Coincidence point theorems for some multi-valued mappings in complete metric spaces endowed with a graph
- Chalongchai Klanarong^{1} and
- Suthep Suantai^{1}Email author
https://doi.org/10.1186/s13663-015-0379-1
© Klanarong and Suantai 2015
Received: 10 February 2015
Accepted: 10 July 2015
Published: 28 July 2015
Abstract
In this paper, we introduce the concepts of weak g-graph-preserving for multi-valued mappings and weak G-contractions in a metric space endowed with a directed graph. We establish the coincidence point theorems for this type of mappings in a complete metric space endowed with a directed graph. Examples illustrating our main results are also presented. Our results extend and generalize various known results in the literature.
Keywords
MSC
1 Introduction
Fixed point theorems for contraction mappings and their generalizations play an important role in the study of theory of equations. The Banach contraction principle [1] is a fundamental result which can be applied widely for solving the existence of solutions of various equations. Over the years, it has been widely extended and generalized in different directions by many authors (see [1–10]).
Zamfirescu [10] proved some fixed point theorems by combining (1.1), (1.2) and (1.3), which is stated as follows.
Theorem 1.1
[10]
- (i)
\(d(Tx,Ty) \leq ad(x,y)\);
- (ii)
\(d(Tx,Ty)\leq bd(x,Tx)+bd(y,Ty)\);
- (iii)
\(d(Tx,Ty)\leq cd(x,Ty)+ cd(y,Tx)\).
In [2] Berinde introduced and studied a weak contraction mapping on a complete metric space which is weaker than the Zamfirescu operators.
Definition 1.2
[2]
The following results can be found in [2].
Proposition 1.3
[2]
- (1)
\(x \preceq x \) (reflexivity);
- (2)
\(x \preceq y \) and \(y \preceq x\), then \(x= y\) (antisymmetry);
- (3)
\(x \preceq y \) and \(y \preceq z\), then \(x\preceq z\) (transitivity)
Fixed point theorems for monotone single-valued mappings have been investigated and studied in partially ordered metric spaces by many mathematicians (see [7–9, 11, 12]). Nieto and Rodriguez-Lopez [7, 8] were the first who studied some fixed point theorems for monotone nondecreasing mappings in partially ordered metric spaces and applied the obtained results to study an existence problem of ordinary differential equations.
The study of fixed point for multi-valued contraction mappings using the Pompeiu-Hausdorff metric was first performed by Nadler [13].
We let \(\operatorname{Comp}(X)\) be the set of all nonempty compact subsets of X. It is clear that \(\operatorname{Comp}(X)\) is included in \(CB(X)\).
- (i)
fixed point of T if \(x \in T(x)\),
- (ii)
coincidence point of a hybrid pair \(\{T, g\}\) if \(g(x) \in T(x)\).
In 1969, Nadler [13] introduced the concept of Banach contraction principle for a multi-valued mapping and proved the existence of fixed point for multi-valued version of the Banach contraction principle. The following theorem is the first well-known theorem of multi-valued contractions studied by Nadler.
Theorem 1.4
[13]
In 2007, Berinde and Berinde [14] provided the new type of contraction which is a generalization of the contraction principle considered by Nadler.
Definition 1.5
[14]
The next definition, G-contraction, was introduced by Jachymski [15] in 2008.
Definition 1.6
[15]
The mapping \(T: X\to X\) satisfying condition (1.5) is called a graph-preserving mapping. Under some additional properties of a metric space X endowed with a directed graph, Jachymski showed that a G-contraction \(T : X\to X\) has a fixed point if and only if there exists \(x \in X\) such that \((x, T(x)) \in E(G)\).
Subsequently, Beg and Butt [16] tried to introduce the concept of G-contraction for multi-valued mappings, but their extension was not carried correctly (see [17, 18]).
In 2011, Nicolae et al. [19] extended the notion of multivalued contraction on a metric space with a graph.
Recently, Dinevari and Frigon [20] introduced a new concept of G-contraction multivalued mappings.
Definition 1.7
[20]
Let \(T : X \to2^{X}\) be a map with nonempty values. We say that T is a G-contraction (in the sense of Dinevari and Frigon) if there exists \(\alpha\in(0,1) \) such that for all \((x, y) \in E(G)\) and \(u \in Tx\), there exists \(v \in Ty\) such that \((u, v) \in E(G)\) and \(d(u, v) \leq\alpha d(x, y)\).
They showed that under some properties, weaker than Property (A), a multi-valued G-contraction with the closed value has a fixed point. Recently, Tiammee and Suantai [21] introduced the concept of graph-preserving for multi-valued mappings and proved their fixed point theorem in a complete metric space endowed with a graph.
Definition 1.8
[21]
Let X be a nonempty set, \(G = (V(G),E(G))\) be a directed graph such that \(V(G) = X\), and \(T : X\to CB(X)\). Then T is said to be graph-preserving if \((x,y) \in E(G) \Rightarrow(u,v) \in E(G) \) for all \(u \in Tx\) and \(v \in Ty\).
Definition 1.9
[21]
Recently, Phon-on et al. [22] introduced a new type of weak G-contraction which is weaker than that of Tiammee and Suantai [21], and they proved some fixed point theorems for this type of mappings with compact values which is a generalization of several known results in a complete metric space endowed with a graph.
Definition 1.10
[22]
Let X be a nonempty set and \(G = (V(G),E(G))\) be a directed graph such that \(V(G) = X\), and \(T : X \to \operatorname{Comp}(X)\). Then T is said to be weak graph-preserving if it satisfies the following: for each \(x, y \in X\), if \((x, y) \in E(G)\), then for each \(u \in Tx\) there is \(v \in P_{Ty}(u)\) such that \((u, v)\in E(G)\), where \(P_{Ty}(u)=\{a \in Ty \mid d(u,a)=D(u,Ty)\}\).
Motivated and inspired by all of those works mentioned above, we aim to introduce a new type of multi-valued contractions which is more general than that of Berinde [2]. These classes of mappings are defined for multi-valued mappings in complete metric spaces endowed with directed graphs. Fixed point theorems for this type of mappings are established. We also apply our main results for proving the existence theorem of a fixed point for a multi-valued mapping defined on a partially ordered metric space. Moreover, we also apply the obtained results to prove the existence theorem of a coupled fixed point. Some examples are given to illustrate our results.
2 Preliminaries
The following lemma is useful for our main results.
Lemma 2.1
[13]
Let \(A,B \in CB(X)\) and \(a\in A\). Then, for \(\varepsilon> 0\), there exists an element \(b \in B\) such that \(d(a,b) \leqslant H(A,B)+\varepsilon\).
Definition 2.2
Let X be a nonempty set, \(G=(V(G),E(G))\) be a directed graph such that \(V(G)=X\) and let \(T:X \to PB(X)\) and \(g:X\to X\). Then T is said to be weak g-graph-preserving if for any \(x,y \in X\) such that \((g(x),g(y)) \in E(G)\), then for each \(u \in Tx\) there exists \(v \in P_{Ty}(u)\) such that \((u,v) \in E(G)\).
Example 2.3
Now we will show that T is weak g-graph-preserving. Let \((g(x),g(y))\in E(G)\).
If \((g(x),g(y))=(1,0)\), then \((x,y)=(1,0)\), \(Tx=\{\frac{1}{2},1\}\), and \(Ty=\{0,1\}\), we have \(P_{Ty}(\frac{1}{2})=\{0,1\}\), \(P_{Ty}(1)=\{1\}\) and \((\frac {1}{2},0),(1,1)\in E(G)\).
If \((g(x),g(y))= (\frac{1}{2^{2n}},\frac{1}{2^{2n}})\), then \((x,y)=(\frac{1}{2^{n}},\frac{1}{2^{n}})\) and \(Tx=\{\frac {1}{2^{n+1}},1\}=Ty\), we have \(P_{Ty}(1)=\{1\}\), \(P_{Ty}(\frac{1}{2^{n+1}})=\{\frac{1}{2^{n+1}}\}\) and \((1,1),(\frac{1}{2^{n+1}},\frac{1}{2^{n+1}})\in E(G)\).
If \((g(x),g(y))=(\frac{1}{2^{2n}},0)\), then \((x,y)=(\frac {1}{2^{n}},0)\), \(Tx=\{\frac{1}{2^{n+1}},1\}\), and \(Ty=\{0,1\}\), we have \(P_{Ty}(\frac{1}{2^{n+1}})=\{0\}\), \(P_{Ty}(1)=\{1\}\) and \((\frac {1}{2^{n+1}},0), (1,1) \in E(G)\).
If \((g(x),g(y))=(1,1)\), then \((x,y)=(1,1)\) and \(Tx=\{\frac{1}{2},1\} =Ty\), we have \(P_{Ty}(\frac{1}{2})=\{\frac{1}{2}\}\), \(P_{Ty}(1)=\{1\} \) and \((\frac {1}{2},\frac{1}{2}), (1,1)\in E(G)\).
Hence T is weak g-graph-preserving.
Definition 2.4
3 Main results
We start with introducing a new type of weak G-contraction with respect to g.
Definition 3.1
Example 3.2
Now we prove that T is a \((k,L,r)\) weak G-contraction with respect to g, where \(k=\frac{1}{2}\), \(\frac{2}{7}\leq r <\frac{1}{2}\), and \(L\geq0\). Let \(x,y \in X\) such that \((x,y)\in E(G)\).
The following property is useful for our main results.
Property (A)
[15]
For any sequence \((x_{n})_{n\in\mathbb{N}}\) in X, if \(x_{n} \to x\) and \((x_{n},x_{n+1}) \in E(G)\) for \(n \in\mathbb{N}\), then there is a subsequence \((x_{k_{n}})_{n\in\mathbb{N}}\) with \((x_{k_{n}},x)\in E(G)\) for \(n \in\mathbb{N}\).
We first prove our main result.
Theorem 3.3
- (i)
T is a weak g-graph-preserving mapping;
- (ii)
there exists \(x_{0} \in X\) such that \((g(x_{0}),y)\in E(G)\) for some \(y\in Tx_{0}\);
- (iii)
X has Property (A);
- (iv)
T is a \((k,L,r)\) weak G-contraction,
Proof
The following result is obtained directly from Theorem 3.3 in case that T is a single-valued mapping.
Theorem 3.4
- (i)
T is a G-edge-preserving mapping with respect to g;
- (ii)
there exists \(x_{0} \in X\) such that \((g(x_{0}),Tx_{0})\in E(G)\);
- (iii)
X has Property (A);
- (iv)there exist \(L\geq0 \) and \(k,r \in[0,1)\) with \(0< k+r < 1\) such thatfor all \(x,y \in X \) with \((g(x),g(y)) \in E(G)\),$$d(Tx,Ty) \leq kd\bigl(g(x),g(y)\bigr)+Ld\bigl(g(y),Tx\bigr)+rd\bigl(g(x),Ty \bigr) $$
Example 3.5
Now we will show that T is weak g-graph-preserving. Let \((g(x),g(y))\in E(G)\).
If \((g(x),g(y))=(0,0)\), then \((x,y)=(1,1)\) and so \(Tx=\{0\}=Ty\). We have \(P_{Ty}(0)=\{0\}\) and \((0,0)\in E(G)\).
If \((g(x),g(y))=(1,0)\), then \((x,y)=(0,1)\) and so \(Tx=\{1\}\), \(Ty=\{0\} \). We have \(P_{Ty}(1)=\{0\}\) and \((1,0)\in E(G)\).
If \((g(x),g(y))=(1,\frac{1}{2^{n}})\), then \((x,y)=(0,\frac{1}{2^{n}})\) and so \(Tx=\{1\}\), \(Ty=\{\frac{1}{2^{n+1}},0\}\). We have \(P_{Ty}(1)=\{\frac {1}{2^{n+1}}\}\) and \((1,\frac{1}{2^{n+1}})\in E(G)\).
If \((g(x),g(y))= (\frac{1}{2^{n}},\frac{1}{2^{n}})\), then \((x,y)=(\frac {1}{2^{n}},\frac{1}{2^{n}})\) and so \(Tx=\{\frac{1}{2^{n+1}},0\}=Ty\). We have \(P_{Ty}(0)=\{0\}\), \(P_{Ty}(\frac{1}{2^{n+1}})=\{\frac{1}{2^{n+1}}\}\) and \((0,0),(\frac{1}{2^{n+1}},\frac{1}{2^{n+1}})\in E(G)\).
If \((g(x),g(y))=(\frac{1}{2^{n}},0)\), then \((x,y)=(\frac{1}{2^{n}},1)\) and so \(Tx=\{\frac{1}{2^{n+1}},0\}\), \(Ty=\{0\}\). We have \(P_{Ty}(0)=\{0\} =P_{Ty}(\frac{1}{2^{n+1}})\) and \((0,0),(\frac{1}{2^{n+1}},0)\in E(G)\).
If \((g(x),g(y))=(\frac{1}{2^{n}},\frac{1}{2^{n+1}})\), then \((x,y)=(\frac {1}{2^{n}},\frac{1}{2^{n+1}})\) and so \(Tx=\{\frac{1}{2^{n+1}},0\}\), \(Ty=\{ \frac{1}{2^{n+2}},0\}\). We have \(P_{Ty}(0)=\{0\}\), \(P_{Ty}(\frac {1}{2^{n+1}})=\{\frac{1}{2^{n+1}}\}\) and \((0,0),(\frac{1}{2^{n+1}},\frac {1}{2^{n+2}})\in E(G)\).
Hence T is weak g-graph-preserving.
Next we prove that T is a \((k,L,r)\) weak G-contraction with respect to g, where \(k=\frac{1}{2}\), \(r =\frac{1}{4}\), and \(L\geq1\). Let \(x,y \in X\) such that \((g(x),g(y))\in E(G)\).
We next apply Theorem 3.3 to obtain fixed point theorems for some contraction mappings in partially ordered metric spaces.
Definition 3.6
Let (X,d) be a metric space endowed with a partial ordering ⪯. For each \(A, B \in PB(X)\), we write \(A \prec_{1} B\) if for each \(a \in A\), there exists \(b \in P_{B}(a)\) such that \(a \prec b\).
Definition 3.7
If g is the identity map and \(T:X\to PB(X)\) is g-increasing, we simply say that T is increasing.
Example 3.8
Moreover, we take \(g(x) = x\), then it is easy to see that T is increasing.
Corollary 3.9
- (1)
T is g-increasing;
- (2)
there exist \(x_{0} \in X \) and \(u \in Tx_{0}\) such that \(g(x_{0}) \prec u\);
- (3)
for each sequence \(\{x_{n}\}\) such that \(g(x_{n}) \prec g(x_{n+1})\) for all \(n \in\mathbb{N}\) and \(g(x_{n})\) converges to \(g(x)\), for some \(x \in X\), then \(g(x_{n})\prec g(x)\) for all \(n \in\mathbb{N}\);
- (4)there exist \(L\geq0 \) and \(k,r\in[0,1 )\) with \(0< k + r < 1\) such thatfor all \(x,y \in X \) with \(g(x) \prec g(y)\).$$H(Tx,Ty) \leq kd\bigl(g(x),g(y)\bigr)+LD\bigl(g(y),Tx\bigr)+rD\bigl(g(x),Ty \bigr) $$
Proof
Define \(G=(V(G),E(G))\), where \(V(G)=X\) and \(E(G)=\{(x,y):x\prec y\}\). Let \(x,y \in X\) be such that \((g(x),g(y))\in E(G)\). Then \(g(x)\prec g(y) \). Since T is g-increasing, we have \(Tx \prec_{1} Ty\). Then, for any \(u\in Tx \), there exists \(v\in P_{Ty}(u)\) such that \(u \prec v\) and so \((u,v)\in E(G)\). Thus T is weak g-graph-preserving. By (2), there exist \(x_{0} \in X\) and \(u \in Tx_{0}\) such that \(g(x_{0}) \prec u\), and so \((g(x_{0}),u)\in E(G)\). So assumption (ii) of Theorem 3.3 is satisfied. It is easy to see that (iii) and (iv) of Theorem 3.3 are also satisfied. Therefore, this corollary is obtained directly by Theorem 3.3. □
If g is the identity map, then the following result is directly obtained by Corollary 3.9.
Corollary 3.10
- (1)
T is increasing;
- (2)
there exist \(x_{0} \in X \) and \(u \in Tx_{0}\) such that \(x_{0} \prec u\);
- (3)
for each sequence \(\{x_{n}\}\) such that \(x_{n} \prec x_{n+1}\) for all \(n \in\mathbb{N}\) and \(x_{n}\) converges to x, for some \(x \in X\), then \(x_{n}\prec x\) for all \(n \in\mathbb{N}\);
- (4)there exist \(L\geq0\) and \(k,r\in[0,\infty)\) with \(0< k + r < 1\) such thatfor all \(x,y \in X \) with \(x \prec y\).$$H(Tx,Ty) \leq kd(x,y)+LD(y,Tx)+rD(x,Ty) $$
4 Applications
In this section, we prove the existence of a coupled fixed point for a single-valued mapping in a complete metric space endowed with a directed graph.
Let X be a nonempty set and \(F : X \times X \to X\) be a single-valued mapping. An element \((x,y)\in X \times X\) is called a coupled fixed point of F if \(x = F(x, y)\) and \(y= F(y, x)\). We denote by \(C\operatorname{Fix}(F)\) the set of all coupled fixed points of the mapping F, i.e., \(C\operatorname{Fix}(F) = \{ (x,y) \in X\times X \mid F(x,y)=x \mbox{ and } F(y,x)=y\}\).
Coupled fixed point theorems and their application were investigated by many authors (see [12, 19, 23–27] for examples).
Definition 4.1
[12]
Recently, Chifu and Petrusel [28] introduced the concept of edge-preserving as follows.
Definition 4.2
[28]
We say that \(F : X \times X \to X\) is edge-preserving if \((x,u) \in E(G)\), \((y,v) \in E(G^{-1})\) implies \((F(x,y),F(u,v))\in E(G)\) and \((F(y,x),F(v,u)) \in E(G^{-1})\).
Note that an element \((x,y)\in X\times X\) is a coupled fixed point of F if and only if \((x,y)\) is a fixed point of \(T_{F}\).
Theorem 4.3
- (i)
there exist \(x_{0},y_{0} \in X\) such that \((x_{0},F(x_{0},y_{0}))\in E(G)\) and \((y_{0},F(y_{0},x_{0})) \in E(G^{-1})\);
- (ii)X has the following property:
- (a)
if any sequence \(\{x_{n}\}\) in X such that \(x_{n} \to x\) and \((x_{n},x_{n+1}) \in E(G)\) for \(n \in\mathbb{N}\), then \((x_{n},x)\in E(G)\) for all \(n \in\mathbb{N}\);
- (b)
if any sequence \(\{y_{n}\}\) in X such that \(y_{n} \to y\) and \((y_{n},y_{n+1}) \in E(G^{-1})\) for \(n \in\mathbb{N}\), then \((y_{n},y)\in E(G^{-1})\) for all \(n \in\mathbb{N}\);
- (a)
- (iii)there exist \(L\geq0 \) and \(k,r \in[0,1)\) with \(0< k+r < 1\) such thatfor all \(x,y,u,v \in X\) with \((x , u) \in E(G)\) and \((y , v) \in E(G^{-1})\).$$\begin{aligned} &d\bigl(F(x,y),F(u,v)\bigr)+d\bigl(F(y,x),F(v,u)\bigr)\\ &\quad \leq k\bigl[d(x,u)+d(y,v)\bigr]+L\bigl[d\bigl(u,F(x,y)\bigr)+d\bigl(v,F(y,x) \bigr)\bigr]\\ &\qquad{} +r\bigl[d\bigl(x,F(u,v)\bigr)+d\bigl(y,F(v,u)\bigr)\bigr] \end{aligned}$$
Proof
The following corollary is a consequence of Theorem 4.3.
Corollary 4.4
- (a)
if a nondecreasing sequence \(\{x_{n}\} \to x\), then \(x_{n} \preceq x\) for all \(n \in\mathbb{N}\);
- (b)
if a nonincreasing sequence \(\{y_{n}\} \to y\), then \(y_{n} \succeq y\) for all \(n \in\mathbb{N}\).
Proof
Let \(G=(V(G),E(G))\), where \(V(G)=X\) and \(E(G)=\{ (x,y) \mid x \preceq y \}\). We can directly check that all conditions of Theorem 4.3 are satisfied. Therefore F has a coupled fixed point. □
Declarations
Acknowledgements
The authors would like to thank the Thailand Research Fund under the project RTA5780007 and Chiang Mai University, Chiang Mai, Thailand for the financial support.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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