Application of measures of noncompactness to the system of integral equations

Abstract

In this paper, by applying a measure of noncompactness in the space L^∞(ℝⁿ) and a new generalization of Darbo fixed point theorem, we study the existence of solutions for a class of the system of integral equations. Our main result is more general than the main result of [2]. Finally, an example is presented to show the usefulness of the outcome.

Keywords:

• Darbo’s fixed point theorem
• measure of noncompactness
• integral equations

PUBLIC INTEREST STATEMENT

Metric fixed point theory is a powerful tool for solving many problems in various parts of mathematics and its applications. In particular, the technique of measure of noncompactness is a very useful tool for studying the existing solutions o integral equations .In this paper, by applying the technique of measure of noncompactness, we study the existence of solutions for a class of the system of integral equations.

1. Introduction

The technique of measure of noncompactness is the main tool to solvability of several types of integral equations, see, for example, (Aghajani, Mursaleen, & Shole Haghighi, 2015; Darwish, 2008; Das, Hazarika, & Mursaleen, 2017; Dhage & Bellale, 2008; Hazarika, Arab, & Mursaleen, 2019; Liu, Guo, Wu, & Wu, 2005; Mohiuddine, Srivastava, & Alotaibi, 2016; Mursaleen & Mohiuddine, 2012; Mursaleen & SYED, 2016; Olszowy, 1980; Srivastava, Das, Hazarika, & Mohiuddine, 2018; Srivastava, Das, Hazarika,S, & Mohiuddine, 2019). Besides, it has been frequently applied in several branches of nonlinear analysis. The first measure of noncompactness was introduced by Kuratowski (Kuratowski, 1934). The most important fixed point theorem of this measure was introduced by Darbo (Darbo, 1955). In the last years, authors have applied this result to establish the existence and uniqueness of solutions for integral equations in Banach spaces. The space $L^{\mathrm{\infty }}({\mathbb{R}}^n)$ is one of the most important spaces where several integral equations have been solved in this apace. The additional advantage of this space depends on the fact that the functions of the space $L^{\mathrm{\infty }}({\mathbb{R}}^n)$ are not necessary to be continuous. Allahyari (Allahyari, 2018) introduced the construction of the measure of noncompactness on the space $L^{\mathrm{\infty }}({\mathbb{R}}^n)$ and studied the existence of solutions for a class of nonlinear functional integral equations of Urysohn type. The aim of this paper is to study the existence of solutions for a class of the system of integral equations of the form

(1.1) $\begin{array}{rl}& \\ & \left\{\begin{array}{c}\mu (x)=F\left(x,h(x,u(x),v(x)){\int }_{{\mathrm{\Omega }}^{\prime }\subseteq {\mathbb{R}}^n}g(x,y,u(\xi (y)),v(\xi (y)))dy,\right.\\ \left.(Tu)(x){\int }_0^{\mathrm{\infty }}V(t,s,u(x))ds\right),\\ v(x)=F\left(x,h(x,v(x),u(x)){\int }_{{\mathrm{\Omega }}^{\prime }\subseteq {\mathbb{R}}^n}g(x,y,v(\xi (y)),u(\xi (y)))dy,\right.\\ \left.(Tv)(x){\int }_0^{\mathrm{\infty }}V(t,s,v(x))ds\right).\end{array}\right.\end{array}$(1.1)

The above system of integral equations is more general than the integral equation studied by Allahyari (2018).

2. Preliminaries

Here, we recall some facts which will be used in our main results. Let $E$ be a real Banach space with norm $\parallel .\parallel$ and zero element $\theta$. Besides, we suppose $\bar{X}$ and Convc(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 2.1 ((Bannas & Goebel, 1980)). A mapping $\mu :{\mathfrak{M}}_E\to {\mathbb{R}}_{+}$ is said to be a measure of noncompactness in $E$ if it satisfies the following conditions:

1° The family $ker\mu =\left\{X\in {\mathfrak{M}}_E:\mu (X)=0\right\}$ is nonempty and $ker\mu \subset {\mathfrak{N}}_E$.

2° $X\subset Y\Rightarrow \mu (X)\le \mu (Y)$.

3° $\mu (\bar{X})=\mu (X)$.

4° $\mu (ConvX)=\mu (X)$.

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

6° If $\{X_n\}$ is a sequence of closed sets from ${\mathfrak{M}}_E$ such that $X_{n+1}\subset X_n$, for $n=1,2,\cdots$ and if $\underset{n\to \mathrm{\infty }}{lim}\mu (X_n)=0$ then $X_{\mathrm{\infty }}={\cap }_{n=1}^{\mathrm{\infty }}{X}_n\ne \mathrm{\varnothing }$.

The following theorem is basic for our main result.

Theorem 2.2 (Darbo (Darbo, 1955)). Let $C$ be a nonempty, bounded, closed and convex subset of a Banach space $E$ and $T:$ $C\to$ $C$ be a continuous mapping. Assume that there exists a constant $K\in [0,1)$ such that $\mu (TX)\le K\mu (X)$ for any nonempty subset $X$ of $C$, where $\mu$ is a $(M.N.C)$ defined in $E$. Then $T$ has at least a fixed point in $C$.

Definition 2.3. (Bhaskar & Lakshmikantham, 2006) An element $(x,y)\in X\times X$ is called coupled fixed point of the mapping $F:X\times X\to X$ if $F(x,y)=x$ and $F(y,x)=y$.

Theorem 2.4. (Aghajani & Allahyari, 2014) Suppose ${\mu }_1,{\mu }_2,\dots ,{\mu }_n$ are measures of noncompactness in Banach spaces $E_1,E_2,\dots ,E_n$ respectively. Moreover assume that the function $F:{\mathbb{R}}_{+}^n\to {\mathbb{R}}_{+}$ is convex and $F(x_1,\dots ,x_n)=0$ if and only if $x_i=0$ for $i=1,2,\dots ,n$. Then

$\tilde{\mu }(X)=F({\mu }_1(X_1),{\mu }_2(X_2),\dots ,{\mu }_n(X_n)),$

defines a measure of noncompactness in $E_1\times E_2\times \dots \times E_n$ where $X_i$ denotes the natural projection of X into $E_i$, for $i=1,2,\dots ,n$.

Samadi (Samadi, 2018) obtained the following result.

Theorem 2.5. Let $C$ be a nonempty bounded, closed and convex subset of a Banach space $E$. Assume $T:C\times C\to C$ be a continuous operator satisfying

(2.2) $\theta (\mu (T(X_1\times X_2)))+f(\mu (T(X_1\times X_2)))\le f(\mu (X_1)+\mu (X_2))$(2.2)

for all nonempty subsets $X_1,X_2\subseteq C$, where $\mu$ is an arbitrary measure of noncompactness defined in $E$ and $(\theta ,f)\in \mathrm{\Delta }$. Then $T$ has at least a coupled fixed point.

Let $L^{\mathrm{\infty }}({\mathbb{R}}^n)$ denote the space of all Lebesgue measurable functions on ${\mathbb{R}}^n$ with the standard norm

$\parallel f{\parallel }_{\mathrm{\infty }}=inf\{C>0:|f(t)|\le Ca.e.on{\mathbb{R}}^n\}.$

The function $f:\mathrm{\Omega }\to \mathbb{R}$ is said to belong to $L_{loc}^{\mathrm{\infty }}({\mathbb{R}}^n)$ if $f{\chi }_K\in L^{\mathrm{\infty }}(\mathrm{\Omega })$ for every compact set $K$ contained in $\mathrm{\Omega }$, where $\mathrm{\Omega }$ is an open subset of ${\mathbb{R}}^n$. Allahyari (Allahyari, 2018) proved the following theorems to characterize the compact subsets of $L^{\mathrm{\infty }}({\mathbb{R}}^n)$.

Theorem 2.6. Let $\mathcal{B}$ be a bounded set in $L^{\mathrm{\infty }}({\mathbb{R}}^n)$. Then $\mathcal{B}$ is relatively compact if the following conditions are satisfied:

(i) $\underset{h\to 0}{lim}\parallel \tau { }_{h}f-f{\parallel }_{L^{\mathrm{\infty }}({\bar{B}}_T)}=0$ uniformly with respect to $f\in \mathcal{B}$ for any $T>0$, where ${\tau }_{h}f(x)=f(x+h)$.

(ii) For $\epsilon >0$ there is some $T>0$ so that for every $f,g\in \mathcal{B}$

We recall the Euclidean norm on the space ${\mathbb{R}}^n$ to be

$\parallel x\parallel ={\left(\sum _{i=1}^n(x_i^2)\right)}^{\frac{1}{2}},$

for $x=(x_1,x_2,\dots ,x_n)$.

Theorem 2.7 ((Allahyari, 2018)). Let $\mathcal{B}$ be a bounded set in $L^{\mathrm{\infty }}({\bar{B}}_T)$. Then $\mathcal{B}$ is relatively compact if and only if

(2.3) $\underset{h\to 0}{lim}\parallel {\tau }_{h}f-f{\parallel }_{L^{\mathrm{\infty }}({\bar{B}}_T)}=0$(2.3)

uniformly with respect to $f\in \mathcal{B}$, where $\tau { }_{h}f(x)=f(x+h)$.

Now we recall the definition of a measure of noncompactness in $L^{\mathrm{\infty }}({\mathbb{R}}^n)$ which has been presented in (Allahyari, 2018).

Suppose $X$ be a bounded subset of the space $L^{\mathrm{\infty }}({\mathbb{R}}^n)$. For $x\in X$ and $\epsilon >0$. Let us denote

$\begin{array}{rl}& {\omega }^T(f,\epsilon )=sup\{\parallel \tau { }_{h}f-f{\parallel }_{L^{\mathrm{\infty }}({\bar{B}}_T)}:\parallel h\parallel <\epsilon \},\\ & {\omega }^T(X,\epsilon )=sup\{{\omega }^T(f,\epsilon ):f\in X\},\\ & {\omega }^T(X)=\underset{\epsilon \to 0}{lim}{\omega }^T(X,\epsilon ),\\ & \omega (X)=\underset{T\to \mathrm{\infty }}{lim}{\omega }^T(X),\end{array}$

and

$\begin{array}{rl}& d(X)=\underset{T\to \mathrm{\infty }}{lim}sup\{\parallel f-g{\parallel }_{L^{\mathrm{\infty }}({\mathbb{R}}^n\mathrm{\setminus }{\bar{B}}_T)}:f,g\in X\},\\ & {\omega }_0(X)=\omega (X)+d(X).\end{array}$

From (Allahyari, 2018) we know that ${\omega }_0$ is a measure of noncompactness in the space $L^{\mathrm{\infty }}({\mathbb{R}}^n).$

The following definition will be used in the next section.

Definition 2.8. A function $f:\mathrm{\Omega }\times {\mathbb{R}}^n\to \mathbb{R}$ is said to have the Caratheodory conditions if

(i) for all $u\in {\mathbb{R}}^n$ the function $x\to f(x,u)$ is measurable on $\mathrm{\Omega }$.

(ii) for almost all $x\in {\mathbb{R}}^n$ the function $u\to f(x,u)$ is continuous on ${\mathbb{R}}^n$

3. Application

In this section, as an application of Theorem 2.5, we study the solvability ofEquation (1.1) on the space $L^{\mathrm{\infty }}(\mathrm{\Omega })$. We need the following conditions:

(H₁) $\xi :{\mathrm{\Omega }}^{\prime }\to \mathrm{\Omega }$ is a measurable function $({\mathrm{\Omega }}^{\prime }\subseteq {\mathbb{R}}^n).$

(H₂) $F:\mathrm{\Omega }\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ satisfies the Caratheodory conditions and $F(.,0,0)\in L^{\mathrm{\infty }}(R^n)$. Besides, there exist positive real numbers $\tau >0$ such that

$|F(x,u_1,v_1)-F(x,u_2,v_2)|\le e^{-\tau }(|u_1-u_2|+|v_1-v_2|)a.e.x\in \mathrm{\Omega }.$

(H₃) $g:\mathrm{\Omega }\times {\mathrm{\Omega }}^{\prime }\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ satisfies the Caratheodory conditions, $g\in L_{loc}^{\mathrm{\infty }}(\mathrm{\Omega }\times {\mathrm{\Omega }}^{\prime }\times \mathbb{R}\times \mathbb{R})$ and there exists nondecreasing function $b:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ such that for all $r>0$ and $u,v\in L^{\mathrm{\infty }}(\mathrm{\Omega })$ with $\parallel u{\parallel }_{\mathrm{\infty }}\le r,\parallel v{\parallel }_{\mathrm{\infty }}\le r$ we have

(3.4) $ess\underset{x\in \mathrm{\Omega }}{sup}|h(x,u(x),v(x))||{\int }_{{\mathrm{\Omega }}^{\prime }}g(x,y,u(\xi (y)),v(\xi (y)))dy|\le b(r).$(3.4)

Besides, for all $r\in {\mathbb{R}}_{+}$

$\underset{T\to \mathrm{\infty }}{lim}ess\underset{\parallel x\parallel >T}{sup}\left|{\int }_{{\mathrm{\Omega }}^{\prime }}g(x,y,u(\xi (y)),v(\xi (y)))dy-g(x,y,v(\xi (y)),u(\xi (y)))dy\right|=0$

uniformly with respect to $u,v\in L^{\mathrm{\infty }}(\mathrm{\Omega })$ such that $\parallel u{\parallel }_{\mathrm{\infty }}\parallel v{\parallel }_{\mathrm{\infty }}\le r.$ Moreover, we have

$\underset{T\to \mathrm{\infty }}{lim}ess\underset{x\in \mathrm{\Omega }}{sup}{\int }_{{\mathrm{\Omega }}^{\prime }-\overline{B_T}}|g(x,y,u(\xi (y)),v(\xi (y)))|dy=0.$

(H₄) $T:L^{\mathrm{\infty }}({\mathbb{R}}^n)\to L^{\mathrm{\infty }}({\mathbb{R}}^n)$ is a continuous operator such that

$\begin{array}{ccc}|(Tu)(x)|& \le & |u(x)|,\\ |(Tu)(x)-(Tv)(x)|& \le & |u(x)-v(x)|,\\ |(Tu)(x_1)-(Tu)(x_2)|& \le & |u(x_1)-u(x_2)|,\end{array}$

for all $u,v\in L^{\mathrm{\infty }}({\mathbb{R}}^n)$ and $x_1,x_2,x\in {\mathbb{R}}^n$.

(H5) $V:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times \mathbb{R}\to \mathbb{R}$ is a continuous function and there exists a continuous function $k:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ such that the function $s\to k(t,s)$ is integrable over ${\mathbb{R}}_{+}$ and the following conditions hold:

$\begin{array}{ccc}|V(t,s,x)|& \le & k(t,s)||x|,\\ |V(t,s,x)-|V(t,s,y)|& \le & k(t,s)|x-y|,\end{array}$

for all $t,s\in {\mathbb{R}}_{+}$ and $x,y\in \mathbb{R}$. Moreover, assume that

$M=\underset{t\in {\mathbb{R}}_{+}}{sup}{\int }_0^{\mathrm{\infty }}k(t,s)ds.$

(H₆) $h:\mathrm{\Omega }\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ satisfies the Caratheodory conditions and $h(.,0,0)\in L^{\mathrm{\infty }}({\mathbb{R}}^n)$. Besides, there exists positive real number $\tau >0$ such that

$|h(x,x_1,x_2)-h(x,y_1,y_2)|\le e^{-\tau }(|x_1-y_1|+|x_2-y_2|),a.eonx\in \mathrm{\Omega }.$

(H₇) There exists a positive solution $r_0$ of the inequality

$e^{\tau }-\tau b\left(r\right)+M{e}^{-\tau }{r}^{2+}∥F\left(.,0,0\right)∥\mathrm{\infty }\le r$

Theorem 3.1. Under the assumptions $({\mathrm{H}}_1)-({\mathrm{H}}_7)$, Equation (1.1) has at least one solution in the space $L^{\mathrm{\infty }}(\mathrm{\Omega })\times L^{\mathrm{\infty }}(\mathrm{\Omega })$.

Proof. Let us define the operator $G$ on the space $L^{\mathrm{\infty }}(\mathrm{\Omega })\times L^{\mathrm{\infty }}(\mathrm{\Omega })$ by

$\begin{array}{rl}& G(u,v)(x)=F\left(x,h(x,u(x),v(x)){\int }_{{\mathrm{\Omega }}^{\prime }\subseteq {\mathbb{R}}^n}g(x,y,u(\xi (y)),v(\xi (y)))dy,(Tu)(x){\int }_0^{\mathrm{\infty }}V(t,s,u(x))ds\right).\end{array}$

The space $L^{\mathrm{\infty }}(\mathrm{\Omega })\times L^{\mathrm{\infty }}(\mathrm{\Omega })$ is equipped with the norm

$\parallel (x,y)\parallel =\parallel x{\parallel }_{\mathrm{\infty }}+\parallel y{\parallel }_{\mathrm{\infty }}.$

We prove that $G(\overline{B_{r_0}}\times \overline{B_{r_0}})\subseteq \overline{B_{r_0}}$. Since the function $F$ satisfies the Caratheodory conditions, so $G(u,v)$ is measurable for any $u,v\in L^{\mathrm{\infty }}(\mathrm{\Omega })$. In view of our assumptions, we obtain the following inequality

$\begin{array}{rl}& |G(u,v)(x)|\\ & \le |F(x,h(x,u(x),v(x)){\int }_{{\mathrm{\Omega }}^{\prime }}g(x,y,u(\xi (y)),v(\xi (y)))dy,\\ & (Tu)(x){\int }_0^{\mathrm{\infty }}V(t,s,u(x))ds)-F(x,0,0)|\\ & +|F(x,0,0)|\le e^{-\tau }|h(x,u(x),v(x)){\int }_{{\mathrm{\Omega }}^{\prime }\subseteq {\mathbb{R}}^n}g(x,y,u(\xi (y)),v(\xi (y)))dy|\\ & +e^{-\tau }|(Tu)(x){\int }_0^{\mathrm{\infty }}V(t,s,u(x))ds|+|F(x,0,0)|\\ & \le e^{-\tau }b(\parallel u{\parallel }_{\mathrm{\infty }})+e^{-\tau }\parallel u{\parallel }_{\mathrm{\infty }}^{2}M+\parallel F(.,0,0){\parallel }_{\mathrm{\infty }}.\end{array}$

Accordingly,

(3.5) $\begin{array}{cc}\parallel G(u,v){\parallel }_{\mathrm{\infty }}& \le e^{-\tau }b(\parallel u{\parallel }_{\mathrm{\infty }})+e^{-\tau }\parallel u{\parallel }_{\mathrm{\infty }}^{2}M+\parallel F(.,0,0){\parallel }_{\mathrm{\infty }}.\end{array}$(3.5)

Consequently, from (3.5) and condition $({\mathrm{H}}_7)$ we infer that $G(\overline{B_{r_0}}\times \overline{B_{r_0}})\subseteq \overline{B_{r_0}}$. Now we prove that the operator $G$ is contnuous on $\overline{B_{r_0}}\times \overline{B_{r_0}}$. To do so, let us $\epsilon >0$ and take arbitrary $(x,y),(u,v)\in \overline{B_{r_0}}\times \overline{B_{r_0}}$ such that $\parallel (u_1,v_1)-(u_2,v_2){\parallel }_{\overline{B_{r_0}}\times \overline{B_{r_0}}}<\epsilon$. Thus, we have

(3.6) $\begin{array}{c}|G(u_1,v_1)(x)-G(u_2,v_2)(x)|\\ \le e^{-\tau }|h(x,u_1(x),v_1(x))||{\int }_{{\mathrm{\Omega }}^{\prime }}g(x,y,u_1(\xi (y)),v_1(\xi (y)))-g(x,y,u_2(\xi (y)),v_2(\xi (y)))dy|\\ +e^{-\tau }|h(x,u_1(x),v_1(x))-h(x,u_2(x),v_2(x))||{\int }_{{\mathrm{\Omega }}^{\prime }}g(x,y,u_2(\xi (y)),v_2(\xi (y)))dy|\\ +e^{-\tau }|(T{u}_1)(x)||{\int }_0^{\mathrm{\infty }}V(t,s,u_1(x))-V(t,s,u_2(x))ds|\\ +e^{-\tau }|(T{u}_1)(x)-(T{u}_2)(x)||{\int }_0^{\mathrm{\infty }}V(t,s,u_2(x))ds|\\ \le (e^{-2\tau }\parallel u_1{\parallel }_{\mathrm{\infty }}+e^{-2\tau }\parallel v_1{\parallel }_{\mathrm{\infty }}){\int }_{\overline{B_T}}|g(x,y,u_1(\xi (y)),v_1(\xi (y)))-g(x,y,u_2(\xi (y)),v_2(\xi (y)))|dy\\ +(e^{-2\tau }\parallel u_1{\parallel }_{\mathrm{\infty }}+e^{-2\tau }\parallel v_1{\parallel }_{\mathrm{\infty }}){\int }_{{\mathrm{\Omega }}^{\prime }-\overline{B_T}}|g(x,y,u_1(\xi (y)),v_1(\xi (y)))-g(x,y,u_2(\xi (y)),v_2(\xi (y)))|dy\\ +e^{-2\tau }\parallel h(x,0,0){\parallel }_{\mathrm{\infty }}{\int }_{\overline{B_T}}|g(x,y,u_1(\xi (y)),v_1(\xi (y)))-g(x,y,u_2(\xi (y)),v_2(\xi (y)))|dy\\ +e^{-2\tau }\parallel h(x,0,0){\parallel }_{\mathrm{\infty }}{\int }_{{\mathrm{\Omega }}^{\prime }-\overline{B_T}}|g(x,y,u_1(\xi (y)),v_1(\xi (y)))-g(x,y,u_2(\xi (y)),v_2(\xi (y)))|dy\\ +e^{-\tau }M\parallel u_1{\parallel }_{\mathrm{\infty }}\parallel u_1-u_2{\parallel }_{\mathrm{\infty }}+e^{-\tau }\parallel u_1-u_2{\parallel }_{\mathrm{\infty }}M\parallel u_2{\parallel }_{\mathrm{\infty }}\\ \le (e^{-2\tau }\parallel u_1{\parallel }_{\mathrm{\infty }}+e^{-2\tau }\parallel v_1{\parallel }_{\mathrm{\infty }})m(\overline{B_T})\omega (\epsilon )\\ +(e^{-2\tau }\parallel u_1{\parallel }_{\mathrm{\infty }}+e^{-2\tau }\parallel v_1{\parallel }_{\mathrm{\infty }})2ess\underset{x\in \mathrm{\Omega }}{sup}{\int }_{{\mathrm{\Omega }}^{\prime }-\overline{B_T}}|g(x,y,u_1(\xi (y)),v_1(\xi (y)))|dy\\ +e^{-2\tau }\parallel h(x,0,0){\parallel }_{\mathrm{\infty }}m(\overline{B_T})\omega (\epsilon )\\ +e^{-2\tau }\parallel h(x,0,0){\parallel }_{\mathrm{\infty }}2ess\underset{x\in \mathrm{\Omega }}{sup}{\int }_{{\mathrm{\Omega }}^{\prime }-\overline{B_T}}||g(x,y,u_1(\xi (y)),v_1(\xi (y)))|dy\\ +e^{-\tau }\parallel u_1{\parallel }_{\mathrm{\infty }})M\parallel u_1-u_2{\parallel }_{\mathrm{\infty }}+e^{-\tau }\parallel u_1-u_2{\parallel }_{\mathrm{\infty }}M\parallel u_2{\parallel }_{\mathrm{\infty }}\end{array}$(3.6)

By applying condition $({\mathrm{H}}_3)$, there exist $T_1,T_2>0$ such that for $T=max\{T_1,T_2\}$ we have

(3.7) $\begin{array}{c}ess{sup}_{\parallel x\parallel >T}\left|{\int }_{{\mathrm{\Omega }}^{\prime }}g(x,y,u_1(\xi (y)),v_1(\xi (y)))-g(x,y,v_1(\xi (y)),u_1(\xi (y)))dy\right|<\epsilon ,\\ ess{sup}_{x\in \mathrm{\Omega }}{\int }_{{\mathrm{\Omega }}^{\prime }-\overline{B_T}}|g(x,y,u(\xi (y)),v(\xi (y)))|dy<\epsilon .\end{array}$(3.7)

From (3.6), we have

(3.8) $\begin{array}{c}ess{sup}_{\parallel x\parallel >T}|G(u_1,v_1)(x)-G(u_2,v_2)(x)|\\ \le (e^{-2\tau }\parallel u_1{\parallel }_{\mathrm{\infty }}+e^{-2\tau }\parallel v_1{\parallel }_{\mathrm{\infty }})\epsilon +e^{-2\tau }\parallel h(x,0,0){\parallel }_{\mathrm{\infty }}\epsilon \\ +e^{-\tau }M\parallel u_1{\parallel }_{\mathrm{\infty }})\epsilon +e^{-\tau }\parallel u_2{\parallel }_{\mathrm{\infty }}M\epsilon \end{array}$(3.8)

Now for almost all $x\in \overline{B_T}{\bigcap }_{}\mathrm{\Omega }$, we have

(3.9) $\begin{array}{c}|G(u_1,v_1)(x)-G(u_2,v_2)(x)|\\ \le (e^{-2\tau }\parallel u_1{\parallel }_{\mathrm{\infty }}+e^{-2\tau }\parallel v_1{\parallel }_{\mathrm{\infty }})m(\overline{B_T})\omega (\epsilon )\\ +(e^{-2\tau }\parallel u_1{\parallel }_{\mathrm{\infty }}+e^{-2\tau }\parallel v_1{\parallel }_{\mathrm{\infty }})2ess{sup}_{x\in \mathrm{\Omega }}{\int }_{{\mathrm{\Omega }}^{\prime }-\overline{B_T}}g(x,y,u_1(\xi (y)),v_1(\xi (y)))dy\\ +e^{-2\tau }\parallel h(x,0,0){\parallel }_{\mathrm{\infty }}m(\overline{B_T})\omega (\epsilon )\\ +e^{-2\tau }\parallel h(x,0,0){\parallel }_{\mathrm{\infty }}2ess{sup}_{x\in \mathrm{\Omega }}{\int }_{{\mathrm{\Omega }}^{\prime }-\overline{B_T}}g(x,y,u_1(\xi (y)),v_1(\xi (y)))dy\\ +e^{-\tau }\parallel u_1{\parallel }_{\mathrm{\infty }}M\epsilon +e^{-\tau }\parallel u_2{\parallel }_{\mathrm{\infty }}M\epsilon \end{array}$(3.9)

where

$\begin{array}{rl}& \omega (\epsilon )=inf\{C\ge 0:|g(x,y,u_1,v_1)-g(x,y,u_2,v_2)|\le Ca.eonx\in \overline{B_T}\subseteq \mathrm{\Omega },\\ & \left.y\in \overline{B_T}\subseteq {\mathrm{\Omega }}^{\prime },u_1,v_1,u_2,v_2\in [-r_0,r_0],|u_1-v_1|<\epsilon ,|u_2-v_2|<\epsilon \right\}\end{array}$

Now by applying the Caratheodory conditions for $g$ on the compact set $\overline{B_T}\times \overline{B_T}\times [-r_0,r_0]\times [-r_0,r_0]$ and Theorem 2.7, we infer that $\omega (\epsilon )\to 0$ as $\epsilon \to 0$. Combining (7), (8) and (9) we conclude that $G$ is continuous on $\overline{B_{r_0}}\times \overline{B_{r_0}}$. Now we prove that for any $X_1,X_2\subseteq \overline{B_{r_0}}$ we have

$\theta (\mu (G(X_1\times X_2)))+f(\mu (G(X_1\times X_2)))\le f(\mu (X_1)\times \mu (X_2)).$

To do so, assume that $T>0$ and $\epsilon >0$ are arbitrary constants. Hence, for almost all $x\in \mathrm{\Omega }$, $\parallel h\parallel \le \epsilon$ and $(u,v)\in X_1\times X_2$, we have

(3.10) $\begin{array}{c}|G(u,v)(x)-G(u,v)(x+h)|\\ \le |F(x,h(x,u(x),v(x)){\int }_{{\mathrm{\Omega }}^{\prime }}g(x,y,u(\xi (y)),v(\xi (y)))dy,(Tu)(x){\int }_0^{\mathrm{\infty }}V(t,s,u(x))ds)\\ -F(x+h,h(x,u(x),v(x)){\int }_{{\mathrm{\Omega }}^{\prime }}g(x,y,u(\xi (y)),v(\xi (y)))dy,(Tu)(x){\int }_0^{\mathrm{\infty }}V(t,s,u(x))ds)|\\ +|F(x+h,h(x,u(x),v(x)){\int }_{{\mathrm{\Omega }}^{\prime }}g(x,y,u(\xi (y)),v(\xi (y)))dy,(Tu)(x){\int }_0^{\mathrm{\infty }}V(t,s,u(x))ds)\\ -F(x+h,h(x+h,u(x+h),v(x+h)){\int }_{{\mathrm{\Omega }}^{\prime }}g(x+h,y,u(\xi (y)),v(\xi (y)))dy,\\ (Tu)(x+h){\int }_0^{\mathrm{\infty }}V(t,s,u(x+h))ds)|\\ \le {\omega }_{r_0}^T(F,\epsilon )+e^{-\tau }|h(x,u(x),v(x)){\int }_{{\mathrm{\Omega }}^{\prime }}g(x,y,u(\xi (y)),v(\xi (y)))dy\\ -h(x+h,u(x+h)){\int }_{{\mathrm{\Omega }}^{\prime }}g(x+h,y,u(\xi (y)),v(\xi (y)))dy|\\ +e^{-\tau }|(Tu)(x){\int }_0^{\mathrm{\infty }}V(t,s,u(x))ds-(Tu)(x+h){\int }_0^{\mathrm{\infty }}V(t,s,u(x+h))ds|\\ \le {\omega }_{r_0}^T(F,\epsilon )+e^{-\tau }{\omega }_{r_0}^T(h,\epsilon )|{\int }_{\overline{B_T}}g(x,y,u(\xi (y)),v(\xi (y)))-g(x+h,y,u\xi (y)),v\xi (y)))dy|\\ +e^{-\tau }{\omega }_{r_0}^T(h,\epsilon )|{\int }_{{\mathrm{\Omega }}^{\prime }-\overline{B_T}}g(x,y,u(\xi (y)),v(\xi (y)))-g(x+h,y,u(\xi (y)),v(\xi (y)))dy|\\ +e^{-\tau }|(Tu)(x)||{\int }_0^{\mathrm{\infty }}V(t,s,u(x))-V(t,s,u(x+h))ds|\\ +e^{-\tau }|(Tu)(x)-(Tu)(x+h)||{\int }_0^{\mathrm{\infty }}V(t,s,u(x+h))ds|\\ \le {\omega }_{r_0}^T(F,\epsilon )+e^{-\tau }{\omega }_{r_0}^T(h,\epsilon )m(\overline{B_T}){\omega }_{r_0}^T(g,\epsilon )+e^{-\tau }{\omega }_{r_0}^T(h,\epsilon )2esssup|{\int }_{{\mathrm{\Omega }}^{\prime }-\overline{B_T}}g(x,y,u(\xi (y)))dy|\\ +e^{-\tau }{\omega }^T(u,\epsilon )\parallel u{\parallel }_{\mathrm{\infty }}M+e^{-\tau }M{\omega }^T(u,\epsilon )\parallel u{\parallel }_{\mathrm{\infty }},\end{array}$(3.10)

where

$\begin{array}{rl}& {\omega }_{r_0}^T(h,\epsilon )=inf\{C\ge 0:|h(x,u,v)-h(x+h,v,v)|\le Ca.eonx\in \overline{B_T},\\ & \parallel h\parallel \le \epsilon ,\parallel u\parallel \le r_0,\parallel v\parallel \le r_0\},\end{array}$

$\begin{array}{rl}& {\omega }_{r_0}^T(F,\epsilon )=inf\{C\ge 0:|F(x,u,v)-F(x+h,u,v)|\le Ca.eonx\in \overline{B_T},\\ & \parallel h\parallel \le \epsilon ,|u|\le b(r_0),|v|<M{r}_0^2\},\end{array}$

$\begin{array}{rl}& {\omega }_{r_0}^T(g,\epsilon )=inf\{C\ge 0:|g(x,y,u,v)-g(x+h,y,u,v)|\le Ca.eonx\in \overline{B_T}\subseteq \mathrm{\Omega },\\ & y\in \overline{B_T}\subseteq {\mathrm{\Omega }}^{\prime },\parallel h\parallel \le \epsilon ,|v|<r_0,|u|\le r_0\}.\end{array}$

Thus, we have

(3.11) $\begin{array}{rl}& {\omega }^T(G(X_1\times X_2),\epsilon )\le {\omega }_{r_0}^T(F,\epsilon )+e^{-}\tau {\omega }_{r_0}^T(h,\epsilon )m(\overline{B_T})\omega {r_0}^T(g,\epsilon )\\ & +e^{-}\tau {\omega }_{r_0}^T(h,\epsilon )2esssup{\int }_{{\mathrm{\Omega }}^{\prime }-\overline{B_T}}g(x,y,u(\xi (y))),v(\xi (y)))dy|\\ & +e^{-}\tau \omega T(X_1,\epsilon )\parallel u{\parallel }_{\mathrm{\infty }}M+e^{-}\text{tauM}{\omega }^T(X_1,\epsilon )\parallel u{\parallel }_{\mathrm{\infty }},\end{array}$(3.11)

The functions $F$, $g$ and $h$ satisfy the Caratheodory conditions on the compact set $\overline{B_T}\times [-b(r_0),b(r_0)]\times [-M{r}_0^2,M{r}_0^2]$, $\overline{B_T}\times \overline{B_T}\times [-r_0,r_0]\times [-r_0,r_0]$ and $\overline{B_T}\times [-r_0,r_0]\times [-r_0,r_0]$, respectively. So, ${\omega }_{r_0}^T(F,\epsilon )\to 0$, ${\omega }_{r_0}^T(g,\epsilon )\to 0$ and ${\omega }_{r_0}^T(h,\epsilon )\to 0$ as $\epsilon \to 0$. Accordingly, thanks to our assumptions, we have

(3.12) ${\omega }^T(G(X_1\times X_2))\le e^{-\tau }{r}_{0}M{\omega }^T(X_1,\epsilon )+e^{-\tau }M{r}_0{\omega }^T(X_1,\epsilon ).$(3.12)

Taking $T\to \mathrm{\infty }$ we get

(3.13) $\omega (G(X_1\times X_2)\le 2e^{-\tau }M{r}_0({\omega }_0(X_1)+{\omega }_0(X_2))$(3.13)

Moreover, for all $(u_1,v_1),(u_2,v_2)\in X_1\times X_2$, we have

(3.14) $\begin{array}{rl}& ess{sup}_{\parallel x\parallel >T}|G(u_1,v_1)(x)-G(u_2,v_2)(x)|\\ & \le e^{-\tau }|h(x,u_1(x),v_1(x)){\int }_{{\mathrm{\Omega }}^{\prime }}g(x,y,u_1(\xi (y)),v_1(\xi (y)))dy\\ & -h(x,u_2(x),v_2(x)){\int }_{{\mathrm{\Omega }}^{\prime }}g(x,y,u_2(\xi (y)),v_2(\xi (y)))dy|\\ & +e^{-\tau }|(T{u}_1)(x){\int }_0^{\mathrm{\infty }}V(t,s,u_1(x))ds-(T{u}_2)(x){\int }_0^{\mathrm{\infty }}V(t,s,u_2(x))ds|\end{array}$(3.14)

Consequently,

(3.15) $\begin{array}{rl}& \parallel G(u_1,v_1)-G(u_2,v_2){\parallel }_{L^{\mathrm{\infty }}(\mathrm{\Omega }-\overline{B_T})}\\ & \le |h(x,u_1(x),v_1(x))|ess{sup}_{\parallel x\parallel >T}{\int }_{{\mathrm{\Omega }}^{\prime }}|g(x,y,u_1(\xi (y)),v_1(\xi (y)))\\ & -g(x,y,u_2(\xi (y)),v_2(\xi (y)))|dy\\ & +|h(x,u_1(x),v_1(x))-h(x,u_2(x),v_2(x))|ess{sup}_{\parallel x\parallel >T}{\int }_{{\mathrm{\Omega }}^{\prime }}|g(x,y,u_2(\xi (y)),v_2(\xi (y)))|dy\\ & +e^{-\tau }M{r}_0|u_1(x)-u_2(x)|+e^{-\tau }M{r}_0|u_1(x)-u_2(x)|\end{array}$(3.15)

Hence,

$\begin{array}{l}\parallel G(u_1,v_1)-G(u_2,v_2){\parallel }_{L^{\infty }(\Omega -\overline{B_T})}\\ \le |h(x,u_1(x),v_1(x))|ess{\sup }_{\parallel x\parallel T}{\displaystyle {\int }_{{\Omega }^{\prime }}g(x,y,u_1(\xi (y)),v_1(\xi (y)))}\end{array}$

$\begin{array}{rl}& -g(x,y,u_2(\xi (y)),v_2(\xi (y)))dy+|h(x,u_1(x),v_1(x))\\ & -h(x,u_2(x),v_2(x))|ess{sup}_{\parallel x\parallel >T}{\int }_{{\mathrm{\Omega }}^{\prime }}|g(x,y,u_2(\xi (y)),v_2(\xi (y)))|dy\end{array}$

(3.16) $+e^{-\tau }M{r}_0(e^{-\tau }M{r}_0\parallel u_1-u_2{\parallel }_{L^{\infty }(\Omega -\overline{B_T})}+e^{-\tau }M{r}_0|u_1(x)-u_2(x){|}_{L^{\infty }(\Omega -\overline{B_T}})$(3.16)

By applying assumption $({\mathrm{H}}_3)$ and taking $T\to \mathrm{\infty }$, we have

(3.17) $d(G(X_1\times X_2))\le 2M{r}_0{e}^{-\tau }(d(X_1)+d(X_2)).$(3.17)

Since $2M{r}_0<1$, by combining (3.13) and (3.17), we get

$\omega (G(X_1\times X_2))+d(G(X_1\times X_2))$

$\le e^{-\tau }(\omega (X_1)+\omega (X_2)+d(X_1)+d(X_2))).$

By passing to logarithms, we get

(3.18) $\tau +ln({\omega }_0(G(X_1\times X_2)))\le ln({\omega }_0(X_1\times X_2)))$(3.18)

Now applying Theorem 2.5 with $f(t)=ln(t)$ and $\theta (t)=\tau$, we obtain that there exists $u,v\in L^{\mathrm{\infty }}(\mathrm{\Omega })$ such that

(3.19) $\begin{array}{rl}& u(x)=F(x,h(x,u(x),v(x)){\int }_{{\mathrm{\Omega }}^{\prime }\subseteq {\mathbb{R}}^n}g(x,y,u(\xi (y)),v(\xi (y)))dy,\\ & (Tu)(x){\int }_0^{\mathrm{\infty }}V(t,s,u(x))ds),\\ & v(x)=F(x,h(x,v(x),u(x)){\int }_{{\mathrm{\Omega }}^{\prime }\subseteq {\mathbb{R}}^n}g(x,y,v(\xi (y)),u(\xi (y)))dy,\\ & (Tv)(x){\int }_0^{\mathrm{\infty }}V(t,s,v(x))ds).\end{array}$(3.19)

Example 3.2. Now, we study the following system of integral equations:

(3.20) $\left\{\begin{array}{c}u(x)=e^{-x_1-x_2-x_3-\tau }sin\left(e^{-\tau -x_1-x_2-x_3}sin\left(|u(x)|+|v(x)|\right)\times \right.\\ {\int }_{{\mathbb{R}}^2}arctan\left(\frac{e^{-3(|x_2|+|y_2|)-|u(y)|-|v(y)|}}{8+|y_1|+|x_1|}\right)\\ +\left.cos\left(\frac{|u(x)|}{1+|u(x)|}\right){\int }_0^{\mathrm{\infty }}\frac{e^{-s}}{1+\frac{t}{8}}sin\left(\frac{|u(x)|}{8}\right)ds\right),\\ v(x)=e^{-x_1-x_2-x_3-\tau }sin\left(e^{-\tau -x_1-x_2-x_3}sin\left(|u(x)|+|v(x)|\right)\times \right.\\ {\int }_{{\mathbb{R}}^2}arctan\left(\frac{e^{-3(|x_2|+|y_2|)-|u(y)|-|v(y)|}}{8+|y_1|+|x_1|}\right)\\ \left.+cos\left(\frac{|v(x)|}{1+|v(x)|}\right){\int }_0^{\mathrm{\infty }}\frac{e^{-}s}{1+\frac{t}{8}}sin\left(\frac{|v(x)|}{8}\right)ds\right)\end{array}\right.$(3.20)

Equation (3.20) is a special case of the functional integral Equation (1.1), where

(3.22) $\begin{array}{rl}& F(x,u,v)=e^{-x_1-x_2-x_3-\tau }sin\left(u+v\right),\\ & g(x_1,x_2,x_3,y_1,y_2,u)=arctan\left(\frac{e^{-3(|x_2|+|y_2|)-|u|-|v|}}{8+|y_1|+|x_1|}\right),\\ & h(x_1,x_2,x_3,u,v)=e^{-x_1-x_2-x_3-\tau }sin\left(|u|+|v|\right),\\ & (Tu)(x)=cos\left(\frac{|u(x)|}{1+|u(x)|}\right),\\ & \xi (y_1,y_2)=(y_1,y_2,0),\end{array}$(3.22)

$V(t,s,x)=\frac{e^{-s}}{8(1+\frac{t}{8})}\sin \left(\frac{\left|x\right|}{8}\right),(\Omega ={ℝ}^3,{\Omega }^{\prime }={ℝ}^2)$

It easily seen that the functions $\xi$ and $F$ satisfy the assumptions $({\mathrm{H}}_1)$ and $({\mathrm{H}}_2)$ respectively. On the other hand $g\in L_{loc}^{\mathrm{\infty }}({\mathbb{R}}^3\times {\mathbb{R}}^2\times \mathbb{R}\times \mathbb{R})$ such that for all $u\in L^{\mathrm{\infty }}({\mathbb{R}}^3)$ we have

(3.23) $\begin{array}{rl}& ess{sup}_{x\in {\mathbb{R}}^3}\left|{\int }_{{\mathbb{R}}^2}g(x,y,u(\xi (y)))dy\right|\\ & =ess{sup}_{x\in {\mathbb{R}}^3}\left|{\int }_{{\mathbb{R}}^2}arctan\left(\frac{e^{-3(|x_2|+|y_2|)-|u(y)|}-|v(y)|}{8+|y_1|+|x_1|}\right)dy\right|\\ & \le ess\underset{x\in {\mathbb{R}}^3}{sup}\left\{\frac{e^{-}|x_2|}{8+|x_1|}\right\}\le \frac{1}{8}=b(r)\end{array}$(3.23)

Moreover, for all $r\in {\mathbb{R}}_{+}$ we have

$\underset{T\to \mathrm{\infty }}{lim}ess{sup}_{\parallel x\parallel >T}\left|{\int }_{{\mathbb{R}}^2}arctan\left(\frac{e^{-3(|x_2|+|y_2|)-|u(y)|}}{8+|y_1|+|x_1|}\right)-arctan\left(\frac{e^{-3(|x_2|+|y_2|)-|v(y)|}}{8+|y_1|+|x_1|}\right)dy\right|=0$

uniformly with respect to $x,y\in L^{\mathrm{\infty }}({\mathbb{R}}^3)$ such that $\parallel x{\parallel }_{\mathrm{\infty }}\parallel y{\parallel }_{\mathrm{\infty }}\le r.$ Moreover,

$\underset{T\to \infty }{\lim }ess{\sup }_{x\in {ℝ}^3}\text{ |}{\displaystyle {\int }_{{ℝ}^2-\overline{B_T}}\arctan }\left(\frac{e^{-3(|x_2|+|y_2|)-|u(y)|-|v(y)|}}{8+\left|y_1\right|+\left|x_1\right|}\right)dy=0.$

Hence condition $({\mathrm{H}}_3)$ is satisfied. In this example $(Tx)(t)=cos\left(\frac{|x(t)|}{1+|x(t)|}\right)$ verifies assumption $({\mathrm{H}}_4)$. Now notice that the function $V$ is continuous and satisfies the assumption $({\mathrm{H}}_5)$ with $k(t,s)=\frac{e^{-}s}{8(1+\frac{t}{8})}$ and $M=\frac{1}{8}$. On the hand obviously the assumption $({\mathrm{H}}_6)$ holds. Finally, the existing inequalities in the assumption $({\mathrm{H}}_7)$ have the form

$\frac{e^{-\tau }}{8}+\frac{e^{-\tau }}{8}{r}^2\le r$

It is easily seen that the last inequality has a positive solution. For example $r_0=1$. So, $2M{r}_0^2<1$ We see that all assumptions of Theorem 3.1 are satisfied. Consequently, from Theorem 3.1 the integral equation (11) has a solution in the space $L^{\mathrm{\infty }}({\mathbb{R}}^3)$.

Additional information

Funding

This work was supported by the Islamic Azad University [1534189084]

Notes on contributors

Ayub Samadi

The research field of authors is fixed point theory with its applications. They are assistant professors and faculty members in Islamic Azad University. The authors have studied the existence of solutions for a class of the systems of integral equations. Also, they give an application of the obtained results. They applied the technique of measure of noncompactness in the main results.