Common fixed points of G-nonexpansive mappings on Banach spaces with a graph
- Orawan Tripak^{1}Email author
https://doi.org/10.1186/s13663-016-0578-4
© Tripak 2016
Received: 12 February 2016
Accepted: 12 August 2016
Published: 1 September 2016
Abstract
In this paper, we prove the weak and strong convergence of a sequence \(\{x_{n}\}\) generated by the Ishikawa iteration to some common fixed points of two G-nonexpansive mappings defined on a Banach space endowed with a graph.
Keywords
MSC
1 Introduction
In 1922, Banach proved a remarkable and powerful result called the Banach contraction principle. Because of its fruitful applications, the principle has been generalized in many directions. The recent version of the theorem was given in Banach spaces endowed with a graph. In 2008, Jachymski [1] gave a generalization of the Banach contraction principle to mappings on a metric space endowed with a graph. In 2012, Aleomraninejad et al. [2] presented some iterative scheme results for G-contractive and G-nonexpansive mappings on graphs. In 2015, Alfuraidan and Khamsi [3] defined the concept of G-monotone nonexpansive multivalued mappings defined on a metric space with a graph. In the same year, Alfuraidan [4] gave a new definition of the G-contraction and obtained sufficient conditions for the existence of fixed points for multivalued mappings on a metric space with a graph, and also in [5], he proved the existence of a fixed point of monotone nonexpansive mapping defined in a Banach space endowed with a graph. Recently, Tiammee et al. [6] proved Browder’s convergence theorem for G-nonexpansive mapping in a Banach space with a directed graph. They also proved the strong convergence of the Halpern iteration for a G-nonexpansive mapping.
Inspired by all aforementioned references, the author proves strong and weak convergence theorems for G-nonexpansive mappings using the Ishikawa iteration generated from arbitrary \(x_{0} \) in a closed convex subset C of a uniformly convex Banach space X endowed with a graph.
2 Preliminaries
In this section, we recall some standard graph notations and terminology and also some needed results.
Let \((X, d)\) be a metric space, and \(\triangle= \{(x, x) | x \in X\}\). Consider a directed graph G for which the set \(V(G)\) of its vertices coincides with X and the set \(E(G)\) of its edges contains all loops. Assume that G has no parallel edges. Then \(G = (V(G), E(G))\), and by assigning to each edge the distance between its vertices, G may be treated as a weighted graph.
Definition 2.1
Definition 2.2
Definition 2.3
A graph G is said to be connected if there is a path between any two vertices of the graph G.
Definition 2.4
A directed graph \(G=(V(G), E(G))\) is said to be transitive if, for any \(x,y,z \in V(G)\) such that \((x,y)\) and \((y,z)\) are in \(E(G)\), we have \((x,z) \in E(G)\).
The definition of a G-nonexpansive mapping is given as follows.
Definition 2.5
- (i)
T is edge-preserving.
- (ii)
\(\|Tx -Ty\| \leq\|x-y\|\) whenever \((x,y) \in E(G)\) for any \(x,y \in C\).
Definition 2.6
([7])
Definition 2.7
([7])
Let C be a subset of a metric space \((X, d)\). A mapping T is semicompact if for a sequence \(\{x_{n}\}\) in C with \(\lim_{n \rightarrow\infty} d(x_{n}, Tx_{n}) = 0\), there exists a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{i}} \rightarrow p \in C\).
Definition 2.8
Definition 2.9
Let X be a Banach space. A mapping T with domain D and range R in X is demiclosed at 0 if, for any sequence \(\{x_{n}\}\) in D such that \(\{x_{n}\}\) converges weakly to \(x \in D\) and \(\{Tx_{n}\} \) converges strongly to 0, we have \(Tx = 0\).
Lemma 2.10
([8])
Let X be a uniformly convex Banach space, and \(\{\alpha_{n}\}\) a sequence in \([\delta, 1 - \delta]\) for some \(\delta\in(0,1)\). Suppose that sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) in X are such that \(\limsup_{n \to\infty} \|x_{n}\| \leq c\), \(\limsup_{n \to\infty} \| y_{n}\| \leq c\) and \(\limsup_{n \to\infty} \|\alpha x_{n} + (1-\alpha _{n})y_{n}\| = c\) for some \(c \geq0\). Then \(\lim_{n \to\infty} \| x_{n} -y_{n}\| = 0\).
Lemma 2.11
([9])
Lemma 2.12
([10])
Let X be a Banach space that satisfies Opial’s property, and let \(\{ x_{n}\}\) be a sequence in X. Let x, y in X be such that \(\lim_{n \to\infty} \|x_{n} - x\|\) and \(\lim_{n \to\infty} \|x_{n} - y\|\) exist. If \(\{x_{n_{j}}\}\) and \(\{x_{n_{k}}\}\) are subsequences of \(\{ x_{n}\}\) that converge weakly to x and y, respectively, then \(x=y\).
3 Main results
We first begin by proving the following useful results.
Proposition 3.1
Let \(z_{0} \in F\) be such that \((x_{0}, z_{0})\), \((y_{0}, z_{0})\), \((z_{0}, x_{0})\), and \((z_{0}, y_{0})\) are in \(E(G)\). Then \((x_{n}, z_{0})\), \((y_{n}, z_{0})\), \((z_{0}, x_{n})\), \((z_{0}, y_{n})\), and \((x_{n}, y_{n})\) are in \(E(G)\).
Proof
Lemma 3.2
Let \(z_{0} \in F\). Suppose that \((x_{0}, z_{0}), (y_{0}, z_{0}), (z_{0}, x_{0}), (z_{0}, y_{0})\in E(G)\) for arbitrary \(x_{0}\) in C. Then \(\lim_{n \to\infty} \|x_{n} -z_{0}\|\) exists.
Proof
Lemma 3.3
Proof
Lemma 3.4
Suppose that X satisfies the Opial’s property and that \((x_{0},z_{0})\), \((y_{0},z_{0})\) are in \(E(G)\) for \(z_{0} \in F\) and arbitrary \(x_{0} \in C\). Then \(I -T_{i}\) (\(i=1,2\)) are demiclosed.
Proof
Theorem 3.5
Suppose X is uniformly convex, \(\{\alpha_{n}\} , \{\beta_{n}\} \subset[\delta, 1-\delta]\) for some \(\delta\in(0,\frac{1}{2})\), \(T_{i}\) (\(i =1,2\)) satisfy Condition B, F is dominated by \(x_{0}\), F dominates \(x_{0}\), and \((x_{0}, z), (y_{0}, z), (z, x_{0}), (z, y_{0})\in E(G)\) for each \(z \in F\) and arbitrary \(x_{0}\in C\). Then \(\{ x_{n}\}\) converges strongly to a common fixed point of \(T_{i}\).
Proof
- (i)
\(\{x_{n}\}\) is bounded;
- (ii)
\(\lim_{n \to\infty} \|x_{n} - z\|\) exists;
- (iii)
\(\|x_{n+1} -z\| \leq\|x_{n} -z\|\) for all \(n \geq1\).
Theorem 3.6
Suppose that X is uniformly convex, \(\{\alpha_{n}\}, \{\beta_{n}\} \subset[\delta, 1-\delta]\) for some \(\delta\in(0,\frac{1}{2})\), one of \(T_{i}\) (\(i =1,2\)) is semicompact, F is dominated by \(x_{0}\), F dominates \(x_{0}\), and \((x_{0}, z_{0}), (y_{0}, z_{0}), (z_{0}, x_{0}), (z_{0}, y_{0})\in E(G)\) for \(z_{0} \in F\) and arbitrary \(x_{0} \in C\). Then \(\{x_{n}\}\) converges strongly to a common fixed point of \(T_{i}\).
Proof
Theorem 3.7
Suppose that X is uniformly convex, \(\{\alpha_{n}\}, \{\beta_{n}\} \subset[\delta, 1-\delta]\) for some \(\delta\in(0,\frac{1}{2})\). If X satisfies Opial’s property, \(I-T_{i}\) is demiclosed at zero for each i, F is dominated by \(x_{0}\), F dominates \(x_{0}\), and \((x_{0}, z_{0}), (y_{0}, z_{0}), (z_{0}, x_{0}), (z_{0}, y_{0})\in E(G)\) for \(z_{0} \in F\) and arbitrary \(x_{0} \in C\), then \(\{x_{n}\}\) converges weakly to a common fixed point of \(T_{i}\).
Proof
Declarations
Acknowledgements
The author is grateful to Professor Suthep Suantai for valuable suggestion and comments. The author would also like to thank the anonymous reviewers for their helpful comments.
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Authors’ Affiliations
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