Reich type weak contractions on metric spaces endowed with a graph
- Aniruth Phon-on^{1},
- Areeyuth Sama-Ae^{1}Email author,
- Nifatamah Makaje^{1} and
- Pakwan Riyapan^{1}
https://doi.org/10.1186/s13663-015-0307-4
© Phon-on et al.; licensee Springer. 2015
Received: 16 December 2014
Accepted: 6 April 2015
Published: 22 April 2015
Abstract
In this paper, we define a new class of Reich type multi-valued contractions on a complete metric space satisfying the g-graph preserving condition and we study the fixed point theorem for such mappings. In addition, we present the existence and uniqueness of the fixed point for at least one of two multi-valued mappings. The results of this paper extend and generalize several well-known results. Some examples illustrate the usability of our results.
Keywords
MSC
1 Introduction
The classical contraction mapping principle of Banach states that if \((X,d)\) is a complete metric space and \(f:X \to X\) such that \(d(f(x),f(y)) \le\alpha d(x,y)\) for all \(x,y \in X\), where \(\alpha \in[0,1)\), then f has a unique fixed point. The Banach fixed point theorem plays an important role in studying the existence of solutions of nonlinear integral equations, system of linear equations, nonlinear differential equations, and proving the convergence of algorithms in computational mathematics. The Banach fixed point theorem has been extended in many directions; see [1–17]. Fixed point theory of multi-valued mappings plays a central role in control theory, optimization, partial differential equations, and economics. For a metric space \((X,d)\), we let \(\operatorname{CB}(X)\) be the set of all nonempty, closed, and bounded subsets of X. A point \(x\in X\) is a fixed point of a multi-valued mapping \(T:X \to 2^{X}\) if \(x \in Tx\). Nadler [18] has proved a multi-valued version of the Banach contraction principle which we state as the following theorem.
Theorem 1.1
Let \((X,d)\) be a complete metric space and \(T:X \to \operatorname{CB}(X)\). Assume that there exists \(k \in[0,1)\) such that \(H(Tx,Ty) \le kd(x,y)\) for all \(x,y\in X\). Then there exists \(z \in X\) such that \(z \in Tz\).
Reich [19] generalized the Banach fixed point theorem for single-valued maps and multi-valued maps as the two following theorems.
Theorem 1.2
Theorem 1.3
In 2008, Jachymski [20] introduced the concept of a contraction concerning a graph, called a G-contraction, and proved some fixed point results of the G-contraction in a complete metric space endowed with a graph and he showed that the results of many authors can be derived by his results.
Definition 1.4
Jachymski showed in [20] that assuming some properties for X, a G-contraction \(f:X \to X\) has a fixed point if and only if there exists \(x\in X\) such that \((x,f(x)) \in E(G)\). The results of Jachymski were generalized by several authors (see, for example, Bojor [3]; Chifu and Petrusel [6]; Samreen and Kamran [13]; Asl et al. [2]; Abbas and Nazir [1]). Recently, Tiammee and Suantai [21] introduced the concept of g-graph preserving for multi-valued mappings and proved their fixed point theorem in a complete metric space endowed with a graph.
Definition 1.5
[21]
Definition 1.6
[21]
By using the concept of ‘g-graph preserving’ introduced by Tiammee and Suantai [21] and the concept of a Reich type multi-valued contraction defined by Reich [19], we define a new class of Reich type multi-valued contraction on a complete metric space satisfying the g-graph preserving condition and then we shall study the fixed point theorem for such mappings. Moreover, we establish some results on common fixed points for two multi-valued mappings. The results of this research extend and generalize several well-known results from previous work.
2 Main results
The mapping H is said to be a Hausdorff metric induced by d. The next lemma will play central roles in our main results.
Lemma 2.1
We start with the new class of Reich type multi-valued \((\alpha,\beta ,\gamma)\)-contraction on a complete metric space.
Definition 2.2
- (1)
T is g-graph preserving;
- (2)there exist nonnegative numbers α, β, γ with \(\alpha+\beta+\gamma< 1\) andfor all \(x, y\in X\) such that \((g(x), g(y))\in E(G)\).$$ H(Tx, Ty) \leq\alpha d\bigl(g(x),g(y)\bigr)+ \beta D\bigl(g(x), Tx\bigr)+ \gamma D\bigl(g(y),Ty\bigr) $$
Example 2.3
Property A
For any sequence \(\{x_{n}\}_{n\in\mathbb{N}}\) in X, if \(x_{n}\rightarrow x\) and \((x_{n}, x_{n+1})\in E(G)\) for \(n\in \mathbb{N}\), then there is a subsequence \(\{x_{n_{k}}\}_{n_{k}\in\mathbb {N}}\) such that \((x_{n_{k}}, x)\in E(G)\) for \(n_{k}\in\mathbb{N}\).
Lemma 2.4
- (A)
there exists \(x_{0}\in X\) such that \((g_{1}(x_{0}), u)\in E(G)\) for some \(u\in T_{1}x_{0}\), and
- (B)
if \((g_{1}(x), g_{2}(y))\in E(G)\), then \((z,w)\in E(G)\) for all \(z\in T_{1}x\), \(w\in T_{2}y\), and if \((g_{2}(x), g_{1}(y))\in E(G)\), then \((b,r)\in E(G)\) for all \(b\in T_{2}x\), \(r\in T_{1}y\),
Proof
Since \(g_{2}\) is surjective, there exists \(x_{1}\in X\) such that \(g_{2}(x_{1})\in T_{1}x_{0}\) and \((g_{1}(x_{0}), g_{2}(x_{1}))\in E(G)\). By Lemma 2.1, there exists \(x_{2}\in X\) such that \(g_{1}(x_{2})\in T_{2}x_{1}\) and \(d(g_{2}(x_{1}), g_{1}(x_{2}))\leq H(T_{1}x_{0}, T_{2}x_{1})+ (\alpha+\beta)\).
Theorem 2.5
- (1)
there exists \(x_{0}\in X\) such that \((g_{1}(x_{0}), u)\in E(G)\) for some \(u\in T_{1}x_{0}\);
- (2)
if \((g_{1}(x), g_{2}(y))\in E(G)\), then \((z,w)\in E(G)\) for all \(z\in T_{1}x\), \(w\in T_{2}y\) and if \((g_{2}(x), g_{1}(y))\in E(G)\), then \((b,r)\in E(G)\) for all \(b\in T_{2}x\), \(r\in T_{1}y\);
- (3)
X has Property A,
Proof
The following example illustrates Theorem 2.5.
Example 2.6
If \((g_{1}(x), g_{2}(y)), (g_{2}(x), g_{1}(y))\in E(G)\setminus\{(1,1)\}\), then \(H(Tx, Sy)=0=H(Sx,Ty)\). So the above inequalities are satisfied.
By Theorem 2.5, there are \(u,v\in X\) such that \(g_{1}(u)\in Tu\) or \(g_{2}(v)\in Sv\). In this example, choose \(u=\frac{1}{4}\). Since \(g_{2}(u)=u\), we have \(u\in Su=\{u\}\).
From Theorem 2.5 one deduces the next two corollaries.
Corollary 2.7
- (1)
there exists \(x_{0}\in X\) such that \((g(x_{0}), u)\in E(G)\) for some \(u\in T_{1}x_{0}\);
- (2)
X has Property A,
Proof
Set \(g_{1}=g_{2}=g\). Then this corollary follows immediately from Theorem 2.5. □
Corollary 2.8
- (1)
T is a \((g, \alpha, \beta, \gamma)\)-G-contraction;
- (2)
the set \(X_{T}=\{x\in X \mid (g(x),y)\in E(G) \textit{ for some } y\in Tx\}\neq \emptyset\);
- (3)
X has Property A,
Proof
Set \(T_{1}=T_{2}=T\) and \(g_{1}=g_{2}=g\). Then the result follows directly from Theorem 2.5. □
- (1)
\(x\leq x\) (reflexivity);
- (2)
if \(x\leq y\) and \(y\leq x\), then \(x=y\) (antisymmetry);
- (3)
if \(x\leq y\) and \(y\leq z\), then \(x\leq z\) (transitivity),
Definition 2.9
Let \((X,\leq)\) be a partially ordered set. For each \(A,B \subset X\), \(A\prec B\) if \(a\leq b\) for any \(a\in A, b\in B\).
Definition 2.10
[21]
In the case g is the identity map, the mapping T is called an increasing mapping.
Theorem 2.11
- (1)
T is g-increasing;
- (2)
there exist \(x_{0}\in X\) and \(u\in Tx_{0}\) such that \(g(x_{0})< u\);
- (3)
for each sequence \(\{x_{k}\}\) such that \(g(x_{k})< g(x_{k+1})\) for all \(k\in\mathbb{N}\) and \(g(x_{k})\) converges to \(g(x)\) for some \(x\in X\), then \(g(x_{k})< g(x)\) for all \(k\in\mathbb{N}\);
- (4)there exist nonnegative numbers α, β, γ with \(\alpha+\beta+\gamma< 1\) such thatfor all \(x, y\in X\) such that \(g(x)< g(y)\);$$H(Tx, Ty) \leq\alpha d\bigl(g(x), g(y)\bigr)+ \beta D\bigl(g(x),Tx\bigr)+\gamma D\bigl(g(y), Ty\bigr) $$
- (5)
the metric d is complete.
Proof
We obtain the following result by considering \(g(x)= x\) for all \(x\in X\).
Corollary 2.12
- (1)
T is increasing;
- (2)
there exist \(x_{0}\in X\) and \(u\in Tx_{0}\) such that \(x_{0}< u\);
- (3)
for each sequence \(\{x_{k}\}\) such that \(x_{k}< x_{k+1}\) for all \(k\in\mathbb{N}\) and \(x_{k}\) converges to x for some \(x\in X\), then \(x_{k}< x\) for all \(k\in\mathbb{N}\);
- (4)there exist nonnegative numbers α, β, γ with \(\alpha+\beta+\gamma< 1\) such thatfor all \(x, y\in X\) such that \(x< y\);$$H(Tx, Ty) \leq\alpha d(x, y)+ \beta D(x,Tx)+\gamma D(y, Ty) $$
- (5)
the metric d is complete.
Declarations
Acknowledgements
This paper was supported by the Faculty of Science and Technology, Prince of Songkla University, Pattani Campus.
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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