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Ulam–Hyers stability of impulsive integrodifferential equations with Riemann–Liouville boundary conditions
Advances in Difference Equations volume 2020, Article number: 64 (2020)
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.
1 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, 28–31, 33, 36, 37, 39, 41–45, 47–49]. The monographs [5, 6, 16, 38] treated fractional differential equations with instantaneous impulses of the following form:
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
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
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
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:
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
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:
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 \(\mathcal{M}=\{\omega:\mathcal{J}\rightarrow \mathbb{R}: \omega \in C(\mathcal{J}_{i},\mathbb{R})\text{ and }\omega (\tau _{i}^{+}), \omega (\tau _{i}^{-})\text{ exist such that }\Delta \omega (\tau _{i})=\omega (\tau _{i}^{+})-\omega (\tau _{i}^{-})\text{ for }i=1,2,\dots,m\}\).
2 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.
2.1 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
where
if and only ifωsatisfies
Proof
Applying Lemma A.3 (see Appendix 1) to (2.1) with \(a_{0}, a_{1}\in \mathbb{R}\), we have
Furthermore, we obtain
For \(\tau \in (\tau _{1},\tau _{2}]\), there are \(b_{0}, b_{1}\in \mathbb{R}\) such that
Hence it follows that
Using
we obtain
Thus
Similarly, we have
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
where
Let
Also, we consider \(\mathcal{M}=\mathrm{PC}(\mathcal{J},\mathbb{R})\) endowed with the norm
We can easily see that \({\mathcal{M}}\) is a Banach space. Further, if ω is a solution of problem (1.3), then
Now, to study (1.3) by fixed point theory, let \(\mathcal{T}: \mathcal{M}\rightarrow \mathcal{M}\) be the operator defined as
where
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
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.
2.2 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
has a solution \((\omega,y)\) if and only if
and
where
and
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
and
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
Thus \((\omega,y)\) is a solution of (1.4). □
For convenience, we use the following notations:
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
and
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. □
3 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].
3.1 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:
and
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
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
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
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
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
- (i)
\(|\varPhi (\tau )|\leq \epsilon _{r}\) and \(|\varPhi _{i}|\leq \epsilon _{r} \text{ for all }\tau \in \mathcal{J}, i=1,2,\dots,m\);
- (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
- (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
- (i)
\(|\varPhi (\tau )|\leq \psi _{r}(\tau )\) and \(|\varPhi _{i}|\leq \phi _{r} \text{ for all }\tau \in \mathcal{J}, i=1,2,\dots,m\);
- (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
- (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
- (i)
\(|\varPhi (\tau )|\leq \psi _{r}(\tau )\) and \(|\varPhi _{i}|\leq \epsilon _{r}\phi _{r}\text{ for all }\tau \in \mathcal{J}, i=1,2, \dots,m\);
- (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
- (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
Proof
Let ω be a solution of inequality (3.1). Then by Remark 3.6ω is also a solution of
that is,
For simplicity, let \(q(\tau )\) denote the terms of \(\omega (\tau )\) that are free from \(\varPhi (\tau )\), that is,
Thus (3.5) can be written as
Using (i) from Remark 3.6, we get
□
Theorem 3.11
If hypothesis\([A_{1}]\)holds and
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.
3.2 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:
and
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
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
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
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
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
- (i)
\(|\mu _{\mathcal{A},\mathcal{B}}(\tau )|\leq \epsilon _{r}, | \varLambda _{\mathcal{A}',\mathcal{B}'}(\tau )|\leq \epsilon _{p} \text{ for all } \tau \in \mathcal{J}\);
- (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
$$\begin{aligned} ^{c}\mathcal{D}^{p}y(\tau )={}&\mathcal{A}\bigl(\tau,\omega ( \tau ),{}^{c} \mathcal{D}^{p}y(\tau )\bigr)\\ &{} + \int _{0}^{\tau }\frac{(\tau -s)^{\sigma -1}}{ \varGamma (\delta )} \mathcal{B}\bigl(s, \omega (s),{}^{c}\mathcal{D}^{r}y(s)\bigr)\,ds +\varLambda _{\mathcal{A}',\mathcal{B}'}(\tau ),\\ & \tau \in \mathcal{J} _{j}; \end{aligned}$$ - (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
Proof
Let \((\omega,y)\) be a solution of inequality (3.7). Then by Remark 3.18\((\omega,y)\) is also a solution of
that is,
and
where
and
From (3.11a) we have
Thus (3.12) becomes
where
Using (i) from Remark 3.18, we obtain
Repeating a similar procedure for (3.11b) together with (i) from Remark 3.18, we have
where
Thus the proof is complete. □
Theorem 3.20
If hypotheses\([\tilde{A}_{1}]\)–\([\tilde{A}_{3}]\)hold with
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.
4 Illustrative examples
We present two examples to demonstrate the existence and stability of our obtained results.
Example 4.1
Consider
where \(r=\frac{3}{2}\), \(\mathcal{J}_{0}=[0,\frac{1}{3}]\), \(\mathcal{J}_{1}=(\frac{1}{3},1]\).
Set
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
and
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
and
Then we have
and
respectively. Hence \(\mathbb{A}=\frac{1}{35}\) and \(\mathbb{B}= \frac{1}{20}\). Thus condition \([A_{2}]\) is satisfied.
Also,
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
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
\(\tau _{i}=\frac{1}{3}\text{ for }i=1,2,3,\dots,60,\text{ and } \tau _{j}=\frac{1}{4}\text{ for } j=1,2,3,\dots,100\).
For any \(\omega, \overline{\omega }, y, \overline{y}\in \mathbb{R}\) and \(\tau \in [0,1]\), we obtain
and
Similarly, for any \(\omega, \overline{\omega }, y, \overline{y}\in \mathbb{R}\), and \(\tau \in [0,1]\), we obtain
and
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
and
Then for \(\omega, \overline{\omega }\in \mathcal{X}\), we have
and
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
then for \(y, \overline{y}\in \mathcal{Y}\), we have
and if
then for \(y, \overline{y}\in \mathcal{Y}\), we have
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
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,
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
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
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
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
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
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
- (i)
\(\mathcal{F}\omega +\mathcal{G}y\in \mathcal{H}\)whenever\(\omega, y\in \mathcal{H}\).
- (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
where \(v_{n}, v\in \mathcal{M}\) are given by
and
respectively. Using hypothesis \([A_{1}]\), we have
Then
Hypotheses \([A_{1}]\), \([A_{2}]\) and inequalities (B.1) and (B.2) lead to
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
and
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
which leads to
For \(\tau \in \mathcal{J}_{i}\), we obtain
Further, using hypothesis \([A_{3}]\), we have
Therefore we get
Now by (B.4) and \([A_{4}]\) relation (B.3) becomes
Thus
Similarly for \(\tau \in \mathcal{J}_{0}\), we can verify that
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
Obviously, the right-hand side of inequality (B.5) tends to zero as \(\tau _{1}\rightarrow \tau _{2}\). Therefore
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
We need to prove that the set Ω is bounded. Suppose \(\omega \in \varOmega \) is such that
Then for each \(\tau \in \mathcal{J}_{i}\), we have
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
where \(v, \overline{v}\in \mathcal{M}\) are given by
and
Using \([A_{1}]\), we have
Thus
Using hypotheses \([A_{1}]\), \([A_{2}]\) and inequalities (B.7) and (B.6), we obtain
Now taking the norm on both sides, we have
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
and
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
Using \([\tilde{A_{4}}]\) for \(\tau \in \mathcal{J}_{i}\), we have
Therefore we get
Using (B.9) and \([\tilde{A_{6}}]\), relation (B.8) becomes
Thus
Similarly, for \(\tau \in \mathcal{J}_{0}\), we can verify that
In the similar manner, we have
Using \([\tilde{A_{4}}]\) for \(\tau \in \mathcal{J}_{i}\), we have
Therefore we get
Using (B.11) and \([\tilde{A_{6}}]\), relation (B.10) becomes
Thus
Similarly, for \(\tau \in \mathcal{J}_{0}\), we can verify that
Hence
Now, for any \((\omega,y)\in \mathscr{B}\), consider
Using \([\tilde{A_{5}}]\) for \(\tau \in \mathcal{J}_{j}\), we have
Therefore we get
Using (B.13) and \([\tilde{A_{7}}]\), relation (B.12) becomes
Thus
Similarly, for \(\tau \in \mathcal{J}_{0}\), we can verify that
In a similar manner, we have
Using \([\tilde{A_{5}}]\) for \(\tau \in \mathcal{J}_{j}\), we have
Therefore we get
Using (B.15) and \([\tilde{A_{7}}]\), relation (B.14) becomes
Thus
Similarly, for \(\tau \in \mathcal{J}_{0}\), we can verify that
Hence
and thus
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
Similarly,
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
By \([\tilde{A_{4}}]\), for \(\tau \in \mathcal{J}_{i}\), we have
Therefore we get
Using (B.17) in (B.16), after simplification, we get
In a similar manner, we have
By \([\tilde{A_{4}}]\), for \(\tau \in \mathcal{J}_{i}\), we have
Therefore we get
Using (B.19) in (B.18), after simplification, we get
Hence
Now for any \((\omega,y)\in \mathscr{B}\), we have
By \([\tilde{A_{5}}]\), for \(\tau \in \mathcal{J}_{j}\), we have
Therefore we get
Using (B.21) in (B.20), after simplification, we get
In a similar manner, we have
By \([\tilde{A_{5}}]\), for \(\tau \in \mathcal{J}_{j}\), we have
Therefore we get
Using (B.23) in (B.22), after simplification, we get
Hence
Thus
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
Obviously, the right-hand side of inequality (B.24) tends to zero as \(\tau _{1}\rightarrow \tau _{2}\).
Therefore
Similarly,
Now for any \(\tau _{1}, \tau _{2}\in \mathcal{J}_{j}\) with \(0\leq \tau _{1}\leq \tau _{2}\leq 1\), we have
Obviously, the right-hand side of inequality (B.25) tends to zero as \(\tau _{1}\rightarrow \tau _{2}\).
Therefore
Similarly,
Thus
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
where
and
Using \([\tilde{A_{1}}]\), we have
Thus
Using hypotheses \([\tilde{A_{1}}]\), and \([\tilde{A_{3}}]\) and inequalities (B.27) and (B.26), we get
Now taking the norm on both sides, we have
In the same way, we can directly verify that
Therefore from (B.28) and (B.29) we get
Now, suppose \(\omega, \overline{\omega }\in \mathcal{Y}\). For \(\tau \in \mathcal{J}_{j}\), we have
where
and
Using \([\tilde{A_{2}}]\), we have
Thus
Using hypotheses \([\tilde{A_{2}}]\), \([\tilde{A_{3}}]\) and inequalities (B.31) and (B.30), we have
Now taking the norm on both sides, we have
In the same way, we can obtain
Thus from (B.32) and (B.33) we get
Hence it follows that
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
Using the inequality
by Theorem 3.10 we have
where \(v, v^{*}\in \mathcal{M}\) are given by
and
Using \([A_{1}]\), we have
Thus
Using hypothesis \([A_{2}]\) and (C.3), by inequality (C.2) we get
By taking the norm and simplifying we get
from which we obtain
Thus
where
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
Then in view of Lemma A.3, the solution of (C.4) is
and
where
and
Consider
where \(v, v^{*}\in \mathcal{X}\) are given by
and
Using \([\tilde{A}_{1}]\), we have
Thus
Using hypothesis \([\tilde{A}_{3}]\) and (C.6), inequality (C.5) implies
By taking the norm and simplifying, we get
For simplicity, we consider
Then (C.7) implies
and, similarly,
Solving the last inequality, we have
where
Further simplification gives
from which we have
Let \(\max \{\epsilon _{r},\epsilon _{p} \}=\epsilon \). Then from (C.10) we get
where
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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DOI: https://doi.org/10.1186/s13662-020-2534-1