On Browder’s convergence theorem and Halpern iteration process for Gnonexpansive mappings in Hilbert spaces endowed with graphs
 Jukrapong Tiammee^{1},
 Attapol Kaewkhao^{1} and
 Suthep Suantai^{1}Email author
https://doi.org/10.1186/s1366301504369
© Tiammee et al. 2015
Received: 6 July 2015
Accepted: 7 October 2015
Published: 19 October 2015
Abstract
In this paper, we prove Browder’s convergence theorem for Gnonexpansive mappings in a Hilbert space with a directed graph. Moreover, we also prove strong convergence of the Halpern iteration process to a fixed point of Gnonexpansive mappings in a Hilbert space endowed with a directed graph. The main results obtained in this paper extend and generalize many wellknown results in the literature.
Keywords
MSC
1 Introduction
Let \((X,d)\) be a metric space. A mapping \(T:X \rightarrow X\) is said to be contraction if there is \(0< k<1\) such that \(d(Tx,Ty) \leq k d(x,y)\) for all \(x,y \in X\). A mapping T is said to be nonexpansive if \(d(Tx,Ty) \leq d(x,y)\) for all \(x,y \in X\). We use the notation \(F(T)\) to stand for the set of all fixed points of T, i.e., \(x \in F(T)\) if and only if \(x=Tx\).
The study of contractivetype mappings is a famous topic in a metric fixed point theory. Banach [1] proved a classical theorem, known as the Banach contraction principle, which is a very important tool for solving existence problems in many branches of mathematics and physics.
Theorem 1.1
([1])
Let \((X,d)\) be a complete metric space and \(T:X \rightarrow X\) a contraction mapping. Then T has a unique fixed point.
There are many generalizations of the Banach contraction principle in the literature (see [2–4]).
If x and y are vertices in G, then a path in G from x to y of length \(n \in\mathbb{N} \cup\{ 0 \}\) is a sequence \(\{ x_{i} \} ^{n}_{i=0}\) of \(n+1\) vertices such that \(x_{0}=x\), \(x_{n}=y\), \((x_{i1},x_{i}) \in E(G)\) for \(i=1,2,\ldots,n\). A graph G is connected if there is a (directed) path between any two vertices of G.
In 2008, Jachymski [5] combined the concept of fixed point theory and graph theory to study fixed point theory in a metric space endowed with a directed graph. He introduced a concept of Gcontraction and generalized the Banach contraction principle in a metric space endowed with a directed graph.
Definition 1.2
([5])
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., \(\bigtriangleup=\{ (x,x) : x \in X \} \subseteq E(G)\).
Theorem 1.3
([5])
The above theorem has been improved and extended in many ways, see [6–8] for examples.
 (1)
T is edgepreserving, i.e., for any \(x,y \in C\) such that \((x,y) \in E(G) \Rightarrow(Tx,Ty) \in E(G)\);
 (2)
\(\ Tx  Ty \ \leq\ xy \\), whenever \((x,y) \in E(G)\) for any \(x,y \in C\).
Example 1.4
We note that \(E(G)\) in the above example is not convex in \(C \times C\), while \(E(G)\) in the following example is convex.
Example 1.5
The study of fixed point theorems for nonexpansive mappings and the structure of their fixed point sets on both Hilbert and Banach spaces were widely investigated by many authors (see [9–18]). In 1967, Browder [9] proved a strong convergence theorem to a fixed point of a nonexpansive mapping in a Hilbert space by using the Banach contraction principle.
Very recently, in 2015, Alfuraidan [10] proved a fixed point theorem for a Gnonexpansive mapping \(T:C \rightarrow C\) in a Banach space X which satisfies the τOpial condition and C is a bounded convex τcompact subset of X.
In this paper, we prove Browder’s convergence theorem for a Gnonexpansive mapping in a Hilbert space endowed with a directed graph and we also prove a strong convergence theorem of the Halpern iteration process for this type of mappings.
2 Preliminaries
In this section, we give some basic and useful definitions and wellknown results that will be used in the other sections.
Proposition 2.1
([11])
Let X be a Hilbert space. For any \(x,y \in X\). If \(\ x+y \ = \ x \ + \ y \\), then there exists \(t \geq0\) such that \(y=tx\) or \(x=ty\).
Definition 2.2
A sequence \(\{ x_{n} \}\) in a Hilbert space X is said to converge weakly to \(x \in X\) if \(\langle x_{n}, y \rangle\rightarrow\langle x,y \rangle\) for all \(y \in X\). In this case, we write \(x_{n} \rightharpoonup x\).
The following useful result is due to [11].
Theorem 2.3
([11])
Let X be a Banach space. Then X is reflexive if and only if every closed convex bounded subset C of X is weakly compact, i.e., every bounded sequence in C has a weakly convergent subsequence.
The following lemma shows some useful properties of \(P_{C}\) on a Hilbert space.
Lemma 2.4
([12], Lemma 3.1.2)
 (1)
\(\ x  y \ = d(x,C)\);
 (2)
\((xy,yz) \geq0\) for every \(z \in C\).
Theorem 2.5
([12])
The following property is useful for our main results.
Property G
Let C be a nonempty subset of a normed space X and let \(G=(V(G),E(G))\), where \(V(G)=C\), be a directed graph. Then C is said to have Property G if every sequence \(\{x_{n} \} \) in C converging weakly to \(x \in C\), there is a subsequence \(\{ x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \((x_{n_{k}},x) \in E(G) \) for all \(k \in\mathbb{N}\).
Definition 2.6
Let C be a nonempty closed convex subset of a Hilbert space H and \(G=(V(G),E(G))\) be a directed graph such that \(V(G)=C\). Then T is said to be Gmonotone if \(\langle TxTy , xy \rangle\geq0\) whenever \((x,y) \in E(G)\) for any \(x,y \in C\).
In order to obtain our main result, we need some basic definitions of domination in graphs [19, 20].
Let \(G=(V(G),E(G))\) be a directed graph. A set \(X \subseteq V(G)\) is called a dominating set if every \(v \in V(G) \setminus X\) there exists \(x \in X\) such that \((x,v) \in E(G)\) and we say that x dominates v or v is dominated by x. Let \(v \in V\), a set \(X \subseteq V\) is dominated by v if \((v,x) \in E(G)\) for any \(x \in X\) and we say that X dominates v if \((x,v) \in E(G)\) for all \(x \in X\). In this paper, we always assume that \(E(G)\) contains all loops.
3 Main result
In this section, we prove a fixed point theorem for Gnonexpansive mapping in a Hilbert space endowed with a directed graph. First, we begin with the property of Gnonexpansive mapping and the structure of its fixed point set.
Lemma 3.1
Let X be a normed space and \(G=(V(G),E(G))\) a directed graph with \(V(G)=X\). Suppose \(T : X \rightarrow X\) is a Gnonexpansive mapping. If X has a Property G , then T is continuous.
Proof
We now discuss the structure of the fixed point set of Gnonexpansive mappings.
Theorem 3.2
Let X be a normed space and C be a subset of X having Property G . Let \(G=(V(G),E(G))\) be a directed graph such that \(V(G)=C\) and \(E(G)\) is convex. Suppose \(T : C \rightarrow C\) is a Gnonexpansive mapping and \(F(T) \times F(T) \subseteq E(G)\). Then \(F(T)\) is closed and convex.
Proof
Proposition 3.3
Let C be a nonempty closed convex subset of a Hilbert space H and \(G=(V(G),E(G))\) a directed graph such that \(V(G)=C\). If T is Gnonexpansive, then \(I  T\) is Gmonotone, where I is the identity mapping on C.
Proof
Next, we prove a Browder’s fixed point theorem for a Gnonexpansive mapping.
Theorem 3.4
 (i)
\(T_{n}\) has a fixed point \(u_{n} \in C\);
 (ii)
\(F(T) \neq\emptyset\);
 (iii)
if \(F(T) \times F(T) \subseteq E(G)\) and \(Px_{0}\) is dominated by \(\{u_{n}\}\), then the sequence \(\{u_{n}\}\) converges strongly to \(w_{0}=Px_{0}\) where P is the metric projection onto \(F(T)\).
Proof
Next, we give an example which supports Theorem 3.4.
Example 3.5
Proof
Open question
It is noted that the set C in the above example has no Property G but we still have the Browder convergence theorem for a Gnonexpansive mapping T. Is it possible to obtain Theorem 3.4 with a property which is weaker than the Property G or without the Property G ?
As a consequence of Theorem 3.4, by putting \(E(G)=C \times C\), we obtain the Browder convergence theorem.
Corollary 3.6
([9])

\(T_{n}\) has a unique fixed point \(u_{n}\) in C;

the sequence \(\{u_{n}\}\) converges strongly to \(Px_{0} \in F(T)\), where P is the metric projection onto \(F(T)\).
4 Convergence of Halpern iteration process
In this section, we prove strong convergence of Halpern iteration process for Gnonexpansive mappings in a Hilbert space endowed with a graph.
Definition 4.1
([13])
Lemma 4.2
([16])
 1.
\((\alpha_{n}) \subset[0,1]\), \(\sum^{\infty}_{n=0} \alpha_{n}= \infty\), or equivalently, \(\prod^{\infty}_{n=1} (1\alpha_{n}) = 0\);
 2.
\(\limsup_{n \rightarrow\infty} \beta_{n} \leq0\);
 3.
\(\gamma_{n} \geq0\) for all \(n \geq0\) and \(\sum^{\infty}_{n=0} \gamma_{n} < \infty\).
Definition 4.3
Let \(G=(V(G),E(G))\) be a directed graph. A graph G is called transitive if for any \(x,y,z \in V(G)\) such that \((x,y)\) and \((y,z)\) are in \(E(G)\), then \((x,z) \in E(G)\).
The following result is needed for proving strong convergence of Halpern iteration process for Gnonexpansive mapping in Hilbert spaces endowed with a directed graph.
Proposition 4.4
Let C be a convex subset of a vector space X and \(G=(V(G),E(G))\) a directed graph such that \(V(G)=C\) and \(E(G)\) is convex. Let G be transitive and \(T:C \rightarrow C\) be edgepreserving. Let \(\{x_{n}\}\) be a sequence defined by (4.1), where \(u=x_{0}\) and \((x_{0},Tx_{0}) \in E(G)\). If \(\{x_{n}\}\) dominates \(x_{0}\), then \((x_{n},x_{n+1})\), \((x_{0},x_{n})\), and \((x_{n},Tx_{n})\) are in \(E(G)\) for any \(n \in\mathbb{N}\).
Proof
We prove by induction. Since \(E(G)\) is convex, \((x_{0},x_{0})\) and \((x_{0},Tx_{0})\) are in \(E(G)\), we have \((x_{0},x_{1}) \in E(G)\). Then \((Tx_{0},Tx_{1}) \in E(G)\), since T is edgepreserving. Because G is transitive, we have \((x_{0},Tx_{1}) \in E(G)\). By convexity of \(E(G)\) and \((x_{0},Tx_{1}), (Tx_{0},Tx_{1}) \in E(G)\), we get \((x_{1},Tx_{1}) \in E(G)\). By assumption, \((x_{1},x_{0}) \in E(G)\). So, by convexity of \(E(G)\), we get \((x_{1},x_{2}) \in E(G)\). Next, assume that \((x_{k},x_{k+1}) \), \((x_{0},Tx_{k})\), and \((x_{k},Tx_{k})\) are in \(E(G)\). Then \((Tx_{k},Tx_{k+1}) \in E(G)\), since T is edgepreserving. By transitivity of G, we have \((x_{0},Tx_{k+1}) \in E(G)\). By convexity of \(E(G)\) and \((x_{0},Tx_{k+1}), (Tx_{k},Tx_{k+1}) \in E(G)\), we get \((x_{k+1},Tx_{k+1}) \in E(G)\). Since \(\{x_{0}\}\) is dominated by \(\{x_{n}\}\), we have \((x_{k+1},x_{0}) \in E(G)\). By convexity of \(E(G)\), we get \((x_{k+1},x_{k+2}) \in E(G)\). So, by induction, we can conclude that \((x_{n},x_{n+1})\), \((x_{0},x_{n})\), and \((x_{n},Tx_{n})\) are in \(E(G)\) for any \(n \in\mathbb{N}\). □
We now ready to prove the strong convergence theorem.
Theorem 4.5
Let C be a nonempty closed convex subset of a Hilbert space H and let \(G=(V(G),E(G))\) be a directed graph such that \(V(G)=C\), \(E(G)\) is convex and G is transitive. Suppose C has Property G . Let \(T:C \rightarrow C\) be a Gnonexpansive mapping. Assume that there exists \(x_{0} \in C\) such that \((x_{0},Tx_{0}) \in E(G)\). Suppose that \(F(T) \neq \emptyset\) and \(F(T) \times F(T) \subseteq E(G)\). Let \(\{\alpha_{n}\}\) be a sequence satisfying (4.2). Let \(\{x_{n}\}\) be a sequence defined by Halpern iteration, where \(u=x_{0}\). If \(\{x_{n}\}\) is dominated by \(Px_{0}\) and \(\{x_{n}\}\) dominates \(x_{0}\), then \(\{x_{n}\}\) converges strongly to \(Px_{0}\), where P is the metric projection on \(F(T)\).
Proof
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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