# Coupled fixed point results on quasi-Banach spaces with application to a system of integral equations

- Nawab Hussain
^{1}, - Peyman Salimi
^{2}and - Saleh Al-Mezel
^{1}Email author

**2013**:261

https://doi.org/10.1186/1687-1812-2013-261

© Hussain et al.; licensee Springer. 2013

**Received: **4 June 2013

**Accepted: **27 August 2013

**Published: **7 November 2013

## Abstract

The aim of this paper is to obtain coupled fixed point theorems for self-mappings defined on an ordered closed and convex subset of a quasi-Banach space. Our method of proof is different and constructive in nature. As an application of our coupled fixed point results, we establish corresponding coupled coincidence point results without any type of commutativity of underlying maps. Moreover, an application to integral equations is given to illustrate the usability of the obtained results.

**MSC:**47H10, 54H25, 55M20.

## Keywords

## Dedication

Dedicated to Professor Wataru Takahashi on the occasion of his seventieth birthday

## 1 Introduction

It is well known that the Banach contraction principle is one of the most important results in classical functional analysis. It is widely considered as the source of metric fixed point theory. Also, its significance lies in its vast applicability in a number of branches of mathematics (see [1–15]). The study of coupled fixed points in partially ordered metric spaces was first investigated in 1987 by Guo and Lakshmikantham [16], and then it attracted many researchers; see, for example, [3, 5] and references therein. Recently, Bhaskar and Lakshmikantham [12] presented coupled fixed point theorems for contractions in partially ordered metric spaces. Luong and Thuan [17] presented nice generalizations of these results. Alsulami *et al.* [3] further extended the work of Luong and Thuan to coupled coincidences in partial metric spaces. For more related work on coupled fixed points and coincidences, we refer the readers to recent results in [13–30].

for any $x,y,z\in X$. In this paper, we adopt a different and constructive method to prove some coupled fixed and coincidence point results for contractive mappings defined on an ordered closed convex subset of a quasi-Banach space. Moreover, an application to integral equations is given to illustrate the usability of the obtained results.

## 2 Preliminaries

The aim of this section is to present some notions and results used in the paper.

**Definition 2.1**Let

*X*be a non-empty set and $d:X\times X\to [0,+\mathrm{\infty})$. $(X,d)$ is a symmetric space (also called

*E*-space) if and only if it satisfies the following conditions:

- (W1)
$d(x,y)=0$ if and only if $x=y$;

- (W2)
$d(x,y)=d(y,x)$ for any $x,y\in X$.

Also, a sequence $\{{x}_{n}\}\subset X$ is said to be a Cauchy sequence if, for every given $\epsilon >0$, there exists a positive integer $n(\epsilon )$ such that $d({x}_{m},{x}_{n})<\epsilon $ for all $m,n\ge n(\epsilon )$. A symmetric space $(X,d)$ is said to be complete if and only if each of its Cauchy sequences converges to some $x\in X$.

**Definition 2.2**Let

*X*be a nonempty set and let $K\ge 1$ be a given real number. A function $d:X\times X\to {\mathbb{R}}_{+}$ is said to be of metric type if and only if for all $x,y,z\in X$ the following conditions are satisfied:

- (1)
$d(x,y)=0$ if and only if $x=y$;

- (2)
$d(x,y)=d(y,x)$;

- (3)
$d(x,z)\le K[d(x,y)+d(y,z)]$.

A triplet $(X,d,K)$ is called a metric-type space.

We observe that a metric-type space is included in the class of symmetric spaces. So the notions of convergent sequence, Cauchy sequence and complete space are defined as in symmetric spaces.

Next, we give some examples of metric-type spaces.

**Example 2.3** [4]

where $x=\{{x}_{n}\},y=\{{y}_{n}\}\in {l}_{p}$, is a metric-type space. By an elementary calculation, we obtain $d(x,y)\le {2}^{1/p}[d(x,y)+d(y,z)]$.

**Example 2.4** [4]

for each $x,y\in {L}^{p}$. The constant *K* is again equal to ${2}^{1/p}$.

*b*-metric) spaces, we refer to [8, 10, 11]. We recall that a quasi-norm $\parallel \cdot \parallel $ defined on a real vector space

*X*is a mapping $X\to {\mathbb{R}}_{+}$ such that:

- (1)
$\parallel x\parallel >0$ if and only if $x\ne 0$;

- (2)
$\parallel \lambda x\parallel =|\lambda |\parallel x\parallel $ for $\lambda \in \mathbb{R}$ and $x\in X$;

- (3)
$\parallel x+y\parallel \le K[\parallel x\parallel +\parallel y\parallel ]$ for all $x,y\in X$, where $K\ge 1$ is a constant independent of

*x*,*y*.

A triplet $(X,\parallel \cdot \parallel ,K)$ is called a quasi-Banach space.

What makes quasi-Banach spaces different from the more classical Banach spaces is the triangle inequality. In quasi-Banach spaces, the triangle inequality is allowed to hold approximately: $\parallel x+y\parallel \le K(\parallel x\parallel +\parallel y\parallel )$ for some constant $K\ge 1$. This relaxation leads to a broad class of spaces. Lebesgue spaces ${L}^{p}$ are Banach spaces for $1\le p\le +\mathrm{\infty}$ and only quasi-Banach spaces for $0<p<1$.

**Remark 2.5** Let $(X,\parallel \cdot \parallel ,K)$ be a quasi-Banach space, then the mapping $d:X\times X\to {\mathbb{R}}_{+}$ defined by $d(x,y)=\parallel x-y\parallel $ for all $x,y\in X$ is a metric-type (*b*-metric). In general, a metric-type (*b*-metric) function *d* is not continuous (see [23, 26]).

The following result is useful for some of the proofs in the paper.

**Lemma 2.6**

*Let*$(X,d,K)$

*be a metric*-

*type space and let*${\{{x}_{k}\}}_{k=0}^{n}\subset X$.

*Then*

From Lemma 2.6, we deduce the following lemma.

**Lemma 2.7**

*Let*$\{{y}_{n}\}$

*be a sequence in a metric*-

*type space*$(X,d,K)$

*such that*

*for some* *λ*, $0<\lambda <1/K$, *and each* $n\in \mathbb{N}$. *Then* $\{{x}_{n}\}$ *and* $\{{y}_{n}\}$ *are two Cauchy sequences in X*.

**Definition 2.8** (Mixed monotone property)

*F*has the mixed monotone property if

*F*is monotone non-decreasing in its first argument and is monotone non-increasing in its second argument. That is, for any $x,y\in X$,

**Definition 2.9** [13]

Let $F:X\times X\to X$. We say that $(x,y)\in X\times X$ is a coupled fixed point of *F* if $F(x,y)=x$ and $F(y,x)=y$.

Lakshmikantham and Ćirić [13] introduced the following concept of a mixed *g*-monotone mapping.

**Definition 2.10** [13]

*F*has the mixed

*g*-monotone property if

*F*is monotone

*g*-non-decreasing in its first argument and is monotone

*g*-non-increasing in its second argument, that is, for any $x,y\in X$,

Note that if *g* is the identity mapping, then this definition reduces to Definition 2.8.

**Definition 2.11** [13]

## 3 Main results

- (A)
If $x\u2aafy$ and $\lambda \in {\mathbb{R}}_{+}$, then $\lambda x\u2aaf\lambda y$;

- (B)
If $x\u2aafy$ and $z\in C$, then $x+z\u2aafy+z$.

The following theorem is our first main result.

**Theorem 3.1**

*Let*$(C,\u2aaf)$

*be an ordered closed and convex subset of a quasi*-

*Banach space*$(X,\parallel \cdot \parallel ,K)$

*where*$1\le K<\sqrt{2}$

*and*$d:X\times X\to {\mathbb{R}}_{+}$

*is such that*$d(x,y)=\parallel x-y\parallel $.

*Assume that*$F:C\times C\to C$

*is a mapping with the mixed monotone property on*

*C*

*and suppose that there exist non*-

*negative real numbers*

*α*,

*β*

*and*

*γ*

*with*$0\le \gamma +2\alpha +2\beta +1<2/{K}^{2}$

*such that*

*for all*$x,y,u,v\in C$,

*for which*$u\u2aafx$

*and*$y\u2aafv$.

*Also suppose that either*

- (a)
*F**is continuous*,*or* - (b)
*C**has the following property*:$\begin{array}{c}\mathit{\text{if a non-decreasing sequence}}\{{x}_{n}\}\to x,\mathit{\text{then}}{x}_{n}\u2aafx\mathit{\text{for all}}n\ge 0,\hfill \\ \mathit{\text{if a non-increasing sequence}}\{{y}_{n}\}\to y,\mathit{\text{then}}y\u2aaf{y}_{n}\mathit{\text{for all}}n\ge 0.\hfill \end{array}$

*If there exist* ${x}_{0},{y}_{0}\in C$ *such that* ${x}_{0}\u2aafF({x}_{0},{y}_{0})$ *and* $F({y}_{0},{x}_{0})\u2aaf{y}_{0}$, *then there exist* $x,y\in C$ *such that* $x=F(x,y)$ *and* $y=F(y,x)$, *that is*, *F* *has a coupled fixed point in* *C*.

*Proof*Let ${x}_{0},{y}_{0}\in C$ be such that ${x}_{0}\u2aafF({x}_{0},{y}_{0})$ and ${y}_{0}\u2ab0F({y}_{0},{x}_{0})$. Then

*F*has the mixed monotone property, so we have

Then, by mathematical induction, it follows that (3.4) hold for all $n\ge 0$.

By Lemma 2.7 we conclude that $\{{x}_{n}\}$ and $\{{y}_{n}\}$ are two Cauchy sequences. Then there exist ${x}^{\ast},{y}^{\ast}\in C$ such that ${x}_{n}\to {x}^{\ast}$ and ${y}_{n}\to {y}^{\ast}$.

*F*is continuous. Hence

That is, *F* has a coupled fixed point.

and hence $d({x}^{\ast},F({x}^{\ast},{y}^{\ast}))=0$. That is, ${x}^{\ast}=F({x}^{\ast},{y}^{\ast})$. Similarly, ${y}^{\ast}=F({y}^{\ast},{x}^{\ast})$ as required. □

If in Theorem 3.1 we take $\alpha =\beta =0$, we obtain following result.

**Corollary 3.2**

*Let*$(C,\u2aaf)$

*be an ordered closed and convex subset of a quasi*-

*Banach space*$(X,\parallel \cdot \parallel ,K)$,

*where*$1\le K<\sqrt{2}$,

*and let*$d:X\times X\to {\mathbb{R}}_{+}$

*be such that*$d(x,y)=\parallel x-y\parallel $.

*Assume that*$F:C\times C\to C$

*is a mapping such that*

*F*

*has the mixed monotone property on*

*X*

*and there exists a non*-

*negative real number*

*γ*

*with*$0\le \gamma +1<2/{K}^{2}$

*such that*

*for all*$x,y,u,v\in X$,

*for which*$u\u2aafx$

*and*$y\u2aafv$.

*Also suppose that either*

- (a)
*F**is continuous*,*or* - (b)
*C**has the following property*:$\begin{array}{c}\mathit{\text{if a non-decreasing sequence}}\{{x}_{n}\}\to x,\mathit{\text{then}}{x}_{n}\u2aafx\mathit{\text{for all}}n\ge 0,\hfill \\ \mathit{\text{if a non-increasing sequence}}\{{y}_{n}\}\to y,\mathit{\text{then}}y\u2aaf{y}_{n}\mathit{\text{for all}}n\ge 0.\hfill \end{array}$

*If there exist* ${x}_{0},{y}_{0}\in C$ *such that* ${x}_{0}\u2aafF({x}_{0},{y}_{0})$ *and* $F({y}_{0},{x}_{0})\u2aaf{y}_{0}$, *then there exist* $x,y\in C$ *such that* $x=F(x,y)$ *and* $y=F(y,x)$, *that is*, *F* *has a coupled fixed point*.

The following theorem is our second main result.

**Theorem 3.3**

*Let*$(C,\u2aaf)$

*be an ordered closed and convex subset of a quasi*-

*Banach space*$(X,\parallel \cdot \parallel ,K)$,

*where*$1\le K<\frac{1+\sqrt{17}}{4}$,

*and let*$d:X\times X\to {\mathbb{R}}_{+}$

*be such that*$d(x,y)=\parallel x-y\parallel $.

*Assume that*$F:C\times C\to C$

*is a mapping such that*

*F*

*has the mixed monotone property on*

*X*

*and there exists a non*-

*negative real number*

*α*

*with*$0\le \alpha +2K-1<2/K$

*such that*

*for all*$x,y,u,v\in C$,

*for which*$u\u2aafx$

*and*$y\u2aafv$.

*Also suppose that either*

- (a)
*F**is continuous*,*or* - (b)
*C**has the following property*:$\begin{array}{c}\mathit{\text{if a non-decreasing sequence}}\{{x}_{n}\}\to x,\mathit{\text{then}}{x}_{n}\u2aafx\mathit{\text{for all}}n\ge 0,\hfill \\ \mathit{\text{if a non-increasing sequence}}\{{y}_{n}\}\to y,\mathit{\text{then}}y\u2aaf{y}_{n}\mathit{\text{for all}}n\ge 0.\hfill \end{array}$

*If there exist* ${x}_{0},{y}_{0}\in X$ *such that* ${x}_{0}\u2aafF({x}_{0},{y}_{0})$ *and* $F({y}_{0},{x}_{0})\u2aaf{y}_{0}$, *then* *F* *has a coupled fixed point*.

*Proof*Let ${x}_{0},{y}_{0}\in X$ be such that ${x}_{0}\u2aafF({x}_{0},{y}_{0})$ and ${y}_{0}\u2ab0F({y}_{0},{x}_{0})$. Then

*F*has the mixed monotone property, so

Then, by mathematical induction, it follows that (3.12) holds for all $n\ge 0$.

By Lemma 2.7, we conclude that $\{{x}_{n}\}$ and $\{{y}_{n}\}$ are Cauchy sequences. Thus, there exist ${x}^{\ast},{y}^{\ast}\in C$ such that ${x}_{n}\to {x}^{\ast}$ and ${y}_{n}\to {y}^{\ast}$.

Now, proceeding as in the proof of Theorem 3.1, we can prove that $({x}^{\ast},{y}^{\ast})$ is a coupled fixed point of *F*. □

so by Theorem 3.3 we obtain the following result.

**Corollary 3.4**

*Let*$(C,\u2aaf)$

*be an ordered closed and convex subset of a quasi*-

*Banach space*$(X,\parallel \cdot \parallel ,K)$,

*where*$1\le K<\frac{1+\sqrt{17}}{4}$,

*and let*$d:X\times X\to {\mathbb{R}}_{+}$

*be such that*$d(x,y)=\parallel x-y\parallel $.

*Assume that*$F:C\times C\to C$,

*F*

*has the mixed monotone property on*

*C*

*and for a non*-

*negative real number*

*α*

*with*$0\le \alpha +2K-1<2/K$,

*F*

*satisfies following inequality*:

*for all*$x,y,u,v\in X$,

*for which*$u\u2aafx$

*and*$y\u2aafv$.

*Also suppose that either*

- (a)
*F**is continuous*,*or* - (b)
*C**has the following property*:

*If there exist* ${x}_{0},{y}_{0}\in C$ *such that* ${x}_{0}\u2aafF({x}_{0},{y}_{0})$ *and* $F({y}_{0},{x}_{0})\u2aaf{y}_{0}$, *then* *F* *has a coupled fixed point*.

By taking $K=1$ in the above proved results, we can obtain the following couple fixed results in Banach spaces.

**Corollary 3.5**

*Let*$(C,\u2aaf)$

*be an ordered closed and convex subset of a Banach space*$(X,\parallel \cdot \parallel )$,

*and let*$F:C\times C\to C$

*be a mapping such that*

*F*

*has the mixed monotone property on*

*C*.

*Suppose that there exist non*-

*negative real numbers*

*α*,

*β*

*and*

*γ*

*with*$0\le \gamma +2\alpha +2\beta <1$

*such that*

*for all*$x,y,u,v\in C$

*with*$u\u2aafx$

*and*$y\u2aafv$.

*Also suppose that either*

- (a)
*F**is continuous*,*or* - (b)
*C**has the following property*:

*If there exist* ${x}_{0},{y}_{0}\in C$ *such that* ${x}_{0}\u2aafF({x}_{0},{y}_{0})$ *and* $F({y}_{0},{x}_{0})\u2aaf{y}_{0}$, *then* *F* *has a coupled fixed point*.

**Corollary 3.6**

*Let*$(C,\u2aaf)$

*be an ordered closed and convex subset of a Banach space*$(X,\parallel \cdot \parallel )$,

*and let*$F:C\times C\to C$

*be a mapping such that*

*F*

*has the mixed monotone property on*

*C*.

*Suppose that there exists a non*-

*negative real number*

*γ*

*with*$0\le \gamma <1$

*such that*

*for all*$x,y,u,v\in C$

*with*$u\u2aafx$

*and*$y\u2aafv$.

*Also suppose that either*

- (a)
*F**is continuous*,*or* - (b)
*C**has the following property*:

*If there exist* ${x}_{0},{y}_{0}\in C$ *such that* ${x}_{0}\u2aafF({x}_{0},{y}_{0})$ *and* $F({y}_{0},{x}_{0})\u2aaf{y}_{0}$, *then* *F* *has a coupled fixed point*.

**Corollary 3.7**

*Let*$(C,\u2aaf)$

*be an ordered closed and convex subset of a Banach space*$(X,\parallel \cdot \parallel )$,

*and let*$F:C\times C\to C$

*be a mapping such that*

*F*

*has the mixed monotone property on*

*C*.

*Suppose that there exists a non*-

*negative real number*

*α*

*with*$0\le \alpha <1$

*such that*

*for all*$x,y,u,v\in C$

*with*$u\u2aafx$

*and*$y\u2aafv$.

*Also suppose that either*

- (a)
*F**is continuous*,*or* - (b)
*C**has the following property*:$\mathit{\text{if a non-decreasing sequence}}\{{x}_{n}\}\to x,\mathit{\text{then}}{x}_{n}\u2aafx\mathit{\text{for all}}n\ge 0,$(3.17)$\mathit{\text{if a non-increasing sequence}}\{{y}_{n}\}\to y,\mathit{\text{then}}y\u2aaf{y}_{n}\mathit{\text{for all}}n\ge 0.$(3.18)

*If there exist* ${x}_{0},{y}_{0}\in C$ *such that* ${x}_{0}\u2aafF({x}_{0},{y}_{0})$ *and* $F({y}_{0},{x}_{0})\u2aaf{y}_{0}$, *then* *F* *has a coupled fixed point*.

**Corollary 3.8**

*Let*$(C,\u2aaf)$

*be an ordered closed and convex subset of a Banach space*$(X,\parallel \cdot \parallel )$,

*and let*$F:C\times C\to C$

*be a mapping such that*

*F*

*has the mixed monotone property on*

*C*.

*Suppose that there exists a non*-

*negative real number*

*α*

*with*$0\le \alpha <1$

*such that*

*for all*$x,y,u,v\in C$

*with*$u\u2aafx$

*and*$y\u2aafv$.

*Also suppose that either*

- (a)
*F**is continuous*,*or* - (b)
*C**has the following property*:

*If there exist* ${x}_{0},{y}_{0}\in C$ *such that* ${x}_{0}\u2aafF({x}_{0},{y}_{0})$ *and* $F({y}_{0},{x}_{0})\u2aaf{y}_{0}$, *then* *F* *has a coupled fixed point*.

The following lemma is an easy consequence of the axiom of choice (see p.5 [25], AC5: For every function $f:X\to X$, there is a function *g* such that $D(g)=R(f)$ and for every $x\in D(g)$ , $f(gx)=x$).

**Lemma 3.9** *Let* *X* *be a nonempty set and* $g:X\to X$ *be a mapping*. *Then there exists a subset* $E\subseteq X$ *such that* $g(E)=g(X)$ *and* $g:E\to X$ *is one*-*to*-*one*.

As an application of Theorem 3.1, we now establish a coupled coincidence point result.

**Theorem 3.10**

*Let*$(C,\u2aaf)$

*be a nonempty ordered subset of a quasi*-

*Banach space*$(X,\parallel \cdot \parallel ,K)$,

*where*$1\le K<\sqrt{2}$,

*and let*$d:X\times X\to {\mathbb{R}}_{+}$

*be such that*$d(x,y)=\parallel x-y\parallel $.

*Assume that*$g:C\to C$

*and*$F:C\times C\to C$

*are mappings where*

*F*

*has the mixed*

*g*-

*monotone property on*

*C*, $g(C)$

*is closed and convex and*$F(C\times C)\subseteq g(C)$.

*Suppose that there exist non*-

*negative real numbers*

*α*,

*β*

*and a real number*

*γ*

*with*$0\le \gamma +2\alpha +2\beta +1<2/{K}^{2}$

*such that*

*for all*$x,y,u,v\in C$,

*for which*$gu\u2aafgx$

*and*$gy\u2aafgv$.

*Also suppose that either*

- (a)
*F**is continuous*,*or* - (b)
*C**has the following property*:$\begin{array}{c}\mathit{\text{if a non-decreasing sequence}}\{g{x}_{n}\}\to gx,\mathit{\text{then}}g{x}_{n}\u2aafgx\mathit{\text{for all}}n\ge 0,\hfill \\ \mathit{\text{if a non-increasing sequence}}\{g{y}_{n}\}\to gy,\mathit{\text{then}}gy\u2aafg{y}_{n}\mathit{\text{for all}}n\ge 0.\hfill \end{array}$

*If there exist* ${x}_{0},{y}_{0}\in C$ *such that* $g({x}_{0})\u2aafF({x}_{0},{y}_{0})$ *and* $F({y}_{0},{x}_{0})\u2aafg({y}_{0})$, *then there exist* $x,y\in C$ *such that* $gx=F(x,y)$ *and* $gy=F(y,x)$, *that is*, *F* *and* *g* *have a coupled coincidence point in* *C*.

*Proof*Using Lemma 3.9, there exists $E\subseteq C$ such that $g(E)=g(C)$ and $g:E\to C$ is one-to-one. We define a mapping $G:g(E)\times g(E)\to g(E)$ by

*g*is one-to-one on $g(E)$ and $F(C\times C)\subseteq g(C)$, so

*G*is well defined. Thus, it follows from (3.19) and (3.20) that

*F*has the mixed

*g*-monotone property, for all $gx,gy\in g(C)$,

*G*has the mixed monotone property. Also, there exist ${x}_{0},{y}_{0}\in C$ such that

Suppose that assumption (a) holds. Since *F* is continuous, *G* is also continuous. Using Theorem 3.1 to the mapping *G*, it follows that *G* has a coupled fixed point $(u,v)\in g(C)\times g(C)$.

*G*has a coupled fixed point $(u,v)\in g(C)\times g(C)$. Finally, we prove that

*F*and

*g*have a coupled coincidence point. Since $(u,v)$ is a coupled fixed point of

*G*, we get

Thus, $({u}_{0},{v}_{0})$ is a required coupled coincidence point of *F* and *g*. This completes the proof. □

Similarly, as an application of Theorem 3.3, we can prove the following coupled coincidence point result.

**Theorem 3.11**

*Let*$(C,\u2aaf)$

*be a nonempty ordered subset of a quasi*-

*Banach space*$(X,\parallel \cdot \parallel ,K)$,

*where*$1\le K<\frac{1+\sqrt{17}}{4}$,

*and let*$d:X\times X\to {\mathbb{R}}_{+}$

*be such that*$d(x,y)=\parallel x-y\parallel $.

*Assume that*$g:C\to C$

*and*$F:C\times C\to C$

*are mappings where*

*F*

*has the mixed*

*g*-

*monotone property on*

*C*, $g(C)$

*is closed and convex and*$F(C\times C)\subseteq g(C)$.

*Suppose that there exists a real number*

*α*

*with*$0\le \alpha +2K-1<2/K$

*such that*

*for all*$x,y,u,v\in C$,

*for which*$gu\u2aafgx$

*and*$gy\u2aafgv$.

*Also suppose that either*

- (a)
*F**is continuous*,*or* - (b)
*C**has the following property*:$\begin{array}{c}\mathit{\text{if a non-decreasing sequence}}\{g{x}_{n}\}\to gx,\mathit{\text{then}}g{x}_{n}\u2aafgx\mathit{\text{for all}}n\ge 0,\hfill \\ \mathit{\text{if a non-increasing sequence}}\{g{y}_{n}\}\to gy,\mathit{\text{then}}gy\u2aafg{y}_{n}\mathit{\text{for all}}n\ge 0.\hfill \end{array}$

*If there exist* ${x}_{0},{y}_{0}\in C$ *such that* $g({x}_{0})\u2aafF({x}_{0},{y}_{0})$ *and* $F({y}_{0},{x}_{0})\u2aafg({y}_{0})$, *then* *F* *and* *g* *have a coupled coincidence point in* *C*.

## 4 Existence of a solution for a system of integral equations

*A*and

*B*. Further, it is known that $(\mathcal{X},d,\u2aaf)$ is regular [24], that is,

Motivated by the work in [1, 17, 18, 26], we study the existence of solutions for a system of nonlinear integral equations using the results proved in the previous section.

- (i)
$f,k:[0,T]\times \mathbb{R}\to \mathbb{R}$ are continuous;

- (ii)
$P:[0,T]\to \mathbb{R}$ is continuous;

- (iii)
$S:[0,T]\times \mathbb{R}\to [0,\mathrm{\infty})$ is continuous;

- (iv)there exist $a,b,c>0$ with $0\le 2a+2b+c<1$ such that for all $r\in [0,T]$ and all $x(r),y(r),u(r),v(r)\in \mathcal{X}$ with $u(r)\u2aafx(r)\u2aafy(r)\u2aafv(r)$, we have$\begin{array}{c}\begin{array}{rl}0\le & f(r,y(r))-f(r,x(r))\le a|x(r)-F(x(r),y(r))|+b|y(r)-F(y(r),x(r))|\\ +\frac{c}{2}[(x(r)-u(r))+(y(r)-v(r))],\end{array}\hfill \\ \begin{array}{rl}0\le & k(r,x(r))-k(r,y(r))\le a|x(r)-F(x(r),y(r))|+b|y(r)-F(y(r),x(r))|\\ +\frac{c}{2}[(x(r)-u(r))+(y(r)-v(r))],\end{array}\hfill \end{array}$

- (v)there exist continuous functions $\alpha ,\gamma :[0,T]\to \mathbb{R}$ such that$\begin{array}{c}\alpha (t)\le P(t)+{\int}_{0}^{T}S(t,r)[f(r,\alpha (r))+k(r,\gamma (r))]\phantom{\rule{0.2em}{0ex}}dr,\hfill \\ \gamma (t)\ge P(t)+{\int}_{0}^{T}S(t,r)[f(r,\gamma (r))+k(r,\alpha (r))]\phantom{\rule{0.2em}{0ex}}dr;\hfill \end{array}$
- (vi)assume that$\underset{t\in [0,T]}{sup}{\int}_{0}^{T}S(t,r)\phantom{\rule{0.2em}{0ex}}dr\le 1/2.$

**Theorem 4.1** *Under assumptions* (i)-(vi), *system* (4.1) *has a solution in* ${\mathcal{X}}^{2}$, *where* $\mathcal{X}=(C([0,T],\mathbb{R}))$ *is defined above*.

*Proof*We consider the operator $F:{\mathcal{X}}^{2}\to \mathcal{X}$ defined by

for all $t,r\in [0,T]$, ${x}_{1},{x}_{2}\in \mathcal{X}$.

Clearly, *F* has the mixed monotone property [26].

*F*has the mixed monotone property, we have

All of the conditions of Corollary 3.5 are satisfied, so we deduce the existence of ${x}_{1},{x}_{2}\in \mathcal{X}$ such that ${x}_{1}=F({x}_{1},{x}_{2})$ and ${x}_{2}=F({x}_{2},{x}_{1})$. □

## Declarations

### Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first and third authors acknowledge with thanks DSR, KAU for financial support. The second author is thankful for the support of Rasht Branch, Islamic Azad University. The authors would like to express their thanks to the referees for their helpful comments and suggestions.

## Authors’ Affiliations

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