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 [2]. 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 [517].

Choudhury and Kundu [7] give a notion of compatibility.

Definition 1.1

([7])

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. [1] 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 [18]. 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 [1923].

Recently, Chifu and Petrusel [24] 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 [4] 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. [1]. 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 [1] 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 [1]. □

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 [1].

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). □

References

1. Kadelburg, Z, Kumam, P, Radenović, S, Sintunavarat, W: Common coupled fixed point theorems for Geraghty-type contraction mappings using monotone property. Fixed Point Theory Appl. 2015, 27 (2015). doi:10.1186/s13663-015-0278-5

2. Lakshmikantham, V, Ćirić, L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. TMA 70, 4341-4349 (2009)

3. Guo, D, Lakshmikantham, V: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal., Theory Methods Appl. 11, 623-632 (1987)

4. Gnana Bhaskar, T, Lakshmikantham, V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. TMA 65, 1379-1393 (2006)

5. Berinde, V: Coupled fixed point theorems for φ-contractive mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal., Theory Methods Appl. 75(6), 3218-3228 (2012)

6. Berinde, V: Coupled coincidence point theorems for mixed monotone nonlinear operators. Comput. Math. Appl. 64(6), 1770-1777 (2012)

7. Choudhury, BS, Kundu, A: A coupled coincidence point result in partially ordered metric spaces for compatible mappings. Nonlinear Anal. TMA 73, 2524-2531 (2010)

8. Sintunavarat, W, Radenović, S, Golubović, Z, Kuman, P: Coupled fixed point theorems for F-invariant set. Appl. Math. Inf. Sci. 7(1), 247-255 (2013)

9. Karapinar, E, Luong, NV, Thuan, NX, Hai, TT: Coupled coincidence points for mixed monotone operators in partially ordered metric spaces. Arab. J. Math. 1, 329-339 (2012)

10. Borcut, M: Tripled fixed point theorems for monotone mappings in partially ordered metric spaces. Carpath. J. Math. 28(2), 207-214 (2012)

11. Borcut, M: Tripled coincidence theorems for monotone mappings in partially ordered metric spaces. Creative Math. Inform. 21(2), 135-142 (2012)

12. Radenović, S: Coupled fixed point theorems for monotone mappings in partially ordered metric spaces. Kragujev. J. Math. 38(2), 249-257 (2014)

13. Radenović, S: Some coupled coincidence points results of monotone mappings in partially ordered metric spaces. Int. J. Anal. Appl. 5(2), 174-184 (2014)

14. Radenović, S: Bhaskar-Lakshmikantham type results for monotone mappings in partially ordered metric spaces. Int. J. Nonlinear Anal. Appl. 5(2), 37-49 (2014)

15. Radenović, S: A note on tripled coincidence and tripled common fixed point theorems in partially ordered metric spaces. Appl. Math. Comput. 236, 367-372 (2014)

16. Fadail, ZM, Rad, GS, Ozturk, V, Radenović, S: Some remarks on coupled, tripled and n-tupled fixed points theorems in ordered abstract metric spaces. Far East J. Math. Sci. 97, 809-839 (2015)

17. Dorić, D, Kadelburg, Z, Radenović, S: Coupled fixed point results for mappings without mixed monotone property. Appl. Math. Lett. (2012). doi:10.1016/j.aml.2012.02.022

18. Jachymski, J: The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 136, 1359-1373 (2008)

19. Beg, I, Butt, AR, Radojević, S: The contraction principle for set valued mappings on a metric space with a graph. Comput. Math. Appl. 60, 1214-1219 (2010)

20. Bojor, F: Fixed point theorems for Reich type contractions on metric spaces with a graph. Nonlinear Anal. 75(9), 3895-3901 (2012)

21. Alfuraidan, MR: The contraction principle for multivalued mappings on a modular metric space with a graph. Can. Math. Bull. 2015, 29 (2015). doi:10.4153/CMB-2015-029-x

22. Alfuraidan, MR: Remarks on monotone multivalued mappings on a metric space with a graph. J. Inequal. Appl. 2015, 202 (2015). doi:10.1186/s13660-015-0712-6

23. Alfuraidan, MR, Khamsi, MA: Caristi fixed point theorem in metric spaces with a graph. Abstr. Appl. Anal. 2014, 303484 (2014). doi:10.1155/2014/303484

24. Chifu, C, Petrusel, G: New results on coupled fixed point theory in metric spaces endowed with a directed graph. Fixed Point Theory Appl. 2014, 151 (2014). doi:10.1186/1687-1812-2014-151

Acknowledgements

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

Author information

Authors

Corresponding author

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.

Rights and permissions

Reprints and Permissions

Suantai, S., Charoensawan, P. & Lampert, T.A. Common coupled fixed point theorems for θ-ψ-contraction mappings endowed with a directed graph. Fixed Point Theory Appl 2015, 224 (2015). https://doi.org/10.1186/s13663-015-0473-4

• Accepted:

• Published:

• DOI: https://doi.org/10.1186/s13663-015-0473-4

MSC

• coupled fixed point
• coupled coincidence point
• common fixed point
• Geraghty-type condition
• edge preserving
• metric spaces
• connected graph
• monotone
• partially ordered set