biggr)-\varUpsilon _{j}\biggl(\overline{y}\biggl( \frac{1}{4} \biggr)\biggr) \biggr\vert &= \biggl\vert \frac{ \vert y(\frac{1}{4}) \vert }{50+ \vert y(\frac{1}{4}) \vert }- \frac{ \vert \overline{y}( \frac{1}{4}) \vert }{50+|\overline{y}(\frac{1}{4})} \biggr\vert \\ &\leq \frac{1}{50} \vert y-\overline{y} \vert , \end{aligned}$$

and if

$$ \hat{\varUpsilon }_{j}\biggl(y\biggl(\frac{1}{4}\biggr)\biggr)= \frac{ \vert y(\frac{1}{4}) \vert }{101+ \vert y( \frac{1}{4}) \vert } \quad\text{for } y\in \mathcal{Y}, $$

then for \(y, \overline{y}\in \mathcal{Y}\), we have

$$\begin{aligned} \biggl\vert \hat{\varUpsilon }_{j}\biggl(y\biggl(\frac{1}{4} \biggr)\biggr)-\hat{\varUpsilon }_{j}\biggl( \overline{y}\biggl( \frac{1}{4}\biggr)\biggr) \biggr\vert & = \biggl\vert \frac{ \vert y(\frac{1}{4}) \vert }{101+ \vert y( \frac{1}{4}) \vert }-\frac{ \vert \overline{y}(\frac{1}{4}) \vert }{101+|\overline{y}( \frac{1}{4})} \biggr\vert \\ &\leq \frac{1}{101} \vert y-\overline{y} \vert . \end{aligned}$$

Thus \(\mathbb{A}_{\varUpsilon _{j}}=\frac{1}{50}\) and \(\mathbb{A}_{ \hat{\varUpsilon }_{j}}=\frac{1}{101}\) satisfy our requirement from \([\tilde{A}_{3}]\).

The condition

$$\begin{aligned} \Delta _{1} ={}& \biggl[ \biggl(\frac{m\mathrm{T}^{r}}{\varGamma (r+1)} +\frac{m \mathrm{T}^{r-1}}{\varGamma (r)} \biggr) \biggl(\frac{\mathrm{M}_{1}}{1- \mathrm{N}_{1}-\mathrm{N}_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{M}_{2}\frac{\mathrm{T}^{\sigma }}{ \sigma \varGamma (\delta )}}{1-\mathrm{N}_{1} -\mathrm{N}_{2}\frac{ \mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} \biggr) \\ &{}+\frac{\xi _{2}\mathrm{T}^{r}}{\eta _{2}\varGamma (r+1)} +m(\mathbb{A} _{\hat{\varUpsilon _{i}}}+\mathbb{A}_{\varUpsilon _{i}}) \biggr]\thickapprox 0.83097< 1 \end{aligned}$$

is valid with \(m=1, \mathrm{T}=1, \xi _{2}=\eta _{2}=1, \sigma = \delta =\frac{5}{2}, r=\frac{1}{2}, \mathrm{M}_{1}=\mathrm{N}_{1}= \mathrm{M}_{2}=\mathrm{N}_{2}=\frac{1}{104e^{5}}, \mathbb{A}_{ \varUpsilon _{i}}=\frac{1}{35}, \mathbb{A}_{\hat{\varUpsilon }_{i}}= \frac{1}{20}\).

Also,

$$\begin{aligned} \Delta _{2} ={}& \biggl[ \biggl(\frac{n\mathrm{T}^{p}}{\varGamma (p+1)} +\frac{n \mathrm{T}^{p-1}}{\varGamma (p)} \biggr) \biggl(\frac{\mathrm{M}'_{1}}{1- \mathrm{N}'_{1}-\mathrm{N}'_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{M}'_{2}\frac{\mathrm{T}^{\sigma }}{ \sigma \varGamma (\delta )}}{1-\mathrm{N}'_{1} -\mathrm{N}'_{2}\frac{ \mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} \biggr) \\ &{} +\frac{\xi _{4}\mathrm{T}^{p}}{\eta _{4}\varGamma (p+1)} +n(\mathbb{A}_{ \hat{\varUpsilon _{j}}}+ \mathbb{A}_{\varUpsilon _{j}}) \biggr]\thickapprox 0.78689< 1 \end{aligned}$$

with \(n=1, \mathrm{T}=1, \xi _{4}=\eta _{4}=1, \sigma =\delta = \frac{5}{2}, p=\frac{1}{2}, \mathrm{M}'_{1}=\mathrm{N}'_{1}= \mathrm{M}'_{2}=\mathrm{N}'_{2}=\frac{1}{70e^{2}}, \mathbb{A}_{ \varUpsilon _{j}}=\frac{1}{50}, \mathbb{A}_{\hat{\varUpsilon }_{j}}= \frac{1}{101}\). Hence \(\Delta =\max (\Delta _{1},\Delta _{2})<1\) is also true.

It is easy to check that

$$ 1-\mathcal{Q}_{r}\mathcal{Q}_{p}\thickapprox 1.00000>0 $$

and condition (3.13) is verified. We conclude that problem (4.2) is Ulam–Hyers stable, generalized Ulam–Hyers stable, Ulam–Hyers–Rassias stable, and generalized Ulam–Hyers–Rassias stable.

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Acknowledgements

The research of the second author is supported by Prince Sultan University through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM), group number RG-DES-2017-01-17.

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Appendices

Appendix 1: Supplementary results

The following definitions are adopted from [15].

Definition A.1

The integral of a function \(u\in L^{1}(\mathcal{J},\mathbb{R})\) of order \(r\in \mathbb{R}^{+}\) is defined by

$$ I^{r}u(\tau )=\frac{1}{\varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1}u(s)\,ds, $$

provided that the integral exists.

Definition A.2

The Caputo derivative of a function \(u\in C^{(\rho )}((0,\infty ), \mathbb{R})\) of arbitrary order r is defined by

$$ ^{c}\mathcal{D}^{r}u(\tau )=\frac{1}{\varGamma (\rho -r)} \int _{0}^{ \tau }(\tau -s)^{\rho -r-1}u^{(\rho )}(s) \,ds, $$

where \(\rho =[r]+1\) in which \([r]\) is the integer part of r.

Lemma A.3

For\(r>0\), the solution of the Caputo fractional differential equation\({}^{c}D_{0,\tau }^{r}u(\tau )=0\)is

$$ u(\tau )=z_{0}+z_{1}\tau +z_{2}\tau ^{2}+\cdots +z_{\rho -1}\tau ^{ \rho -1}, $$

where\(z_{i} \in \mathbb{R}\), \(i=0,1,\dots,\rho -1\), and\(\rho =[r]+1\).

Lemma A.4

For\(r>0\), the solution of\({}^{c}\mathcal{D}^{r}u(\tau )=\beta (\tau )\)is given by

$$ u(\tau )=I^{r}\beta (\tau )+\sum_{\rho =0}^{n-1} \frac{u^{(\rho )}(0)}{ \rho !}\tau ^{\rho }, $$

where\(\rho =[r]+1\).

Theorem A.5

([10])

Let\(\mathcal{M}\)be a Banach space, let\(\mathcal{T}:\mathcal{M}\rightarrow \mathcal{M}\)be a completely continuous operator, and let the set\(\varOmega =\{\omega \in \mathcal{M}: \omega =\aleph \mathcal{T}(\omega ), 0<\aleph <1\}\)be bounded. Then\(\mathcal{T}\)has at least one fixed point in\(\mathcal{M}\).

Theorem A.6

([10])

Let\(\mathcal{T}: \mathcal{S} \rightarrow \mathcal{S}\)be a contraction on a nonempty closed subset of a Banach space\(\mathcal{M}\). Then\(\mathcal{T}\)has a unique fixed point.

Theorem A.7

([40])

Let\(\mathcal{H}\)be a convex, closed, and nonempty subset of Banach space\(\mathcal{X}\times \mathcal{Y}\), and let\(\mathcal{F}, \mathcal{G}\)be the operators such that

  1. (i)

    \(\mathcal{F}\omega +\mathcal{G}y\in \mathcal{H}\)whenever\(\omega, y\in \mathcal{H}\).

  2. (ii)

    \(\mathcal{F}\)is compact and continuous, and\(\mathcal{G}\)is a contraction mapping.

Then there exists\(z\in \mathcal{H}\)such that\(z=\mathcal{F}z+ \mathcal{G}z\), where\(z=(\omega,y)\in \mathcal{X}\times \mathcal{Y}\).

Appendix 2

Proof of Theorem 2.3

Consider the operator \(\mathcal{T}\) defined in (2.5). We have to show that problem (1.3) has at least one solution.

We show the operator \(\mathcal{T}\) is continuous. Consider the sequence \(\{\omega _{n}\}\) such that \(\omega _{n}\rightarrow \omega \in \mathcal{M}, \tau \in \mathcal{J}\). Then

$$\begin{aligned} & \bigl\vert (\mathcal{T}\omega _{n}) ( \tau )-(\mathcal{T}\omega ) (\tau ) \bigr\vert \\ &\quad\leq \frac{1}{\varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1} \bigl\vert v_{n}(s)-v(s) \bigr\vert \,ds -\frac{ \tau }{\mathrm{T}}\frac{\xi _{2}}{\eta _{2}\varGamma (r)} \int _{0}^{ \mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert \omega _{n}(s)-\omega (s) \bigr\vert \,ds \\ &\qquad{} -\frac{\tau }{\mathrm{T}}\sum_{i=1}^{m} \biggl[\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert v_{n}(s)-v(s) \bigr\vert \,ds +\frac{1}{ \varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \bigl\vert v_{n}(s)-v(s) \bigr\vert \,ds \\ &\qquad{} +\frac{\mathrm{T}-\tau _{i}}{\varGamma (r-1)} \int _{\tau _{i-1}}^{\tau _{i}}( \tau _{i}-s)^{r-2} \bigl\vert v_{n}(s)-v(s) \bigr\vert \,ds +(\mathrm{T}-\tau _{i}) \bigl\vert \hat{\varUpsilon _{i}}\bigl(\omega _{n}(\tau _{i})\bigr)-\hat{\varUpsilon _{i}}\bigl( \omega (\tau _{i})\bigr) \bigr\vert \\ &\qquad{}+ \bigl\vert \varUpsilon _{i}\bigl(\omega _{n}(\tau _{i})\bigr)-\varUpsilon _{i}\bigl(\omega (\tau _{i})\bigr) \bigr\vert \biggr], \end{aligned}$$
(B.1)

where \(v_{n}, v\in \mathcal{M}\) are given by

$$ v_{n}(\tau )=\mathcal{A}\bigl(\tau,\omega _{n}(\tau ),v_{n}(\tau )\bigr) + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )}\mathcal{B}\bigl(s, \omega _{n}(s),v_{n}(s)\bigr)\,ds $$

and

$$ v(\tau )=\mathcal{A}\bigl(\tau,\omega (\tau ),v(\tau )\bigr) + \int _{0}^{\tau }\frac{( \tau -s)^{\sigma -1}}{\varGamma (\delta )}\mathcal{B}\bigl(s, \omega (s),v(s)\bigr)\,ds, $$

respectively. Using hypothesis \([A_{1}]\), we have

$$\begin{aligned} &\bigl\vert v_{n}(\tau )-v(\tau ) \bigr\vert \\ & \quad= \biggl\vert \mathcal{A}\bigl(\tau,\omega _{n}(\tau ),v _{n}(\tau )\bigr) + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma ( \delta )} \mathcal{B}\bigl(s, \omega _{n}(s),v_{n}(s)\bigr)\,ds \\ &\qquad{}-\mathcal{A}\bigl(\tau,\omega (\tau ),v(\tau )\bigr) - \int _{0}^{\tau }\frac{( \tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B}\bigl(s, \omega (s),v(s)\bigr)\,ds \biggr\vert \\ &\quad\leq \bigl\vert \mathcal{A}\bigl(\tau,\omega _{n}(\tau ),v_{n}(\tau )\bigr) - \mathcal{A}\bigl(\tau,\omega (\tau ),v(\tau ) \bigr) \bigr\vert \\ &\qquad{}+ \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \bigl\vert \mathcal{B}\bigl(s,\omega _{n}(s),v_{n}(s)\bigr) - \mathcal{B}\bigl(s,\omega (s),v(s)\bigr) \bigr\vert \,ds \\ &\quad\leq \mathrm{M}_{1} \bigl\vert \omega _{n}(\tau )- \omega (\tau ) \bigr\vert +\mathrm{N} _{1} \bigl\vert v_{n}(\tau )-v(\tau ) \bigr\vert \\ &\qquad{}+\frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )} \bigl(\mathrm{M}_{2} \bigl\vert \omega _{n}(\tau )-\omega (\tau ) \bigr\vert +\mathrm{N}_{2} \bigl\vert v_{n}(\tau )-v( \tau ) \bigr\vert \bigr). \end{aligned}$$

Then

$$ \bigl\vert v_{n}(\tau )-v(\tau ) \bigr\vert \leq \biggl(\frac{\mathrm{M}_{1}}{1-\mathrm{N} _{1}-\mathrm{N}_{2}\frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{ \mathrm{M}_{2}\frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )}}{1- \mathrm{N}_{1} -\mathrm{N}_{2}\frac{\tau ^{\sigma }}{\sigma \varGamma ( \delta )}} \biggr) \bigl\vert \omega _{n}(\tau )-\omega (\tau ) \bigr\vert . $$
(B.2)

Hypotheses \([A_{1}]\), \([A_{2}]\) and inequalities (B.1) and (B.2) lead to

$$\begin{aligned} & \bigl\vert (\mathcal{T}\omega _{n}) (\tau )-(\mathcal{T}\omega ) (\tau ) \bigr\vert \\ &\quad \leq \biggl[ \biggl(\frac{\tau ^{r}}{\varGamma (r+1)}-\frac{m\tau \mathrm{T}^{r-1}}{ \varGamma (r+1)}- \frac{m\tau ^{r+1}}{\mathrm{T}\varGamma (r+1)} -\frac{m\tau ^{r}}{\mathrm{T}\varGamma (r)} \biggr) \\ &\qquad{}\times \biggl(\frac{\mathrm{M}_{1}}{1-\mathrm{N}_{1}-\mathrm{N}_{2}\frac{ \tau ^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{M}_{2}\frac{ \tau ^{\sigma }}{\sigma \varGamma (\delta )}}{1-\mathrm{N}_{1} - \mathrm{N}_{2}\frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )}} \biggr) -\frac{ \xi _{2}\tau \mathrm{T}^{r-1}}{\eta _{2}\varGamma (r+1)}- \frac{\tau }{ \mathrm{T}}m(\mathbb{A}+\mathbb{B}) \biggr]\\ &\qquad{}\times \bigl\vert \omega _{n}(\tau )-\omega ( \tau ) \bigr\vert . \end{aligned}$$

For each \(\tau \in \mathcal{J}\), the sequence \(\omega _{n}\rightarrow \omega \) as \(n\rightarrow \infty \), and hence by the Lebesgue dominated convergence theorem inequality (B.1) implies that

$$ \bigl\vert (\mathcal{T}\omega _{n}) (\tau )-(\mathcal{T}\omega ) ( \tau ) \bigr\vert \rightarrow 0 \quad\text{as } n\rightarrow \infty $$

and

$$ \Vert \mathcal{T}\omega _{n}-\mathcal{T}\omega \Vert _{\mathcal{M}}\rightarrow 0 \quad\text{as } n\rightarrow \infty. $$

Hence \(\mathcal{T}\) is continuous on \(\mathcal{J}\).

Now we have to show that \(\mathcal{T}\) is bounded in \(\mathcal{M}\). For any \(\wp >0\), there is \(\mathrm{R}_{\mathbf{E}}>0\) such that

$$ \mathbf{E}=\bigl\{ \omega \in \mathcal{M}: \Vert \omega \Vert _{\mathcal{M}} \leq \wp \bigr\} , $$

which leads to

$$ \Vert \mathcal{T}\omega \Vert _{\mathcal{M}}\leq \mathrm{R}_{\mathbf{E}}. $$

For \(\tau \in \mathcal{J}_{i}\), we obtain

$$\begin{aligned} & \bigl\vert (\mathcal{T}\omega ) (\tau ) \bigr\vert \\ &\quad\leq \frac{1}{\varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds+\frac{\nu _{1}}{\eta _{1}} -\frac{\tau }{\mathrm{T}} \biggl[ \frac{\xi _{2}}{\eta _{2}\varGamma (r)} \int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert \omega (s) \bigr\vert \,ds \\ &\qquad{} +\frac{\nu _{1}}{\eta _{1}}-\frac{\nu _{2}}{\eta _{2}} \biggr] -\frac{ \tau }{\mathrm{T}} \sum_{i=1}^{m} \biggl[\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds +\frac{1}{\varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\mathrm{T}-\tau _{i}}{\varGamma (r-1)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-2} \bigl\vert v(s) \bigr\vert \,ds +(\mathrm{T}-\tau _{i}) \bigl\vert \hat{\varUpsilon _{i}}\bigl(\omega (\tau _{i})\bigr) \bigr\vert + \bigl\vert \varUpsilon _{i}\bigl(\omega (\tau _{i})\bigr) \bigr\vert \biggr]. \end{aligned}$$
(B.3)

Further, using hypothesis \([A_{3}]\), we have

$$\begin{aligned} \bigl\vert v(\tau ) \bigr\vert &\leq \bigl\vert \mathcal{A}\bigl(\tau, \omega (\tau ),v(\tau )\bigr) \bigr\vert + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \bigl\vert \mathcal{B}\bigl(s,\omega (s),v(s)\bigr) \bigr\vert \,ds \\ &\leq l_{1}(\tau )+m_{1}(\tau ) \bigl\vert \omega (\tau ) \bigr\vert +n_{1}(\tau ) \bigl\vert v( \tau ) \bigr\vert + \frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )} \bigl(l_{2}( \tau )+m_{2}(\tau ) \bigl\vert \omega (\tau ) \bigr\vert +n_{2}(\tau ) \bigl\vert v(\tau ) \bigr\vert \bigr) \\ &\leq l_{1}^{*}+m_{1}^{*} \Vert \omega \Vert _{\mathcal{M}}+n_{1}^{*} \Vert v \Vert _{\mathcal{M}} +\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )} \bigl(l_{2}^{*}+m_{2}^{*} \Vert \omega \Vert _{\mathcal{M}}+n_{2}^{*} \Vert v \Vert _{ \mathcal{M}} \bigr). \end{aligned}$$

Therefore we get

$$ \bigl\vert v(\tau ) \bigr\vert \leq \Vert v \Vert _{\mathcal{M}}\leq \frac{l_{1}^{*}+m_{1}^{*} \Vert \omega \Vert _{\mathcal{M}}}{1-n_{1}^{*}-n_{2}^{*}\frac{\mathrm{T}^{ \sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{T}^{\sigma }}{ \sigma \varGamma (\delta )}\frac{l_{2}^{*} +m_{2}^{*} \Vert \omega \Vert _{ \mathcal{M}}}{1-n_{1}^{*}-n_{2}^{*}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}}= \hbar. $$
(B.4)

Now by (B.4) and \([A_{4}]\) relation (B.3) becomes

$$\begin{aligned} \bigl\vert \mathcal{T}\omega (\tau ) \bigr\vert \leq{}& \frac{\hbar \tau ^{r}}{\varGamma (r+1)}+ \frac{ \nu _{1}}{\eta _{1}} \\ &{}-\frac{\tau }{\mathrm{T}} \biggl[\frac{\xi _{2} \mathrm{T}^{r}}{\eta _{2}\varGamma (r+1)}+ \frac{\nu _{1}}{\eta _{1}}-\frac{ \nu _{2}}{\eta _{2}} +\frac{m\hbar \mathrm{T}^{r}}{\varGamma (r+1)}+\frac{m \hbar \tau ^{r}}{\varGamma (r+1)} + \frac{m\hbar \tau ^{r-1}}{\varGamma (r)} \\ &{}+m\bigl(\mathcal{K}'_{\hat{\varUpsilon _{i}}}+\mathcal{K}_{\varUpsilon _{i}} \bigr) \bigl\vert \omega (\tau ) \bigr\vert +m\bigl(\mathcal{L}'_{\hat{\varUpsilon _{i}}}+ \mathcal{L}_{ \varUpsilon _{i}}\bigr) \biggr] \\ = {}&\mathbf{C}. \end{aligned}$$

Thus

$$ \Vert \mathcal{T}\omega \Vert _{\mathcal{M}}\leq \mathbf{C}. $$

Similarly for \(\tau \in \mathcal{J}_{0}\), we can verify that

$$ \Vert \mathcal{T}\omega \Vert _{\mathcal{M}}\leq \mathbf{C}. $$

Now we have to show that the operator \(\mathcal{T}\) is equicontinuous in E. Let \(\tau _{1}, \tau _{2}\in \mathcal{J}_{i}\) be such that \(0<\tau _{1}<\tau _{2}<\mathrm{T}\), and let \(\omega \in \mathbf{E}\). Then

$$\begin{aligned} & \bigl\vert \mathcal{T}\omega (\tau _{2})-\mathcal{T}\omega (\tau _{1}) \bigr\vert \\ &\quad\leq \frac{1}{\varGamma (r)} \int _{0}^{\tau _{2}}(\tau _{2}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds +\frac{1}{ \varGamma (r)} \int _{0}^{\tau _{1}}(\tau _{1}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds \\ &\qquad{} -\frac{\tau }{\mathrm{T}}\sum_{0< \tau _{i}< \tau _{2}-\tau _{1}} \biggl[ \frac{1}{ \varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds+ \frac{( \tau _{i-1}-\tau _{i})}{\varGamma (r-1)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-2} \bigl\vert v(s) \bigr\vert \,ds \\ &\qquad{} +\frac{1}{\varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds +(\mathrm{T}-\tau _{i}) \bigl\vert \hat{\varUpsilon _{i}}\bigl(\omega (\tau _{i})\bigr) \bigr\vert + \bigl\vert \varUpsilon _{i}\bigl(\omega (\tau _{i})\bigr) \bigr\vert \biggr] \\ &\quad\leq \frac{1}{\varGamma (r)} \int _{0}^{\tau _{2}}\bigl[(\tau _{2}-s)^{r-1}-( \tau _{1}-s)^{r-1}\bigr] \bigl\vert v(s) \bigr\vert \,ds + \frac{1}{\varGamma (r)} \int _{0}^{\tau _{1}}(\tau _{1}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds \\ &\qquad{} -\frac{\tau }{\mathrm{T}}\sum_{0< \tau _{i}< \tau _{2}-\tau _{1}} \biggl[ \frac{1}{ \varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds +\frac{( \tau _{i-1}-\tau _{i})}{\varGamma (r-1)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-2} \bigl\vert v(s) \bigr\vert \,ds \\ &\qquad{}+\frac{1}{\varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds +(\mathrm{T}-\tau _{i}) \bigl\vert \hat{\varUpsilon _{i}}\bigl(\omega (\tau _{i})\bigr) \bigr\vert + \bigl\vert \varUpsilon _{i}\bigl(\omega (\tau _{i})\bigr) \bigr\vert \biggr]. \end{aligned}$$
(B.5)

Obviously, the right-hand side of inequality (B.5) tends to zero as \(\tau _{1}\rightarrow \tau _{2}\). Therefore

$$ \bigl\vert \mathcal{T}\omega (\tau _{2})-\mathcal{T}\omega (\tau _{1}) \bigr\vert \rightarrow 0 \quad\text{as } \tau _{1} \rightarrow \tau _{2}. $$

Similarly, for \(\tau \in \mathcal{J}_{0}\). Thus \(\mathcal{T}\) is equicontinuous and therefore completely continuous. Further, we consider a set \(\varOmega \subset \mathcal{M}\) defined as

$$ \varOmega =\bigl\{ \omega \in \mathcal{M}: \omega =\aleph \mathcal{T}(\omega ), 0< \aleph < 1\bigr\} . $$

We need to prove that the set Ω is bounded. Suppose \(\omega \in \varOmega \) is such that

$$ \omega (\tau )=\aleph \mathcal{T}\bigl(\omega (\tau )\bigr), \quad\text{where } 0< \aleph < 1. $$

Then for each \(\tau \in \mathcal{J}_{i}\), we have

$$\begin{aligned} \bigl\vert \omega (\tau ) \bigr\vert ={}& \Biggl\vert \frac{\aleph }{\varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1}v(s)\,ds + \frac{\aleph \nu _{1}}{\eta _{1}} -\frac{\aleph \tau }{\mathrm{T}} \biggl[\frac{\xi _{2}}{\eta _{2}\varGamma (r)} \int _{0}^{\mathrm{T}}( \mathrm{T}-s)^{r-1}\omega (s)\,ds \\ &{}+\frac{\nu _{1}}{\eta _{1}}-\frac{\nu _{2}}{\eta _{2}} \biggr] -\frac{ \aleph \tau }{\mathrm{T}}\sum _{i=1}^{m} \biggl[\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}v(s)\,ds +\frac{1}{ \varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1}v(s) \,ds \\ &{}+\frac{\mathrm{T}-\tau _{i}}{\varGamma (r-1)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-2}v(s) \,ds +(\mathrm{T}-\tau _{i}) \hat{\varUpsilon _{i}}\bigl(\omega (\tau _{i})\bigr)+\varUpsilon _{i}\bigl(\omega (\tau _{i})\bigr) \biggr] \Biggr\vert \\ \leq{}& \frac{1}{\varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds+\frac{\nu _{1}}{\eta _{1}} -\frac{\tau }{\mathrm{T}} \biggl[ \frac{\xi _{2}}{\eta _{2}\varGamma (r)} \int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert \omega (s) \bigr\vert \,ds \\ &{}+\frac{\nu _{1}}{\eta _{1}}-\frac{\nu _{2}}{\eta _{2}} \biggr] -\frac{ \tau }{\mathrm{T}}\sum _{i=1}^{m} \biggl[\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds\\ &{} +\frac{1}{\varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds \\ &{}+\frac{\mathrm{T}-\tau _{i}}{\varGamma (r-1)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-2} \bigl\vert v(s) \bigr\vert \,ds +(\mathrm{T}-\tau _{i}) \bigl\vert \hat{\varUpsilon _{i}}\bigl(\omega (\tau _{i})\bigr) \bigr\vert + \bigl\vert \varUpsilon _{i}\bigl(\omega (\tau _{i})\bigr) \bigr\vert \biggr] \\ \leq {}&\frac{\tau ^{r}}{\varGamma (r+1)}+\frac{\nu _{1}}{\eta _{1}} -\frac{ \tau }{\mathrm{T}} \biggl[ \frac{\xi _{2}\mathrm{T}^{r}}{\eta _{2}\varGamma (r+1)} +\frac{\nu _{1}}{ \eta _{1}} -\frac{\nu _{2}}{\eta _{2}}+ \frac{m\mathrm{T}^{r}}{\varGamma (r+1)}+ \frac{m\tau ^{r}}{\varGamma (r+1)}+\frac{m \tau ^{r-1}}{\varGamma (r)} \\ &{}+m\bigl(\mathcal{K}'_{\hat{\varUpsilon _{i}}}+\mathcal{K}_{\varUpsilon _{i}} \bigr) \bigl\vert \omega (\tau ) \bigr\vert +m\bigl(\mathcal{L}'_{\hat{\varUpsilon _{i}}}+ \mathcal{L}_{ \varUpsilon _{i}}\bigr) \biggr]. \end{aligned}$$

Taking the norm on both sides, we get \(\|\omega \|_{\mathcal{M}} \leq \mathcal{Q}\). Also, for \(\tau \in \mathcal{J}_{0}\), we can show that \(\|\omega \|_{\mathcal{M}}\leq \mathcal{Q}\). Thus, Ω is bounded. By Schaefer’s fixed point theorem we conclude that \(\mathcal{T}\) has at least one fixed point. Hence, the considered problem (1.3) has at least one solution in \(\mathcal{M}\). The proof is complete. □

Proof of Theorem 2.4

For \(\omega, \overline{\omega }\in \mathcal{M}\) and \(\tau \in \mathcal{J}_{i}\), we have

$$\begin{aligned} & \bigl\vert (\mathcal{T}\overline{\omega }) (\tau )-(\mathcal{T}\omega ) (\tau ) \bigr\vert \\ &\quad\leq \frac{1}{\varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1} \bigl\vert \overline{v}(s)-v(s) \bigr\vert \,ds -\frac{\tau }{\mathrm{T}}\frac{\xi _{2}}{\eta _{2}\varGamma (r)} \int _{0} ^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert \overline{\omega }(s)-\omega (s) \bigr\vert \,ds \\ &\qquad{} -\frac{\tau }{\mathrm{T}}\sum_{i=1}^{m} \biggl[\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert \overline{v}(s)-v(s) \bigr\vert \,ds +\frac{1}{\varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \bigl\vert \overline{v}(s)-v(s) \bigr\vert \,ds \\ &\qquad{} +\frac{\mathrm{T}-\tau _{i}}{\varGamma (r-1)} \int _{\tau _{i-1}}^{\tau _{i}}( \tau _{i}-s)^{r-2} \bigl\vert \overline{v}(s)-v(s) \bigr\vert \,ds +(\mathrm{T}-\tau _{i}) \bigl\vert \hat{\varUpsilon _{i}}\bigl(\overline{ \omega }(\tau _{i})\bigr)-\hat{\varUpsilon _{i}}\bigl( \omega ( \tau _{i})\bigr) \bigr\vert \\ &\qquad{}+ \bigl\vert \varUpsilon _{i}\bigl(\overline{\omega }(\tau _{i})\bigr)-\varUpsilon _{i}\bigl(\omega ( \tau _{i})\bigr) \bigr\vert \biggr], \end{aligned}$$
(B.6)

where \(v, \overline{v}\in \mathcal{M}\) are given by

$$ v(\tau )=\mathcal{A}\bigl(\tau,\omega (\tau ),v(\tau )\bigr) + \int _{0}^{\tau }\frac{( \tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B}\bigl(s, \omega (s),v(s)\bigr)\,ds $$

and

$$ \overline{v}(\tau )=\mathcal{A}\bigl(\tau,\overline{\omega }(\tau ), \overline{v}(\tau )\bigr) + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{ \varGamma (\delta )} \mathcal{B}\bigl(s, \overline{\omega }(s),\overline{v}(s)\bigr)\,ds. $$

Using \([A_{1}]\), we have

$$\begin{aligned} &\bigl\vert v(\tau )-\overline{v}(\tau ) \bigr\vert \\ & \quad= \biggl\vert \mathcal{A}\bigl(\tau,\omega ( \tau ),v(\tau )\bigr) + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B}\bigl(s, \omega (s),v(s)\bigr)\,ds \\ &\qquad{}-\mathcal{A}\bigl(\tau,\overline{\omega }(\tau ),\overline{v}(\tau )\bigr) - \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B}\bigl(s, \overline{\omega }(s),\overline{v}(s)\bigr)\,ds \biggr\vert \\ &\quad\leq \bigl\vert \mathcal{A}\bigl(t,\omega (\tau ),v(\tau )\bigr)- \mathcal{A}\bigl(t,\overline{ \omega }(\tau ),\overline{v}(\tau )\bigr) \bigr\vert \\ &\qquad{}+ \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \bigl\vert \mathcal{B}\bigl(s,\omega (s),v(s)\bigr)-\mathcal{B}\bigl(s,\overline{\omega }(s), \overline{v}(s)\bigr) \bigr\vert \,ds \\ &\quad\leq \mathrm{M}_{1} \bigl\vert \omega (\tau )-\overline{\omega }(\tau ) \bigr\vert + \mathrm{N}_{1} \bigl\vert v(\tau )- \overline{v}(\tau ) \bigr\vert \\ &\qquad{}+\frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )} \bigl(\mathrm{M}_{2} \bigl\vert \omega (\tau )-\overline{\omega }(\tau ) \bigr\vert +\mathrm{N}_{2} \bigl\vert v(\tau )- \overline{v}(\tau ) \bigr\vert \bigr). \end{aligned}$$

Thus

$$ \bigl\vert v(\tau )-\overline{v}(\tau ) \bigr\vert \leq \biggl(\frac{\mathrm{M}_{1}}{1- \mathrm{N}_{1}-\mathrm{N}_{2}\frac{\tau ^{\sigma }}{\sigma \varGamma ( \delta )}} +\frac{\mathrm{M}_{2}\frac{\tau ^{\sigma }}{\sigma \varGamma ( \delta )}}{1-\mathrm{N}_{1} -\mathrm{N}_{2}\frac{\tau ^{\sigma }}{ \sigma \varGamma (\delta )}} \biggr) \bigl\vert \omega (\tau )-\overline{\omega }( \tau ) \bigr\vert . $$
(B.7)

Using hypotheses \([A_{1}]\), \([A_{2}]\) and inequalities (B.7) and (B.6), we obtain

$$\begin{aligned} & \bigl\vert (\mathcal{T}\omega ) (\tau )-(\mathcal{T}\overline{\omega }) ( \tau ) \bigr\vert \\ &\quad \leq \biggl[ \biggl(\frac{\tau ^{r}}{\varGamma (r+1)}-\frac{m\tau \mathrm{T} ^{r-1}}{\varGamma (r+1)}- \frac{m\tau ^{r+1}}{\mathrm{T}\varGamma (r+1)} -\frac{m \tau ^{r}}{\mathrm{T}\varGamma (r)} \biggr) \\ &\qquad{} \times\biggl(\frac{\mathrm{M}_{1}}{1-\mathrm{N}_{1}-\mathrm{N}_{2}\frac{ \tau ^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{M}_{2}\frac{ \tau ^{\sigma }}{\sigma \varGamma (\delta )}}{1-\mathrm{N}_{1} - \mathrm{N}_{2}\frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )}} \biggr) -\frac{ \xi _{2}\tau \mathrm{T}^{r-1}}{\eta _{2}\varGamma (r+1)}- \frac{\tau }{ \mathrm{T}}m(\mathbb{A}+\mathbb{B}) \biggr]\\ &\qquad{}\times \bigl\vert \omega (\tau )- \overline{ \omega }(\tau ) \bigr\vert . \end{aligned}$$

Now taking the norm on both sides, we have

$$\begin{aligned} & \Vert \mathcal{T}\omega -\mathcal{T}\overline{\omega } \Vert _{\mathcal{M}} \\ &\quad \leq \biggl[ \biggl(\frac{m\mathrm{T}^{r}}{\varGamma (r+1)} +\frac{m\mathrm{T} ^{r-1}}{\varGamma (r)} \biggr) \biggl(\frac{\mathrm{M}_{1}}{1-\mathrm{N} _{1}-\mathrm{N}_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{M}_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma ( \delta )}}{1-\mathrm{N}_{1} -\mathrm{N}_{2}\frac{\mathrm{T}^{\sigma }}{ \sigma \varGamma (\delta )}} \biggr) \\ &\qquad{} +\frac{\xi _{2}\mathrm{T}^{r}}{\eta _{2}\varGamma (r+1)}+m(\mathbb{A}+ \mathbb{B}) \biggr] \Vert \omega -\overline{\omega } \Vert _{\mathcal{M}}. \end{aligned}$$

Hence, the operator \(\mathcal{T}\) is a contraction. Thus \(\mathcal{T}\) has a unique fixed point, so the problem (1.3) has a unique solution. □

Proof of Theorem 2.7

Construct the closed ball \(\mathscr{B}=\{(\omega,y)\in \mathcal{X} \times \mathcal{Y}: \|(\omega,y)\|\leq \mathbf{R}\}\). Split the operator \(\mathcal{T}\) into two parts as \(\mathcal{T}=\mathcal{F}+ \mathcal{G}\) with \(\mathcal{F}=(\mathcal{F}_{r},\mathcal{F}_{p})\) and \(\mathcal{G}=(\mathcal{G}_{r},\mathcal{G}_{p})\), where

$$\begin{aligned} &\mathcal{F}_{r}(\omega,y) (\tau ) = \frac{1}{\varGamma (r)} \int _{0}^{ \tau }(\tau -s)^{r-1}v(s)\,ds - \frac{\tau }{\mathrm{T}}\frac{\xi _{2}}{ \eta _{2}\varGamma (r)} \int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}\omega (s)\,ds \\ &\phantom{\mathcal{F}_{r}(\omega,y) (\tau ) =}{}-\frac{\tau }{\mathrm{T}}\sum_{i=1}^{m} \biggl[\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}v(s)\,ds +\frac{1}{ \varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1}v(s) \,ds \\ &\phantom{\mathcal{F}_{r}(\omega,y) (\tau ) =}{}+\frac{\mathrm{T}-\tau _{i}}{\varGamma (r-1)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-2}v(s) \,ds \biggr], \\ &\mathcal{F}_{p}(\omega,y) (\tau ) =\frac{1}{\varGamma (p)} \int _{0}^{ \tau }(\tau -s)^{p-1}z(s)\,ds - \frac{\tau }{\mathrm{T}}\frac{\xi _{4}}{ \eta _{4}\varGamma (p)} \int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{p-1}y(s)\,ds \\ &\phantom{\mathcal{F}_{p}(\omega,y) (\tau ) =}{}-\frac{\tau }{\mathrm{T}}\sum_{j=1}^{n} \biggl[\frac{1}{\varGamma (p)} \int _{\tau _{j}}^{\mathrm{T}}(\mathrm{T}-s)^{p-1}z(s)\,ds +\frac{1}{ \varGamma (p)} \int _{\tau _{j-1}}^{\tau _{j}}(\tau _{j}-s)^{p-1}z(s) \,ds \\ &\phantom{\mathcal{F}_{p}(\omega,y) (\tau ) =}{}+\frac{\mathrm{T}-\tau _{j}}{\varGamma (p-1)} \int _{\tau _{j-1}}^{\tau _{j}}(\tau _{j}-s)^{p-2}z(s) \,ds \biggr], \\ &\mathcal{G}_{r}(\omega ) (\tau )=\frac{\nu _{1}}{\eta _{1}}+ \frac{\tau }{ \mathrm{T}} \biggl[\frac{\nu _{2}}{\eta _{2}} -\frac{\nu _{1}}{\eta _{1}} \biggr] - \frac{\tau }{\mathrm{T}}\sum_{i=1}^{m} \bigl[( \mathrm{T}-\tau _{i})\hat{\varUpsilon _{i}}\bigl(\omega (\tau _{i})\bigr)+\varUpsilon _{i}\bigl(\omega (\tau _{i})\bigr) \bigr], \end{aligned}$$

and

$$ \mathcal{G}_{p}(y) (\tau )=\frac{\nu _{3}}{\eta _{3}}+\frac{\tau }{ \mathrm{T}} \biggl[\frac{\nu _{4}}{\eta _{4}} -\frac{\nu _{3}}{\eta _{3}} \biggr] -\frac{\tau }{\mathrm{T}}\sum _{j=1}^{n} \bigl[(\mathrm{T}-\tau _{j})\hat{\varUpsilon _{j}}\bigl(y(\tau _{j}) \bigr)+\varUpsilon _{j}\bigl(y(\tau _{j})\bigr) \bigr]. $$

Clearly, \(\mathcal{T}_{r}=\mathcal{F}_{r}+\mathcal{G}_{r} \text{ and } \mathcal{T}_{p}=\mathcal{F}_{p}+\mathcal{G}_{p}\).

The first step is to show that \(\mathcal{T}(\omega,y)(\tau )= \mathcal{F}(\omega,y)(\tau )+\mathcal{G}(\omega,y)(\tau )\in \mathscr{B}\) for all \((\omega,y)\in \mathscr{B}\).

For any \((\omega,y)\in \mathscr{B}\), consider

$$\begin{aligned} &\bigl\vert (\mathcal{T}_{r}\omega ) (\tau ) \bigr\vert \\ &\quad \leq \frac{1}{\varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds+\frac{\nu _{1}}{\eta _{1}} \\ &\qquad{} -\frac{\tau }{\mathrm{T}} \biggl[ \frac{\xi _{2}}{\eta _{2}\varGamma (r)} \int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert \omega (s) \bigr\vert \,ds +\frac{\nu _{1}}{\eta _{1}}-\frac{\nu _{2}}{\eta _{2}} \biggr] \\ &\qquad{} -\frac{\tau }{\mathrm{T}}\sum_{i=1}^{m} \biggl[\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds +\frac{1}{ \varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\mathrm{T}-\tau _{i}}{\varGamma (r-1)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-2} \bigl\vert v(s) \bigr\vert \,ds +(\mathrm{T}-\tau _{i}) \bigl\vert \hat{\varUpsilon _{i}}\bigl(\omega (\tau _{i})\bigr) \bigr\vert + \bigl\vert \varUpsilon _{i}\bigl(\omega (\tau _{i})\bigr) \bigr\vert \biggr]. \end{aligned}$$
(B.8)

Using \([\tilde{A_{4}}]\) for \(\tau \in \mathcal{J}_{i}\), we have

$$\begin{aligned} \bigl\vert v(\tau ) \bigr\vert &\leq \bigl\vert \mathcal{A}\bigl(\tau,y( \tau ),v(\tau )\bigr) \bigr\vert + \int _{0}^{ \tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \bigl\vert \mathcal{B}\bigl(s,y(s),v(s)\bigr) \bigr\vert \,ds \\ &\leq a_{1}(\tau )+b_{1}(\tau ) \bigl\vert y(\tau ) \bigr\vert +c_{1}(\tau ) \bigl\vert v(\tau ) \bigr\vert + \frac{ \tau ^{\sigma }}{\sigma \varGamma (\delta )} \bigl(a_{2}(\tau )+b_{2}( \tau ) \bigl\vert y(\tau ) \bigr\vert +c_{2}(\tau ) \bigl\vert v(\tau ) \bigr\vert \bigr) \\ &\leq a_{1}^{*}+b_{1}^{*} \Vert y \Vert _{\mathcal{X}}+c_{1}^{*} \Vert v \Vert _{ \mathcal{X}} +\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )} \bigl(a_{2}^{*}+b_{2}^{*} \Vert y \Vert _{\mathcal{X}}+c_{2}^{*} \Vert v \Vert _{ \mathcal{X}} \bigr). \end{aligned}$$

Therefore we get

$$ \bigl\vert v(\tau ) \bigr\vert \leq \Vert v \Vert _{\mathcal{X}}\leq \frac{a_{1}^{*}+b_{1}^{*} \Vert y \Vert _{\mathcal{X}}}{1-c_{1}^{*}-c_{2}^{*}\frac{\mathrm{T}^{\sigma }}{ \sigma \varGamma (\delta )}} +\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}\frac{a_{2}^{*} +b_{2}^{*} \Vert y \Vert _{\mathcal{X}}}{1-c_{1}^{*}-c _{2}^{*}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}}= \hbar. $$
(B.9)

Using (B.9) and \([\tilde{A_{6}}]\), relation (B.8) becomes

$$\begin{aligned} \bigl\vert \mathcal{T}_{r}\omega (\tau ) \bigr\vert \leq{}& \frac{\hbar \tau ^{r}}{\varGamma (r+1)}+\frac{\nu _{1}}{\eta _{1}} -\frac{ \xi _{2}\tau \mathrm{T}^{r-1}}{\varGamma (r+1)}-\frac{\tau \nu _{1}}{ \mathrm{T}\eta _{1}}+ \frac{\tau \nu _{2}}{\mathrm{T}\eta _{2}} -\frac{m \tau \hbar \mathrm{T}^{r-1}}{\varGamma (r+1)}\\ &{}-\frac{m\hbar \tau ^{r+1}}{ \mathrm{T}\varGamma (r+1)} -\frac{m\hbar \tau ^{r-1}}{\varGamma (r)} \\ &{}-\bigl(\mathcal{K}'_{\hat{\varUpsilon _{i}}}+\mathcal{K}_{\varUpsilon _{i}} \bigr) \bigl\vert \omega (\tau ) \bigr\vert -\bigl(\mathcal{L}'_{\hat{\varUpsilon _{i}}}+ \mathcal{L}_{ \varUpsilon _{i}}\bigr) \\ ={}& \mathbf{C}. \end{aligned}$$

Thus

$$ \Vert \mathcal{T}_{r}\omega \Vert _{\mathcal{X}}\leq \mathbf{C}. $$

Similarly, for \(\tau \in \mathcal{J}_{0}\), we can verify that

$$ \Vert \mathcal{T}_{r}\omega \Vert _{\mathcal{X}}\leq \mathbf{C}. $$

In the similar manner, we have

$$\begin{aligned} &\bigl\vert (\mathcal{T}_{r}y) (\tau ) \bigr\vert \\ &\quad \leq \frac{1}{\varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds+\frac{\nu _{1}}{\eta _{1}} \\ &\qquad{}-\frac{\tau }{\mathrm{T}} \biggl[ \frac{\xi _{2}}{\eta _{2}\varGamma (r)} \int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert \omega (s) \bigr\vert \,ds +\frac{\nu _{1}}{\eta _{1}}-\frac{\nu _{2}}{\eta _{2}} \biggr] \\ &\qquad{} -\frac{\tau }{\mathrm{T}}\sum_{i=1}^{m} \biggl[\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds +\frac{1}{ \varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\mathrm{T}-\tau _{i}}{\varGamma (r-1)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-2} \bigl\vert v(s) \bigr\vert \,ds +(\mathrm{T}-\tau _{i}) \bigl\vert \hat{\varUpsilon _{i}}\bigl(\omega (\tau _{i})\bigr) \bigr\vert + \bigl\vert \varUpsilon _{i}\bigl(\omega (\tau _{i})\bigr) \bigr\vert \biggr]. \end{aligned}$$
(B.10)

Using \([\tilde{A_{4}}]\) for \(\tau \in \mathcal{J}_{i}\), we have

$$\begin{aligned} \bigl\vert v(\tau ) \bigr\vert &\leq \bigl\vert \mathcal{A}\bigl(\tau,y( \tau ),v(\tau )\bigr) \bigr\vert + \int _{0}^{ \tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \bigl\vert \mathcal{B}\bigl(s,y(s),v(s)\bigr) \bigr\vert \,ds \\ &\leq a_{1}(\tau )+b_{1}(\tau ) \bigl\vert y(\tau ) \bigr\vert +c_{1}(\tau ) \bigl\vert v(\tau ) \bigr\vert + \frac{ \tau ^{\sigma }}{\sigma \varGamma (\delta )} \bigl(a_{2}(\tau )+b_{2}( \tau ) \bigl\vert y(\tau ) \bigr\vert +c_{2}(\tau ) \bigl\vert v(\tau ) \bigr\vert \bigr) \\ &\leq a_{1}^{*}+b_{1}^{*} \Vert y \Vert _{\mathcal{X}}+c_{1}^{*} \Vert v \Vert _{ \mathcal{X}} +\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )} \bigl(a_{2}^{*}+b_{2}^{*} \Vert y \Vert _{\mathcal{X}}+c_{2}^{*} \Vert v \Vert _{ \mathcal{X}} \bigr). \end{aligned}$$

Therefore we get

$$ \bigl\vert v(\tau ) \bigr\vert \leq \Vert v \Vert _{\mathcal{X}}\leq \frac{a_{1}^{*}+b_{1}^{*} \Vert y \Vert _{\mathcal{X}}}{1-c_{1}^{*}-c_{2}^{*}\frac{\mathrm{T}^{\sigma }}{ \sigma \varGamma (\delta )}} +\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}\frac{a_{2}^{*} +b_{2}^{*} \Vert y \Vert _{\mathcal{X}}}{1-c_{1}^{*}-c _{2}^{*}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}}= \hbar. $$
(B.11)

Using (B.11) and \([\tilde{A_{6}}]\), relation (B.10) becomes

$$\begin{aligned} \bigl\vert \mathcal{T}_{r}y(\tau ) \bigr\vert \leq{}& \frac{\hbar \tau ^{r}}{\varGamma (r+1)}+\frac{ \nu _{1}}{\eta _{1}} -\frac{\xi _{2}\tau \mathrm{T}^{r-1}}{\varGamma (r+1)}-\frac{ \tau \nu _{1}}{\mathrm{T}\eta _{1}}+ \frac{\tau \nu _{2}}{\mathrm{T}\eta _{2}} -\frac{\tau \hbar \mathrm{T}^{r-1}}{\varGamma (r+1)}-\frac{m\hbar \tau ^{r+1}}{\mathrm{T}\varGamma (r+1)} -\frac{m\hbar \tau ^{r-1}}{\varGamma (r)} \\ &{}-\bigl(\mathcal{K}'_{\hat{\varUpsilon _{i}}}+\mathcal{K}_{\varUpsilon _{i}} \bigr) \bigl\vert \omega (\tau ) \bigr\vert -\bigl(\mathcal{L}'_{\hat{\varUpsilon _{i}}}+ \mathcal{L}_{ \varUpsilon _{i}}\bigr) \\ ={}& \mathbf{C}. \end{aligned}$$

Thus

$$ \Vert \mathcal{T}_{r}y \Vert _{\mathcal{X}}\leq \mathbf{C}. $$

Similarly, for \(\tau \in \mathcal{J}_{0}\), we can verify that

$$ \Vert \mathcal{T}_{r}y \Vert _{\mathcal{X}}\leq \mathbf{C}. $$

Hence

$$ \bigl\Vert \mathcal{T}_{r}(\omega,y) \bigr\Vert _{\mathcal{X}} \leq \mathbf{C}. $$

Now, for any \((\omega,y)\in \mathscr{B}\), consider

$$\begin{aligned} &\bigl\vert (\mathcal{T}_{p}\omega ) (\tau ) \bigr\vert \\ &\quad \leq \frac{1}{\varGamma (p)} \int _{0}^{\tau }(\tau -s)^{p-1} \bigl\vert z(s) \bigr\vert \,ds+\frac{\nu _{3}}{\eta _{3}} \\ &\qquad{} -\frac{\tau }{\mathrm{T}} \biggl[ \frac{\xi _{4}}{\eta _{4}\varGamma (p)} \int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{p-1} \bigl\vert y(s) \bigr\vert \,ds +\frac{ \nu _{3}}{\eta _{3}}-\frac{\nu _{4}}{\eta _{4}} \biggr] \\ &\qquad{} -\frac{\tau }{\mathrm{T}}\sum_{j=1}^{n} \biggl[\frac{1}{\varGamma (p)} \int _{\tau _{j}}^{\mathrm{T}}(\mathrm{T}-s)^{p-1} \bigl\vert z(s) \bigr\vert \,ds +\frac{1}{ \varGamma (p)} \int _{\tau _{j-1}}^{\tau _{j}}(\tau _{j}-s)^{p-1} \bigl\vert z(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\mathrm{T}-\tau _{j}}{\varGamma (p-1)} \int _{\tau _{j-1}}^{\tau _{j}}(\tau _{j}-s)^{p-2} \bigl\vert z(s) \bigr\vert \,ds +(\mathrm{T}-\tau _{j}) \bigl\vert \hat{\varUpsilon _{j}}\bigl(y(\tau _{j})\bigr) \bigr\vert + \bigl\vert \varUpsilon _{j}\bigl(y(\tau _{j})\bigr) \bigr\vert \biggr]. \end{aligned}$$
(B.12)

Using \([\tilde{A_{5}}]\) for \(\tau \in \mathcal{J}_{j}\), we have

$$\begin{aligned} \bigl\vert z(\tau ) \bigr\vert &\leq \bigl\vert \mathcal{A}' \bigl(\tau,\omega (\tau ),y(\tau )\bigr) \bigr\vert + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \bigl\vert \mathcal{B}'\bigl(s,\omega (s),y(s)\bigr) \bigr\vert \,ds \\ &\leq l_{1}(\tau )+m_{1}(\tau ) \bigl\vert \omega (\tau ) \bigr\vert +n_{1}(\tau ) \bigl\vert y( \tau ) \bigr\vert + \frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )} \bigl(l_{2}( \tau )+m_{2}(\tau ) \bigl\vert \omega (\tau ) \bigr\vert +n_{2}(\tau ) \bigl\vert y(\tau ) \bigr\vert \bigr) \\ &\leq l_{1}^{*}+m_{1}^{*} \Vert \omega \Vert _{\mathcal{Y}}+n_{1}^{*} \Vert y \Vert _{\mathcal{Y}} +\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )} \bigl(l_{2}^{*}+m_{2}^{*} \Vert \omega \Vert _{\mathcal{Y}}+n_{2}^{*} \Vert y \Vert _{ \mathcal{Y}} \bigr). \end{aligned}$$

Therefore we get

$$ \bigl\vert z(\tau ) \bigr\vert \leq \Vert z \Vert _{\mathcal{Y}}\leq \frac{l_{1}^{*}+m_{1}^{*} \Vert \omega \Vert _{\mathcal{Y}}}{1-n_{1}^{*}-n_{2}^{*}\frac{\mathrm{T}^{ \sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{T}^{\sigma }}{ \sigma \varGamma (\delta )}\frac{l_{2}^{*} +m_{2}^{*} \Vert \omega \Vert _{ \mathcal{Y}}}{1-n_{1}^{*}-n_{2}^{*}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}}= \hbar ^{*}. $$
(B.13)

Using (B.13) and \([\tilde{A_{7}}]\), relation (B.12) becomes

$$\begin{aligned} \bigl\vert \mathcal{T}_{p}\omega (\tau ) \bigr\vert \leq{}& \frac{\hbar ^{*} \tau ^{p}}{ \varGamma (p+1)}+\frac{\nu _{3}}{\eta _{3}} -\frac{\xi _{4}\tau \mathrm{T} ^{p-1}}{\varGamma (p+1)}-\frac{\tau \nu _{3}}{\mathrm{T}\eta _{3}}+ \frac{ \tau \nu _{4}}{\mathrm{T}\eta _{4}} -\frac{n\tau \hbar ^{*} \mathrm{T} ^{p-1}}{\varGamma (p+1)}\\ &{}- \frac{n\hbar ^{*} \tau ^{p+1}}{\mathrm{T}\varGamma (p+1)} -\frac{n\hbar ^{*} \tau ^{p-1}}{\varGamma (p)} \\ &{}-\bigl(\mathcal{K}'_{\hat{\varUpsilon _{j}}}+\mathcal{K}_{\varUpsilon _{j}} \bigr) \bigl\vert \omega (\tau ) \bigr\vert -\bigl(\mathcal{L}'_{\hat{\varUpsilon _{j}}}+ \mathcal{L}_{ \varUpsilon _{j}}\bigr) \\ ={}& \mathbf{C}^{*}. \end{aligned}$$

Thus

$$ \Vert \mathcal{T}_{p}\omega \Vert _{\mathcal{Y}}\leq \mathbf{C}^{*}. $$

Similarly, for \(\tau \in \mathcal{J}_{0}\), we can verify that

$$ \Vert \mathcal{T}_{p}\omega \Vert _{\mathcal{Y}}\leq \mathbf{C}^{*}. $$

In a similar manner, we have

$$\begin{aligned} &\bigl\vert (\mathcal{T}_{p}y) (\tau ) \bigr\vert \\ &\quad \leq \frac{1}{\varGamma (p)} \int _{0}^{\tau }(\tau -s)^{p-1} \bigl\vert z(s) \bigr\vert \,ds+\frac{\nu _{3}}{\eta _{3}} \\ &\qquad{}-\frac{\tau }{\mathrm{T}} \biggl[ \frac{\xi _{4}}{\eta _{4}\varGamma (p)} \int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{p-1} \bigl\vert y(s) \bigr\vert \,ds +\frac{ \nu _{3}}{\eta _{3}}-\frac{\nu _{4}}{\eta _{4}} \biggr] \\ &\qquad{} -\frac{\tau }{\mathrm{T}}\sum_{j=1}^{n} \biggl[\frac{1}{\varGamma (p)} \int _{\tau _{j}}^{\mathrm{T}}(\mathrm{T}-s)^{p-1} \bigl\vert z(s) \bigr\vert \,ds +\frac{1}{ \varGamma (p)} \int _{\tau _{j-1}}^{\tau _{j}}(\tau _{j}-s)^{p-1} \bigl\vert z(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\mathrm{T}-\tau _{j}}{\varGamma (p-1)} \int _{\tau _{j-1}}^{\tau _{j}}(\tau _{j}-s)^{p-2} \bigl\vert z(s) \bigr\vert \,ds +(\mathrm{T}-\tau _{j}) \bigl\vert \hat{\varUpsilon _{j}}\bigl(y(\tau _{j})\bigr) \bigr\vert + \bigl\vert \varUpsilon _{j}\bigl(y(\tau _{j})\bigr) \bigr\vert \biggr]. \end{aligned}$$
(B.14)

Using \([\tilde{A_{5}}]\) for \(\tau \in \mathcal{J}_{j}\), we have

$$\begin{aligned} \bigl\vert z(\tau ) \bigr\vert \leq{}& \bigl\vert \mathcal{A}' \bigl(\tau,\omega (\tau ),y(\tau )\bigr) \bigr\vert + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \bigl\vert \mathcal{B}'\bigl(s,\omega (s),y(s)\bigr) \bigr\vert \,ds \\ \leq{}& l_{1}(\tau )+m_{1}(\tau ) \bigl\vert \omega (\tau ) \bigr\vert +n_{1}(\tau ) \bigl\vert y( \tau ) \bigr\vert + \frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )} \bigl(l_{2}( \tau )+m_{2}(\tau ) \bigl\vert \omega (\tau ) \bigr\vert +n_{2}(\tau ) \bigl\vert y(\tau ) \bigr\vert \bigr) \\ \leq{} &l_{1}^{*}+m_{1}^{*} \Vert \omega \Vert _{\mathcal{Y}}+n_{1}^{*} \Vert y \Vert _{\mathcal{Y}} +\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )} \bigl(l_{2}^{*}+m_{2}^{*} \Vert \omega \Vert _{\mathcal{Y}}+n_{2}^{*} \Vert y \Vert _{ \mathcal{Y}} \bigr). \end{aligned}$$

Therefore we get

$$ \bigl\vert z(\tau ) \bigr\vert \leq \Vert z \Vert _{\mathcal{Y}}\leq \frac{l_{1}^{*}+m_{1}^{*} \Vert \omega \Vert _{\mathcal{Y}}}{1-n_{1}^{*}-n_{2}^{*}\frac{\mathrm{T}^{ \sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{T}^{\sigma }}{ \sigma \varGamma (\delta )}\frac{l_{2}^{*} +m_{2}^{*} \Vert \omega \Vert _{ \mathcal{Y}}}{1-n_{1}^{*}-n_{2}^{*}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}}= \hbar ^{*}. $$
(B.15)

Using (B.15) and \([\tilde{A_{7}}]\), relation (B.14) becomes

$$\begin{aligned} \bigl\vert \mathcal{T}_{p}y(\tau ) \bigr\vert \leq{}& \frac{\hbar ^{*} \tau ^{p}}{\varGamma (p+1)}+\frac{\nu _{3}}{\eta _{3}} -\frac{ \xi _{4}\tau \mathrm{T}^{p-1}}{\varGamma (p+1)}-\frac{\tau \nu _{3}}{ \mathrm{T}\eta _{3}}+ \frac{\tau \nu _{4}}{\mathrm{T}\eta _{4}} -\frac{n \tau \hbar ^{*} \mathrm{T}^{p-1}}{\varGamma (p+1)}\\ &{}-\frac{n\hbar ^{*} \tau ^{p+1}}{\mathrm{T}\varGamma (p+1)} -\frac{n\hbar ^{*} \tau ^{p-1}}{\varGamma (p)} \\ &{}-\bigl(\mathcal{K}'_{\hat{\varUpsilon _{j}}}+\mathcal{K}_{\varUpsilon _{j}} \bigr) \bigl\vert \omega (\tau ) \bigr\vert -\bigl(\mathcal{L}'_{\hat{\varUpsilon _{j}}}+ \mathcal{L}_{ \varUpsilon _{j}}\bigr) \\ ={}& \mathbf{C}^{*}. \end{aligned}$$

Thus

$$ \Vert \mathcal{T}_{p}y \Vert _{\mathcal{Y}}\leq \mathbf{C}^{*}. $$

Similarly, for \(\tau \in \mathcal{J}_{0}\), we can verify that

$$ \Vert \mathcal{T}_{p}y \Vert _{\mathcal{Y}}\leq \mathbf{C}^{*}. $$

Hence

$$ \bigl\Vert \mathcal{T}_{p}(\omega,y) \bigr\Vert _{\mathcal{Y}} \leq \mathbf{C}^{*}, $$

and thus

$$ \bigl\Vert \mathcal{T}(\omega,y) \bigr\Vert _{\mathcal{X}\times \mathcal{Y}}\leq \bigl\Vert \mathcal{T}_{r}(\omega,y)+\mathcal{T}_{p}(\omega,y) \bigr\Vert _{\mathcal{X} \times \mathcal{Y}}\leq \mathbf{C}+\mathbf{C}^{*}=\mathbf{R}, $$

which implies that \(\mathcal{T}(\mathscr{B})\subseteq \mathscr{B}\).

Second, we show that \(\mathcal{G}\) is a contraction. For any \((\omega,y), (\overline{\omega },\overline{y})\in \mathscr{B}\), we have

$$\begin{aligned} \bigl\vert \mathcal{G}_{r}(\omega )-\mathcal{G}_{r}( \overline{\omega }) \bigr\vert &\leq \frac{ \tau }{\mathrm{T}}\sum _{i=1}^{m} \bigl[(\mathrm{T}-\tau _{i}) \bigl\vert \hat{\varUpsilon _{i}}\bigl(\omega (\tau _{i}) \bigr)-\hat{\varUpsilon _{i}}\bigl(\overline{ \omega }(\tau _{i})\bigr) \bigr\vert + \bigl\vert \varUpsilon _{i}\bigl( \omega (\tau _{i})\bigr)-\varUpsilon _{i}\bigl(\overline{ \omega }(\tau _{i})\bigr) \bigr\vert \bigr] \\ &\leq m(\mathbb{A}_{\hat{\varUpsilon _{i}}}+\mathbb{A}_{\varUpsilon _{i}}) \Vert \omega - \overline{\omega } \Vert _{\mathcal{X}}. \end{aligned}$$

Similarly,

$$\begin{aligned} \bigl\vert \mathcal{G}_{p}(y)-\mathcal{G}_{p}( \overline{y}) \bigr\vert &\leq \frac{\tau }{ \mathrm{T}}\sum _{j=1}^{n} \bigl[(\mathrm{T}-\tau _{j}) \bigl\vert \hat{\varUpsilon _{j}}\bigl(y(\tau _{j})\bigr)- \hat{\varUpsilon _{j}}\bigl(\overline{y}(\tau _{j})\bigr) \bigr\vert + \bigl\vert \varUpsilon _{j}\bigl(y(\tau _{j}) \bigr)-\varUpsilon _{j}\bigl(\overline{y}(\tau _{j})\bigr) \bigr\vert \bigr] \\ &\leq n(\mathbb{A}_{\hat{\varUpsilon _{j}}}+\mathbb{A}_{\varUpsilon _{j}}) \Vert y- \overline{y} \Vert _{\mathcal{Y}}. \end{aligned}$$

From the assumptions \(m(\mathbb{A}_{\hat{\varUpsilon _{i}}}+\mathbb{A} _{\varUpsilon _{i}})<1\) and \(n(\mathbb{A}_{\hat{\varUpsilon _{j}}}+ \mathbb{A}_{\varUpsilon _{j}})<1\) it follows that \(\mathcal{G}\) is a contraction.

Our final step is to show that \(\mathcal{F}=(\mathcal{F}_{r}+ \mathcal{F}_{p})\) is compact. The continuity of \(\mathcal{F}\) follows from the continuity of \(\mathcal{A}, \mathcal{B}, \mathcal{A}', \mathcal{B}'\). For \((\omega,y)\in \mathscr{B}\), we have

$$\begin{aligned} \bigl\vert \mathcal{F}_{r}\omega (\tau ) \bigr\vert \leq{}& \frac{1}{\varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds -\frac{ \tau }{\mathrm{T}}\frac{\xi _{2}}{\eta _{2}\varGamma (r)} \int _{0}^{ \mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert \omega (s) \bigr\vert \,ds \\ &{} -\frac{\tau }{\mathrm{T}}\sum_{i=1}^{m} \biggl[\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds +\frac{1}{ \varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds \\ &{} +\frac{\mathrm{T}-\tau _{i}}{\varGamma (r-1)} \int _{\tau _{i-1}}^{\tau _{i}}( \tau _{i}-s)^{r-2} \bigl\vert v(s) \bigr\vert \,ds \biggr]. \end{aligned}$$
(B.16)

By \([\tilde{A_{4}}]\), for \(\tau \in \mathcal{J}_{i}\), we have

$$\begin{aligned} \bigl\vert v(\tau ) \bigr\vert &\leq \bigl\vert \mathcal{A}\bigl(\tau, \omega (\tau ),v(\tau )\bigr) \bigr\vert + \int _{0}^{\tau }\frac{(\tau -\xi )^{\sigma -1}}{\varGamma (\delta )} \bigl\vert \mathcal{B}\bigl(s,\omega (s),v(s)\bigr) \bigr\vert \,ds \\ &\leq a_{1}(\tau )+b_{1}(\tau ) \bigl\vert \omega (\tau ) \bigr\vert +c_{1}(\tau ) \bigl\vert v( \tau ) \bigr\vert + \frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )} \bigl(a_{2}( \tau )+b_{2}(\tau ) \bigl\vert \omega (\tau ) \bigr\vert +c_{2}(\tau ) \bigl\vert v(\tau ) \bigr\vert \bigr) \\ &\leq a_{1}^{*}+b_{1}^{*} \Vert \omega \Vert _{\mathcal{X}}+c_{1}^{*} \Vert v \Vert _{\mathcal{X}} +\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )} \bigl(a_{2}^{*}+b_{2}^{*} \Vert \omega \Vert _{\mathcal{X}}+c_{2}^{*} \Vert v \Vert _{ \mathcal{X}} \bigr). \end{aligned}$$

Therefore we get

$$ \bigl\vert v(\tau ) \bigr\vert \leq \Vert v \Vert _{\mathcal{X}}\leq \frac{a_{1}^{*}+b_{1}^{*} \Vert \omega \Vert _{\mathcal{X}}}{1-c_{1}^{*}-c_{2}^{*}\frac{\mathrm{T}^{ \sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{T}^{\sigma }}{ \sigma \varGamma (\delta )}\frac{a_{2}^{*} +b_{2}^{*} \Vert \omega \Vert _{ \mathcal{X}}}{1-c_{1}^{*}-c_{2}^{*}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}}= \hbar. $$
(B.17)

Using (B.17) in (B.16), after simplification, we get

$$ \bigl\vert \mathcal{F}_{r}\omega (\tau ) \bigr\vert \leq \wp _{1}. $$

In a similar manner, we have

$$\begin{aligned} \bigl\vert \mathcal{F}_{r}y(\tau ) \bigr\vert \leq{}& \frac{1}{\varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds -\frac{ \tau }{\mathrm{T}}\frac{\xi _{2}}{\eta _{2}\varGamma (r)} \int _{0}^{ \mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert \omega (s) \bigr\vert \,ds \\ &{} -\frac{\tau }{\mathrm{T}}\sum_{i=1}^{m} \biggl[\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds +\frac{1}{ \varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds \\ &{} +\frac{\mathrm{T}-\tau _{i}}{\varGamma (r-1)} \int _{\tau _{i-1}}^{\tau _{i}}( \tau _{i}-s)^{r-2} \bigl\vert v(s) \bigr\vert \,ds \biggr]. \end{aligned}$$
(B.18)

By \([\tilde{A_{4}}]\), for \(\tau \in \mathcal{J}_{i}\), we have

$$\begin{aligned} \bigl\vert v(\tau ) \bigr\vert &\leq \bigl\vert \mathcal{A}\bigl(\tau, \omega (\tau ),v(\tau )\bigr) \bigr\vert + \int _{0}^{\tau }\frac{(\tau -\xi )^{\sigma -1}}{\varGamma (\delta )} \bigl\vert \mathcal{B}\bigl(s,\omega (s),v(s)\bigr) \bigr\vert \,ds \\ &\leq a_{1}(\tau )+b_{1}(\tau ) \bigl\vert \omega (\tau ) \bigr\vert +c_{1}(\tau ) \bigl\vert v( \tau ) \bigr\vert + \frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )} \bigl(a_{2}( \tau )+b_{2}(\tau ) \bigl\vert \omega (\tau ) \bigr\vert +c_{2}(\tau ) \bigl\vert v(\tau ) \bigr\vert \bigr) \\ &\leq a_{1}^{*}+b_{1}^{*} \Vert \omega \Vert _{\mathcal{X}}+c_{1}^{*} \Vert v \Vert _{\mathcal{X}} +\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )} \bigl(a_{2}^{*}+b_{2}^{*} \Vert \omega \Vert _{\mathcal{X}}+c_{2}^{*} \Vert v \Vert _{ \mathcal{X}} \bigr). \end{aligned}$$

Therefore we get

$$ \bigl\vert v(\tau ) \bigr\vert \leq \Vert v \Vert _{\mathcal{X}}\leq \frac{a_{1}^{*}+b_{1}^{*} \Vert \omega \Vert _{\mathcal{X}}}{1-c_{1}^{*}-c_{2}^{*}\frac{T^{\sigma }}{ \sigma \varGamma (\delta )}} +\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}\frac{a_{2}^{*} +b_{2}^{*} \Vert \omega \Vert _{\mathcal{X}}}{1-c _{1}^{*}-c_{2}^{*} \frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}}= \hbar. $$
(B.19)

Using (B.19) in (B.18), after simplification, we get

$$ \bigl\vert \mathcal{F}_{r}\omega (\tau ) \bigr\vert \leq \wp _{1}. $$

Hence

$$ \bigl\Vert \mathcal{F}_{r}(\omega,y) \bigr\Vert _{\mathcal{X}} \leq \wp _{1}. $$

Now for any \((\omega,y)\in \mathscr{B}\), we have

$$\begin{aligned} \bigl\vert \mathcal{F}_{p}\omega (\tau ) \bigr\vert \leq{}& \frac{1}{\varGamma (p)} \int _{0}^{\tau }(\tau -s)^{p-1} \bigl\vert z(s) \bigr\vert \,ds -\frac{ \tau }{\mathrm{T}}\frac{\xi _{4}}{\eta _{4}\varGamma (p)} \int _{0}^{ \mathrm{T}}(\mathrm{T}-s)^{p-1} \bigl\vert y(s) \bigr\vert \,ds \\ &{} -\frac{\tau }{\mathrm{T}}\sum_{j=1}^{n} \biggl[\frac{1}{\varGamma (p)} \int _{\tau _{j}}^{\mathrm{T}}(\mathrm{T}-s)^{p-1} \bigl\vert z(s) \bigr\vert \,ds +\frac{1}{ \varGamma (p)} \int _{\tau _{j-1}}^{\tau _{j}}(\tau _{j}-s)^{p-1} \bigl\vert z(s) \bigr\vert \,ds \\ &{}+\frac{\mathrm{T}-\tau _{j}}{\varGamma (p-1)} \int _{\tau _{j-1}}^{\tau _{j}}(\tau _{j}-s)^{p-2} \bigl\vert z(s) \bigr\vert \,ds \biggr]. \end{aligned}$$
(B.20)

By \([\tilde{A_{5}}]\), for \(\tau \in \mathcal{J}_{j}\), we have

$$\begin{aligned} \bigl\vert z(\tau ) \bigr\vert &\leq \bigl\vert \mathcal{A}' \bigl(\tau,\omega (\tau ),z(\tau )\bigr) \bigr\vert + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \bigl\vert \mathcal{B}'\bigl(s,\omega (s),z(s)\bigr) \bigr\vert \,ds \\ &\leq l_{1}(\tau )+m_{1}(\tau ) \bigl\vert x(\tau ) \bigr\vert +n_{1}(\tau ) \bigl\vert z(\tau ) \bigr\vert + \frac{ \tau ^{\sigma }}{\sigma \varGamma (\delta )} \bigl(l_{2}(\tau )+m_{2}( \tau ) \bigl\vert \omega (\tau ) \bigr\vert +n_{2}(\tau ) \bigl\vert z(\tau ) \bigr\vert \bigr) \\ &\leq l_{1}^{*}+m_{1}^{*} \Vert \omega \Vert _{\mathcal{Y}}+n_{1}^{*} \Vert z \Vert _{\mathcal{Y}} +\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )} \bigl(l_{2}^{*}+m_{2}^{*} \Vert \omega \Vert _{\mathcal{Y}}+n_{2}^{*} \Vert z \Vert _{ \mathcal{Y}} \bigr). \end{aligned}$$

Therefore we get

$$ \bigl\vert z(\tau ) \bigr\vert \leq \Vert z \Vert _{\mathcal{Y}}\leq \frac{l_{1}^{*}+m_{1}^{*} \Vert \omega \Vert _{\mathcal{Y}}}{1-n_{1}^{*}-n_{2}^{*}\frac{\mathrm{T}^{ \sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{T}^{\sigma }}{ \sigma \varGamma (\delta )}\frac{l_{2}^{*} +m_{2}^{*} \Vert \omega \Vert _{ \mathcal{Y}}}{1-n_{1}^{*}-n_{2}^{*}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}}= \hbar. $$
(B.21)

Using (B.21) in (B.20), after simplification, we get

$$ \Vert \mathcal{F}_{p}\omega \Vert _{\mathcal{Y}}\leq \wp _{2}. $$

In a similar manner, we have

$$\begin{aligned} \bigl\vert \mathcal{F}_{p}y(\tau ) \bigr\vert \leq{} & \frac{1}{\varGamma (p)} \int _{0}^{\tau }(\tau -s)^{p-1} \bigl\vert z(s) \bigr\vert \,ds -\frac{ \tau }{\mathrm{T}}\frac{\xi _{4}}{\eta _{4}\varGamma (p)} \int _{0}^{ \mathrm{T}}(\mathrm{T}-s)^{p-1} \bigl\vert y(s) \bigr\vert \,ds \\ &{} -\frac{\tau }{\mathrm{T}}\sum_{j=1}^{n} \biggl[\frac{1}{\varGamma (p)} \int _{\tau _{j}}^{\mathrm{T}}(\mathrm{T}-s)^{p-1} \bigl\vert z(s) \bigr\vert \,ds +\frac{1}{ \varGamma (p)} \int _{\tau _{j-1}}^{\tau _{j}}(\tau _{j}-s)^{p-1} \bigl\vert z(s) \bigr\vert \,ds \\ &{}+\frac{\mathrm{T}-\tau _{j}}{\varGamma (p-1)} \int _{\tau _{j-1}}^{\tau _{j}}(\tau _{j}-s)^{p-2} \bigl\vert z(s) \bigr\vert \,ds \biggr]. \end{aligned}$$
(B.22)

By \([\tilde{A_{5}}]\), for \(\tau \in \mathcal{J}_{j}\), we have

$$\begin{aligned} \bigl\vert z(\tau ) \bigr\vert &\leq \bigl\vert \mathcal{A}' \bigl(\tau,\omega (\tau ),z(\tau )\bigr) \bigr\vert + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \bigl\vert \mathcal{B}'\bigl(s,\omega (s),z(s)\bigr) \bigr\vert \,ds \\ &\leq l_{1}(\tau )+m_{1}(\tau ) \bigl\vert x(\tau ) \bigr\vert +n_{1}(\tau ) \bigl\vert z(\tau ) \bigr\vert + \frac{ \tau ^{\sigma }}{\sigma \varGamma (\delta )} \bigl(l_{2}(\tau )+m_{2}( \tau ) \bigl\vert \omega (\tau ) \bigr\vert +n_{2}(\tau ) \bigl\vert z(\tau ) \bigr\vert \bigr) \\ &\leq l_{1}^{*}+m_{1}^{*} \Vert \omega \Vert _{\mathcal{Y}}+n_{1}^{*} \Vert z \Vert _{\mathcal{Y}} +\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )} \bigl(l_{2}^{*}+m_{2}^{*} \Vert \omega \Vert _{\mathcal{Y}}+n_{2}^{*} \Vert z \Vert _{ \mathcal{Y}} \bigr). \end{aligned}$$

Therefore we get

$$ \bigl\vert z(\tau ) \bigr\vert \leq \Vert z \Vert _{\mathcal{Y}}\leq \frac{l_{1}^{*}+m_{1}^{*} \Vert \omega \Vert _{\mathcal{Y}}}{1-n_{1}^{*}-n_{2}^{*}\frac{\mathrm{T}^{ \sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{T}^{\sigma }}{ \sigma \varGamma (\delta )}\frac{l_{2}^{*} +m_{2}^{*} \Vert \omega \Vert _{ \mathcal{Y}}}{1-n_{1}^{*}-n_{2}^{*}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}}= \hbar. $$
(B.23)

Using (B.23) in (B.22), after simplification, we get

$$ \Vert \mathcal{F}_{p}y \Vert _{\mathcal{Y}}\leq \wp _{2}. $$

Hence

$$ \bigl\Vert \mathcal{F}_{p}(\omega,y) \bigr\Vert _{\mathcal{Y}} \leq \wp _{2}. $$

Thus

$$ \bigl\Vert \mathcal{F}(\omega,y) \bigr\Vert _{\mathcal{X}\times \mathcal{Y}}\leq \bigl\Vert \mathcal{F}_{r}(\omega,y)+\mathcal{F}_{p}(\omega,y) \bigr\Vert _{\mathcal{X} \times \mathcal{Y}}\leq \wp _{1}+\wp _{2}= \mathbf{R}_{1}, $$

which implies that \(\mathcal{F}\) is uniformly bounded on \(\mathscr{B}\).

Take a bounded subset \(\mathbb{C}\) of \(\mathscr{B}\) and \((\omega,y) \in \mathbb{C}\). Then for \(\tau _{1}, \tau _{2}\in \mathcal{J}_{i}\) with \(0\leq \tau _{1}\leq \tau _{2}\leq 1\), we have

$$\begin{aligned} & \bigl\vert \mathcal{F}_{r}\omega (\tau _{2})-\mathcal{F}_{r}\omega (\tau _{1}) \bigr\vert \\ &\quad\leq \frac{1}{\varGamma (r)} \int _{0}^{\tau _{2}}(\tau _{2}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds +\frac{1}{ \varGamma (r)} \int _{0}^{\tau _{1}}(\tau _{1}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds \\ &\qquad{} -\frac{\tau }{\mathrm{T}}\sum_{0< \tau _{i}< \tau _{2}-\tau _{1}} \biggl[ \frac{1}{ \varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds +\frac{( \tau _{i-1}-\tau _{i})}{\varGamma (r-1)} \int _{\tau _{i}}^{\tau _{i-1}}(\tau _{i}-s)^{r-2} \bigl\vert v(s) \bigr\vert \,ds \\ &\qquad{} -\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\tau _{i-1}}(\tau _{i}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds \biggr] \\ &\quad\leq \frac{1}{\varGamma (r)} \int _{0}^{\tau _{2}}\bigl[(\tau _{2}-s)^{r-1}-( \tau _{1}-s)^{r-1}\bigr] \bigl\vert v(s) \bigr\vert \,ds + \frac{1}{\varGamma (r)} \int _{0}^{\tau _{1}}(\tau _{1}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds \\ &\qquad{} -\frac{\tau }{\mathrm{T}}\sum_{0< \tau _{i}< \tau _{2}-\tau _{1}} \biggl[ \frac{1}{ \varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds +\frac{( \tau _{i-1}-\tau _{i})}{\varGamma (r-1)} \int _{\tau _{i}}^{\tau _{i-1}}(\tau _{i}-s)^{r-2} \bigl\vert v(s) \bigr\vert \,ds \\ &\qquad{}-\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\tau _{i-1}}(\tau _{i}-s)^{r-1} \bigl\vert v(s) \bigr\vert \,ds \biggr]. \end{aligned}$$
(B.24)

Obviously, the right-hand side of inequality (B.24) tends to zero as \(\tau _{1}\rightarrow \tau _{2}\).

Therefore

$$ \bigl\vert \mathcal{F}_{r}\omega (\tau _{2})- \mathcal{F}_{r}\omega (\tau _{1}) \bigr\vert \rightarrow 0 \quad\text{as } \tau _{1}\rightarrow \tau _{2}. $$

Similarly,

$$ \bigl\vert \mathcal{F}_{r}y(\tau _{2})- \mathcal{F}_{r}y(\tau _{1}) \bigr\vert \rightarrow 0 \quad\text{as } \tau _{1}\rightarrow \tau _{2}. $$

Now for any \(\tau _{1}, \tau _{2}\in \mathcal{J}_{j}\) with \(0\leq \tau _{1}\leq \tau _{2}\leq 1\), we have

$$\begin{aligned} & \bigl\vert \mathcal{F}_{p}\omega (\tau _{2})-\mathcal{F}_{p}\omega (\tau _{1}) \bigr\vert \\ &\quad\leq \frac{1}{\varGamma (p)} \int _{0}^{\tau _{2}}(\tau _{2}-s)^{p-1} \bigl\vert z(s) \bigr\vert \,ds +\frac{1}{ \varGamma (p)} \int _{0}^{\tau _{1}}(\tau _{1}-s)^{p-1} \bigl\vert z(s) \bigr\vert \,ds \\ &\qquad{} -\frac{\tau }{\mathrm{T}}\sum_{0< \tau _{j}< \tau _{2}-\tau _{1}} \biggl[ \frac{1}{ \varGamma (p)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{p-1} \bigl\vert z(s) \bigr\vert \,ds +\frac{( \tau _{j-1}-\tau _{j})}{\varGamma (p-1)} \int _{\tau _{j}}^{\tau _{j-1}}(\tau _{j}-s)^{p-2} \bigl\vert z(s) \bigr\vert \,ds \\ &\qquad{} -\frac{1}{\varGamma (p)} \int _{\tau _{j}}^{\tau _{j-1}}(\tau _{j}-s)^{p-1} \bigl\vert z(s) \bigr\vert \,ds \biggr] \\ &\quad\leq \frac{1}{\varGamma (p)} \int _{0}^{\tau _{2}}\bigl[(\tau _{2}-s)^{p-1}-( \tau _{1}-s)^{p-1}\bigr] \bigl\vert z(s) \bigr\vert \,ds + \frac{1}{\varGamma (p)} \int _{0}^{\tau _{1}}(\tau _{1}-s)^{p-1} \bigl\vert z(s) \bigr\vert \,ds \\ &\qquad{} -\frac{\tau }{\mathrm{T}}\sum_{0< \tau _{j}< \tau _{2}-\tau _{1}} \biggl[ \frac{1}{ \varGamma (p)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{p-1} \bigl\vert z(s) \bigr\vert \,ds +\frac{( \tau _{j-1}-\tau _{j})}{\varGamma (p-1)} \int _{\tau _{j}}^{\tau _{j-1}}(\tau _{j}-s)^{p-2} \bigl\vert z(s) \bigr\vert \,ds \\ &\qquad{}-\frac{1}{\varGamma (p)} \int _{\tau _{j}}^{\tau _{j-1}}(\tau _{j}-s)^{p-1} \bigl\vert z(s) \bigr\vert \,ds \biggr]. \end{aligned}$$
(B.25)

Obviously, the right-hand side of inequality (B.25) tends to zero as \(\tau _{1}\rightarrow \tau _{2}\).

Therefore

$$ \bigl\vert \mathcal{F}_{p}\omega (\tau _{2})- \mathcal{F}_{p}\omega (\tau _{1}) \bigr\vert \rightarrow 0 \quad\text{as } \tau _{1}\rightarrow \tau _{2}. $$

Similarly,

$$ \bigl\vert \mathcal{F}_{p}y(\tau _{2})- \mathcal{F}_{p}y(\tau _{1}) \bigr\vert \rightarrow 0 \quad\text{as } \tau _{1}\rightarrow \tau _{2}. $$

Thus

$$ \bigl\vert \mathcal{F}(x,y) (\tau _{2})-\mathcal{F}(x,y) (\tau _{1}) \bigr\vert \rightarrow 0 \quad\text{as } \tau _{1} \rightarrow \tau _{2}. $$

Hence \(\mathcal{F}\) is equicontinuous, and by the Arzelà–Ascoli theorem we obtain that \(\mathcal{F}\) is compact. Finally, by Theorem A.7 system (1.4) has at least one solution, which completes the proof. □

Proof of Theorem 2.8

Suppose \(\omega, \overline{\omega }\in \mathcal{X}\). For \(\tau \in \mathcal{J}_{i}\), we have

$$\begin{aligned} & \bigl\vert \mathcal{T}_{r}\omega (\tau )-\mathcal{T}_{r}\overline{\omega }( \tau ) \bigr\vert \\ &\quad\leq \frac{1}{\varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1} \bigl\vert v(s)-\overline{v}(s) \bigr\vert \,ds -\frac{\tau }{\mathrm{T}}\frac{\xi _{2}}{\eta _{2}\varGamma (r)} \int _{0} ^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert \omega (s)-\overline{\omega }(s) \bigr\vert \,ds \\ &\qquad{} -\frac{\tau }{\mathrm{T}}\sum_{i=1}^{m} \biggl[\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert v(s)-\overline{v}(s) \bigr\vert \,ds +\frac{1}{\varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \bigl\vert v(s)- \overline{v}(s) \bigr\vert \,ds \\ &\qquad{} +\frac{\mathrm{T}-\tau _{i}}{\varGamma (r-1)} \int _{\tau _{i-1}}^{\tau _{i}}( \tau _{i}-s)^{r-2} \bigl\vert v(s)-\overline{v}(s) \bigr\vert \,ds +(\mathrm{T}-\tau _{i}) \bigl\vert \hat{\varUpsilon _{i}}\bigl(\omega (\tau _{i})\bigr)-\hat{\varUpsilon _{i}}\bigl(\overline{ \omega }( \tau _{i})\bigr) \bigr\vert \\ &\qquad{}+ \bigl\vert \varUpsilon _{i}\bigl(\omega (\tau _{i}) \bigr)-\varUpsilon _{i}\bigl(\overline{\omega }( \tau _{i}) \bigr) \bigr\vert \biggr], \end{aligned}$$
(B.26)

where

$$ v(\tau )=\mathcal{A}\bigl(\tau,y(\tau ),v(\tau )\bigr) + \int _{0}^{\tau }\frac{( \tau -\xi )^{\sigma -1}}{\varGamma (\delta )}\mathcal{B} \bigl(s,y(s),v(s)\bigr)\,ds $$

and

$$ \overline{v}(\tau )=\mathcal{A}\bigl(\tau,\overline{y}(\tau ), \overline{v}(\tau )\bigr) + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{ \varGamma (\delta )}\mathcal{B}\bigl(s, \overline{y}(s),\overline{v}(s)\bigr)\,ds. $$

Using \([\tilde{A_{1}}]\), we have

$$\begin{aligned} &\bigl\vert v(\tau )-\overline{v}(\tau ) \bigr\vert \\ & \quad= \biggl\vert \mathcal{A}\bigl(\tau,y(\tau ),v( \tau )\bigr) + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B} \bigl(s,y(s),v(s)\bigr)\,ds \\ &\qquad{}-\mathcal{A}\bigl(\tau,\overline{y}(\tau ),\overline{v}(\tau )\bigr) - \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )}\mathcal{B}\bigl(s, \overline{y}(s),\overline{v}(s)\bigr)\,ds \biggr\vert \\ &\quad\leq \bigl\vert \mathcal{A}\bigl(\tau,y(\tau ),v(\tau )\bigr)-\mathcal{A} \bigl(\tau, \overline{y}(\tau ),\overline{v}(\tau )\bigr) \bigr\vert \\ &\qquad{}+ \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \bigl\vert \mathcal{B}\bigl(s,y(s),v(s)\bigr)-\mathcal{B}\bigl(s,\overline{y}(s), \overline{v}(s)\bigr) \bigr\vert \,ds \\ &\quad\leq \mathrm{M}_{1} \bigl\vert y(\tau )-\overline{y}(\tau ) \bigr\vert +\mathrm{N}_{1} \bigl\vert v( \tau )-\overline{v}(\tau ) \bigr\vert \\ &\qquad{}+\frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )} \bigl(\mathrm{M}_{2} \bigl\vert y( \tau )- \overline{y}(\tau ) \bigr\vert +\mathrm{N}_{2} \bigl\vert v(\tau )- \overline{v}( \tau ) \bigr\vert \bigr). \end{aligned}$$

Thus

$$ \bigl\vert v(\tau )-\overline{v}(\tau ) \bigr\vert \leq \biggl(\frac{\mathrm{M}_{1}}{1- \mathrm{N}_{1}-\mathrm{N}_{2}\frac{\tau ^{\sigma }}{\sigma \varGamma ( \delta )}} +\frac{\mathrm{M}_{2}\frac{\tau ^{\sigma }}{\sigma \varGamma ( \delta )}}{1-\mathrm{N}_{1}-\mathrm{N}_{2}\frac{\tau ^{\sigma }}{ \sigma \varGamma (\delta )}} \biggr) \bigl\vert y(\tau )- \overline{y}(\tau ) \bigr\vert . $$
(B.27)

Using hypotheses \([\tilde{A_{1}}]\), and \([\tilde{A_{3}}]\) and inequalities (B.27) and (B.26), we get

$$\begin{aligned} &\bigl\vert (\mathcal{T}_{r}\omega ) (\tau )-(\mathcal{T}_{r} \overline{\omega }) ( \tau ) \bigr\vert \\ &\quad\leq \biggl[ \biggl( \frac{\tau ^{r}}{\varGamma (r+1)} -\frac{m \tau \mathrm{T}^{r-1}}{\varGamma (r+1)}-\frac{m\tau ^{r+1}}{\mathrm{T} \varGamma (r+1)} -\frac{m\tau ^{r}}{\mathrm{T}\varGamma (r)} \biggr) \\ &\qquad{}\times \biggl(\frac{\mathrm{M}_{1}}{1-\mathrm{N}_{1}-\mathrm{N}_{2}\frac{ \tau ^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{M}_{2}\frac{ \tau ^{\sigma }}{\sigma \varGamma (\delta )}}{1-\mathrm{N}_{1} - \mathrm{N}_{2}\frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )}} \biggr) \biggr] \bigl\vert y(\tau )-\overline{y}(\tau ) \bigr\vert \\ &\qquad{}- \biggl[\frac{\xi _{2}\tau \mathrm{T}^{r-1}}{\eta _{2}\varGamma (r+1)} +\frac{ \tau }{\mathrm{T}}m(\mathbb{A}_{\hat{\varUpsilon _{i}}}+ \mathbb{A}_{ \varUpsilon _{i}}) \biggr] \bigl\vert \omega (\tau )-\overline{\omega }(\tau ) \bigr\vert . \end{aligned}$$

Now taking the norm on both sides, we have

$$\begin{aligned} &\Vert \mathcal{T}_{r}\omega -\mathcal{T}_{r} \overline{\omega } \Vert _{ \mathcal{X}} \\ &\quad\leq \biggl[ \biggl( \frac{m\mathrm{T}^{r}}{\varGamma (r+1)} +\frac{m \mathrm{T}^{r-1}}{\varGamma (r)} \biggr) \biggl(\frac{\mathrm{M}_{1}}{1- \mathrm{N}_{1}-\mathrm{N}_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} + \frac{\mathrm{M}_{2}\frac{\mathrm{T}^{\sigma }}{ \sigma \varGamma (\delta )}}{1-\mathrm{N}_{1} -\mathrm{N}_{2}\frac{ \mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} \biggr) \biggr] \Vert y- \overline{y} \Vert _{\mathcal{X}} \\ &\qquad{}+ \biggl[\frac{\xi _{2}\mathrm{T}^{r}}{\eta _{2}\varGamma (r+1)} +m( \mathbb{A}_{\hat{\varUpsilon _{i}}}+ \mathbb{A}_{\varUpsilon _{i}}) \biggr] \Vert \omega -\overline{\omega } \Vert _{\mathcal{X}}. \end{aligned}$$
(B.28)

In the same way, we can directly verify that

$$\begin{aligned} & \Vert \mathcal{T}_{r}y-\mathcal{T}_{r} \overline{y} \Vert _{\mathcal{X}} \\ &\quad \leq \biggl[ \biggl(\frac{m\mathrm{T}^{r}}{\varGamma (r+1)} + \frac{m \mathrm{T}^{r-1}}{\varGamma (r)} \biggr) \biggl(\frac{\mathrm{M}_{1}}{1- \mathrm{N}_{1}-\mathrm{N}_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{M}_{2}\frac{\mathrm{T}^{\sigma }}{ \sigma \varGamma (\delta )}}{1-\mathrm{N}_{1} -\mathrm{N}_{2}\frac{ \mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} \biggr) \biggr] \Vert y- \overline{y} \Vert _{\mathcal{X}} \\ &\qquad{}+ \biggl[\frac{\xi _{2}\mathrm{T}^{r}}{\eta _{2}\varGamma (r+1)} +m( \mathbb{A}_{\hat{\varUpsilon _{i}}}+ \mathbb{A}_{\varUpsilon _{i}}) \biggr] \Vert \omega -\overline{\omega } \Vert _{\mathcal{X}}. \end{aligned}$$
(B.29)

Therefore from (B.28) and (B.29) we get

$$ \bigl\Vert \mathcal{T}_{r}(\omega,y)-\mathcal{T}_{r}( \overline{\omega }, \overline{y}) \bigr\Vert _{\mathcal{X}} \leq \Delta _{1} \bigl\Vert (\omega,y)-(\overline{ \omega },\overline{y}) \bigr\Vert _{\mathcal{X}}. $$

Now, suppose \(\omega, \overline{\omega }\in \mathcal{Y}\). For \(\tau \in \mathcal{J}_{j}\), we have

$$\begin{aligned} & \bigl\vert \mathcal{T}_{p}\omega (\tau )-\mathcal{T}_{p}\overline{\omega }( \tau ) \bigr\vert \\ &\quad\leq \frac{1}{\varGamma (p)} \int _{0}^{\tau }(\tau -s)^{p-1} \bigl\vert z(s)-\overline{z}(s) \bigr\vert \,ds -\frac{\tau }{\mathrm{T}}\frac{\xi _{4}}{\eta _{4}\varGamma (p)} \int _{0} ^{\mathrm{T}}(\mathrm{T}-s)^{p-1} \bigl\vert y(s)-\overline{y}(s) \bigr\vert \,ds \\ &\qquad{} -\frac{\tau }{\mathrm{T}}\sum_{j=1}^{n} \biggl[\frac{1}{\varGamma (p)} \int _{\tau _{j}}^{\mathrm{T}}(\mathrm{T}-s)^{p-1} \bigl\vert z(s)-\overline{z}(s) \bigr\vert \,ds +\frac{1}{\varGamma (p)} \int _{\tau _{j-1}}^{\tau _{j}}(\tau _{j}-s)^{p-1} \bigl\vert z(s)- \overline{z}(s) \bigr\vert \,ds \\ &\qquad{} +\frac{\mathrm{T}-\tau _{j}}{\varGamma (p-1)} \int _{\tau _{j-1}}^{\tau _{j}}( \tau _{j}-s)^{p-2} \bigl\vert z(s)-\overline{z}(s) \bigr\vert \,ds +(\mathrm{T}-\tau _{j}) \bigl\vert \hat{\varUpsilon _{j}}\bigl(y(\tau _{j})\bigr)-\hat{\varUpsilon _{j}}\bigl(\overline{y}(\tau _{j})\bigr) \bigr\vert \\ &\qquad{}+ \bigl\vert \varUpsilon _{j}\bigl(y(\tau _{j})\bigr)- \varUpsilon _{j}\bigl(\overline{y}(\tau _{j})\bigr) \bigr\vert \biggr], \end{aligned}$$
(B.30)

where

$$ z(\tau )=\mathcal{A}'\bigl(\tau,\omega (\tau ),z(\tau )\bigr) + \int _{0}^{ \tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B}'\bigl(s, \omega (s),z(s)\bigr)\,ds $$

and

$$ \overline{z}(\tau )=\mathcal{A}'\bigl(\tau,\overline{\omega }(\tau ), \overline{z}(\tau )\bigr) + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{ \varGamma (\delta )}\mathcal{B}' \bigl(s,\overline{\omega }(s),\overline{z}(s)\bigr)\,ds. $$

Using \([\tilde{A_{2}}]\), we have

$$\begin{aligned} &\bigl\vert z(\tau )-\overline{z}(\tau ) \bigr\vert \\ &\quad = \biggl\vert \mathcal{A}'\bigl(\tau,\omega ( \tau ),z(\tau )\bigr) + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )}\mathcal{B}' \bigl(s,\omega (s),z(s)\bigr)\,ds \\ &\qquad{}-\mathcal{A}'\bigl(\tau,\overline{\omega }(\tau ),\overline{z}( \tau )\bigr) - \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B}'\bigl(s,\overline{\omega }(s),\overline{z}(s)\bigr)\,ds \biggr\vert \\ &\quad\leq \bigl\vert \mathcal{A}'\bigl(\tau,\omega (\tau ),z(\tau ) \bigr)-\mathcal{A}'\bigl( \tau,\overline{\omega }(\tau ), \overline{z}(\tau )\bigr) \bigr\vert \\ &\qquad{}+ \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \bigl\vert \mathcal{B}'\bigl(s,\omega (s),z(s)\bigr)-\mathcal{B}' \bigl(s,\overline{\omega }(s), \overline{z}(s)\bigr) \bigr\vert \,ds \\ &\quad\leq \mathrm{M}'_{1} \bigl\vert \omega (\tau )- \overline{\omega }(\tau ) \bigr\vert + \mathrm{N}'_{1} \bigl\vert z(\tau )-\overline{z}(\tau ) \bigr\vert \\ &\qquad{}+\frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )} \bigl(\mathrm{M}'_{2} \bigl\vert \omega (\tau )-\overline{\omega }(\tau ) \bigr\vert + \mathrm{N}'_{2} \bigl\vert z(\tau )- \overline{z}(\tau ) \bigr\vert \bigr). \end{aligned}$$

Thus

$$ \bigl\vert z(\tau )-\overline{z}(\tau ) \bigr\vert \leq \biggl(\frac{\mathrm{M}'_{1}}{1- \mathrm{N}'_{1}-\mathrm{N}'_{2}\frac{\tau ^{\sigma }}{\sigma \varGamma ( \delta )}} +\frac{\mathrm{M}'_{2}\frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )}}{1-\mathrm{N}'_{1}-\mathrm{N}'_{2}\frac{\tau ^{\sigma }}{ \sigma \varGamma (\delta )}} \biggr) \bigl\vert \omega (\tau )-\overline{\omega }( \tau ) \bigr\vert . $$
(B.31)

Using hypotheses \([\tilde{A_{2}}]\), \([\tilde{A_{3}}]\) and inequalities (B.31) and (B.30), we have

$$\begin{aligned} &\bigl\vert (\mathcal{T}_{p}\omega ) (\tau )-(\mathcal{T}_{p} \overline{\omega }) ( \tau ) \bigr\vert \\ &\quad\leq \biggl[ \biggl(\frac{\tau ^{p}}{\varGamma (p+1)}- \frac{n\tau \mathrm{T}^{p-1}}{\varGamma (p+1)}- \frac{n\tau ^{p+1}}{\mathrm{T}\varGamma (p+1)} -\frac{n\tau ^{p}}{ \mathrm{T}\varGamma (p)} \biggr) \\ &\qquad{}\times \biggl(\frac{\mathrm{M}_{1}}{1-\mathrm{N}_{1}-\mathrm{N}_{2}\frac{ \tau ^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{M}_{2}\frac{ \tau ^{\sigma }}{\sigma \varGamma (\delta )}}{1-\mathrm{N}_{1} - \mathrm{N}_{2}\frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )}} \biggr) \biggr] \bigl\vert \omega (\tau )-\overline{\omega }(\tau ) \bigr\vert \\ &\qquad{}- \biggl[\frac{\xi _{4}\tau \mathrm{T}^{p-1}}{\eta _{4}\varGamma (p+1)} +\frac{ \tau }{\mathrm{T}}n(\mathbb{A}_{\hat{\varUpsilon _{j}}}+ \mathbb{A}_{ \varUpsilon _{j}}) \biggr] \bigl\vert y(\tau )-\overline{y}(\tau ) \bigr\vert . \end{aligned}$$

Now taking the norm on both sides, we have

$$\begin{aligned} & \Vert \mathcal{T}_{p}\omega - \mathcal{T}_{p}\overline{\omega } \Vert _{ \mathcal{Y}} \\ &\quad\leq \biggl[ \biggl(\frac{n\mathrm{T}^{p}}{\varGamma (p+1)} +\frac{n\mathrm{T} ^{p-1}}{\varGamma (p)} \biggr) \biggl(\frac{\mathrm{M}'_{1}}{1-\mathrm{N}'_{1}- \mathrm{N}'_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{ \mathrm{M}'_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}}{1 -\mathrm{N}'_{1}-\mathrm{N}'_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} \biggr) \biggr] \Vert \omega - \overline{\omega } \Vert _{ \mathcal{Y}} \\ &\qquad{} + \biggl[\frac{\xi _{4}\mathrm{T}^{p}}{\eta _{4}\varGamma (p+1)} +n( \mathbb{A}_{\hat{\varUpsilon _{j}}}+ \mathbb{A}_{\varUpsilon _{j}}) \biggr] \Vert y- \overline{y} \Vert _{\mathcal{Y}}. \end{aligned}$$
(B.32)

In the same way, we can obtain

$$\begin{aligned} & \Vert \mathcal{T}_{p}y- \mathcal{T}_{p}\overline{y} \Vert _{\mathcal{Y}} \\ &\quad\leq \biggl[ \biggl(\frac{n\mathrm{T}^{p}}{\varGamma (p+1)} +\frac{n\mathrm{T} ^{p-1}}{\varGamma (p)} \biggr) \biggl(\frac{\mathrm{M}'_{1}}{1-\mathrm{N}'_{1}- \mathrm{N}'_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{ \mathrm{M}'_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}}{1 -\mathrm{N}'_{1}-\mathrm{N}'_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} \biggr) \biggr] \Vert y- \overline{y} \Vert _{\mathcal{Y}} \\ &\qquad{} + \biggl[\frac{\xi _{4}\mathrm{T}^{p}}{\eta _{4}\varGamma (p+1)} +n( \mathbb{A}_{\hat{\varUpsilon _{j}}}+ \mathbb{A}_{\varUpsilon _{j}}) \biggr] \Vert \omega -\overline{\omega } \Vert _{\mathcal{Y}}. \end{aligned}$$
(B.33)

Thus from (B.32) and (B.33) we get

$$ \bigl\Vert \mathcal{T}_{p}(\omega,y)-\mathcal{T}_{p}( \overline{\omega }, \overline{y}) \bigr\Vert _{\mathcal{Y}} \leq \Delta _{2} \bigl\Vert (\omega,y)-(\overline{ \omega },\overline{y}) \bigr\Vert _{\mathcal{Y}}. $$

Hence it follows that

$$ \bigl\Vert \mathcal{T}(\omega,y)-\mathcal{T}(\overline{\omega },\overline{y}) \bigr\Vert _{\mathcal{X}\times \mathcal{Y}} \leq \max (\Delta _{1},\Delta _{2}) \bigl( \Vert \omega -\overline{\omega } \Vert _{\mathcal{X}\times \mathcal{Y}}+ \Vert y- \overline{y} \Vert _{\mathcal{X}\times \mathcal{Y}}\bigr). $$

This implies that \(\mathcal{T}\) is a contraction and hence has a unique fixed point. This completes the proof. □

Appendix 3

Proof of Theorem 3.11

Let \(\omega \in \mathcal{M}\) be a solution of inequality (3.1), and let \(\omega ^{*}\) be a solution of the considered problem (1.3). Then

$$ \textstyle\begin{cases} {}^{c}\mathcal{D}^{r}\omega ^{*}(\tau )=\mathcal{A}(\tau,\omega ^{*}( \tau ),{}^{c}\mathcal{D}^{r}\omega ^{*}(\tau )) \\ \phantom{{}^{c}\mathcal{D}^{r}\omega ^{*}(\tau )=}{}+\int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B}(s,\omega ^{*}(s),{}^{c}\mathcal{D}^{r}\omega ^{*}(s))\,ds, \quad \tau \in \mathcal{J}, \tau \neq \tau _{i}, i=1,2,\dots,m, \\ \Delta \omega ^{*}(\tau _{i})=\varUpsilon _{i}(\omega ^{*}(\tau _{i})), \qquad \Delta \omega ^{*'}(\tau _{i})=\hat{\varUpsilon _{i}}(\omega ^{*}(\tau _{i})),\quad i=1,2,\dots,m, \\ \eta _{1}\omega ^{*}(0)+\xi _{1}I^{r}\omega ^{*}(0)=\nu _{1}, \qquad \eta _{2} \omega ^{*}(\mathrm{T})+\xi _{2}I^{r}\omega ^{*}(\mathrm{T})=\nu _{2}. \end{cases} $$

Using the inequality

$$ \bigl\vert \omega (\tau )-\omega ^{*}(\tau ) \bigr\vert \leq \bigl\vert \omega (\tau )-q(\tau ) \bigr\vert + \bigl\vert q( \tau )+\omega ^{*}(\tau ) \bigr\vert , $$
(C.1)

by Theorem 3.10 we have

$$\begin{aligned} & \bigl\vert \omega (\tau )-\omega ^{*}(\tau ) \bigr\vert \\ &\quad\leq \biggl[\frac{\tau ^{r}}{\varGamma (r+1)}-\frac{m\tau ^{r+1}}{\mathrm{T} \varGamma (r+1)}- \frac{\tau m}{\mathrm{T}} \biggr]\epsilon _{r} +\frac{1}{ \varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1} \bigl\vert v(s)-v^{*}(s) \bigr\vert \,ds \\ &\qquad{} -\frac{\tau \xi _{2}}{\mathrm{T}\eta _{2}\varGamma (r)} \int _{0}^{ \mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert \omega (s)-\omega ^{*}(s) \bigr\vert \,ds \\ &\qquad{} -\frac{ \tau }{\mathrm{T}} \sum_{i=1}^{m} \biggl[\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert v(s)-v^{*}(s) \bigr\vert \,ds \\ &\qquad{} +\frac{1}{\varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \bigl\vert v(s)-v ^{*}(s) \bigr\vert \,ds +\frac{\mathrm{T}-\tau _{i}}{\varGamma (r-1)} \int _{\tau _{i-1}} ^{\tau _{i}}(\tau _{i}-s)^{r-2} \bigl\vert v(s)-v^{*}(s) \bigr\vert \,ds \\ &\qquad{}+(\mathrm{T}-\tau _{i}) \bigl\vert \hat{\varUpsilon _{i}} \bigl(\omega (\tau _{i})\bigr)- \hat{\varUpsilon _{i}}\bigl( \omega ^{*}(\tau _{i})\bigr) \bigr\vert + \bigl\vert \varUpsilon _{i}\bigl(\omega (\tau _{i})\bigr)-\varUpsilon _{i}\bigl(\omega ^{*}(\tau _{i})\bigr) \bigr\vert \biggr], \end{aligned}$$
(C.2)

where \(v, v^{*}\in \mathcal{M}\) are given by

$$ v(\tau )=\mathcal{A}\bigl(\tau,\omega (\tau ),v(\tau )\bigr) + \int _{0}^{\tau }\frac{( \tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B}\bigl(s, \omega (s),v(s)\bigr)\,ds $$

and

$$ v^{*}(\tau )=\mathcal{A}\bigl(\tau,\omega ^{*}(\tau ),v^{*}(\tau )\bigr) + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B}\bigl(s, \omega ^{*}(s),v^{*}(s)\bigr)\,ds. $$

Using \([A_{1}]\), we have

$$\begin{aligned} &\bigl\vert v(\tau )-v^{*}(\tau ) \bigr\vert \\ &\quad = \biggl\vert \mathcal{A}\bigl(\tau,\omega (\tau ),v( \tau )\bigr) + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B}\bigl(s, \omega (s),v(s)\bigr)\,ds \\ &\qquad{}-\mathcal{A}\bigl(\tau,\omega ^{*}(\tau ),v^{*}(\tau ) \bigr) - \int _{0}^{ \tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B}\bigl(s, \omega ^{*}(s),v^{*}(s)\bigr)\,ds \biggr\vert \\ &\quad\leq \bigl\vert \mathcal{A}\bigl(\tau,\omega (\tau ),v(\tau )\bigr)- \mathcal{A}\bigl( \tau,\omega ^{*}(\tau ),v^{*}(\tau )\bigr) \bigr\vert \\ &\qquad{}+ \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \bigl\vert \mathcal{B}\bigl(s,\omega (s),v(s)\bigr) -\mathcal{B}\bigl(s,\omega ^{*}(s),v^{*}(s)\bigr) \bigr\vert \,ds \\ &\quad\leq \mathrm{M}_{1} \bigl\vert \omega (\tau )-\omega ^{*}(\tau ) \bigr\vert +\mathrm{N} _{1} \bigl\vert v( \tau )-v^{*}(\tau ) \bigr\vert \\ &\qquad{}+\frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )} \bigl(\mathrm{M}_{2} \bigl\vert \omega (\tau )-\omega ^{*}(\tau ) \bigr\vert +\mathrm{N}_{2} \bigl\vert v(\tau )-v^{*}( \tau ) \bigr\vert \bigr). \end{aligned}$$

Thus

$$ \bigl\vert v(\tau )-v^{*}(\tau ) \bigr\vert \leq \biggl(\frac{\mathrm{M}_{1}}{1-\mathrm{N} _{1}-\mathrm{N}_{2}\frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{ \mathrm{M}_{2}\frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )}}{1- \mathrm{N}_{1} -\mathrm{N}_{2}\frac{\tau ^{\sigma }}{\sigma \varGamma ( \delta )}} \biggr) \bigl\vert \omega (\tau )-\omega ^{*}(\tau ) \bigr\vert . $$
(C.3)

Using hypothesis \([A_{2}]\) and (C.3), by inequality (C.2) we get

$$\begin{aligned} & \bigl\vert \omega (\tau )-\omega ^{*}(\tau ) \bigr\vert \\ &\quad\leq \biggl[\frac{\tau ^{r}}{\varGamma (r+1)}-\frac{m\tau ^{r+1}}{ \mathrm{T}\varGamma (r+1)}-\frac{\tau m}{\mathrm{T}} \biggr] \epsilon _{r} \\ &\qquad{} + \biggl[ \biggl(\frac{\tau ^{r}}{\varGamma (r+1)}-\frac{m\tau \mathrm{T}^{r-1}}{ \varGamma (r+1)}- \frac{m\tau ^{r+1}}{\mathrm{T}\varGamma (r+1)} -\frac{m\tau ^{r}}{\mathrm{T}\varGamma (r)} \biggr) \\ & \qquad{}\times\biggl(\frac{\mathrm{M}_{1}}{1-\mathrm{N}_{1}-\mathrm{N}_{2}\frac{ \tau ^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{M}_{2}\frac{ \tau ^{\sigma }}{\sigma \varGamma (\delta )}}{1-\mathrm{N}_{1} - \mathrm{N}_{2}\frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )}} \biggr) -\frac{ \xi _{2}\tau \mathrm{T}^{r-1}}{\eta _{2}\varGamma (r+1)}- \frac{\tau }{ \mathrm{T}}m(\mathbb{A}+\mathbb{B}) \biggr] \\ &\qquad{}\times \bigl\vert \omega (\tau )- \omega ^{*}( \tau ) \bigr\vert . \end{aligned}$$

By taking the norm and simplifying we get

$$\begin{aligned} & \bigl\Vert \omega -\omega ^{*} \bigr\Vert _{\mathcal{M}} \\ &\quad \leq \biggl[\frac{\mathrm{T}^{r}}{\varGamma (r+1)}-\frac{m\mathrm{T} ^{r}}{\varGamma (r+1)}-m \biggr]\epsilon _{r} + \biggl[ \biggl(\frac{m \mathrm{T}^{r}}{\varGamma (r+1)}+\frac{m\mathrm{T}^{r-1}}{\varGamma (r)} \biggr) \\ &\qquad{}\times \biggl(\frac{\mathrm{M}_{1}}{1 -\mathrm{N}_{1}-\mathrm{N}_{2}\frac{ \mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{M}_{2}\frac{ \mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}}{1-\mathrm{N}_{1} - \mathrm{N}_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} \biggr)+\frac{\xi _{2}\mathrm{T}^{r}}{\eta _{2}\varGamma (r+1)} +m( \mathbb{A}+\mathbb{B}) \biggr]\\ &\qquad{}\times \bigl\Vert \omega -\omega ^{*} \bigr\Vert _{\mathcal{M}}, \end{aligned}$$

from which we obtain

$$\begin{aligned} &\bigl\Vert \omega -\omega ^{*} \bigr\Vert _{\mathcal{M}}\\ &\quad \leq \frac{ [\frac{\mathrm{T} ^{r}}{\varGamma (r+1)}-\frac{m\mathrm{T}^{r}}{\varGamma (r+1)}-m ] \epsilon _{r}}{1- [ (\frac{m\mathrm{T}^{r}}{\varGamma (r+1)}+\frac{m \mathrm{T}^{r-1}}{\varGamma (r)} ) (\frac{\mathrm{M}_{1}}{1 - \mathrm{N}_{1}-\mathrm{N}_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{M}_{2}\frac{\mathrm{T}^{\sigma }}{ \sigma \varGamma (\delta )}}{1-\mathrm{N}_{1} -\mathrm{N}_{2}\frac{ \mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} )+\frac{\xi _{2} \mathrm{T}^{r}}{\eta _{2}\varGamma (r+1)} +m(\mathbb{A}+\mathbb{B}) ]}. \end{aligned}$$

Thus

$$\begin{aligned} \bigl\Vert \omega -\omega ^{*} \bigr\Vert _{\mathcal{M}} &\leq \mathbf{C}_{r}\epsilon _{r}, \end{aligned}$$

where

$$\begin{aligned} \mathbf{C}_{r}=\frac{ [\frac{\mathrm{T}^{r}}{\varGamma (r+1)}-\frac{m \mathrm{T}^{r}}{\varGamma (r+1)}-m ]}{1- [ (\frac{m \mathrm{T}^{r}}{\varGamma (r+1)}+\frac{m\mathrm{T}^{r-1}}{\varGamma (r)} ) (\frac{\mathrm{M}_{1}}{1 -\mathrm{N}_{1}-\mathrm{N}_{2}\frac{ \mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{M}_{2}\frac{ \mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}}{1-\mathrm{N}_{1} - \mathrm{N}_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} )+\frac{\xi _{2}\mathrm{T}^{r}}{\eta _{2}\varGamma (r+1)} +m( \mathbb{A}+\mathbb{B}) ]}, \end{aligned}$$

that is, problem (1.3) is Ulam–Hyers stable. Now putting \(\vartheta (\epsilon )=\mathbf{C}_{r}\epsilon _{r} \vartheta (0)=0\) yields that problem (1.3) is generalized Ulam–Hyers stable. □

Proof of Theorem 3.20

Let \((\omega,y)\in \mathcal{X}\times \mathcal{Y}\) be a solution of inequality (3.9), and let \((\omega ^{*},y^{*})\in \mathcal{X} \times \mathcal{Y}\) be a solution of the system

$$ \textstyle\begin{cases} ^{c}\mathcal{D}^{r}\omega ^{*}(\tau )=\mathcal{A}(\tau,y^{*}(\tau ),{}^{c} \mathcal{D}^{r}\omega ^{*}(\tau )) +\int _{0}^{\tau }\frac{(\tau -s)^{ \sigma -1}}{\varGamma (\delta )} \mathcal{B}(s,y^{*}(s),{}^{c}\mathcal{D} ^{r}\omega ^{*}(s))\,ds \\ \quad\text{where } \tau \in \mathcal{J}, \tau \neq \tau _{i} \text{ for } i=1,2,\dots,m, \\ ^{c}\mathcal{D}^{p}y^{*}(\tau )=\mathcal{A}'(\tau,\omega ^{*}(\tau ),{}^{c} \mathcal{D}^{p}y^{*}(\tau )) +\int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B}'(s,\omega ^{*}(s),{}^{c}\mathcal{D} ^{p}y^{*}(s))\,ds \\ \quad\text{where } \tau \in \mathcal{J}, \tau \neq \tau _{j} \text{ for } j=1,2,\dots,n, \\ \Delta \omega ^{*}(\tau _{i})=\varUpsilon _{i}(\omega ^{*}(\tau _{i})), \qquad\Delta \omega ^{\prime *}(\tau _{i})=\hat{\varUpsilon _{i}}(\omega ^{*}(\tau _{i})), \quad i=1,2,\dots,m, \\ \Delta y^{*}(\tau _{j})=\varUpsilon _{j}(y^{*}(\tau _{j})),\qquad \Delta y^{\prime *}( \tau _{j})=\hat{\varUpsilon _{j}}(y^{*}(\tau _{j})),\quad j=1,2,\dots,n, \\ \eta _{1}\omega ^{*}(0)+\xi _{1}I^{r}\omega ^{*}(0)=\nu _{1}, \qquad \eta _{2} \omega ^{*}(\mathrm{T})+\xi _{2}I^{r}\omega ^{*}(\mathrm{T})=\nu _{2}, \\ \eta _{3}y^{*}(0)+\xi _{3}I^{p}y^{*}(0)=\nu _{3}, \qquad \eta _{4}y^{*}( \mathrm{T})+\xi _{4}I^{p}y^{*}(\mathrm{T})=\nu _{4}. \end{cases} $$
(C.4)

Then in view of Lemma A.3, the solution of (C.4) is

$$\begin{aligned} \omega ^{*}(\tau ) ={}& \frac{1}{\varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1}v(s)\,ds + \frac{\nu _{1}}{\eta _{1}} -\frac{\tau }{\mathrm{T}} \biggl[\frac{\nu _{1}}{\eta _{1}}- \frac{\nu _{2}}{\eta _{2}} +\frac{\xi _{2}}{\eta _{2}\varGamma (r)} \int _{0}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}\omega ^{*}(s)\,ds \biggr] \\ &{} -\frac{\tau }{\mathrm{T}}\sum_{i=1}^{m} \biggl[\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1}v(s)\,ds +\frac{1}{ \varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1}v(s) \,ds \\ &{} +\frac{\mathrm{T}-\tau _{i}}{\varGamma (r-1)} \int _{\tau _{i-1}}^{\tau _{i}}( \tau _{i}-s)^{r-2}v(s) \,ds +(\mathrm{T}-\tau _{i})\hat{\varUpsilon _{i}}\bigl( \omega ^{*}(\tau _{i})\bigr)+\varUpsilon _{i}\bigl(\omega ^{*}(\tau _{i})\bigr) \biggr] \end{aligned}$$

and

$$\begin{aligned} y^{*}(\tau ) ={}& \frac{1}{\varGamma (p)} \int _{0}^{\tau }(\tau -s)^{p-1}z(s)\,ds - \frac{ \tau }{\mathrm{T}} \biggl[\frac{\nu _{3}}{\eta _{3}}-\frac{\nu _{4}}{\eta _{4}} + \frac{\xi _{4}}{\eta _{4}\varGamma (p)} \int _{0}^{\mathrm{T}}( \mathrm{T}-s)^{p-1}y^{*}(s) \,ds \biggr] \\ &{} -\frac{\tau }{\mathrm{T}}\sum_{j=1}^{n} \biggl[\frac{1}{\varGamma (p)} \int _{\tau _{j}}^{\mathrm{T}}(\mathrm{T}-s)^{p-1}z(s)\,ds +\frac{1}{ \varGamma (p)} \int _{\tau _{j-1}}^{\tau _{j}}(\tau _{j}-s)^{p-1}z(s) \,ds \\ &{} +\frac{\mathrm{T}-\tau _{j}}{\varGamma (p-1)} \int _{\tau _{j-1}}^{\tau _{j}}( \tau _{j}-s)^{p-2}z(s) \,ds +(\mathrm{T}-\tau _{j})\hat{\varUpsilon _{j}}\bigl(y ^{*}(\tau _{j})\bigr)+\varUpsilon _{j} \bigl(y^{*}(\tau _{j})\bigr) \biggr], \end{aligned}$$

where

$$\begin{aligned} v(\tau ) = \mathcal{A}\bigl(\tau,y(\tau ),v(\tau )\bigr) + \int _{0}^{\tau }\frac{( \tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B} \bigl(s,y(s),v(s)\bigr)\,ds \end{aligned}$$

and

$$\begin{aligned} z(\tau ) = \mathcal{A}'\bigl(\tau,\omega (\tau ),z(\tau )\bigr) + \int _{0}^{ \tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B}'\bigl(s, \omega (s),z(s)\bigr)\,ds. \end{aligned}$$

Consider

$$\begin{aligned} & \bigl\vert \omega (\tau )-\omega ^{*}(\tau ) \bigr\vert \\ &\quad\leq \bigl\vert \omega (\tau )-q(\tau ) \bigr\vert + \bigl\vert q(\tau )-\omega ^{*}(\tau ) \bigr\vert \\ &\quad\leq \biggl[\frac{\tau ^{r}}{\varGamma (r+1)}-\frac{m\tau ^{r+1}}{\mathrm{T} \varGamma (r+1)}- \frac{\tau m}{\mathrm{T}} \biggr]\epsilon _{r} +\frac{1}{ \varGamma (r)} \int _{0}^{\tau }(\tau -s)^{r-1} \bigl\vert v(s)-v^{*}(s) \bigr\vert \,ds \\ &\qquad{} -\frac{\tau \xi _{2}}{\mathrm{T}\eta _{2}\varGamma (r)} \int _{0}^{ \mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert \omega (s)-\omega ^{*}(s) \bigr\vert \,ds \\ &\qquad{}-\frac{ \tau }{\mathrm{T}} \sum_{i=1}^{m} \biggl[\frac{1}{\varGamma (r)} \int _{\tau _{i}}^{\mathrm{T}}(\mathrm{T}-s)^{r-1} \bigl\vert v(s)-v^{*}(s) \bigr\vert \,ds \\ &\qquad{} +\frac{1}{\varGamma (r)} \int _{\tau _{i-1}}^{\tau _{i}}(\tau _{i}-s)^{r-1} \bigl\vert v(s)-v ^{*}(s) \bigr\vert \,ds +\frac{\mathrm{T}-\tau _{i}}{\varGamma (r-1)} \int _{\tau _{i-1}} ^{\tau _{i}}(\tau _{i}-s)^{r-2} \bigl\vert v(s)-v^{*}(s) \bigr\vert \,ds \\ &\qquad{}+(\mathrm{T}-\tau _{i}) \bigl\vert \hat{\varUpsilon _{i}} \bigl(\omega (\tau _{i})\bigr)- \hat{\varUpsilon _{i}}\bigl( \omega ^{*}(\tau _{i})\bigr) \bigr\vert + \bigl\vert \varUpsilon _{i}\bigl(\omega (\tau _{i})\bigr)-\varUpsilon _{i}\bigl(\omega ^{*}(\tau _{i})\bigr) \bigr\vert \biggr], \end{aligned}$$
(C.5)

where \(v, v^{*}\in \mathcal{X}\) are given by

$$ v(\tau )=\mathcal{A}\bigl(\tau,y(\tau ),v(\tau )\bigr) + \int _{0}^{\tau }\frac{(t-s)^{ \sigma -1}}{\varGamma (\delta )} \mathcal{B} \bigl(s,y(s),v(s)\bigr)\,ds $$

and

$$ v^{*}(\tau )=\mathcal{A}\bigl(\tau,y^{*}(\tau ),v^{*}(\tau )\bigr) + \int _{0} ^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B} \bigl(s,y ^{*}(s),v^{*}(s)\bigr)\,ds. $$

Using \([\tilde{A}_{1}]\), we have

$$\begin{aligned} &\bigl\vert v(\tau )-v^{*}(\tau ) \bigr\vert \\ & \quad= \biggl\vert \mathcal{A}\bigl(\tau,y(\tau ),v(\tau )\bigr) + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B} \bigl(s,y(s),v(s)\bigr)\,ds \\ &\qquad{}-\mathcal{A}\bigl(\tau,y^{*}(\tau ),v^{*}(\tau )\bigr) - \int _{0}^{\tau }\frac{( \tau -s)^{\sigma -1}}{\varGamma (\delta )} \mathcal{B} \bigl(s,y^{*}(s),v^{*}(s)\bigr)\,ds \biggr\vert \\ &\quad\leq \bigl\vert \mathcal{A}\bigl(\tau,y(\tau ),v(\tau )\bigr)-\mathcal{A} \bigl(\tau,y^{*}( \tau ),v^{*}(\tau )\bigr) \bigr\vert \\ &\qquad{}+ \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{\varGamma (\delta )} \bigl\vert \mathcal{B}\bigl(s,y(s),v(s)\bigr)-\mathcal{B}\bigl(s,y^{*}(s),v^{*}(s) \bigr) \bigr\vert \,ds \\ &\quad\leq \mathrm{M}_{1} \bigl\vert y(\tau )-y^{*}(\tau ) \bigr\vert +\mathrm{N}_{1} \bigl\vert v(\tau )-v ^{*}( \tau ) \bigr\vert \\ &\qquad{}+\frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )} \bigl(\mathrm{M}_{2} \bigl\vert y( \tau )-y^{*}(\tau ) \bigr\vert +\mathrm{N}_{2} \bigl\vert v( \tau )-v^{*}(\tau ) \bigr\vert \bigr). \end{aligned}$$

Thus

$$ \bigl\vert v(\tau )-v^{*}(\tau ) \bigr\vert \leq \biggl(\frac{\mathrm{M}_{1}}{1-\mathrm{N} _{1} -\mathrm{N}_{2}\frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{ \mathrm{M}_{2}\frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )}}{1- \mathrm{N}_{1} -\mathrm{N}_{2}\frac{\tau ^{\sigma }}{\sigma \varGamma ( \delta )}} \biggr) \bigl\vert y(\tau )-y^{*}(\tau ) \bigr\vert . $$
(C.6)

Using hypothesis \([\tilde{A}_{3}]\) and (C.6), inequality (C.5) implies

$$\begin{aligned} & \bigl\vert \omega (\tau )-\omega ^{*}(\tau ) \bigr\vert \\ &\quad\leq \biggl[\frac{\tau ^{r}}{\varGamma (r+1)}-\frac{m\tau ^{r+1}}{ \mathrm{T}\varGamma (r+1)}-\frac{\tau m}{\mathrm{T}} \biggr] \epsilon _{r} \\ &\qquad{}+ \biggl[ \biggl(\frac{\tau ^{r}}{\varGamma (r+1)}-\frac{m\tau \mathrm{T}^{r-1}}{ \varGamma (r+1)} - \frac{m\tau ^{r+1}}{\mathrm{T}\varGamma (r+1)}-\frac{m\tau ^{r}}{\mathrm{T}\varGamma (r)} \biggr) \\ &\qquad{}\times \biggl(\frac{\mathrm{M}_{1}}{1 -\mathrm{N}_{1}-\mathrm{N}_{2}\frac{ \tau ^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{M}_{2}\frac{ \tau ^{\sigma }}{\sigma \varGamma (\delta )}}{1-\mathrm{N}_{1} - \mathrm{N}_{2}\frac{\tau ^{\sigma }}{\sigma \varGamma (\delta )}} \biggr) \bigl\vert y( \tau )-y^{*}(\tau ) \bigr\vert \biggr] \\ &\qquad{}- \biggl[\frac{\tau \xi _{2}\mathrm{T}^{r-1}}{\eta _{2}\varGamma (r+1)} +\frac{ \tau }{\mathrm{T}}m(\mathbb{A}_{\hat{\varUpsilon _{i}}}+ \mathbb{A}_{ \varUpsilon _{i}}) \biggr] \bigl\vert \omega (\tau )-\omega ^{*}(\tau ) \bigr\vert . \end{aligned}$$

By taking the norm and simplifying, we get

$$\begin{aligned} \bigl\Vert \omega -\omega ^{*} \bigr\Vert _{\mathcal{X}}\leq{}& \biggl[\frac{\mathrm{T}^{r}}{\varGamma (r+1)}-\frac{m\mathrm{T}^{r}}{ \varGamma (r+1)}-m \biggr]\epsilon _{r} + \biggl[ \biggl(\frac{m\mathrm{T}^{r}}{ \varGamma (r+1)}+ \frac{m\mathrm{T}^{r-1}}{\varGamma (r)} \biggr) \\ &{}\times \biggl(\frac{\mathrm{M}_{1}}{1 -\mathrm{N}_{1}-\mathrm{N}_{2}\frac{ \mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{M}_{2}\frac{ \mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}}{1-\mathrm{N}_{1} - \mathrm{N}_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} \biggr) \bigl\Vert y-y^{*} \bigr\Vert _{\mathcal{X}} \biggr] \\ &{}- \biggl[\frac{\xi _{2}\mathrm{T}^{r}}{\eta _{2}\varGamma (r+1)} +m( \mathbb{A}_{\hat{\varUpsilon _{i}}}+\mathbb{A}_{\varUpsilon _{i}}) \biggr] \bigl\Vert \omega -\omega ^{*} \bigr\Vert _{\mathcal{X}}. \end{aligned}$$
(C.7)

For simplicity, we consider

$$\begin{aligned} &\mathcal{S}_{r}=\frac{\frac{\mathrm{T}^{r}}{\varGamma (r+1)}-\frac{m \mathrm{T}^{r}}{\varGamma (r+1)}-m}{1+\frac{\xi _{2}\mathrm{T}^{r}}{\eta _{2}\varGamma (r+1)}+m(\mathbb{A}_{\hat{\varUpsilon _{i}}}+\mathbb{A}_{ \varUpsilon _{i}})}, \\ &\mathcal{Q}_{r}=\frac{ (\frac{m\mathrm{T}^{r}}{\varGamma (r+1)}+\frac{m \mathrm{T}^{r-1}}{\varGamma (r)} ) (\frac{\mathrm{M}_{1}}{1 - \mathrm{N}_{1}-\mathrm{N}_{2}\frac{\mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} +\frac{\mathrm{M}_{2}\frac{\mathrm{T}^{\sigma }}{ \sigma \varGamma (\delta )}}{1-\mathrm{N}_{1} -\mathrm{N}_{2}\frac{ \mathrm{T}^{\sigma }}{\sigma \varGamma (\delta )}} )}{1+\frac{\xi _{2}\mathrm{T}^{r}}{\eta _{2}\varGamma (r+1)}+m(\mathbb{A}_{ \hat{\varUpsilon _{i}}}+\mathbb{A}_{\varUpsilon _{i}})}. \end{aligned}$$

Then (C.7) implies

$$ \bigl\Vert \omega -\omega ^{*} \bigr\Vert _{\mathcal{X}}\leq \mathcal{S}_{r}\epsilon _{r}+ \mathcal{Q}_{r} \bigl\Vert y-y^{*} \bigr\Vert _{\mathcal{X}} $$
(C.8)

and, similarly,

$$ \bigl\Vert y-y^{*} \bigr\Vert _{\mathcal{Y}} \leq \mathcal{S}_{p}\epsilon _{p}+\mathcal{Q} _{p} \bigl\Vert \omega -\omega ^{*} \bigr\Vert _{\mathcal{Y}}. $$
(C.9)

From (C.8) and (C.9) we write

$$\begin{aligned} &\bigl\Vert \omega -\omega ^{*} \bigr\Vert _{\mathcal{X}}- \mathcal{Q}_{r} \bigl\Vert y-y^{*} \bigr\Vert _{ \mathcal{X}}\leq \mathcal{S}_{r}\epsilon _{r}, \\ &\bigl\Vert y-y^{*} \bigr\Vert _{\mathcal{Y}}- \mathcal{Q}_{p} \bigl\Vert \omega -\omega ^{*} \bigr\Vert _{ \mathcal{Y}}\leq \mathcal{S}_{p}\epsilon _{p}, \\ & \begin{bmatrix} 1 & -\mathcal{Q}_{r} \\ -\mathcal{Q}_{p} & 1 \end{bmatrix} \begin{bmatrix} \Vert \omega -\omega ^{*} \Vert _{\mathcal{X}\times \mathcal{Y}} \\ \Vert y-y^{*} \Vert _{\mathcal{X}\times \mathcal{Y}} \end{bmatrix}\leq \begin{bmatrix} \mathcal{S}_{r}\epsilon _{r} \\ \mathcal{S}_{p}\epsilon _{p} \end{bmatrix}. \end{aligned}$$

Solving the last inequality, we have

$$ \begin{bmatrix} \Vert \omega -\omega ^{*} \Vert _{\mathcal{X}\times \mathcal{Y}} \\ \Vert y-y^{*} \Vert _{\mathcal{X}\times \mathcal{Y}} \end{bmatrix}\leq \begin{bmatrix} \frac{1}{\Delta } & \frac{\mathcal{Q}_{r}}{\Delta } \\ \frac{\mathcal{Q}_{p}}{\Delta } & \frac{1}{\Delta } \end{bmatrix} \begin{bmatrix} \mathcal{S}_{r}\epsilon _{r} \\ \mathcal{S}_{p}\epsilon _{p} \end{bmatrix}, $$

where

$$ \Delta =1-\mathcal{Q}_{r}\mathcal{Q}_{p}> 0. $$

Further simplification gives

$$ \begin{aligned} & \bigl\Vert \omega -\omega ^{*} \bigr\Vert _{\mathcal{X}\times \mathcal{Y}}\leq \frac{ \mathcal{S}_{r}\epsilon _{r}}{\Delta }+\frac{\mathcal{Q}_{r} \mathcal{S}_{p}\epsilon _{p}}{\Delta }, \\ & \bigl\Vert y-y^{*} \bigr\Vert _{\mathcal{X}\times \mathcal{Y}}\leq \frac{\mathcal{S} _{p}\epsilon _{p}}{\Delta }+\frac{\mathcal{Q}_{r}\mathcal{S}_{r} \epsilon _{r}}{\Delta }, \end{aligned} $$

from which we have

$$ \bigl\Vert \omega -\omega ^{*} \bigr\Vert _{\mathcal{X}\times \mathcal{Y}}+ \bigl\Vert y-y^{*} \bigr\Vert _{\mathcal{X}\times \mathcal{Y}} \leq \frac{\mathcal{S}_{r}\epsilon _{r}}{\Delta }+\frac{\mathcal{S}_{p}\epsilon _{p}}{\Delta } +\frac{ \mathcal{Q}_{r}\mathcal{S}_{p}\epsilon _{p}}{\Delta } + \frac{ \mathcal{Q}_{r}\mathcal{S}_{r}\epsilon _{r}}{\Delta }. $$
(C.10)

Let \(\max \{\epsilon _{r},\epsilon _{p} \}=\epsilon \). Then from (C.10) we get

$$ \bigl\Vert (\omega,y)-\bigl(\omega ^{*}, y^{*}\bigr) \bigr\Vert _{\mathcal{X}\times \mathcal{Y}} \leq C_{r,p}\epsilon, $$

where

$$ C_{r,p}= \biggl[\frac{\mathcal{S}_{r}}{\Delta }+\frac{\mathcal{S}_{p}}{ \Delta } + \frac{\mathcal{Q}_{r}\mathcal{S}_{p}}{\Delta } +\frac{ \mathcal{Q}_{r}\mathcal{S}_{r}}{\Delta } \biggr]. $$

This completes the proof. □

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Zada, A., Alzabut, J., Waheed, H. et al. Ulam–Hyers stability of impulsive integrodifferential equations with Riemann–Liouville boundary conditions. Adv Differ Equ 2020, 64 (2020). https://doi.org/10.1186/s13662-020-2534-1

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MSC

  • 34A08
  • 34B37

Keywords

  • Caputo derivative
  • Riemann–Liouville integral
  • Impulse
  • Ulam–Hyers stability
  • Fixed point theory