An extension of Darbo’s theorem and its application to existence of solution for a system of integral equations

Abstract

In this paper, we extend Darbo’s fixed point theorem in Banach space via the concept of the class of operators O(f;.) and obtain a tripled fixed point theorem. The main tool applied in our investigation is the technique of measures of noncompactness. Finally, as an application of our obtained results, we analyze the existence of solutions for a system of integral equations.

Keywords:

• measure of noncompactness
• fixed point theorem
• integral equations

Subjects:

• 47H09
• 47H10

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 of integral equations. In this paper, we extend the Darbo’s fixed point theorem and analyze the existence of solutions for a system of integral equations.

1. Introduction and preliminaries

The concept of a measure of noncompactness (MNC) was introduced by Kuratowski (1930). These measures are very useful tools in the wide area of nonlinear functional analysis. Darbo’s fixed point theorem (Darbo, 1955) which ensures the existence of fixed point is an important application of this measure, since it generalizes both the classical Schauder fixed point and Banach contraction principle. Thereafter, many papers have been studied in the MNC and it’s application, see for example (Aghajani et al., 2014; Aghajani, Banás, & Jalilian, 2011; Aghajani, Banás, & Sabzali, 2013; Banaei, 2018; Banaei, & Ghaemi, 2019; Banaei, Ghaemi, & Saadati, 2017; Habib, & Benaicha, 2018; Ilkhan, & Kara; Mondal, 2019) . The aim of this paper is to generalize the Darbo’s fixed point theorem via the concept of the class of operators $O(f;.)$ and extend the results of the theorems in, (Samadi, & Ghaemi, 2014). In this work, first, some definitions and results are recalled. In the second section, main results and an extension of Darbo fixed point theorem are presented. finally, we apply our extension to study the existence of solutions for the system

(1.1) $\left\{\begin{array}{c}x(t,s,r)=a(t,s,r)+f(t,s,r,x(t,s,r),y(t,s,r),z(t,s,r))+g(t,s,r,x(t,s,r),y(t,s,r),z(t,s,r))\\ {\int }_0^{{\alpha }_1(t)}{\int }_0^{{\alpha }_2(s)}{\int }_0^{{\alpha }_3(r)}k(t,s,r,u,v,w,x(u,v,w),y(u,v,w),z(u,v,w)dudvdw\\ y(t,s,r)=a(t,s,r)+f(t,s,r,y(t,s,r),x(t,s,r),z(t,s,r))+g(t,s,r,y(t,s,r),x(t,s,r),z(t,s,r))\\ {\int }_0^{{\alpha }_1(t)}{\int }_0^{{\alpha }_2(s)}{\int }_0^{{\alpha }_3(r)}k(t,s,r,u,v,w,y(u,v,w),x(u,v,w),z(u,v,w)dudvdw\\ z(t,s,r)=a(t,s,r)+f(t,s,r,z(t,s,r),y(t,s,r),x(t,s,r))+g(t,s,r,z(t,s,r),y(t,s,r),x(t,s,r))\\ {\int }_0^{{\alpha }_1(t)}{\int }_0^{{\alpha }_2(s)}{\int }_0^{{\alpha }_3(r)}k(t,s,r,u,v,w,z(u,v,w),y(u,v,w),x(u,v,w)dudvdw\end{array}\right.$(1.1)

where $t,s,r\in {\mathbb{R}}_{+},x,y,z\in E$.

Now, we introduce some notations and definitions which are used throughout this paper. Let $\mathbb{R}$ be the set of real numbers, ${\mathbb{R}}_{+}=[0,\mathrm{\infty })$ and $(E,\parallel .\parallel )$ be a real Banach space with the zero element 0. We write $\bar{B}(x,r)$ to denote the closed ball centered at $x$ with radius $r$. If $X$ be a nonempty subset of $E$ then the symbols $\bar{X}$ and $ConvX$ stand for the closure and closed convex hull of $X$, respectively. Moreover, ${\mathfrak{M}}_E$ is the family of nonempty bounded subset of $E$ and ${\mathfrak{N}}_E$ denote its subfamily consisting of all relatively compact sets.

Definition 1.1 (Banas & Goebel, 1980). A mapping $\mu :{\mathfrak{M}}_E\to [0,\mathrm{\infty })$ is said to be a measure of noncompactness in $E$ if it satisfies the following conditions:

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

$2\ast$ $X\subseteq Y\Rightarrow \mu (X)\le \mu (Y)$.

$3\ast$ $\mu (\bar{X})=\mu (X)$.

$4\ast$ $\mu (ConvX)=\mu (X)$.

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

$6\ast$ If $(X_n)\in {\mathfrak{M}}_E$ such that $X_{n+1}\subseteq X_{n}forn=1,2,3,...$ and ${lim}_{n\to \mathrm{\infty }}\mu (X_n)=0$, then $X_{\mathrm{\infty }}={\cap }_{n=1}^{\mathrm{\infty }}{X}_n$ is nonempty.

The following concept of $O(f;.)$ was given by Altun and Turkoglu (Altun & Turkoglu, 2007).

Let $F([0,\mathrm{\infty }))$ be the class of all functions $f:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ and let $\mathrm{\Theta }$ be the class of all operators

$O(\bullet ;\cdot ):F([0,\mathrm{\infty }))\to F([0,\mathrm{\infty })),f\to O(f;\cdot )$

satisfying the following conditions:

$(1)$ $O(f;t)>0$ for $t>0$ and $O(f;0)=0$.

$(2)$ $O(f;t)\le O(f;s)$ for $t\le s$.

$(3)$ ${lim}_{n\to \mathrm{\infty }}O(f;t_n)=O(f;{lim}_{n\to \mathrm{\infty }}{t}_n)$.

$(4)$ $O(f;max\{t,s\})=max\{O(f;t),O(f;s)\}$ for some $f\in F([0,\mathrm{\infty }))$.

Definition 1.2. (Banaei, Ghaemi, & Saadati, 2017). A triple $(x,y,z)$ of a mapping $T:$ $X\times X\times X\to$ $X$ is called a triple fixed pointed if

$T(x,y,z)=x,T(y,x,z)=y,T(z,y,x)=z.$

The following theorems are basic for our main results.

Theorem 1.1. (Schauder (Akmerov, Kamenski, Potapov, Rodkina, & Sadovskii, 1992)). Let $C$ be a nonempty, bounded, closed and convex subset of a Banach space $E$. Then every compact and continuous map $T:$ $C\to$ $C$ has at least one fixed point.

Theorem 1.2 .(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 MNC defined in $E$. Then $T$ has at least a fixed point in $C$.

Theorem 1.3. (Aghajani, 2014). Let ${\mu }_1,{\mu }_2,{\mu }_3,...,{\mu }_n$ are measures of noncompactness in $E_1,E_2,E_3,...,E_n$ respectively. Suppose $F:[0,\mathrm{\infty }{)}^n\to [0,\mathrm{\infty })$ is a convex function and $F(x_1,x_2,x_3,...,x_n)=0$ if and only if $x_i$ = 0 for $i=1,2,3,...,n.$ Then, the function $\tilde{\mu }(X)$ = $F({\mu }_1(X_1),{\mu }_2(X_2),{\mu }_3(X_3),...,{\mu }_n(X_n))$ defines a measure of noncompactness in $E_1\times E_2\times E_3,...\times E_n$, where $X_i$ denote the natural projection of $X$ into $E_i$ for $i=1,2,3,...,n$.

Remark 1.1. Aghajani et al. (2013) illustrated the Theorem 1.3 by the following example. Let $F$ be as follows:

$F(x,y,z)=max\{x,y,z\},orF(x,y,z)=x+y+z,forany(x,y,z)\in [0,\mathrm{\infty }{)}^3.$

They showed that

$\tilde{\mu }(X)=max({\mu }_1(X_1),{\mu }_2(X_2),{\mu }_3(X_3)),or\tilde{\mu }(X)={\mu }_1(X_1)+{\mu }_2(X_2)+{\mu }_3(X_3),$

defines an MNC in the space $E_1\times E_2\times E_3$ where $X_i$, $(i=1,2,3)$ are the natural projection of $X$ into $E_i$.

Samadi and Ghaemi (Samadi, 2014) generalized Darbo’s fixed point theorem as fallow.

Theorem 1.4. Let $U$ be a nonempty, bounded, closed and convex subset of a Banach space $E$. Assume $F:$ $U\to$ $U$ be a continuous operator such that satisfying

$\psi (\mu (F(X)))\le \varphi (\psi (\mu (X)))\psi (\mu (X))$

for all nonempty subset $X$ of $U$, where $\mu$ is an arbitrary MNC in $E$, $\psi$: ${\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is nondecreasing function such that $\psi (t)=0$ if and only if $t$=0 and $\varphi :[0,\mathrm{\infty })\to [0,1)$ is a continuous function such that ${limsup}_{t\to r}\varphi (t)<1$ for all $r\ge 0$. Then F has a fixed point in U.

The Darbo’s fixed point theorem is followed if $\psi =I$ (identity map) and $\varphi =K$ in the Theorem 1.4.

Now we recall one of the theorems in this paper which extends and generalizes Darbo’s fixed point theorem by using the concept of $O(f;.)$.

Theorem 1.5. Let $C$ be a nonempty, bounded, closed and convex subset of a Banach space $E$ and $T:$ $C\to$ $C$ be a continuous operator such that

$\psi [O(f;\mu (TX))]\le \mathrm{\Phi }[\psi (O(f;\mu (X)))]\psi (\mu (X)),$

for $X$ of $C$, $O(\bullet ;\cdot )\in \mathrm{\Theta }$ and $\psi$: ${\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is a nondecreasing function such that $\psi (t)=0$ if and only if $t$=0. Let $\mathrm{\Phi }:[0,\mathrm{\infty })\to [0,1)$ is a continuous function such that ${lim}_{n\to \mathrm{\infty }}{\mathrm{\Phi }}^n(t)=0$ for each $t\ge 0$, where $\mu$ is an arbitrary MNC. Then $T$ has at least one fixed point in $C$.

Proof. (Banaei, Ghaemi, & Saadati, 2017).

Remark 1.2. The Theorem 1.4 is followed if $O(f;t)=t$ and $f=I$ in Theorem 1.5.

The following Corollary is immediate of Theorem 1.5.

Corollary 1.6. Let $C$ be a nonempty, bounded, closed and convex subset of a Banach space $E$, $T:$ $C\to$ $C$ and $\psi$: ${\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ are continuous functions. Suppose that there exists a constant $0<\lambda <1$ such that for all $X\subseteq C$,

$\psi [O(f;\mu (TX))]\le \lambda [\psi (O(f;\mu (X)))]\psi (\mu (X),$

where $\mu$ is an arbitrary measure of noncompactness and $O(\bullet ;\cdot )\in \mathrm{\Theta }$. Then $T$ has at least one fixed point in $C$.

Remark 1.3. The Darbo’s fixed point theorem is followed if $O(f;t)=t$, $f=I$ and $\psi =I$ in Corollary 1.6.

2. Main results

In this section, we state one of the main results in this paper which extends and generalizes Darbo’s fixed point theorem by using the concept of $O(f;.)$.

(2.1)

Theorem 2.1. Let $C$ be a nonempty, bounded, closed and convex subset of a Banach space $E$ and $T:C\times C\times C\to C$ be a continuous function such that

(2.1) $\begin{array}{rl}& \psi [O(f;\mu (T(X_1\times X_2\times X_3)))]\le \frac{1}{3}\mathrm{\Phi }[\psi (O(f;max\{\mu (x_1),\mu (x_2),\mu (x_3)\})]\\ & \times \psi [max\{\mu (x_1),\mu (x_2),\mu (x_3)\}]\end{array}$(2.1)

for any subset $X_1,X_2,X_3$ of $C$, where $\mu$ is an arbitrary MNC, $\mathrm{\Phi }:[0,\mathrm{\infty })\to [0,1)$ is nondecreasing function such that ${lim}_{n\to \mathrm{\infty }}{\mathrm{\Phi }}^n(t)=0$ for $t\ge 0$ and $\psi$: ${\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is a continuous function such that $\psi (t)=0$ if and only if $t$=0. Also, $O(\bullet ;\cdot )\in \mathrm{\Theta }$ and $O(f;max\{t,r,s\})=max\{O(f;t),O(f;s),O(f;r)\}$ for all $t,s,r\ge 0$. Then $T$ has at least a tripled fixed point.

Proof. From Remark 1.1 we have $\tilde{\mu }(X)=max\{\mu (x_1),\mu (x_2),\mu (x_3)\}$. Now we define a mapping $\tilde{T}:C\times C\times C\to C\times C\times C$ by

$\tilde{T}(x,y,z)=(T(x,y,z),T(y,x,z),T(z,y,x)).$

It is clear that $\tilde{T}$ is continuous. We prove that $\tilde{T}$ satisfies all the conditions of Theorem 1.5. Let $X\subseteq C\times C\times C$ be any nonempty subset. By (2.1) and $2\ast$ we obtain

$\begin{array}{rl}& \psi [O(f;\tilde{\mu }(\tilde{T}(X))]\le \psi [O(f;\tilde{\mu }(T(x_1\times x_2\times x_3))\times T(x_2\times x_1\times x_3)\times T(x_3\times x_2\times x_1)]\\ & =\psi [O(f;\mu (T(x_1\times x_2\times x_3)+\mu (T(x_2\times x_1\times x_3)+\mu (T(x_3\times x_2\times x_1)]\\ & =\psi [O(f;\mu (T(x_1\times x_2\times x_3)))]+\psi [O(f;\mu (T(x_2\times x_1\times x_3)))]\\ & +\psi [O(f;\mu (T(x_3\times x_2\times x_1)))]\\ & \le \frac{1}{3}\mathrm{\Phi }[\psi (O(f;max\{\mu (x_1),\mu (x_2),\mu (x_3)\})]\psi [max\{\mu (x_1),\mu (x_2),\mu (x_3)\})]\\ & +\frac{1}{3}\mathrm{\Phi }[\psi (O(f;max\{\mu (x_2),\mu (x_1),\mu (x_3)\})]\psi [max\{\mu (x_2),\mu (x_1),\mu (x_3)\})]\\ & +\frac{1}{3}\mathrm{\Phi }[\psi (O(f;max\{\mu (x_3),\mu (x_2),\mu (x_1)\})]\psi [max\{\mu (x_3),\mu (x_2),\mu (x_1)\})]\\ & =\mathrm{\Phi }[\psi (O(f;max\{\mu (x_1),\mu (x_2),\mu (x_3)\})]\psi [max\{\mu (x_1),\mu (x_2),\mu (x_3)]\\ & =\mathrm{\Phi }[\psi (O(f;\tilde{\mu }(X)))]\psi (\tilde{\mu }(X)).\end{array}$

From Theorem 1.5 deduce that $\tilde{T}$ has at least a fixed point in $C\times C\times C$ and $T$ has at least a tripled fixed point.

As application of Theorem 2.1, we can get the following theorem.

(2.2)

Theorem 2.2 Let $C$ be a nonempty, bounded, closed and convex subset of a Banach space $E$ and $T:C\times C\times C\to C$ be a continuous function such that

(2.2) $\begin{array}{rl}& \psi [O(f;\mu (T(x_1\times x_2\times x_3)))]\le \frac{1}{3}\mathrm{\Phi }[\psi (O(f;\mu (x_1)+\mu (x_2)+\mu (x_3))]\\ & \times \psi [\mu (x_1)+\mu (x_2)+\mu (x_3)]\end{array}$(2.2)

for any subset $x_1,x_2,x_3$ of $C$, where $\mu$ is an arbitrary MNC, $\mathrm{\Phi }:[0,\mathrm{\infty })\to [0,1)$ is nondecreasing function such that ${lim}_{n\to \mathrm{\infty }}{\mathrm{\Phi }}^n(t)=0$ for $t\ge 0$ and $\psi$: $R_{+}\to R_{+}$ is a continuous function such that $\psi (t)=0$ if and only if $t$=0. Also, $O(\bullet ;\cdot )\in \mathrm{\Theta }$ and $O(f;t+r+s)\le O(f;t)+O(f;s)+O(f;r)$ for all $t,s,r\ge 0$. Then $T$ has at least a tripled fixed point.

Proof. From Remark 1.1, we know that $\tilde{\mu }(X)=\mu (x_1)+\mu (x_2)+\mu (x_3)$ is a MNC. Now we define a mapping $\tilde{T}:C\times C\times C\to C\times C\times C$ by

$\tilde{T}(x,y,z)=(T(x,y,z),T(y,x,z),T(z,y,x)).$

It is clear that $\tilde{T}$ satisfies all the conditions of Theorem.

Corollary 2.3. Let $C$ be a nonempty, bounded, closed and convex subset of a Banach space $E$ and $T:C\times C\times C\to C$ be a continuous function such that

$\mu (T(X_1\times X_2\times X_3))\le \frac{1}{3}\mathrm{\Phi }(\mu (x_1)+\mu (x_2)+\mu (x_3)),$

for any subset $X_1,X_2,X_3$ of $C$, where $\mu$ is an arbitrary MNC. Also, $\mathrm{\Phi }:[0,\mathrm{\infty })\to [0,1)$ is a nondecreasing function such that ${lim}_{n\to \mathrm{\infty }}{\mathrm{\Phi }}^n(t)=0$ for $t\ge 0$. Then $T$ has at least a tripled fixed point.

3. Application

In this section as an application of Theorem 2.2, we study the existence of solutions for the system of integral Equations (1.1). Let the Banach space $BC({\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+})$ consisting of all real functions defined, bounded and continuous on ${\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}$ equipped with the standard norm

$\parallel x\parallel =sup\{|x(t,s,r)|:t,s,r\ge 0\}.$

We will use a measure of noncompactness in the space $BC({\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+})$, which is given in (Banás & Goebel, 1980). Let $X$ be fix a nonempty bounded subset of $BC({\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+})$ and a positive number $T>0$. For $x\in X$ and $ϵ>0$ we denote

${\omega }^T(x,ϵ)=sup\{|x(t,s,r)-x(u,v,w)|:t,s,r,u,v,w\in [0,T],|t-u|\le ϵ,|s-v|\le ϵ,|r-w|\le ϵ\}$

and

${\omega }^T(X,ϵ)=sup\{{\omega }^T(x,ϵ):x\in X\},$

${\omega }_0^T(X)=li{m}_{\epsilon \to 0}{\omega }^T(X,ϵ),$

and

${\omega }_0(X)=li{m}_{T\to \mathrm{\infty }}{\omega }_0^T(X).$

Moreover, for three fixed numbers $t,s,r\in R_{+}$ the function $\mu$ on the family ${\mathfrak{M}}_{BC({\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+})}$ is defined as the following formula

$\mu (X)={\omega }_0(X)+\alpha (X),$

where $\alpha (X)={limsup}_{t,s,r\to \mathrm{\infty }}diamX(t,s,r)$, $X(t,s,r)=\{x(t,s,r):x\in X\}$

and

$diamX(t,s,r)=sup\{|x(t,s,r)-y(t,s,r)|:x,y\in X\}.$

Now consider the following assumptions:

$(A_1)$ ${\alpha }_i:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ are continuous, nondecreasing and ${lim}_{t\to \mathrm{\infty }}{\alpha }_i(t)=\mathrm{\infty },i=1,2,3.$

$(A_2)$ The function $a:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is continuous and bounded.

$(A_3)$ $k:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is continuous and there exists a positive constant $M$ such that .

$\begin{array}{rl}& M=sup\left\{{\int }_0^{{\alpha }_1(t)}{\int }_0^{{\alpha }_2(s)}{\int }_0^{{\alpha }_3(r)}|k(t,s,r,u,v,w,x(u,v,w),y(u,v,w),z(u,v,w))|\right.\\ & dudvdw:t,s,r\in R_{+},x,y,z\in E\}\end{array}$

Moreover,

$\begin{array}{rl}& li{m}_{t,s,r\to \mathrm{\infty }}\left|{\int }_0^{{\alpha }_1(t)}{\int }_0^{{\alpha }_2(s)}{\int }_0^{{\alpha }_3(r)}[k(t,s,r,u,v,w,x_2(u,v,w),y_2(u,v,w),z_2(u,v,w)\right.\\ & -k(t,s,r,u,v,w,x_1(u,v,w),y_1(u,v,w),z_1(u,v,w))]dudvdw|=0\end{array}$

uniformly respect to $x_1,y_1,z_1,x_2,y_2,z_2\in E.$

$(A_4)$ The functions $f,g:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are continuous and there exists an upper semicontinuous and nondecreasing function $\mathrm{\Phi }:{\mathbb{R}}_{+}\to \mathbb{R}$ with ${lim}_{n\to \mathrm{\infty }}{\mathrm{\Phi }}^n(t)=0$ for each $t\ge 0$. Also, there exists two bounded functions $a_1,a_2,:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to \mathbb{R}$ with bound

$K=max\{su{p}_{(t,s,r)\in {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}}{a}_i(t,s,r),i=1,2\}$

and a positive constant $D$ such that

$|f(t,s,r,x_2,y_2,z_2)-f(t,s,r,x_1,y_1,z_1)|\le \frac{a_1(t,s,r)\mathrm{\Phi }(|x_2-x_1|+|y_2-y_1|+|z_2-z_1|)}{D+\mathrm{\Phi }(|x_2-x_1|+|y_2-y_1|+|z_2-z_1|)},$

and

$|g(t,s,r,x_2,y_2,z_2)-g(t,s,r,x_1,y_1,z_1)|\le \frac{a_2(t,s,r)\mathrm{\Phi }(|x_2-x_1|+|y_2-y_1|+|z_2-z_1|)}{D+\mathrm{\Phi }(|x_2-x_1|+|y_2-y_1|+|z_2-z_1|)},$

for all $t,s,r\in {\mathbb{R}}_{+}$and $x_1,y_1,z_1,x_2,y_2,z_2\in \mathbb{R}$. We suppose that $\mathrm{\Phi }$ has the property

$\mathrm{\Phi }(t)+\mathrm{\Phi }(s)+\mathrm{\Phi }(r)\le \mathrm{\Phi }(t+s+r)$

for all $t,s,r\in {\mathbb{R}}_{+}$. Moreover, we assume that $3K(1+M)\le D$.

$(A_5)$ The functions $H_1,H_2:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ defined by $H_1(t,s,r)=|f(t,s,r,0,0,0)|$ and $H_2(t,s,r)=|g(t,s,r,0,0,0)|$ are bounded on ${\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}$ with

$H_0=max\left\{sup{H}_i\left(t,s,r\right):\left(t,s,r\right)\in {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+},i=1,2\right\}.$

(3.1)

Theorem 3.1 If the assumptions $(A_1)$-$(A_5)$ are satisfied, then the system of integral Equations (1.1) has at least one solution $(x,y,z)\in E\times E\times E$.

$Proof.$ Define the operator $T:E\times E\times E\to E$ associated with the system of integral Equations (1.1) by

(3.1) $T(x,y,z)(t,s,r)=a(t,s,r)+f(t,s,r,x(t,s,r),y(t,s,r),z(t,s,r))$(3.1)

(3.2) $+g(t,s,r,x(t,s,r),y(t,s,r),z(t,s,r))[F(x,y,z)(t,s,r)]$(3.2)

where,

$F(x,y,z)(t,s,r)={\int }_0^{{\alpha }_1(t)}{\int }_0^{{\alpha }_2(s)}{\int }_0^{{\alpha }_3(r)}k(t,s,r,u,v,w,x(u,v,w),y(u,v,w),z(u,v,w))dudvdw.$

We divide our proof in three steps.

$H_0=max\{sup{H}_i(t,s,r):(t,s,r)\in {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+},i=1,2\}.$

Step 1: In the first step we prove that $T$ transforms the space $E\times E\times E$ into $E$.

Obviously, that $T(x,y,z)$ is continuous on ${\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}$. Now we show that $T(x,y,z)\in E.$ For arbitrarily fixed $(t,s,r)\in R_{+}\times R_{+}\times {\mathbb{R}}_{+}$ we have

$\begin{array}{rl}& |(T(x,y,z))(t,s,r)|\le |a(t,s,r)|+|f(t,s,r,x(t,s,r),y(t,s,r),z(t,s,r))|\\ & +|g(t,s,r,x(t,s,r),y(t,s,r),z(t,s,r))||F(x,y,z)(t,s,r)|\\ & \le |a(t,s,r)|+\frac{K\mathrm{\Phi }(|x(t,s,r)|+|y(t,s,r)|+|z(t,s,r)|)}{D+\mathrm{\Phi }(|x(t,s,r)|+|y(t,s,r)|+|z(t,s,r)|)}\\ & +H_0+[\frac{K\mathrm{\Phi }(|x(t,s,r)|+|y(t,s,r)|+|z(t,s,r)|)}{D+\mathrm{\Phi }(|x(t,s,r)|+|y(t,s,r)|+|z(t,s,r)|)}+H_0]M_0\end{array}$

Indeed,

$\begin{array}{rl}& |f(t,s,r,x(t,s,r),y(t,s,r),z(t,s,r))|\le |f(t,s,r,x(t,s,r),y(t,s,r),z(t,s,r))\\ & -f(t,s,r,0,0,0)|+|f(t,s,r,0,0,0)|\\ & \le \frac{a_1(t,s,r)\mathrm{\Phi }(|x(t,s,r)|+|y(t,s,r)|+|z(t,s,r)|)}{D+\mathrm{\Phi }(|x(t,s,r)|+|y(t,s,r)|+|z(t,s,r)|)}\\ & +H_1(t,s,r)\\ & \le \frac{K\mathrm{\Phi }(|x(t,s,r)|+|y(t,s,r)|+|z(t,s,r)|)}{D+\mathrm{\Phi }(|x(t,s,r)|+|y(t,s,r)|+|z(t,s,r)|)}+H_0,\end{array}$

and

$\begin{array}{rl}& |g(t,s,r,x(t,s,r),y(t,s,r),z(t,s,r))|\le |g(t,s,r,x(t,s,r),y(t,s,r),z(t,s,r))\\ & -g(t,s,r,0,0,0)|+|g(t,s,r,0,0,0)|\\ & \le \frac{a_2(t,s,r)\mathrm{\Phi }(|x(t,s,r)|+|y(t,s,r)|+|z(t,s,r)|)}{D+\mathrm{\Phi }(|x(t,s,r)|+|y(t,s,r)|+|z(t,s,r)|)}\\ & +H_2(t,s,r)\\ & \le \frac{K\mathrm{\Phi }(|x(t,s,r)|+|y(t,s,r)|+|z(t,s,r)|)}{D+\mathrm{\Phi }(|x(t,s,r)|+|y(t,s,r)|+|z(t,s,r)|)}+H_0,\end{array}$

$|F(x,y,z)(t,s,r)|=\left|{\int }_0^{{\alpha }_1(t)}{\int }_0^{{\alpha }_2(s)}{\int }_0^{{\alpha }_3(r)}k(t,s,r,u,v,w,x(u,v,w),y(u,v,w),z(u,v,w))dudvdw\right|$

$\le {\int }_0^{{\alpha }_1(t)}{\int }_0^{{\alpha }_2(s)}{\int }_0^{{\alpha }_3(r)}|k(t,s,r,u,v,w,x(u,v,w),y(u,v,w),z(u,v,w))|dudvdw\le M.$

By $(A_4)$, we have

(3.3) $\begin{array}{rl}& \parallel T(x,y,z)\parallel \le \parallel a\parallel +(\frac{K\mathrm{\Phi }(\parallel x\parallel +\parallel y\parallel +\parallel z\parallel )}{D+\mathrm{\Phi }(\parallel x\parallel +\parallel y\parallel +\parallel z\parallel ))}+H_0)(1+M)\\ & \le \parallel a\parallel +(K+H_0)(1+M).\end{array}$(3.3)

Therefore, $T$ maps the space $E\times E\times E$ into $E$ and $T(\overline{B_r}\times \overline{B_r}\times \overline{B_r})\subseteq \overline{B_r}$,

where $r=\parallel a\parallel +(K+H_0)(1+M).$

Step 2: We prove that the map $T:\overline{B_r}\times \overline{B_r}\times \overline{B_r}\to \overline{B_r}$ is continuous.

Let us fix arbitrarily $ϵ>0$ and take $(x,y,z),(m,n,p)\in \overline{B_r}\times \overline{B_r}\times \overline{B_r}$ such that

$\parallel (x,y,z)-(m,n,p)\parallel \le ϵ.$

Then

(3.4) $\begin{array}{rl}& |(T(x,y,z)(t,s,r))-(T(m,n,p)(t,s,r))|\\ & =|f(t,s,r,x(t,s,r),y(t,s,r),z(t,s,r))+g(t,s,r,x(t,s,r),y(t,s,r),z(t,s,r))[F(x,y,z)(t,s,r)]\\ & -f(t,s,r,m(t,s,r),n(t,s,r),p(t,s,r))-g(t,s,r,m(t,s,r),n(t,s,r),p(t,s,r))[F(m,n,p)(t,s,r)]|\\ & \le |f(t,s,r,x(t,s,r),y(t,s,r),z(t,s,r))-f(t,s,r,m(t,s,r),n(t,s,r),p(t,s,r))|\\ & +|g(t,s,r,x(t,s,r),y(t,s,r),z(t,s,r))||F(x,y,z)(t,s,r)-F(m,n,p)(t,s,r)|\\ & +|g(t,s,r,x(t,s,r),y(t,s,r),z(t,s,r))-g(t,s,r,m(t,s,r),n(t,s,r),p(t,s,r))||F(m,n,p)(t,s,r)|\\ & \le \frac{a_1(t,s,r)\mathrm{\Phi }(|x(t,s,r)-m(t,s,r)|+|y(t,s,r)-n(t,s,r)|+|z(t,s,r)-p(t,s,r)|)}{D+\mathrm{\Phi }(|x(t,s,r)-m(t,s,r)|+|y(t,s,r)-n(t,s,r)|+|z(t,s,r)-p(t,s,r)|)}\\ & +[\frac{K\mathrm{\Phi }(|x(t,s,r)|+|y(t,s,r)|+|z(t,s,r)|)}{D+\mathrm{\Phi }(|x(t,s,r)|+|y(t,s,r)|+|z(t,s,r)|)}+H_0]|F(x,y,z)(t,s,r)-F(m,n,p)(t,s,r)|\\ & +[\frac{a_2(t,s,r)\mathrm{\Phi }(|x(t,s,r)-m(t,s,r)|+|y(t,s,r)-n(t,s,r)|+|z(t,s,r)-p(t,s,r)|)}{D+\mathrm{\Phi }(|x(t,s,r)-m(t,s,r)|+|y(t,s,r)-n(t,s,r)|+|z(t,s,r)-p(t,s,r)|)}]M\\ & \le \frac{K(1+M)\mathrm{\Phi }(\parallel x-m\parallel +\parallel y-n\parallel +\parallel z-p\parallel )}{D+\mathrm{\Phi }(\parallel x-m\parallel +\parallel y-n\parallel +\parallel z-p\parallel )}\\ & +\left[\frac{K\mathrm{\Phi }(\parallel x\parallel +\parallel y\parallel +\parallel z\parallel )}{D+\mathrm{\Phi }(\parallel x\parallel +\parallel y\parallel +\parallel z\parallel )}+H_0\right]|F(x,y,z)(t,s,r)-F(m,n,p)(t,s,r)|.\end{array}$(3.4)

By using condition $(A_2)$ there exist $T>0$ such that for $t>T$

(3.5) $\begin{array}{rl}& |F(x,y,z)(t,s,r)-F(m,n,p)(t,s,r)|\\ & =\left|{\int }_0^{{\alpha }_1(t)}{\int }_0^{{\alpha }_2(s)}{\int }_0^{{\alpha }_3(r)}[k(t,s,r,u,v,w,x(u,v,w),y(u,v,w),z(u,v,w))\right.\\ & -k(t,s,r,u,v,w,m(u,v,w),n(u,v,w),p(u,v,w))]dudvdw|\\ & \le ϵ.\end{array}$(3.5)

Suppose that $t,s,r>T.$ We have from (3.4) and (3.5)

(3.6) $|T(x,y,z)(t,s,r)-T(m,n,p)(t,s,r)|<ϵ.$(3.6)

If $t,s,r\in [0,T]$, then we obtain

(3.7) $|F(x,y,z)(t,s,r)-F(m,n,p)(t,s,r)|\le {\alpha }_T^3{\omega }_1(k,ϵ).$(3.7)

where ${\alpha }_T=sup\{{\alpha }_i(t):t\in [0,T],i=1,2,3\},$

$\begin{array}{rl}& {\omega }_1(k,ϵ)=sup\{|k(t,s,r,u,v,w,x,y,z)-k(t,s,r,u,v,w,m,n,p)|:t,s,r\in [0,T],\\ & u,v,w\in [0,{\alpha }_T],x,y,z,m,n,p\in [-r,r],\parallel (x,y,z)-(m,n,p)\parallel \le ϵ\}.\end{array}$

By using continuity of $k$ on $[0,T]\times [0,T]\times [0,T]\times [0,{\alpha }_T]\times [0,{\alpha }_T]\times [0,{\alpha }_T]\times [-r,r]\times [-r,r]\times [-r,r],$ we have ${\omega }_1(k,ϵ)\to 0$ as $ϵ\to 0.$

From inequalities (3.4) and (3.7) we deduce that

(3.8) $|T(x,y,z)(t,s,r)-T(m,n,p)(t,s,r)|\le ϵ+[K+H_0]{\alpha }_T^3{\omega }_1(k,ϵ).$(3.8)

We conclude that $T$ is continuous on $\overline{B_r}\times \overline{B_r}\times \overline{B_r}.$

Step 3: Now we prove that for any nonempty set $X_1,X_2,X_3\subseteq \overline{B_r}$,

$\mu (T(X_1\times X_2\times X_3))\le \frac{1}{3}\mathrm{\Phi }(\mu (X_1)+\mu (X_2)+\mu (X_3)).$

By using assumptions $(A_1)-(A_5)$ for any $(x,y,z),(m,n,p)\in X_1\times X_2\times X_3$ and $t,s,r\in R_{+},$

$\begin{array}{rl}& |(T(x,y,z))(t,s,r)-(T(m,n,p))(t,s,r)|\\ & \le \frac{K(1+M)\mathrm{\Phi }(|x(t,s,r)-m(t,s,r)|+|y(t,s,r)-n(t,s,r)|+|z(t,s,r)-p(t,s,r)|)}{D+\mathrm{\Phi }(|x(t,s,r)-m(t,s,r)|+|y(t,s,r)-n(t,s,r)|+|z(t,s,r)-p(t,s,r)|)}\\ & +\left[\frac{K\mathrm{\Phi }(|x(t,s,r)|+|y(t,s,r)|+|z(t,s,r)|)}{D+\mathrm{\Phi }(|x(t,s,r)|+|y(t,s,r)|+|z(t,s,r)|)}+H_0\right]\beta (t,s,r)\\ & \le \frac{1}{3}\mathrm{\Phi }(|x(t,s,r)-m(t,s,r)|+|y(t,s,r)-n(t,s,r)|+|z(t,s,r)-p(t,s,r)|)\\ & +\left[\frac{K\mathrm{\Phi }(|x(t,s,r)|+|y(t,s,r)|+|z(t,s,r)|)}{D+\mathrm{\Phi }(|x(t,s,r)|+|y(t,s,r)|+|z(t,s,r)|)}+H_0\right]\beta (t,s,r),\end{array}$

where

$\begin{array}{rl}& \beta (t,s,r)=sup\left\{\left|{\int }_0^{{\alpha }_1(t)}{\int }_0^{{\alpha }_2(s)}{\int }_0^{{\alpha }_3(r)}[k(t,s,r,u,v,w,x(u,v,w),y(u,v,w),z(u,v,w))\right.\right.\\ & -k(t,s,r,u,v,w,m(u,v,w),n(u,v,w),p(u,v,w))]dudvdw|:x,y,z\in E\}.\end{array}$

Now we have

(3.9) $\begin{array}{rl}& diam(T(X_1\times X_2\times X_3))(t,s,r)\le \frac{1}{3}\mathrm{\Phi }(diam{X}_1(t,s,r)+diam{X}_2(t,s,r)+diam{X}_3(t,s,r))\\ & +\left[\frac{K\mathrm{\Phi }(|x(t,s,r)|+|y(t,s,r)|+|z(t,s,r)|)}{D+\mathrm{\Phi }(|x(t,s,r)|+|y(t,s,r)|+|z(t,s,r)|)}+H_0\right]\beta (t,s,r).\end{array}$(3.9)

By considering upper semicontinuity of the function $\mathrm{\Phi }$, from (3.9) and $(A_3)$ we have

(3.10) $\begin{array}{rl}& \underset{t,s,r\to \mathrm{\infty }}{limsup}diam(T(X_1\times X_2\times X_3))(t,s,r))\le \frac{1}{3}\mathrm{\Phi }(\underset{t,s,r\to \mathrm{\infty }}{limsup}diam{X}_1(t,s,r)\\ & +\underset{t,s,r\to \mathrm{\infty }}{limsup}diam{X}_2(t,s,r)+\underset{t,s,r\to \mathrm{\infty }}{limsup}diam{X}_3(t,s,r))\end{array}$(3.10)

Fix arbitrary $T>0$ and $ϵ>0.$ Choose $t_1,t_2,s_1,s_2,r_1,r_2\in [0,T]$ such that $|t_2-t_1|\le ϵ,$ $|s_2-s_1|\le ϵ$, $|r_2-r_1|\le ϵ.$ Without loss of generality, suppose that $t_1\le t_2,$ $s_1\le s_2$ and $r_1\le r_2.$ Then, for $(x,y,z)\in X_1\times X_2\times X_3$ we have

$\begin{array}{rl}& |f(t_2,s_2,r_2,x(t_2,s_2,r_2),y(t_2,s_2,r_2),z(t_2,s_2,r_2))-f(t_1,s_1,r_1,x(t_1,s_1,r_1),y(t_1,s_1,r_1),z(t_1,s_1,r_1))|\\ & \le |f(t_2,s_2,r_2,x(t_2,s_2,r_2),y(t_2,s_2,r_2),z(t_2,s_2,r_2))-f(t_2,s_2,r_2,x(t_1,s_1,r_1),y(t_1,s_1,r_1),z(t_1,s_1,r_1))|\\ & +|f(t_2,s_2,r_2,x(t_1,s_1,r_1),y(t_1,s_1,r_1),z(t_1,s_1,r_1))-f(t_1,s_1,r_1,x(t_1,s_1,r_1),y(t_1,s_1,r_1),z(t_1,s_1,r_1))|\\ & \le \frac{K\mathrm{\Phi }(|x(t_2,s_2,r_2)-x(t_1,s_1,r_1)|+|y(t_2,s_2,r_2)-y(t_1,s_1,r_1)|+|z(t_2,s_2,r_2)-z(t_1,s_1,r_1)|)}{D+\mathrm{\Phi }(|x(t_2,s_2,r_2)-x(t_1,s_1,r_1)|+|y(t_2,s_2,r_2)-y(t_1,s_1,r_1)|+|z(t_2,s_2,r_2)-z(t_1,s_1,r_1)|)}\\ & +|f(t_2,s_2,r_2,x(t_1,s_1,r_1),y(t_1,s_1,r_1),z(t_1,s_1,r_1))-f(t_1,s_1,r_1,x(t_1,s_1,r_1),y(t_1,s_1,r_1),z(t_1,s_1,r_1))|\\ & \le \frac{1}{3(1+M)}\mathrm{\Phi }({\omega }^T(x,ϵ)+{\omega }^T(y,ϵ)+{\omega }^T(z,ϵ))+{\omega }^T(f,ϵ),\end{array}$

and

$\begin{array}{rl}& |F(x,y,z)(t_2,s_2,r_2)-F(x,y,z)(t_1,s_1,r_1)|\\ & \le {\int }_0^{{\alpha }_1(t_2)}{\int }_0^{{\alpha }_2(s_2)}{\int }_0^{{\alpha }_3(r_2)}|k(t_2,s_2,r_2,u,v,w,x(u,v,w),y(u,v,w),z(u,v,w))\\ & -k(t_1,s_1,r_1,u,v,w,x(u,v,w),y(u,v,w),z(u,v,w))|dudvdw\\ & +{\int }_{{\alpha }_1(t_1)}^{{\alpha }_1(t_2)}{\int }_{{\alpha }_2(s_1)}^{{\alpha }_2(s_2)}{\int }_{{\alpha }_3(r_1)}^{{\alpha }_3(r_2)}|k(t_1,s_1,r_1,u,v,w,x(u,v,w),y(u,v,w),z(u,v,w))|dudvdw\\ & \le {\int }_0^{{\alpha }_1(t_2)}{\int }_0^{{\alpha }_2(s_2)}{\int }_0^{{\alpha }_3(r_2)}{\omega }^T(k,ϵ)dudvdw+{\int }_{{\alpha }_1(t_1)}^{{\alpha }_1(t_2)}{\int }_{{\alpha }_2(s_1)}^{{\alpha }_2(s_2)}{\int }_{{\alpha }_3(r_1)}^{{\alpha }_3(r_2)}{K}^{T}dudvdw\\ & \le {\alpha }_T^3{\omega }^T(k,ϵ)+{\omega }^T({\alpha }_1,ϵ){\omega }^T({\alpha }_2,ϵ){\omega }^T({\alpha }_3,ϵ)K^T,\end{array}$

and

$|g(t_2,s_2,r_2,,x(t_2,s_2,r_2),y(t_2,s_2,r_2),z(t_2,s_2,r_2))F(x,y,z)(t_2,s_2,r_2)$

$-g(t_1,s_1,r_1,,x(t_1,s_1,r_1),y(t_1,s_1,r_1),z(t_1,s_1,r_1))F(x,y,z)(t_1,s_1,r_1)|$

$\le |g(t_2,s_2,r_2,,x(t_2,s_2,r_2),y(t_2,s_2,r_2),z(t_2,s_2,r_2))F(x,y,z)(t_2,s_2,r_2)$

$-g(t_1,s_1,r_1,,x(t_1,s_1,r_1),y(t_1,s_1,r_1),z(t_1,s_1,r_1))F(x,y,z)(t_2,s_2,r_2)|$

$+|g(t_1,s_1,r_1,,x(t_1,s_1,r_1),y(t_1,s_1,r_1),z(t_1,s_1,r_1))F(x,y,z)(t_2,s_2,r_2)$

$-g(t_1,s_1,r_1,,x(t_1,s_1,r_1),y(t_1,s_1,r_1),z(t_1,s_1,r_1))F(x,y,z)(t_1,s_1,r_1)|$

$\le \frac{K\mathrm{\Phi }(|x(t_2,s_2,r_2)-x(t_1,s_1,r_1)|+|y(t_2,s_2,r_2)-y(t_1,s_1,r_1)|+|z(t_2,s_2,r_2)-z(t_1,s_1,r_1)|)}{D+\mathrm{\Phi }(|x(t_2,s_2,r_2)-x(t_1,s_1,r_1)|+|y(t_2,s_2,r_2)-y(t_1,s_1,r_1)|+|z(t_2,s_2,r_2)-z(t_1,s_1,r_1)|)}$

$\times |F(x,y,z)(t_2,s_2,r_2)|+[\frac{K\mathrm{\Phi }(|x(t_1,s_1,r_1)|+|y(t_1,s_1,r_1)|+|z(t_1,s_1,r_1)|)}{D+\mathrm{\Phi }(|x(t_1,s_1,r_1)|+|y(t_1,s_1,r_1)|+|z(t_1,s_1,r_1)|)}+H_0]$

$\times |F(x,y,z)(t_2,s_2,r_2)-F(x,y,z)(t_1,s_1,r_1)|$

$\le \frac{M}{3(1+M)}\mathrm{\Phi }({\omega }^T(x,ϵ)+{\omega }^T(y,ϵ)+{\omega }^T(z,ϵ))+(K+H_0)[{\alpha }_T^3{\omega }^T(k,ϵ)+{\omega }^T({\alpha }_1,ϵ){\omega }^T({\alpha }_2,ϵ){\omega }^T({\alpha }_3,ϵ)K^T].$

Therefore,

(3.11) $\begin{array}{rl}& |(T(x,y,z))(t_2,s_2,r_2)-(T(x,y,z))(t_1,s_1,r_1)|\\ & \le |a(t_2,s_2,r_2)-a(t_1,s_1,r_1)|+|f(t_2,s_2,r_2,,x(t_2,s_2,r_2),y(t_2,s_2,r_2),z(t_2,s_2,r_2))\\ & -f(t_1,s_1,r_1,x(t_1,s_1,r_1),y(t_1,s_1,r_1),z(t_1,s_1,r_1))|\\ & +|g(t_2,s_2,r_2,x(t_2,s_2,r_2),y(t_2,s_2,r_2),z(t_2,s_2,r_2))(F(x,y,z))(t_2,s_2,r_2)\\ & -g(t_1,s_1,r_1,x(t_1,s_1,r_1),y(t_1,s_1,r_1),z(t_1,s_1,r_1))(F(x,y,z))(t_1,s_1,r_1)|\\ & \le {\omega }^T(a,ϵ)+\frac{M}{3(1+M)}({\omega }^T(x,ϵ)+{\omega }^T(y,ϵ)+{\omega }^T(z,ϵ))+{\omega }^T(f,ϵ)\\ & +\frac{M}{3(1+M)}\mathrm{\Phi }({\omega }^T(x,ϵ)+{\omega }^T(y,ϵ)+{\omega }^T(z,ϵ))+(K+H_0)\\ & \times [{\alpha }_T^3{\omega }^T(k,ϵ)+{\omega }^T({\alpha }_1,ϵ){\omega }^T({\alpha }_2,ϵ){\omega }^T({\alpha }_3,ϵ)K^T],\end{array}$(3.11)

where

${\omega }^T(f,ϵ)=sup\{|f(t_2,s_2,r_2,x.y,z)-f(t_1,s_1,r_1,x,y,z)|:t_1,t_2,s_1,s_2,r_1,r_2\in [0,T],$

$|t_2-t_1|\le ϵ,|s_2-s_1|\le ϵ,|r_2-r_1|\le ϵ,x,y,z\in [-r,r]\},$

${\omega }^T(k,ϵ)=sup\{|k(t_2,s_2,r_2,u,v,w,x,y,z)-k(t_1,s_1,r_1,u,v,w,x,y,z)|:$

$t_1,t_2,s_1,s_2,r_1,r_2\in [0,T],|t_2-t_1|\le ϵ,|s_2-s_1|\le ϵ,|r_2-r_1|\le ϵ,u,v,w\in [0,{\alpha }_T],$

$x,y,z\in [-r,r]\},$

${\omega }^T({\alpha }_i,ϵ)=sup\{|{\alpha }_i(t)-{\alpha }_i(s)|,|{\alpha }_i(s)-{\alpha }_i(r)|,|{\alpha }_i(t)-{\alpha }_i(r)|:t,s,r\in [0,T],$

$|t-s|\le ϵ,|s-r|\le ϵ,i=1,2,3\},$

${\omega }^T(x,ϵ)=sup\{|x(t_2,s_2,r_2)-x(t_1,s_1,r_1)|:t_1,t_2,s_1,s_2,r_1,r_2\in [0,T],|t_2-t_1|\le ϵ,$

$|s_2-s_1|\le ϵ,|r_2-r_1|\le ϵ\},$

$K^T=sup\{|k(t,s,r,u,v,w,x,y,z)|:t,s,r\in [0,T],u,v,w\in [0,{\alpha }_T],x,y,z\in [-r,r]\},$

${\omega }^T(a,ϵ)=sup\{|a(t_2,s_2,r_2)-a(t_1,s_1,r_1)|:t_1,t_2,s_1,s_2,r_1,r_2\in [0,T],$

$|t_2-t_1|\le ϵ,|s_2-s_1|\le ϵ,|r_2-r_1|\le ϵ\}.$

Since $(x,y,z)$ is an arbitrary element of $X_1\times X_2\times X_3,$ the inequality (3.11) implies that

(3.12) $\begin{array}{rl}& {\omega }^T(T(X_1\times X_2\times X_3),ϵ)\\ & \le {\omega }^T(a,ϵ)+\frac{1}{3}\mathrm{\Phi }({\omega }^T(X_1,ϵ)+{\omega }^T(X_2,ϵ)+{\omega }^T(X_3,ϵ))+{\omega }^T(f,ϵ)\\ & +(K+H_0)[{\alpha }_T^3{\omega }^T(k,ϵ)+{\omega }^T({\alpha }_1,ϵ){\omega }^T({\alpha }_2,ϵ){\omega }^T({\alpha }_3,ϵ)K^T].\end{array}$(3.12)

By considering the uniform continuity of the functions $a$, $f$ and $k$ on $[0,T]\times [0,T]\times [0,T]$ and $[0,T]\times [0,T]\times [0,T]\times [-r,r]$ and $[0,T]\times [0,T]\times [0,T]\times [0,{\alpha }_T]\times [0,{\alpha }_T]\times [0,{\alpha }_T]\times [-r,r]\times [-r,r]\times [-r,r]$ respectively, we conclude that ${\omega }^T(a,ϵ)\to 0,$ ${\omega }^T(f,ϵ)\to 0$ and ${\omega }^T(k,ϵ)\to 0.$ It is obvious that constant $K^T$ is finite and ${\omega }^T({\alpha }_i,ϵ)\to 0$, $i=1,2,3$ as $ϵ\to 0$. Thus, we get

(3.13) ${\omega }_0(T(X_1\times X_2\times X_3))\le \frac{1}{3}\mathrm{\Phi }({\omega }_0(X_1)+{\omega }_0(X_2)+{\omega }_0(X_3)).$(3.13)

From inequality (3.10), (3.13) and definition measure of noncompactness $\mu$ we have

$\begin{array}{rl}& \mu (T(X_1\times X_2\times X_3))={\omega }_0(T(X_1\times X_2\times X_3))+\underset{t,s,r\to \mathrm{\infty }}{limsup}diam(T(X_1\times X_2\times X_3))(t,s,r)\\ & \le \frac{1}{3}\mathrm{\Phi }({\omega }_0(X_1)+{\omega }_0(X_2)+{\omega }_0(X_3))+\frac{1}{3}\mathrm{\Phi }(\underset{t,s,r\to \mathrm{\infty }}{limsup}diam{X}_1(t,s,r)\\ & +\underset{t,s,r\to \mathrm{\infty }}{limsup}diam{X}_2(t,s,r)+\underset{t,s,r\to \mathrm{\infty }}{limsup}diam{X}_3(t,s,r))\\ & \le \frac{1}{3}\mathrm{\Phi }({\omega }_0(X_1)+\underset{t,s,r\to \mathrm{\infty }}{limsup}diam{X}_1(t,s,r)+{\omega }_0(X_2)+\underset{t,s,r\to \mathrm{\infty }}{limsup}diam{X}_2(t,s,r)\\ & +{\omega }_0(X_3)+\underset{t,s,r\to \mathrm{\infty }}{limsup}diam{X}_3(t,s,r))\\ & =\frac{1}{3}\mathrm{\Phi }(\mu (x_1)+\mu (x_2)+\mu (x_3)).\end{array}$

By applying Corollary 2.3 the desired result is obtained.

Additional information

Funding

The author received no direct funding for this research.

Notes on contributors

Shahram Banaei

Shahram Banaei The research field of author is fixed point theory with its applications. He is assistant professor and faculty member in Islamic Azad university. The author has studied Darbo’s fixed point theorem in Banach space. Also, he give an application of their obtained results and analyze the existence of solutions for a system of integral equations by using the technique of measure of noncompactness.