On the qualitative behaviors of Volterra-Fredholm integro differential equations with multiple time-varying delays

Abstract

This article considers a Volterra-Fredholm integro-differential equation including multiple time-varying delays. The aim of this article is to study the uniqueness of solution, the Ulam–Hyers–Rassias stability and the Ulam–Hyers stability of the Volterra-Fredholm integro-differential equation including multiple time-varying delays. We prove four new results in connection with the uniqueness of solution, the Ulam–Hyers–Rassias stability and the Ulam–Hyers stability of the considered Volterra-Fredholm integro-differential equation, respectively. The new results of this article involve sufficient conditions. The techniques of the proofs depend on the fixed point method according to the definitions of a suitable metric, operators and the related calculations. In particular case of the considered Volterra-Fredholm integro-differential equation, two illustrative examples are presented to verify the applications of the results. This article also involves some new complementary outcomes in connection with qualitative theory of Volterra-Fredholm integro-differential equations with delays.

Keywords:

• İntegro-differential equation
• Ulam–Hyers–Rassias stability
• Volterra-Fredholm
• Ulam-Hyers stability
• contraction mapping principle
• generalized complete metric

2020 MSC:

• 45D05
• 45G10
• 45J05
• 45M10
• 47H10

1. Introduction

The field of time-delay mathematical models as a whole has its beginning dated back to the eighteenth century, and it received substantial attention in the early twentieth century in works devoted to the modeling of biological, ecological, as well as engineering systems. A time-delay mathematical model is the feature that the mathematical model’s future evolution depends not only on its present state, but also on a period of its history. This particular cause–effect relationship can be traditionally modeled as FDEs, in particular, as delay differential and integro-differential equations. While in practice numerous dynamical systems are described by ODEs alone, for which the systems future evolution depends solely on its current state, there are times when delay effect cannot be neglected, or it will be more beneficial for it to be accounted for. Hence, time-delay mathematical models have been studied long and well and are still being investigated very effectively. They have for decades been an active area of scientific research in mathematics, biology, ecology, economics, and in engineering, under such terms as hereditary systems, systems with after effect, or systems with time-lag, and more generally as a subclass of FDEs and infinite dimensional systems (Gu et al., 2003).

As we also know ODEs, FrODEs, Volterra IEs, Volterra IDEs, Volterra-Fredholm IDEs, etc., which involve delay or without delay, are very effective and important mathematical tools to model and then investigate various real world problems in physics, engineering, medicine, mechanics, economy, biology and so on. In general, most of times it is very hard and sometimes impossible to solve explicitly these kinds of equations excepting numerically. After modeling a real world phenomena as an appropriate equation, it can be discussed qualitative properties of that equation using suitable mathematical techniques without solving the equation understudy.

In the relevant literature, among the known qualitative properties of the above equations, Ulam type stabilities have essential roles during the investigations and applications. In fact, it is seen from the relevant literature that the first and key work on the Ulam stability is related to the FEs, which also belongs to Ulam (1964). Later, in the time, this qualitative concept was extended and applied to ODEs, Volterra IEs, Volterra IDEs, PDEs, DDEs, FrDEs, etc., as the new qualitative concepts called UHR stability, generalized UHR stability, UH stability, $\sigma -$ semi UH stability, $\sigma -$ semi UHR stability, etc., by researchers.

Indeed, today, in the relevant literature, the subject of the Ulam type stabilities is a very attractive and interesting field of the research, and this subject is also effectively investigated for various kind of the mathematical models of science, engineering, etc. By this way, without loss of generality, it is said that a Volterra-Fredholm IDE admits the UH stability if for each function satisfying the Volterra-Fredholm IDE approximately such that there exists an exact solution of the Volterra-Fredholm IDE, it is close to the approximation solution. This means that the Ulam stability case of the Volterra-Fredholm IDE is how will the solutions of the inequality vary from those of the considered the Volterra-Fredholm IDE? As for the UHR stability, there is no reason for the Cauchy difference to be bounded (Jung, 2011). Toward this point, Rassias (1978) tried to weaken the condition for the Cauchy difference and succeeded in proving what is now known to be the UHR stability for the additive Cauchy equation. Namely, the possibility to use functions for that bounds (and not simply constants) was initially proposed by Rassias (1978), originating, therefore, a generalization of the initial concept, and making appear the so-called UHR stability This terminology is justified because the theorem of Rassias (1978, Theorem 2.5) has strongly influenced mathematicians studying stability problems of FEs. Similar comments can be given for all kind of the Ulam type stabilities.

For a comprehensive treatment of some related qualitative properties of ODEs, FrODEs, Volterra IEs, Volterra IDEs, Volterra-Fredholm IDEs, etc., we refer the readers for the existence and computational results in connection with Volterra-Fredholm IDEs with delay, see Amin et al. (2021), the existence and Ulam type results in connection with Volterra-Fredholm IDEs with delay, see Kucche and Shikhare (2018b), Miah et al. (2024), Tunç and Tunç (2024), the Ulam type stability of ODEs with and without delay, see Aslıyüce and Öğrekçi (2023), Jung (2010), Jung and Rezaei (2015), Onitsuka (2024), the Ulam stability for impulsive IDEs, see Bensalem et al. (2024), existence of solutions and Ulam stability for IDEs and IEs, see Burton (2016, 2019), Castro and Ramos (2009, 2010), Castro and Simões (2017, 2018), Chauhan et al. (2022), Ciplea et al. (2022), Deep et al. (2020), Graef et al. (2023), Ilea and Otrocol (2020), Jung (2007), Kucche and Shikhare (2018a), Öğrekçi et al. (2023), Tunç and Tunç (2023), Tunç et al. (2024), Tunç et al. (2024), the UHR stability of FrDEs with and without delay, see Benzarouala and Tunç (2024), Khan et al. (2018), Xu and Farman (2023), for the Mittag–Leffler–Hyers–Ulam stability of ODEs, see Eghbali and Kalvandi (2018), the UHR stability of FEs, see Jung (2011), the existence and stability of operator equations, see Petruşel et al. (2019), the stability of linear mappings in Banach spaces, see Rassias (1978), existence and the UH stability for neutral stochastic FDEs, see Selvam et al. (2024), the global existence and uniqueness of the solution of the fractional delay Volterra IDEs in the Atangana–Beleanu–Caputo sense, see Sweis et al. (2023a), some qualitative results in connection with DDEs including Mittag-Leffler kernel, IDEs with time delays and Hilfer fractional DDEs, see Sweis et al. (2022, 2023b, 2024), respectively, the existence and uniqueness of a mild solution, etc. of a mixed Volterra-Fredholm-type third-order dispersion system, see Patel et al. (2023), the existence and approximate controllability for certain fractional stochastic Volterra-Fredholm IDEs, see Dineshkumar et al. (2023), for the approximate controllability of a stochastic Volterra-Fredholm IDE of second order including delay and impulses, see Ma et al. (2023) and some other qualitative works, see Bohner et al. (2021), Graef and Tunç (2015), Miahi et al. (2021), Tunç and Tunç (2019) and Shukla et al. (2015).

In particular, we will now also present only a few earlier results in connection with the qualitative properties of some Volterra IEs, Volterra IDEs and Volterra-Fredholm IDEs.

Jung (2007) studied the UHR and the UH stabilities of the Volterra IE: $$y(x)=\underset{c}{\overset{x}{\int }}f(\tau ,y(\tau ))d\tau$$ by using the FP method. To the best of information, the results of Jung (2007, Theorem 2.1, Theorem 3.1) can be considered as the first and key results in connection with the UHR stability and UH stability for Volterra IEs in the relevant literature. The work of Jung (2007) is the first key and significant paper on the Ulam type of stabilities of IEs and Volterra IDEs, in which the Banach CMP was used as main tool in the proofs.

Castro and Ramos (2009, 2010) derived sufficient conditions with regard to the UH stability and the UHR stability of two kind of Volterra IEs: $$y(x)=\underset{a}{\overset{x}{\int }}f(x,\tau ,y(\tau ))d\tau$$ and $$y(x)=\underset{a}{\overset{x}{\int }}f(x,\tau ,y(\tau ),y(\alpha (\tau )))d\tau .$$

The first IE above does not have delay and the second IE above has a variable delay.

Castro and Ramos (2009, 2010) presented a simple way, which is called “progressive contractions”, to establish global existence and uniqueness of a solution on ${\mathrm{R}}^{+},{\text{R}}^{+}=[0,\infty ),$ of the following the Volterra IDE and IE, respectively: $$x^{\prime }(t)=g(t,x(t))+\underset{0}{\overset{t}{\int }}A(t-s)f(s,x(s))\mathit{\text{ds}}$$ and $$x(t)=g(t,x(t))+\underset{0}{\overset{t}{\int }}A(t-s)f(s,x(s))\mathit{\text{ds}}.$$

Ilea and Otrocol (2020) applied the Burton method and obtained similar results for the following Volterra IEs in a Banach space: $$x(t)=\underset{0}{\overset{t}{\int }}K(t,s,x(s))\mathit{\text{ds}}$$ and $$x(t)=g(t,x(t))+\underset{0}{\overset{t}{\int }}f(t,s,x(s))\mathit{\text{ds}}.$$

Tunç et al. (2024) considered the following Volterra IE with N-variable time delays and the IDE without delay, respectively: $$x(t)=q(t)+r(x(t))+h(t,x(t))+g(t,x(t))\underset{0}{\overset{t}{\int }}A(t-s)f(s,x(s))\mathit{\text{ds}}$$ $$+\sum _{i=1}^N\underset{0}{\overset{t}{\int }}{A}_i(t-s)f_i(s,x(s),x(s-{\tau }_i(s)))\mathit{\text{ds}}$$ and $$x^{\prime }(t)=r(x(t))+g(t,x(t))+h(t,x(t))\underset{0}{\overset{t}{\int }}A(t-s)f(s,x(s))\mathit{\text{ds}}.$$

Tunç et al. (2024) dealt with global existence and uniqueness of solutions of the above Volterra IE and the Volterra IDE by fixed point method according to the progressive contractions, which belong to T.A. Burton.

Now, before summarizing some qualitative results in connection with certain Volterra-Fredholm IDEs, we introduce some brief and useful information as follows: A Volterra-Fredholm IDE, which is a combination of disjoint Volterra and Fredholm integrals and differential operator, may appear in one integral equation. Volterra-Fredholm IDEs arise from many physical and chemical applications similar to the Volterra-Fredholm IEs. In the Volterra IDEs, the upper limit of integration is the variable x, while in the Fredholm IDEs, the upper limit of integration is a fixed constant. Volterra (1959), in the early 1900, studied the population growth, where new type of equations have been developed and was termed as IDEs. Indeed, an initial value problem can be transformed to an equivalent Volterra IE or Volterra IDE. However, a boundary value problem can be converted to an equivalent Fredholm IE or Fredholm IDE. It is also important to point out here that the procedure of reducing a boundary value problem to a Fredholm IE or a Fredholm IDE is complicated and rarely used.

Kucche and Shikhare (2018a, b) considered the below Volterra IDEs without and with delay and the Volterra-Fredholm delay IDE, respectively: $$x^{\prime }(t)=\mathit{\text{Ax}}(t)+f\left(t,x(t),\underset{0}{\overset{t}{\int }}g(t,s,x(s))\mathit{\text{ds}}\right),$$ $$x^{\prime }(t)=\mathit{\text{Ax}}(t)+f\left(t,x_t,\underset{0}{\overset{t}{\int }}g(t,s,x_s)\mathit{\text{ds}}\right)$$ and $$x^{\prime }(t)=\mathit{\text{Ax}}(t)+f\left(t,x_t,\underset{0}{\overset{t}{\int }}{g}_1(t,s,x_s)\mathit{\text{ds}},\underset{0}{\overset{b}{\int }}{g}_1(t,s,x_s)\mathit{\text{ds}}\right),t\in J=[0,b].$$

The authors studied the UH stabilities the first two the IDEs, and the UH stability and UHR stability of the second Volterra-Fredholm delay IDE in the completed metric spaces using Banach’s FPT, Pachpatte’s inequality and some suitable norms.

The motivation behind this article, in particular, is the key and more recently article of Miah et al. (2024). In Miah et al. (2024), the authors established results for the UHR stability and the UH stability of the following Volterra-Freedholm IDE involving a variable delay: (1) $${\theta }^{\prime }(t)=\psi \left(t,\theta (t),\theta \left(\alpha (t)\right)\right)+\underset{0}{\overset{t}{\int }}{\chi }_1\left(t,\gamma ,\theta (\gamma ),\theta \left(\alpha (\gamma )\right)\right)d\gamma +\underset{0}{\overset{T}{\int }}{\chi }_2\left(t,\gamma ,\theta (\gamma ),\theta \left(\alpha (\gamma )\right)\right)d\gamma$$(1) with the initial condition $\theta (0)=\beta .$ Additionally, Miah et al. (2024) dealt with the existence of unique solution for Volterra-Freedholm IDE (1)(1) $${\theta }^{\prime }(t)=\psi \left(t,\theta (t),\theta \left(\alpha (t)\right)\right)+\underset{0}{\overset{t}{\int }}{\chi }_1\left(t,\gamma ,\theta (\gamma ),\theta \left(\alpha (\gamma )\right)\right)d\gamma +\underset{0}{\overset{T}{\int }}{\chi }_2\left(t,\gamma ,\theta (\gamma ),\theta \left(\alpha (\gamma )\right)\right)d\gamma$$(1) by employing the Banach CMP.

Inspired by the works of Burton (2016, 2019), Kucche and Shikhare (2018b), Miah et al. (2024) and that given above, in this article, we will focus on the following Volterra-Freedholm IDE involving multiple variable time delays: (2) $${\vartheta }^{\prime }(t)=\sum _{k=1}^N{F}_k\left(t,\vartheta (t),\vartheta \left({\beta }_k(t)\right)\right)+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma +\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma$$(2) with $\vartheta (0)={\alpha }_0,$ ${\alpha }_0\in R,$ where $\vartheta \in C^1(R,R),$ $F_k\in C(\wp \times \mathrm{R}\mathrm{\times }\text{R,R}),\wp =[0,T],$ $0<T<\infty ,G_k,H_k\in C(\wp \times \wp \times \mathrm{R}\mathrm{\times }\text{R,R}),G_k,H_k\in C(\wp \times \wp \times \mathrm{R}\mathrm{\times }\text{R,R}),0\le {\beta }_k(t)\le t,{\beta }_k\in C[\wp ,\wp ],$ $k=1,2,\dots N.$ We will here investigate three qualitative concepts by employing the Banach CMP: the first one is existence of the unique solution of the Volterra-Fredholm IDE (2)(2) $${\vartheta }^{\prime }(t)=\sum _{k=1}^N{F}_k\left(t,\vartheta (t),\vartheta \left({\beta }_k(t)\right)\right)+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma +\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma$$(2) , the second and third ones are the UHR stability and the UH stability of the Volterra-Fredholm IDE (2)(2) $${\vartheta }^{\prime }(t)=\sum _{k=1}^N{F}_k\left(t,\vartheta (t),\vartheta \left({\beta }_k(t)\right)\right)+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma +\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma$$(2) , respectively.

We note that a large number of applications in the theory of artificial neural networks, networks known as bidirectional associative memory with leakage delays, numerous models for some population dynamics, and ecology problems, etc., can be denoted by FDEs and IDEs with multiple delays (Bohner et al., 2021). Thereby, since our Volterra-Freedholm IDE (2)(2) $${\vartheta }^{\prime }(t)=\sum _{k=1}^N{F}_k\left(t,\vartheta (t),\vartheta \left({\beta }_k(t)\right)\right)+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma +\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma$$(2) is nonlinear and has multiple variable delays, it can be useful for researchers working in the fields of artificial neural networks, population dynamics, ecology problems, and so on. This is one of the potential impacts of our work and the difference it will make.

We should also mention that despite the existence much works with regard to the existence, the uniqueness of solutions and the Ulam type stabilities of various mathematical models of ODEs, FEs, Volterra IEs, Volterra IDEs, PDEs, DDEs, FrDEs, there is only a few works in connection with that concepts on Volterra-Freedholm IDEs without or with delay (see Kucche & Shikhare, 2018b; Miah et al., 2024; Tunç & Tunç, 2024). Additionally, we did not find any work dealt with the existence, the uniqueness of solutions and the Ulam type stabilities of Volterra-Freedholm IDEs involving multiple variable time delays. Thereby, by virtue of the above information, it deserves to discuss the mentioned concepts for the Volterra-Freedholm IDE (2)(2) $${\vartheta }^{\prime }(t)=\sum _{k=1}^N{F}_k\left(t,\vartheta (t),\vartheta \left({\beta }_k(t)\right)\right)+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma +\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma$$(2) . Furthermore, this article is the first work on the mentioned concepts for the Volterra-Freedholm IDE (2)(2) $${\vartheta }^{\prime }(t)=\sum _{k=1}^N{F}_k\left(t,\vartheta (t),\vartheta \left({\beta }_k(t)\right)\right)+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma +\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma$$(2) involving multiple variable time delays. The novelty of this article is that the delay Volterra-Freedholm IDE (2)(2) $${\vartheta }^{\prime }(t)=\sum _{k=1}^N{F}_k\left(t,\vartheta (t),\vartheta \left({\beta }_k(t)\right)\right)+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma +\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma$$(2) is a novel and general mathematical model. Applying the Banach CMP and the generalized metric, we obtain new results on the existence, the uniqueness of solutions and the Ulam type stabilities with the only Lipschitz type conditions. The imposed conditions are not strict assumptions on the functions included in the Volterra-Freedholm IDE (2)(2) $${\vartheta }^{\prime }(t)=\sum _{k=1}^N{F}_k\left(t,\vartheta (t),\vartheta \left({\beta }_k(t)\right)\right)+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma +\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma$$(2) .

The study is structured as follows. In Sect. 2, we give some basic information such as definitions, two theorems, etc., that are used in this article. Section 3 involves a new theorem on the existence and the uniqueness of solutions of the Volterra-Freedholm IDE (2)(2) $${\vartheta }^{\prime }(t)=\sum _{k=1}^N{F}_k\left(t,\vartheta (t),\vartheta \left({\beta }_k(t)\right)\right)+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma +\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma$$(2) . In Sect. 4, we give two new theorems on the UHR stability of the Volterra-Freedholm IDE (2)(2) $${\vartheta }^{\prime }(t)=\sum _{k=1}^N{F}_k\left(t,\vartheta (t),\vartheta \left({\beta }_k(t)\right)\right)+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma +\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma$$(2) . Section 5 involves a new theorem for the UH stability of the same equation. Then, in Sect. 6, two examples are given to explain our main results. Finally, Sect. 7 involves the conclusion of this article.

2. Basic information

We will now present basic concepts and a result that are needed in the main results.

Definition 2.1.

(Miah et al., 2024). Let $\Omega \ne \varnothing$ be a set and $\Delta :\Omega \times \Omega \to [0,\infty )$ be a distance function described by

1. $\Delta (\Xi ,\Xi )=0\iff$ ${\Xi }_1={\Xi }_2;$

2. $\Delta ({\Xi }_1,{\Xi }_2)=\Delta ({\Xi }_2,{\Xi }_1);$

3. $\Delta ({\Xi }_1,{\Xi }_2)\le \Delta ({\Xi }_1,{\Xi }_3)+\Delta ({\Xi }_3,{\Xi }_2);$

4. every $\Delta -$Cauchy sequences in $\Omega$ is $\Delta -$convergent, i.e., $\underset{p,q\to \infty }{\mathrm{\text{lim}}}\Delta ({\Xi }_p,{\Xi }_q)=0$ for an element ${\Xi }_n\in \Omega ,$ and there exists an element $\Xi \in \Omega$ such that $\underset{p\to \infty }{\mathrm{\text{lim}}}\Delta ({\Xi }_p,\Xi )=0.$

Then $(\Omega ,\Delta )$ is said a generalized complete metric space.

We will now give a theorem, which is known as a variant of the Banach FPT.

Theorem 2.1

(Diaz & Margolis, 1968; Miahi et al., 2021). Let $(X,d)$ be a complete generalized metric space and $\Theta :X\to X$ be a strictly contractive mapping with the Lipschitz constant $L<1.$ Then for each given element $x\in X,$ either $d({\Theta }^{n}x,{\Theta }^{n+1}x)=\infty$ for all nonnegative integers $n$ or there exists an $n_0>0,$ $n_0\in N,$ such that

1. $d({\Theta }^{k}x,{\Theta }^{k+1}x)<\infty$ for all $n\ge n_0;$

2. the sequence ${\left\{{\Theta }^{n}x\right\}}_{n\in \mathbb{N}}$ converges to a fixed point $y*$ of $\Theta ;$

3. $y*$ is the unique fixed point of $\Theta$ in the set $\Delta =\left\{y\in X:d({\Theta }^{n_0}x,y)<\infty \right\};$

4. $d(y,y*)\le \frac{1}{1-L}d(y,\Theta y)$ for all $y\in \Delta .$

Definition 2.2.

If for each function $\chi \in C^1(\wp ,R)$ satisfying (3) $$\begin{array}{c}|{\chi }^{\prime }(t)-\sum _{k=1}^N{F}_k(t,\chi (t),\chi ({\beta }_k(t)))-\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\chi (\gamma ),\chi ({\beta }_k(\gamma ))\right)\right]d\gamma \\ -\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\chi (\gamma ),\chi ({\beta }_k(\gamma ))\right)\right]d\gamma |\le \phi (t),\end{array}$$(3) where $\phi (t)\ge 0$ for all $t\in \wp ,$ there exists a solution ${\chi }_0(t)$ of the Volterra-Freedholm IDE (2)(2) $${\vartheta }^{\prime }(t)=\sum _{k=1}^N{F}_k\left(t,\vartheta (t),\vartheta \left({\beta }_k(t)\right)\right)+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma +\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma$$(2) and a constant ${\rm B}\in R,{\rm B}>0,$ which does not dependent $\chi (t)$ and ${\chi }_0(t),$ such that (4) $$\left|\chi (t)-{\chi }_0(t)\right|\le {\rm B}\phi (t),$$(4) for all $t\in \wp ,$ then we call that the Volterra-Freedholm IDE (2)(2) $${\vartheta }^{\prime }(t)=\sum _{k=1}^N{F}_k\left(t,\vartheta (t),\vartheta \left({\beta }_k(t)\right)\right)+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma +\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma$$(2) involving multiple variable time delays admits the UHR stability.

Definition 2.3.

If for each function $\chi \in C^1(\wp ,R)$ satisfying $$\begin{array}{l}|{\chi }^{\prime }(t)-\sum _{k=1}^N{F}_k(t,\chi (t),\chi ({\beta }_k(t)))-\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\chi (\gamma ),\chi ({\beta }_k(\gamma ))\right)\right]d\gamma \\ -\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\chi (\gamma ),\chi ({\beta }_k(\gamma ))\right)\right]d\gamma |\le \theta ,\end{array}$$ where $\theta \ge 0,$ $\theta \in R,$ there exists a solution ${\chi }_0(t)$ of the Volterra-Freedholm IDE (2)(2) $${\vartheta }^{\prime }(t)=\sum _{k=1}^N{F}_k\left(t,\vartheta (t),\vartheta \left({\beta }_k(t)\right)\right)+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma +\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma$$(2) and a constant ${\rm B}\in R,{\rm B}>0,$ which does not dependent $\chi (t)$ and ${\chi }_0(t),$ such that $$\left|\chi (t)-{\chi }_0(t)\right|\le {\rm B}\theta ,$$ for all $t\in \wp ,$ then we call that the Volterra-Freedholm IDE (2)(2) $${\vartheta }^{\prime }(t)=\sum _{k=1}^N{F}_k\left(t,\vartheta (t),\vartheta \left({\beta }_k(t)\right)\right)+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma +\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma$$(2) involving multiple variable time delays admits the UH stability.

Theorem 2.2

(Burton, 2006, the Banach CMP). Let $(S,\rho )$ be a complete metric space and let $P:S\to S.$ If there is a positive constant $\alpha <1$ such that for each pair ${\varphi }_1,{\varphi }_2\in S$ we have $$\rho \left(P{\varphi }_1,P{\varphi }_2\right)\le \alpha \rho \left({\varphi }_1,{\varphi }_2\right),$$ then there is one and only one point $\varphi \in S$ with $P\varphi =\varphi .$

3. Existence and uniqueness

The first new qualitative is presented in the below by Theorem 3.1.

Theorem 3.1.

Let $0<T<\infty ,\wp =[0,T],L_{F_k},L_{G_k}$ and $L_{H_k}$ be positive constants. We also assume that the following conditions $(C1)$ and $(C2)$hold true: (C1) $$F_k\in C\left(\wp \times \mathrm{R}\mathrm{\times }\text{R,R}\right),G_k,H_k\in C\left(\wp \times \wp \times \mathrm{R}\mathrm{\times }\text{R,R}\right),0\le {\beta }_k(t)\le t,{\beta }_k\in C[\wp ,\wp ],k=1,2,\dots N;$$(C1) (C2) $$\left|F_k\left(t,{\sigma }_1,{\sigma }_1\left({\beta }_k(t)\right)\right)-F_k\left(t,{\sigma }_2,{\sigma }_2\left({\beta }_k(t)\right)\right)\right|\le L_{F_k}\left|{\sigma }_1-{\sigma }_2\right|,\left|G_k\left(t,\gamma ,{\sigma }_1,{\sigma }_1\left({\beta }_k(\gamma )\right)\right)-G_k\left(t,\gamma ,{\sigma }_2,{\sigma }_2\left({\beta }_k(\gamma )\right)\right)\right|\le L_{G_k}\left|{\sigma }_1-{\sigma }_2\right|,\left|H_k\left(t,\gamma ,{\sigma }_1,{\sigma }_2\left({\beta }_k(\gamma )\right)\right)-H_k\left(t,\gamma ,{\sigma }_2,{\sigma }_2\left({\beta }_k(\gamma )\right)\right)\right|\le L_{H_k}\left|{\sigma }_1-{\sigma }_2\right| \mathit{\text{with}} k=1,2,\dots ,N\text{for all}\gamma ,t\in \mathrm{\wp }\mathit{\text{and}} {\sigma }_1,{\sigma }_2\in R.$$(C2)

If (5) $$0<\sum _{k=1}^N\left(L_{F_k}T+2^{-1}{L}_{G_k}{T}^2+L_{H_k}{T}^2\right)<1,$$(5) then there exists a unique solution of the Volterra-Freedholm IDE (2)(2) $${\vartheta }^{\prime }(t)=\sum _{k=1}^N{F}_k\left(t,\vartheta (t),\vartheta \left({\beta }_k(t)\right)\right)+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma +\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma$$(2) involving multiple variable time delays with $\vartheta (0)={\alpha }_0.$

Proof.

Let $\wp =[0,T]$ and ${\rm X}=C(\wp ,\mathrm{R}).$ We define the following metric: (6) $$\mathrm{d}(\mathcal{l},\kappa )=\mathrm{\text{inf}}\left\{\mathrm{B}\in [0,\infty ):\left|\mathcal{l}(t)-\kappa (t)\right|\le \mathrm{B},\forall t\in \wp \right\},$$(6) where $\mathcal{l},\kappa \in {\rm X}.$ Considering (6)(6) $$\mathrm{d}(\mathcal{l},\kappa )=\mathrm{\text{inf}}\left\{\mathrm{B}\in [0,\infty ):\left|\mathcal{l}(t)-\kappa (t)\right|\le \mathrm{B},\forall t\in \wp \right\},$$(6) and following the way of Jung (2010), it can be easily shown that $({\rm X},\mathrm{d})$ is a generalized complete metric space.

We now define an operator $\mathfrak{I}:{\rm X}\to {\rm X}$ by (7) $$\begin{array}{l}(\mathfrak{I}\mathcal{l})(t)={\alpha }_0+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{F}_k\left(\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)\right]d\mu +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\gamma }{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}\\ +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}},\forall \mathcal{l}\in {\rm X}.\end{array}$$(7)

For any $\mathcal{l},\kappa \in {\rm X},$ let ${\mathrm{B}}_{\mathcal{l}\kappa }\in {\mathrm{R}}^{+},{\text{R}}^{+}=[0,\infty ),$ be a random constant such that $$d(\mathcal{l},\kappa )\le {\mathrm{B}}_{\mathcal{l}\kappa },\forall t\in \wp ,$$ that is, (8) $$\left|\mathcal{l}(t)-\kappa (t)\right|\le {\mathrm{B}}_{\mathcal{l}\kappa }\text{,}\forall t\in \wp .$$(8)

By applying the conditions of Theorem 3.1 and the above operator, we have to show that the operator $\mathfrak{I}$ is strictly contractive according to the metric given by (6)(6) $$\mathrm{d}(\mathcal{l},\kappa )=\mathrm{\text{inf}}\left\{\mathrm{B}\in [0,\infty ):\left|\mathcal{l}(t)-\kappa (t)\right|\le \mathrm{B},\forall t\in \wp \right\},$$(6) . Hence, by employing the conditions $(C1)$ and $(C2),$ we get from (7)(7) $$\begin{array}{l}(\mathfrak{I}\mathcal{l})(t)={\alpha }_0+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{F}_k\left(\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)\right]d\mu +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\gamma }{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}\\ +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}},\forall \mathcal{l}\in {\rm X}.\end{array}$$(7) that $$\left|(\mathfrak{I}\mathcal{l})(t)-(\mathfrak{I}\kappa )(t)\right|$$ $$\le \underset{0}{\overset{t}{\int }}\left|\sum _{k=1}^N{F}_k\left(\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)-\sum _{k=1}^N{F}_k\left(\mu ,\kappa (\mu ),\kappa ({\beta }_k(\mu ))\right)\right|d\mu$$ $$+\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\gamma }{\int }}\left|\sum _{k=1}^N\left[G_k\left(t,\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)-G_k\left(t,\mu ,\kappa (\mu ),\kappa ({\beta }_k(\mu ))\right)\right]\right|\mathit{\text{dμdγ}}$$ $$+\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left|\sum _{k=1}^N\left[H_k\left(t,\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)-H_k\left(t,\mu ,\kappa (\mu ),\kappa ({\beta }_k(\mu ))\right)\right]\right|\mathit{\text{dμdγ}}$$ $$\le \underset{0}{\overset{t}{\int }}\sum _{k=1}^N\left(L_{F_k}\right)\left|\mathcal{l}(\mu )-\kappa (\mu )\right|d\mu +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\gamma }{\int }}\left[\sum _{k=1}^N\left(L_{G_k}\right)\left|\mathcal{l}(\mu )-\kappa (\mu )\right|\right]\mathit{\text{dμdγ}}$$ $$+\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N\left(L_{H_k}\right)\left|\mathcal{l}(\mu )-\kappa (\mu )\right|\right]\mathit{\text{dμdγ}}$$ $$\begin{array}{c}\le \sum _{k=1}^N\left(L_{F_k}\right){\mathrm{B}}_{\mathcal{l}\kappa }t+\underset{0}{\overset{t}{\int }}\sum _{k=1}^N\left(L_{G_k}\right){\mathrm{B}}_{\mathcal{l}\kappa }\gamma d\gamma +\underset{0}{\overset{t}{\int }}\sum _{k=1}^N\left(L_{H_k}\right){\mathrm{B}}_{\mathcal{l}\kappa }Td\gamma \\ =\sum _{k=1}^N\left(L_{F_k}\right){\mathrm{B}}_{\mathcal{l}\kappa }t+\frac{1}{2}\sum _{k=1}^N\left(L_{G_k}\right){\mathrm{B}}_{\mathcal{l}\kappa }{t}^2+\sum _{k=1}^N\left(L_{H_k}\right){\mathrm{B}}_{\mathcal{l}\kappa }\mathit{\text{tT}}\\ \le \sum _{k=1}^N\left(L_{F_k}\right){\mathrm{B}}_{\mathcal{l}\kappa }T+\frac{1}{2}\sum _{k=1}^N\left(L_{G_k}\right){\mathrm{B}}_{\mathcal{l}\kappa }{T}^2+\sum _{k=1}^N\left(L_{H_k}\right){\mathrm{B}}_{\mathcal{l}\kappa }{T}^2\\ =\sum _{k=1}^N\left(L_{F_k}+2^{-1}{L}_{G_k}T+L_{H_k}T\right)T{\mathrm{B}}_{\mathcal{l}\kappa },\forall t\in \wp .\end{array}$$

Then, we have (9) $$\mathrm{d}(\mathfrak{I}\mathcal{l},\mathfrak{I}\kappa )\le \left[\sum _{k=1}^N\left(L_{F_k}+2^{-1}{L}_{G_k}T+L_{H_k}T\right)T\right]\mathrm{d}(\mathcal{l},\kappa )$$(9) for all $\mathcal{l},\kappa \in {\rm X},$ where due to (5)(5) $$0<\sum _{k=1}^N\left(L_{F_k}T+2^{-1}{L}_{G_k}{T}^2+L_{H_k}{T}^2\right)<1,$$(5) $$0<\sum _{k=1}^N\left(L_{F_k}T+2^{-1}{L}_{G_k}{T}^2+L_{H_k}{T}^2\right)<1.$$

Thereby, according to the above outcomes, we conclude that $\mathfrak{I}$ is a contractive operator. Hence, taking into account Theorem 2.1, there exists a unique continuous function ${\chi }^{\ast }$ such that $\mathfrak{I}{\chi }^{\ast }={\chi }^{\ast },$ which implies that $${\chi }^{\ast }(t)={\alpha }_0+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{F}_k\left(\mu ,{\chi }^{\ast }(\mu ),{\chi }^{\ast }({\beta }_k(\mu ))\right)\right]d\mu$$ $$+\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\gamma }{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\mu ,{\chi }^{\ast }(\mu ),{\chi }^{\ast }({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}$$ (10) $$+\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,{\chi }^{\ast }(\mu ),{\chi }^{\ast }({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}.$$(10)

If we differentiate both sides of the integral Eq. (10)(10) $$+\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,{\chi }^{\ast }(\mu ),{\chi }^{\ast }({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}.$$(10) with respect to $t,$ then we obtain $$\begin{array}{l}\left(\frac{d}{\mathit{\text{dt}}}{\chi }^{\ast }(t)\right)=\sum _{k=1}^N{F}_k\left(t,{\chi }^{\ast }(t),{\chi }^{\ast }({\beta }_k(t))\right)+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,{\chi }^{\ast }(\gamma ),{\chi }^{\ast }({\beta }_k(\gamma ))\right)\right]d\gamma \\ +\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,{\chi }^{\ast }(\gamma ),{\chi }^{\ast }({\beta }_k(\gamma ))\right)\right]d\gamma \end{array}$$ with ${\chi }^{\ast }(0)={\alpha }_0.$ This result confirms that ${\chi }^{\ast }$ is the unique solution of Volterra-Freedholm IDE (2)(2) $${\vartheta }^{\prime }(t)=\sum _{k=1}^N{F}_k\left(t,\vartheta (t),\vartheta \left({\beta }_k(t)\right)\right)+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma +\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma$$(2) . Hence, the poof is completed.

4. UHR stability

Within this section, we will focus on the UHR stability via two new theorems for the Volterra-Freedholm IDE (2)(2) $${\vartheta }^{\prime }(t)=\sum _{k=1}^N{F}_k\left(t,\vartheta (t),\vartheta \left({\beta }_k(t)\right)\right)+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma +\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma$$(2) involving multiple variable delays. The first one is given by Theorem 4.1.

Theorem 4.1.

Let $0<T<\infty ,$ $\wp =[0,T],$ $L_{F_k},L_{G_k},L_{H_k}$ and $N_{\ast }$ be positive constants such that in addition to $(C1),(C2)$ of Theorem 3.1, the following condition $$0<\sum _{k=1}^N\left(L_{F_k}{N}_{\ast }+L_{G_k}{N}_{\ast }^2+L_{H_k}{N}_{\ast }^2\right)<1$$ is satisfied.

If a function $\chi \in C^1(\wp ,\mathrm{R})$ satisfies (11) $$\begin{array}{c}|{\chi }^{\prime }(t)-\sum _{k=1}^N{F}_k(t,\chi (t),\chi ({\beta }_k(t)))-\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\chi (\gamma ),\chi ({\beta }_k(\gamma ))\right)\right]d\gamma \\ -\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\chi (\gamma ),\chi ({\beta }_k(\gamma ))\right)\right]d\gamma |\le \phi (t),\forall t\in \wp ,\end{array}$$(11) where $\phi \in C(\wp ,(0,\infty ))$ with (12) $$\underset{0}{\overset{t}{\int }}\phi (\mu )d\mu \le N_{\ast }\phi (t),\forall t\in \wp ,$$(12) then there exists a unique function ${\chi }_0\in C(\wp ,R)$ such that (13) $$\begin{array}{l}{\chi }_0(t)={\alpha }_0+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{F}_k(\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu )))\right]d\mu \\ +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\gamma }{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}\\ +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}\end{array}$$(13) and (14) $$\left|\chi (t)-{\chi }_0(t)\right|\le \frac{N_{\ast }}{1-\underset{k=1}{\overset{N}{\Sigma }}\left(L_{F_k}{N}_{\ast }+L_{G_k}{N}_{\ast }^2+L_{H_k}{N}_{\ast }^2\right)}\phi (t),\forall t\in \wp .$$(14)

Proof.

Let $\wp =[0,T]$ and ${\rm X}=C(\wp ,\mathrm{R}).$ For $\mathcal{l},\kappa \in {\rm X},$ we define the following metric: (15) $$\mathrm{d}(\mathcal{l},\kappa )=\mathrm{\text{inf}}\left\{\mathrm{B}\in [0,\infty ]:\left|\mathcal{l}(t)-\kappa (t)\right|\le \mathrm{B}\phi (t),\forall t\in \wp \right\}.$$(15)

Following the procedure in Jung (2010), it can be verified that $({\rm X},\mathrm{d})$ is a generalized complete metric space.

We now describe the same operator $\mathfrak{I}:{\rm X}\to {\rm X}$ as given by (7)(7) $$\begin{array}{l}(\mathfrak{I}\mathcal{l})(t)={\alpha }_0+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{F}_k\left(\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)\right]d\mu +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\gamma }{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}\\ +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}},\forall \mathcal{l}\in {\rm X}.\end{array}$$(7) . We will show that $\mathfrak{I}$ is a strictly contractive operator on ${\rm X}.$

For any $\mathcal{l},\kappa \in {\rm X},$ let ${\mathrm{B}}_{\mathcal{l}\kappa }$ be an arbitrary constant such that ${\mathrm{B}}_{\mathcal{l}\kappa }\in [0,\infty )$ with

$\mathrm{d}(\mathcal{l},\kappa )\le {\mathrm{B}}_{\mathcal{l}\kappa }$ for any $\mathcal{l},\kappa \in {\rm X},$

that is, by virtue of (15)(15) $$\mathrm{d}(\mathcal{l},\kappa )=\mathrm{\text{inf}}\left\{\mathrm{B}\in [0,\infty ]:\left|\mathcal{l}(t)-\kappa (t)\right|\le \mathrm{B}\phi (t),\forall t\in \wp \right\}.$$(15) , it follows that (16) $$\left|\mathcal{l}(t)-\kappa (t)\right|\le {\mathrm{B}}_{\mathcal{l}\kappa }\phi (t),\forall t\in \wp .$$(16)

Thereby, according to the operator (7)(7) $$\begin{array}{l}(\mathfrak{I}\mathcal{l})(t)={\alpha }_0+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{F}_k\left(\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)\right]d\mu +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\gamma }{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}\\ +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}},\forall \mathcal{l}\in {\rm X}.\end{array}$$(7) , in the light of the conditions (C1)(C1) $$F_k\in C\left(\wp \times \mathrm{R}\mathrm{\times }\text{R,R}\right),G_k,H_k\in C\left(\wp \times \wp \times \mathrm{R}\mathrm{\times }\text{R,R}\right),0\le {\beta }_k(t)\le t,{\beta }_k\in C[\wp ,\wp ],k=1,2,\dots N;$$(C1) , (C2)(C2) $$\left|F_k\left(t,{\sigma }_1,{\sigma }_1\left({\beta }_k(t)\right)\right)-F_k\left(t,{\sigma }_2,{\sigma }_2\left({\beta }_k(t)\right)\right)\right|\le L_{F_k}\left|{\sigma }_1-{\sigma }_2\right|,\left|G_k\left(t,\gamma ,{\sigma }_1,{\sigma }_1\left({\beta }_k(\gamma )\right)\right)-G_k\left(t,\gamma ,{\sigma }_2,{\sigma }_2\left({\beta }_k(\gamma )\right)\right)\right|\le L_{G_k}\left|{\sigma }_1-{\sigma }_2\right|,\left|H_k\left(t,\gamma ,{\sigma }_1,{\sigma }_2\left({\beta }_k(\gamma )\right)\right)-H_k\left(t,\gamma ,{\sigma }_2,{\sigma }_2\left({\beta }_k(\gamma )\right)\right)\right|\le L_{H_k}\left|{\sigma }_1-{\sigma }_2\right| \mathit{\text{with}} k=1,2,\dots ,N\text{for all}\gamma ,t\in \mathrm{\wp }\mathit{\text{and}} {\sigma }_1,{\sigma }_2\in R.$$(C2) , (12)(12) $$\underset{0}{\overset{t}{\int }}\phi (\mu )d\mu \le N_{\ast }\phi (t),\forall t\in \wp ,$$(12) and (16)(16) $$\left|\mathcal{l}(t)-\kappa (t)\right|\le {\mathrm{B}}_{\mathcal{l}\kappa }\phi (t),\forall t\in \wp .$$(16) , we acquire that $$\begin{array}{l}\left|(\mathfrak{I}\mathcal{l})(t)-(\mathfrak{I}\kappa )(t)\right|\le \underset{0}{\overset{t}{\int }}\sum _{k=1}^N\left(L_{F_k}\right)\left|\mathcal{l}(\mu )-\kappa (\mu )\right|d\mu +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\gamma }{\int }}\left[\sum _{k=1}^N\left(L_{G_k}\right)\left|\mathcal{l}(\mu )-\kappa (\mu )\right|\right]\mathit{\text{dμdγ}}\\ +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N\left(L_{H_k}\right)\left|\mathcal{l}(\mu )-\kappa (\mu )\right|\right]\mathit{\text{dμdγ}}\\ \le {\mathrm{B}}_{\mathcal{l}\kappa }\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N\left(L_{F_k}\right)\phi (\mu )\right]d\mu +{\mathrm{B}}_{\mathcal{l}\kappa }\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\gamma }{\int }}\left[\sum _{k=1}^N\left(L_{G_k}\right)\phi (\mu )\right]\mathit{\text{dμdγ}}\\ +{\mathrm{B}}_{\mathcal{l}\kappa }\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N\left(L_{H_k}\right)\phi (\mu )\right]\mathit{\text{dμdγ}}\\ \le \sum _{k=1}^N\left(L_{F_k}\right){\mathrm{B}}_{\mathcal{l}\kappa }{N}_{\ast }\phi (t)+\sum _{k=1}^N\left(L_{G_k}\right){\mathrm{B}}_{\mathcal{l}\kappa }{N}_{\ast }^2\phi (t)+\sum _{k=1}^N\left(L_{H_k}\right){\mathrm{B}}_{\mathcal{l}\kappa }{N}_{\ast }^2\phi (t)\\ =\sum _{k=1}^N\left(L_{F_k}{N}_{\ast }+L_{G_k}{N}_{\ast }^2+L_{H_k}{N}_{\ast }^2\right){\mathrm{B}}_{\mathcal{l}\kappa }\phi (t),\forall \in \wp .\end{array}$$

Hence, we have (17) $$d(\mathfrak{I}\mathcal{l},\mathfrak{I}\kappa )\le \left[\sum _{k=1}^N\left(L_{F_k}{N}_{\ast }+L_{G_k}{N}_{\ast }^2+L_{H_k}{N}_{\ast }^2\right)\right]d(\mathcal{l},\kappa )$$(17) for all $\mathcal{l},\kappa \in {\rm X},$ where $0<{\sum }_{k=1}^N(L_{F_k}{N}_{\ast }+L_{G_k}{N}_{\ast }^2+L_{H_k}{N}_{\ast }^2)<1.$

Thereby, we deduce that $\mathfrak{I}$ is a contractive operator.

Then, according to the operator (7)(7) $$\begin{array}{l}(\mathfrak{I}\mathcal{l})(t)={\alpha }_0+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{F}_k\left(\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)\right]d\mu +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\gamma }{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}\\ +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}},\forall \mathcal{l}\in {\rm X}.\end{array}$$(7) , it follows that for a random ${\kappa }_0\in {\rm X},$ there exists a nonzero constant ${\rm B}$ with $0<{\rm B}<\infty$ such that $$\begin{array}{l}\left|(\mathfrak{I}{\kappa }_0)(t)-{\kappa }_0(t)\right|=|{\alpha }_0+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{F}_k(\mu ,{\kappa }_0(\mu ),{\kappa }_0({\beta }_k(\mu )))\right]d\mu \\ +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\gamma }{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\mu ,{\kappa }_0(\mu ),{\kappa }_0({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}\\ +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,{\kappa }_0(\mu ),{\kappa }_0({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}-{\kappa }_0(t)|,\end{array}$$ which implies that (18) $$\left|(\mathfrak{I}{\kappa }_0)(t)-{\kappa }_0(t)\right|\le {\rm B}\phi (t),{\chi }_0\in \left(\wp ,\mathrm{R}\right)$$(18)

Next, since $F_k(\mu ,{\kappa }_0(\mu ),{\kappa }_0({\beta }_k(\mu ))),G_k(t,\mu ,{\kappa }_0(\mu ),{\kappa }_0({\beta }_k(\mu ))),H_k(t,\mu ,{\kappa }_0(\mu ),{\kappa }_0({\beta }_k(\mu ))),k=1,2,\dots ,N,$ and ${\kappa }_0(t)$ are bounded in their respective domains and ${\mathrm{\text{min}}}_{t\in \wp }\phi (t)>0,$ then (15)(15) $$\mathrm{d}(\mathcal{l},\kappa )=\mathrm{\text{inf}}\left\{\mathrm{B}\in [0,\infty ]:\left|\mathcal{l}(t)-\kappa (t)\right|\le \mathrm{B}\phi (t),\forall t\in \wp \right\}.$$(15) and (18)(18) $$\left|(\mathfrak{I}{\kappa }_0)(t)-{\kappa }_0(t)\right|\le {\rm B}\phi (t),{\chi }_0\in \left(\wp ,\mathrm{R}\right)$$(18) imply that $$\mathrm{d}(\mathfrak{I}{\kappa }_0,{\kappa }_0)<\infty ,t\in \wp .$$

Hence, taking into account Theorem 2.1, there exists a function ${\chi }_0\in (\wp ,\mathrm{R})$ such that ${\mathfrak{I}}^m{\kappa }_0\to {\chi }_0$ in the space $({\rm X},\mathrm{d})$ and $\mathfrak{I}{\chi }_0={\chi }_0.$ According to this outcome, ${\chi }_0$ satisfies Eq. (13)(13) $$\begin{array}{l}{\chi }_0(t)={\alpha }_0+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{F}_k(\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu )))\right]d\mu \\ +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\gamma }{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}\\ +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}\end{array}$$(13) for all $t\in \wp .$

Since $\kappa$ and ${\kappa }_0$ are bounded on the set $\wp$ for all $\kappa \in {\rm X}$ and ${\mathrm{\text{min}}}_{t\in \wp }\phi (t)>0,$ then there exists a constant $0<{\mathrm{B}}_{\kappa }<\infty$ such that (19) $$\left|{\kappa }_0(t)-\kappa (t)\right|\le {\mathrm{B}}_{\kappa }\phi (t),\forall t\in \wp .$$(19)

We also have $$\mathrm{d}({\kappa }_0,\kappa )<\infty ,\forall \kappa \in {\rm X}.$$

Hence, we get $$\left\{\kappa \in {\rm X}:\mathrm{d}({\kappa }_0,\kappa )<\infty \right\}={\rm X}.$$

Thereby, according to Theorem 2.1, we conclude that ${\chi }_0,$ which is given in Eq. (13)(13) $$\begin{array}{l}{\chi }_0(t)={\alpha }_0+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{F}_k(\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu )))\right]d\mu \\ +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\gamma }{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}\\ +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}\end{array}$$(13) , is the unique continuous function.

Also, the inequality (11)(11) $$\begin{array}{c}|{\chi }^{\prime }(t)-\sum _{k=1}^N{F}_k(t,\chi (t),\chi ({\beta }_k(t)))-\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\chi (\gamma ),\chi ({\beta }_k(\gamma ))\right)\right]d\gamma \\ -\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\chi (\gamma ),\chi ({\beta }_k(\gamma ))\right)\right]d\gamma |\le \phi (t),\forall t\in \wp ,\end{array}$$(11) can be extended as (20) $$\begin{array}{l}-\phi (t)\le {\chi }^{\prime }(t)-\sum _{k=1}^N{F}_k(t,\chi (t),\chi ({\beta }_k(t)))-\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\chi (\gamma ),\chi ({\beta }_k(\gamma ))\right)\right]d\gamma \\ -\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\chi (\gamma ),\chi ({\beta }_k(\gamma ))\right)\right]d\gamma \le \phi (t),\forall t\in \wp .\end{array}$$(20)

Integrating the terms of the inequality (20)(20) $$\begin{array}{l}-\phi (t)\le {\chi }^{\prime }(t)-\sum _{k=1}^N{F}_k(t,\chi (t),\chi ({\beta }_k(t)))-\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\chi (\gamma ),\chi ({\beta }_k(\gamma ))\right)\right]d\gamma \\ -\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\chi (\gamma ),\chi ({\beta }_k(\gamma ))\right)\right]d\gamma \le \phi (t),\forall t\in \wp .\end{array}$$(20) from $0$ to $t,$ we arrive that (21) $$\begin{array}{c}|\chi (t)-{\alpha }_0-\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{F}_k(\mu ,\chi (\mu ),\chi ({\beta }_k(\mu )))\right]d\mu -\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\gamma }{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\mu ,\chi (\mu ),\chi ({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}\\ -\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,\chi (\mu ),\chi ({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}|\le \underset{0}{\overset{t}{\int }}\phi (\mu )d\mu ,\forall t\in \wp .\end{array}$$(21)

Hence, using the inequality (12)(12) $$\underset{0}{\overset{t}{\int }}\phi (\mu )d\mu \le N_{\ast }\phi (t),\forall t\in \wp ,$$(12) and the operator (7)(7) $$\begin{array}{l}(\mathfrak{I}\mathcal{l})(t)={\alpha }_0+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{F}_k\left(\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)\right]d\mu +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\gamma }{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}\\ +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}},\forall \mathcal{l}\in {\rm X}.\end{array}$$(7) , we obtain (22) $$\left|\chi (t)-(\mathfrak{I}\chi )(t)\right|\le \underset{0}{\overset{t}{\int }}\phi (\mu )d\mu \le N_{\ast }\phi (t),\forall t\in \wp .$$(22)

Thus (23) $$d\left(\chi ,\mathfrak{I}\chi \right)\le N_{\ast }.$$(23)

Thereby, using Theorem 2.1 and (23)(23) $$d\left(\chi ,\mathfrak{I}\chi \right)\le N_{\ast }.$$(23) , we get $$d\left(\chi ,{\chi }_0\right)\le \frac{d\left(\chi ,\mathfrak{I}\chi \right)}{1-\underset{k=1}{\overset{N}{\Sigma }}\left(L_{F_k}{N}_{\ast }+L_{G_k}{N}_{\ast }^2+L_{H_k}{N}_{\ast }^2\right)}$$ $$\le \frac{N_{\ast }}{1-\underset{k=1}{\overset{N}{\Sigma }}\left(L_{F_k}{N}_{\ast }+L_{G_k}{N}_{\ast }^2+L_{H_k}{N}_{\ast }^2\right)}.$$

Then, we can obtain $$\left|\chi (t)-{\chi }_0(t)\right|\le \frac{N_{\ast }}{1-\underset{k=1}{\overset{N}{\Sigma }}\left(L_{F_k}{N}_{\ast }+L_{G_k}{N}_{\ast }^2+L_{H_k}{N}_{\ast }^2\right)}\phi (t),\forall t\in \wp .$$

This inequality ends the proof of Theorem 4.1.

We now give our third result in connection with the UHR stability of the nonlinear Volterra-Freedholm IDE (2)(2) $${\vartheta }^{\prime }(t)=\sum _{k=1}^N{F}_k\left(t,\vartheta (t),\vartheta \left({\beta }_k(t)\right)\right)+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma +\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma$$(2) with multiple variable delays in the real line, $R=(-\infty ,\infty ).$

Theorem 4.2.

Let $T\ge 0,T\in \mathrm{R},$ and $\wp =(-\infty ,0]$ or $\wp =[0,\infty )$or $\wp \text{=R=}\mathrm{(}‐\infty ,\infty ),L_{F_k},L_{G_k},L_{H_k}$ and $N_{\ast }$ be positive constants with $$0<\sum _{k=1}^N\left(L_{F_k}{N}_{\ast }+L_{G_k}{N}_{\ast }^2+L_{H_k}{N}_{\ast }^2\right)<1.$$

We also assume that $(C1)$ and $(C2)$ of Theorem 3.1 hold true for all $t\in \wp ,\wp =(-\infty ,0]$ or $t\in \wp ,\wp =[0,\infty )$or $t\in \wp ,\wp =\mathrm{R}$ and ${\sigma }_1,{\sigma }_2\in R.$

If a function $\chi \in C^1(\wp ,\mathrm{R})$ satisfies the inequality (24) $$\begin{array}{l}|{\chi }^{\prime }(t)-\sum _{k=1}^N{F}_k(t,\chi (t),\chi ({\beta }_k(t)))-\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\chi (\gamma ),\chi ({\beta }_k(\gamma ))\right)\right]d\gamma \\ -\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\chi (\gamma ),\chi ({\beta }_k(\gamma ))\right)\right]d\gamma |\le \phi (t),\forall t\in \wp ,\end{array}$$(24)

where $\phi \in C(\wp ,(0,\infty ))$ with $$\underset{0}{\overset{t}{\int }}\phi (\mu )d\mu \le N_{\ast }\phi (t),\forall t\in \wp ,$$

then there is a unique function ${\chi }_0\in C(\wp ,\mathrm{R})$ such that this function satisfies $${\chi }_0(t)={\alpha }_0+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{F}_k(\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu )))\right]d\mu$$ $$+\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\gamma }{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}$$ (25) $$+\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}$$(25) and (26) $$|\chi (t)-{\chi }_0(t)|\le \frac{N_{\ast }}{1-\underset{k=1}{\overset{N}{\Sigma }}\left(L_{F_k}{N}_{\ast }+L_{G_k}{N}_{\ast }^2+L_{H_k}{N}_{\ast }^2\right)}\phi (t),\forall t\in \wp .$$(26)

Proof.

Let $\wp =\mathrm{R}.$ We will first show that ${\chi }_0$ is continuous. Let $m\in \mathrm{N}$ and define ${\wp }_m=[-m,m].$ According to Theorem 4.1, there exists a unique ${\chi }_m\in C({\wp }_m,\mathrm{R})$ such that $${\chi }_m(t)={\alpha }_0+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{F}_k(\mu ,{\chi }_m(\mu ),{\chi }_m({\beta }_k(\mu )))\right]d\mu$$ $$+\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\gamma }{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\mu ,{\chi }_m(\mu ),{\chi }_m({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}$$ (27) $$+\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,{\chi }_m(\mu ),{\chi }_m({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}$$(27) and (28) $$\left|\chi (t)-{\chi }_m(t)\right|\le \frac{N_{\ast }}{1-\sum _{k=1}^N\left(L_{F_k}{N}_{\ast }+L_{G_k}{N}_{\ast }^2+L_{H_k}{N}_{\ast }^2\right)}\phi (t),\forall t\in {\wp }_m.$$(28)

If $t\in {\wp }_m,$ then the uniqueness of ${\chi }_m$ implies that (29) $${\chi }_m(t)={\chi }_{m+1}(t)={\chi }_{m+2}(t)={\chi }_{m+3}(t)=\dots .$$(29)

For any $t\in \mathrm{R},$ we define $m(t)\in \mathrm{N}$ as $$m(t)=\mathrm{\text{min}}\{m\in \mathrm{N}:t\in {\wp }_m\}.$$

Next, we also define a function ${\chi }_0:\mathrm{R}\to \mathrm{R}$ such that (30) $${\chi }_0(t)={\chi }_{m(t)}(t).$$(30)

Then, we claim that ${\chi }_0$ is continuous.

Let $t_1\in \mathrm{R},t_1$ is arbitrary, and $m_1=m(t_1).$ Hence, $t_1$ is in the interior points of ${\wp }_{m+1}$ and there exists an $\epsilon >0$ such that ${\chi }_0(t)={\chi }_{m+1}(t)$ with $\left|t-t_1\right|<\epsilon .$ Since ${\chi }_{m+1}$ is a continuous function at $t_1,$ then ${\chi }_0$ is also continuous at $t_1,$ $\forall t_1\in R.$

We now have to show that ${\chi }_0$ satisfies Eqs. (25)(25) $$+\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}$$(25) and (15)(15) $$\mathrm{d}(\mathcal{l},\kappa )=\mathrm{\text{inf}}\left\{\mathrm{B}\in [0,\infty ]:\left|\mathcal{l}(t)-\kappa (t)\right|\le \mathrm{B}\phi (t),\forall t\in \wp \right\}.$$(15) for all $t\in \text{R}.$ Let $m(t)\in \mathrm{N}$ for some random $t\in R.$ Then, according to (27)(27) $$+\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,{\chi }_m(\mu ),{\chi }_m({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}$$(27) and (30)(30) $${\chi }_0(t)={\chi }_{m(t)}(t).$$(30) , we have $t\in {\wp }_{m(t)}$ and $${\chi }_0(t)={\chi }_{m(t)}(t)={\alpha }_0+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{F}_k\left(\mu ,{\chi }_{m(t)}(\mu ),{\chi }_{m(t)}({\beta }_k(\mu ))\right)\right]d\mu$$ $$+\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\gamma }{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\mu ,{\chi }_{m(t)}(\mu ),{\chi }_{m(t)}({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}$$ (31) $$+\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,{\chi }_{m(t)}(\mu ),{\chi }_{m(t)}({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}.$$(31)

Thereby, we get $${\chi }_0(t)={\alpha }_0+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{F}_k\left(\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu ))\right)\right]d\mu$$ $$+\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\gamma }{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}$$ (32) $$+\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}.$$(32)

Since $m(\mu )\le m(t)$ for all $t\in {\wp }_{m(t)},$ then (32)(32) $$+\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}.$$(32) is true, and in the light of (29)(29) $${\chi }_m(t)={\chi }_{m+1}(t)={\chi }_{m+2}(t)={\chi }_{m+3}(t)=\dots .$$(29) and (30)(30) $${\chi }_0(t)={\chi }_{m(t)}(t).$$(30) we get $${\chi }_{m(t)}(\mu )={\chi }_{m(\mu )}(\mu )={\chi }_0(\mu ).$$

Since $t\in {\wp }_{m(t)},$ then, according to (28)(28) $$\left|\chi (t)-{\chi }_m(t)\right|\le \frac{N_{\ast }}{1-\sum _{k=1}^N\left(L_{F_k}{N}_{\ast }+L_{G_k}{N}_{\ast }^2+L_{H_k}{N}_{\ast }^2\right)}\phi (t),\forall t\in {\wp }_m.$$(28) and (30)(30) $${\chi }_0(t)={\chi }_{m(t)}(t).$$(30) , it follows that $$\left|\chi (t)-{\chi }_0(t)\right|=\left|\chi (t)-{\chi }_m(t)\right|\le \frac{N_{\ast }}{1-\sum _{k=1}^N\left(L_{F_k}{N}_{\ast }+L_{G_k}{N}_{\ast }^2+L_{H_k}{N}_{\ast }^2\right)}\phi (t)$$ for all $t\in \mathrm{R}.$

Next, we now have to prove that ${\chi }_0$ is unique.

Consider ${\chi }_0^{\ast }\in C(\text{R,R})$ which satisfies Eqs. (32)(32) $$+\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}.$$(32) and (15)(15) $$\mathrm{d}(\mathcal{l},\kappa )=\mathrm{\text{inf}}\left\{\mathrm{B}\in [0,\infty ]:\left|\mathcal{l}(t)-\kappa (t)\right|\le \mathrm{B}\phi (t),\forall t\in \wp \right\}.$$(15) with ${\chi }_0^{\ast }$ in place of ${\chi }_0$ for all $t\in R.$ Let $T\in \mathrm{R}$ be a random real number. Since both ${\chi }_0$ and ${\chi }_0^{\ast }$ satisfy Eqs. (32)(32) $$+\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}.$$(32) and (20)(20) $$\begin{array}{l}-\phi (t)\le {\chi }^{\prime }(t)-\sum _{k=1}^N{F}_k(t,\chi (t),\chi ({\beta }_k(t)))-\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\chi (\gamma ),\chi ({\beta }_k(\gamma ))\right)\right]d\gamma \\ -\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\chi (\gamma ),\chi ({\beta }_k(\gamma ))\right)\right]d\gamma \le \phi (t),\forall t\in \wp .\end{array}$$(20) for all $t\in {\wp }_{m(t)},$ then the uniqueness of ${\chi }_{m(t)}={\chi }_0$ on ${\wp }_{m(t)}$ implies that $${\chi }_0(t)={\chi }_0|{\gamma }_{m(t)}={\chi }_0^{\ast }|{\wp }_{m(t)}={\chi }_0^{\ast }(t).$$

Thereby, ${\chi }_0$ is unique.

For the cases $\wp =(-\infty ,0]$ or $\wp =\text{[0,}\infty \mathrm{)}\text{,}$ the proofs can be provided. We ignore the details of them.

5. UH stability

Within Sect. 5, we will focus on the UH stability in connection with the nonlinear Volterra-Freedholm IDE (2)(2) $${\vartheta }^{\prime }(t)=\sum _{k=1}^N{F}_k\left(t,\vartheta (t),\vartheta \left({\beta }_k(t)\right)\right)+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma +\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma$$(2) with multiple variable delays.

Theorem 5.1.

Let $T>0,$ $\wp =[0,T]$ and $L_{F_k},L_{G_k}$ and $L_{H_k}$ be positive constants such that $(C1),(C2)$ of Theorem 2.1 and the following condition $$0<\sum _{k=1}^N\left(L_{F_k}T+2^{-1}{L}_{G_k}{T}^2+L_{H_k}{T}^2\right)<1$$ are satisfied.

If for any $\epsilon \ge 0$ there exists a function $\chi \in C^1(\wp ,\mathrm{R})$ which satisfies the inequality (33) $$\begin{array}{l}|{\chi }^{\prime }(t)-\sum _{k=1}^N{F}_k(t,\chi (t),\chi ({\beta }_k(t)))-\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\chi (\gamma ),\chi ({\beta }_k(\gamma ))\right)\right]d\gamma \\ -\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\chi (\gamma ),\chi ({\beta }_k(\gamma ))\right)\right]d\gamma |\le \epsilon ,\forall t\in \wp ,\end{array}$$(33) then there exists a unique function ${\chi }_0\in C(\wp ,\mathrm{R})$ which satisfies Eq. (13)(13) $$\begin{array}{l}{\chi }_0(t)={\alpha }_0+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{F}_k(\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu )))\right]d\mu \\ +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\gamma }{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}\\ +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}\end{array}$$(13) and (34) $$\left|\chi (t)-{\chi }_0(t)\right|\le \frac{T}{1-\sum _{k=1}^N\left(L_{F_k}T+2^{-1}{L}_{G_k}{T}^2+L_{H_k}{T}^2\right)}\epsilon ,\forall t\in \wp .$$(34)

Proof.

Let $\wp =[0,T]$ and ${\rm X}=C(\wp ,\mathrm{R}).$ We define the following metric: (35) $$\mathrm{d}(\mathcal{l},\kappa )=\mathrm{\text{inf}}\left\{\mathrm{B}\in [0,\infty ):\left|\mathcal{l}(t)-\kappa (t)\right|\le \mathrm{B},\forall t\in \wp \right\},$$(35) where $\mathcal{l},\kappa \in {\rm X}.$ Then $({\rm X},\mathrm{d})$ is a generalized complete metric space.

We now take into consideration the operator $\mathfrak{I}:{\rm X}\to {\rm X},$ which is described by (7)(7) $$\begin{array}{l}(\mathfrak{I}\mathcal{l})(t)={\alpha }_0+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{F}_k\left(\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)\right]d\mu +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\gamma }{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}\\ +\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,\mathcal{l}(\mu ),\mathcal{l}({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}},\forall \mathcal{l}\in {\rm X}.\end{array}$$(7) . Then, depending upon the conditions of Theorem 5.1, it is clear that $\mathfrak{I}$ is a strictly contractive operator on ${\rm X}.$ Let ${\kappa }_0$ be an arbitrary element of ${\rm X}.$ Then, there exists a constant ${\rm B}$ with $0<{\rm B}<\infty$ such that (36) $$\left|(\mathfrak{I}{\kappa }_0)(t)-{\kappa }_0(t)\right|\le {\rm B},\forall t\in \wp .$$(36)

By virtue of (35)(35) $$\mathrm{d}(\mathcal{l},\kappa )=\mathrm{\text{inf}}\left\{\mathrm{B}\in [0,\infty ):\left|\mathcal{l}(t)-\kappa (t)\right|\le \mathrm{B},\forall t\in \wp \right\},$$(35) and (36)(36) $$\left|(\mathfrak{I}{\kappa }_0)(t)-{\kappa }_0(t)\right|\le {\rm B},\forall t\in \wp .$$(36) , we derive that $$\mathrm{d}(\mathfrak{I}{\kappa }_0,{\kappa }_0)<\infty .$$

Thereby, according to Theorem 2.1, there exists a ${\chi }_0\in C(\wp ,R)$ such that ${\mathfrak{I}}^m{\kappa }_0\to {\chi }_0$ as $m\to \infty$ in $({\rm X},\mathrm{d})$ and $\mathfrak{I}{\chi }_0\to {\chi }_0.$ Hence, ${\chi }_0$ satisfies Eq. (32)(32) $$+\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}.$$(32) for all $t\in \wp .$ Next, as in the proof of Theorem 3.1, it can be shown that $\left\{\kappa \in {\rm X}:\mathrm{d}({\kappa }_0,\kappa )<\infty \right\}={\rm X}.$ Due to Theorem 2.1, ${\chi }_0$ is the unique continuous solution of Eq. (32)(32) $$+\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,{\chi }_0(\mu ),{\chi }_0({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}.$$(32) .

If we expand the modulus of the inequality (33)(33) $$\begin{array}{l}|{\chi }^{\prime }(t)-\sum _{k=1}^N{F}_k(t,\chi (t),\chi ({\beta }_k(t)))-\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\chi (\gamma ),\chi ({\beta }_k(\gamma ))\right)\right]d\gamma \\ -\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\chi (\gamma ),\chi ({\beta }_k(\gamma ))\right)\right]d\gamma |\le \epsilon ,\forall t\in \wp ,\end{array}$$(33) and then integrating the expanded inequality term by term from $0$ to $t,$ we obtain (37) $$\begin{array}{l}|\chi (t)-{\alpha }_0-\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{F}_k(\mu ,\chi (\mu ),\chi ({\beta }_k(\mu )))\right]d\mu -\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\gamma }{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\mu ,\chi (\mu ),\chi ({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}\\ -\underset{0}{\overset{t}{\int }}\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\mu ,\chi (\mu ),\chi ({\beta }_k(\mu ))\right)\right]\mathit{\text{dμdγ}}|\le \epsilon T,\forall t\in \wp .\end{array}$$(37)

Thereby, we get (38) $$\mathrm{d}\left(\chi (t),(\mathfrak{I}\chi )(t)\right)\le \epsilon T,\forall t\in \wp .$$(38)

According to Theorem 2.1, the conditions $(C1),(C2)$ of Theorem 5.1 and (38)(38) $$\mathrm{d}\left(\chi (t),(\mathfrak{I}\chi )(t)\right)\le \epsilon T,\forall t\in \wp .$$(38) , we obtain $$\left|\chi (t)-{\chi }_0(t)\right|\le \frac{T}{1-\sum _{k=1}^N\left(L_{F_k}T+2^{-1}{L}_{G_k}{T}^2+L_{H_k}{T}^2\right)}\epsilon ,\forall t\in \wp .$$

This inequality ends the proof of Theorem 5.1.

6. Applications

Within this section, we have two examples as the numerical applications for the UH stability of two particular nonlinear Volterra-Freedholm IDEs involving a variable delay.

Example 6.1

We will now consider the following the Volterra-Fredholm IDE involving a variable delay: $${\vartheta }^{\prime }(t)=\frac{1}{100}\mathrm{\text{cos}}\hspace{0.17em}\left(t\right)+\frac{1}{100}\vartheta \left(t\right)+\frac{1}{50}\mathrm{\text{sin}}\hspace{0.17em}\left(\vartheta \left({\beta }_1(t)\right)\right)$$ $$+\frac{1}{100}\underset{0}{\overset{t}{\int }}\left[\vartheta (\gamma )+t\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}\left(\vartheta \left({\beta }_1(\gamma )\right)\right)\right]d\gamma$$ (39) $$+\frac{1}{200}\underset{0}{\overset{\frac{\pi }{2}}{\int }}\left[\vartheta (\gamma )+\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}(\gamma )\vartheta \left({\beta }_1(\gamma )\right)\right]d\gamma ,$$(39) and the inequality $$|{\vartheta }^{\prime }(t)-\frac{1}{100}\mathrm{\text{cos}}\hspace{0.17em}\left(t\right)-\frac{1}{100}\vartheta \left(t\right)-\frac{1}{50}\mathrm{\text{sin}}\hspace{0.17em}\left(\vartheta \left({\beta }_1(t)\right)\right)-\frac{1}{100}\underset{0}{\overset{t}{\int }}\left[\vartheta (\gamma )+t\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}\left(\vartheta \left({\beta }_1(\gamma )\right)\right)\right]d\gamma -\frac{1}{200}\underset{0}{\overset{\frac{\pi }{2}}{\int }}\left[\vartheta (\gamma )+\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}(\gamma )\vartheta \left({\beta }_1(\gamma )\right)\right]d\gamma |\le \epsilon ,$$ where $${\beta }_1(t)=\frac{1}{2}t,{\beta }_1(t)\le t,\wp =\left[0,\frac{\pi }{2}\right],t\in \wp ,T=\frac{\pi }{2},\epsilon >0.$$

Hence, it is seen that the Volterra-Fredholm IDE (39)(39) $$+\frac{1}{200}\underset{0}{\overset{\frac{\pi }{2}}{\int }}\left[\vartheta (\gamma )+\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}(\gamma )\vartheta \left({\beta }_1(\gamma )\right)\right]d\gamma ,$$(39) has the form of the Volterra-Fredholm IDE (2)(2) $${\vartheta }^{\prime }(t)=\sum _{k=1}^N{F}_k\left(t,\vartheta (t),\vartheta \left({\beta }_k(t)\right)\right)+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma +\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma$$(2) with the arguments as the follows: $$F_1\left(t,\vartheta (t),\vartheta \left({\beta }_1(t)\right)\right)=\frac{1}{100}\mathrm{\text{cos}}\hspace{0.17em}\left(t\right)+\frac{1}{100}\vartheta \left(t\right)+\frac{1}{50}\mathrm{\text{sin}}\hspace{0.17em}\left(\vartheta \left({\beta }_1(t)\right)\right),$$ $$G_1\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_1(\gamma )\right)\right)=\frac{1}{100}\left[\vartheta (\gamma )+t\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}\left(\vartheta \left({\beta }_1(\gamma )\right)\right)\right],$$ $$H_1\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_1(\gamma )\right)\right)=\frac{1}{200}\left[\vartheta (\gamma )+\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}(\gamma )\vartheta \left({\beta }_1(\gamma )\right)\right].$$

We will now approve that the items of Theorem 3.1 and Theorem 4.1 are satisfied.

It can be seen that the functions $F_1,G_1$ and $H_1$ are continuous.

According to the above data, we obtain the following outcomes, respectively: $$\left|F_1\left(t,{\sigma }_1,{\sigma }_1\left({\beta }_1(t)\right)\right)-F_1\left(t,{\sigma }_2,{\sigma }_2\left({\beta }_1(t)\right)\right)\right|$$ $$=\frac{1}{100}\left|{\sigma }_1-{\sigma }_2+2\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}\left({\sigma }_1\left({\beta }_1(t)\right)\right)-2\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}\left({\sigma }_2\left({\beta }_1(t)\right)\right)\right|$$ $$\le \frac{1}{100}\left|{\sigma }_1-{\sigma }_2\right|+\frac{1}{50}\left|\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}\left({\sigma }_1\left({\beta }_1(t)\right)\right)-\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}\left({\sigma }_2\left({\beta }_1(t)\right)\right)\right|$$ $$\le \frac{1}{100}\left|{\sigma }_1-{\sigma }_2\right|+\frac{1}{100}\left|{\sigma }_1\left({\beta }_1(t)\right)-{\sigma }_2\left({\beta }_1(t)\right)\right|$$ $$\le \frac{1}{50}\left|{\sigma }_1-{\sigma }_2\right|,L_{F_1}=\frac{1}{50};$$ $$\left|G_1\left(t,\gamma ,{\sigma }_1,{\sigma }_1\left({\beta }_1(\gamma )\right)\right)-G_1\left(t,\gamma ,{\sigma }_2,{\sigma }_2\left({\beta }_1(\gamma )\right)\right)\right|$$ $$=\frac{1}{100}\left|{\sigma }_1-{\sigma }_2+t\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}\left({\sigma }_1\left({\beta }_1(t)\right)\right)-t\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}\left({\sigma }_2\left({\beta }_1(t)\right)\right)\right|$$ $$\le \frac{1}{100}\left|{\sigma }_1-{\sigma }_2\right|+\frac{t}{100}\left|\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}\left({\sigma }_1\left({\beta }_1(t)\right)\right)-\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}\left({\sigma }_2\left({\beta }_1(t)\right)\right)\right|$$ $$\le \frac{1}{100}\left|{\sigma }_1-{\sigma }_2\right|+\frac{\pi }{200}\left|\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}\left({\sigma }_1\left({\beta }_1(t)\right)\right)-\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}\left({\sigma }_2\left({\beta }_1(t)\right)\right)\right|$$ $$\le \frac{1}{100}\left|{\sigma }_1-{\sigma }_2\right|+\frac{\pi }{400}\left|{\sigma }_1\left({\beta }_1(t)\right)-{\sigma }_2\left({\beta }_1(t)\right)\right|$$ $$=\frac{(\pi +4)}{400}\left|{\sigma }_1-{\sigma }_2\right|,L_{G_1}=\frac{(\pi +4)}{400};$$ $$\left|H_1\left(t,\gamma ,{\sigma }_1,{\sigma }_2\left({\beta }_1(\gamma )\right)\right)-H_1\left(t,\gamma ,{\sigma }_2,{\sigma }_2\left({\beta }_1(\gamma )\right)\right)\right|$$ $$=\frac{1}{200}\left|{\sigma }_1-{\sigma }_2+\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}\left(\gamma \right){\sigma }_1\left({\beta }_1(\gamma )\right)-\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}\left(\gamma \right){\sigma }_2\left({\beta }_1(\gamma )\right)\right|$$ $$\le \frac{1}{200}\left|{\sigma }_1-{\sigma }_2\right|+\frac{1}{200}\left|{\sigma }_1\left({\beta }_1(t)\right)-{\sigma }_2\left({\beta }_1(t)\right)\right|$$ $$\le \frac{1}{100}\left|{\sigma }_1-{\sigma }_2\right|,L_{H_1}=\frac{1}{100};$$ $$0<L_{F_1}T+2^{-1}{L}_{G_1}{T}^2+L_{H_1}{T}^2=\frac{1}{50}+\frac{3}{400}+\frac{1}{100}=\frac{3}{80}<1.$$

Thereby, all the items of Theorem 3.1 and Theorem 4.1 are fulfilled. Then the nonlinear Volterra-Freedholm IDE (39)(39) $$+\frac{1}{200}\underset{0}{\overset{\frac{\pi }{2}}{\int }}\left[\vartheta (\gamma )+\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}(\gamma )\vartheta \left({\beta }_1(\gamma )\right)\right]d\gamma ,$$(39) involving a variable delay has a unique solution and it also admits the UH stability.

Example 6.2

We will now consider the following the Volterra-Fredholm IDE including a variable delay: $${\vartheta }^{\prime }(t)=\frac{1}{100}+\frac{1}{20}t+\frac{1}{50}{t}^2+\frac{1}{100}\vartheta (t)-\frac{1}{100}\vartheta \left({\beta }_1(t)\right)$$ $$+\frac{1}{200}\underset{0}{\overset{t}{\int }}\left[t+\vartheta (\gamma )+\gamma {\vartheta }^2\left({\beta }_1(\gamma )\right)\right]d\gamma$$ (40) $$+\frac{1}{400}\underset{0}{\overset{1}{\int }}\left[\gamma {\vartheta }^3(\gamma )+\vartheta \left({\beta }_1(\gamma )\right)\right]d\gamma ,$$(40) and the inequality: $$\begin{array}{l}|{\vartheta }^{\prime }(t)-\frac{1}{100}-\frac{1}{20}t-\frac{1}{50}{t}^2-\frac{1}{100}\vartheta (t)+\frac{1}{100}\vartheta \left({\beta }_1(t)\right)-\frac{1}{200}\underset{0}{\overset{t}{\int }}\left[t+\vartheta (\gamma )+\gamma {\vartheta }^2\left({\beta }_1(\gamma )\right)\right]d\gamma \\ -\frac{1}{400}\underset{0}{\overset{1}{\int }}\left[\gamma {\vartheta }^3(\gamma )+\vartheta \left({\beta }_1(\gamma )\right)\right]d\gamma |\le \epsilon ,\end{array}$$ where $${\beta }_1(t)=\frac{1}{2}t,{\beta }_1(t)\le t,\wp =[0,1],t\in \wp ,T=1,\epsilon >0.$$

It is obvious that the Volterra-Fredholm IDE (40)(40) $$+\frac{1}{400}\underset{0}{\overset{1}{\int }}\left[\gamma {\vartheta }^3(\gamma )+\vartheta \left({\beta }_1(\gamma )\right)\right]d\gamma ,$$(40) has the form of the Volterra-Fredholm IDE (2)(2) $${\vartheta }^{\prime }(t)=\sum _{k=1}^N{F}_k\left(t,\vartheta (t),\vartheta \left({\beta }_k(t)\right)\right)+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^N{G}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma +\underset{0}{\overset{T}{\int }}\left[\sum _{k=1}^N{H}_k\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_k(\gamma )\right)\right)\right]d\gamma$$(2) with the arguments as follows: $$F_1\left(t,\vartheta (t),{\vartheta }_1\left({\beta }_1(t)\right)\right)=\frac{1}{100}+\frac{1}{20}t+\frac{1}{50}{t}^2+\frac{1}{100}\vartheta (t)-\frac{1}{100}\vartheta \left({\beta }_1(t)\right),$$ $$G_1\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_1(\gamma )\right)\right)=\frac{1}{200}t+\frac{1}{200}\vartheta (\gamma )+\frac{1}{200}\gamma {\vartheta }^2\left({\beta }_1(\gamma )\right),$$ $$H_1\left(t,\gamma ,\vartheta (\gamma ),\vartheta \left({\beta }_1(\gamma )\right)\right)=\frac{1}{400}\gamma {\vartheta }^3(\gamma )+\frac{1}{400}\vartheta \left({\beta }_1(\gamma )\right).$$

We will now approve that the conditions of Theorem 3.1 and Theorem 4.1 are satisfied.

It is clear that the functions $F_1,G_1$ and $H_1$ are continuous.

According to the data above, we carry out the following relations, respectively: $$\begin{array}{l}\left|F_1\left(t,{\sigma }_1,{\sigma }_1\left({\beta }_1(t)\right)\right)-F_1\left(t,{\sigma }_2,{\sigma }_2\left({\beta }_1(t)\right)\right)\right|=\frac{1}{100}\left|{\sigma }_1-{\sigma }_1\left({\beta }_1(t)\right)-{\sigma }_2+{\sigma }_2\left({\beta }_1(t)\right)\right|\\ \le \frac{1}{100}\left|{\sigma }_1-{\sigma }_2\right|+\frac{1}{100}\left|{\sigma }_1\left({\beta }_1(t)\right)-{\sigma }_2\left({\beta }_1(t)\right)\right|\\ \le \frac{1}{50}\left|{\sigma }_1-{\sigma }_2\right|,L_{F_1}=\frac{1}{50};\end{array}$$ $$\begin{array}{c}\left|G_1\left(t,\gamma ,{\sigma }_1,{\sigma }_1\left({\beta }_1(\gamma )\right)\right)\text{\hspace{0.05em}\hspace{0.05em}}-G_1\left(t,\gamma ,{\sigma }_2,{\sigma }_2\left({\beta }_1(\gamma )\right)\right)\right|\\ =\frac{1}{200}\left|{\sigma }_1-{\sigma }_2+\gamma {\sigma }_1^2\left({\beta }_1(\gamma )\right)-\gamma {\sigma }_2^2\left({\beta }_1(\gamma )\right)\right|\\ \le \frac{1}{200}\left|{\sigma }_1-{\sigma }_2\right|+\frac{1}{200}\left|\gamma {\sigma }_1^2\left({\beta }_1(\gamma )\right)-\gamma {\sigma }_2^2\left({\beta }_1(\gamma )\right)\right|\\ \le \frac{3}{200}\left|{\sigma }_1-{\sigma }_2\right|,L_{G_1}=\frac{3}{200};\end{array}$$ $$\left|H_1\left(t,\gamma ,{\sigma }_1,{\sigma }_2\left({\beta }_1(\gamma )\right)\right)-H_1\left(t,\gamma ,{\sigma }_2,{\sigma }_2\left({\beta }_1(\gamma )\right)\right)\right|$$ $$=\frac{1}{400}\left|\gamma {\sigma }_1^3+{\sigma }_1\left({\beta }_1(\gamma )\right)-\gamma {\sigma }_2^3+{\sigma }_2\left({\beta }_1(\gamma )\right)\right|$$ $$\le \frac{1}{400}\left|\gamma {\sigma }_1^3-\gamma {\sigma }_2^3\right|+\frac{1}{400}\left|{\sigma }_1\left({\beta }_1(\gamma )\right)-{\sigma }_2\left({\beta }_1(\gamma )\right)\right|$$ $$\le \frac{1}{100}\left|{\sigma }_1-{\sigma }_2\right|,L_{H_1}=\frac{1}{100};$$ $$0<L_{F_1}T+2^{-1}{L}_{G_1}{T}^2+L_{H_1}{T}^2=\frac{\pi }{100}+\frac{3{\pi }^2}{1600}+\frac{{\pi }^2}{400}=\frac{(16+7\pi )\pi }{1600}<1.$$

Thereby, all the items of Theorem 3.1 and Theorem 4.1 are fulfilled. Then the nonlinear Volterra-Freedholm IDE (40)(40) $$+\frac{1}{400}\underset{0}{\overset{1}{\int }}\left[\gamma {\vartheta }^3(\gamma )+\vartheta \left({\beta }_1(\gamma )\right)\right]d\gamma ,$$(40) with a variable delay has a unique solution and it also admits the UH stability.

Remark 6.1

We should note that Example 6.1 and Example 6.2 can be updated accordingly for the UHR stability of the Volterra-Freedholm IDEs (39)(39) $$+\frac{1}{200}\underset{0}{\overset{\frac{\pi }{2}}{\int }}\left[\vartheta (\gamma )+\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}(\gamma )\vartheta \left({\beta }_1(\gamma )\right)\right]d\gamma ,$$(39) and (40)(40) $$+\frac{1}{400}\underset{0}{\overset{1}{\int }}\left[\gamma {\vartheta }^3(\gamma )+\vartheta \left({\beta }_1(\gamma )\right)\right]d\gamma ,$$(40) , respectively. We ignore the details of the discussions.

7. Conclusion

According to the data from the earlier literature, no work is available in connection with the existence, the unique of solutions and the Ulam type stabilities of Volterra-Freedholm IDEs involving multiple variable time delays. In this article, a new mathematical model of that kind of multiple delay Volterra-Freedholm IDEs is considered and the mentioned qualitative concepts are studied. Hence, this article includes the first and new results with connection to the existence, the unique of solutions, the UHR and the UH stabilities of nonlinear Volterra-Freedholm IDEs involving multiple variable time delays. Thereby, this work allows some new qualitative contributions to the qualitative theory of Volterra-Freedholm IDEs. In this article, we provide four new theorems in the cases of finite or infinite intervals by employing the Banach FPT. The new theorems of the present study involve sufficient conditions on the mentioned concepts for the Volterra-Freedholm IDE taken under study. The outcomes of this article allow essential advances and contributions from the Volterra-Freedholm IDEs including a variable delay to the more general and nonlinear Volterra-Freedholm IDEs including multiple variable time delays. The techniques utilized in the proofs employ the fixed point method depending on the use of the Banach CMP, the generalized complete metric and mathematical inequalities. Thereby, we obtain new contributions to the qualitative theory of the delay Volterra-Freedholm IDEs. As proper future suggestions, the HU stability, HUR stability, etc. of nonlinear Caputo fractional Volterra-Freedholm IDEs including multiple delays, Caputo–Hadamard fractional Volterra-Freedholm IDEs including multiple delays, Riemann–Liouville fractional Volterra-Freedholm IDEs including multiple delays, etc., can be considered as new open problems.

Abbreviations
Banach CMP	=	Banach contraction mapping principle
Banach’s FPT	=	Banach’s fixed point theorem
DDE	=	Delay differential equation
FEs	=	Functional equation
FDE	=	Functional differential equation
FP method	=	Fixed point method
FrDE	=	Fractional differential equation
FrODE	=	Fractional ordinary differential equation
IDE	=	Integro-differential equation
IE	=	Integral equation
ODE	=	Ordinary differential equation
PDE	=	Partial differential equation
UH stability	=	Ulam–Hyers stability
UHR stability	=	Ulam–Hyers–Rassias stability
Volterra-Fredholm IDE	=	Volterra-Fredholm integro differential equation

Acknowledgments

The authors would like to thank the anonymous referees and the handling editor for many useful comments and suggestions, leading to a substantial improvement in the presentation of this article.

Disclosure statement

The authors declare no conflict of interest.