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On Browder’s convergence theorem and Halpern iteration process for Gnonexpansive mappings in Hilbert spaces endowed with graphs
Fixed Point Theory and Applications volume 2015, Article number: 187 (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.
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]).
Let \(G=(V(G),E(G))\) be a directed graph where \(V(G)\) is a set of vertices of graph and \(E(G)\) be a set of its edges. Assume that G has no parallel edges. We denote by \(G^{1}\) the directed graph obtained from G by reversing the direction of edges. That is,
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., \(\bigtriangleup=\{ (x,x) : x \in X \} \subseteq E(G)\).
We say that a mapping \(f:X \rightarrow X\) is a Gcontraction if f preserves edges of G, i.e.,
and there exists \(\alpha\in(0,1)\) such that for any \(x,y \in X\),
Using this concept, he proved in [5] the following theorem.
Theorem 1.3
([5])
Let \((X,d)\) be complete, and let a triple \((X,d,G)\) have the following property:
Let f be a Gcontraction, and \(X_{f} = \{ x \in X : (x,f(x)) \in E(G) \}\). Then \(F(T) \neq\emptyset\) if and only if \(X_{f} \neq\emptyset\).
The above theorem has been improved and extended in many ways, see [6–8] for examples.
Let C be a nonempty convex subset of a Banach space, \(G=(V(G),E(G))\) be a directed graph such that \(V(G)=C\) and \(T:C \rightarrow C\). Then T is said to be Gnonexpansive if the following conditions hold:

(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
Let c be the Banach space of convergent sequences and \(k>1\). Let \(G=(X,E(G))\), where \(X=c\) and
Define a mapping \(T:X \rightarrow X\) by
Note that T is Gnonexpansive, but it is not nonexpansive.
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
Let c be a closed unit ball of the space \(l_{1}\) with the norm \(\ \{ x_{k}\} \ = \sum_{k} x_{k}\). Let \(G=(C,E(G))\) be the graph on C defined by
It is easy to show that \(E(G)\) is convex. Now let \(T:C \rightarrow C\) be defined by
We can easily show that T is Gnonexpansive. However, it is not nonexpansive because \(\ Tx  Ty \ > \ xy \\) where \(\{x\}= \{\frac {1}{2},0,0,\ldots\}\) and \(\{y\}= \{1,0,0, \ldots\}\).
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.
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.
Let C be a nonempty closed convex subset of a real Hilbert space X. For every point \(x \in X\), there exists a unique nearest point in C, denoted by \(P_{C}x\), such that
\(P_{C}\) is called the metric projection of X onto C.
The following lemma shows some useful properties of \(P_{C}\) on a Hilbert space.
Lemma 2.4
([12], Lemma 3.1.2)
Let C be a convex subset of a Hilbert space H and let \(x \in H\) and \(y \in C\). Then the following are equivalent:

(1)
\(\ x  y \ = d(x,C)\);

(2)
\((xy,yz) \geq0\) for every \(z \in C\).
Theorem 2.5
([12])
Let X be a Hilbert space. Let \(\{x_{n}\}\) be a sequence of X with \(x_{n} \rightharpoonup x\). If \(x \neq y\), then
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.
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
Let \(\{x_{n}\}\) be a sequence in X such that \(x_{n} \rightarrow x\). We will show that \(Tx_{n} \rightarrow Tx\). To show this, let \(\{Tx_{n_{k}}\}\) be a subsequence of \(\{Tx_{n}\}\). Since \(x_{n_{k}} \rightarrow x\), by Property G , there is a subsequence \((x_{m_{k}})\) such that \((x_{m_{k}},x) \in E(G)\) for each \(k \in\mathbb{N}\). Since T is Gnonexpansive and \((x_{m_{k}},x) \in E(G)\), we obtain
Hence \(Tx_{m_{k}} \rightarrow Tx\). By the double extract subsequence principle, we conclude that \(Tx_{n} \rightarrow Tx\). Therefore T is continuous. □
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
Suppose \(F(T) \neq\emptyset\). Let \(\{ x_{n}\}\) be a sequence in \(F(T)\) such that \(x_{n} \rightarrow x\). Since C has Property G , there is a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \((x_{n_{k}},x) \in E(G)\) for all \(k \in\mathbb{N}\). Since T is Gnonexpansive, we obtain
Therefore \(x=Tx\), i.e., \(x \in F(T)\). This shows that \(F(T)\) is closed.
Next, we will show that \(F(T)\) is convex. Let \(x,y \in F(T)\) and \(\lambda\in[0,1]\). Then \((x,x), (x,y) \in E(G)\). Denote \(z=\lambda x + (1\lambda) y\). Since \(E(G) \) is convex, we obtain
Similarly, we also have \((y,z) \in E(G)\). Since T is Gnonexpansive, we obtain
and
Hence
This implies that \(\ xy \ = \ x  Tz \ + \ Tz  y \ = \ xz \ + \ yz \\) and
By (3.1) and (3.2), we can conclude that
By Proposition 2.1, there exists \(t \geq0\) such that \(xTz = t(Tz  y)\), so
Hence \(x  Tz = (1  \beta) (xy) = \frac{1  \beta}{1  \lambda} (xz)\), which implies that \(x  Tz = x  z\). Therefore \(z=Tz\), i.e., \(z \in F(T)\). Thus \(F(T)\) is convex. □
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
Let \(x,y \in C\) be such that \((x,y) \in E(G)\). By the CauchySchwarz inequality and Gnonexpansiveness of T, we have
Hence \(IT\) is Gmonotone. □
Next, we prove a Browder’s fixed point theorem for a Gnonexpansive mapping.
Theorem 3.4
Let C be a bounded closed convex subset of a Hilbert space H and let \(G=(V(G),E(G))\) a directed graph such that \(V(G)=C\) and \(E(G)\) is convex. Suppose C has Property G . Let \(T:C \rightarrow C\) be a Gnonexpansive. Assume that there exists \(x_{0} \in C\) such that \((x_{0},Tx_{0}) \in E(G)\). Define \(T_{n}:C \rightarrow C\) by
for each \(x \in C\) and \(n \in\mathbb{N}\), where \(\{\alpha_{n}\}\) is a sequence in \((0,1)\) such that \(\alpha_{n} \rightarrow0\). Then the following hold:

(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
(i) Let \(x_{0}\) be such that \((x_{0},Tx_{0}) \in E(G)\). We first show that \(T_{n}\) is Gcontraction for all \(n \in\mathbb{N}\). Let \(n \in \mathbb{N}\) and \(x,y \in C\) such that \((x,y) \in E(G)\). Since T is Gnonexpansive, we obtain
Since T is edgepreserving, \((Tx,Ty) \in E(G)\). By convexity of \(E(G)\), we have
Therefore \(T_{n}\) is Gcontraction. For any sequence \(\{x_{n}\}\) in C such that \(x_{n} \rightarrow x\) and \((x_{n},x_{n+1}) \in E(G)\), by Property G of C, there is a subsequence \((x_{n_{k}})\) such that \((x_{n_{k}},x) \in E(G)\) for \(k \in\mathbb{N}\). Since \(E(G)\) is convex and \((x_{0},x_{0}) \in E(G)\), we have
Therefore all conditions of Theorem 1.3 are satisfied, so \(T_{n}\) has a fixed point, i.e., \(u_{n}=T_{n}u_{n}\).
(ii) We will show that \(F(T) \neq\emptyset\). Since \(\{u_{n}\} \) is bounded, by Theorem 2.3, there is a subsequence \(\{ u_{n_{i}}\}\) of \(\{u_{n}\}\) such that \(u_{n_{i}} \rightharpoonup v\) for some \(v \in C\). Suppose \(Tv \neq v\). By Property G , without loss of generality, we may assume that \((u_{n_{i}},v) \in E(G)\) for all \(i \in \mathbb{N}\). Since \(u_{n_{i}}Tu_{n_{i}} \rightarrow0 \) as \(i \rightarrow \infty\), by Theorem 2.5, we have
which is a contradiction. Hence \(Tv=v\).
(iii) Next, assume that \(F(T) \times F(T) \subseteq E(G)\) and \(\{Px_{0}\} \) is dominated by \(\{u_{n}\}\). We will show that \(u_{n} \rightarrow w_{0} =Px_{0}\). Let \(\{u_{n_{i}}\}\) be a subsequence of \(\{u_{n}\}\), we denote \(v_{i}=u_{n_{i}}\). For each i, \(v_{i}\) is a fixed point of \(T_{n_{i}}\). Hence we have
Since \(w_{0}\) is a fixed point of T, we have
If we subtract these two equations and take the inner product of the difference with \(v_{i}w_{0}\), we obtain
where \(U=IT\) and I is the identity map. Since \(Px_{0}\) is dominated by \(\{u_{n}\}\), we obtain \((v_{i},w_{0}) \in E(G)\) for all \(i \in\mathbb{N}\). By Proposition 3.3, U is Gmonotone, so \(\langle Uv_{i}Uw_{0}, v_{1}w_{0}\rangle\geq0\) for all \(i \in\mathbb{N}\). This together with (3.1) shows
Hence
By Lemma 2.4, we know that \(\langle x_{0}w_{0},vw_{0} \rangle\leq0\), so we get
because \(v_{i} \rightharpoonup v\). Hence \(v_{i} \rightarrow w_{0}=Px_{0}\). By the double extract subsequence principle, we can conclude that \(u_{n} \rightarrow w_{0}=Px_{0}\). □
Next, we give an example which supports Theorem 3.4.
Example 3.5
Let \(H=\mathbb{R}\) and \(C=[0,\frac{1}{2}]\) with the usual norm \(\ xy \ = xy\) and let \(G=(V(G),E(G))\) be such that \(V(G)=C\), \(E(G)= \{ (x,y) : x,y \in[0,\frac{3}{8}] \text{ such that } xy \leq \frac{1}{8} \}\). Define \(T:C \rightarrow C\) by
Proof
We see that \(F(T)= \{0\}\). Choose \(x_{0}=\frac{1}{8}\), so \((x_{0},Tx_{0}) \in E(G)\). It is easy to see that \(E(G)\) is convex. Let \((x,y) \in E(G)\). Then \(x,y \in[0,\frac{3}{8}]\) and \(xy \leq\frac{1}{8}\). So, we have \(TxTy = \frac{8}{6}x^{2}y^{2} \leq\frac {8}{6}x+yxy \leqxy \leq\frac{1}{8}\), which implies that \((Tx,Ty) \in E(G)\) and \(\ Tx  Ty \ \leq\ xy \\). Thus T is Gnonexpansive. Next, for each \(n \in\mathbb{N}\), define \(T_{n}:C \rightarrow C\) by
Then the unique fixed point of \(T_{n}\) is \(u_{n}=\frac{3 n+15\sqrt{3} \sqrt{3 n^{2}+28 n+67}}{8 (n+4)}\). By using elementary calculus, we can show that \(u_{n} \leq\frac{1}{8}\) for all \(n \in\mathbb{N}\). Thus \((u_{n},Px_{0}) = (u_{n},0) \in E(G)\), i.e., \(Px_{0}\) is dominated by \(\{u_{n}\}\) and \(u_{n} \rightarrow0=Px_{0}\) as \(n \rightarrow\infty\). □
It is noted that T is not nonexpansive because
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])
Let C be a bounded closed convex subset of a Hilbert space H and let T be a nonexpansive mapping of C into itself. Let \(x_{0}\) be an arbitrary point of C and define \(T_{n}:C \rightarrow C\) by
for each \(x \in C\) and \(n \in\mathbb{N}\). Then the following hold:

\(T_{n}\) 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 \(F(T)\).
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])
Let C be a nonempty convex subset of a linear space and \(T:C \rightarrow C\) a mapping. Let \(u \in C\) and \(\{\alpha_{n}\}\) be a sequence in \([0,1]\). Then a sequence \(\{x_{n}\}\) defined by
is called the Halpern iteration.
In 1992, Wittmann [14] proved the strong convergence of the Halpern iteration for a nonexpansive mapping in a Hilbert space and \(\{ \alpha_{n}\}\) satisfies
The following is also useful for proving our main result.
Lemma 4.2
([16])
Let \((s_{n})\) be a sequence of nonnegative real numbers satisfying
where \((\alpha_{n})\), \((\beta_{n})\), and \((\gamma_{n})\) satisfy the conditions:

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 \(\sum^{\infty}_{n=0} \gamma_{n} < \infty\).
Then \(\lim_{n \rightarrow\infty} s_{n} = 0\).
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
Let \(z_{0}=Px_{0}\). From Proposition 4.4, \((x_{n},x_{n+1}) \in E(G)\) for all \(n \in\mathbb{N}\). First we will show that \(\{x_{n}\}\) is bounded. Since \(z_{0} \in F(T)\) and \(z_{0}=Px_{0}\) is dominated by \(\{x_{n}\}\), we have \((x_{n},z_{0}) \in E(G)\), we get
for all \(n \in\mathbb{N}\). Therefore \(\{x_{n}\}\) is bounded. Moreover, \(\{Tx_{n}\}\) is bounded. By (4.1) and \((x_{n},x_{n+1}) \in E(G)\), we have
where \(K=\sup\{\ x_{0} \ + \ Tx_{n} \ : n \in\mathbb{N}\}\). By using (4.3), for \(m,n \in\mathbb{N}\), we have
Since \(\{x_{n}\}\) is bounded and \(\sum^{\infty}_{k=0}\alpha_{k} = \infty \), we obtain
for all \(m \in\mathbb{N}\). Hence, by \(\sum^{\infty}_{n=0}  \alpha _{n+1}  \alpha_{n}  < \infty\), we get
For each \(n \in\mathbb{N}\), we have
Because \(\{Tx_{n}\}\) is bounded with (4.4), we obtain
as \(n \rightarrow\infty\). We next show that
Indeed, take a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that
Because all the \(x_{n_{k}}\) lie in the weakly compact set C and C has Property G , we may assume without loss of generality that \(x_{n_{k}} \rightharpoonup y\) for some \(y \in C\) and \((x_{n_{k}},y) \in E(G)\). Suppose \(y \neq Ty\). By Theorem 2.5, (4.5), and Gnonexpansiveness of T, we get
which is a contradiction. So \(y=Ty\). Hence, by Lemma 2.4, we get
Therefore \(\limsup_{n \rightarrow\infty} \langle x_{n}  z_{0} , x_{0}  z_{0} \rangle\leq0\).
Since \((1\alpha_{n}) (Tx_{n}  z_{0}) = (x_{n+1}  z_{0})  \alpha_{n} (x_{0}  z_{0})\), we have
This implies, by Gnonexpansiveness of T and \((z_{0},x_{n}) \in E(G)\), that
for each \(n \in\mathbb{N}\). By Lemma 4.2, we can conclude that
Therefore \(\{x_{n}\}\) converges strongly to \(z_{0}=Px_{0}\). □
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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.
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Tiammee, J., Kaewkhao, A. & Suantai, S. On Browder’s convergence theorem and Halpern iteration process for Gnonexpansive mappings in Hilbert spaces endowed with graphs. Fixed Point Theory Appl 2015, 187 (2015). https://doi.org/10.1186/s1366301504369
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DOI: https://doi.org/10.1186/s1366301504369
MSC
 47H04
 47H10
Keywords
 fixed point theorems
 nonexpansive mappings
 Browder’s convergence theorem
 edgepreserving
 directed graph