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# Common coupled fixed point theorems for θ-ψ-contraction mappings endowed with a directed graph

## Abstract

In this paper, we present some existence and uniqueness results for coupled coincidence point and common fixed point of θ-ψ-contraction mappings in complete metric spaces endowed with a directed graph. Our results generalize the results obtained by Kadelburg et al. (Fixed Point Theory Appl. 2015:27, 2015, doi:10.1007/s11590-013-0708-4). We also have an application to some integral system to support the results.

## Introduction and preliminaries

For $$F:X\times X\rightarrow X$$ and $$g:X\rightarrow X$$, a concept of coupled coincidence point $$(x,y)\in X\times X$$ such that $$gx=F(x,y)$$ and $$gy=F(y,x)$$ was first introduced by Lakshimikantham and Ćirić . Their results extended the result in [3, 4]. Also, the existence and uniqueness of a coupled coincidence point for such a mapping that satisfies the mixed monotone property in a partially ordered metric space were studied. Consequently, a number of coupled fixed point and coupled coincidence point results have been shown recently. For example, see .

Choudhury and Kundu  give a notion of compatibility.

### Definition 1.1

()

Let $$(X,d)$$ be a metric space, and let $$g:X\to X$$ and $$F:X\times X\to X$$. The mappings g and F are said to be compatible if

$$\lim_{n\to\infty}d\bigl(gF(x_{n},y_{n}),F(gx_{n},gy_{n}) \bigr)=0 \quad \text{and}\quad \lim_{n\to\infty}d\bigl(gF(y_{n},x_{n}),F(gy_{n},gx_{n}) \bigr)=0$$

whenever $$\{x_{n}\}$$ and $$\{y_{n}\}$$ are sequences in X such that $$\lim_{n\to\infty}F(x_{n},y_{n})=\lim_{n\to\infty}gx_{n}$$ and $$\lim_{n\to\infty}F(y_{n},x_{n})=\lim_{n\to\infty}gy_{n}$$.

Let Θ denote the class of all functions $$\theta :[0,\infty)\times[0,\infty)\rightarrow[0,1)$$ that satisfy the following conditions:

($$\theta_{1}$$):

$$\theta(s,t)=\theta(t,s)$$ for all $$s,t\in[0,\infty)$$;

($$\theta_{2}$$):

for any two sequences $$\{s_{n}\}$$ and $$\{ t_{n}\}$$ of nonnegative real numbers,

$$\theta(s_{n},t_{n})\rightarrow1\quad \Rightarrow \quad s_{n},t_{n}\rightarrow0.$$

In 2015, Kadelburg et al.  used the monotone and g-monotone properties to obtained common coupled fixed point theorems for Geraghty-type contraction with compatibility of F and g.

Let $$(X,d)$$ be a metric space, Δ be a diagonal of $$X\times X$$, and G be a directed graph with no parallel edges such that the set $$V(G)$$ of its vertices coincides with X and $$\Delta\subseteq E(G)$$, where $$E(G)$$ is the set of the edges of the graph. That is, G is determined by $$(V(G), E(G))$$. We will use this notation of G throughout this work.

The fixed point theorem using the context of metric spaces endowed with a graph was first studied by Jachymski . The result generalized the Banach contraction principle to mappings on metric spaces with a graph. Since then, many authors studied the problem of existence of fixed points for single-valued mappings and multivalued mappings in several spaces with graphs; see .

Recently, Chifu and Petrusel  give the concept of G-continuity for a mapping $$F:X^{2}\to X$$ and the property A as follows.

### Definition 1.2

Let $$(X, d)$$ be a complete metric space, G be a directed graph, and $$F:X^{2}\to X$$ be a mapping. Then

1. (i)

F is called G-continuous if for all $$(x^{*}, y^{*})\in X^{2}$$ and for any sequence $$(n_{i})_{i}\in\mathbb{N}$$ of positive integers such that $$F(x_{n_{i}}, y_{n_{i}})\to x^{*}$$, $$F(y_{n_{i}}, x_{n_{i}})\to y^{*}$$ as $$i\to \infty$$ and $$( F(x_{n_{i}}, y_{n_{i}}), F(x_{n_{i}+1}, y_{n_{i}+1}) ), ( F(y_{n_{i}}, x_{n_{i}}), F(y_{n_{i}+1}, x_{n_{i}+1}) )\in E(G)$$, we have that

\begin{aligned}& F\bigl( F(x_{n_{i}}, y_{n_{i}}), F(y_{n_{i}}, x_{n_{i}})\bigr)\to F\bigl(x^{*}, y^{*}\bigr) \quad \text{and} \\& F\bigl(F(y_{n_{i}}, x_{n_{i}}), F(x_{n_{i}}, y_{n_{i}})\bigr)\to F\bigl(y^{*}, x^{*}\bigr)\quad \text{as } i\to\infty; \end{aligned}
2. (ii)

$$(X, d, G)$$ has property A if for any sequence $$(x_{n})_{n\in\mathbb{N}}\subset X$$ with $$x_{n}\to x$$ as $$n\to\infty$$ and $$(x_{n}, x_{n+1})\in E(G)$$ for $$n\in\mathbb{N}$$, then $$(x_{n}, x)\in E(G)$$.

Their results generalized the result in  by using the context of metric spaces endowed with a directed graph.

The aim of this work is to prove some existence and uniqueness results for a coupled coincidence point and a common fixed point of θ-ψ contraction mappings in complete metric spaces endowed with a directed graph. The results generalize the results obtained by Kadelburg et al. . An application to some integral system is provided to support the results.

## Common coupled fixed point

We define the set $$\operatorname{CcFix}(F)$$ of all coupled coincidence points of mappings $$F:X^{2}\to X$$ and $$g:X\to X$$ and the set $$(X^{2})^{F}_{g}$$ as follows:

$$\operatorname{CcFix}(F)=\bigl\{ (x, y)\in X^{2} : F(x, y)=gx \text{ and } F(y, x)=gy \bigr\}$$

and

$$\bigl(X^{2}\bigr)^{F}_{g}= \bigl\{ (x, y)\in X^{2} : \bigl(gx, F(x, y)\bigr), \bigl(gy, F(y, x)\bigr)\in E(G)\bigr\} .$$

Now, we give some definitions that are useful for our main results.

### Definition 2.1

We say that $$F:X^{2}\to X$$ and $$g:X\to X$$ are G-edge preserving if

$$\bigl[(gx, gu), (gy, gv)\in E( G ) \bigr]\quad \Rightarrow\quad \bigl[\bigl( F(x, y), F(u, v)\bigr), \bigl( F(y, x), F(v, u)\bigr)\in E(G)\bigr].$$

### Definition 2.2

Let $$(X, d)$$ be a complete metric space, and $$E(G)$$ be the set of the edges of the graph. We say that $$E(G)$$ satisfies the transitivity property if and only if, for all $$x,y,a \in X$$,

$$(x,a),(a,y)\in E(G)\rightarrow(x,y)\in E(G).$$

Let Ψ denote the class of all functions $$\psi :[0,\infty)\rightarrow[0,\infty)$$ that satisfy the following conditions:

($$\psi_{1}$$):

ψ is nondecreasing;

($$\psi_{2}$$):

$$\psi(s+t)\leq\psi(s)+\psi(t)$$;

($$\psi_{3}$$):

ψ is continuous;

($$\psi_{4}$$):

$$\psi(t)=0 \Leftrightarrow t=0$$.

### Definition 2.3

Let $$(X, d)$$ be a complete metric space endowed with a directed graph G. The mappings $$F:X^{2}\to X$$ and $$g:X\to X$$ are called a θ-ψ-contraction if:

1. (1)

F and g is G-edge preserving;

2. (2)

there exist $$\theta\in\Theta$$ and $$\psi\in\Psi$$ such that for all $$x,y,u,v\in X$$ satisfying $$(gx, gu), (gy, gv)\in E(G)$$,

$$\psi\bigl(d\bigl(F(x,y),F(u,v)\bigr)\bigr)\leq\theta \bigl(d(gx,gu),d(gy,gv)\bigr)\psi\bigl(M(gx,gu,gy,gv)\bigr),$$
(1)

where $$M(gx,gu,gy,gv)=\max\{d(gx,gu),d(gy,gv)\}$$.

### Lemma 2.4

Let $$(X, d)$$ be a complete metric space endowed with a directed graph G, and let $$F:X^{2}\to X$$ and $$g:X\to X$$ be a θ-ψ-contraction. Assume that there exist $$x_{0}, y_{0}, a_{0},b_{0} \in X$$ and $$F(X\times X)\subset g(X)$$. Then:

1. (i)

There exists sequences $$\{x_{n}\}$$, $$\{y_{n}\}$$, $$\{a_{n}\}$$, $$\{ b_{n}\}$$ in X for which

\begin{aligned} &gx_{n}=F(x_{n-1},y_{n-1}) \quad \textit{and} \quad gy_{n}=F(y_{n-1},x_{n-1}), \\ &ga_{n}=F(a_{n-1},b_{n-1}) \quad \textit{and}\quad gb_{n}=F(b_{n-1},a_{n-1}) \quad \textit {for } n=1,2, \ldots. \end{aligned}
(2)
2. (ii)

If $$(gx_{n},ga_{n})$$ and $$(gy_{n},gb_{n})\in E(G)$$ for all $$n\in \mathbb{N}$$, then

$$\lim_{n\to\infty}d_{n}=\lim_{n\to\infty}M(gx_{n},ga_{n},gy_{n},gb_{n})= 0.$$

### Proof

(i) Let $$x_{0}, y_{0}, a_{0}, b_{0}\in X$$. By the assumption that $$F(X\times X)\subset g(X)$$ and $$F(x_{0}, y_{0}), F(y_{0},x_{0}), F(a_{0},b_{0}), F(b_{0},a_{0}) \in X$$, it easy to construct sequences $$\{x_{n}\}$$, $$\{y_{n}\}$$, $$\{a_{n}\}$$, and $$\{b_{n}\}$$ in X for which

\begin{aligned}& gx_{n}=F(x_{n-1},y_{n-1}) \quad \text{and} \quad gy_{n}=F(y_{n-1},x_{n-1}), \\& ga_{n}=F(a_{n-1},b_{n-1})\quad \text{and} \quad gb_{n}=F(b_{n-1},a_{n-1})\quad \text{for } n=1,2, \ldots. \end{aligned}

(ii) Let $$(gx_{n},ga_{n})$$ and $$(gy_{n},gb_{n})\in E(G)$$ for all $$n\in\mathbb {N}$$. Using the θ-ψ-contraction (1) and (2), we obtain that

\begin{aligned} \psi\bigl( d(gx_{n+1},ga_{n+1}) \bigr) &= \psi \bigl(d\bigl(F(x_{n},y_{n}),F(a_{n},b_{n}) \bigr) \bigr) \\ &\leq \theta\bigl( d(gx_{n},ga_{n}),d(gy_{n},gb_{n}) \bigr) \psi\bigl( M(gx_{n},ga_{n},gy_{n},gb_{n}) \bigr) \end{aligned}
(3)

and

\begin{aligned} \psi\bigl(d(gy_{n+1},gb_{n+1}) \bigr)&= \psi \bigl(d\bigl(F(y_{n},x_{n}),F(b_{n},a_{n}) \bigr)\bigr) \\ &\leq \theta\bigl( d(gy_{n},gb_{n}),d(gx_{n},ga_{n}) \bigr) \psi \bigl(M(gy_{n},gb_{n},gx_{n},ga_{n}) \bigr) \\ &=\theta\bigl( d(gx_{n},ga_{n}),d(gy_{n},gb_{n}) \bigr)\psi\bigl( M(gx_{n},ga_{n},gy_{n},gb_{n}) \bigr) \end{aligned}
(4)

for all $$n\in\mathbb{N}$$. From (3) and (4) we get

\begin{aligned}& \psi\bigl(M(gx_{n+1},ga_{n+1},gy_{n+1},gb_{n+1}) \bigr) \\& \quad = \psi\bigl(\max\bigl\{ d(gx_{n+1},ga_{n+1}),d(gy_{n+1},gb_{n+1}) \bigr\} \bigr) \\& \quad \leq\theta\bigl( d(gx_{n},ga_{n}),d(gy_{n},gb_{n}) \bigr)\psi \bigl(M(gx_{n},ga_{n},gy_{n},gb_{n}) \bigr) \\& \quad < \psi\bigl( M(gx_{n},ga_{n},gy_{n},gb_{n}) \bigr) \end{aligned}
(5)

for all $$n\in\mathbb{N}$$, that is,

$$\psi\bigl(M(gx_{n+1},ga_{n+1},gy_{n+1},gb_{n+1}) \bigr)< \psi\bigl( M(gx_{n},ga_{n},gy_{n},gb_{n}) \bigr).$$

Regarding the properties of ψ, we conclude that

$$M(gx_{n+1},ga_{n+1},gy_{n+1},gb_{n+1})< M(gx_{n},ga_{n},gy_{n},gb_{n}).$$

It follows that $$d_{n}:=M(gx_{n},ga_{n},gy_{n},gb_{n})$$ is decreasing. Then $$d_{n}\rightarrow d$$ as $$n\rightarrow \infty$$ for some $$d\geq0$$. We claim that $$d=0$$. Suppose not. Using (5), we have

$$\frac{\psi(M(gx_{n+1},ga_{n+1},gy_{n+1},gb_{n+1}))}{\psi( M(gx_{n},ga_{n},gy_{n},gb_{n}))}\leq \theta\bigl( d(gx_{n},ga_{n}),d(gy_{n},gb_{n}) \bigr) < 1.$$

Taking the limit as $$n\rightarrow\infty$$, we have

$$\theta\bigl( d(gx_{n},ga_{n}),d(gy_{n},gb_{n}) \bigr) \rightarrow1.$$

Since $$\theta\in\Theta$$,

$$d(gx_{n},ga_{n})\rightarrow0\quad \text{and}\quad d(gy_{n},gb_{n}) \rightarrow0$$

as $$n\rightarrow\infty$$. Therefore,

$$\lim_{n\to\infty}d_{n}=\lim_{n\to\infty}M(gx_{n},ga_{n},gy_{n},gb_{n})= 0,$$

$$\lim_{n\to\infty}d_{n}=\lim_{n\to\infty}M(gx_{n},ga_{n},gy_{n},gb_{n})= 0$$

□

Next, we will prove the main result.

### Theorem 2.5

Let $$(X, d)$$ be a complete metric space endowed with a directed graph G, and let $$F:X^{2}\to X$$ and $$g:X\to X$$ be a θ-ψ-contraction. Suppose that:

1. (i)

g is continuous, and $$g(X)$$ is closed;

2. (ii)

$$F(X\times X)\subset g(X)$$, and g and F are compatible;

3. (iii)

F is G-continuous, or the tripled $$(X, d, G)$$ has property A;

4. (iv)

$$E(G)$$ satisfies the transitivity property.

Under these conditions, $$\operatorname{CcFix}(F)\neq\emptyset$$ if and only if $$(X^{2})^{F}_{g}\neq\emptyset$$.

### Proof

Let $$\operatorname{CcFix}(F)\neq\emptyset$$. Then there exists $$(u, v)\in \operatorname{CcFix}(F)$$ such that $$(gu, F(u, v))=(gu, gu)$$ and $$(gv,F(v,u))=(gv, gv)\in\Delta \subset E(G)$$. Thus, $$(gu, F(u, v))$$ and $$(gv,F(v,u))\in E(G)$$. It follows that $$(u, v)\in(X^{2})^{F}_{g}$$, so that $$(X^{2})^{F}_{g}\neq\emptyset$$.

Now, suppose that $$(X^{2})^{F}_{g}\neq\emptyset$$. Let $$x_{0}, y_{0}\in X$$ be such that $$(x_{0}, y_{0})\in(X^{2})^{F}_{g}$$. Then $$(gx_{0}, F(x_{0}, y_{0}))$$ and $$(gy_{0},F(y_{0},x_{0}))\in E(G)$$. From Lemma 2.4(i) we have sequences $$\{x_{n}\}$$ and $$\{y_{n}\}$$ in X for which

$$gx_{n}=F(x_{n-1},y_{n-1})\quad \text{and}\quad gy_{n}=F(y_{n-1},x_{n-1}) \quad \text{for } n=1,2, \ldots.$$

Since $$(gx_{0}, F(x_{0}, y_{0}))=(gx_{0}, gx_{1})$$ and $$(gy_{0},F(y_{0},x_{0}))=(gy_{0},gy_{1})\in E(G)$$ and F and g are G-edge preserving, we have $$(F(x_{0},y_{0}), F(x_{1}, y_{1}))=(gx_{1}, gx_{2})$$ and $$(F(y_{0},x_{0}), F(y_{1},x_{1}))=(gy_{1},gy_{2})\in E(G)$$. By induction we shall obtain $$(gx_{n-1}, gx_{n})$$ and $$(gy_{n-1},gy_{n})\in E(G)$$ for each $$n\in\mathbb {N}$$. By Lemma 2.4(ii) we have

$$d_{n}:=M(gx_{n-1},gx_{n},gy_{n-1},gy_{n}) \rightarrow0 \quad \text{as } n\rightarrow\infty.$$
(6)

Now, we shall show that $$\{gx_{n}\}$$ and $$\{gy_{n}\}$$ are Cauchy sequences. Applying a similar argument as in the proof of Theorem 3.1 in  and using (6), condition (iv), and property of ψ, it follows that $$\{gx_{n}\}$$ and $$\{gy_{n}\}$$ are Cauchy sequences. By condition (i) there exist $$u,v\in g(X)$$ such that

$$\lim_{n\to\infty}gx_{n}=\lim_{n\to\infty}F(x_{n},y_{n})=u \quad \text{and}\quad \lim_{n\to\infty}gy_{n}=\lim _{n\to\infty}F(y_{n},x_{n})=v.$$

By the compatibility of g and F we have that

$$\lim_{n\to\infty}d\bigl(gF(x_{n},y_{n}),F(gx_{n},gy_{n}) \bigr)=0 \quad \text{and}\quad \lim_{n\to\infty}d\bigl(gF(y_{n},x_{n}),F(gy_{n},gx_{n}) \bigr)=0.$$
(7)

Now, suppose that (a) F is G-continuous. It is easy to see that

$$d\bigl(gu,F(gx_{n},gy_{n})\bigr)\leq d\bigl(gu,gF(x_{n},y_{n}) \bigr)+d\bigl(gF(x_{n},y_{n}),F(gx_{n},gy_{n}) \bigr).$$

Taking the limit as $$n\to\infty$$ and using (7), the continuity of g, and G-continuity of F, we have that $$d(gu,F(u,v))=0$$, that is, $$gu=F(u,v)$$. Using a similar idea, we also have that $$gv=F(v,u)$$. Therefore, $$\operatorname{CcFix}(F)\neq\emptyset$$.

Suppose now that (b) the tripled $$(X, d, G)$$ with property A. Let $$gx=u$$ and $$gy=v$$ for some $$x, y\in X$$. Then we have $$(gx_{n},gx)$$ and $$(gy_{n},gy)\in E(G)$$ for each $$n\in\mathbb{N}$$. By (1) we have

\begin{aligned}& \psi\bigl(d\bigl(gx,F(x,y)\bigr)+d\bigl(gy,F(y,x)\bigr)\bigr) \\& \quad \leq\psi \bigl(d(gx,gx_{n+1})+d\bigl(gx_{n+1},F(x,y) \bigr)+d(gy,gy_{n+1})+d\bigl(gy_{n+1},F(y,x)\bigr)\bigr) \\& \quad \leq\psi\bigl(d\bigl(F(x_{n},y_{n}), F(x,y)\bigr) \bigr)+\psi\bigl(d\bigl(F(y_{n},x_{n}), F(y,x)\bigr)\bigr) \\& \qquad {} +\psi\bigl(d(gx,gx_{n+1})\bigr)+\psi\bigl(d(gy,gy_{n+1}) \bigr) \\& \quad \leq2\theta\bigl(d(gx_{n},gx),d(gy_{n},gy)\bigr)\psi \bigl(M(gx_{n},gx,gy_{n},gy)\bigr) \\& \qquad {} +\psi\bigl(d(gx,gx_{n+1})\bigr)+\psi\bigl(d(gy,gy_{n+1}) \bigr). \end{aligned}

Letting $$n\to\infty$$, we have $$\psi(d(gx,F(x,y))+d(gy,F(y,x)))= 0$$. By properties of ψ, we can see that $$d(gx,F(x,y))+d(gy,F(y,x))= 0$$. Finally, $$gx=F(x,y)$$ and $$gy=F(y,x)$$. □

We denote by $$\operatorname{CmFix}(F)$$ the set of all common fixed points of mappings $$F:X^{2}\to X$$ and $$g:X\to X$$, that is,

$$\operatorname{CmFix}(F)=\bigl\{ (x, y)\in X^{2} : F(x, y)=gx=x \text{ and } F(y, x)=gy=y \bigr\} .$$

### Theorem 2.6

In addition to hypotheses of Theorem  2.5, assume that

1. (vi)

for any two elements $$(x,y),(u,v)\in X\times X$$, there exists $$(a,b)\in X\times X$$ such that $$(gx,ga), (gu, ga), (gy,gb), (gv, gb)\in E(G)$$.

Then, $$\operatorname{CmFix}(F)\neq\emptyset$$ if and only if $$(X^{2})^{F}_{g}\neq\emptyset$$.

### Proof

Theorem 2.5 implies that there exists $$(x,y)\in X\times X$$ such that $$gx=F(x,y)$$ and $$gy=F(y,x)$$. Suppose that there exists another $$(u,v)\in X\times X$$ such that $$gu=F(u,v)$$ and $$gv=F(v,u)$$. We will show that $$gx=gu$$ and $$gy=gv$$.

By condition (vi) there exists $$(a,b)\in X\times X$$ such that $$(gx,ga), (gu, ga), (gy,gb), (gv, gb)\in E(G)$$. Set $$a_{0}=a$$, $$b_{0}=b$$, $$x_{0}=x$$, $$y_{0}=y$$, $$u_{0}=u$$, and $$v_{0}=v$$. By Lemma 2.4(i) we have sequences $$\{a_{n}\}$$, $$\{b_{n}\}$$ $$\{x_{n}\}$$, $$\{y_{n}\}$$, $$\{u_{n}\}$$, and $$\{v_{n}\}$$ in X for which

\begin{aligned}& ga_{n}=F(a_{n-1},b_{n-1}) \quad \text{and} \quad gb_{n}=F(b_{n-1},a_{n-1}), \\& gx_{n}=F(x_{n-1},y_{n-1})\quad \text{and} \quad gy_{n}=F(y_{n-1},x_{n-1}), \\& gu_{n}=F(u_{n-1},v_{n-1}) \quad \text{and}\quad gv_{n}=F(v_{n-1},u_{n-1}) \end{aligned}

for $$n\in\mathbb{N}$$. By the properties of coincidence points, $$x_{n}=x$$, $$y_{n}=y$$ and $$u_{n}=u$$, $$v_{n}=v$$, that is,

$$gx_{n}=F(x,y),\qquad gy_{n}=F(y,x)\quad \text{and}\quad gu_{n}=F(u,v),\qquad gv_{n}=F(v,u) \quad \text{for all } n\in \mathbb{N}.$$

Since $$(gx,ga), (gy,gb) \in E(G)$$, we have $$(gx, ga_{0})$$ and $$(gy,gb_{0})\in E(G)$$. Because F and g are G-edge preserving, we have $$(F(x,y),F(a_{0},b_{0}))=(gx,ga_{1})$$ and $$(F(y,x),F(b_{0},a_{0}))=(gy,gb_{1})\in E(G)$$. Similarly, $$(gx, ga_{n})$$ and $$(gy,gb_{n})\in E(G)$$. By Lemma 2.4(ii) we obtain

$$\lim_{n\to\infty}d_{n}=\lim_{n\to\infty}M(gx,ga_{n},gy,gb_{n})= 0,$$

and then

$$\lim_{n\to\infty}d(gx,ga_{n})=0 \text{and} \lim _{n\to\infty}d(gy,gb_{n})=0.$$

Similarly, from $$(gu, ga), (gv, gb)\in E(G)$$ we have

$$\lim_{n\to\infty}d(gu,ga_{n})=0\quad \text{and} \quad \lim _{n\to\infty}d(gv,gb_{n})=0.$$

By the triangle inequality we have

$$d(gx,gu) \leq d(gx,ga_{n})+d(ga_{n},gu ) \quad\text{and} \quad d(gy,gv) \leq d(gy,gb_{n})+d(gb_{n},gv)$$

for all $$n\in\mathbb{N}$$. Letting $$n\rightarrow\infty$$ in these two inequalities, we get that $$d(gx,gu)=0$$ and $$d(gy,gv)=0$$. Therefore, we have $$gx=gu$$ and $$gy=gv$$.

The proof of the existence and uniqueness of a common fixed point can be derived using a similar argument as in Theorem 3.7 in . □

### Remark 2.1

In the case where $$(X,d,\preceq)$$ is a partially ordered complete metric space, letting $$E(G)=\{ (x,y)\in X\times X : x\preceq y\}$$ and $$\psi(t)=t$$, we obtain Theorem 3.1 and Theorem 3.7 in .

## Applications

In this section, we apply our theorem to the existence theorem for a solution of the following integral system:

\begin{aligned} &x(t)= \int_{0}^{T} f\bigl(t,s,x(s),y(s)\bigr)\, ds+h(t), \\ &y(t)= \int_{0}^{T} f\bigl(t,s,y(s),x(s)\bigr)\, ds+h(t), \end{aligned}
(8)

where $$t\in[0,T]$$ with $$T >0$$.

Let $$X:=C([0,T],\mathbb{R}^{n})$$ with $$\|x\| =\max_{t\in[0,T]}|x(t)|$$, for $$x\in C([0,T],\mathbb{R}^{n})$$.

We define the graph G with partial order relation by

$$x,y\in X,\quad x\leq y\quad \Leftrightarrow \quad x(t)\leq y(t)\quad \text{for any } t\in[0,T].$$

Thus, $$(X,\| \cdot\|)$$ is a complete metric space endowed with a directed graph G.

Let $$E(G)=\{(x,y)\in X\times X : x\leq y\}$$. Then $$E(G)$$ satisfies the transitivity property, and $$(X, {\|\cdot\|}, G)$$ has property A.

### Theorem 3.1

Consider system (8). Suppose that

1. (i)

$$f:[0,T]\times[0,T]\times\mathbb{R}^{n}\times\mathbb {R}^{n}\to\mathbb{R}^{n}$$ and $$h:[0,T]\to\mathbb{R}^{n}$$ are continuous;

2. (ii)

for all $$x,y,u,v \in\mathbb{R}^{n}$$ with $$x\leq u$$, $$y\leq v$$, we have $$f(t,s,x,y)\leq f(t,s,u,v)$$ for all $$t,s\in[0,T]$$;

3. (iii)

there exist $$0\leq k<1$$ and $$T >0$$ such that

$$\bigl\vert f(t,s,x,y) - f(t,s,u,v)\bigr\vert \leq\frac{k}{T} \bigl( \vert x-u\vert +\vert y-v\vert \bigr)$$

for all $$t,s\in[0,T]$$, $$x,y,u,v \in\mathbb{R}^{n}$$, $$x\leq u$$, $$y\leq v$$;

4. (iv)

there exists $$(x_{0},y_{0})\in X\times X$$ such that

\begin{aligned}& x_{0}(t)\leq \int_{0}^{T} f\bigl(t,s,x_{0}(s),y_{0}(s) \bigr)\, ds+h(t) \quad \textit{and} \\& y_{0}(t)\leq \int_{0}^{T} f\bigl(t,s,y_{0}(s),x_{0}(s) \bigr)\, ds+h(t), \end{aligned}

where $$t\in[0, T]$$.

Then there exists at least one solution of the integral system (8).

### Proof

Let $$F:X\times X\to X$$, $$(x,y)\mapsto F(x,y)$$, where

$$F(x,y) (t)= \int_{0}^{T} f\bigl(t,s,x(s),y(s)\bigr)\, ds+h(t), \quad t\in[0,T],$$

and define $$g:X\to X$$ by $$gx(t)=\frac{x(t)}{2}$$.

System (8) can be written as

$$x=F(x,y) \quad \text{and}\quad y=F(y,x).$$

Let $$x,y,u,v \in X$$ be such that $$gx\leq gu$$ and $$gy\leq gv$$. We have $$x\leq u$$, $$y\leq v$$ and

\begin{aligned} F(x,y) (t)&= \int_{0}^{T} f\bigl(t,s,x(s),y(s)\bigr)\, ds+h(t) \\ &\leq \int_{0}^{T} f\bigl(t,s,u(s),v(s)\bigr)\, ds+h(t)=F(u,v) (t)\quad \text{for all }t\in[0,T] \end{aligned}

and

\begin{aligned} F(y,x) (t)&= \int_{0}^{T} f\bigl(t,s,y(s),x(s)\bigr)\, ds+h(t) \\ &\leq \int_{0}^{T} f\bigl(t,s,v(s),u(s) \bigr)\, ds+h(t)=F(v,u) (t) \quad \text{for all }t\in[0,T]. \end{aligned}

Thus, F and g are G-edge preserving.

By condition (iv) it follows that $$(X^{2})^{F}_{g}=\{ (x, y)\in X\times X : gx \leq F(x, y) \text{ and } gy \leq F(y, x) \}\neq\emptyset$$.

On the other hand,

\begin{aligned} \begin{aligned} &\bigl\vert F(x,y) (t)-F(u,v) (t)\bigr\vert \\ &\quad \leq \int_{0}^{T} \bigl\vert f\bigl(t,s,x(s),y(s) \bigr)-f\bigl(t,s,u(s),v(s)\bigr)\bigr\vert \, ds \\ &\quad = \int_{0}^{T} \bigl\vert f\bigl(t,s,x(s),y(s) \bigr)-f\bigl(t,s,u(s),v(s)\bigr)\bigr\vert \, ds \\ &\quad \leq\frac{k}{T} \int_{0}^{T}\bigl(\bigl\vert x(s) - u(s)\bigr\vert + \bigl\vert y(s) - v(s)\bigr\vert \bigr)\, ds \\ &\quad \leq k\biggl( \frac{\|gx-gu\|+\|gy-gv\|}{2}\biggr) \\ &\quad \leq kM(gx,gu,gy,gv) \quad \text{for all }t\in[0,T]. \end{aligned} \end{aligned}

Then, there exist $$\psi(t)=t$$ and $$\theta\in\Theta$$, where $$\theta (s,t)=k$$ for $$s,t\in[0,\infty )$$ with $$k\in[0,1)$$, such that

$$\psi\bigl(\bigl\Vert F(x,y)-F(u,v)\bigr\Vert \bigr)\leq\theta\bigl( \Vert gx-gu\Vert ,\Vert gy-gv\Vert \bigr)\psi\bigl(M(gx,gu,gy,gv)\bigr),$$

where $$M(gx,gu,gy,gv)=\max\{ \|gx-gu\|,\|gy-gv\|\}$$. Hence, F and g are a θ-ψ-contraction.

Thus, there exists a coupled common fixed point $$(x^{*}, y^{*})\in X\times X$$ of the mapping F and g, which is the solution of the integral system (8). □

### Theorem 3.2

Consider system (8). Suppose that

1. (i)

$$f:[0,T]\times[0,T]\times\mathbb{R}^{n}\times\mathbb {R}^{n}\to\mathbb{R}^{n}$$ and $$h:[0,T]\to\mathbb{R}^{n}$$ are continuous;

2. (ii)

for all $$x,y,u,v \in\mathbb{R}^{n}$$ with $$x\leq u$$, $$y\leq v$$, we have $$f(t,s,x,y)\leq f(t,s,u,v)$$ for all $$t,s\in[0,T]$$;

3. (iii)

for all $$t,s\in[0,T]$$, $$x,y,u,v \in\mathbb{R}^{n}$$, $$x\leq u$$, $$y\leq v$$,

$$\bigl\vert f(t,s,x,y) - f(t,s,u,v)\bigr\vert \leq\frac{1}{T} \ln\bigl(1+ \max\bigl\{ \vert x-u\vert ,\vert y-v\vert \bigr\} \bigr);$$
4. (iv)

there exists $$(x_{0},y_{0})\in X\times X$$ such that

\begin{aligned}& x_{0}(t)\leq \int_{0}^{T} f\bigl(t,s,x_{0}(s),y_{0}(s) \bigr)\, ds+h(t), \\& y_{0}(t)\leq \int_{0}^{T} f\bigl(t,s,y_{0}(s),x_{0}(s) \bigr)\, ds+h(t), \end{aligned}

where $$t\in[0, T]$$.

Then there exists at least one solution of the integral system (8).

### Proof

Let $$F:X\times X\to X$$, $$(x,y)\mapsto F(x,y)$$, where

$$F(x,y) (t)= \int_{0}^{T} f\bigl(t,s,x(s),y(s)\bigr)\, ds+h(t), \quad t\in[0,T],$$

and define $$g:X\to X$$ by $$gx(t)=x(t)$$. As in Theorem 3.1, we have that F and g are G-edge preserving.

By condition (iv) it follows that $$(X^{2})^{F}_{g}=\{ (x, y)\in X\times X : gx \leq F(x, y) \text{ and } gy \leq F(y, x) \}\neq\emptyset$$.

On the other hand,

\begin{aligned}& \bigl\vert F(x,y) (t)-F(u,v) (t)\bigr\vert \\& \quad \leq \int_{0}^{T} \bigl\vert f\bigl(t,s,x(s),y(s) \bigr)-f\bigl(t,s,u(s),v(s)\bigr)\bigr\vert \, ds \\& \quad = \int_{0}^{T} \bigl\vert f\bigl(t,s,x(s),y(s) \bigr)-f\bigl(t,s,u(s),v(s)\bigr)\bigr\vert \, ds \\& \quad \leq\frac{1}{T} \int_{0}^{T} \ln\bigl(1+\max\bigl\{ \bigl\vert x(s)-u(s)\bigr\vert ,\bigl\vert y(s)-v(s)\bigr\vert \bigr\} \bigr)\, ds \\& \quad \leq \ln\Bigl(1+\max\Bigl\{ \max_{t\in[0,T]}\bigl|x(t)-u(t)\bigr|,\max _{t\in [0,T]}\bigl|y(t)-v(t)\bigr| \Bigr\} \Bigr) \\& \quad \leq \ln\bigl(1+\max\bigl\{ \Vert x-u\Vert ,\Vert y-v\Vert \bigr\} \bigr) \\& \quad = \ln\bigl(1+M(gx,gu,gy,gv)\bigr)\quad \text{for all }t\in[0,T], \end{aligned}

where $$M(gx,gu,gy,gv)=\max\{ \|gx-gu\|,\|gy-gv\|\}$$, which yields

\begin{aligned}& \ln\bigl( \bigl\vert F(x,y) (t)-F(u,v) (t)\bigr\vert +1\bigr) \\& \quad \leq\ln\bigl( \ln\bigl(1+M(gx,gu,gy,gv)\bigr)+1\bigr) \\& \quad =\frac{\ln( \ln(1+M(gx,gu,gy,gv))+1)}{\ln(1+M(gx,gu,gy,gv))}\ln \bigl(1+M(gx,gu,gy,gv)\bigr). \end{aligned}

Hence, there exist $$\psi(x)=\ln(x+1)$$ and $$\theta\in\Theta$$ defined by

$$\theta(s,t)= \textstyle\begin{cases} \frac{\ln(\ln(1+\max\{s,t\}))}{\ln(1+\max\{s,t \})}, & s>0\mbox{ or }t>0, \\ r\in[0,1), & s=0,t=0, \end{cases}$$

such that

\begin{aligned} \psi\bigl(d\bigl(F(x,y),F(u,v)\bigr)\bigr)&=\psi\bigl(\bigl\Vert F(x,y)-F(u,v) \bigr\Vert \bigr) \\ &\leq\theta\bigl( d(gx,gu), d(gy,gv)\bigr)\psi\bigl(M(gx,gu,gy,gv)\bigr). \end{aligned}

Hence, we see that F and g are a θ-ψ-contraction. Thus, there exists a coupled common fixed point $$(x^{*}, y^{*})\in X\times X$$ of the mapping F and g, which is a solution for the integral system (8). □

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## Acknowledgements

This research was supported by Thailand Research Fund under the project RTA5780007 and Chiang Mai University.

## Author information

Correspondence to Phakdi Charoensawan.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The author read and approved the final manuscript.

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