About this Journal Submit a Manuscript Table of Contents
Journal of Applied Mathematics
Volume 2012 (2012), Article ID 902437, 12 pages
http://dx.doi.org/10.1155/2012/902437
Research Article

Approximation of Common Fixed Points of a Sequence of Nearly Nonexpansive Mappings and Solutions of Variational Inequality Problems

1Department of Mathematics, Banaras Hindu University, Varanasi 221005, India
2Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 10 April 2012; Accepted 14 May 2012

Academic Editor: Yonghong Yao

Copyright © 2012 D. R. Sahu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce an explicit iterative scheme for computing a common fixed point of a sequence of nearly nonexpansive mappings defined on a closed convex subset of a real Hilbert space which is also a solution of a variational inequality problem. We prove a strong convergence theorem for a sequence generated by the considered iterative scheme under suitable conditions. Our strong convergence theorem extends and improves several corresponding results in the context of nearly nonexpansive mappings.

1. Introduction

Let 𝐶 be a nonempty subset of a real Hilbert space 𝐻 with inner product , and norm , respectively. A mapping 𝑇𝐶𝐻 is called the following:(1)monotone if 𝑇𝑥𝑇𝑦,𝑥𝑦0𝑥,𝑦𝐶,(1.1)(2)𝜂-strongly monotone if there exists a positive real number 𝜂 such that 𝑇𝑥𝑇𝑦,𝑥𝑦𝜂𝑥𝑦2𝑥,𝑦𝐶,(1.2)(3)𝑘-Lipschitzian if there exists a constant 𝑘>0 such that 𝑇𝑥𝑇𝑦𝑘𝑥𝑦𝑥,𝑦𝐶,(1.3)(4)nonexpansive if 𝑇𝑥𝑇𝑦𝑥𝑦𝑥,𝑦𝐶,(1.4)(5)nearly nonexpansive [1, 2] with respect to a fixed sequence {𝑎n} in [0,) with 𝑎𝑛0 if 𝑇𝑛𝑥𝑇𝑛𝑦𝑥𝑦+𝑎𝑛𝑥,𝑦𝐶and𝑛.(1.5)

In [3], Moudafi proposed viscosity approximation methods of selecting a particular fixed point of a given nonexpansive mapping in Hilbert spaces (see [4] for further developments in both Hilbert and Banach spaces). Let 𝑓 be a contraction on 𝐻. Starting with an arbitrary initial 𝑥1𝐻, define a sequence {𝑥𝑛} recursively by 𝑥𝑛+1=𝛼𝑛𝑓𝑥𝑛+1𝛼𝑛𝑇𝑥𝑛𝑛,(1.6) where {𝛼𝑛} is a sequence in (0,1). It is proved in [4] that under appropriate conditions imposed on {𝛼𝑛}, the sequence {𝑥𝑛} generated by (1.6) strongly converges to the unique solution 𝑥𝐶 of the variational inequality (𝐼𝑓)𝑥,𝑥𝑥0𝑥𝐶,(1.7) where 𝐶=𝐹(𝑇), the set of fixed points of 𝑇.

In 2006, Marino and Xu [5] introduced the viscosity iterative method for nonexpansive mappings. Starting with an arbitrary initial 𝑥1𝐻, define a sequence {𝑥𝑛} recursively by 𝑥𝑛+1=𝛼𝑛𝑥𝛾𝑓𝑛+𝐼𝛼𝑛𝐴𝑇𝑥𝑛𝑛.(1.8) They proved that the sequence {𝑥𝑛} generated by (1.8) converges strongly to the unique solution of the variational inequality (𝐴𝛾𝑓)𝑥,𝑥𝑥0𝑥𝐶,(1.9) which is the optimality condition for the minimization problem min𝑥𝐶12𝐴𝑥,𝑥(𝑥),(1.10) where is a potential function for 𝛾𝑓 (i.e., (𝑥)=𝛾𝑓(𝑥) for all 𝑥𝐻), and 𝐴 is a strongly positive bounded linear operator on 𝐻; that is, there is a constant 𝛾>0 such that 𝐴𝑥,𝑥𝛾𝑥2𝑥𝐻.(1.11) The applications of the iterative method (1.8) have been studied by some researchers (see [6, 7]).

Also, Wang [8, 9] and Wang and Hu [10] introduced the iterative method for nonexpansive mappings.

Recently, Tian [11] proposed an implicit and an explicit schemes on combining the iterative methods of Marino and Xu [5] and Yamada [12]. He also proved the strong convergence of these two schemes to a fixed point of a nonexpansive mapping 𝑇 defined on a real Hilbert space under suitable conditions.

More recently, Ceng et al. [13] introduced an implicit and an explicit schemes using the properties of projection for finding the fixed points of a nonexpansive mapping defined on the closed convex subset of a real Hilbert space. They also proved the strong convergence of the sequences generated by the proposed schemes to a fixed point of a nonexpansive mapping which is also a solution of a variational inequality defined on the set of fixed points.

Aoyama et al. [14] proved strong convergence of an iterative scheme for a sequence of nonexpansive mappings as follows.

Theorem 1.1. Let 𝑋 be a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable and C be a nonempty closed convex subset of 𝑋. Let {𝑇𝑛} be a sequence of nonexpansive mappings from 𝐶 into itself such that 𝑛=1𝐹(𝑇𝑛). Let 𝑇 be a mapping from 𝐶 into itself defined by 𝑇𝑥=lim𝑛𝑇𝑛𝑥 for all 𝑥𝐶 and 𝐹(𝑇)=𝑛=1𝐹(𝑇𝑛). Let {𝑥𝑛} be a sequence in 𝐶 generated by the following iterative process: 𝑥1𝑥=𝑥𝐶,𝑛+1=𝛼𝑛𝑥+1𝛼𝑛𝑇𝑛𝑥𝑛𝑛,(1.12) where {𝛼𝑛} is a sequence in [0,1] satisfying the following conditions:(a)lim𝑛𝛼𝑛=0 and 𝑛=1𝛼𝑛=;(b) either 𝑛=1|𝛼𝑛+1𝛼𝑛|< or 𝛼𝑛(0,1] and lim𝑛𝛼𝑛+1/𝛼𝑛=1;(c)𝑛=1sup{𝑇𝑛𝑧𝑇𝑛+1𝑧𝑧𝐵}< for any bounded subset 𝐵 of 𝐶.
Then, the sequence {𝑥𝑛} converges strongly to 𝑄𝑥, where 𝑄 is the sunny nonexpansive retraction from 𝑋 onto 𝐹(𝑇).

Let 𝐶 be a nonempty subset of a real Hilbert space 𝐻. Let 𝒯={𝑇𝑛} be a sequence of mappings from 𝐶 into itself. We denote by 𝐹(𝒯) the set of common fixed points of the sequence 𝒯, that is, 𝐹(𝒯)=𝑛=1𝐹(𝑇𝑛). Fix a sequence {𝑎𝑛} in [0,) with 𝑎𝑛0, and let {𝑇𝑛} be a sequence of mappings from 𝐶 into 𝐻. Then, the sequence {𝑇𝑛} is called a sequence of nearly nonexpansive mappings [15] with respect to a sequence {𝑎𝑛} if 𝑇𝑛𝑥𝑇𝑛𝑦𝑥𝑦+𝑎𝑛𝑥,𝑦𝐶,𝑛.(1.13)

It is obvious that the sequence of nearly nonexpansive mappings is a wider class of sequence of nonexpansive mappings.

In this paper, inspired by Aoyama et al. [14], Ceng et al. [13], and Sahu et al. [15], we introduce an explicit iterative scheme and prove a strong convergence theorem for computing an element of 𝐹(𝒯), the set of common fixed points of a sequence 𝒯={𝑇𝑛} of nearly nonexpansive mappings which is also a solution of a variational inequality over 𝐹(𝒯). Our result generalizes and improves the results of Ceng et al. [13], Tian [11], and many other related works.

2. Preliminaries

Throughout this paper, we denote by 𝐼 the identity operator of 𝐻. Also, we denote by and the strong convergence and weak convergence, respectively. The symbol stands for the set of all natural numbers.

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Then, for any 𝑥𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃𝐶(𝑥), such that 𝑥𝑃𝐶(𝑥)𝑥𝑦𝑦𝐶.(2.1) The mapping 𝑃𝐶 is called the metric projection from 𝐻 onto 𝐶 (see [1]).

Let 𝐶 be a nonempty subset of a real Hilbert space 𝐻 and 𝑇1,𝑇2𝐶𝐻 be two mappings. We denote (𝐶), the collection of all bounded subsets of 𝐶. The deviation between 𝑇1 and 𝑇2 on 𝐵(𝐶), denoted by 𝒟𝐵(𝑇1,𝑇2), is defined by 𝒟𝐵𝑇1,𝑇2𝑇=sup1𝑥𝑇2𝑥𝑥𝐵.(2.2)

The following lemmas will be needed to prove our main result.

Lemma 2.1 (see [16]). The metric projection mapping 𝑃𝐶 is characterized by the following properties:(a)𝑃𝐶(𝑥)𝐶 for all 𝑥𝐻;(b)𝑥𝑃𝐶(𝑥),𝑃𝐶(𝑥)𝑦0 for all 𝑥𝐻 and 𝑦𝐶;(c)𝑥𝑦2𝑥𝑃𝐶(𝑥)2+𝑦𝑃𝐶(𝑥)2 for all 𝑥𝐻 and 𝑦𝐶;(d)𝑃𝐶(𝑥)𝑃𝐶(𝑦),𝑥𝑦𝑃𝐶(𝑥)𝑃𝐶(𝑦)2 for all 𝑥,𝑦𝐻.

Lemma 2.2 (see [13]). Let 𝑉𝐶𝐻 be an 𝐿-Lipschitzian mapping and 𝐹𝐶𝐻 be a 𝑘-Lipschitzian and 𝜂-strongly monotone operator. Then, for 0𝛾𝐿<𝜇𝜂, 𝑥𝑦,(𝜇𝐹𝛾𝑉)𝑥(𝜇𝐹𝛾𝑉)𝑦(𝜇𝜂𝛾𝐿)𝑥𝑦2𝑥,𝑦𝐶.(2.3) That is, 𝜇𝐹𝛾𝑉 is strongly monotone with coefficient 𝜇𝜂𝛾𝐿.

Lemma 2.3 (see [12]). Let 𝐶 be a nonempty subset of a real Hilbert space 𝐻. Suppose that 𝜆(0,1) and 𝜇>0. Let 𝐹𝐶𝐻 be a 𝑘-Lipschitzian and 𝜂-strongly monotone operator on 𝐶. Define the mapping 𝐺𝐶𝐻 by 𝐺𝑥=𝑥𝜆𝜇𝐹𝑥𝑥𝐶.(2.4) Then 𝐺 is a contraction that provided 𝜇<2𝜂/𝑘2. More precisely, for 𝜇(0,2𝜂/𝑘2), 𝐺𝑥𝐺𝑦(1𝜆𝜏)𝑥𝑦𝑥,𝑦𝐶,(2.5) where 𝜏=11𝜇(2𝜂𝜇𝑘2)(0,1].

Lemma 2.4 (see [1]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇𝐶𝐶 be a nonexpansive mapping. Then 𝐼𝑇 is demiclosed at zero; that is, if {𝑥𝑛} is a sequence in 𝐶 weakly converging to some 𝑥𝐶 and the sequence {(𝐼𝑇)𝑥𝑛} converges strongly to 0, then 𝑥𝐹(𝑇).

Lemma 2.5 (see [17]). Assume that {𝑡𝑛} is a sequence of nonnegative real numbers such that 𝑡𝑛+11𝛼𝑛𝑡𝑛+𝛼𝑛𝛽𝑛𝑛,(2.6) where {𝛼𝑛} and {𝛽𝑛} are sequences of nonnegative real numbers which satisfy the following conditions:(a){𝛼𝑛}𝑛=1(0,1) and 𝑛=1𝛼𝑛=;(b)limsup𝑛𝛽𝑛0, or(b’)𝑛=1𝛼𝑛𝛽𝑛 is convergent. Then lim𝑛𝑡𝑛=0.

Lemma 2.6 (see [18]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝜆𝑖>0.(𝑖=1,2,3,,𝑁) such that 𝑁𝑖=1𝜆𝑖=1. Let 𝑇1,𝑇2,𝑇3,,𝑇𝑁𝐶𝐶 be nonexpansive mappings such that 𝑁𝑖=1𝐹(𝑇𝑖), and let 𝑇=𝑁𝑖=1𝜆𝑖𝑇𝑖. Then 𝑇 is nonexpansive from 𝐶 into itself and 𝐹(𝑇)=𝑁𝑖=1𝐹(𝑇𝑖).

3. Main Result

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐹𝐶𝐻 be a 𝑘-Lipschitzian and 𝜂-strongly monotone operator and 𝑉𝐶𝐻 be an 𝐿-Lipschitzian mapping. Let 𝒯={𝑇𝑛} be a sequence of nearly nonexpansive mappings from 𝐶 into itself with respect to a sequence {𝑎𝑛} such that 𝐹(𝒯) and 𝑇 be a mapping from 𝐶 into itself defined by 𝑇𝑥=lim𝑛𝑇𝑛𝑥 for all 𝑥𝐶. Suppose that 𝐹(𝑇)=𝐹(𝒯),0<𝜇<2𝜂/𝑘2 and 0𝛾𝐿<𝜏, where 𝜏=11𝜇(2𝜂𝜇𝑘2). For an arbitrary 𝑥1𝐶, consider the sequence {𝑥𝑛} in 𝐶 generated by the following iterative process: 𝑥1𝑥𝐶,𝑛+1=𝑃𝐶𝛼𝑛𝛾𝑉𝑥𝑛+𝐼𝛼𝑛𝑇𝜇𝐹𝑛𝑥𝑛𝑛,(3.1) where {𝛼𝑛} is a sequence in (0,1) satisfying the conditions:(a)lim𝑛𝛼𝑛=0 and 𝑛=1𝛼𝑛=;(b)either 𝑛=1|𝛼𝑛+1𝛼𝑛|< or lim𝑛𝛼𝑛+1/𝛼𝑛=1;(c)either 𝑛=1𝒟𝐵(𝑇𝑛,𝑇𝑛+1)< or lim𝑛𝒟𝐵(𝑇𝑛,𝑇𝑛+1)/𝛼𝑛+1=0 for each 𝐵(𝐶);(d)lim𝑛𝑎n/𝛼𝑛=0. Then, the sequence {𝑥𝑛} converges strongly to ̃𝑥𝐹(𝒯), where ̃𝑥 is the unique solution of variational inequality (𝜇𝐹𝛾𝑉)̃𝑥,̃𝑥𝑦0𝑦𝐹(𝒯).(3.2)

Proof. Let 𝑇 be a mapping from 𝐶 into itself defined by 𝑇𝑥=lim𝑛𝑇𝑛𝑥 for all 𝑥𝐶. It is clear that 𝑇 is a nonexpansive mapping. So, we have 𝐹(𝑇). Now, we proceed with the following steps.
Step 1. ({𝑥𝑛} is bounded). Let 𝑧𝐹(𝒯). From (3.1), we have 𝑥𝑛+1=𝑃𝑧𝐶𝛼𝑛𝛾𝑉𝑥𝑛+𝐼𝛼𝑛𝑇𝜇𝐹𝑛𝑥𝑛𝑃𝐶𝛼(𝑧)𝑛𝛾𝑉𝑥𝑛+𝐼𝛼𝑛𝑇𝜇𝐹𝑛𝑥𝑛𝛼𝑧𝑛𝛾𝑉𝑥𝑛+𝜇𝐹𝑧𝐼𝛼𝑛𝑇𝜇𝐹𝑛𝑥𝑛𝐼𝛼𝑛𝑇𝜇𝐹𝑛𝑧𝛼𝑛𝑥𝛾𝐿𝑛𝑧+𝛼𝑛(𝛾𝑉𝜇𝐹)𝑧+1𝛼𝑛𝜏𝑥𝑛𝑧+𝑎𝑛1𝛼𝑛𝑥(𝜏𝛾𝐿)𝑛𝑧+𝛼𝑛(𝛾𝑉𝜇𝐹)𝑧+1𝛼𝑛𝜏𝑎𝑛1𝛼𝑛(𝑥𝜏𝛾𝐿)𝑛𝑧+𝛼𝑛(𝛾𝑉𝜇𝐹)𝑧+𝑎𝑛.(3.3) Note that lim𝑛𝑎𝑛/𝛼𝑛=0, so there exists a constant 𝐾>0 such that 𝛼𝑛(𝛾𝑉𝜇𝐹)𝑧+𝑎𝑛𝛼𝑛𝐾𝑛.(3.4) Thus, we have 𝑥𝑛+1𝑧1𝛼𝑛𝑥(𝜏𝛾𝐿)𝑛𝑧+𝛼𝑛𝐾𝑥max𝑛,𝐾𝑧𝜏𝛾𝐿𝑛.(3.5) Hence, {𝑥𝑛} is bounded. So {𝑇𝑛𝑥𝑛} and {𝑉𝑥𝑛} are bounded.
Step 2. (𝑥𝑛+1𝑥𝑛0 as 𝑛). From (3.1), we have 𝑥𝑛+1𝑥𝑛=𝑃𝐶𝛼𝑛𝛾𝑉𝑥𝑛+𝐼𝛼𝑛𝑇𝜇𝐹𝑛𝑥𝑛𝑃𝐶𝛼𝑛1𝛾𝑉𝑥𝑛1+𝐼𝛼𝑛1𝑇𝜇𝐹𝑛1𝑥𝑛1𝛼𝑛𝛾𝑉𝑥𝑛+𝐼𝛼𝑛𝑇𝜇𝐹𝑛𝑥𝑛𝛼𝑛1𝛾𝑉𝑥𝑛1+𝐼𝛼𝑛1𝑇𝜇𝐹𝑛1𝑥𝑛1𝛼𝑛𝛾𝑉𝑥𝑛𝑉𝑥𝑛1𝛼+𝛾𝑛𝛼𝑛1𝑉𝑥𝑛1+𝐼𝛼𝑛𝑇𝜇𝐹𝑛𝑥𝑛𝐼𝛼𝑛𝑇𝜇𝐹𝑛𝑥𝑛1+𝑇𝑛𝑥𝑛1𝑇𝑛1𝑥𝑛1+𝛼𝑛1𝜇𝐹𝑇𝑛1𝑥𝑛1𝛼𝑛𝜇𝐹𝑇𝑛𝑥𝑛1𝛼𝑛𝑥𝛾𝐿𝑛𝑥𝑛1+𝛾𝛼𝑛𝛼𝑛1𝑉𝑥𝑛1+1𝛼𝑛𝜏𝑇𝑛𝑥𝑛𝑇𝑛𝑥𝑛1+𝑇𝑛𝑥𝑛1𝑇𝑛1𝑥𝑛1𝛼+𝜇𝑛1𝐹𝑇𝑛1𝑥𝑛1𝛼𝑛𝐹𝑇𝑛𝑥𝑛11𝛼𝑛(𝑥𝜏𝛾𝐿)𝑛𝑥𝑛1+𝒟𝐵𝑇𝑛,𝑇𝑛1+1𝛼𝑛𝜏𝑎𝑛+𝛾𝛼𝑛𝛼𝑛1𝑉𝑥𝑛1𝛼+𝜇𝑛1𝐹𝑇𝑛1𝑥𝑛1𝐹𝑇𝑛𝑥𝑛1𝛼𝑛𝛼𝑛1𝐹𝑇𝑛𝑥𝑛11𝛼𝑛𝑥(𝜏𝛾𝐿)𝑛𝑥𝑛1+𝒟𝐵𝑇𝑛,𝑇𝑛11+𝜇𝛼𝑛1𝑘||𝛼+𝑀𝑛𝛼𝑛1||+𝑎𝑛,(3.6) for some constant 𝑀>0. Thus, using Lemma 2.5, we get 𝑥𝑛+1𝑥𝑛0 as 𝑛.
Step 3. We have (𝑥𝑛𝑇𝑥𝑛0 as 𝑛). Note that 𝑥𝑛𝑇𝑛𝑥𝑛𝑥𝑛𝑥𝑛+1+𝑥𝑛+1𝑇𝑛𝑥𝑛=𝑥𝑛𝑥𝑛+1+𝑃𝐶𝛼𝑛𝛾𝑉𝑥𝑛+𝐼𝛼𝑛𝑇𝜇𝐹𝑛𝑥𝑛𝑃𝐶𝑇𝑛𝑥𝑛𝑥𝑛𝑥𝑛+1+𝛼𝑛𝛾𝑉𝑥𝑛+𝐼𝛼𝑛𝑇𝜇𝐹𝑛𝑥𝑛𝑇𝑛𝑥𝑛=𝑥𝑛𝑥𝑛+1+𝛼𝑛𝛾𝑉𝑥𝑛𝜇𝐹𝑇𝑛𝑥𝑛0as𝑛.(3.7) Since 𝑥𝑛𝑇𝑥𝑛𝑥𝑛𝑇𝑛𝑥𝑛+𝑇𝑛𝑥𝑛𝑇𝑥𝑛𝑥𝑛𝑇𝑛𝑥𝑛+𝒟𝐵𝑇𝑛,,𝑇(3.8) it follows that lim𝑛𝑥𝑛𝑇𝑥𝑛=0.
Step 4. We have (limsup𝑛𝑥𝑛̃𝑥,(𝛾𝑉𝜇𝐹)̃𝑥0). Let us choose a subsequence {𝑥𝑛𝑘} of {𝑥𝑛} such that limsup𝑛𝑥𝑛̃𝑥,(𝛾𝑉𝜇𝐹)̃𝑥=lim𝑘𝑥𝑛𝑘.̃𝑥,(𝛾𝑉𝜇𝐹)̃𝑥(3.9) Without loss of generality, we may assume that 𝑥𝑛𝑘𝑧𝐶. By using Lemma 2.4, we get that 𝑧𝐹(𝑇). Note that 𝐹(𝑇)=𝐹(𝒯), it follows that 𝑧𝐹(𝒯). Hence from (3.2), we get the following: limsup𝑛𝑥𝑛̃𝑥,(𝛾𝑉𝜇𝐹)̃𝑥=𝑧̃𝑥,(𝛾𝑉𝜇𝐹)̃𝑥0.(3.10)
Step 5. We have (𝑥𝑛̃𝑥 as 𝑛). Set 𝑦𝑛=𝛼𝑛𝛾𝑉𝑥𝑛+(𝐼𝛼𝑛𝜇𝐹)𝑇𝑛𝑥𝑛 and 𝛾𝑛=𝛼𝑛(𝜏𝛾𝐿). Noticing that 𝑥𝑛+1=𝑃𝐶(𝑦𝑛). From (3.1), we have 𝑥𝑛+1̃𝑥2=𝑦𝑛̃𝑥,𝑥𝑛+1+𝑃̃𝑥𝐶𝑦𝑛𝑦𝑛,𝑃𝐶𝑦𝑛𝑦̃𝑥𝑛̃𝑥,𝑥𝑛+1̃𝑥=𝛼𝑛𝛾𝑉𝑥𝑛𝜇𝐹̃𝑥,𝑥𝑛+1+̃𝑥𝐼𝛼𝑛𝑇𝜇𝐹𝑛𝑥𝑛𝐼𝛼𝑛𝑇𝜇𝐹𝑛̃𝑥,𝑥𝑛+1̃𝑥=𝛼𝑛𝛾𝑉𝑥𝑛𝑉̃𝑥,𝑥𝑛+1̃𝑥+𝛼𝑛𝛾𝑉̃𝑥𝜇𝐹̃𝑥,𝑥𝑛+1+̃𝑥𝐼𝛼𝑛𝑇𝜇𝐹𝑛𝑥𝑛𝐼𝛼𝑛𝑇𝜇𝐹𝑛̃𝑥,𝑥𝑛+1̃𝑥𝛼𝑛𝑥𝛾𝐿𝑛𝑥̃𝑥𝑛+1̃𝑥+𝛼𝑛(𝛾𝑉𝜇𝐹)̃𝑥,𝑥𝑛+1+̃𝑥1𝛼𝑛𝜏𝑥𝑛̃𝑥+𝑎𝑛𝑥𝑛+1=̃𝑥1𝛼𝑛(𝑥𝜏𝛾𝐿)𝑛𝑥̃𝑥𝑛+1̃𝑥+𝛼𝑛(𝛾𝑉𝜇𝐹)̃𝑥,𝑥𝑛+1+̃𝑥1𝛼𝑛𝜏𝑎𝑛𝑥𝑛+1̃𝑥1𝛼𝑛1(𝜏𝛾𝐿)2𝑥𝑛̃𝑥2+𝑥𝑛+1̃𝑥2+𝛼𝑛(𝛾𝑉𝜇𝐹)̃𝑥,𝑥𝑛+1̃𝑥+𝑎𝑛𝑥𝑛+1.̃𝑥(3.11) Hence, we have 𝑥𝑛+1̃𝑥21𝛼𝑛(𝜏𝛾𝐿)1+𝛼𝑛(𝑥𝜏𝛾𝐿)𝑛̃𝑥2+2𝛼𝑛1+𝛾𝑛(𝛾𝑉𝜇𝐹)̃𝑥,𝑥𝑛+1+̃𝑥2𝑎𝑛1+𝛾𝑛𝑥𝑛+1̃𝑥1𝛼𝑛(𝑥𝜏𝛾𝐿)𝑛̃𝑥2+2𝛼𝑛1+𝛾𝑛(𝛾𝑉𝜇𝐹)̃𝑥,𝑥𝑛+1+̃𝑥2𝑎𝑛1+𝛾𝑛𝑥𝑛+1=̃𝑥1𝛾𝑛𝑥𝑛̃𝑥2+𝛾𝑛𝛿𝑛+2𝑎𝑛1+𝛾𝑛𝑥𝑛+1,̃𝑥(3.12) where 𝛿𝑛=21+𝛾𝑛(𝜏𝛾𝐿)(𝛾𝑉𝜇𝐹)̃𝑥,𝑥𝑛+1̃𝑥.(3.13) Noticing that lim𝑛𝑎𝑛/𝛼𝑛=0, it follows from Lemma 2.5 that lim𝑛𝑥𝑛=̃𝑥. This completes the proof.
Now, we derive the main result of Ceng et al. ([13], Theorem  3.2) as the following corollary.

Corollary 3.2. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐹𝐶𝐻 be a 𝑘-Lipschitzian and 𝜂-strongly monotone operator and 𝑉𝐶𝐻 be an 𝐿-Lipschitzian mapping. Let 𝑇𝐶𝐶 be a nonexpansive mapping such that 𝐹(𝑇). Suppose that 0<𝜇<2𝜂/𝑘2 and 0𝛾𝐿<𝜏, where 𝜏=11𝜇(2𝜂𝜇𝑘2). For an arbitrary 𝑥1𝐶, consider the sequence {𝑥𝑛} generated by the following iterative process: 𝑥1𝑥𝐶,𝑛+1=𝑃𝐶𝛼𝑛𝛾𝑉𝑥𝑛+𝐼𝛼𝑛𝜇𝐹𝑇𝑥𝑛𝑛,(3.14) where {𝛼𝑛} is a sequence in (0,1) satisfying the conditions (a) and (b) of Theorem 3.1.
Then, the sequence {𝑥𝑛} converges strongly to ̃𝑥𝐹(𝑇), where ̃𝑥 is the unique solution of the following variational inequality: (𝜇𝐹𝛾𝑉)̃𝑥,̃𝑥𝑦0𝑦𝐹(𝑇).(3.15)

Again, we derive the result of Tian ([11], Theorem  3.2) as the following corollary.

Corollary 3.3. Let 𝐻 be a real Hilbert space. Let 𝑓 be an 𝛼-contraction on 𝐻 and 𝐹𝐻𝐻 be a 𝑘-Lipschitzian and 𝜂-strongly monotone operator. Let 𝑇𝐻𝐻 be a nonexpansive mapping such that 𝐹(𝑇). Suppose that 0<𝜇<2𝜂/𝑘2 and 0<𝛾𝛼<𝜏, where 𝜏=𝜇(𝜂𝜇𝑘2/2). For an arbitrary 𝑥1𝐻, consider the sequence {𝑥𝑛} generated by the following iterative process: 𝑥1𝑥𝐻,𝑛+1=𝛼𝑛𝑥𝛾𝑓𝑛+𝐼𝛼𝑛𝜇𝐹𝑇𝑥𝑛𝑛,(3.16) where {𝛼𝑛} is a sequence in (0,1) satisfying the conditions (𝑎) and (𝑏) of Theorem 3.1.
Then, the sequence {𝑥𝑛} converges strongly to ̃𝑥𝐹(𝑇), where ̃𝑥 is the unique solution of the following variational inequality: (𝜇𝐹𝛾𝑓)̃𝑥,̃𝑥𝑦0𝑦𝐹(𝑇).(3.17)

The following result obtains immediately from Theorem 3.1.

Corollary 3.4. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐹𝐶𝐻 be a 𝑘-Lipschitzian and 𝜂-strongly monotone operator and 𝑉𝐶𝐻 be an 𝐿-Lipschitzian mapping. Let {𝑇𝑛} be a sequence of nonexpansive mappings from 𝐶 into itself such that 𝑛=1𝐹(𝑇𝑛) and 𝑇 be a mapping from 𝐶 into itself defined by 𝑇𝑥=lim𝑛𝑇𝑛𝑥 for all 𝑥𝐶. Suppose that 0<𝜇<2𝜂/𝑘2 and 0𝛾𝐿<𝜏, where 𝜏=11𝜇(2𝜂𝜇𝑘2). For an arbitrary 𝑥1𝐶, consider the sequence {𝑥𝑛} in 𝐶 generated by the following iterative process: 𝑥1𝑥𝐶,𝑛+1=𝑃𝐶𝛼𝑛𝛾𝑉𝑥𝑛+𝐼𝛼𝑛𝑇𝜇𝐹𝑛𝑥𝑛𝑛,(3.18) where {𝛼𝑛} is a sequence in (0,1) satisfying the conditions (a)–(c) of Theorem 3.1.
Then, the sequence {𝑥𝑛} converges strongly to ̃𝑥𝑛=1𝐹(𝑇𝑛), where ̃𝑥 is the unique solution of the following variational inequality: (𝜇𝐹𝛾𝑉)̃𝑥,̃𝑥𝑦0𝑦𝑛=1𝐹𝑇𝑛.(3.19)

4. Application

Recall that the so-called problem of image recovery is essentially to find a common element of finitely many nonexpansive retracts 𝐶1,𝐶2,,𝐶𝑟 of 𝐶 with 𝑟𝑖=1𝐶𝑖. It is easy to see that every nonexpansive retraction 𝑃𝑖 of 𝐶 onto 𝐶𝑖 is a nonexpansive mapping of C into itself. There is no doubt that the problem of image recovery is equivalent to finding a common fixed point of finitely many nonexpansive mappings 𝑃1,𝑃2,,𝑃𝑟 of 𝐶 into itself. Applying our main result, we obtain the following result which improves a number of results connected to the problem of image recovery.

Theorem 4.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐹𝐶𝐻 be a 𝑘-Lipschitzian and 𝜂-strongly monotone operator and 𝑉𝐶𝐻 be an 𝐿-Lipschitzian mapping. Let 𝜆𝑖>0(𝑖=1,2,3,,𝑁) such that 𝑁𝑖=1𝜆𝑖=1 and 𝑇1,𝑇2,𝑇3,,𝑇𝑁𝐶𝐶 be nonexpansive mappings such that 𝑁𝑖=1𝐹(𝑇𝑖). Suppose that 0<𝜇<2𝜂/𝑘2 and 0𝛾𝐿<𝜏, where 𝜏=11𝜇(2𝜂𝜇𝑘2). For an arbitrary 𝑥1𝐶, consider the sequence {𝑥𝑛} in 𝐶 generated by the following iterative process:𝑥1𝑥𝐶,𝑛+1=𝑃𝐶𝛼𝑛𝛾𝑉𝑥𝑛+𝐼𝛼𝑛𝜇𝐹𝑁𝑖=1𝜆𝑖𝑇𝑖𝑥𝑛𝑛,(4.1) where {𝛼𝑛} is a sequence in (0,1) satisfying the conditions (𝑎) and (𝑏) of Theorem 3.1.
Then, the sequence {𝑥𝑛} converges strongly to ̃𝑥𝑁𝑖=1𝐹(𝑇𝑖), where ̃𝑥 is the unique solution of the following variational inequality: (𝜇𝐹𝛾𝑉)̃𝑥,̃𝑥𝑦0𝑦𝑁𝑖=1𝐹𝑇𝑖.(4.2)

Proof. Define 𝑇=𝑁𝑖=1𝜆𝑖𝑇𝑖. Then 𝑇 is nonexpansive mapping from 𝐶 into itself. Thus, using Lemma 2.6, we get 𝐹(𝑇)=𝑁𝑖=1𝐹(𝑇𝑖). Therefore, the proof follows from Corollary 3.2.

5. Numerical Example

For showing the effectiveness and convergence of the sequence generated by the considered iterative scheme, we discuss the following example.

Example 5.1. Let 𝐻= and 𝐶=[0,1]. Let 𝑇 be a self-mapping defined by 𝑇𝑥=1𝑥 for all 𝑥𝐶. Let 𝐹,𝑉𝐶𝐻 be two mappings defined by 𝐹𝑥=4𝑥 and 𝑉𝑥=2𝑥 for all 𝑥𝐶, where 𝐹 is a 𝑘-Lipschitzian and 𝜂-strongly monotone, and 𝑉 is an 𝐿-Lipschitzian mapping. We take 0<𝜇<2𝜂/𝑘2 and 0𝛾𝐿<𝜏, and we have 𝜇=1/4,𝜏=1 and 𝛾=1/4. Define {𝛼𝑛} in (0,1) by 𝛼𝑛=1/𝑛+1. Without loss of generality, we may assume that 𝑎𝑛=1/𝑛3/2 for all 𝑛. For each 𝑛, define 𝑇𝑛𝐶𝐶 by 𝑇𝑛[𝑎𝑥=1𝑥,if𝑥0,1),𝑛,if𝑥=1.(5.1)
In [15], it is proved that 𝒯={𝑇𝑛} is a sequence of nearly nonexpansive mappings from 𝐶 into itself such that 𝐹(𝒯)={1/2} and 𝑇𝑥=lim𝑛𝑇𝑛𝑥 for all 𝑥𝐶, where 𝑇 is nonexpansive mapping.

It can be observed that all the assumptions of Theorem 3.1 are satisfied and the sequence {𝑥𝑛} generated by (3.1) converges to a unique solution 1/2 of variational inequality (3.2) over 𝐹(𝒯). The graphical presentation of the convergence of the sequence {𝑥𝑛} generated by the iterative scheme (3.1) is given in Figure 1.

902437.fig.001
Figure 1

Acknowledgments

The authors would like to thank the referees for useful comments and suggestions.

References

  1. R. P. Agarwal, D. O'Regan, and D. R. Sahu, Fixed Point Theory for Lipschitzian-Type Mappings with Applications, Topological Fixed Point Theory and Its Applications, Springer, New York, NY, USA, 2009.
  2. D. R. Sahu, “Fixed points of demicontinuous nearly Lipschitzian mappings in Banach spaces,” Commentationes Mathematicae Universitatis Carolinae, vol. 46, no. 4, pp. 653–666, 2005. View at Publisher · View at Google Scholar · View at Scopus
  3. A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000. View at Scopus
  4. H. K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004. View at Publisher · View at Google Scholar · View at Scopus
  5. G. Marino and H. K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006. View at Publisher · View at Google Scholar · View at Scopus
  6. Y. Liu, “A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert spaces,” Nonlinear Analysis, Theory, Methods and Applications, vol. 71, no. 10, pp. 4852–4861, 2009. View at Publisher · View at Google Scholar · View at Scopus
  7. X. Qin, M. Shang, and S. M. Kang, “Strong convergence theorems of modified Mann iterative process for strict pseudo-contractions in Hilbert spaces,” Nonlinear Analysis, Theory, Methods and Applications, vol. 70, no. 3, pp. 1257–1264, 2009. View at Publisher · View at Google Scholar · View at Scopus
  8. S. Wang, “Convergence and weaker control conditions for hybrid iterative algorithms,” Fixed Point Theory and Applications, vol. 2011, no. 1, article 3, 14 pages, 2011. View at Publisher · View at Google Scholar
  9. S. Wang, “Two general algorithms for computing fixed points of nonexpansive mappings in Banach spaces,” Journal of Applied Mathematics, vol. 2012, Article ID 658905, 11 pages, 2012.
  10. S. Wang and C. Hu, “Two new iterative methods for a countable family of nonexpansive mappings in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 852030, 12 pages, 2010. View at Publisher · View at Google Scholar · View at Scopus
  11. M. Tian, “A general iterative algorithm for nonexpansive mappings in Hilbert spaces,” Nonlinear Analysis, Theory, Methods and Applications, vol. 73, no. 3, pp. 689–694, 2010. View at Publisher · View at Google Scholar · View at Scopus
  12. I. Yamada, “The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), D. Butnariu, Y. Censor, and S. Reich, Eds., vol. 8, pp. 473–504, Studies in Computational Mathematics, Amsterdam, The Netherlands, 2001.
  13. L. C. Ceng, Q. H. Ansari, and J. C. Yao, “Some iterative methods for finding fixed points and for solving constrained convex minimization problems,” Nonlinear Analysis, Theory, Methods and Applications, vol. 74, no. 16, pp. 5286–5302, 2011. View at Publisher · View at Google Scholar · View at Scopus
  14. K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,” Nonlinear Analysis, Theory, Methods and Applications, vol. 67, no. 8, pp. 2350–2360, 2007. View at Publisher · View at Google Scholar · View at Scopus
  15. N. C. Wong, D. R. Sahu, and J. C. Yao, “A generalized hybrid steepest-descent method for variational inequalities in Banach spaces,” Fixed Point Theory and Applications, vol. 2011, Article ID 754702, 28 pages, 2011. View at Publisher · View at Google Scholar · View at Scopus
  16. K. Goebel and W. A. Kirk, Topics on Metric Fixed Point Theory, Cambridge University Press, Cambridge, UK, 1990.
  17. H. K. Xu and T. H. Kim, “Convergence of hybrid steepest-descent methods for variational inequalities,” Journal of Optimization Theory and Applications, vol. 119, no. 1, pp. 185–201, 2003. View at Publisher · View at Google Scholar · View at Scopus
  18. N. C. Wong, D. R. Sahu, and J. C. Yao, “Solving variational inequalities involving nonexpansive type mappings,” Nonlinear Analysis, Theory, Methods and Applications, vol. 69, no. 12, pp. 4732–4753, 2008. View at Publisher · View at Google Scholar · View at Scopus