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.
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1 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 [5–17].
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
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 [19–23].
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
-
(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
$$\begin{aligned}& F\bigl( F(x_{n_{i}}, y_{n_{i}}), F(y_{n_{i}}, x_{n_{i}})\bigr)\to F\bigl(x^{*}, y^{*}\bigr) \quad \text{and} \\& F\bigl(F(y_{n_{i}}, x_{n_{i}}), F(x_{n_{i}}, y_{n_{i}})\bigr)\to F\bigl(y^{*}, x^{*}\bigr)\quad \text{as } i\to\infty; \end{aligned}$$ -
(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.
2 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:
and
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
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 \(x,y,a \in X\),
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)
F and g is G-edge preserving;
-
(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:
-
(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) -
(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
(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
and
for all \(n\in\mathbb{N}\). From (3) and (4) we get
for all \(n\in\mathbb{N}\), that is,
Regarding the properties of ψ, we conclude that
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
Taking the limit as \(n\rightarrow\infty\), we have
Since \(\theta\in\Theta\),
as \(n\rightarrow\infty\). Therefore,
which is a contradiction. Hence,
□
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:
-
(i)
g is continuous, and \(g(X)\) is closed;
-
(ii)
\(F(X\times X)\subset g(X)\), and g and F are compatible;
-
(iii)
F is G-continuous, or the tripled \((X, d, G)\) has property A;
-
(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
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
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
By the compatibility of g and F we have that
Now, suppose that (a) F is G-continuous. It is easy to see that
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
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,
Theorem 2.6
In addition to hypotheses of Theorem 2.5, assume that
-
(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
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,
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
and then
Similarly, from \((gu, ga), (gv, gb)\in E(G)\) we have
By the triangle inequality we have
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].
3 Applications
In this section, we apply our theorem to the existence theorem for a solution of the following integral system:
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
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
-
(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;
-
(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 \(t,s\in[0,T]\);
-
(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 \(t,s\in[0,T]\), \(x,y,u,v \in\mathbb{R}^{n}\), \(x\leq u\), \(y\leq v\);
-
(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
and define \(g:X\to X\) by \(gx(t)=\frac{x(t)}{2}\).
System (8) can be written as
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
and
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,
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
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
-
(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;
-
(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 \(t,s\in[0,T]\);
-
(iii)
for all \(t,s\in[0,T]\), \(x,y,u,v \in\mathbb{R}^{n}\), \(x\leq u\), \(y\leq v\),
$$ \bigl\vert f(t,s,x,y) - f(t,s,u,v)\bigr\vert \leq\frac{1}{T} \ln\bigl(1+ \max\bigl\{ \vert x-u\vert ,\vert y-v\vert \bigr\} \bigr); $$ -
(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
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,
where \(M(gx,gu,gy,gv)=\max\{ \|gx-gu\|,\|gy-gv\|\}\), which yields
Hence, there exist \(\psi(x)=\ln(x+1)\) and \(\theta\in\Theta\) defined by
such that
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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This research was supported by Thailand Research Fund under the project RTA5780007 and Chiang Mai University.
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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
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DOI: https://doi.org/10.1186/s13663-015-0473-4