# On Browder’s convergence theorem and Halpern iteration process for G-nonexpansive mappings in Hilbert spaces endowed with graphs

## Abstract

In this paper, we prove Browder’s convergence theorem for G-nonexpansive 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 G-nonexpansive mappings in a Hilbert space endowed with a directed graph. The main results obtained in this paper extend and generalize many well-known 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 contractive-type 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 [24]).

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,

$$E\bigl(G^{-1}\bigr) = \bigl\{ (x,y) : (y,x) \in E(G) \bigr\} .$$

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 $$x_{0}=x$$, $$x_{n}=y$$, $$(x_{i-1},x_{i}) \in E(G)$$ 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 G-contraction 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 G-contraction if f preserves edges of G, i.e.,

$$x,y \in X, \quad (x,y) \in E(G)\quad \Rightarrow\quad \bigl(f(x),f(y)\bigr) \in E(G)$$
(1.1)

and there exists $$\alpha\in(0,1)$$ such that for any $$x,y \in X$$,

$$(x,y) \in E(G)\quad \Rightarrow \quad d\bigl(f(x),f(y)\bigr) \leq\alpha d(x,y).$$

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:

\begin{aligned}& \textit{for any } (x_{n})_{n \in\mathbb{N}} \textit{ if } x_{n} \rightarrow x \textit{ and } (x_{n},x_{n+1}) \in E(G) \textit{ for } n \in \mathbb{N} \\& \textit{and there is a subsequence } (x_{k_{n}})_{n \in\mathbb{N}} \textit{ with } (x_{k_{n}},x) \in E(G) \textit{ for } n \in\mathbb{N} . \end{aligned}

Let f be a G-contraction, 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 [68] 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 G-nonexpansive if the following conditions hold:

1. (1)

T is edge-preserving, i.e., for any $$x,y \in C$$ such that $$(x,y) \in E(G) \Rightarrow(Tx,Ty) \in E(G)$$;

2. (2)

$$\| Tx - Ty \| \leq\| x-y \|$$, 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

$$E(G) = \bigl\{ \bigl((x_{n}),(y_{n})\bigr) \mid\text{for all } n \in\mathbb{N}, x_{n},y_{n} \in\mathbb{Z} \text{ and } y_{n}=x_{n}+1, n \geq2 \bigr\} .$$

Define a mapping $$T:X \rightarrow X$$ by

$$T(x_{1},x_{2},\ldots,x_{i},\ldots)= \textstyle\begin{cases} (0,x_{2},x_{3},x_{4},\ldots) &\mbox{if } x_{n} \in\mathbb{Z} \text{ for all } n \in\mathbb{Z} , \\ (kx_{1},kx_{2},kx_{3},\ldots) & \mbox{if } x_{n} \notin\mathbb{Z} \text{ for some } n \in\mathbb{Z}. \end{cases}$$

Note that T is G-nonexpansive, 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

$$E(G) = \biggl\{ \bigl( \{x\}_{k},\{y_{k} \}\bigr) : |x_{k}| + |y_{k}| \leq1 \text{ and } \bigl\Vert \{ x_{k}\} - \{y_{k}\} \bigr\Vert \leq\frac{3}{8} \biggr\} .$$

It is easy to show that $$E(G)$$ is convex. Now let $$T:C \rightarrow C$$ be defined by

$$T\bigl(\{x_{k}\}\bigr) = \bigl\{ x^{2}_{k} \bigr\} , \quad \{x_{k}\} \in C.$$

We can easily show that T is G-nonexpansive. However, it is not nonexpansive because $$\| Tx - Ty \| > \| x-y \|$$ 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 [918]). 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 G-nonexpansive mapping $$T:C \rightarrow C$$ in a Banach space X which satisfies the τ-Opial condition and C is a bounded convex τ-compact subset of X.

In this paper, we prove Browder’s convergence theorem for a G-nonexpansive 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 well-known 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

$$\| x - P_{C}x \| \leq\| x - y \| \quad \text{for all } y \in C.$$

$$P_{C}$$ 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. (1)

$$\| x - y \| = d(x,C)$$;

2. (2)

$$(x-y,y-z) \geq0$$ 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

$$\liminf_{n \rightarrow\infty} \| x_{n} - x \| < \liminf _{n \rightarrow\infty} \| x_{n} - y \|.$$

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 G-monotone if $$\langle Tx-Ty , x-y \rangle\geq0$$ 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 G-nonexpansive mapping in a Hilbert space endowed with a directed graph. First, we begin with the property of G-nonexpansive 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 G-nonexpansive 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 G-nonexpansive and $$(x_{m_{k}},x) \in E(G)$$, we obtain

$$\| Tx_{m_{k}} - Tx \| \leq\| x_{m_{k}} - x \| \rightarrow0 \quad \text{as } k \rightarrow\infty.$$

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 G-nonexpansive 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 G-nonexpansive 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 G-nonexpansive, we obtain

\begin{aligned} \| x - Tx \| &\leq\| x - x_{n_{k}} \| + \| x_{n_{k}} - Tx \| \\ &= \| x - x_{n_{k}} \| + \| Tx_{n_{k}} - Tx \| \\ &= \| x - x_{n_{k}} \| + \| x_{n_{k}} - x \| \rightarrow0. \end{aligned}

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

$$(x,z) = \bigl(\lambda x + (1- \lambda) x, \lambda x + (1-\lambda) y\bigr) \in E(G).$$

Similarly, we also have $$(y,z) \in E(G)$$. Since T is G-nonexpansive, we obtain

$$\| x - Tz \| = \| Tx - Tz \| \leq\| x-z \| = (1- \lambda) \| x-y \|$$
(3.1)

and

$$\| y - Tz \| = \| Ty - Tz \| \leq\| y-z \| = \lambda\| x-y \|.$$
(3.2)

Hence

\begin{aligned} \| x - y \| &= \bigl\Vert (x-Tz) + (Tz-y) \bigr\Vert \\ & \leq\| x - Tz \| + \| Tz - y \| \\ &\leq\| x - z \| + \| y - z \| = \| x-y \|. \end{aligned}

This implies that $$\| x-y \| = \| x - Tz \| + \| Tz - y \| = \| x-z \| + \| y-z \|$$ and

$$\bigl\Vert (x-Tz) + (Tz-y) \bigr\Vert = \| x - Tz \| + \| Tz - y \|.$$

By (3.1) and (3.2), we can conclude that

$$\| x -Tz \| = \| x-z \| \quad \text{and}\quad \| y -Tz \| = \|y - z \|.$$

By Proposition 2.1, there exists $$t \geq0$$ such that $$x-Tz = t(Tz - y)$$, so

$$Tz = \beta x +(1 - \beta) y,\quad \text{where } \beta=\frac{1}{1+t}.$$

Hence $$x - Tz = (1 - \beta) (x-y) = \frac{1 - \beta}{1 - \lambda} (x-z)$$, 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 G-nonexpansive, then $$I - T$$ is G-monotone, where I is the identity mapping on C.

### Proof

Let $$x,y \in C$$ be such that $$(x,y) \in E(G)$$. By the Cauchy-Schwarz inequality and G-nonexpansiveness of T, we have

\begin{aligned} 0 &\leq\| Tx - Ty \| \| x-y \| - \langle Tx-Ty, x-y \rangle \\ &\leq\| x-y \|^{2} - \langle Tx-Ty, x-y \rangle \\ &= \langle x-y, x-y \rangle- \langle Tx-Ty, x-y \rangle \\ &= \bigl\langle (x-y) - (Tx-Ty) , x-y \bigr\rangle \\ &= \bigl\langle (I-T)x - (I-T)y , x-y \bigr\rangle . \end{aligned}

Hence $$I-T$$ is G-monotone. □

Next, we prove a Browder’s fixed point theorem for a G-nonexpansive 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 G-nonexpansive. Assume that there exists $$x_{0} \in C$$ such that $$(x_{0},Tx_{0}) \in E(G)$$. Define $$T_{n}:C \rightarrow C$$ by

$$T_{n}x= ( 1-\alpha_{n} ) Tx+\alpha_{n} x_{0}$$

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:

1. (i)

$$T_{n}$$ has a fixed point $$u_{n} \in C$$;

2. (ii)

$$F(T) \neq\emptyset$$;

3. (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 G-contraction 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 G-nonexpansive, we obtain

$$\|T_{n}x - T_{n}y \| = ( 1-\alpha_{n} ) \| Tx -Ty \| \leq ( 1-\alpha_{n} ) \| x -y \|.$$

Since T is edge-preserving, $$(Tx,Ty) \in E(G)$$. By convexity of $$E(G)$$, we have

$$(T_{n}x,T_{n}y)= \bigl( ( 1-\alpha_{n} ) Tx + \alpha_{n}x_{0} , ( 1-\alpha_{n} ) Ty + \alpha_{n}x_{0} \bigr) \in E(G).$$

Therefore $$T_{n}$$ is G-contraction. 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

$$( x_{0}, T_{n}x_{0}) = \bigl( ( 1- \alpha_{n} ) x_{0} + \alpha _{n}x_{0} , ( 1-\alpha_{n} ) Tx_{0} +\alpha_{n}x_{0} \bigr) \in E(G) .$$

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

\begin{aligned} \liminf_{i\rightarrow\infty}\|u_{n_{i}}-v\|&< \liminf _{i\rightarrow \infty}\|u_{n_{i}}-Tv\| \\ &= \liminf_{i\rightarrow\infty}\|u_{n_{i}}-Tu_{n_{i}}+Tu_{n_{i}}-Tv \| \\ &= \liminf_{i\rightarrow\infty}\|Tu_{n_{i}}-Tv\| \\ &\leq\liminf_{i\rightarrow\infty}\|u_{n_{i}}-v\|, \end{aligned}

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

$$\alpha_{n_{i}} v_{i} + (1-\alpha_{n_{i}} ) (v_{i}-Tv_{i}) =\alpha_{n_{i}}x_{0}.$$

Since $$w_{0}$$ is a fixed point of T, we have

$$\alpha_{n_{i}} w_{0} + (1-\alpha_{n_{i}} ) (w_{0}-Tw_{0}) =\alpha_{n_{i}}w_{0}.$$

If we subtract these two equations and take the inner product of the difference with $$v_{i}-w_{0}$$, we obtain

$$\alpha_{n_{i}} \langle v_{i}-w_{0},v_{i}-w_{0} \rangle+ (1-\alpha _{n_{i}} ) \langle Uv_{i}-Uw_{0},v_{i}-w_{0} \rangle=\alpha_{n_{i}} \langle x_{0}-w_{0},v_{i}-w_{0} \rangle,$$
(3.3)

where $$U=I-T$$ 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 G-monotone, so $$\langle Uv_{i}-Uw_{0}, v_{1}-w_{0}\rangle\geq0$$ for all $$i \in\mathbb{N}$$. This together with (3.1) shows

$$\alpha_{n_{i}} \|v_{i}-w_{0}\|^{2} \leq \alpha_{n_{i}}\langle x_{0}-w_{0},v_{i}-w_{0} \rangle.$$

Hence

\begin{aligned} \|v_{i}-w_{0}\|^{2} &\leq\langle x_{0}-w_{0},v_{i}-w_{0} \rangle \\ &= \langle x_{0}-w_{0},v-w_{0} \rangle+\langle x_{0}-w_{0},v_{i}-v \rangle. \end{aligned}

By Lemma 2.4, we know that $$\langle x_{0}-w_{0},v-w_{0} \rangle\leq0$$, so we get

$$\|v_{i}-w_{0}\|^{2} \leq\langle x_{0}-w_{0},v_{i}-v \rangle\rightarrow0\quad \text{as } i \rightarrow\infty,$$

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 $$\| x-y \| = |x-y|$$ and let $$G=(V(G),E(G))$$ be such that $$V(G)=C$$, $$E(G)= \{ (x,y) : x,y \in[0,\frac{3}{8}] \text{ such that } |x-y| \leq \frac{1}{8} \}$$. Define $$T:C \rightarrow C$$ by

$$Tx= \textstyle\begin{cases} \frac{8}{6}x^{2} &\mbox{if } x \in[0,\frac{1}{2}), \\ \frac{25}{64} & \mbox{if } x=\frac{1}{2}. \end{cases}$$

### 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 $$|x-y| \leq\frac{1}{8}$$. So, we have $$|Tx-Ty| = \frac{8}{6}|x^{2}-y^{2}| \leq\frac {8}{6}|x+y||x-y| \leq|x-y| \leq\frac{1}{8}$$, which implies that $$(Tx,Ty) \in E(G)$$ and $$\| Tx - Ty \| \leq\| x-y \|$$. Thus T is G-nonexpansive. Next, for each $$n \in\mathbb{N}$$, define $$T_{n}:C \rightarrow C$$ by

$$T_{n}x= \frac{1}{8(n+5)} + \biggl(1-\frac{1}{n+5} \biggr)Tx.$$

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

$$\biggl\Vert T\biggl(\frac{1}{2}\biggr) -T\biggl(\frac{3}{8}\biggr) \biggr\Vert = \biggl\Vert \frac{25}{64} - \frac {3}{16} \biggr\Vert = \frac{13}{64} > \frac{1}{8}=\biggl\Vert \frac{1}{2} - \frac {3}{8} \biggr\Vert .$$

### 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 G-nonexpansive 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

$$T_{n} = \biggl( 1-\frac{1}{n} \biggr)Tx + \frac{1}{n}x_{0}$$

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 G-nonexpansive 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

$$\textstyle\begin{cases} x_{0} \in C , \\ x_{n+1} = \alpha_{n} u + (1- \alpha_{n}) Tx_{n}, \quad n\geq0, \end{cases}$$
(4.1)

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

$$\alpha_{n} \in[0,1],\qquad \sum ^{\infty}_{n=0}\alpha_{n} = \infty, \qquad \lim _{n \rightarrow\infty} \alpha_{n} = 0 \quad \text{and}\quad \sum ^{\infty}_{n=0} | \alpha_{n+1} - \alpha_{n} | < \infty.$$
(4.2)

The following is also useful for proving our main result.

### Lemma 4.2

([16])

Let $$(s_{n})$$ be a sequence of non-negative real numbers satisfying

$$s_{n+1} \leq(1 - \alpha_{n}) s_{n} + \alpha_{n} \beta_{n} + \gamma_{n},\quad n\geq0,$$

where $$(\alpha_{n})$$, $$(\beta_{n})$$, and $$(\gamma_{n})$$ satisfy the conditions:

1. 1.

$$(\alpha_{n}) \subset[0,1]$$, $$\sum^{\infty}_{n=0} \alpha_{n}= \infty$$, or equivalently, $$\prod^{\infty}_{n=1} (1-\alpha_{n}) = 0$$;

2. 2.

$$\limsup_{n \rightarrow\infty} \beta_{n} \leq0$$;

3. 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 G-nonexpansive 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 edge-preserving. 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 edge-preserving. 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 edge-preserving. 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 G-nonexpansive 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

\begin{aligned} \| x_{n+1} - z_{0} \| &\leq(1 - \alpha_{n}) \| Tx_{n} - z_{0} \| + \alpha_{n} \| x_{0} - z_{0} \| \\ &= (1 - \alpha_{n}) \| Tx_{n} - Tz_{0} \| + \alpha_{n} \| x_{0} - z_{0} \| \\ &\leq(1 - \alpha_{n}) \| x_{n} - z_{0} \| + \alpha_{n} \| x_{0} - z_{0} \| \\ &\leq\max\bigl\{ \Vert x_{n} - z_{0} \Vert , \| x_{0} - z_{0} \| \bigr\} \end{aligned}

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

\begin{aligned} \| x_{n+1} - x_{n} \| &\leq| \alpha_{n} - \alpha_{n-1}\| \bigl(\Vert x_{0} \Vert + \| Tx_{n-1} \|\bigr) + (1-\alpha_{n}) \| x_{n} - x_{n-1} \| \\ &\leq\| \alpha_{n} - \alpha_{n-1}\|K + (1- \alpha_{n}) \| x_{n} - x_{n-1} \|, \end{aligned}
(4.3)

where $$K=\sup\{\| x_{0} \| + \| Tx_{n} \| : n \in\mathbb{N}\}$$. By using (4.3), for $$m,n \in\mathbb{N}$$, we have

\begin{aligned}& \| x_{n+m+1} - x_{n+m} \| \\ & \quad \leq \Biggl( \sum^{n+m-1}_{k=m} | \alpha_{k+1} - \alpha_{k}| \Biggr) K + \Biggl( \prod ^{n+m-1}_{k=m} |1 - \alpha_{k+1}| \Biggr) \| x_{m+1} - x_{m} \| \\ & \quad \leq \Biggl( \sum^{n+m-1}_{k=m} | \alpha_{k+1} - \alpha_{k}| \Biggr) K + \exp \Biggl( - \sum ^{n+m-1}_{k=m} \alpha_{k+1} \Biggr) \| x_{m+1} - x_{m} \|. \end{aligned}

Since $$\{x_{n}\}$$ is bounded and $$\sum^{\infty}_{k=0}\alpha_{k} = \infty$$, we obtain

$$\limsup_{n \rightarrow\infty} \| x_{n+1} - x_{n} \| = \limsup_{n \rightarrow\infty} \| x_{n+m+1} - x_{n+m} \| \leq \Biggl( \sum^{\infty}_{k=m} |\alpha_{k+1} - \alpha_{k}| \Biggr) K$$

for all $$m \in\mathbb{N}$$. Hence, by $$\sum^{\infty}_{n=0} | \alpha _{n+1} - \alpha_{n} | < \infty$$, we get

$$\lim_{n \rightarrow\infty} \| x_{n+1} - x_{n} \| = 0.$$
(4.4)

For each $$n \in\mathbb{N}$$, we have

\begin{aligned} \| x_{n} - Tx_{n} \| &\leq\| x_{n} - x_{n+1} \| + \| x_{n+1} - Tx_{n} \| \\ &= \| x_{n} - x_{n+1} \| + \alpha_{n} \| x_{0} - Tx_{n} \|. \end{aligned}

Because $$\{Tx_{n}\}$$ is bounded with (4.4), we obtain

$$\| x_{n} - Tx_{n} \| \rightarrow0$$
(4.5)

as $$n \rightarrow\infty$$. We next show that

$$\limsup_{n \rightarrow\infty} \langle x_{n} - z_{0} , x_{0} - z_{0} \rangle \leq0.$$

Indeed, take a subsequence $$\{x_{n_{k}}\}$$ of $$\{x_{n}\}$$ such that

$$\limsup_{n \rightarrow\infty} \langle x_{n} - z_{0} , x_{0} - z_{0} \rangle = \lim_{k \rightarrow\infty} \langle x_{n_{k}} - z_{0} , x_{0} - z_{0} \rangle.$$

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 G-nonexpansiveness of T, we get

\begin{aligned} \liminf_{k \rightarrow\infty} \| x_{n_{k}} - y \| &< \liminf _{k \rightarrow\infty} \| x_{n_{k}} - Ty \| \\ &\leq\liminf_{k \rightarrow\infty} \bigl( \Vert x_{n_{k}} - Tx_{n_{k}} \Vert + \| Tx_{n_{k}} - Ty \| \bigr) \\ &= \liminf_{k \rightarrow\infty} \| Tx_{n_{k}} - Ty \| \\ &\leq\liminf_{k \rightarrow\infty} \| x_{n_{k}} - y \|, \end{aligned}

which is a contradiction. So $$y=Ty$$. Hence, by Lemma 2.4, we get

$$\lim_{k \rightarrow\infty} \langle x_{n_{k}} - z_{0} , x_{0} - z_{0} \rangle = \langle y - z_{0} , x_{0} - z_{0} \rangle\leq0 .$$
(4.6)

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

\begin{aligned} \bigl\Vert (1 - \alpha_{n}) ( Tx_{n} - z_{0}) \bigr\Vert ^{2} &= \| x_{n+1} - z_{0} \|^{2} + \alpha ^{2}_{n} \| x_{0} - z_{0} \|^{2} - 2 \alpha_{n} \langle x_{n+1} - z_{0} , x_{0} - z_{0} \rangle \\ &\geq\| x_{n+1} - z_{0} \|^{2} - 2 \alpha_{n} \langle x_{n+1} - z_{0} , x_{0} - z_{0} \rangle. \end{aligned}

This implies, by G-nonexpansiveness of T and $$(z_{0},x_{n}) \in E(G)$$, that

$$\| x_{n+1} - z_{0} \|^{2} \leq(1- \alpha_{n})\| x_{n} - z_{0} \|^{2} + 2 \alpha _{n}\langle x_{n+1} - z_{0} , x_{0} - z_{0} \rangle$$

for each $$n \in\mathbb{N}$$. By Lemma 4.2, we can conclude that

$$\lim_{n \rightarrow\infty} \| x_{n} - z_{0} \|^{2} = 0.$$

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.

## Author information

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Correspondence to Suthep Suantai.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

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Tiammee, J., Kaewkhao, A. & Suantai, S. On Browder’s convergence theorem and Halpern iteration process for G-nonexpansive mappings in Hilbert spaces endowed with graphs. Fixed Point Theory Appl 2015, 187 (2015). https://doi.org/10.1186/s13663-015-0436-9

• Accepted:

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• DOI: https://doi.org/10.1186/s13663-015-0436-9

• 47H04
• 47H10

### Keywords

• fixed point theorems
• nonexpansive mappings
• Browder’s convergence theorem
• edge-preserving
• directed graph