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- Open Access
Coincidence point and fixed point theorems for a new type of G-contraction multivalued mappings on a metric space endowed with a graph
- Adisak Hanjing^{1} and
- Suthep Suantai^{1}Email author
https://doi.org/10.1186/s13663-015-0420-4
© Hanjing and Suantai 2015
- Received: 30 June 2015
- Accepted: 8 September 2015
- Published: 21 September 2015
Abstract
In this paper, a new type of G-contraction multivalued mappings in a metric space endowed with a directed graph is introduced and studied. This type of mappings is more general than that of Mizoguchi and Takahashi (J. Math. Anal. Appl. 141:177-188, 1989), Berinde and Berinde (J. Math. Anal. Appl. 326:772-782, 2007), Du (Topol. Appl. 159:49-56, 2012), and Sultana and Vetrivel (J. Math. Anal. Appl. 417:336-344, 2014). A fixed point and coincidence point theorem for this type of mappings is established. Some examples illustrating our main results are also given. The main results obtained in this paper extend and generalize those in (Tiammee and Suantai in Fixed Point Theory Appl. 2014:70, 2014) and many well-known results in the literature.
Keywords
- fixed point
- multivalued mappings
- Mizoguchi-Takahashi G-contraction
MSC
- 47H10
- 54H25
1 Introduction
Fixed point theory plays a very important role in nonlinear analysis and applications. It is well known that many metric fixed point theorems were motivated from the Banach contraction principle.
Theorem 1.1
In 1969, Nadler [1] extended the Banach contraction principle for multivalued mappings.
Theorem 1.2
([1])
Nadler’s fixed point theorem for multivalued contractive mappings has been extended in many directions (see [2–5]). Reich [6] proved the following fixed point theorem for multivalued φ-contraction mappings.
Theorem 1.3
([6])
In 1989, Mizoguchi and Takahashi [7] relaxed the compactness assumption on T to closed and bounded subsets of X. They proved the following theorem, which is a generalization of Nadler’s theorem.
Theorem 1.4
([7])
In 2007, Berinde and Berinde [4] extended Theorem 1.1 to the class of multivalued weak contractions.
Definition 1.5
([4])
Definition 1.6
([4])
They proved that in a complete metric space, every multivalued \((\theta, L) \)-weak contraction has a fixed point. In the same paper, they also proved that every generalized multivalued \((\alpha, L) \)-weak contraction has a fixed point (see [4]). This result was generalized by Du [8] in 2012 as in the following theorem.
Theorem 1.7
([8])
- (a)
\(T(x) \) is g-invariant (i.e., \(g(T(x)) \subseteq T(x)\) for each \(x \in X \)),
- (b)there exists a function \(h:X\to[0,\infty) \) such that$$H\bigl(T(x),T(y)\bigr) \leq\alpha\bigl(d(x,y)\bigr)d(x,y) + h\bigl(g(y)\bigr)d \bigl(g(y),T(x)\bigr),\quad \textit {for all } x, y \in X . $$
In 2008, Jachymski [9] introduced the concept of a G-contraction and proved some fixed point results of G-contractions in a complete metric space endowed with a directed graph.
Let \((X, d) \) be a metric space and let \(G=(V(G),E(G)) \) be a directed graph such that \(V(G)=X \) and \(E(G) \) contains all loops, i.e., \(\Delta= \{(x,x): x\in X\}\subseteq E(G)\).
Definition 1.8
([9])
He showed that in the case that there are certain properties on \((X,d,G) \) a G-contraction \(f:X\to X \) has a fixed point if and only if \(X_{f}=\{x\in X : (x,f(x))\in E(G)\} \) is nonempty. The mapping \(f : X\to X \) satisfying condition (1.1) is also called a graph-preserving mapping.
In 2010, Beg and Butt [5] introduced the concept of G-contraction for a multivalued mapping \(T : X\to \operatorname{CB}(X) \) as follows.
Definition 1.9
([5])
Recently, in 2015, Alfuraidan [10] pointed out that the above definition of a G-contraction is flawed and the argument behind the proof of the main result of [5] fails.
By using the idea of multivalued contraction mappings in [11, 12], Alfuraidan introduced the following concept of a G-contraction.
Definition 1.10
([10])
A multivalued mapping \(T:X\to2^{X} \) is said to be a monotone increasing G-contraction if there exists \(\alpha\in[0,1) \) such that, for any \(u,v\in X \) with \((u,v)\in E(G) \) and any \(U\in T(u) \), there exists \(V\in T(v) \) such that \((U,V)\in E(G) \) and \(d(U,V)\leq\alpha d(u,v) \).
He showed that under some properties on a metric space, a monotone increasing G-contraction multivalued mappings has a fixed point (see [10], Theorem 3.1).
In 2014, Tiammee and Suantai [13] introduced the concept of graph-preserving for multivalued mappings as follows.
Definition 1.11
([13])
In the same year, Sultana and Vetrivel [14] introduced a concept of a Mizoguchi-Takahashi G-contraction as follows.
Definition 1.12
([14])
- (i)
\(H(T(x),T(y)) \leq\alpha(d(x,y))d(x,y) \),
- (ii)
if \(u \in T(x) \) and \(v \in T(y) \) are such that \(d(u,v) \leq d(x,y) \), then \((u,v) \in E(G) \).
They showed that if there are some properties on a metric space, a multivalued Mizoguchi-Takahashi G-contraction has a fixed point (see [14], Theorem 3).
In this paper, we introduce a new concept of a G-contraction in a metric space endowed with a directed graph which is more general than the Mizoguchi-Takahashi G-contraction for multivalued mappings. We establish some coincidence point and fixed point theorems for this type of mappings and give some examples illustrating our main results.
2 Preliminaries
Let \(g:X\to X \) be a self-map and \(T:X\to2^{X} \) be a multivalued map. A point x in X is a coincidence point of g and T if \(g(x)\in T(x) \). If g is the identity map on X, then \(x=g(x)\in T(x) \) and we call x a fixed point of T. The set of all fixed points of T and the set of all coincidence points of g and T are denoted by \(\mathcal{F}(T)\) and \(\mathcal {COP}(g,T) \), respectively.
The following lemmas are useful for our main results.
Lemma 2.1
([1])
Let \((X,d) \) be a metric space. If \(A,B \in \operatorname{CB}(X) \) and \(a \in A\), then, for each \(\varepsilon> 0\), there exists \(b \in B \) such that \(d(a,b) \leq H(A,B) + \varepsilon\).
Lemma 2.2
([6])
3 Main results
We first introduce a concept of weak G-contraction on a metric space endowed with a directed graph.
Definition 3.1
- (i)
\(H(T(x),T(y)) \leq\alpha(d(x,y))d(x,y) + h(g(y))d(g(y),T(x)) \),
- (ii)
if \(u \in T(x) \) and \(v \in T(y) \) are such that \(d(u,v) \leq d(x,y) \), then \((u,v) \in E(G) \).
Example 3.2
We show that \(T : X \to \operatorname{CB}(X) \) is a weak G-contraction with respect to g. Let \((x,y)\in E(G) \).
Case 1. \((x,y)=(\frac{1}{2^{n}},0) \) for some \(n \in \mathbb{N}\cup\{0\} \).
Therefore \(T : X \to \operatorname{CB}(X) \) is a weak G-contraction with respect to g.
Theorem 3.3
- (1)
there is \(x_{0} \in X \) such that \((x_{0},y) \in E(G) \) for some \(y \in T(x_{0}) \),
- (2)
\(T(x) \) is g-invariant (i.e., \(g(T(x)) \subseteq T(x) \) for each \(x \in X \)),
- (3)
for any sequence \(\{x_{n}\} \) in X, if \(x_{n} \to x \) and \((x_{n},x_{n+1})\in E(G) \) for all \(n \in\mathbb{N} \), then there is a subsequence \(\{ x_{n_{k}} \} \) of \(\{ x_{n} \} \) with \((x_{n_{k}},x) \in E(G) \) for some \(k \in\mathbb{N} \).
Proof
Let \(x_{0} \in X \) and \(x_{1} \in T(x_{0}) \) such that \((x_{0},x_{1}) \in E(G) \).
Example 3.4
Let X, \(E(G)\), d, α, T, g, and h be the same as in Example 3.2. Then T is a weak G-contraction with respect to g, and g is continuous. It is easy to see that the conditions (1)-(3) of Theorem 3.3 hold. Hence all conditions of Theorem 3.3 are satisfied and we see that \(\mathcal{COP}(g,T) \cap\mathcal{F}(T)=\{0,1\}\).
Remark 3.5
- (i)
In Theorem 3.3, if we take a directed graph G with \(E(G)= X\times X\), we obtain Theorem 2.2 of Du [8] immediately.
- (ii)
In Theorem 3.3, if we take a function \(h=0 \), then we obtain the existence result which is similar to Theorem 3 of Sultana and Vetrivel [14].
- (iii)
In Theorem 3.3, if we take a directed graph G with \(E(G)= X \times X \) and a function \(h=0 \), we obtain immediately the Mizoguchi-Takahashi theorem [7].
- (iv)
In Theorem 3.3, if we take a directed graph G with \(E(G)= X\times X\), a function g that is the identity mapping on X, and a function \(h=L \), for some \(L\geq0\), we obtain the Berinde and Berinde theorem ([4], Theorem 4).
Example 3.6
The following result is also immediately obtained by Theorem 3.3, by setting \(h(x)=L \) for all \(x\in X \) and some \(L\geq0\).
Corollary 3.7
- (1)
there is \(x_{0} \in X \) such that \((x_{0},y) \in E(G) \) for some \(y \in T(x_{0}) \),
- (2)
\(T(x) \) is g-invariant (i.e., \(g(T(x)) \subseteq T(x) \) for each \(x \in X \)),
- (3)
for any sequence \(\{x_{n}\}\) in X, if \(x_{n} \to x \) and \((x_{n},x_{n+1}) \in E(G) \) for all \(n \in\mathbb{N} \), then there is a subsequence \(\{ x_{n_{k}} \} \) of \(\{ x_{n} \} \) with \((x_{n_{k}},x) \in E(G) \) for some \(k \in\mathbb{N} \).
- (4)there exists \(L\geq0 \) such that\(x \neq y \) with \((x,y) \in E(G) \),$$H\bigl(T(x),T(y)\bigr) \leq\alpha\bigl(d(x,y)\bigr)d(x,y) + Ld\bigl(g(y),T(x) \bigr) , \quad \textit{for all } x,y \in X, $$
- (5)
if \(u \in T(x) \) and \(v \in T(y) \) are such that \(d(u,v) \leq d(x,y) \), then \((u,v) \in E(G)\).
If we set g in Theorem 3.3 to be the identity map on X, then we obtain the following result.
Corollary 3.8
- (1)
there is \(x_{0} \in X \) such that \((x_{0},y) \in E(G) \) for some \(y \in T(x_{0}) \),
- (2)
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 all \(n \in\mathbb {N} \), then there is a subsequence \(\{ x_{n_{k}} \}\) of \(\{ x_{n} \} \) with \((x_{n_{k}},x) \in E(G) \) for some \(k \in\mathbb{N} \).
- (3)there exists a function \(h : X \to[0,\infty) \) such that\(x \neq y \) with \((x,y) \in E(G) \),$$H\bigl(T(x),T(y)\bigr) \leq\alpha\bigl(d(x,y)\bigr)d(x,y) + h(y)d\bigl(y,T(x) \bigr) ,\quad \textit{for all } x,y \in X, $$
- (4)
if \(u \in T(x) \) and \(v \in T(y) \) are such that \(d(u,v) \leq d(x,y) \), then \((u,v) \in E(G)\).
Declarations
Acknowledgements
The authors would like to thank the referees for valuable comments and suggestions for improving this work and the Thailand Research Fund under the project RTA 5780007 and Chiang Mai University for the financial support. The first author was supported by the Science Achievement Scholarship of Thailand.
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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