Stability in the Sense of Hyers–Ulam–Rassias for the Impulsive Volterra Equation

Abstract

This article aims to use various fixed-point techniques to study the stability issue of the impulsive Volterra integral equation in the sense of Ulam–Hyers (sometimes known as Hyers–Ulam) and Hyers–Ulam–Rassias. By eliminating key assumptions, we are able to expand upon and enhance some recent findings.

1. Introduction

For each physical model, the exact solution is crucial for applied research. This kind of exact answer provides the right physical interpretation and is helpful in confirming the approximations made by analytical or numerical techniques. If there is an exact solution that is close (in some way) to each approximate (in a particular sense) solution to a given (integral, functional, differential, difference, or fractional differential) equation, the equation is called stable (in the sense of Ulam). In his renowned speech at the University of Wisconsin in 1940, S. M. Ulam raised an open question that served as the basis for the theory of stability (see, e.g., [1,2] for more details). The intriguing open issue that gave rise to the stability hypothesis is best expressed as follows. Let $G,G^{*}$ represent some groups, with $G^{*}$ being a metric group with a metric $\chi$. Given $\epsilon >0$, Ulam asked if, there is $\delta >0$: if $\mathfrak{h}:G\to G^{*}$: ($\forall x_1,x_2\in G$)

$$\chi (\mathfrak{h}\left(x_1{x}_2\right),\mathfrak{h}\left(x_1\right)\mathfrak{h}\left(x_2\right))<\delta ,$$

then $\mathfrak{g}:G\to G^{*}$ (a homomorphism) exists:

$$\chi (\mathfrak{h}\left(x_1\right),\mathfrak{g}\left(x_1\right))<\epsilon ?$$

The stability question has been a major topic of research in mathematical analysis for the last few decades, and we direct the reader to [3] for further references.

There are some responses available for the Ulam inquiry. For instance, in 1941, D. H. Hyers positively answered in the case of Banach spaces (see [1]). T. Aoki (see [4]) treated this problem for mappings (additive only). T. M. Rassias, in 1978 (see [5]), gave a generalized version of the theorem above by studying the unbounded Cauchy differences case. The stability problem pioneered by Rassias (see [5]) is nowadays called the U–HR stability. The well-known outcome obtained by Rassias is as follows [5].

Theorem 1. 

Suppose Banach spaces $B_1,B_2$, and a mapping $f_1:B_1\to B_2$: the function $t\mapsto f_1\left(t{e}_1\right):\mathbb{R}\to B_2$ is continuous $\forall e_1\in B_1$. Let $\nu \ge 0$ and $\iota \in [0,1)$:

$$\parallel f_1(e_1+e_2)-f_1\left(e_1\right)-f_1\left(e_2\right)\parallel \le \nu (\parallel e_1{\parallel }^{\iota }+\parallel e_2{\parallel }^{\iota }),e_1,e_2\in B_1\backslash \left\{0\right\}.$$

(1)

Then, ∃ a unique solution $f_2:B_1\to B_2$ of $f_1(e_1+e_2)=f_1\left(e_1\right)+f_1\left(e_2\right)$ with

$$\parallel f_1\left(e_1\right)-f_2\left(e_1\right)\parallel \le \frac{2\nu \parallel e_1{\parallel }^{\iota }}{|2-2^{\iota }|},e_1\in B_1\backslash \left\{0\right\}.$$

(2)

The findings of Hyers and Rassias have been applied to various situations in a variety of ways. Many authors have looked into the stability of differential equations (DEs) (see [6,7]).

There are numerous viewpoints regarding Ulam’s stability. For instance, the idea of Ulam stability in the case of a general equation is further clarified by the following abstract definition in k variables. From now on, we use ${\mathbb{R}}_{+}$ to denote the nonnegative reals, ${\mathfrak{A}}^{\mathfrak{B}}$ to denote a set of all functions mapping a set $\mathfrak{B}\ne \varnothing$ into a set $\mathfrak{A}\ne \varnothing$, and $\mathbb{R}$ and $\mathbb{C}$ to represent the sets of real and complex numbers, respectively.

Definition 1. 

Assume that a nonempty set X, $k\in \mathbb{N}$, $(E,\rho )$ is a metric space, ${\mathcal{D}}_0\subset \mathcal{D}\subset E^X$, $\varnothing \ne \mathcal{Y}\subset {\mathbb{R}}_{+}^{X^k}$, $\mathcal{S}:\mathcal{Y}\to {\mathbb{R}}_{+}^X$, and ${\mathcal{A}}_1,{\mathcal{A}}_2:\mathcal{D}\to E^{X^k}$. The equation

$$\begin{array}{c}\hfill \left({\mathcal{A}}_1\psi \right)(a_1,\dots ,a_k)=\left({\mathcal{A}}_2\psi \right)(a_1,\dots ,a_k)\end{array}$$

(3)

is called $\mathcal{S}$-stable in ${\mathcal{D}}_0$ if, ∀$\psi \in {\mathcal{D}}_0$ and $\delta \in \mathcal{Y}$ with

$$\rho (\left({\mathcal{A}}_1\psi \right)(a_1,\dots ,a_k),\left({\mathcal{A}}_2\psi \right)(a_1,\dots ,a_k))\le \delta (a_1,\dots ,a_k),a_1,\dots ,a_k\in X,$$

a mapping $\varphi \in \mathcal{D}$ exists; (3) for all $a_1,\dots ,a_k\in X$; and with $\rho \left(\varphi \right(a),\psi (a\left)\right)\le \left(\mathcal{S}\delta \right)\left(a\right)$ for $a\in X.$

If $\left(\mathcal{S}\delta \right)\left(a\right)=0$ for $\delta \in \mathcal{Y}$ and $a\in X$, then the equation is called hyperstable in ${\mathcal{D}}_0$.

Note that (3) is the Cauchy equation ($f_1(a+b)=f_1\left(a\right)+f_1\left(b\right)$) with $k=2$, $X=W$, $\left({\mathcal{A}}_1\alpha \right)(a,b)=\alpha (a+b)$, and $\left({\mathcal{A}}_2\alpha \right)(a,b)=\alpha \left(a\right)+\alpha \left(b\right)$ for $\alpha \in \mathcal{D}$ and $a,b\in X=W$.

Evidently, Theorem 1 says that (∀ real number $r\ne 1$) the Cauchy equation is $\mathcal{S}$-stable in ${\mathcal{D}}_0=\mathcal{D}=B^W$, with $\mathcal{S}:\mathcal{Y}\to {\mathbb{R}}_{+}^X$ defined by

$$\left(\mathcal{S}{\delta }_{\eta }\right)\left(a\right)=\frac{{\eta \parallel a\parallel }^r}{|1-2^{r-1}|},{\delta }_{\eta }\in \mathcal{Y},a\in W,$$

where

$${\delta }_{\eta }(a,b)={\eta (\parallel a\parallel }^r+{\parallel b\parallel }^r),a,b\in W,\eta \in {\mathbb{R}}_{+},$$

and

$$\mathcal{Y}=\{{\delta }_{\eta }\in {\mathbb{R}}_{+}^{W\times W}:\eta \in {\mathbb{R}}_{+}\}.$$

However, if $r<0$, then a stronger property holds (see [8]), which states that the Cauchy equation is hyperstable in ${\mathcal{D}}_0=B^W$.

Much more accurate results (for real-variable mappings) have just been demonstrated in [9] using the technique of the Banach limit, namely, the following result (see [9], Theorem 8, and Remark 7).

Theorem 2. 

Take a normed space $\mathfrak{W}$, ${\mathfrak{W}}_0:=\mathfrak{W}\backslash \left\{0\right\}$, $\lambda ,{\sigma }_1,{\sigma }_2\in \mathbb{R}$, $\lambda \ne 1$, and ${\sigma }_1\le {\sigma }_2$. Let $\mathfrak{h}:\mathfrak{W}\to \mathbb{R}$ satisfy

$$\begin{array}{c}\hfill {\sigma }_1{(\parallel a\parallel }^{\lambda }+{\parallel b\parallel }^{\lambda }) \le \mathfrak{h}(a+b)-\mathfrak{h}\left(a\right)-\mathfrak{h}\left(b\right)\le {\sigma }_2{(\parallel a\parallel }^{\lambda }+{\parallel b\parallel }^{\lambda }),a,b\in {\mathfrak{W}}_0.\end{array}$$

Then, a unique additive mapping exists $\alpha :\mathfrak{W}\to \mathbb{R}$: (when $\lambda <1$),

$$\begin{array}{c}\hfill \frac{{\sigma }_1}{1-2^{\lambda -1}}{\parallel a\parallel }^{\lambda }\le \alpha \left(a\right)-\mathfrak{h}\left(a\right)\le \frac{{\sigma }_2}{1-2^{\lambda -1}}{\parallel a\parallel }^{\lambda },a\in {\mathfrak{W}}_0,\end{array}$$

(4)

and (when $\lambda >1$),

$$\begin{array}{c}\hfill \frac{{\sigma }_1}{2^{\lambda -1}-1}{\parallel a\parallel }^{\lambda }\le \mathfrak{h}\left(a\right)-\alpha \left(a\right)\le \frac{{\sigma }_2}{2^{\lambda -1}-1}{\parallel a\parallel }^{\lambda },a\in {\mathfrak{W}}_0.\end{array}$$

(5)

Additionally, α is continuous if $\mathfrak{h}$ is continuous at some point.

Of course, condition (1) can be replaced by

$$\parallel \mathfrak{O}(a+b)-\mathfrak{O}\left(a\right)-\mathfrak{O}\left(b\right)\parallel \le \mathsf{\Pi }(a,b),a,b\in \mathfrak{W},$$

(6)

and we mention that the inequality

$$\parallel \mathfrak{O}(a+b)-\mathfrak{O}\left(a\right)-\mathfrak{O}\left(b\right)\parallel \le {\eta \parallel a\parallel }^p{\parallel b\parallel }^q,a,b\in \mathfrak{W}\backslash \left\{0\right\},$$

(7)

with $p,q\in \mathbb{R}$ and $\eta >0$, was studied in [10,11] (see also [12]). Additionally, see [13,14,15] for more details and examples.

It is obvious that there are several interpretations of what an approximate solution and the proximity of two mappings mean. Consequently, it makes sense to take the Ulam stability into account for different distance measurement techniques. The idea of 2-norms, first put forth by Gähler in 1964, can be used to introduce one of the non-classical distance measuring techniques (see [16,17]). It should be noted that a logical extension of this idea is n-normed space (see, e.g., [18]), i.e., the 2-normed space is an n-normed space, with $n=2$. For more stability results (especially in 2-normed spaces), readers are advised to read [19].

Numerous methodologies have been used to explore stability in general, including the direct method (see [20]), the fixed-point theory (FPT) (see [21,22]), the invariant means method (see [23]), Grönwall’s inequality (GI), and many more (see, e.g., [24]). The FPT is the second most frequent technique for showing that functional equations (FEs) are stable. In 1991, Baker (see [21]) pioneered the FPT in the investigations of stability of FEs. To determine the Ulam–Hyers (U–H) stability of FEs, he used a variation of the Banach FPT.

FPT occupies a significant position in both pure and applied mathematics because of its numerous applications in areas such as differential and integral equations, approximation theory, variational inequalities, equilibrium issues, fractal generation, and many more. One of the key conclusions in FPT is the Banach contraction theorem, which Banach first proposed in 1921. In metric FPT, it ensures the existence and uniqueness of FPs for contraction mappings. It also enables different iterative algorithms to converge to an FP. In many mathematical disciplines, the theorem is now a frequently employed technique for resolving existence-related issues.

FPT plays a crucial role in solving many problems such as the following. In [25], the authors employed the Schaefer and Banach FPT to study the existence and uniqueness results for nonlinear Hadamard-type nonlocal turbulent flow models in a porous medium involving the p-Laplacian operator. The U–H stability of this problem has been investigated in the Banach space. In [26], FPTs played a basic role in proving the stability of many fractional-order systems, and in proving the existence and uniqueness results for some of these systems. In [27], the authors employed FPTs to study the existence and uniqueness of the solution of some fractional integral DEs involving non-instantaneous impulsive boundary conditions (N-IIBC). Known FPTs have been used in [28] to study some fractional DEs with fractional boundary conditions (FBCs). In [29], the authors utilized some FPTs to investigate the stability of some fractional neutral equations in the sense of Ulam–Hyers–Rassias (U–HR).

Volterra integral equations (VIEs) are essential for resolving a wide range of issues that crop up in mathematical physics and engineering (see, e.g., [30]). VIEs can be used to represent a wide range of starting and boundary value issues, such as diffraction problems, conformal maps, water waves, etc. VIEs are also used in population dynamics, elasticity, fluid movement, and chemical reactions.

Numerous systems in the realm of biological control evolve progressively over a predetermined amount of time. However, the system state will be subjected to some relatively short time interference owing to some natural or human intervention behavior, causing the system state to change instantly. Because of this abrupt change in state, the system is now defined by an impulse dynamical system rather than only by continuous or discrete dynamical systems. With significant applications in the biological sciences, impulsive differential equations (IDEs) are a vital tool for researching process state changes (see, e.g., [31]). An IDE can more properly and honestly represent the motion processes seen in nature and science fields than can a differential equation without a pulse. As a result, it is frequently utilized in neural network systems, infectious illness dynamic models, microbiological models, chemotherapeutic models, and population dynamic models. Numerous research studies have demonstrated the tremendous practical value of understanding the biological regulation of these systems and of utilizing IDEs to forecast the future development trend of biological systems.

Numerous writers have researched the stability of integral equations. For instance, in 2007, S. M. Jung introduced an FPT to study the stability of a Volterra integral equation (see [32]), and in 2011, M. Akkouchi (see [33]) established both the U–H and the U–HR stability for some integral equations using some alternative FPTs. The authors in [34] examined the H–U stability of linear impulsive Volterra integral equations using the open mapping theorem. In their analysis, they investigated the existence and uniqueness of the solutions of a class of nonlinear impulsive integral equations of the form

$$\{\begin{array}{c}w\left(\xi \right)=f\left(\xi \right)+\mu {\int }_a^{\xi }K(\xi ,\tau )w\left(\tau \right)d\tau +\sum _{s=0}^n{I}_s\left(\xi \right)w\left({\xi }_s\right)\hfill \\ K(\xi ,\tau )\mathrm{is}\mathrm{continuous}\mathrm{on}a\le \tau \le \xi \hfill \\ a\le \xi \le b\hfill \end{array}.$$

Moreover, using GI, they studied the H–U and H–U–R stability of the same class of integral equations.

In [35], the authors investigated the H–U and U–HR stability of nonlinear impulsive system on time scales via the FPT

$$\left\{\begin{array}{c}{z}^{\mathsf{\Delta }}\left(\xi \right)=M\left(\xi \right)z\left(\xi \right)+{\int }_{t_0}^{\xi }K(\xi ,s,z\left(s\right),z\left(h\left(s\right)\right)\mathsf{\Delta }s\hfill \\ \mathsf{\Delta }z\left({\xi }_k\right)=z\left({\xi }_k^{+}\right)-z\left({\xi }_k^{-}\right)={\gamma }_k\left(z\left({\xi }_k^{-}\right)\right)\hfill \\ z\left(\xi \right)=\alpha \left(\xi \right),\xi \in [{\xi }_0-\lambda ,{\xi }_0]\hfill \\ z\left({\xi }_0\right)=\alpha \left({\xi }_0\right)=z_0\hfill \end{array}\right..$$

The uniqueness and existence of the solution to nonlinear impulsive Volterra integral-delay dynamic systems is proved with the help of the Picard operator. The main tools for proving the results were abstract GI and Banach FPT. In [36], the authors presented H–U and U–HR stability results for the nonlinear Volterra system on time scales with fractional integrable impulses.

$$\left\{\begin{array}{c}{w}^{\mathsf{\Delta }}\left(\xi \right)=M\left(\xi \right)w\left(\xi \right)+{\int }_{t_0}^{\xi }K(\xi ,s,w\left(s\right),w\left(h\left(s\right)\right)\mathsf{\Delta }s\hfill \\ \xi \in (s_i,{\xi }_{i+1}]\cap T_S,i=0,1,\dots ,m\hfill \\ w\left(\xi \right)=I_{{\xi }_i,t}^{\alpha }{g}_i(\xi ,w\left(\xi \right),w\left(h\left(\xi \right)\right)),\xi \in ({\xi }_i,s_i]\cap T_S,i=1,\dots ,m\hfill \\ w\left(\xi \right)=\alpha \left(\xi \right),\xi \in [s_0-\lambda ,s_0]\cap T_S\hfill \\ w\left({\xi }_0\right)=\alpha \left({\xi }_0\right)=w_0\hfill \end{array}\right.$$

The Picard FPT is used for obtaining the existence and uniqueness of solutions. By means of abstract GI and GI on time scales, they established H–U stability and U–HR stability results.

In [37], the authors generalized two types of VIEs given on time scales

$$\left\{\begin{array}{c}x\left(\xi \right)=f\left(\xi \right)+{\int }_a^{\xi }K(\xi ,s,x\left(s\right),x\left(\sigma \left(s\right)\right)\mathsf{\Delta }s\hfill \\ f\in C_{rd}(I_T,{\mathbb{R}}^n)\hfill \\ K\in C_{rd}(I_T\times I_T\times {\mathbb{R}}^n\times {\mathbb{R}}^n,{\mathbb{R}}^n)\hfill \end{array}\right.$$

and examined their H–U and U–HR stabilities. Additionally, they proved stability results for the inhomogeneous, nonlinear VIE on time scales and provided an example to support the results. Moreover, they proved that the general Volterra type integral equation given on time scales has U–HR stability.

In [38], the authors examined the stability and existence findings for impulsive integro-differential equations. The results obtained in [39] have been generalized in [40], in which they discussed H–U stability of higher-order nonlinear DEs with fractional integrable impulses.

Diaz and Margolis (see [41]) established a well-known FPT on a GM space that is complete. Numerous authors have made substantial use of the Diaz and Margolis theorem (see, e.g., [33]). In 2012, K. Ciepliński (see [42]) surveyed some applications of various FPTs used in the U–H stability of FEs. An FPT for (not necessarily) linear operators has been proved by J. Brzdęk (see [43]).

In 2003, researchers employed some FPTs to study the stability of some FEs (see [44,45]), and they introduced proofs for U–HR stability. In that way, they combined the findings of Hyers, Rassias, and Gajda [46].

In this article, we examine the U–H and U–HR stability of some integral equations. FPT is the major instrument utilized in the analysis.

Main Contributions:

• Study the stability in the sense of U–H and U–HR for some impulsive VIEs.
• Obtain new results by dropping essential conditions in some recent interesting publications.
• Employ a known FPT as the main instrument in our analysis.

2. Preliminaries

This paper’s objective is to examine the U–H and the U–HR stability of the following impulsive Volterra integral equation (of second kind)

$$\mathfrak{H}\left(\lambda \right)={\int }_0^{\lambda }\mathfrak{f}(\alpha ,\mathfrak{H}\left(\alpha \right))d\alpha +\sum _{0<{\lambda }_k<\lambda }{\mathfrak{I}}_{\mathfrak{k}}\left(\mathfrak{H}\left({\lambda }_k^{-}\right)\right),$$

(8)

where ${\mathfrak{I}}_{\mathfrak{k}}:\mathbb{C}\to \mathbb{C}$, $\mathfrak{H}\left({\lambda }_k^{-}\right)$ represents the left limit of $\mathfrak{H}\left(t\right)$ at $\lambda ={\lambda }_k$, $k=1,2,\dots ,m$, and $\mathfrak{f}$ is a continuous function.

Definition 2. 

Suppose that, for all $\mathfrak{H}\left(\lambda \right)$:

$$|\mathfrak{H}\left(\lambda \right)-{\int }_0^{\lambda }\mathfrak{f}(\alpha ,\mathfrak{H}\left(\alpha \right))d\alpha -\sum _{0<{\lambda }_k<\lambda }{\mathfrak{I}}_{\mathfrak{k}}\left(\mathfrak{H}\left({\lambda }_k^{-}\right)\right)|\le \phi \left(\lambda \right),$$

where $\phi \left(\lambda \right)\ge 0\forall \lambda \in I$, if there is a solution ${\mathfrak{H}}_0\left(\lambda \right)$ to (8)

$$|\mathfrak{H}\left(\lambda \right)-{\mathfrak{H}}_0\left(\lambda \right)|\le K\phi \left(\lambda \right)\forall \lambda \in I,$$

where $K>0$ is (some constant) independent of $\mathfrak{H}\left(\lambda \right)$ and ${\mathfrak{H}}_0\left(\lambda \right)$, then we say that (8) is U–HR stable. Additionally, (8) is U–H stable if $\phi \left(\lambda \right)$ is a constant in the above inequalities.

A basic concept in our analysis is the generalized metric (GM), which is defined as follows.

Definition 3. 

Let $\mathfrak{Y}$ be a non-empty set. A mapping $\gamma :\mathfrak{Y}\times \mathfrak{Y}\to [0,\infty ]$ is called a GM on $\mathfrak{Y}$ iff:

G1 

$\gamma (q_1,q_2)=0$ if and only if $q_1=q_2$;

G2 

$\gamma (q_1,q_2)=\gamma (q_2,q_1)$ for all $q_1,q_2\in \mathfrak{Y}$;

G3 

$\gamma (q_1,q_3)\le \gamma (q_1,q_2)+\gamma (q_2,q_3)$ for all $q_1,q_2,q_3\in \mathfrak{Y}$;

One of the fundamental results of FPT that plays a crucial role in demonstrating our primary findings is the theorem below (see [41]).

Theorem 3. 

Let $(\mathfrak{Y},\gamma )$ be a GM that is complete. Assume a strictly contractive operator $\theta :\mathfrak{Y}\to \mathfrak{Y}$ with (a Lipschitz constant) $A<1$. If ∃$k\ge 0$: $\gamma ({\theta }^{k+1}y,{\theta }^{k}y)<\infty$ for some $y\in \mathfrak{Y}$, then:

(a) 

The sequence ${\theta }^{n}y$ converges to an FP $y^{*}$ of θ;

(b) 

$y^{*}$ is the unique FP of θ in $Y^{*}:=\{y_1\in Y:\gamma ({\theta }^{k}y,y_1)<\infty \}$;

(c) 

If $y_1\in Y^{*}$, then $\gamma (y_1,y^{*})\le \frac{1}{1-A}\gamma (\theta y_1,y_1)$.

In this article, using Theorem 3, we study the U–HR and the U–H stability of (8).

3. Main Results

We start by investigating the U–HR stability of the impulsive Volterra integral Equation (8). We introduce the following theorem to serve this aim.

Theorem 4. 

If r is a fixed real number, suppose that $I=[0,r]$ is given. Let $\mathfrak{f}:I\times \mathbb{C}\to \mathbb{C}$ be continuous and satisfy

$$|\mathfrak{f}(t,u_1)-\mathfrak{f}(t,v_1)| \le A_1|u_1-v_1|,\forall t\in I,$$

(9)

for $u_1,v_1\in \mathbb{C}$, and some positive constant $A_1$. Moreover, let ${\mathfrak{I}}_{\mathfrak{k}}:\mathbb{C}\to \mathbb{C}$, and there exists a positive constant $A_2<1$:

$$|{\mathfrak{I}}_{\mathfrak{k}}\left(u_1\right)-I_k\left(v_1\right)| \le A_2|u_1-v_1|,\forall u_1,v_1\in \mathbb{C}.$$

(10)

Let a continuous function $\mathfrak{H}\left(t\right):I\to \mathbb{C}$:

$$|\mathfrak{H}\left(\lambda \right)-{\int }_0^{\lambda }f(\alpha ,\mathfrak{H}\left(\alpha \right))d\alpha -\sum _{0<{\lambda }_k<\lambda }{\mathfrak{I}}_{\mathfrak{k}}\left(\mathfrak{H}\left({\lambda }_k^{-}\right)\right)| \le \phi \left(\lambda \right)$$

(11)

for all $\lambda \in I$, where ${\mathfrak{I}}_{\mathfrak{k}}:\mathbb{C}\to \mathbb{C},$ $\mathfrak{H}\left({\lambda }_k^{-}\right)$ represents the left limit of $\mathfrak{H}\left(t\right)$ at $\lambda ={\lambda }_k$, $k=1,2,\dots ,m$, where $\phi :I\to (0,\infty )$ is a nondecreasing continuous function. Then, a unique continuous function ${\mathfrak{H}}_0:I\to \mathbb{C}$ exists:

$${\mathfrak{H}}_0\left(\lambda \right)={\int }_0^{\lambda }f(\alpha ,\mathfrak{H}\left(\alpha \right))d\alpha +\sum _{0<{\lambda }_k<\lambda }{\mathfrak{I}}_{\mathfrak{k}}\left(\mathfrak{H}\left({\lambda }_k^{-}\right)\right),$$

(12)

and

$$|\mathfrak{H}\left(\lambda \right)-{\mathfrak{H}}_0\left(\lambda \right)|\le \frac{e^{\beta r}}{1-\left(\frac{A_1}{\beta }+A_2\right)}\phi \left(t\lambda \right),$$

(13)

for all $\lambda \in I$ and for the arbitrary fixed real number $\beta >A_1/(1-A_2)$.

Proof. 

We start by defining a set $\mathfrak{Y}:=\{h_0:I\to \mathbb{C}|h_0\mathrm{is}\mathrm{continuous}\}$ and introduce a GM on $\mathfrak{Y}$ as follows:

$$d(f,h_0)=inf\{C\in [0,\infty ]:|f\left(\lambda \right)-h_0\left(\lambda \right)|e^{-\beta \lambda }\le C\phi \left(\lambda \right),\lambda \in I\}.$$

(14)

It is easy to prove that $(\mathfrak{Y},d)$ is a complete GM space (see, e.g., [32]). Now, define the operator $\theta :\mathfrak{Y}\to \mathfrak{Y}$ by

$$\left(\theta h_1\right)\left(\lambda \right)={\int }_0^{\lambda }f(\alpha ,h_1\left(\alpha \right))d\alpha +\sum _{0<t_k<t}{\mathfrak{I}}_{\mathfrak{k}}\left(h_1\left({\lambda }_k^{-}\right)\right),\forall h_1\in \mathfrak{Y}\mathrm{and}\lambda \in I.$$

(15)

We prove that the operator $\theta$ is strictly contractive on the set $\mathfrak{Y}$.

Suppose $g_1,h_1\in \mathfrak{Y}$, and let $C_{g_1{h}_1}\in [0,\infty ]$ be a constant with $d(g_1,h_1)\le C_{g_1{h}_1}$ for any $g_1,h_1\in \mathfrak{Y}$. By (14), we can write

$$|g_1\left(\lambda \right)-h_1\left(\lambda \right)|e^{-\beta \lambda }\le C_{g_1{h}_1}\phi \left(\lambda \right),$$

(16)

for all $\lambda \in I$. Then, using (9)–(11), (15), and (16) we obtain

$$\begin{array}{ccc}\hfill |\theta g_1\left(\lambda \right)-\theta h_1\left(\lambda \right)|& \le & \left|{\int }_0^{\lambda }\left\{\mathfrak{f}(s,g_1\left(s\right))-\mathfrak{f}(s,h_1\left(s\right))\right\}ds\right|\hfill \\ & +& \left|\sum _{0<{\lambda }_k<\lambda }\left\{{\mathfrak{I}}_{\mathfrak{k}}\left(g_1\left({\lambda }_k^{-}\right)\right)-{\mathfrak{I}}_{\mathfrak{k}}\left(h_1\left({\lambda }_k^{-}\right)\right)\right\}\right|\hfill \\ & \le & {\int }_0^{\lambda }\left|\left\{\mathfrak{f}(s,g_1\left(s\right))-\mathfrak{f}(s,h_1\left(s\right))\right\}\right|ds\hfill \\ & +& \sum _{0<{\lambda }_k<\lambda }\left|\left\{{\mathfrak{I}}_{\mathfrak{k}}\left(g_1\left({\lambda }_k^{-}\right)\right)-{\mathfrak{I}}_{\mathfrak{k}}\left(h_1\left({\lambda }_k^{-}\right)\right)\right\}\right|\hfill \\ & \le & A_1{\int }_0^{\lambda }|\{g_1\left(s\right)-h_1\left(s\right)\}|ds+A_2\sum _{0<{\lambda }_k<\lambda }|g_1\left({\lambda }_k^{-}\right)-h_1\left({\lambda }_k^{-}\right)|\hfill \\ & \le & A_1{\int }_0^{\lambda }|\{g_1\left(s\right)-h_1\left(s\right)\}|e^{-\beta s}{e}^{\beta s}ds+A_2\sum _{0<{\lambda }_k<\lambda }|g_1\left({\lambda }_k^{-}\right)-h_1\left({\lambda }_k^{-}\right)|\hfill \\ & \le & A_1{C}_{g_1{h}_1}{\int }_0^{\lambda }\phi \left(s\right)e^{\beta s}ds+A_2\sum _{0<{\lambda }_k<\lambda }|g_1\left({\lambda }_k^{-}\right)-h_1\left({\lambda }_k^{-}\right)|e^{-\beta {\lambda }_k^{-}}{e}^{\beta {\lambda }_k^{-}}\hfill \\ & \le & A_1{C}_{g_1{h}_1}\phi \left(t\right){\int }_0^{\lambda }{e}^{\beta s}ds+A_2{C}_{g_1{h}_1}\sum _{0<{\lambda }_k<\lambda }\phi \left({\lambda }_k^{-}\right)e^{\beta {\lambda }_k^{-}}\hfill \\ & =& A_1{C}_{g_1{h}_1}\phi \left(\lambda \right)\frac{1}{\beta }(e^{\beta \lambda }-1)+A_2{C}_{g_1{h}_1}\phi \left(\lambda \right)e^{\beta \lambda }\hfill \\ & \le & A_1{C}_{g_1{h}_1}\phi \left(\lambda \right)\frac{1}{\beta }{e}^{\beta \lambda }+A_2{C}_{g_1{h}_1}\phi \left(\lambda \right)e^{\beta \lambda }\hfill \\ & \le & \left(\frac{A_1}{\beta }+A_2\right)C_{g_1{h}_1}\phi \left(\lambda \right)e^{\beta \lambda },\lambda \in I.\hfill \end{array}$$

Therefore, we have

$$|\theta g_1\left(\lambda \right)-\theta h_1\left(\lambda \right)|e^{-\beta \lambda }\le \left(\frac{A_1}{\beta }+A_2\right)C_{g_1{h}_1}\phi \left(\lambda \right),$$

for all $\lambda \in I$, i.e.,

$$d(\theta g_1,\theta h_1)\le \left(\frac{A_1}{\beta }+A_2\right)C_{g_1{h}_1}\phi \left(\lambda \right)\Rightarrow d(\theta g_1,\theta h_1)\le \left(\frac{A_1}{\beta }+A_2\right)d(g_1,h_1)\forall g_1,h_1\in \mathfrak{Y}.$$

Note that we have $A_1/\beta +A_2<1$, since $\beta >A_1/(1-A_2)$; thus, we conclude that the operator $\theta :\mathfrak{Y}\to \mathfrak{Y}$ is strictly contractive. It follows from (15) that, for any arbitrary $h_0\in \mathfrak{Y}$, ∃ a constant $C\in [0,\infty ]$ with

$$|\theta h_0\left(\lambda \right)-h_0\left(\lambda \right)|=|{\int }_0^{\lambda }{g}_1(\alpha ,h_0\left(\alpha \right))d\alpha -\sum _{0<{\lambda }_k<\lambda }{\mathfrak{I}}_{\mathfrak{k}}\left(h_0\left({\lambda }_k^{-}\right)\right)-h_0\left(\lambda \right)|e^{-\beta \lambda }\le C\phi \left(\lambda \right)$$

for all $\lambda \in I$. Since $g_1,h_0$ are bounded and ${min}_{\lambda \in I}\phi \left(\lambda \right)>0$, then (14) implies that

$$d(\theta h_0,h_0)<\infty .$$

(17)

Therefore, according to Theorem 3(a), a continuous function $v_0:I\to \mathbb{C}$ exists such that ${\theta }^n{v}_0\to v_0$ in $(\mathfrak{Y},d)$ and $\theta v_0=v_0$, i.e., $v_0$ satisfies (12) $\forall \lambda \in I$.

Next, we show that $\{g_1\in \mathfrak{Y}|d(h_0,g_1)<\infty \}=\mathfrak{Y}$, where $h_0$ satisfies (17). Letting $g_1\in \mathfrak{Y}$, since $g_1$ and $h_0$ are bounded on I and ${min}_{\lambda \in I}\phi \left(\lambda \right)>0$, then ∃ some constant $C_{g_1}\in [0,\infty ]$: $|h_0\left(\lambda \right)-g_1\left(\lambda \right)|e^{-\beta \lambda }\le C_{g_1}\phi \left(\lambda \right)\forall \lambda \in I$. Thus, $d(h_0,g_1)<\infty$ for any $g_1\in \mathfrak{Y}$. Therefore, $\{g_1\in \mathfrak{Y}|d(h_0,g_1)<\infty \}=\mathfrak{Y}$. From Theorem 3(b), $v_0$, given by (12), is the unique continuous function. Lastly, Theorem 3(c) implies that

$$d(\mathfrak{H},{\mathfrak{H}}_0)\le \frac{1}{1-\left(\frac{A_1}{\beta }+A_2\right)}d(\theta \mathfrak{H},\mathfrak{H})\le \frac{1}{1-\left(\frac{A_1}{\beta }+A_2\right)}.$$

(18)

From the definition of the metric $d(·,·)$, we have

$$\left|\mathfrak{H}\left(\lambda \right)-{\mathfrak{H}}_0\left(\lambda \right)\right|e^{-\beta \lambda }\le \frac{1}{1-\left(\frac{A_1}{\beta }+A_2\right)}\phi \left(\lambda \right)$$

for all $\lambda \in I$. Therefore, we obtain

$$\left|\mathfrak{H}\left(\lambda \right)-{\mathfrak{H}}_0\left(\lambda \right)\right|\le \frac{e^{\beta \lambda }}{1-\left(\frac{A_1}{\beta }+A_2\right)}\phi \left(\lambda \right)$$

and thus

$$\left|\mathfrak{H}\left(\lambda \right)-{\mathfrak{H}}_0\left(\lambda \right)\right|\le \frac{e^{\beta r}}{1-\left(\frac{A_1}{\beta }+A_2\right)}\phi \left(\lambda \right).$$

(19)

□

Remark 1. 

We drop the assumption that $0<K{A}_1+A_2<1$ in Theorem 4, whereas it has been required in Theorem 2.1 in the article of R. Shah and A. Zada [47].

Remark 2. 

We drop the assumption that $|{\int }_c^t\phi \left(\zeta \right)d\zeta |\le K\phi \left(t\right)$ in Theorem 4, whereas it is essential in Theorem 2.1 in [47].

Considering Definition 2, the following corollary gives the U–H stability result of (8).

Corollary 1. 

Given a fixed real number r, let $I=[0,r]$. Let a continuous function $\mathfrak{f}:I\times \mathbb{C}\to \mathbb{C}$ satisfying (9) $\forall t\in I$ and $u_1,v_1\in \mathbb{C}$, and some positive constant $A_1$.

Moreover, let $I_k:\mathbb{C}\to \mathbb{C}$ satisfy condition (10) for all $u_1,v_1\in \mathbb{C}$, and a positive constant $A_2<1$. Further, if $ϵ\ge 0$ and $\mathfrak{H}:I\to \mathbb{C}$ (a continuous function):

$$|\mathfrak{H}\left(\lambda \right)-{\int }_0^{\lambda }\mathfrak{f}(\alpha ,\mathfrak{H}\left(\alpha \right))d\alpha -\sum _{0<{\lambda }_k<\lambda }{\mathfrak{I}}_{\mathfrak{k}}\left(\mathfrak{H}\left({\lambda }_k^{-}\right)\right)|\le ϵ,\forall \lambda \in I,$$

(20)

${\mathfrak{I}}_{\mathfrak{k}}:\mathbb{C}\to \mathbb{C}$, and $\mathfrak{H}\left({\lambda }_k^{-}\right)$ represents the left limit of $\mathfrak{H}\left(\lambda \right)$ at $\lambda ={\lambda }_k$, $k=1,2,\dots ,m$, then a unique continuous function ${\mathfrak{H}}_0:I\to \mathbb{C}$ exists:

$${\mathfrak{H}}_0\left(\lambda \right)={\int }_0^{\lambda }\mathfrak{f}(\alpha ,\mathfrak{H}\left(\alpha \right))d\alpha +\sum _{0<{\lambda }_k<\lambda }{\mathfrak{I}}_{\mathfrak{k}}\left(\mathfrak{H}\left({\lambda }_k^{-}\right)\right),$$

(21)

and

$$|\mathfrak{H}\left(\lambda \right)-{\mathfrak{H}}_0\left(\lambda \right)|\le \frac{e^{\beta r}}{1-\left(\frac{A_1}{\beta }+A_2\right)}ϵ,\forall t\in I,$$

(22)

and the arbitrary fixed real number $\beta >A_1/(1-A_2)$.

Proof. 

The proof is a simple application of Theorem 4 when the function $\phi \left(\lambda \right):=ϵ$. □

4. Conclusions

In our work, we studied the U–HR and U–H stability of the impulsive Volterra integral equation. We abandoned certain fundamental presumptions of several recent, intriguing discoveries in our research. By doing this, we enhanced a number of earlier results. Potential future work could be to investigate the U–HR and U–H stability for much more complicated integral equations. Another direction is to investigate the stability using some novel techniques. The authors also intend to examine the U–HR and U–H stability of some complicated BVPs involving impulsive effects with applications.