1 Introduction

We call that a Fredholm or Volterra IE is stable in the HU sense if for every function satisfying that the Fredholm IE or the Volterra IE approximately such that there is a solution of the equation that is close to it. In the relevant literature, the first result with regard to the Ulam stability of functional equations (FEs) has been originated and constructed from a significant question belonging to S.M. Ulam, (see, [28]). Following this inaugural research work of Ulam [28], up to now, several kinds of Ulam type stabilities were defined and many important and attractive results on them were formulated and proved with regard to functional differential equations (FDEs), Volterra and Fredholm IEs, Volterra and Fredholm integro-differential equations (Volterra and Fredholm IDEs), etc., in particular, see, Abbas et al. [1], Akkouchi [2], Castro and Simões [6,7,8], Chauhan et al. [9], Deep et al. [10], Egri [11], Găvruţa [12], Graef et al. [15], Janfada and Sadeghi [17], Jung [18,19,20], Khan et al. [21], Öğrekçi et al. [22], Petruşel et al. [23], Radu [24], Rassias [25], Shah et al. [26], Tunç and Biçer [27], Shah and Zada [29] and the citied references therein.

According to the data bases and the documents of the current literature, several results related to the content of this study are reported shortly in the following lines.

Egri [11] dealt with the Ulam stabilities of an iterative FDE.

Tunç and Tunç [32] dealt with the Ulam stabilities of an iterative functional IDE. In [32], the authors focused on two new results with regard to the HU and the HUR stabilities of the considered equation via the Banach FPT.

Specifically, Jung [18] interested in the Volterra IE in the below line:

$$ y(x) = \int_{c}^{x} f(\tau, y(\tau))d\tau. $$

The author utilized the fixed point method (FPM) for proving the uniqueness of solution and the HUR stability of this Volterra IE of the second kind. Indeed, the work of Jung [18] has been a milestone in the literature for the coming works with regard to the Ulam stabilities of IEs.

The paper of Jung [19] was devoted to an IE, where the unknown function depends on two independent variables. The author showed that the considered IE is firmly connected to the wave equation and confirmed the generalized HU stability of the equation. The main tool of [19] is the FPM.

The target of the book of Jung [20] has been to provide an overview of the stability theory of the FEs, and Jung [20] considered and discussed the stability of the several kind of the FEs.

In 2009, Castro and Ramos [6] interested in the nonlinear Volterra IE in the below:

$$ y(x) = \int_a^x {f(x,\tau ,y(\tau ))d\tau } . $$

In [6], the HU and the HUR stabilities of this IE were obtained in finite and infinite intervals cases.

In 2011, Akkouchi [2] took into consideration the Volterra IE depicted by

$$ f(x) = h(x) + \lambda \int_a^x {G(x,y,f(y))dy} . $$

Using the FPM, the author established new results with regard to the HU and the HUR stabilities of this Volterra IE in the Banach spaces.

In 2010, Castro and Ramos [7] proposed a study on both the HU and the HUR stabilities for the delay Volterra IEs via the FPM:

$$ y(x) = \int_a^x {f(x,\tau ,y(\tau ),y(\alpha (\tau )))d\tau } . $$

In 2013, Castro and Guerra [5] studied the nonlinear Volterra IE including a variable delay:

$$ y(x) = g(x) + \Psi \left( {\int_a^x {k(x,t,y(t),y(\alpha (t)))d\tau } } \right). $$

The authors of [5] addressed the problem of HUR stability for this Volterra IE and obtained conditions using the Banach FPT via suitable complete metric space and the Bielecki metric. In [5], some illustrative examples are also given.

Janfada and Sadeghi [17] and Öğrekçi et al. [22] investigated the HU and the HUR stabilities of the Volterra IE depicted by

$$ x(t) = g(t,x(t)) + \int_0^t {K(t,s,x(s))ds} $$

using the FPM.

In 2015, Abbas and Benchohra [1] focused on the HUR stability of the quadratic IE in a complex Hilbert space:

$$ u(t) = f(t,u(t))\int_0^t {k(t,s)g(s,u(s))ds} . $$

In [1], the existence result was also proved by using the Schauder FPT.

As for the key reference paper of this study, in 2018, Castro and Simões [8] considered the nonlinear Fredholm and Volterra IEs in the following lines, respectively:

$$ y(x) = f\left( {x,y(x),y(\alpha (x)),\int_a^b {k(x,\tau ,y(\tau ),y(\beta (\tau )))d\tau } } \right) $$
(1)

and

$$ y(x) = f\left( {x,y(x),y(\alpha (x)),\int_a^x {k(x,\tau ,y(\tau ),y(\beta (\tau )))d\tau } } \right). $$
(2)

In Castro and Simões [8], some very interesting and attractive results with regard to the HUR, the \(\sigma\)-semi HU and the HU stabilities of the Fredholm IE (1) and the Volterra IE (2) were built on the bounded and the unbounded intervals cases, which are integration domains. The constructed conditions of [8] were based on the use of the FPM and the Bielecki metric.

In addition, some very interesting stability problems in sense of Lyapunov for various IDEs and some other mathematical models have attracted extensively researchers’ attentions and significant results have been obtained, for example, see, Bohner and Tunç [3], Bohner et al. [4], Graef and Tunç [13, 14], Hammami and Hnia [16], Janfada and Sadeghi [17], Tunç and Tunç [30, 31], Tunç et al. [33, 34], in sense of Ulam, see, Graef et al. [15], Shah and Zada [29] and reference therein. We also refer the readers to the following very recent works on existence and uniqueness of solutions, Ulam stabilities of certain integral equations and integro-differential equations (see Tunç and Tunç ([35, 36]), Tunç et al. ([37,38,39]) and some others Khan et al. ([40, 41])

In the present paper, inspired by the study of Castro and Simões [8] and that are presented above, we will first consider the following nonlinear iterative Fredholm and Volterra IEs with variable delays, respectively:

$$ y(t) = F\left( {\ldots} \right), $$
(3)

with

$$ F\left( {\ldots} \right) = F\left( {t,y^{[1]} (t),y^{[1]} (\tau (t)),\ldots,y^{[m]} (\tau (t)),\int_a^b {K\left( {t,s,y^{[1]} (s),\ldots,y^{[m]} (\alpha (s))} \right)ds} } \right) $$

and

$$ y(x) = G\left( {\ldots} \right), $$
(4)

with

$$ G\left( {\ldots} \right) = G\left( {x,y^{[1]} (x),y^{[1]} (\tau (x)),\ldots,y^{[m]} (\tau (x)),\int_a^x {K\left( {x,s,y^{[1]} (s),\ldots,y^{[m]} (\alpha (s))} \right)ds} } \right), $$

where \(t, \, s, \, x \in [a, \, b],\) \(F, \, G \in C\left( {[a, \, b] \times {\mathbb{C}}^{2m + 1} ,{\mathbb{C}}} \right),\) \(K \in \left( {[a, \, b] \times [a, \, b] \times {\mathbb{C}}^{2m} ,{\mathbb{C}}} \right),\) \(a, \, b \in {\mathbb{R}},\) \(a < b,\) and \(y^{[m]}\) symbolizes mth iterate of the function \(y\) and denotes

$$ y^{[1]} (t) = y(t)\ , \ldots y^{[m]} (t) = \underbrace {y\left( {y( \ldots y(y(t))\ )} \right)}_m. $$
(5)

Remark 1

In the rest of the paper, for the sake of the brevity without mentioning, instead of the representations above, we will use the following depictions and some similar ones:

$$ y(t) = F\left( {t,y^{[1]} (t),\ldots,\int_a^b {K(t,s,y^{[1]} (s),\ldots)ds} } \right) $$

and

$$ y(x) = G\left( {x,y^{[1]} (x),\ldots,\int_a^x {K(x,s,y^{[1]} (s),\ldots)ds} } \right). $$

To our knowledge, as it is seen from the literature review above, there are limited literature concentrate on the uniqueness of solutions and the Ulam type stabilities of integral equations. According to the data above, it is also easy to see that the study of the concepts mentioned deserve exploring for delay iterative Fredholm and Volterra IEs. In addition, to our knowledge, there is not any present study with regard to the same concepts for iterative Fredholm and Volterra IEs. The outcomes of this study with regard to the Ulam type stabilities are first time obtained for the iterative Fredholm and Volterra IEs including multiple variable time-delays. Therefore, our results improve or complement the results in the papers of Abbas and Benchohra [1], Akkouchi [2], Castro and Guerra [5], Castro and Ramos [6, 7], Castro and Simões [8], Janfada and Sadeghi [17], Jung [18], Öğrekçi et al. [22], etc., in some senses.

The study is systematized as in the following lines. In Sect. 2, the background definitions of the study and a reference theorem of the fixed point theory are given. In Sect. 3, the leading results of the study as the uniqueness of solutions, the HUR stability and the \(\sigma\)-semi HU stability of the iterative Fredholm IE (3) are presented. In Sect. 4, two new results of the study with regard to the uniqueness of solutions, the HUR stability and the \(\sigma\)-semi HU stability of the iterative Volterra IE (4) are established. In Sect. 5, the final main result of the study with regard to the uniqueness of solutions and the HU stability of the iterative Volterra IE (4) is provided in the infinite intervals cases. Section 6 provides an example as the numerical aptitude of the new main results of the study. Section 7 outlines the contributions of the study and allows some related statements. At the end, in Sect. 8, the conclusion of the study is designed.

2 Basic information

We now begin with some definitions with regard to the primary concepts of this study. Hence, the following definitions and the theorem are beneficial to confirm the Ulam type stability results of the study.

Definition 1.

Let \(t \in [a, \, b].\) If for each function \(y\) satisfying

$$ \left| {y(t) - F(\ldots)} \right| \le \sigma (t), $$
(6)

where \(F\left( {\ldots} \right)\) is given together with (3) and \(\sigma \ge 0\) is a function, there is a solution \(y_0\) of the iterative Fredholm IE (3) and a \(C \in {\mathbb{R}},\) \(C > 0,\) which does not dependent \(y_0\) and \(y,\) such that

$$ \left| {y(t) - y_0 (t)} \right| \le C\sigma (t), $$

for all \(t\) in the given interval, then we say that the iterative Fredholm IE (3) has HUR stability.

Definition 2.

Let \(x \in [a, \, b].\) If for each function \(y\) satisfying.

$$ \left| {y(x) - G(\ldots)} \right| \le \sigma (x), $$
(7)

where \(G\left( {\ldots} \right)\) is given together with (4) and \(\sigma \ge 0\) is a function, there is a solution \(y_0\) of the iterative Volterra IE (4) and a \(C \in {\mathbb{R}},\) \(C > 0,\) which does not dependent of \(y_0\) and \(y,\) such that

$$ \left| {y(x) - y_0 (x)} \right| \le C\sigma (x), $$

for all \(x\) in the given interval, then we say that the iterative Volterra IE (4) has HUR stability.

Definition 3.

Let \(t \in [a, \, b].\) If for each function \(y\) satisfying

$$ \left| {y(t) - F(\ldots)} \right| \le \theta , $$
(8)

where \(F\left( {\ldots} \right)\) is given together with (3) and \(\theta \ge 0,\) \(\theta \in {\mathbb{R}},\) there is a solution \(y_0\) of the iterative Fredholm IE (3) and a \(C \in {\mathbb{R}},\) \(C > 0,\) which does not dependent of \(y_0\) and \(y,\) so that

$$ \left| {y(t) - y_0 (t)} \right| \le C\theta , $$

for all \(t\) in the given interval, then we state that the iterative Fredholm IE (3) has HU stability.

Definition 4.

Let \(t \in [a, \, b]\) and \(\sigma \ge 0\) be a function defined on the given interval. If for each function \(y\) satisfying

$$ \left| {y(t) - F(\ldots)} \right| \le \theta , $$
(9)

where \(F(\ldots)\) is given together with (3) and \(\theta \ge 0,\) \(\theta \in {\mathbb{R}},\) there is a solution \(y_0\) of the iterative Fredholm IE (3) and a \(C \in {\mathbb{R}},\) \(C > 0,\) which does not dependent \(y_0\) and \(y,\) such that

$$ \left| {y(t) - y_0 (t)} \right| \le C\sigma (t), $$

for all \(t\) in the given interval, then we state that the iterative Fredholm IE (3) has the \(\sigma\)‐semi‐HU stability.

Remark 1.

Definitions 3 and 4 can be arranged easily for the iterative Volterra IE (4). We leave out to give that definitions.

It is known that one of an effective basic technique to study the Ulam stabilities is the FPM, which depends upon suitable fixed point results and metrics. Hence, we now recall a definition of this metric on a set \(X,\) \(X \ne \emptyset .\)

Definition 5.

Let \(X\) be a nonempty set. A function \(d\) from \(X \times X\) to \([0, + \infty ]\) is called a generalized metric on the set \(X \Leftrightarrow\) the function \(d\) satisfies:

  1. (M1)

    \(d(x,y) = 0 \Leftrightarrow x = y;\)

  2. (M2)

    \(d(x,y) = d(y,x),\;\forall \;x, \, y \in X;\)

  3. (M3)

    \(d(x,z) \le d(x,y) + d(y,z),\;\forall \;x, \, y, \, z \in X.\)

From the information above, it is clear that the range of this metric is allowed to include the infinity, i.e., “\(\infty\)”. However, this case cannot be allowed for the usual metric. This is an important difference among the generalized and the usual metrics.

Let \(C\left( {[.],[.]} \right)\) and \(C\left( {[.]} \right)\) denote \(C\left( {[a,b],[a,b]} \right)\) and \(C\left( {[a,b]} \right),\) throughout the study, respectively.

A key fixed point background result of this work is presented in Theorem 1.

Theorem 1

(Castro and Simões [7]). Let \((X,d)\) be a generalized compete metric space and \(T\) be a strictly contractive operator from \(X\) to \(X\) with a Lipschitz constant \(L < 1.\) If there exists a \(k \in {\mathbb{Z}},\) \(k \ge 0,\) such that \(d(T^k x,T^k x) < \infty\) for some \(x \in X,\) then the following propositions can be held:

  • \(1^0 )\) The sequence \((T^n x)_{n \in {\mathbb{N}}}\) converges to a fixed point \(x*\) of \(T;\)

  • \(2^0 )\) \(x*\) is the unique fixed point of \(T\) in set \(X* = \left\{ {y \in X:d(T^k x,y) < \infty } \right\};\)

  • \(3^0 )\) if \(y \in X*,\) then

    $$ d\left( {y,x*} \right) \le (1 - L)^{ - 1} d(Ty,y). $$
    (10)

Consider now the space continuous functions, which is represented by \(C\left( {[.]} \right)\) on \([a,b],\) equipped with a generalization of the Bielecki metric

$$ d(u,v) = {\mathop {\sup }\limits_{t \in [a,b]}} \frac{{\left| {u(t) - v(t)} \right|}}{\sigma (t)}, $$
(11)

where \(\sigma\) is a nondecreasing continuous function from the interval \([a,b]\) to the interval \((0,\infty ).\) We remember that \(\left( {C([.]),d} \right)\) is a complete metric space (see, Castro and Simões [8]).

3 The uniqueness and the HUR stability of the iterative Fredholm IE

Section 3 builds new conditions for the uniqueness of solutions, the HUR stability and the \(\sigma\)-semi HU stability of the iterative Fredholm IE (3) on a finite interval case. Hence, the first new result with regard to the uniqueness of solutions and the HUR stability of the iterative Fredholm IE (3) is presented in coming lines.

Theorem 2.

We assume that the conditions in the following lines hold:

\((As1)\) \(\tau ,\alpha \in C\left( {[.],[.]} \right)\) are variable time delay functions such that \(\tau (t) \le t\) and \(\alpha (t) \le t,\) \(\forall t \in [a,b]\) and \(\sigma \in C\left( {[.],(0,\infty )} \right).\) Furthermore, for \(\forall t \in [a,b],\) we have that there is \(\eta \in {\mathbb{R}}\) such that

$$ \int_a^b {\sigma (s)ds \le } \eta \sigma (t). $$

\((As2)\) The continuous functions \(F:[a,b] \times {\mathbb{C}}^{2m + 1} \to {\mathbb{C}}\) and \(K:[a,b] \times [a,b] \times {\mathbb{C}}^{2m} \to {\mathbb{C}}\) allow that the following the Lipschitz conditions are held:

$$ \begin{aligned} &\left| F(t,u_1 (t), \ldots ,u_m (\alpha (t)),g(t)) - F(t,v_1 (t), \ldots ,v_m (\alpha (t)),h(t)) \right| \\ & \quad \le M_F \sum_{k = 1}^m {\left[ {\left| {u_k (t) - v_k (t)} \right| + \left| {u_k (\alpha (t)) - v_k (\alpha (t))} \right|} \right]} + M_F \left| {g(t) - h(t)} \right| \\ \end{aligned} $$

and

$$ \begin{aligned} &\left| K(t,s,u_1 (s), \ldots ,u_m (\alpha (s))) - K(t,s,v_1 (s), \ldots ,v_m (\alpha (s))) \right| \\ & \quad \le M_K \sum_{k = 1}^m {\left[ {\left| {u_k (s) - v_k (s)} \right| + \left| {u_k (\alpha (s)) - v_k (\alpha (s))} \right|} \right]} , \\ \end{aligned} $$

respectively, where \(M_F\) and \(M_K\) are positive constants.

Let \(y \in C([.]),\)

$$ \left| {y(t) - F\left( {t,y^{[1]} (t), \ldots ,\int_a^b {K(t,s,y^{[1]} (s), \ldots )ds} } \right)} \right| \le \sigma (t),\quad t \in [a, \, b], $$
(12)

and

$$ 2M_F \left[ {1 + (1 + \eta M_K )\Delta } \right] < 1 $$

with

$$ \Delta = \left[ {1 + \sum_{k = 1}^2 {L^{k - 1} } + \sum_{k = 1}^3 {L^{k - 1} } + \cdots + \sum_{k = 1}^m {L^{k - 1} } } \right]. $$

Then, there is a unique function \(y_0 \in C([.])\) such that

$$ y_0 (t) = F\left( {t,y_0^{[1]} (t),\ldots,\int_a^b {K(t,s,y_0^{[1]} (s),\ldots)ds} } \right) $$

and

$$ \left| {y(t) - y_0 (t)} \right| \le \frac{1}{{1 - 2M_F \left[ {1 + (1 + \eta M_K )\Delta } \right]}}\sigma (t), $$
(13)
$$ \forall \;t \in [a, \, b]. $$

Hence, we infer from the conditions above that the iterative Fredholm IE (3) has the HUR stability.

Proof.

We now keep in view the operator

$$ T: = C\left( {[.]} \right) \to C\left( {[.]} \right) $$

depicted by

$$ \begin{aligned} (Tu)(t) = F\left( {t,u^{[1]} (t), \ldots ,\int_a^b {K\left( {t,s,u^{[1]} (s), \ldots } \right)ds} } \right), \quad \forall t \in [a,b]\;{\text{and}}\;\forall u \in C\left( {[.]} \right). \\ \end{aligned} $$

According to the data of Theorem 2, we will now confirm that \(T\) is strictly contractive with regard to the metric (11). Letting \(u, \, v \in C\left( {[.]} \right),\) we get

$$ \begin{aligned} & d(Tu,Tv) = {\mathop {\sup }\limits_{t \in [a,b]}} \frac{{\left| {(Tu)(t) - (Tv)(t)} \right|}}{\sigma (t)} \\ & \quad = {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {F\left( {t,u^{[1]} (t),\ldots,\int_a^b {K(t,s,u^{[1]} (s),\ldots)ds} } \right) - F\left( {t,v^{[1]} (t),\ldots,\int_a^b {K(t,s,v^{[1]} (s),\ldots)ds} } \right)} \right| \\ & \quad \le M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[1]} (t) - v^{[1]} (t)} \right| + M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[1]} (\tau (t)) - v^{[1]} (\tau (t))} \right| \\ & \quad \quad + M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[2]} (t) - v^{[2]} (t)} \right| + M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[2]} (\tau (t)) - v^{[2]} (\tau (t))} \right| \\ & \quad \quad + \cdots + M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[m]} (t) - v^{[m]} (t)} \right| + M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[m]} (\tau (t)) - v^{[m]} (\tau (t))} \right| \\ & \quad \quad + M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {\int_a^b {K\left( {t,s,u^{[1]} (s),\ldots,u^{[m]} (\alpha (s))} \right)ds - \int_a^b {K\left( {t,s,v^{[1]} (s),\ldots,v^{[m]} (\alpha (s))} \right)ds} } } \right| \\ & \quad \le M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[1]} (t) - v^{[1]} (t)} \right| + M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[1]} (\tau (t)) - v^{[1]} (\tau (t))} \right| \\ & \quad \quad + M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[2]} (t) - v^{[2]} (t)} \right| + M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[2]} (\tau (t)) - v^{[2]} (\tau (t))} \right| \\ & \quad \quad + \cdots + M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[m]} (t) - v^{[m]} (t)} \right| + M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[m]} (\tau (t)) - v^{[m]} (\tau (t))} \right| \\ & \quad \quad + M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\int_a^b {\left| {K\left( {t,s,u^{[1]} (s),\ldots,u^{[m]} (\alpha (s))} \right) - K\left( {t,s,v^{[1]} (s),\ldots,v^{[m]} (\alpha (s))} \right)} \right|} ds \\ & \quad \le M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[1]} (t) - v^{[1]} (t)} \right| + M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[1]} (\tau (t)) - v^{[1]} (\tau (t))} \right| \\ & \quad \quad + M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[2]} (t) - v^{[2]} (t)} \right| + M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[2]} (\tau (t)) - v^{[2]} (\tau (t))} \right| \\ & \quad \quad + \cdots + M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[m]} (t) - v^{[m]} (t)} \right| + M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[m]} (\tau (t)) - v^{[m]} (\tau (t))} \right| \\ & \quad \quad + M_F M_K {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\int_a^b {\left| {u^{[1]} (s) - v^{[1]} (s)} \right|} ds + M_F M_K {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\int_a^b {\left| {u^{[1]} (\alpha (s)) - v^{[1]} (\alpha (s))} \right|} ds \\ & \quad \quad + M_F M_K {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\int_a^b {\left| {u^{[2]} (s) - v^{[2]} (s)} \right|} ds + M_F M_K {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\int_a^b {\left| {u^{[2]} (\alpha (s)) - v^{[2]} (\alpha (s))} \right|} ds \\ & \quad \quad + \cdots \\ & \quad \quad + M_F M_K {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\int_a^b {\left| {u^{[m]} (s) - v^{[m]} (s)} \right|} ds + M_F M_K {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\int_a^b {\left| {u^{[m]} (\alpha (s)) - v^{[m]} (\alpha (s))} \right|} ds. \\ \end{aligned} $$

For the next steps, letting \(u, \, v \in C\left( {[.]} \right)\) and benefiting from the mathematical induction, we derive the following outcomes, respectively:

$$ \begin{aligned} & M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[1]} (t) - v^{[1]} (t)} \right| = M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{{\left| {u(t) - v(t)} \right|}}{\sigma (t)} = M_F d(u,v); \\ & M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[1]} (\tau (t)) - v^{[1]} (\tau (t))} \right| = M_F d(u,v); \\ & M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[2]} (s) - v^{[2]} (s)} \right| = M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{{\left| {u(u(t)) - u(v(t)) + u(v(t)) - v(v(t))} \right|}}{\sigma (t)} \\ & \quad \le LM_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{{\left| {u(t) - v(t)} \right|}}{\sigma (t)} + M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{{\left| {u(v(t)) - v(v(t))} \right|}}{\sigma (t)} \\ & \quad = \left( {1 + L} \right)M_F d(u,v); \\ & M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[2]} (\tau (t)) - v^{[2]} (\tau (t))} \right| \le \left( {1 + L} \right)M_F d(u,v); \\ & M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[3]} (t) - v^{[3]} (t)} \right| \\ & \quad = M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{{\left| {u(u^{[2]} )(t) - u(v^{[2]} )(t) + u(v^{[2]} )(t) - v(v^{[2]} )(t)} \right|}}{\sigma (t)} \\ & \quad \le M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{{\left| {u(u^{[2]} )(t) - u(v^{[2]} )(t)} \right|}}{\sigma (t)} + M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{{\left| {u(v^{[2]} )(t) - v(v^{[2]} )(t)} \right|}}{\sigma (t)} \\ & \quad \le LM_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{{\left| {u^{[2]} (t) - v^{[2]} (t)} \right|}}{\sigma (t)} + M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{{\left| {u(v^{[2]} )(t) - v(v^{[2]} )(t)} \right|}}{\sigma (t)} \\ & \quad \le L^2 M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{{\left| {u(t) - v(t)} \right|}}{\sigma (t)} + LM_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{{\left| {u(v(t)) - v(v(t))} \right|}}{\sigma (t)} \\ & \quad \quad + M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{{\left| {u(v^{[2]} )(t) - v(v^{[2]} )(t)} \right|}}{\sigma (t)} \\ & \quad \le \left( {1 + L + L^2 } \right)M_F d(u,v); \\ & M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[3]} (\tau (t)) - v^{[3]} (\tau (t))} \right| \le \left( {1 + L + L^2 } \right)M_F d(u,v); \\ & \quad \quad \quad \quad \quad \quad \quad \quad \ldots \\ & M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[m]} (t) - v^{[m]} (t)} \right| \le \left( {1 + L + L^2 + \cdots + L^{m - 1} } \right)M_F d(u,v); \\ & M_F {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\left| {u^{[m]} (\alpha (t)) - v^{[m]} (\alpha (t))} \right| \le \left( {1 + L + L^2 + \cdots + L^{m - 1} } \right)M_F d(u,v); \\ & M_F M_K {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\int_a^b {\left| {u^{[1]} (s) - v^{[1]} (s)} \right|} ds = M_F M_K {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\int_a^b {\sigma (s)\frac{{\left| {u(s) - v(s)} \right|}}{\sigma (s)}} ds \\ & \quad \le M_F M_K {\mathop {\sup }\limits_{s \in [a,b]}} \frac{{\left| {u(s) - v(s)} \right|}}{\sigma (s)}{\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\int_a^b {\sigma (s)} ds \\ & \quad \le M_F M_K d(u,v){\mathop {\sup }\limits_{t \in [a,b]}} \frac{\eta \sigma (t)}{{\sigma (t)}} = \eta M_F M_K d(u,v); \\ & M_F M_K {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\int_a^b {\left| {u^{[1]} (\alpha (s)) - v^{[1]} (\alpha (s))} \right|} ds \le \eta M_F M_K d(u,v); \\ & M_F M_K {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\int_a^b {\left| {u^{[2]} (s) - v^{[2]} (s)} \right|} ds \\ & \quad = M_F M_K {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\int_a^b {\sigma (s)\frac{{\left| {u(u(s)) - u(v(s)) + u(v(s)) - v(v(s))} \right|}}{\sigma (s)}} ds \\ & \quad \le M_F M_K {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\int_a^b {\sigma (s)\frac{{\left| {u(u(s)) - u(v(s))} \right|}}{\sigma (s)}} ds \\ & \quad \quad + M_F M_K {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\int_a^b {\sigma (s)\frac{{\left| {u(v(s)) - v(v(s))} \right|}}{\sigma (s)}} ds \\ & \quad \le M_F M_K {\mathop {\sup }\limits_{s \in [a,b]}} \frac{{\left| {u(u(s)) - u(v(s))} \right|}}{\sigma (s)}{\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\int_a^b {\sigma (s)} ds \\ & \quad \quad + M_F M_K {\mathop {\sup }\limits_{s \in [a,b]}} \frac{{\left| {u(v(s)) - v(v(s))} \right|}}{\sigma (s)}{\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\int_a^b {\sigma (s)} ds \\ & \quad \le \eta M_F M_K \left( {1 + L} \right)d(u,v); \\ & M_F M_K {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\int_a^b {\left| {u^{[2]} (\alpha (s)) - v^{[2]} (\alpha (s))} \right|} ds \le \eta M_F M_K \left( {1 + L} \right)d(u,v); \\ & M_F M_K {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\int_a^b {\left| {u^{[3]} (s) - v^{[3]} (s)} \right|} ds \le M_F M_K \eta \left( {1 + L + L^2 } \right)d(u,v); \\ & M_F M_K {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\int_a^b {\left| {u^{[3]} (\alpha (s)) - v^{[3]} (\alpha (s))} \right|} ds \le M_F M_K \eta \left( {1 + L + L^2 } \right)d(u,v); \\ & \quad \quad \quad \quad \quad \quad \quad \quad \ldots \\ & M_F M_K {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\int_a^b {\left| {u^{[m]} (s) - v^{[m]} (s)} \right|} ds \le \eta M_F M_K \left( {1 + L + \cdots + L^{m - 1} } \right)d(u,v); \\ & M_F M_K {\mathop {\sup }\limits_{t \in [a,b]}} \frac{1}{\sigma (t)}\int_a^b {\left| {u^{[m]} (\alpha (s)) - v^{[m]} (\alpha (s))} \right|} ds \le \eta M_F M_K \left( {1 + L + \cdots + L^{m - 1} } \right)d(u,v). \\ \end{aligned} $$

Hence, according to the above outcomes, we can derive that

$$ \begin{aligned} & d(Tu,Tv) = {\mathop {\sup }\limits_{t \in [a,b]}} \frac{{\left| {(Tu)(t) - (Tv)(t)} \right|}}{\sigma (t)} \\ & \quad \le 2M_F d(u,v) + 2\left( {1 + L} \right)M_F d(u,v) + 2\left( {1 + L + L^2 } \right)M_F d(u,v) + \cdots + \\ & \quad \quad + 2\left( {1 + L + L^2 + \cdots + L^{m - 1} } \right)M_F d(u,v) + 2\eta M_F M_K d(u,v) \\ & \quad \quad + 2\eta M_F M_K \left( {1 + L} \right)d(u,v) + 2M_F M_K \eta \left( {1 + L + L^2 } \right)d(u,v) + \cdots \\ & \quad \quad + 2\eta M_F M_K \left( {1 + L + \cdots + L^{m - 1} } \right)d(u,v) \\ & \quad = 2M_F d(u,v) + 2M_F (1 + \eta M_K )\left[ {1 + \sum_{k = 1}^2 {L^{k - 1} } + \sum_{k = 1}^3 {L^{k - 1} } + \cdots + \sum_{k = 1}^m {L^{k - 1} } } \right]d(u,v) \\ & \quad = 2M_F \left[ {1 + (1 + \eta M_K )\Delta } \right]d(u,v). \\ \end{aligned} $$

Since

$$ 2M_F \left[ {(1 + \eta M_K )\Delta + 1} \right] < 1, $$

then \(T\) is strictly contractive. In addition, when we utilize the Banca FPT, it follows that

$$ d(y,y_0 ) \le \frac{1}{{1 - 2M_F \left[ {1 + (1 + \eta M_K )\Delta } \right]}}d(Ty,y), $$

which allows that the iterative Fredholm IE (3) admits the HUR.

Indeed, according to (11) and (12), we have

$$ {\mathop {\sup }\limits_{t \in [a,b]}} \frac{{\left| {y(t) - y_0 (t)} \right|}}{\sigma (t)} \le \frac{1}{{1 - 2M_F \left[ {1 + (1 + \eta M_K )\Delta } \right]}}. $$

Hence,

$$ \frac{{\left| {y(t) - y_0 (t)} \right|}}{\sigma (t)} \le \frac{1}{{1 - 2M_F \left[ {(1 + \eta M_K )\Delta + 1} \right]}}, $$

and consequently (13) holds. This outcome ends the proof.

The second new outcome of this study with regard to the uniqueness of solutions and the \(\sigma\)-semi HU stability of the iterative Fredholm IE (3) is given in the below theorem.

Theorem 3.

Let \((As1)\) and \((As2)\) of Theorem 2 hold. In addition, if \(y \in C([.]),\)

$$ \left| {y(t) - F\left( {\ldots} \right)} \right| \le \theta ,\quad t \in [a, \, b], $$
(14)

where \(F\left( {\ldots} \right)\) is given together with (3), and

$$ 2M_F \left[ {1 + (1 + \eta M_K )\Delta } \right] < 1, $$

then there is a unique function \(y_0 \in C([.])\) such that

$$ y_0 (t) = F\left( {t,y_0^{[1]} (t),\ldots,\int_a^b {K(t,s,y_0^{[1]} (s),\ldots)ds} } \right), $$

with

$$ \left| {y(t) - y_0 (t)} \right| \le \frac{\theta }{{1 - 2M_F \left[ {1 + (1 + \eta M_K )\Delta } \right]\sigma (a)}}\sigma (t) $$
(15)

for all \(t \in [a, \, b].\)

This outcome allows that in the light of the conditions above the iterative Fredholm IE (3) has the \(\sigma\)-semi HU stability.

Proof.

We now keep in view the operator

$$ T: = C\left( {[.]} \right) \to C\left( {[.]} \right), $$

which is depicted by

$$ \begin{aligned} & (Tu)(t) = F\left( {t,u^{[1]} (t),\ldots,\int_a^b {K\left( {t,s,u^{[1]} (s),\ldots} \right)ds} } \right), \\ & \quad \forall u \in C([.]),\;\forall \;t \in [a,b]. \\ \end{aligned} $$

As the next steps, according to the conditions of Theorem 3, one can confirm that \(T\) is strictly contractive depending upon the metric (11) since

$$ 2M_F \left[ {1 + (1 + \eta M_K )\Delta } \right] < 1. $$

Hence, once again we can utilize the Banach FPT, which allows the \(\sigma\)-semi HU stability of the iterative Fredholm IE (3) with (15). This outcome is obtained via the definition of the metric (11), using (10) and (14). Thus, we verify the proof of Theorem 3.

4 The uniqueness and the HUR stability of the iterative Volterra IE in the finite interval

Section 4 deals with to build appropriate conditions for the HUR stability and the \(\sigma\)-semi HU stability of the iterative Volterra IE (4) on a finite interval, \([a,b],\) \(a,\) \(b \in {\mathbb{R}} \, {.}\) The third new outcome of this study with regard to the uniqueness of solutions and the HUR stability of iterative Volterra IE (4) is introduced in Theorem 4.

Theorem 4.

Let the conditions in the following lines hold.

\((As3)\) \(\tau ,\alpha \in C\left( {[.],[.]} \right)\) are variable delay functions such that \(\tau (t) \le t,\) \(\alpha (t) \le t,\) \(\forall t \in [a,b]\) and \(\sigma \in C\left( {[.],(0,\infty )} \right)\) is a non-decreasing function. Furthermore, we assume that there is \(\eta \in {\mathbb{R}}\) such that

$$ \int_a^x {\sigma (s)ds \le } \eta \sigma (x),\;\forall \;x \in [a,b]. $$

\((As4)\) \(G:[a,b] \times {\mathbb{C}}^{2m + 1} \to {\mathbb{C}}\) and \(K:[a,b] \times [a,b] \times {\mathbb{C}}^{2m} \to {\mathbb{C}}\) are continuous functions allowing the Lipschitz condition:

$$ \begin{aligned} &\left| G(x,u_1 (x),\ldots,u_m (\alpha (x)),g(x)) - G(x,v_1 (x),\ldots,v_m (\alpha (x)),h(x)) \right| \\ & \quad \le M_G \sum_{k = 1}^m {\left[ {\left| {u_k (x) - v_k (x)} \right| + \left| {u_k (\alpha (x)) - v_k (\alpha (x))} \right|} \right]} + M_G \left| {g(x) - h(x)} \right| \\ \end{aligned} $$

and

$$ \begin{aligned} & \left| {K(x,s,u_1 (s),\ldots,u_m (\alpha (s))) - K(x,s,v_1 (s),\ldots,v_m (\alpha (s)))} \right| \\ & \quad \le M_K \sum_{k = 1}^m {\left[ {\left| {u_k (s) - v_k (s)} \right| + \left| {u_k (\alpha (s)) - v_k (\alpha (s))} \right|} \right]} , \\ \end{aligned} $$

respectively, where \(M_F\) and \(M_K\) are positive constants.

Let \(x \in [a, \, b],\)\(y \in C\left( {[.]} \right),\)

$$ \left| {y(x) - G\left( {\ldots} \right)} \right| \le \sigma (x), $$
(16)

where \(G\left( {\ldots} \right)\) is given together with (4), and

$$ 2M_F \left[ {1 + (1 + \eta M_K )\Delta } \right] < 1. $$

Then, there is a unique function \(y_0 \in C\left( {[.]} \right)\) such that

$$ y_0 (x) = G\left( {x,y_0^{[1]} (x),\ldots),\int_a^x {K(x,s,y_0^{[1]} (s),\ldots)ds} } \right), $$

and in additon

$$ \begin{aligned} & \left| {y(x) - y_0 (x)} \right| \le \frac{1}{{1 - 2M_F \left[ {(1 + \eta M_K )\Delta + 1} \right]}}\sigma (x), \\ & \quad \forall \;x \in [a, \, b]. \\ \end{aligned} $$
(17)

Depending on the conditions of Theorem 4, these outcomes allows that the iterative Volterra IE (4) has the HUR stability.

Proof.

Take into consideration the operator

$$ T: = C\left( {[.]} \right) \to C\left( {[.]} \right), $$

which is depicted by

$$ (Tu)(x) = G\left( {x,u^{[1]} (x),\ldots,\int_a^x {K(x,s,u^{[1]} (s),\ldots)ds} } \right),\quad \forall x \in [a, \, b],\;\forall u \in C\left( {[.]} \right). $$

By utilizing the conditions of Theorem 4, keeping a similar way as in the proof of Theorem 2, we can conclude that the operator \(T\) is strictly contractive depending upon the metric (11). Next, letting the metric \(d\) and (16), we deduce

$$ {\mathop {\sup }\limits_{x \in [a,b]}} \frac{{\left| {y(x) - y_0 (x)} \right|}}{\sigma (x)} \le \frac{1}{{1 - 2M_F \left[ {1 + (1 + \eta M_K )\Delta } \right]}} $$

and consequently (17) holds.

The fourth new outcome of this study with regard to the uniqueness of solutions and the \(\sigma\)-semi HU stability of the iterative Volterra IE (4) is given in Theorem 5.

Theorem 5.

We assume that \((As3)\) and \((As4)\) of Theorem 4 hold. In addition, if \(y \in C\left( {[.]} \right)\) with.

$$ \left| {y(x) - G\left( {\ldots} \right)} \right| \le \theta ,\quad x \in [a, \, b], $$
(18)

where \(G\left( {\ldots} \right)\) is given together with (4), \(\theta > 0,\) \(\theta \in {\mathbb{R}},\) and

$$ 2M_F \left[ {1 + (1 + \eta M_K )\Delta } \right] < 1, $$

then there is a unique function \(y_0 \in C\left( {[.]} \right)\) such that

$$ y_0 (x) = G\left( {x,y_0^{[1]} (x),\ldots,\int_a^x {K(x,s,y_0^{[1]} (s),\ldots)ds} } \right) $$

with

$$ \begin{aligned} & \left| {y(x) - y_0 (x)} \right| \le \frac{\theta }{{\left( {1 - 2M_F \left[ {(1 + \eta M_K )\Delta + 1} \right]} \right)\sigma (a)}}\sigma (x), \\ & \quad \forall \;x \in [a, \, b]. \\ \end{aligned} $$
(19)

According to the conditions above, these outcomes allow that the iterative Volterra IE (4) admits the \(\sigma\)-semi HU stability.

Proof.

Take into consideration the operator

$$ T: = C\left( {[.]} \right) \to C\left( {[.]} \right), $$

which is depicted by

$$ (Tu)(x) = G\left( {x,u^{[1]} (x),\ldots,\int_a^x {K(x,s,u^{[1]} (s),\ldots)ds} } \right),\;\forall u \in C\left( {[.]} \right),\;\forall x \in [a, \, b]. $$

By virtue of the conditions of Theorem 5, similarly keeping in mind the way in the proof of Theorem 2, we can conclude that the operator \(T\) is strictly contractive with respect to the metric (11). Hence, we can again apply the Banach FPT, which allows that the iterative Volterra IE (4) admits the \(\sigma\)-semi HU stability, i.e., the inequality (19) is satisfied via the metric \(d\) of (11), using (10) and (18). Thus, the proof of Theorem 5 is done.

5 The uniqueness and the HUR stability of the iterative Volterra IE in the infinite intervals

In Sect. 5, we will study the HU stability of the iterative Volterra IE (4) in the infinite intervals cases. Then, we will now take into consideration the intervals \([a,\infty ),\) \({ ( - }\infty {,}b{],}\) for some fixed \( \, a{, }b \in {\mathbb{R}},\) and \({\mathbb{R}} \, = { ( - }\infty {,}\infty {)}.\) Hence, we will deal with the iterative Volterra IE (4) in the infinite interval cases:

$$ y(x) = G\left( {x,y^{[1]} (x),\ldots),\int_a^x {K(t,s,y^{[1]} (s),\ldots)ds} } \right),\;x \in [a, \, \infty {)}, $$
(20)

Let \(C^b \left( {\left[ {a,\infty } \right)} \right)\) be the space of bounded and continuous functions, which is equipped with the metric

$$ d^b (u,v) = {\mathop {\sup }\limits_{t \in [a,\infty )}} \frac{{\left| {u(x) - v(x)} \right|}}{\sigma (x)}. $$

The fifth and last new result of this study with regard to the HUR stability of the iterative Volterra IE (4) in the infinite interval \([a, \, \infty {)}\) is presented in Theorem 6.

Theorem 6.

Let the following conditions are fulfilled:

\((As5)\) \(\tau ,\alpha \in C\left( {\left[ {a,\infty } \right),\left[ {a,\infty } \right)} \right)\) are variable delay functions such that \(\tau (t) \le t\) and \(\alpha (t) \le t,\) \(\forall t \in \left[ {a,\infty } \right)\) and \(\sigma \in C\left( {\left[ {a,\infty } \right),(\varepsilon ,\omega )} \right)\) is a nondecreasing function with \(\varepsilon > 0, \, \omega > 0,\) \(\varepsilon , \, \omega \in {\mathbb{R}} \, {.}\) Moreover, we assume that there is \(\eta \in {\mathbb{R}}\) such that

$$ \int_a^x {\sigma (s)ds \le } \eta \sigma (x) \;{\text{for}}\;{\text{all}}\;x \in [a,\infty ). $$

\((As6)\) \(G:[a,\infty ) \times {\mathbb{C}}^{2m + 1} \to {\mathbb{C}}\) and \(K:[a,\infty ) \times [a,\infty ) \times {\mathbb{C}}^{2m} \to {\mathbb{C}}\) are continuous functions satisfying the Lipschitz condition in the following lines

$$ \begin{aligned} & \left| G (x,u_1 (x),\ldots,u_m (\alpha (x)),g(x)) - G(x,v_1 (x),\ldots,v_m (\alpha (x)),h(x)) \right| \\ & \quad \le M_G \sum_{k = 1}^m {\left| {u_k (x) - v_k (x)} \right|} + M_G \sum_{k = 1}^m {\left| {u_k (\alpha (x)) - v_k (\alpha (x))} \right| + M_G } \left| {g(x) - h(x)} \right| \\ \end{aligned} $$

and

$$ \begin{aligned} &\left| K(x,s,u_1 (s),\ldots,u_m (\alpha (s))) - K(x,s,v_1 (s),\ldots,v_m (\alpha (s))) \right| \\ & \quad \le M_K \sum_{k = 1}^m {\left[ {\left| {u_k (s) - v_k (s)} \right| + \left| {u_k (\alpha (s)) - v_k (\alpha (s))} \right|} \right]} , \\ \end{aligned} $$

respectively, where \(M_F\) and \(M_K\) are positive constants and the kernel \(\int_a^x {K(x,s,y^{[1]} (s),\ldots)ds}\) is a bounded and continuous function for any bounded continuous function \(y.\)

If \(y \in C^b \left( {\left[ {a,\infty } \right)} \right),\)

$$ \left| {y(x) - G\left( {x,y^{[1]} (x),\ldots,\int_a^x {K(x,s,y^{[1]} (s),\ldots)ds} } \right)} \right| \le \sigma (x),\;x \in [a, \, \infty {)}, $$

and

$$ 2M_F \left[ {(1 + \eta M_K )\Delta + 1} \right] < 1, $$

then there is a unique function \(y_0 \in C^b \left( {\left[ {a,\infty } \right)} \right)\) with

$$ y_0 (x) = G\left( {x,y_0^{[1]} (x),\ldots,\int_a^x {K(x,s,y_0^{[1]} (s),\ldots)ds} } \right) $$
(21)

and

$$ \left| {y(x) - y_0 (x)} \right| \le \frac{1}{{1 - 2M_F \left[ {1 + (1 + \eta M_K )\Delta } \right]}}\sigma (x), $$
(22)

for all \(x \in [a, \, \infty {)}.\)

According to these outcomes, depending upon the conditions above, the iterative Volterra IE (4) has the HUR stability.

Proof.

Let \(I = [a,\infty ), \, a \in {\mathbb{R}},\) and \(I_n = [a,a + n],\) \(n \in {\mathbb{N}}.\) By virtue of Theorem 4, there exists a unique and bounded function \(y_{0,n} \in C[I_n ,{\mathbb{C}}]\) such that

$$ y_{0,n} (x) = G\left( {x,y_{0,n}^{[1]} (x),\ldots),\int_a^x {K(x,s,y_{0,n}^{[1]} (s),\ldots)ds} } \right) $$
(23)

and

$$ \left| {y(x) - y_{0,n} (x)} \right| \le \frac{1}{{1 - 2M_F \left[ {1 + (1 + \eta M_K )\Delta } \right]}}\sigma (x),\quad x \in I_n . $$
(24)

Since \(y_{0,n}\) is unique, if \(x \in I_n ,\) then

$$ y_{0,n} (x) = y_{0,n + 1} (x) = y_{0,n + 2} (x) = \ldots \, {.} $$
(25)

Letting \(x \in [a,\infty ),\) we define \(n(x) \in {\mathbb{N}}\) by

$$ n(x) = \min \{ n \in {\mathbb{N}}:x \in I_n \} . $$

Next, let the function \(y_0 :[a,\infty ) \to {\mathbb{C}}\) be depicted by

$$ y_0 (x) = y_{0,n(x)} (x). $$
(26)

Hence, we claim that \(y_0\) is continuous. Following similar mathematical calculations as in Castro and Simões [8, Theorem 8], it can be shown that \(y_0 (x) = y_{0,n_1 + 1} (x)\) for all \(x \in (x_1 - \varepsilon , \, x_1 + \varepsilon )\) and since \(y_{0,n_1 + 1}\) is continuous at \(x_1 ,\) (see, Theorem 4), thus \(y_0\) is also continuous at \(x_1 ,\) \(x_1 \in {\mathbb{R}} \, .\)

We will now confirm that \(y_0\) satisfiesand in addition

$$ y_0 (x) = G\left( {x,y_0^{[1]} (x),\ldots),\int_a^x {K(x,s,y_0^{[1]} (s),\ldots)ds} } \right), $$
$$ \begin{aligned} & \left| {y(x) - y_0 (x)} \right| \le \frac{1}{{1 - 2M_F \left[ {(1 + \eta M_K )\Delta + 1} \right]}}\sigma (x), \\ & \quad \forall \;x \in [a, \, \infty {)}. \\ \end{aligned} $$
(27)

For an arbitrary \(x \in [a, \, \infty {)}\) we choose \(n(x)\) such that \(x \in I_{n(x)} .\) Next, by (23) and (26), we find

$$ \begin{aligned} y_0 (x) &= y_{0,n(x)} (x) = G\left( {x,y_{0,n(x)}^{[1]} (x),\ldots,\int_a^x {K(x,s,y_{0,n(x)}^{[1]} (s),\ldots)ds} } \right) \\ & = G\left( {x,y_0^{[1]} (x),\ldots),\int_a^x {K(x,s,y_0^{[1]} (s),\ldots)ds} } \right). \\ \end{aligned} $$
(28)

Since \(n(s) \, \le \, n(x)\) for all \(x \in I_{n(x)} ,\) then from (25) we derive that

$$ y_0 (x) = y_{0,n(x)} (x) = y_{0,n(x)} (s). $$

Hence, the last equality of (28) holds true. Next, from (24) and (26), it follows that

$$ \left| {y(x) - y_0 (x)} \right| = \left| {y(x) - y_{0,n(x)} (x)} \right| \le \frac{1}{{1 - 2M_F \left[ {1 + (1 + \eta M_K )\Delta } \right]}}\sigma (x), $$

\(\forall x \in [a,\infty ),\) which is (22).

At the last step, we will prove that \(y_0\) is unique. Let \(y_1\) be another continuous and bounded function such that it satisfies (21) and (22), \(\forall x \in [a,\infty ).\) By the uniqueness of the solution on \(I_{n(x)}\) for any \( \, n(x) \in {\mathbb{N}},\) we have \(y_0 (x) = y_0 |_{I_{n(x)} } (x)\) and \(y_1 (x)\) satisfies (21) and (22), \(\forall x \in I_{n(x)} .\) Hence, we conclude that

$$ y_0 (x) = y_0 |_{I_{n(x)} } (x) = y_1 |_{I_{n(x)} } (x) = y_1 (x). $$

This is the end of the proof.

Remark 2.

We can also proceed and demonstrate the idea of Theorem 6 similarly for the infinite interval cases \(I \, = { ( - }\infty {,}b{], }b \in {\mathbb{R}},\) and \({\mathbb{R}} \, = { ( - }\infty {,}\infty {)}.\)

6 Numerical application

In a particular case, we now give an example to show applications of the results.

Example 1

(Castro and Simões [8]). As a specific case of the nonlinear Volterra IE (4) with variable delays, we consider the Volterra IE:

$$ y(x) = G\left( {x,y^{[1]} (x),y^{[1]} (\tau (x)),\int_0^x {K\left( {x,s,y^{[1]} (s),y^{[1]} (\alpha (s))} \right)ds} } \right), $$
(29)

where

$$ G\left( {\ldots} \right) = \frac{1}{60}x^5 - \frac{1}{5}x^2 + x + \frac{1}{5}y(x^2 ) + \frac{1}{3}\int_0^x {(s - x)y(s^3 )} ds.\;\forall x \in [0,1]. $$
(30)

We can confirm that the function \(G\left( {\ldots} \right)\) is continuous.

Let \(y \in C\left( {[0,1],{\mathbb{R}}} \right)\) and \(\sigma \in C\left( {[0,1],(0,\infty )} \right),\) which is a non-decreasing and defined by \(\sigma (x) = 0.0083x + 0.0005.\) Let \(\tau , \, \alpha \in C\left( {[0,1],[0,1]} \right).\) It is clear from (29) and (30) that \(\tau (x) = x^2 ,\) \(\alpha (x) = x^3\) such that \(\tau (x) \le x,\) \(\alpha (x) \le x,\) \(\forall x \in [0,1].\) Letting \(\eta = 0.52841,\) we derive

$$ \begin{aligned} \int_a^x \sigma (s)ds &= \int_0^x {(0.0083s + 0.0005} )ds \\ & \le \eta \left( {\frac{83}{{10000}}x + \frac{5}{10000}} \right) = 0.52841\sigma (x),\;\forall \;x \in [0,1]. \\ \end{aligned} $$

As for the next step, we have

$$ \begin{aligned} &\left| G(x,u_1 (x),u_1 (\alpha (x)),g(x)) - G(x,v_1 (x),v_2 (\alpha (x)),h(x)) \right| \\ & \quad \le \left| {u_1 (x) - v_1 (x)} \right| + \left| {u_1 (\alpha (x)) - v_1 (\alpha (x))} \right| + \left| {g(x) - h(x)} \right| \\ & \quad \le \frac{1}{3}\left( {\left| {u_1 (x^2 ) - v_1 (x^2 )} \right| + \left| {g(x) - h(x)} \right|} \right), \\ \end{aligned} $$
$$ M_G = \frac{1}{3},\;x \in [0,1] $$

and

$$ \begin{aligned}& K(x,s,u_1 (s),u_1 (\alpha (s))) = (s - x)y(s^3 ), \\ &\left| K(x,s,u_1 (s),u_1 (\alpha (s))) - K(x,s,v_1 (s),v_1 (\alpha (s))) \right| \\ & \quad \le \left( {\left| {u_1 (s) - v_1 (s)} \right| + \left| {u_1 (\alpha (s)) - v_1 (\alpha (s))} \right|} \right), \\ & \quad = \left| {s - x} \right| \, \left| {u_1 (s^3 ) - v_1 (s^3 )} \right| \\ & \quad \le \left| {u_1 (s^3 ) - v_1 (s^3 )} \right|, \\ \end{aligned} $$
$$ M_K = 1,\;s \in [0,x],\;x \in [0,1], $$
$$ \Delta = M_G (2 + \eta M_K ) = \frac{252841}{{300000}} < 1. $$

Letting \(y(x) = \frac{101x}{{100}},\) hence, we get

$$ \begin{aligned} & \left| {y(x) - G\left( {\ldots} \right)} \right| = \left| {\frac{1}{6000}x^5 - \frac{1}{500}x^2 + \frac{1}{100}x} \right| \\ & \quad \le \left( {\frac{83}{{10000}}x + \frac{5}{10000}} \right) = \sigma (x),\;x \in [0,1]. \\ \end{aligned} $$

It can also be tested that the Volterra IE (29) has the exact solution \(y_0 (x) = x\) and the function \(y = \frac{101}{{100}}x\) approximately satisfies the Volterra IE (29). Indeed, according to the present data, it follows that

$$ \left| {y(x) - y_0 (x)} \right| = 100^{ - 1} \left| x \right| \le \frac{1}{1 - M_G (2 + \eta M_K )}\sigma (x) = \frac{300000}{{47159}},\;x \in [0,1]. $$

Then, according to Theorem 4, the Volterra IE (29) admits the HUR stability.

For \(x \in [0,1],\) we also have

$$ \left| {y(x) - G\left( {\ldots} \right)} \right| = \left| {\frac{1}{6000}x^5 - \frac{1}{500}x^2 + \frac{1}{100}x} \right| \le \frac{11}{{1250}} = \theta . $$

Finally, letting \(x \in [0,1],\) we also derive that

$$ \left| {y(x) - y_0 (x)} \right| = (0,01)\left| x \right| \le \frac{\theta }{1 - M_G (2 + \eta M_K )\sigma (0)}\sigma (x) = \frac{5280000}{{47159}}\sigma (x) $$

and

$$ \left| {y(x) - y_0 (x)} \right| = (0.01)\left| x \right| \le \frac{\theta }{1 - M_G (2 + \eta M_K )\sigma (0)}\sigma (1) \approx 0.98527, $$

respectively.

Then, according to Theorem 5, the Volterra IE (29) admits the \(\sigma\)‐semi‐HU and the HU stabilities.

7 Discussions

We will now clarify this study as summarily.

\({\text{I}}^0 )\) To our knowledge, according to the materials of the literature, there is no result on the uniqueness of solutions and Ulam type stabilities of iterative Fredholm and Volterra IEs including multiple variable time delays. This study is the first pioneer work with regard to these classes of the IEs and the concepts metioned.

  • \({\text{II}}^0 )\) In this study, we built five new theorems in connection with the HUR stability and the \(\sigma\)-semi HU stability, etc., of nonlinear iterative Fredholm and Volterra IEs, which including multiple variable time delays, on finite or infinite intervals cases. In the proof of the new outcomes we utilized the Banach FPT and the generalized Bielecki metric. The findings of this study have essential contributions to the results summarized in the begin of the study, Ulam’s qualitative theory with regard to the IEs without and with time delays and allow new advantageous for researchers working on these concepts.

  • \({\text{III}}^0 )\) We should mention that the Banach FPT and the generalized Bielecki metric are very effective tools to investigate the Ulam type stabilities of iterative Fredholm and Volterra IEs. Regarding the possible future recommendations, Ulam type stabilities of iterative fractional Fredholm and Volterra IEs are suggested as proper open problems.

  • \({\text{IV}}^0 )\) In particular cases of the considered iterative Fredholm and Volterra IEs, a numerical example was taken from the earlier data bases to provide the applications and illustrations of the results.

8 Conclusion

In this study, certain nonlinear iterative Fredholm and Volterra integral equations including variable time delays were considered. We built new sufficient conditions for the uniqueness of solutions, the HUR stability and the \(\sigma\)-semi HU stability of that iterative Fredholm and Volterra integral equations on a finite interval case. We also constructed a new qualitative result including sufficient conditions with regard to the uniqueness of solutions and the HUR stability of the iterative Volterra IE (4) on the infinite intervals cases. The outcomes of this study include five new theorems. The proofs of the new five theorems were done using the Banach FPT and the generalized Bielecki metric. In particular case, a specific example was taken from the former data bases to show the availability of the results of this study. The outcomes of this study are new and they have new contributions to the Ulam stabilities of integral equations.