Open Access

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

Fixed Point Theory and Applications20152015:187

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

Received: 6 July 2015

Accepted: 7 October 2015

Published: 19 October 2015

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.

Keywords

fixed point theoremsnonexpansive mappingsBrowder’s convergence theoremedge-preservingdirected graph

MSC

47H0447H10

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 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.

2 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.

3 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 \(F(T)\).

4 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 \(z_{0}=Px_{0}\). □

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

(1)
Department of Mathematics, Faculty of Science, Chiang Mai University

References

  1. Banach, S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3, 133-181 (1922) MATHGoogle Scholar
  2. Nadler, SB Jr.: Multi-valued contraction mappings. Pac. J. Math. 30, 475-488 (1969) MATHMathSciNetView ArticleGoogle Scholar
  3. Kirk, WA: A fixed point theorem for mappings which do not increase distances. Am. Math. Mon. 72, 1004-1006 (1965) MATHMathSciNetView ArticleGoogle Scholar
  4. Zamfirescu, T: Fixed point theorems in metric spaces. Arch. Math. 23(1), 292-298 (1972) MATHMathSciNetView ArticleGoogle Scholar
  5. Jachymski, J: The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 136(4), 1359-1373 (2008) MATHMathSciNetView ArticleGoogle Scholar
  6. Bojor, F: Fixed point theorems for Reich type contractions on metric spaces with a graph. Nonlinear Anal. 75, 3895-3901 (2012) MATHMathSciNetView ArticleGoogle Scholar
  7. Dinevari, T, Frigon, M: Fixed point results for multivalued contractions on a metric space with a graph. J. Math. Anal. Appl. 405, 507-517 (2013) MATHMathSciNetView ArticleGoogle Scholar
  8. Sultana, A, Vetrivel, V: Fixed points of Mizoguchi-Takahashi contraction on a metric space with a graph and applications. J. Math. Anal. Appl. 417, 336-344 (2014) MATHMathSciNetView ArticleGoogle Scholar
  9. Browder, FE: Convergence of approximants to fixed points of non-expansive maps in Banach spaces. Arch. Ration. Mech. Anal. 24, 82-90 (1967) MATHMathSciNetView ArticleGoogle Scholar
  10. Alfuraidan, MR: Fixed points of monotone nonexpansive mappings with a graph. Fixed Point Theory Appl. 2015, 49 (2015). doi:10.1186/s13663-015-0299-0 View ArticleGoogle Scholar
  11. Agarwal, RP, O’Regan, D, Sahu, DR: Fixed Point Theory for Lipschitzian Type-Mappings with Applications. Springer, New York (2009) MATHGoogle Scholar
  12. Takahashi, W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000) MATHGoogle Scholar
  13. Halpern, B: Fixed points of nonexpansive maps. Bull. Am. Math. Soc. 73, 957-961 (1967) MATHView ArticleGoogle Scholar
  14. Wittmann, R: Approximation of fixed points of nonexpansive mappings. Arch. Math. 58, 486-491 (1992) MATHMathSciNetView ArticleGoogle Scholar
  15. Lorentz, GG: A contribution to the theory of divergent series. Acta Math. 80, 167-190 (1948) MATHMathSciNetView ArticleGoogle Scholar
  16. Xu, HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240-256 (2002) MATHView ArticleGoogle Scholar
  17. Yao, Z, Zhu, LJ, Liou, YC: Strong convergence of a Halpern-type iteration algorithm for fixed point problems in Banach spaces. J. Nonlinear Sci. Appl. 8(5), 489-495 (2015) MathSciNetGoogle Scholar
  18. Li, P, Kang, SM, Zhu, LJ: Visco-resolvent algorithms for monotone operators and nonexpansive mappings. J. Nonlinear Sci. Appl. 7(5), 325-344 (2014) MATHMathSciNetGoogle Scholar
  19. Pang, C, Zhang, R, Zhang, Q, Wang, J: Dominating sets in directed graphs. Inf. Sci. 180, 3647-3652 (2010) MATHMathSciNetView ArticleGoogle Scholar
  20. Bang-Jensen, J, Gutin, G: Digraphs Theory, Algorithms and Applications. Springer Monographs in Mathematics. Springer, London (2007) Google Scholar

Copyright

© Tiammee et al. 2015