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1.   Introduction

The interest for studying the theory of infinite systems of integral equations is based on the fact that the theory of infinite systems of integral equations is a branch of nonlinear analysis which has been applied in various fields of science and numerous applications. In fact, most physical and engineering problems are formed by infinite systems of integral equations, see for example ^[1,2,3,4]. The problem of the existence of solutions for infinite systems of integral equations plays a significant role in the investigation of these types of equations and it is important to apply original studies in our investigations (cf.^[5,6,7]). In some papers, integral equations of Volterra type have been converted in the form of integral equations of Volterra-Stieltjes type and numerous results have been obtained on the existence of solutions of nonlinear integral equations (cf.^[8,9]). The aim of this paper is to present some results on the existence of solutions for an infinite system of integral equations of Volterra-Stieltjes type of the form

where $BC(\mathbb{R_{+}}\times \mathbb{R_{+}}, \mathbb{R})$ is the space of all real functions $u (t, x) = u : \mathbb{R_{+}} \times \mathbb{R_{+}}\longrightarrow \mathbb{R}$, which are defined, continuous and bounded on the set $\mathbb{R_{+}} \times \mathbb{R_{+}}$ with a supremum norm $\Vert u \Vert = \sup\bigg\{|u (t, x)| : (t, x) \in \mathbb{R_{+}} \times \mathbb{R_{+}}\bigg\}$. The obtained results extend and generalize the results of ^[6,8,9] in the Banach spaces $c_{0}$ and $\ell_{p}$. In our approach, this is done by applying the measure of noncompactness and Darbo fixed point theorem.

2.   Preliminaries

In future, we apply some notations, definitions and preliminary facts to obtain our main results.

For a bounded subset $S$ of a metric space $X$, Kuratowski ^[10] defined the function $\alpha(S)$ by the formula

known as the Kuratowski measure of noncompactness. Another measure of noncompactness is the Hausdorff measure of noncompactness given by:

Let $E$ be a real Banach space with norm $\|.\|$ and zero element $\theta$. Besides, we suppose $\overline{X}$ and Conv(X) denote the closure and convex hull of $X$, respectively. Moreover, let us denote by $M_{E}$ the family of all nonempty and bounded subsets of $E$ and by $N_{E}$ its subfamily consisting of all relatively compact sets.

Definition 1. ^[11] A mapping $\mu: M_{E}\longrightarrow [0, \infty)$ is called a measure of noncompactness if it satisfies the following conditions:

(1) The set Ker$\mu = \{X\in M_{E}: \mu(X) = 0\}$ is nonempty and Ker$\mu\subseteq N_{E}$.

(2) $X\subseteq Y\Longrightarrow \mu(X)\leq \mu(Y).$

(3) $\mu(\overline{X}) = \mu(X).$

(4) $\mu(Conv(X)) = \mu(X)$.

(5) $\mu(\lambda X + (1 - \lambda)Y)\leq \lambda \mu(X) + (1 - \lambda)\mu(Y)$ for $\lambda\in [0, 1]$.

(6) If $\{X_{n}\}$ is a sequence of closed sets from $M_{E}$ such that $X_{n+1}\subseteq X_{n}$ for $n = 1, 2, \dots$ and ${\lim\limits_{n\rightarrow\infty}}\mu(X_{n}) = 0$, then $\bigcap_{n = 1}^{\infty}X_{n}$ is nonempty.

We will apply the following theorem as the main tool in our investigations.

Theorem 1. (Darbo^[12]) Let $C$ be a nonempty, bounded, closed and convex subset of a Banach space $E$ and $T: \; \; C\rightarrow \; \; C$ be a continuous mapping. Assume that there exists a constant $K \in [0, 1)$ such that $\mu (TX)\leq K\mu (X)$ for any nonempty subset $X$ of $C$, where $\mu$ is a measure of noncompactness defined in $E$. Then $T$ has at least a fixed point in $C$.

Samadi ^[13] extended Darbo's fixed point theorem as follows.

Theorem 2. Let $C$ be a nonempty bounded, closed and convex subset of a Banach space $E$. Assume $T:C\longrightarrow C$ be a continuous operator satisfying

for all nonempty subsets $X$ of $C$, where $\mu$ is an arbitrary measure of noncompactness defined in $E$ and $(\theta, f)\in \Delta =$. Then $T$ has a fixed point in $C$.

In Theorem 2, $\Delta$ is the set of all pairs $(\theta, f)$ satisfying the following:

($\Delta_1$) $\theta(t_n)\nrightarrow 0$ for each strictly increasing sequence $\{t_n\};$

($\Delta_2$) $f$ is strictly increasing function;

($\Delta_3$) for each sequence $\{\alpha_n\}$ of positive numbers, $\lim_{n\to \infty}\alpha_n = 0$ if and only if $\lim_{n\to \infty}f(\alpha_n) = -\infty.$

($\Delta_4$) If $\{t_n\}$ is a decreasing sequence such that $t_n\to 0$ and $\theta(t_n) < f(t_n)-f(t_{n+1}),$ then we have $\sum_{n = 1}^{\infty}t_n < \infty.$

We know that the Hausdorff measure of noncompactness $\chi$ in the Banach space $\ell_{p}$ can be defined as follows:

where $B\in M_{\ell_p}$ and $x = (x_{k})\in \ell_{p}$. For the Banach space $(c_{0} \; , \; \Vert. \Vert_{c_{0}})$, the Hausdorff measure of noncompactnes $\chi$ is given by (cf. Definition 1):

where $B\in M_{c_{0}}$ and $u = (u_{k})\in c_{0}$.

Now, we recall some basic facts concerning the concept of the variation of a function (cf.^[14,15]).

Assume that $f$ is a real function defined on the interval $[a, b]$. The variation of the function $f$ will be denoted by $\bigvee_{a}^{b}f$. If $\bigvee_{a}^{b}f$ is finite, the function $f$ has bounded variation on the interval $[a, b]$. Similarly, if $g:[a, b]\times [c, d]\longrightarrow \mathrm{R}$ is a real function of two variables, then the variation of the function $t\longrightarrow g(t, s)$ on the interval $[p, q]\subseteq [a, b]$ will be denoted by $\bigvee_{s = p}^{q}g(t, s)$. Analogously, we can define $\bigvee_{t = p}^{q}g(t, s)$. Assume that $f$ and $g$ are two real functions defined on the interval $[a, b]$, then under appropriate conditions we can define the Steiltjes integral $\int_{a}^{b}f(t)dg(t)$ of the function $f$ with respect to the function $g$. If the integral $\int_{a}^{b}f(t)dg(t)$ is finite, then $f$ is Stieltjes integrable on the interval $[a, b]$.

The following lemmas will be applied in our investigations.

Lemma 1. If $f$ is Stieltjes integrable on the interval $[a, b]$ with respect to a function $g$ of bounded variation, then

Lemma 2. Let $f_{1}$ and $f_{2}$ be Stieltjes integrable functions on the interval $[a, b]$ with respect to a nondecreasing function $g$ such that $f_{1}(t)\leq f_{2}(t)$ for $t\in [a, b]$. Then,

3.   Existence of solutions for infinite systems of integral equations

In this section, as an application of Theorem 2, the existence of solutions for the infinite system (1.1) is studied in the spaces $\ell_{p}$ and $c_{0}$. First, we show that infinite system (1.1) has a solution that belongs to the space $\ell_{p}$.

We consider the following conditions:

$\mathrm{(H_1)} \; \; F_{n}:\mathbb{R_{+}}\times \mathbb{R_{+}} \times \mathbb{R} \times \mathbb{R}\longrightarrow \mathbb{R}$ is continuous and there exist positive real numbers $\tau > 0$ such that

for all $t, s\in \mathbb{R_{+}}$ and $x_{1}, x_{2}, y_{1}, y_{2}\in \mathbb{R}$. Moreover, we have

$\mathrm{(H_2)} \; \; f_{1}:\mathbb{R_{+}} \times \mathbb{R}^{\infty}\longrightarrow \mathbb{R}$ is continuos with $f_{0} = \sup_{t\in \mathbb{R_{+}}}\vert f(t, 0)\vert$ and there exist positive real numbers $\tau > 0$ such that

for all $t, x\in \mathbb{R_{+}}$ and $u(t, x) = \bigg\{u_{i}(t, x)\bigg\}_{i = 1}^{\infty}, v(t, x) = \bigg\{v_{i}(t, x)\bigg\}_{i = 1}^{\infty}\in \ell_{p}$.

$\mathrm{(H_3)} \; \; T:BC(\mathbb{R_{+}} \times \mathbb{R_{+}}, \ell_{p})\longrightarrow BC(\mathbb{R_{+}} \times \mathbb{R_{+}}, \mathbb{R})$ is a continuos operator such that

for all $u, v\in BC(\mathbb{R_{+}} \times \mathbb{R_{+}}, \ell_{p})$ and $t, x\in\mathbb{R_{+}}$.

$\mathrm{(H_4)}$ For any fixed $t > 0$ the function $s\longrightarrow g_{i}(t, s)$ has a bounded variation on the interval $[0, t]$ and the function $t\longrightarrow \bigvee_{s = 0}^{t}g_{i}(t, s)$ is bounded over $\mathbb{R_{+}}$.

$\mathrm{(H_5)} \; \; g_{n}:\mathbb{R_{+}}\times \mathbb{R_{+}}\times \mathbb{R_{+}}\times \mathbb{R_{+}}\times \mathbb{R}^{\infty} \longrightarrow \mathbb{R}$ is continuous and there exist continuous functions $a_{n}:\mathbb{R_{+}} \times \mathbb{R_{+}}\longrightarrow \mathbb{R_{+}}$ such that

Moreover, assume that

$\mathrm{(H_6)} \; \; V_{n}:\mathbb{R_{+}}\times \mathbb{R_{+}}\times \mathbb{R}^{\infty}\longrightarrow \mathbb{R}$ is a continuous function and there exists continuous function $k:\mathbb{R_{+}}\times \mathbb{R_{+}}\longrightarrow \mathbb{R_{+}}$ such that the function $s\longrightarrow k(t, s)$ is integrable over $\mathbb{R_{+}}$ and the following conditions hold:

for all $t, s, x\in \mathbb{R_{+}}$ and $u, v\in \ell_{p}$. Moreover, assume that

$\mathrm{(H_7)}$ There exists a positive solution $r_{0}$ such that

Moreover, assume that $2^pM < 1$.

Theorem 3. Under the assumptions $\mathrm{(H_{1})}-\mathrm{(H_{7})}$, Eq (1.1) has at least one solution $u(t, x) = \bigg\{u_{i}(t, x)\bigg\}_{i = 1}^{\infty}$ in the space $\ell_{p}$.

Proof. Let us define the operator $G$ on $BC\bigg(\mathbb{R_{+}}\times \mathbb{R_{+}}, \ell_{p}\bigg)$ by

In view of our assumptions, for all $t, x\in \mathbb{R_{+}}$, we get

Thus, by applying the last estimates and assumption $\mathrm{(H_{7})}$ one can easily seen that $G$ maps $\overline{B_{r_{0}}}$ into itself, where

Next, we prove that the operator $G$ is a continuous operator on the Ball $\overline{B_{r_{0}}}$. For this, take $\varepsilon > 0$ arbitrarily and $u(t, x) = \bigg\{u_{i}(t, x)\bigg\}_{i = 1}^{\infty}, v(t, x) = \bigg\{v_{i}(t, x)\bigg\}_{i = 1}^{\infty}\in \overline{B_{r_{0}} }$ with $\Vert u -v \Vert_{ BC(\mathbb{R_{+}} \times \mathbb{R_{+}}, \ell_{p})} < \varepsilon.$ Acordingly, taking into account our assumptions, for $(t, x) \in \mathbb{R_{+}}\times \mathbb{R_{+}}$ we have

On the other hand, we have

Further, by applying our assumptions, we arrive that

Combining (3.2), (3.3) and (3.4), we conclude that

Using $\mathrm{(H_{5})}$, there exists $T > 0$ such that for $t > T$, we get

Hence, by (3.5), we conclude that

For $t\in[0, T]$ we have

where

and $\omega(g, \varepsilon)\longrightarrow 0$ as $\varepsilon \longrightarrow 0$. Consequently, $G$ is continuous on the ball $\overline{B_{r_{0}}}$. To finish the proof, we prove that the condition (2.1) of Theorem 2 is fulfilled. Let $X$ be a nonempty and bounded subset of the ball $\overline{B_{r_{0}}}$. Assume that

Thus, by applying our assumptions, we infer that

Hence,

Consequently,

By passing to logarithms, we get

Now applying Theorem 2 with $f(t) = \ln(t)$ and $\theta(t) = \tau$, we obtain that $G$ has a fixed point and the proof is completed.

Example 1. Now, we investigate the following system of integral equations:

Observe that Eq (3.11) is a special case of the infinite system (1.1) if we put

Thus, it is easily seen that $F_{n}$ and $f_{1}$ satisfy assumptions $\mathrm{(H_1)}$ and $\mathrm{(H_2)}$ with $N_{1} = 0$ and $f_{0 } = 0$. Further, the operator $T$ satisfies hypothesis $\mathrm{(H_3)}$. To justify assumption $\mathrm{(H_5)}$, let $t, s x, y\in \mathbb{R_{+}}$ and $u, u\in \ell_{p}$. Then, we have

Since $\frac{\partial g_{1}}{\partial s} = \frac{e^t}{1 + t^2} > 0$, then $\bigvee_{q = 0}^{s}g_{1}(t, q) = g_{1}(t, s)-g_{1}(t, 0) = \frac{se^t}{1 + t^2}$. Consequently, we have

Inconsequence,

So, $\varphi_{k}\longrightarrow 0$. On the other hand the function $V_{n}(t, s, u(t, x)) = \frac{e^{-s}}{1 + \frac{t}{8}}\sin\bigg(\vert u_{n}(t, x) \vert\bigg)$ verifies assumption $\mathrm{(H_6)}$ with $k(t, s) = \frac{e^{-s}}{1 + \frac{t}{8}}$ and $M = 1$. To show that the functions $g_{1}$ and $g_{2}$ satisfy assumption $\mathrm{(H_4)}$, let first note that the functions $g_{1}$ and $g_{2}$ are increasing on every interval of the form $[0, t]$ and $g_{2}$ is bounded on the triangle $\triangle_{2}$. Consequently, the function $y\longrightarrow g_{2}(x, y)$ has bounded variation on the interval $[0, x]$ and we have

So, $G\leq \frac{\pi}{4}$. We can take $G = \frac{\pi}{4}$. Consequently, all conditions of Theorem 3 are satisfied and Theorem 3 implies that the infinite system (3.11) has at least one solution which belongs to the space $\ell_{p}$.

4.   $c_{0}$-solvability of the infinite system (1.1)

Now the existence of solutions of the system (1.1) is studied in the space $c_{0}$. In this case, we need the following assumptions.

$\mathrm{(D_1)} \; \; F_{n}:\mathbb{R_{+}}\times \mathbb{R_{+}} \times \mathbb{R} \times \mathbb{R}\longrightarrow \mathbb{R}$ is continuous and there exist positive real numbers $\tau > 0$ such that

for all $t, s\in \mathbb{R_{+}}$ and $x_{1}, x_{2}, y_{1}, y_{2}\in \mathbb{R}$. Moreover, assume that

$\mathrm{(D_2)} \; \; f_{1}:\mathbb{R_{+}} \times \mathbb{R}^{\infty}\longrightarrow \mathbb{R}$ is continuous with $f_{0} = \sup_{t\in \mathbb{R_{+}}} \vert f(t, 0) \vert$ and there exist positive real numbers $\tau > 0$ such that

for all $t, x\in \mathbb{R_{+}}$ and $u(t, x) = \bigg\{u_{i}(t, x)\bigg\}, v(t, x) = \bigg\{v_{i}(t, x)\bigg\}\in c_{0}$

$\mathrm{(D_3)} \; \; T:BC(\mathbb{R_{+}} \times \mathbb{R_{+}}, c_{0})\longrightarrow BC(\mathbb{R_{+}} \times \mathbb{R_{+}}, \mathbb{R})$ is a continuous operator such that

for all $u, v\in BC(\mathbb{R_{+}} \times \mathbb{R_{+}}, c_{0})$ and $t, x\in\mathbb{R_{+}}$.

$\mathrm{(D_4)}$ For any fixed $t > 0$ the function $s\longrightarrow g_{i}(t, s)$ has a bounded variation on the interval $[0, t]$ and the functions $t\longrightarrow \bigvee_{s = 0}^{t}g_{i}(t, s)$ are bounded on $\mathbb{R_{+}}$. Moreover, for arbitrarily fixed $T > 0$ the function $w\longrightarrow \bigvee_{z = 0}^{w}g_{i}(w, z)$ is continuous on the interval $[0, T]$ for $i = 1, 2$.

$\mathrm{(D_5)} \; \; g_{n}:\mathbb{R_{+}}\times \mathbb{R_{+}}\times \mathbb{R_{+}}\times \mathbb{R_{+}}\times\mathbb{R}^{\infty}\longrightarrow \mathbb{R}$ is continuous and there exist continuous functions $a_{n}:\mathbb{R_{+}} \times \mathbb{R_{+}}\longrightarrow \mathbb{R_{+}}$ such that

for all $t, s, x, y\in \mathbb{R_{+}}$ and $u, v\in \mathbb{R}^{\infty}$. Moreover, assume that

where $T > 0$ is arbitrarily fixed.

$\mathrm{(D_6)} \; \; V_{n}:\mathbb{R_{+}}\times \mathbb{R_{+}}\times \mathbb{R}^{\infty}\longrightarrow \mathbb{R}$ is a continuous function and there exists continuous function $k:\mathbb{R_{+}}\times \mathbb{R_{+}}\longrightarrow \mathbb{R_{+}}$ such that the function $s\longrightarrow k(t, s)$ is integrable over $\mathbb{R_{+}}$ and the following conditions hold:

for all $t, s, x\in \mathbb{R_{+}}$ and $u, v\in c_{0}$. Moreover, assume that

Theorem 4. Under assumptions $\mathrm{(D_1)}-\mathrm{(D_6)}$, the infinite system (1.1) has at least one solution $u(t) = \bigg\{u_{i}(t, x)\bigg\}_{i = 1}^{\infty}$ belonging to the space $c_{0}$.

Proof. Define the operator $G$ on the space $BC\bigg(\mathbb{R_{+}}\times \mathbb{R_{+}}, c_{0}\bigg)$ as

where $t, x\in \mathbb{R_{+}}$. We show that

is G-invariant where $i = 1, 2, ...$ and $t, x\in\mathbb{R_{+}}$. Assume that

For arbitrarily fixed $(t, x)\in \mathbb{R_{+}}\times \mathbb{R_{+}}$, we have

Consequently,

By applying (4.1), one can easily seen that $G$ maps the ball $\overline{B_{r_{0}}}$ into itself. Next, the continuity property of the operator $G$ will be proved on the ball $\overline{B_{r_{0}}}$. Let $u, v\in B_{r_{0}}$ and $\varepsilon > 0$ such that $\Vert u-v\Vert_{BC\bigg(\mathbb{R_{+}}\times \mathbb{R_{+}}, c_{0}\bigg)} < \varepsilon$. Thus for all $t, x\in \mathbb{R_{+}}$, we have

Besides, we have

By assumption $\mathrm{(D_5)}$, there exists $T > 0$ such that for $t > T$, we have

Further, the assumptions $\mathrm{(D_3)}$ and $\mathrm{(D_6)}$ give us the following eastimates

Applying (4.2), (4.3) and (4.4), we have

For $t\in [0, T]$, we have

where

Moreover, in light of the continuity of $V$ on $\bigtriangleup_{1}\times \bigtriangleup_{2}\times \mathbb{R}^{\infty}$, we have $\omega(g_{n}, \varepsilon)\longrightarrow 0$. Now, combining (4.5) and (4.6) implies that $G$ is continuous on the Ball $\overline{B_{r_{0}}}$. In what follows let $X$ be a nonempty subset of the ball $\overline{B_{r_{0}}}$, In view of the formula (2.3) and our assumptions, we have

Consequently,

As, $M < 1$, by passing to logarithms, we have

Thus all conditions of Theorem 2 hold true with $f(t) = \ln(t)$ and $\theta(t) = \tau$ and by Theorem 2 there exists $\bigg\{u_{i}(t, x)\bigg\}_{i = 1}^{\infty}\in c_{0}$ such that

Hence, the proof is completed.

Example 2.

on the space $c_{0}$. Taking

in the system (1.1), the system of integral Eq (4.8) is obtained. Note that the functions $F_{n}$ and $f_{1}$ satisfy conditions $\mathrm{(D_1)}$ and $\mathrm{(D_2)}$. Indeed, we have

Also, it can easily be seen that the operator $T$ satisfies assumption $\mathrm{(D_3)}$ and

Moreover, since $\frac{\partial g_{1}}{\partial s} = \frac{e^{-2t}}{1 + t^2} > 0$, so $g_{1}$ is increasing and we have

Consequently,

Again, we have

So, $G = \frac{\pi}{4}$ and $A < \infty$. On the other hand the function $V_{n}(t, s, u(t, x)) = e^{-t-s-n}\Sigma_{k\geq n}\frac{\vert u_{k}(t, x) \vert}{10^n(1 + k^2)}$ verifies assumption $\mathrm{(D_6)}$ with $k(t, s) = e^{-t-s}$ and $M = 1$. By applying the continuity of the function $h\longrightarrow \bigvee_{z = 0 }^{w}g_{i}(h, z)$ on the interval $[0, T]$ we can take $G_{1} = \sup\bigg\{\bigvee_{z = 0}^{w}g_{1}(w, z):w\in[0, T]\bigg\}$ where $T > 0$ is arbitrarily fixed. Thus all conditions of Theorem 4 are satisfied and by applying Theorem 4, infinite system (4) has at least one solution in the space $c_{0}$

5.   Conclusions

We studied the existence of solutions for an infinite system of integral equations of Volterra-Stieltjes type of the following form in the Banach sequence spaces $\ell_{p}$ and $c_{0}$ via the techniques of measures of noncompactness and Darbo's fixed point theorem.

where $BC(\mathbb{R_{+}}\times \mathbb{R_{+}}, \mathbb{R})$ is the space of all real functions $u (t, x) = u : \mathbb{R_{+}} \times \mathbb{R_{+}}\longrightarrow \mathbb{R}$, which are defined, continuous and bounded on the set $\mathbb{R_{+}} \times \mathbb{R_{+}}$ with a supremum norm $\Vert u \Vert = \sup\bigg\{|u (t, x)| : (t, x) \in \mathbb{R_{+}} \times \mathbb{R_{+}}\bigg\}$. Some examples in the Banach sequence spaces $\ell_{p}$ and $c_{0}$ are also given to ascertain the usefulness of our main result.

Acknowledgement

Research of the author M. Mursaleen was supported by SERB Core Research Grant, DST, New Delhi, under grant NO. EMR/2017/000340.

Conflict of interest

All authors declare no conflicts of interest in this paper.