On Solvability for Some Classes of System of Non-Linear Integral Equations in Two Dimensions via Measure of Non-Compactness

Abstract

In this paper, we present some results of coupled fixed points for the system of non-linear integral equations in Banach space. Our results enlarge the results of newer papers. Additionally, we prove the applicability of those results to the solvability of the system of non-linear integral equations. Finally, we give an example to validate the applicability of our results.

1. Introduction

The MNC is one of the usual famous and helpful ideas in non-linear analysis. This discussion was started by the fundamental article of Kuratowski in [1] and has given powerful tools for obtaining the solutions of a wide variety of integral equations and systems of integral equations. One of the most valuable results in the fixed point theory is due to G. Darbo [2]. Considerably, several scholars have presented generalizations of Darbo’s theorem with MNC and have been supported in solving the integral equations (for example, see [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]). In this work, we used the technique of MNC and the generalization of Darbo’s fixed point theorem to the existence result of a system of non-linear integral equations in two variables in Banach space.

Assume that H is a real Banach space with the norm $\left|\right|.\left|\right|$. Let $B(x,\sigma )$ be the closed ball with radius $\sigma$ and centered at x. Let J be a non-empty subset of H, and $\overline{J}$ and ConvJ denote the closure and the convex closure of J, respectively. Let $M_H$ be the family of all non-empty and bounded subsets of H and $N_H$ be the subfamily consisting of all relatively compact subsets of $H.$

Definition 1

([20]). Let $J\in M_H$ and

$$\nu \left(J\right)=inf\left\{ϵ>0:J=\bigcup _{i=1}^n{J}_{i}withdiam{J}_i\le ϵ,i=1,...,n\right\},$$

where

$$diamJ=sup\{\parallel x_1-x_2\parallel :x_1,x_2\in J\},$$

$\nu \left(J\right)$ is called the Kuratowski MNC.

Definition 2

([20]). A mapping $\nu :M_H\to R_{+}$ is called the MNC in H if it meets the criteria listed below:

$\left(i\right)$

Ker$\nu =\{J\in M_H:\nu \left(J\right)=0\}$ is non-empty and Ker$\nu \subset N_H,$

$\left(ii\right)$

$\nu \overline{\left(S\right)}=\nu \left(J\right)$;

$\left(iii\right)$

$\nu \left(ConvS\right)=\nu \left(J\right)$;

$\left(iv\right)$

$\nu (\lambda S+(1-\lambda )S_1)\le \lambda \nu \left(J\right)+(1-\lambda )\nu \left(J_1\right)$ for $\lambda \in [0,1]$ for $J,J_1\in M_H,$

$\left(v\right)$

If $J_i$ is a decreasing sequence of sets in $M_H$ such that ${lim}_{i\to \infty }\nu \left(J_i\right)=0,$ then the intersection set $J_{\infty }={\cap }_{i=1}^{\infty }{J}_i$ is non-empty;

$\left(vi\right)$

$J\subset J_1\Rightarrow \nu \left(J\right)\le \nu \left(J_1\right).$

The family Ker ν is said to be the kernel of the MNC ν and intersection set $J_{\infty }\in Ker\nu$. Since $\nu \left(J_{\infty }\right)\le \nu \left(J_i\right)$ for any i, we have $\nu \left(J_{\infty }\right)=0$. By $\left(i\right)$, we obtain $J_{\infty }\in Ker\nu$.

Definition 3

([21]). An element $(x_1,x_2)\in J\times J$ is called a CFP of a mapping $T:J\times J\to J$ if $T(x_1,x_2)=x_1$ and $T(x_2,x_1)=x_2.$

Theorem 1

([20]). Suppose ${\nu }_1,...,{\nu }_n$ is the MNC in Banach spaces $H_1,...,H_n,$ respectively. Let $F(x_1,...,x_n)=0$ and $F:{[0,\infty )}^n\to [0,\infty )$ be convex iff $x_i=0$ for $i=1,...,n.$ Then,

$$\overline{\nu }\left(J\right)=F({\nu }_1\left(J_1\right),...,{\nu }_n\left(J_n\right))$$

defines an MNC in $H_1\times ...\times H_n$. Denote by $J_i$ the natural projection of J into $H_i$ for each i.

Example 1

([5]). Assume ν is an MNC on H and $F(x_1,x_2)=x_1+x_2$ for any $(x_1,x_2)\in {[0,\infty )}^2.$ Then, we obtain that F is convex and $F(x_1,x_2)=0$ iff $x_1=x_2=0,$ so all the conditions of Theorem 1 are satisfied. Since $\overline{\nu }\left(J\right)=\nu \left(J_1\right)+\nu \left(J_2\right)$, define an MNC in $H\times H,$ where $J_1,$$J_2$ are the natural projections of J into $H.$

Theorem 2

(Schauder [5]). Let Γ be a convex closed subset of $H.$ Then, each continuous, compact map $W:\mathsf{\Gamma }\to \mathsf{\Gamma }$ possesses at least one fixed point.

Theorem 3

(Darbo [22]). Let the non-empty subset Γ of H be convex, bounded and closed, and let $W:\mathsf{\Gamma }\to \mathsf{\Gamma }$ be a continuous mapping. Suppose that there ∃ a constant $0\le k<1$ such that

$$\nu \left(W\right(J\left)\right)\le k\nu \left(J\right),$$

for any $J\in \mathsf{\Gamma }.$ Then, W has a fixed point.

Lemma 1

([23]). Let $\chi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ be a non-decreasing and upper semi-continuous function. Then, the following conditions are equivalent:

$\left(i\right)$

${lim}_{n\to \infty }{\chi }^n\left(\varrho \right)=0,\varrho >0.$

$\left(ii\right)$

$\chi \left(\varrho \right)<\varrho ,\varrho >0.$

2. Main Results

In this section, let $\widehat{\chi }$ be the class of all functions $\chi :{\mathbb{R}}_{+}^2\to {\mathbb{R}}_{+}$ and satisfy the following conditions:

$\left(i\right)$$\chi$ is a non-decreasing continuous function on ${\mathbb{R}}_{+}^2.$

$\left(ii\right)$${lim}_{n\to \infty }{\theta }^n\left(\varrho \right)=0$ for every $\varrho >0,$ where $\theta \left(\varrho \right)=\chi (\varrho ,\varrho ).$

$\left(iii\right)$$\frac{1}{2}\chi ({\varrho }_1,{\tau }_1)+\frac{1}{2}\chi ({\varrho }_2,{\tau }_2)\le \chi (\frac{{\varrho }_1+{\varrho }_2}{2},\frac{{\tau }_1+{\tau }_2}{2})$ for all ${\varrho }_1,{\varrho }_2,{\tau }_1,{\tau }_2\in {\mathbb{R}}_{+}.$

• For example, $\chi (\varrho ,\tau )=ln(1+\frac{\varrho +\tau }{2})\in \widehat{\chi }.$

Theorem 4

([8]). Let Γ be a non-empty, bounded, convex and closed subset of $H,$ and let the continuous function $W:\mathsf{\Gamma }\times \mathsf{\Gamma }\to \mathsf{\Gamma }$ satisfy

$$\overline{\nu }\left(W\left(J\right)\right)\le \chi (\overline{\nu }\left(J\right),\overline{\nu }\left(J\right)),$$

for an arbitrary non-empty subset $J\subseteq \mathsf{\Gamma }\times \mathsf{\Gamma },$ where $\overline{\nu }$ is defined in Example 1 and $\chi \in \widehat{\chi }.$ Then, W has a CFP in $\mathsf{\Gamma }\times \mathsf{\Gamma }.$

Theorem 5

([8]). Let the non-empty subset Γ of H be convex, bounded and closed, and ν be an arbitrary MNC. Further, let continuous function $W:\mathsf{\Gamma }\times \mathsf{\Gamma }\to \mathsf{\Gamma }$ satisfy

$$\overline{\nu }\left(W(J_1\times J_2)\right)\le \chi (\overline{\nu }\left(J_1\right),\overline{\nu }\left(J_2\right))$$

for non-empty subsets $J_1,J_2\subseteq \mathsf{\Gamma },$ where $\chi \in \widehat{\chi }.$ Then, W has a CFP.

For CFP theorems and its application in integral equations, readers can see [24].

Now, let $H=BC({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ be a Banach space consisting of all real valued continuous bounded functions on ${\mathbb{R}}_{+}^2$ with max norm

$$\parallel x\parallel =max\left\{\right|x(\varrho ,\tau )|:\varrho ,\tau \in {\mathbb{R}}_{+}\}.$$

Let J be a fixed, non-empty and bounded subset of $H=BC({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ and $U>0.$ For given $ϵ>0$ and $x\in J$, the modulus of continuity of the function x on the bounded-closed interval [0, U] is denoted by ${\omega }^U(x,ϵ)$ and defined as

$${\omega }^U(x,ϵ)=sup\left\{\right|x(\varrho ,\tau )-x(\widehat{\varrho },\widehat{\tau })|:\varrho ,\tau ,\widehat{\varrho },\widehat{\tau }\in [0,U],|\varrho -\widehat{\varrho }|\le ϵ,|\tau -\widehat{\tau }|\le ϵ\}.$$

Next, we have

$${\omega }^U(J,ϵ)=sup\{{\omega }^U(x,ϵ):x\in J\},$$

$${\omega }_0^U\left(J\right)=\underset{ϵ\to 0}{lim}{\omega }^U(J,ϵ),$$

and

$${\omega }_0\left(J\right)=\underset{U\to \infty }{lim}{\omega }_0^U\left(J\right).$$

Now, for fixed numbers $\varrho ,\tau \in {\mathbb{R}}_{+}$, let us define

$$J(\varrho ,\tau )=\left\{x\right(\varrho ,\tau ):x\in J\},$$

$$diamJ(\varrho ,\tau )=sup\left\{\right|x_1(\varrho ,\tau )-x_2(\varrho ,\tau )|:x_1,x_2\in J\}.$$

Finally,

$$\nu \left(J\right)={\omega }_0\left(J\right)+\underset{\varrho ,\tau \to \infty }{lim}supdiam\left(J(\varrho ,\tau )\right)$$

The function $\nu$ is an MNC in H.

Now, we will analyze the existence of a solution for the non-linear functional integral equations system

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \left\{x\right(\varrho ,\tau )=C(\varrho ,\tau )+g(\varrho ,\tau ,x(\varrho ,\tau ),y(\varrho ,\tau ))\hfill \\ & +P\left(\varrho ,\tau ,x(\varrho ,\tau ),y(\varrho ,\tau ),{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \right)\hfill \\ & y(\varrho ,\tau )=C(\varrho ,\tau )+g(\varrho ,\tau ,y(\varrho ,\tau ),x(\varrho ,\tau \left)\right)\hfill \\ & +P\left(\varrho ,\tau ,y(\varrho ,\tau ),x(\varrho ,\tau ),{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}h(\varrho ,\tau ,\eta ,\mu ,y(\eta ,\mu ),x(\eta ,\mu ))d\mu d\eta \right)\}.\hfill \end{array}\end{array}$$

(1)

Suppose that

$\left(1\right)$

$C:{\mathbb{R}}_{+}^2\to \mathbb{R}$ is continuous with $\widehat{C}=sup\left\{\right|C(\varrho ,\tau )|:\varrho ,\tau \in {\mathbb{R}}_{+}\}.$

$\left(2\right)$

$a,b:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ are continuous.

$\left(3\right)$

$P:{\mathbb{R}}_{+}^2\times {\mathbb{R}}^3\to \mathbb{R}$ and $g:{\mathbb{R}}_{+}^2\times {\mathbb{R}}^2\to \mathbb{R}$ are continuous and ∃ a $\theta \in \widehat{\chi },$ and a continuous non-decreasing function $\widehat{\theta }:{\mathbb{R}}_{+}\to \mathbb{R}$ such that $\widehat{\theta }\left(0\right)=0$ and

$$|g(\varrho ,\tau ,x,y)-g(\varrho ,\tau ,\widehat{x},\widehat{y})|\le \frac{1}{2}\theta \left(\right|x-\widehat{x}|,|y-\widehat{y}\left|\right),$$

$$|P(\varrho ,\tau ,x,y,z)-P(\varrho ,\tau ,\widehat{x},\widehat{y},\widehat{z})|\le \frac{1}{2}\theta \left(\right|x-\widehat{x}|,|y-\widehat{y}\left|\right)+\widehat{\theta }\left(\right|z-\widehat{z}\left|\right),$$

for any $\varrho ,\tau \ge 0$ and each $x,y,z,\widehat{x},\widehat{y},\widehat{z}\in \mathbb{R}.$

$\left(4\right)$

$P(\varrho ,\tau ,0,0,0)$ and $g(\varrho ,\tau ,0,0)$ are bounded on ${\mathbb{R}}_{+}$, i.e.,

$$G_1=sup\left\{\right|P(\varrho ,\tau ,0,0,0)|:\varrho ,\tau \in {\mathbb{R}}_{+}\}<\infty .$$

$$G_2=sup\left\{\right|g(\varrho ,\tau ,0,0,)|:\varrho ,\tau ,\in {\mathbb{R}}_{+}\}<\infty .$$

$\left(5\right)$

$h:{\mathbb{R}}_{+}^4\times {\mathbb{R}}^2\to \mathbb{R}$ is a continuous function such that

$$G=sup\left\{\left|{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \right|:\varrho ,\tau ,\eta ,\mu \in {\mathbb{R}}_{+},x,y\in H\right\}$$

is finite. Moreover,

$$\underset{\varrho ,\tau \to \infty }{lim}{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}|h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))-h(\varrho ,\tau ,\eta ,\mu ,\widehat{x}(\eta ,\mu ),\widehat{y}(\eta ,\mu ))|d\mu d\eta =0$$

for each $x,y,\widehat{x},\widehat{y}\in H.$

$\left(6\right)$

There ∃ a positive solution $\rho$ of the inequality

$$\widehat{C}+\theta (\sigma ,\sigma )+G_1+G_2+\widehat{\theta }\left(G\right)\le \sigma .$$

Theorem 6

With the assumptions $\left(1\right)-\left(6\right)$ Equation (1) has at least one solution in $H\times H.$

Proof. 

Let $W:H\times H\to H$ be defined as

$$\begin{array}{c}\hfill \begin{array}{cc}& W(x,y)(\varrho ,\tau )=C(\varrho ,\tau )+g(\varrho ,\tau ,x(\varrho ,\tau ),y(\varrho ,\tau \left)\right)\hfill \\ & +P\left(\varrho ,\tau ,x(\varrho ,\tau ),y(\varrho ,\tau ),{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \right).\hfill \end{array}\end{array}$$

Moreover, the space $H\times H$ is a Banach space with the usual norm ${\parallel (x,y)\parallel }_{H\times H}={\parallel x\parallel }_H+{\parallel y\parallel }_H,$ where ${\parallel x\parallel }_H=sup\left\{\right|x(\varrho ,\tau )|:\varrho ,\tau \ge {0\},\parallel y\parallel }_H=sup\left\{\right|y(\varrho ,\tau )|:\varrho ,\tau \ge 0\}.$ It is clear that $W(x,y)(\varrho ,\tau )$ is continuous for each $x,y\in H.$ Let $B_{\rho }={\{x\in H:\parallel x\parallel }_H\le \rho \}$. Then,

$$\begin{array}{c}\hfill \begin{array}{cc}& \left|W\right(x,y\left)\right(\varrho ,\tau \left)\right|\le \left|C\right(\varrho ,\tau \left)\right|+\left|g\right(\varrho ,\tau ,x(\varrho ,\tau ),y(\varrho ,\tau ))-g(\varrho ,\tau ,0,0\left)\right|\hfill \\ & +\left|P\left(\varrho ,\tau ,x(\varrho ,\tau ),y(\varrho ,\tau ),{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \right)-P(\varrho ,\tau ,0,0,0)\right|\hfill \\ & +\left|P\right(\varrho ,\tau ,0,0,0\left)\right|+\left|g\right(\varrho ,\tau ,0,0\left)\right|\hfill \\ & \le \widehat{C}+\frac{1}{2}\theta \left(\right|x(\varrho ,\tau )|,|y(\varrho ,\tau )\left|\right)+\frac{1}{2}\theta \left(\right|x(\varrho ,\tau )|,|y(\varrho ,\tau )\left|\right)\hfill \\ & +\widehat{\theta }\left(\left|{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \right|\right)+G_1+G_2\hfill \\ & \le \widehat{C}+\theta \left(\right|x(\varrho ,\tau )|,|y(\varrho ,\tau )\left|\right)+\widehat{\theta }\left(\left|{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \right|\right)+G_1+G_2\hfill \\ & \le \widehat{C}+\theta (\parallel x\parallel ,\parallel y\parallel )+\widehat{\theta }\left(G\right)+G_1+G_2.\hfill \end{array}\end{array}$$

Thus, W is well defined and by $\left(6\right)$ we obtain $W(B_{\rho }\times B_{\rho })\subseteq B_{\rho }.$

Now, it is shown that W is continuous on $B_{\rho }\times B_{\rho }.$ Let $(x,y),(q,p)\in B_{\rho }\times B_{\rho }$ with

$${\parallel (x,y)-(q,p)\left|\right|}_{H\times H}<\frac{ϵ}{2}.$$

Now, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& \left|W\right(x,y\left)\right(\varrho ,\tau )-W(q,p\left)\right(\varrho ,\tau \left)\right|\le \left|g\right(\varrho ,\tau ,x(\varrho ,\tau ),y(\varrho ,\tau ))-g(\varrho ,\tau ,q(\varrho ,\tau ),p(\varrho ,\tau )\left)\right|\hfill \\ & +|P\left(\varrho ,\tau ,x(\varrho ,\tau ),y(\varrho ,\tau ),{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \right)\hfill \\ & -P\left(\eta ,\mu ,q(\eta ,\mu ),p(\eta ,\mu ),{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}h(\varrho ,\tau ,\eta ,\mu ,q(\eta ,\mu ),p(\eta ,\mu ))d\mu d\eta \right)|\hfill \\ & \le \frac{1}{2}\theta \left(\right|x(\varrho ,\tau )-q(\varrho ,\tau )|,|y(\varrho ,\tau )-p(\varrho ,\tau )\left|\right)+\frac{1}{2}\theta \left(\right|x(\varrho ,\tau )-q(\varrho ,\tau )|,|y(\varrho ,\tau )-p(\varrho ,\tau )\left|\right)\hfill \\ & +\widehat{\theta }\left(\left|{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}\left(h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu \left)\right)-h(\varrho ,\tau ,\eta ,\mu ,q(\eta ,\mu ),p(\eta ,\mu \left)\right)\right)d\mu d\eta \right|\right)\hfill \\ & \le \theta (\parallel x-q\parallel ,\parallel y-p\parallel )\hfill \\ & +\widehat{\theta }\left(\left|{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}\left(h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu \left)\right)-h(\varrho ,\tau ,\eta ,\mu ,q(\eta ,\mu ),p(\eta ,\mu \left)\right)\right)d\mu d\eta \right|\right).\hfill \end{array}\end{array}$$

By $\left(3\right)$ and $\left(5\right),$ for $ϵ>0$$\exists U>0$ such that $\varrho ,\tau >U$ and

$$\begin{array}{c}\hfill \widehat{\theta }\left(\left|{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}\left(h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu \left)\right)-h(\varrho ,\tau ,\eta ,\mu ,q(\eta ,\mu ),p(\eta ,\mu \left)\right)\right)d\mu d\eta \right|\right)\le \frac{ϵ}{2}\end{array}$$

for each $x,y,q,p\in H.$

Case $1:$ If $\varrho ,\tau >U,$ then

$$\begin{array}{c}\hfill |W(x,y)(\varrho ,\tau )-W(q,v)(\varrho ,\tau )|\le \theta \left(\frac{ϵ}{2},\frac{ϵ}{2}\right)+\frac{ϵ}{2}<\frac{ϵ}{2}+\frac{ϵ}{2}=ϵ.\end{array}$$

Case $2:$ If $\varrho ,\tau \in [0,U]$, then

$$\begin{array}{c}\hfill |W(x,y)(\varrho ,\tau )-W(q,v)(\varrho ,\tau )|\le \theta \left(\frac{ϵ}{2},\frac{ϵ}{2}\right)+\widehat{\theta }\left(\widehat{a}\widehat{b}\widehat{w}\right)<\frac{ϵ}{2}+\widehat{\theta }\left(\widehat{a}\widehat{b}\widehat{w}\right),\end{array}$$

where $\widehat{a}=sup\{a\left(\varrho \right):\varrho \in [0,U]\}$, $\widehat{b}=sup\{b\left(\tau \right):\tau \in [0,U]\}$ and

$$\begin{array}{c}\hfill \begin{array}{cc}& \widehat{w}=sup\{|h(\varrho ,\tau ,\eta ,\mu ,x,y)-h(\varrho ,\tau ,\eta ,\mu ,q,p)|:\varrho ,\tau \in [0,U],\eta \in [0,\widehat{a}],\mu \in [0,\widehat{b}],x,y,q,p\in [-\rho ,\rho ],\hfill \\ & {\parallel (x,y)-(q,p)\left|\right|}_{H\times H}<\frac{ϵ}{2}\}.\hfill \end{array}\end{array}$$

Using the continuity of h on $[0,U]\times [0,U]\times [0,\widehat{a}]\times [0,\widehat{b}]\times [-\rho ,\rho ]\times [-\rho ,\rho ],$ we obtain $\widehat{w}\to 0$ as $ϵ\to 0$ i.e., since $ϵ\to 0$ gives $\widehat{a}\widehat{b}\widehat{w}\to 0$, so $\widehat{\theta }\left(\widehat{a}\widehat{b}\overline{w}\right)\to 0.$ Therefore, W is a continuous mapping from $B_{\rho }\times B_{\rho }$ into $B_{\rho }.$ Now, we prove that W satisfies the condition of the Theorem $.$ Let $U,ϵ\in {\mathbb{R}}_{+}$ and $J_1,J_2$ be arbitrary non-empty subsets of $B_{\rho }$ and suppose ${\varrho }_1,{\varrho }_2,{\tau }_1,{\tau }_2\in [0,U],$ such that $|{\varrho }_1-{\varrho }_2|\le ϵ,|{\tau }_1-{\tau }_2|\le ϵ.$

Now, we can suppose that $a\left({\varrho }_1\right)<a\left({\varrho }_2\right)$ and $b\left({\tau }_1\right)<b\left({\tau }_2\right).$ Then,

$$\begin{array}{c}\hfill \begin{array}{cc}& |W(x,y)({\varrho }_2,{\tau }_2)-W(x,y)({\varrho }_1,{\tau }_1)|\le |C({\varrho }_2,{\tau }_2)-C({\varrho }_1,{\tau }_1)|+|g({\varrho }_2,{\tau }_2,x({\varrho }_2,{\tau }_2),y({\varrho }_2,{\tau }_2))\hfill \\ & -g({\varrho }_2,{\tau }_2,x({\varrho }_1,{\tau }_1),y({\varrho }_1,{\tau }_1))|+|g({\varrho }_2,{\tau }_2,x({\varrho }_1,{\tau }_1),y({\varrho }_1,{\tau }_1))-g({\varrho }_1,{\tau }_1,x({\varrho }_1,{\tau }_1),y({\varrho }_1,{\tau }_1))|\hfill \\ & +|P\left({\varrho }_2,{\tau }_2,x({\varrho }_2,{\tau }_2),y({\varrho }_2,{\tau }_2),{\int }_0^{a\left({\varrho }_2\right)}{\int }_0^{b\left({\tau }_2\right)}h({\varrho }_2,{\tau }_2,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \right)\hfill \\ & -P\left({\varrho }_2,{\tau }_2,x({\varrho }_1,{\tau }_1),y({\varrho }_1,{\tau }_1),{\int }_0^{a\left({\varrho }_2\right)}{\int }_0^{b\left({\tau }_2\right)}h({\varrho }_2,{\tau }_2,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \right)|\hfill \\ & +|P\left({\varrho }_2,{\tau }_2,x({\varrho }_1,{\tau }_1),y({\varrho }_1,{\tau }_1),{\int }_0^{a\left({\varrho }_2\right)}{\int }_0^{b\left({\tau }_2\right)}h({\varrho }_2,{\tau }_2,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \right)\hfill \\ & -P\left({\varrho }_1,{\tau }_1,x({\varrho }_1,{\tau }_1),y({\varrho }_1,{\tau }_1),{\int }_0^{a\left({\varrho }_2\right)}{\int }_0^{b\left({\tau }_2\right)}h({\varrho }_2,{\tau }_2,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \right)|\hfill \\ & +|P\left({\varrho }_1,{\tau }_1,x({\varrho }_1,{\tau }_1),y({\varrho }_1,{\tau }_1),{\int }_0^{a\left({\varrho }_2\right)}{\int }_0^{b\left({\tau }_2\right)}h({\varrho }_2,{\tau }_2,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \right)\hfill \\ & -P\left({\varrho }_1,{\tau }_1,x({\varrho }_1,{\tau }_1),y({\varrho }_1,{\tau }_1),{\int }_0^{a\left({\varrho }_2\right)}{\int }_0^{b\left({\tau }_2\right)}h({\varrho }_1,{\tau }_1,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \right)|\hfill \\ & +|P\left({\varrho }_1,{\tau }_1,x({\varrho }_1,{\tau }_1),y({\varrho }_1,{\tau }_1),{\int }_0^{a\left({\varrho }_2\right)}{\int }_0^{b\left({\tau }_2\right)}h({\varrho }_1,{\tau }_1,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \right)\hfill \\ & -P\left({\varrho }_1,{\tau }_1,x({\varrho }_1,{\tau }_1),y({\varrho }_1,{\tau }_1),{\int }_0^{a\left({\varrho }_1\right)}{\int }_0^{b\left({\tau }_1\right)}h({\varrho }_1,{\tau }_1,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \right)|\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}& \le {\omega }^U(C,ϵ)+\frac{1}{2}\theta \left(\right|x({\varrho }_2,{\tau }_2)-x({\varrho }_1,{\tau }_1)|,|y({\varrho }_2,{\tau }_2)-y({\varrho }_1,{\tau }_1)\left|\right)+{\omega }_{\rho }^U(g,ϵ)\hfill \\ & +\frac{1}{2}\theta \left(\right|x({\varrho }_2,{\tau }_2)-x({\varrho }_1,{\tau }_1)|,|y({\varrho }_2,{\tau }_2)-y({\varrho }_1,{\tau }_1)\left|\right)+{\omega }_{\rho }^U(P,ϵ)\hfill \\ & +\widehat{\theta }\left(\left|{\int }_0^{a\left({\varrho }_2\right)}{\int }_0^{b\left({\tau }_2\right)}(h({\varrho }_2,{\tau }_2,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))-h({\varrho }_1,{\tau }_1,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu )))d\mu d\eta \right|\right)\hfill \\ & +\widehat{\theta }\left(\right|{\int }_0^{a\left({\varrho }_2\right)}{\int }_0^{b\left({\tau }_2\right)}h({\varrho }_1,{\tau }_1,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \hfill \\ & -{\int }_0^{a\left({\varrho }_1\right)}{\int }_0^{b\left({\tau }_1\right)}h({\varrho }_1,{\tau }_1,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \left|\right)\hfill \\ & \le {\omega }^U(C,ϵ)+{\omega }_{\rho }^U(g,ϵ)+\theta ({\omega }^U(x,ϵ),{\omega }^U(y,ϵ))+{\omega }_{\rho }^U(P,ϵ)+\widehat{\theta }\left(\widehat{a}\widehat{b}{\omega }_{\rho }^U(h,ϵ)\right)\hfill \\ & +\widehat{\theta }\left(|{\int }_0^{a\left({\varrho }_2\right)}{\int }_0^{b\left({\tau }_2\right)}h({\varrho }_1,{\tau }_1,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta -{\int }_0^{a\left({\varrho }_1\right)}{\int }_0^{b\left({\tau }_1\right)}h({\varrho }_1,{\tau }_1,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta |\right),\hfill \end{array}\end{array}$$

where

$${\omega }^U(C,ϵ)=sup\left\{\right|C({\varrho }_2,{\tau }_2)-C({\varrho }_2,{\tau }_2)|:{\varrho }_1,{\varrho }_2,{\tau }_1,{\tau }_2\in [0,U],|{\varrho }_1-{\varrho }_2|\le ϵ,|{\tau }_1-{\tau }_2|\le ϵ\},$$

${\omega }_{\rho }^U(g,ϵ)=sup\left\{\right|g({\varrho }_2,{\tau }_2,x,y)-g({\varrho }_1,{\tau }_1,x,y)|:{\varrho }_1,{\varrho }_2,{\tau }_1,{\tau }_2\in [0,U],|{\varrho }_1-{\varrho }_2|\le ϵ,|{\tau }_1-{\tau }_2|\le ϵ,x,y\in [-\rho ,\rho ]\},$

$${\omega }^U(x,ϵ)=sup\left\{\right|x({\varrho }_2,{\tau }_2)-x({\varrho }_1,{\tau }_1)|:{\varrho }_1,{\varrho }_2,{\tau }_1,{\tau }_2\in [0,U],|{\varrho }_1-{\varrho }_2|\le ϵ,|{\tau }_1-{\tau }_2|\le ϵ\},$$

$${\omega }^U(y,ϵ)=sup\left\{\right|y({\varrho }_2,{\tau }_2)-y({\varrho }_1,{\tau }_1)|:{\varrho }_1,{\varrho }_2,{\tau }_1,{\tau }_2\in [0,U],|{\varrho }_1-{\varrho }_2|\le ϵ,|{\tau }_1-{\tau }_2|\le ϵ\},$$

$$G=\widehat{a}\widehat{b}sup\left\{\right|h({\varrho }_1,{\tau }_1,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))|:\varrho ,\tau \in [0,U],\eta \in [0,\widehat{a}],\mu \in [0,\widehat{b}],x,y\in [-\rho ,\rho ]\},$$

$${\omega }_{\rho }^U(P,ϵ)=sup\left\{\right|P({\varrho }_2,{\tau }_2,x,y,z)-P({\varrho }_1,{\tau }_1,x,y,z)|:{\varrho }_1,{\varrho }_2,{\tau }_1,{\tau }_2\in [0,U],|{\varrho }_1-{\varrho }_2|\le ϵ,$$

$$|{\tau }_1-{\tau }_2|\le ϵ,x,y\in [-\rho ,\rho ],z\in [-G,G]\},$$

$${\omega }_{\rho }^U(h,ϵ)=sup\left\{\right|h({\varrho }_2,{\tau }_2,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))-h({\varrho }_1,{\tau }_1,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu )):{\varrho }_1,{\varrho }_2,{\mu }_1,{\mu }_2\in [0,U],$$

$$|{\varrho }_1-{\varrho }_2|\le ϵ,|{\tau }_1-{\tau }_2|\le ϵ,x,y\in [-\rho ,\rho ],\eta \in [0,\widehat{a}],\mu \in [0,\widehat{b}]\}.$$

We know that $(x,y)$ is an arbitrary element of $J_1\times J_2,$ then

$$\begin{array}{c}\hfill \begin{array}{cc}& {\omega }^U(W(J_1\times J_2),ϵ)\le {\omega }^U(C,ϵ)+{\omega }_{\rho }^U(g,ϵ)+\theta ({\omega }^U(x,ϵ),{\omega }^U(y,ϵ)+{\omega }_{\rho }^U(P,ϵ)\hfill \\ & +\widehat{\theta }\left(\widehat{a}\widehat{b}{\omega }_{\rho }^U(h,ϵ)\right)+\widehat{\theta }\left(\right|{\int }_0^{a\left({\varrho }_2\right)}{\int }_0^{b\left({\tau }_2\right)}h({\varrho }_1,{\tau }_1,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \hfill \\ & -{\int }_0^{a\left({\varrho }_1\right)}{\int }_0^{b\left({\tau }_1\right)}h({\varrho }_1,{\tau }_1,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \left|\right).\hfill \end{array}\end{array}$$

(2)

From uniform continuity of g on $[0,U]\times [0,U]\times [-\rho ,\rho ]\times [-\rho ,\rho ],$ P on $[0,U]\times [0,U]\times [-\rho ,\rho ]\times [-\rho ,\rho ]\times [-G,G],$ and h on $[0,U]\times [0,U]\times [0,\widehat{a}]\times [0,\widehat{b}]\times [-\rho ,\rho ]\times [-\rho ,\rho ],$ for $ϵ\to 0,$ we obtain ${\omega }^U(C,ϵ)\to 0,{\omega }_{\rho }^U(g,ϵ)\to 0,{\omega }_{\rho }^U(P,ϵ)\to 0$ and ${\omega }_{\rho }^U(h,ϵ))\to 0.$

Again, by uniform continuity of a and b on $[0,U]$ and $ϵ\to 0$, we have $a\left({\varrho }_2\right)\to a\left({\varrho }_1\right)$ and $b\left({\tau }_2\right)\to b\left({\tau }_1\right).$ Then,

$$\begin{array}{c}\hfill \begin{array}{cc}& |{\int }_0^{a\left({\varrho }_2\right)}{\int }_0^{b\left({\tau }_2\right)}h({\varrho }_1,{\tau }_1,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \hfill \\ & -{\int }_0^{a\left({\varrho }_1\right)}{\int }_0^{b\left({\tau }_1\right)}h({\varrho }_1,{\tau }_1,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta |\to 0,\hfill \end{array}\end{array}$$

which gives

$$\begin{array}{c}\hfill \begin{array}{cc}& \widehat{\theta }\left(\right|{\int }_0^{a\left({\varrho }_2\right)}{\int }_0^{b\left({\tau }_2\right)}h({\varrho }_1,{\tau }_1,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \hfill \\ & -{\int }_0^{a\left({\varrho }_1\right)}{\int }_0^{b\left({\tau }_1\right)}h({\varrho }_1,{\tau }_1,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \left|\right)\to 0.\hfill \end{array}\end{array}$$

Apply the limit as $ϵ\to 0$, then

$${\omega }_0^U\left(W(J_1\times J_2)\right)\le \theta ({\omega }_0^U\left(J_1\right),{\omega }_0^U\left(J_2\right)).$$

As $U\to \infty ,$ we obtain

$${\omega }_0\left(W(J_1\times J_2)\right)\le \theta ({\omega }_0\left(J_1\right),{\omega }_0\left(J_2\right)).$$

(3)

For arbitrary $(x,y),(q,p)\in J_1\times J_2$ and $\varrho ,\tau \in {\mathbb{R}}_{+}$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}& \left|W\right(x,y\left)\right(\varrho ,\tau )-W(q,p\left)\right(\varrho ,\tau \left)\right|\le \left|h\right(\varrho ,\tau ,x(\varrho ,\tau ),y(\varrho ,\tau ))-h(\varrho ,\tau ,q(\varrho ,\tau ),p(\varrho ,\tau )\left)\right|\hfill \\ & +|P\left(\varrho ,\tau ,x(\varrho ,\tau ),y(\varrho ,\tau ),{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \right)\hfill \\ & -P\left(\varrho ,\tau ,q(\varrho ,\tau ),p(\varrho ,\tau ),{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}h(\varrho ,\tau ,\eta ,\mu ,q(\eta ,\mu ),p(\eta ,\mu ))d\mu d\eta \right)|\hfill \\ & \le \frac{1}{2}\theta \left(\right|x(\varrho ,\tau )-q(\varrho ,\tau )|,|y(\varrho ,\tau )-p(\varrho ,\tau )\left|\right)+\frac{1}{2}\widehat{\theta }\left(\right|x(\varrho ,\tau )-q(\varrho ,\tau )|,|y(\varrho ,\tau )-p(\varrho ,\tau )\left|\right)\hfill \\ & +\widehat{\theta }\left(\left|{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}\left(h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu \left)\right)-h(\varrho ,\tau ,\eta ,\mu ,q(\eta ,\mu ),p(\eta ,\mu \left)\right)\right)d\mu d\eta \right|\right)\hfill \\ & \le \varrho \left(diam\right(J_1(\varrho ,\tau ),diam\left(J_2(\varrho ,\tau )\right)\hfill \\ & +\widehat{\theta }\left(\left|{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}\left(h(\varrho ,\tau ,\eta ,\mu ,q(\eta ,\mu ),p(\eta ,\mu \left)\right)-h(\varrho ,\tau ,\eta ,\mu ,q(\eta ,\mu ),p(\eta ,\mu \left)\right)\right)d\mu d\eta \right|\right).\hfill \end{array}\end{array}$$

Since $(x,y),(q,p)$ and $(\varrho ,\tau )$ are arbitrary,

$$\begin{array}{c}\hfill \begin{array}{cc}& diamW(J_1\times J_2)(\varrho ,\tau )\le \varrho \left(diam\right(J_1(\varrho ,\tau ),diam\left(J_2(\varrho ,\tau )\right)\hfill \\ & +\widehat{\theta }\left(\left|{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}\left(h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu \left)\right)-h(\varrho ,\tau ,\eta ,\mu ,q(\eta ,\mu ),p(\eta ,\mu \left)\right)\right)d\mu d\eta \right|\right).\hfill \end{array}\end{array}$$

As $\varrho ,\tau \to \infty ,$ by $\left(5\right)$

$$\begin{array}{c}\hfill \underset{\varrho ,\tau \to \infty }{lim}supdiamW(J_1\times J_2)(\varrho ,\tau )\le \theta \left(\underset{\varrho ,\tau \to \infty }{lim}supdiam\left(J_1(\varrho ,\tau )\right),\underset{\varrho ,\tau \to \infty }{lim}supdiam\left(J_2(\varrho ,\tau )\right)\right).\end{array}$$

(4)

From 3 and 4, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& {\omega }_0\left(W(J_1\times J_2)\right)+\underset{\varrho ,\tau \to \infty }{lim}supdiamW(J_1\times J_2)(\varrho ,\tau )\hfill \\ & \le \theta ({\omega }_0\left(J_1\right),{\omega }_0\left(J_2\right))+\theta \left(\underset{\varrho ,\tau \to \infty }{lim}supdiam\left(J_1(\varrho ,\tau )\right),\underset{\varrho ,\tau \to \infty }{lim}supdiam\left(J_2(\varrho ,\tau )\right)\right)\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \le 2\nu \left(\frac{w_0\left(J_1\right)+{lim}_{\varrho ,\tau \to \infty }supdiam\left(J_1(\varrho ,\tau )\right)}{2},\frac{w_0\left(J_2\right))+{lim}_{\varrho ,\tau \to \infty }supdiam\left(J_2(\varrho ,\tau )\right)}{2}\right).\end{array}$$

Therefore,

$$\begin{array}{c}\hfill \frac{1}{2}\nu \left(W(J_1\times J_2)\right)\le \theta \left(\frac{\nu \left(J_1\right)}{2},\frac{\nu \left(J_2\right)}{2}\right).\end{array}$$

Putting $\nu =\frac{1}{2}{\nu }_1$, we have

$$\begin{array}{c}\hfill {\nu }_1\left(W(J_1\times J_2)\right)\le \theta \left(\nu \left(J_1\right),\nu \left(J_2\right)\right).\end{array}$$

By Theorem 5, J has a CFP in $H\times H.$ □

Corollary 1

([8]). Let

$\left(C_1\right)$

$a,b:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ are continuous.

$\left(C_2\right)$

$P:{\mathbb{R}}_{+}^2\times {\mathbb{R}}^3\to \mathbb{R}$ is continuous and ∃ a function $\theta \in \widehat{\chi },$ and a continuous non-decreasing function $\widehat{\theta }:{\mathbb{R}}_{+}\to \mathbb{R}$ with $\widehat{\theta }\left(0\right)=0$ such that

$$|P(\varrho ,\tau ,x,y,z)-P(\varrho ,\tau ,\widehat{x},\widehat{y},\widehat{z})|\le \theta (|x-\widehat{x}|,|y-\widehat{y}\left|\right)+\widehat{\theta }\left(\right|z-\widehat{z}\left|\right),$$

for each $\varrho ,\tau \ge 0$ and for every $x,y,z,\widehat{x},\widehat{y},\widehat{z}\in H.$

$\left(C_3\right)$

$P(\varrho ,\tau ,0,0,0)$ is bounded, i.e.,

$$G_1=sup\left\{\right|P(\varrho ,\tau ,0,0,0)|:\varrho ,\tau \in {\mathbb{R}}_{+}\}<\infty .$$

$\left(C_4\right)$

$h:{\mathbb{R}}_{+}^4\times {\mathbb{R}}^2\to \mathbb{R}$ is a continuous function such that the constant

$$G=sup\left\{\left|{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \right|:\varrho ,\tau ,\eta ,\mu \in {\mathbb{R}}_{+},x,y\in H\right\}$$

is finite. Moreover,

$$\underset{\varrho ,\tau \to \infty }{lim}{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}|h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))-h(\varrho ,\tau ,\eta ,\mu ,\widehat{x}(\eta ,\mu ),\widehat{y}(\eta ,\mu ))|d\mu d\eta =0$$

for all $x,y,\widehat{x},\widehat{y}\in H.$

$\left(C_5\right)$

∃ a positive solution ρ of the inequality

$$\theta (\sigma ,\sigma )+\widehat{C}+\widehat{\theta }\left(G\right)\le \sigma .$$

If $C(\varrho ,\tau )=0$ and $g(\varrho ,\tau ,x(\varrho ,\tau ),y(\varrho ,\tau \left)\right)=0$, then Equation (1) has the following form

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \{x(\varrho ,\tau )=P\left(\varrho ,\tau ,x(\varrho ,\tau ),y(\varrho ,\tau ),{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \right)\\ \hfill y(\varrho ,\tau )=P\left(\varrho ,\tau ,x(\varrho ,\tau ),y(\varrho ,\tau ),{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \right)\}.\end{array}\end{array}$$

(5)

By assumptions $\left(C_1\right)-\left(C_5\right)$, Equation (5) has at least one solution in $H\times H$ [8].

Example 2.

Consider the SIE

$$\begin{array}{cc}\hfill & x(\varrho ,\tau )=\frac{1}{7}{e}^{-({\varrho }^2+{\tau }^2)}+\frac{{\varrho }^4}{6(1+{\varrho }^4)}+\left(\frac{1}{4}{e}^{-({\varrho }^2+{\tau }^2)}+\frac{{\varrho }^2}{2(1+{\varrho }^4)}\right)ln(1+|x(\varrho ,\tau )\left|\right)\hfill \\ & +\left(\frac{2{\varrho }^2}{2+6{\varrho }^4}+\frac{1}{12}{e}^{-\left({\varrho }^2\right)}\right)ln(1+|y(\varrho ,\tau )\left|\right)+\frac{1}{14}{e}^{-({\varrho }^4+{\tau }^4)}\hfill \\ & +sin\left({\int }_0^{\varrho }{\int }_0^{\tau }\left(\frac{si{n}^2(1+2{\eta }^{2}y(\eta ,\mu ))+cos\left({\mu }^{3}x(\eta ,\mu )\right)d\mu d\eta }{e^{{\varrho }^2{\tau }^2}}\right)\right),\hfill \\ \hfill & y(\varrho ,\tau )=\frac{1}{7}{e}^{-({\varrho }^2+{\tau }^2)}+\frac{{\varrho }^3+{\tau }^3}{3(1+{\varrho }^3)(1+{\tau }^3)}+\left(\frac{1}{4}{e}^{-({\varrho }^2+{\tau }^2)}+\frac{{\varrho }^2}{2(1+{\varrho }^4)}\right)ln(1+|y(\varrho ,\tau )\left|\right)\hfill \\ & +\left(\frac{2{\varrho }^2}{2+6{\varrho }^4}+\frac{1}{12}{e}^{-\left({\varrho }^2\right)}\right)ln(1+|x(\varrho ,\tau )\left|\right)+\frac{1}{14}{e}^{-({\varrho }^4+{\tau }^4)}\hfill \\ & +sin\left({\int }_0^{\varrho }{\int }_0^{\tau }\left(\frac{si{n}^2(1+2{\eta }^{2}x(\eta ,\mu ))+cos\left({\varrho }^{3}y(\eta ,\mu )\right)d\mu d\eta }{e^{{\varrho }^2{\tau }^2}}\right)\right).\hfill \end{array}$$

(6)

Here, Equation (6) is a particular case of Equation (1) with

$$C(\varrho ,\tau )=\frac{1}{7}{e}^{-({\varrho }^2+{\tau }^2)},$$

$$g(u,v,x,y)=\frac{u^4}{6(1+u^4)}+\frac{1}{12}{e}^{-(u^2+v^2)}ln(1+|x(u,v)\left|\right)+\frac{2{\varrho }^2}{2+6{\varrho }^4}ln(1+|y(\varrho ,\tau )\left|\right),$$

$$P(\varrho ,\tau ,x,y,z)=\frac{{\varrho }^2}{2(1+{\varrho }^4)}ln(1+|x(\varrho ,\tau )\left|\right)+\frac{1}{12}{e}^{-\left({\varrho }^2\right)}ln(1+|y(\varrho ,\tau )\left|\right)+\frac{1}{14}{e}^{-({\varrho }^4+{\tau }^4)}+sin\left(z\right),$$

$$h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))=\frac{si{n}^2(1+2{\eta }^{2}x(\eta ,\mu ))+cos\left({\mu }^{3}y(\eta ,\mu )\right)}{e^{{\varrho }^2{\tau }^2}},$$

$$\theta (\varrho ,\tau )=ln\left(1+\frac{\varrho +\tau }{2}\right),a\left(\varrho \right)=\varrho ,b\left(\tau \right)=\tau ,\widehat{\theta }\left(\varrho \right)=\varrho .$$

Based on these values, the operator W is defined as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}& W(x,y)(\varrho ,\tau )=C(\varrho ,\tau )+g(\varrho ,\tau ,x(\varrho ,\tau ),y(\varrho ,\tau \left)\right)\hfill \\ & +P\left(\varrho ,\tau ,x(\varrho ,\tau ),y(\varrho ,\tau ),{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \right).\hfill \end{array}\end{array}$$

By the similar argument as that used in Theorem 6, we can prove that W is continuous on $H\times H.$

Now, we show that conditions of Theorem 6 are holding for SIE (6).

$Step\left(1\right)$: C is continuous and bounded. We have that $\left|C(\varrho ,\tau )\right|=|\frac{1}{7}{e}^{-({\varrho }^2+{\tau }^2)}|$ is bounded for $\varrho ,\tau \in {\mathbb{R}}_{+}$ and $\widehat{C}=\frac{1}{7}.$

$Step\left(2\right)$: $a,b,g,P$ are continuous, $\left|g(\varrho ,\tau ,0,0)\right|=\frac{{\varrho }^4}{6(1+{\varrho }^4)}$ is bounded for $\varrho ,\tau \in {\mathbb{R}}_{+}$ and $G_1=\frac{1}{6}.$ Further, $\left|P(\varrho ,\tau ,0,0,0)\right|=\frac{1}{14}{e}^{-({\varrho }^4+{\tau }^4)}$ is bounded for $\varrho ,\tau \in {\mathbb{R}}_{+}$ and $G_2=\frac{1}{14}.$

$Step\left(3\right)$: Suppose $\varrho ,\tau \in {\mathbb{R}}_{+}$ and $x,y,z,\widehat{x},\widehat{y},\widehat{z}\in H$ with $\left|\right|x\left|\right|\ge \left|\right|\widehat{x}\left|\right|,\left|\right|y\left|\right|\ge \left|\right|\widehat{y}\left|\right|.$ By the Mean Value Theorem for $sin\left(z\right)$ and $ln\left(1+\frac{\varrho +\tau }{2}\right)\in \widehat{\chi },$ we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& |g(\varrho ,\tau ,x,y)-g(\varrho ,\tau ,\widehat{x},\widehat{y})|\hfill \\ & \le \frac{1}{12}{e}^{-({\varrho }^2+{\tau }^2)}|ln(1+\left|\right|x\left|\right|)-ln(1+\widehat{\left|\right|x\left|\right|})|+\frac{2{\varrho }^2}{2+6{\varrho }^4}|ln(1+\left|\right|y\left|\right|)-ln(1+\widehat{\left|\right|y\left|\right|})|\hfill \\ & \le \frac{1}{12}{e}^{-({\varrho }^2+{\tau }^2)}|ln\left(\frac{1+\left|\right|x\left|\right|}{1+\widehat{\left|\right|x\left|\right|}}\right)|+\frac{2{\varrho }^2}{2+6{\varrho }^4}|ln\left(\frac{1+\left|\right|y\left|\right|}{1+\widehat{\left|\right|y\left|\right|}}\right)|\hfill \\ & \le \frac{1}{4}|ln\left(\frac{1+\left|\right|x\left|\right|}{1+\widehat{\left|\right|x\left|\right|}}\right)|+\frac{1}{4}|ln\left(\frac{1+\left|\right|y\left|\right|}{1+\widehat{\left|\right|y\left|\right|}}\right)|\hfill \\ & \le \frac{1}{4}|ln\left(1+\frac{\left|\right|x\left|\right|-\widehat{\left|\right|x\left|\right|}}{1+\widehat{\left|\right|x\left|\right|}}\right)|+\frac{1}{4}|ln\left(1+\frac{\left|\right|y\left|\right|-\widehat{\left|\right|y\left|\right|}}{1+\widehat{\left|\right|y\left|\right|}}\right)|\hfill \\ & \le \frac{1}{4}ln(1+||x-\widehat{x}\left|\right|)+\frac{1}{4}ln(1+||y-\widehat{y}\left|\right|)\hfill \\ & \le \frac{1}{2}ln\left(1+\frac{\left|\right|x-\widehat{x}\left|\right|+\left|\right|y-\widehat{y}\left|\right|}{2}\right)\hfill \\ & =\frac{1}{2}\theta \left(\right||x-\widehat{x}\left|\right|,\left|\right|y-\widehat{y}\left|\right|).\hfill \end{array}\end{array}$$

Again,

$$\begin{array}{c}\hfill \begin{array}{cc}& |P(\varrho ,\tau ,x,y,z)-P(\varrho ,\tau ,\widehat{x},\widehat{y},\widehat{z})|\le \frac{{\varrho }^2}{2(1+{\varrho }^4)}|ln(1+\left|\right|x\left|\right|)-ln(1+\widehat{\left|\right|x\left|\right|})|\hfill \\ & +\frac{1}{12}{e}^{-\left({\varrho }^2\right)}|ln(1+\left|\right|y\left|\right|)-ln(1+\widehat{\left|\right|y\left|\right|})|+||sinz-sin\widehat{z}\left|\right|\hfill \end{array}\\ \hfill \le \frac{{\varrho }^2}{2(1+{\varrho }^4)}|ln\left(\frac{1+\left|\right|x\left|\right|}{1+\widehat{\left|\right|x\left|\right|}}\right)|+\frac{1}{12}{e}^{-\left({\varrho }^2\right)}|ln\left(\frac{1+\left|\right|y\left|\right|}{1+\widehat{\left|\right|y\left|\right|}}\right)|+\left|\right|z-\widehat{z}\left|\right|\\ \hfill \le \frac{1}{4}|ln\left(\frac{1+\left|\right|x\left|\right|}{1+\widehat{\left|\right|x\left|\right|}}\right)|+\frac{1}{4}|ln\left(\frac{1+\left|\right|y\left|\right|}{1+\widehat{\left|\right|y\left|\right|}}\right)|+\left|\right|z-\widehat{z}\left|\right|\\ \hfill \le \frac{1}{4}|ln\left(1+\frac{\left|\right|x\left|\right|-\widehat{\left|\right|x\left|\right|}}{1+\widehat{\left|\right|x\left|\right|}}\right)|+\frac{1}{4}|ln\left(1+\frac{\left|\right|y\left|\right|-\widehat{\left|\right|y\left|\right|}}{1+\widehat{\left|\right|y\left|\right|}}\right)|+\left|\right|z-\widehat{z}\left|\right|\\ \hfill \le \frac{1}{4}ln(1+||x-\widehat{x}\left|\right|)+\frac{1}{4}ln(1+||y-\widehat{y}\left|\right|)+\left|\right|z-\widehat{z}\left|\right|\\ \hfill \le \frac{1}{2}ln\left(1+\frac{\left|\right|x-\widehat{x}\left|\right|+\left|\right|y-\widehat{y}\left|\right|}{2}\right)+\left|\right|z-\widehat{z}\left|\right|\\ \hfill =\frac{1}{2}\theta \left(\right||x-\widehat{x}\left|\right|,\left|\right|y-\widehat{y}\left|\right|)+\eta \left(\right||z-\widehat{z}\left|\right|).\end{array}$$

$Step\left(4\right)$: Since h is continuous, for every $\varrho ,\tau ,\eta ,\mu \in {\mathbb{R}}_{+},$ and $x,y,\widehat{x},\widehat{y}\in H,$ we obtain

$$|h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))-h(\varrho ,\tau ,\eta ,\mu ,\widehat{x}(\eta ,\mu ),\widehat{y}(\eta ,\mu ))|\le \frac{4}{e^{{\varrho }^2{\tau }^2}},$$

$${\int }_0^{\varrho }{\int }_0^{\mu }|h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))-h(\varrho ,\tau ,\eta ,\mu ,\widehat{x}(\eta ,\mu ),\widehat{y}(\eta ,\mu ))|d\mu d\eta \le \frac{4\varrho \tau }{e^{{\varrho }^2{\tau }^2}},$$

$$\underset{\varrho ,\tau \to \infty }{lim}{\int }_0^{a\left(\varrho \right)}{\int }_0^{b\left(\tau \right)}|h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))-h(\varrho ,\tau ,\eta ,\mu ,\widehat{x}(\eta ,\mu ),\widehat{y}(\eta ,\mu ))|d\mu d\eta \le \underset{\varrho ,\tau \to \infty }{lim}\frac{4\varrho \tau }{e^{{\varrho }^2{\tau }^2}}=0$$

for every $x,\widehat{x},y,\widehat{y}\in H.$ Moreover,

$$\left|h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))\right|\le \frac{2}{e^{{\varrho }^2{\tau }^2}}.$$

Additionally,

$${\int }_0^{\varrho }{\int }_0^{\tau }\left|h(\varrho ,\tau ,\eta ,\mu ,x(\eta ,\mu ),y(\eta ,\mu ))d\mu d\eta \right|\le \frac{2\varrho \tau }{e^{{\varrho }^2{\tau }^2}},$$

for any $\varrho ,\tau ,\eta ,\mu \in {\mathbb{R}}_{+}$. Thus,

$$G=sup\left\{\frac{2\varrho \tau }{e^{{\varrho }^2{\tau }^2}}:\varrho ,\tau \ge 0\right\}=\frac{\sqrt{2}}{\sqrt{e}}=0.85776.$$

$Step\left(5\right):$ Put all the value of $\widehat{C},G_1,G_2,\theta$ and $\widehat{\theta }$ in the inequality $\left(6\right).$ Then,

$$\frac{16}{42}+ln(1+\sigma )+0.85776\le \sigma .$$

For $\sigma \ge 3,$ we have

$$\sigma -\frac{16}{42}-ln(1+\sigma )-0.85776>0.$$

Thus, $\rho =3.$ Hence, all the estimates of the Theorem are satisfied. So, Equation (6) has a solution in space $H\times H$.

3. Conclusions

Various generalizations of DFPT are available. Some writers have formed generalizations via MNC. Many other writers have extended DFPT by modifying the domain of mappings that holds a fixed point. In the present article, we utilized contractions to prove that a mapping defined on a non-empty, bounded, closed and convex subset of a given Banach space contains at least one fixed point. We establish the existence result for a SNLIEs in two dimensions.