International Journal of Stochastic Analysis

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International Journal of Stochastic Analysis / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 320820 | 11 pages | https://doi.org/10.1155/2009/320820

Modified Iterative Algorithms for Nonexpansive Mappings

Yonghong Yao,1 Muhammad Aslam Noor ,2 and Syed Tauseef Mohyud-Din2

1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China

2Mathematics Department, Faculty of Sciences, COMSATS Institute of Information Technology, Islamabad, Pakistan

Academic Editor: Hong Kun Xu
Received22 Nov 2008
Accepted02 Mar 2009
Published03 Jun 2009

Abstract

Let 𝐻 be a real Hilbert space, let 𝑆, 𝑇 be two nonexpansive mappings such that 𝐹(𝑆)∩𝐹(𝑇)≠∅, let 𝑓 be a contractive mapping, and let 𝐴 be a strongly positive linear bounded operator on 𝐻. In this paper, we suggest and consider the strong converegence analysis of a new two-step iterative algorithms for finding the approximate solution of two nonexpansive mappings as 𝑥𝑛+1=𝛽𝑛𝑥𝑛+(1−𝛽𝑛)𝑆𝑦𝑛, 𝑦𝑛=𝛼𝑛𝛾𝑓(𝑥𝑛)+(𝐼−𝛼𝑛𝐴)𝑇𝑥𝑛, 𝑛≥0 is a real number and {𝛼𝑛}, {𝛽𝑛} are two sequences in (0,1) satisfying the following control conditions: (C1) lim𝑛→∞𝛼𝑛=0, (C3) 0<liminf𝑛→∞𝛽𝑛≤limsup𝑛→∞𝛽𝑛<1, then ‖𝑥𝑛+1−𝑥𝑛‖→0. We also discuss several special cases of this iterative algorithm.

1. Introduction

Let 𝐻 be a real Hilbert space. Recall that a mapping 𝑓∶𝐻→𝐻 is a contractive mapping on 𝐻 if there exists a constant 𝛼∈(0,1) such that

‖𝑓(𝑥)−𝑓(𝑦)‖≤𝛼‖𝑥−𝑦‖,𝑥,𝑦∈𝐻.(1.1) We denote by Π the collection of all contractive mappings on 𝐻, that is,

={𝑓∶𝐻⟶𝐻isacontractivemapping}.(1.2)

Let 𝑇∶𝐻→𝐻 be a nonexpansive mapping, namely,

‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖,𝑥,𝑦∈𝐻.(1.3)

Iterative algorithms for nonexpansive mappings have recently been applied to solve convex minimization problems (see [1–4] and the references therein).

A typical problem is to minimize a quadratic function over the closed convex set of the fixed points of a nonexpansive mapping 𝑇 on a real Hilbert space 𝐻:

min𝑥∈𝐶12⟨𝐴𝑥,𝑥⟩−⟨𝑥,𝑏⟩,(1.4) where 𝐶 is a closed convex set of the fixed points a nonexpansive mapping 𝑇 on 𝐻, 𝑏 is a given point in 𝐻 and 𝐴 is a linear, symmetric and positive operator.

In [5] (see also [6]), the author proved that the sequence {𝑥𝑛} defined by the iterative method below with the initial point 𝑥0∈𝐻 chosen arbitrarily

𝑥𝑛+1=1−𝛼𝑛𝐴𝑇𝑥𝑛+𝛼𝑛𝑏,𝑛≥0,(1.5) converges strongly to the unique solution of the minimization problem (1.4) provided the sequence {𝛼𝑛} satisfies certain control conditions.

On the other hand, Moudafi [3] introduced the viscosity approximation method for nonexpansive mappings (see also [7] for further developments in both Hilbert and Banach spaces). Let 𝑓 be a contractive mapping on 𝐻. Starting with an arbitrary initial point 𝑥0∈𝐻, define a sequence {𝑥𝑛} in 𝐻 recursively by

𝑥𝑛+1=1−𝛼𝑛𝑇𝑥𝑛+𝛼𝑛𝑓𝑥𝑛,𝑛≥0,(1.6) where {𝛼𝑛} is a sequence in (0,1), which satisfies some suitable control conditions.

Recently, Marino and Xu [8] combined the iterative algorithm (1.5) with the viscosity approximation algorithm (1.6), considering the following general iterative algorithm:

𝑥𝑛+1=𝐼−𝛼𝑛𝐴𝑇𝑥𝑛+𝛼𝑛𝑥𝛾𝑓𝑛,𝑛≥0,(1.7) where 0<𝛾<𝛾/𝛼.

In this paper, we suggest a new iterative method for finding the pair of nonexpansive mappings. As an application and as special cases, we also obtain some new iterative algorithms which can be viewed as an improvement of the algorithm of Xu [7] and Marino and Xu [8]. Also we show that the convergence of the proposed algorithms can be proved under weaker conditions on the parameter {𝛼𝑛}. In this respect, our results can be considered as an improvement of the many known results.

2. Preliminaries

In the sequel, we will make use of the following for our main results:

Lemma 2.1 (see [4]). Let {𝑠𝑛} be a sequence of nonnegative numbers satisfying the condition 𝑠𝑛+1≤1−𝛼𝑛𝑠𝑛+𝛼𝑛𝛽𝑛,𝑛≥0,(2.1) where {𝛼𝑛}, {𝛽𝑛} are sequences of real numbers such that (i){𝛼𝑛}⊂[0,1] and ∑∞𝑛=0𝛼𝑛=∞,(ii)lim𝑛→∞𝛽𝑛≤0 or ∑∞𝑛=0𝛼𝑛𝛽𝑛 is convergent.Then lim𝑛→∞𝑠𝑛=0.

Lemma 2.2 (see [9, 10]). Let {𝑥𝑛} and {𝑦𝑛} be bounded sequences in a Banach space 𝑋 and {𝛽𝑛} be a sequence in [0,1] with 0<liminf𝑛→∞𝛽𝑛≤limsup𝑛→∞𝛽𝑛<1.(2.2) Suppose that 𝑥𝑛+1=(1−𝛽𝑛)𝑦𝑛+𝛽𝑛𝑥𝑛 for all 𝑛≥0 and limsup𝑛→∞(‖𝑦𝑛+1−𝑦𝑛‖−‖𝑥𝑛+1−𝑥𝑛‖)≤0. Then lim𝑛→∞‖𝑦𝑛−𝑥𝑛‖=0.

Lemma 2.3 (see [2] (demiclosedness Principle)). Assume that 𝑇 is a nonexpansive self-mapping of a closed convex subset 𝐶 of a Hilbert space 𝐻. If 𝑇 has a fixed point, then 𝐼−𝑇 is demiclosed, that is, whenever {𝑥𝑛} is a sequence in 𝐶 weakly converging to some 𝑥∈𝐶 and the sequence {(𝐼−𝑇)𝑥𝑛} strongly converges to some 𝑦, it follows that (𝐼−𝑇)𝑥=𝑦, where 𝐼 is the identity operator of 𝐻.

Lemma 2.4 (see [8]). Let {𝑥𝑡} be generated by the algorithm 𝑥𝑡=𝑡𝛾𝑓(𝑥𝑡)+(𝐼−𝑡𝐴)𝑇𝑥𝑡. Then {𝑥𝑡} converges strongly as 𝑡→0 to a fixed point 𝑥∗ of 𝑇 which solves the variational inequality ⟨(𝐴−𝛾𝑓)𝑥∗,𝑥∗−𝑥⟩≤0,𝑥∈𝐹(𝑇).(2.3)

Lemma 2.5 (see [8]). Assume 𝐴 is a strong positive linear bounded operator on a Hilbert space 𝐻 with coefficient 𝛾>0 and 0<𝜌≤‖𝐴‖−1. Then ‖𝐼−𝜌𝐴‖≤1−𝜌𝛾.

3. Main Results

Let 𝐻 be a real Hilbert space, let 𝐴 be a bounded linear operator on 𝐻, and let 𝑆, 𝑇 be two nonexpansive mappings on 𝐻 such that 𝐹(𝑆)∩𝐹(𝑇)≠∅. Throughout the rest of this paper, we always assume that 𝐴 is strongly positive.

Now, let 𝑓∈Π with the contraction coefficient 0<𝛼<1 and let 𝐴 be a strongly positive linear bounded operator with coefficient 𝛾>0 satisfying 0<𝛾<𝛾/𝛼. We consider the following modified iterative algorithm:

𝑥𝑛+1=𝛽𝑛𝑥𝑛+1−𝛽𝑛𝑆𝑦𝑛,𝑦𝑛=𝛼𝑛𝑥𝛾𝑓𝑛+𝐼−𝛼𝑛𝐴𝑇𝑥𝑛,𝑛≥0,(3.1) where 𝛾>0 is a real number and {𝛼𝑛}, {𝛽𝑛} are two sequences in (0,1).

First, we prove a useful result concerning iterative algorithm (3.1) as follows.

Lemma 3.1. Let {𝑥𝑛} be a sequence in 𝐻 generated by the algorithm (3.1) with the sequences {𝛼𝑛} and {𝛽𝑛} satisfying the following control conditions: (C1)lim𝑛→∞𝛼𝑛=0,(C3)0<liminf𝑛→∞𝛽𝑛≤limsup𝑛→∞𝛽𝑛<1.Then ‖𝑥𝑛+1−𝑥𝑛‖→0.

Proof. From the control condition (C1), without loss of generality, we may assume that 𝛼𝑛≤‖𝐴‖−1. First observe that ‖𝐼−𝛼𝑛𝐴‖≤1−𝛼𝑛𝛾 by Lemma 2.5.
Now we show that {𝑥𝑛} is bounded. Indeed, for any 𝑝∈𝐹(𝑆)∩𝐹(𝑇), ‖‖𝑦𝑛‖‖=‖‖𝛼−𝑝𝑛𝑥𝛾𝑓𝑛+−𝐴𝑝𝐼−𝛼𝑛𝐴𝑇𝑥𝑛‖‖−𝑝≤𝛼𝑛‖‖𝑥𝛾𝑓𝑛‖‖−𝛾𝑓(𝑝)+𝛼𝑛(‖𝛾𝑓𝑝)−𝐴𝑝‖+1−𝛼𝑛𝛾‖‖𝑇𝑥𝑛‖‖−𝑝≤𝛼𝑛‖‖𝑥𝛾𝛼𝑛‖‖−𝑝+𝛼𝑛‖𝛾𝑓(𝑝)−𝐴𝑝‖+1−𝛼𝑛𝛾‖‖𝑥𝑛‖‖=−𝑝1−𝛼𝛾−𝛾𝛼𝑛‖‖𝑥𝑛‖‖−𝑝+𝛼𝑛(‖𝛾𝑓𝑝)−𝐴𝑝‖.(3.2) At the same time, ‖‖𝑥𝑛+1‖‖=‖‖𝛽−𝑝𝑛𝑥𝑛+−𝑝1−𝛽𝑛𝑆𝑦𝑛‖‖−𝑝≤𝛽𝑛‖‖𝑥𝑛‖‖+−𝑝1−𝛽𝑛‖‖𝑆𝑦𝑛‖‖−𝑝≤𝛽𝑛‖‖𝑥𝑛‖‖+−𝑝1−𝛽𝑛‖‖𝑦𝑛‖‖.−𝑝(3.3) It follows from (3.2) and (3.3) that ‖‖𝑥𝑛+1‖‖−𝑝≤𝛽𝑛‖‖𝑥𝑛‖‖+−𝑝1−𝛽𝑛1−𝛼𝛾−𝛾𝛼𝑛‖‖𝑥𝑛‖‖−𝑝+𝛼𝑛1−𝛽𝑛(=‖𝛾𝑓𝑝)−𝐴𝑝‖1−𝛼𝛾−𝛾𝛼𝑛1−𝛽𝑛‖‖𝑥𝑛‖‖+−𝑝𝛼𝛾−𝛾𝛼𝑛1−𝛽𝑛‖𝛾𝑓(𝑝)−𝐴𝑝‖,𝛾−𝛾𝛼(3.4) which implies that ‖‖𝑥𝑛‖‖‖‖𝑥−𝑝≤max0‖‖,(−𝑝‖𝛾𝑓𝑝)−𝐴𝑝‖𝛾−𝛾𝛼,𝑛≥0.(3.5) Hence {𝑥𝑛} is bounded and so are {𝐴𝑇𝑥𝑛} and {𝑓(𝑥𝑛)}.
From (3.1), we observe that ‖‖𝑦𝑛+1−𝑦𝑛‖‖=‖‖𝛼𝑛+1𝑥𝛾𝑓𝑛+1+𝐼−𝛼𝑛+1𝐴𝑇𝑥𝑛+1−𝛼𝑛𝑥𝛾𝑓𝑛−𝐼−𝛼𝑛𝐴𝑇𝑥𝑛‖‖=‖‖𝛼𝑛+1𝛾𝑓𝑥𝑛+1𝑥−𝑓𝑛+𝛼𝑛+1−𝛼𝑛𝑥𝛾𝑓𝑛+𝐼−𝛼𝑛+1𝐴𝑇𝑥𝑛+1−𝑇𝑥𝑛+𝛼𝑛−𝛼𝑛+1𝐴𝑇𝑥𝑛‖‖≤𝛼𝑛+1𝛾‖‖𝑓𝑥𝑛+1𝑥−𝑓𝑛‖‖+1−𝛼𝑛+1𝛾‖‖𝑇𝑥𝑛+1−𝑇𝑥𝑛‖‖+||𝛼𝑛+1−𝛼𝑛||‖‖𝑥𝛾𝑓𝑛‖‖+‖‖𝐴𝑇𝑥𝑛‖‖≤𝛼𝑛+1‖‖𝑥𝛾𝛼𝑛+1−𝑥𝑛‖‖+1−𝛼𝑛+1𝛾‖‖𝑥𝑛+1−𝑥𝑛‖‖+||𝛼𝑛+1−𝛼𝑛||‖‖𝑥𝛾𝑓𝑛‖‖+‖‖𝐴𝑇𝑥𝑛‖‖=1−𝛼𝛾−𝛾𝛼𝑛+1‖‖𝑥𝑛+1−𝑥𝑛‖‖+||𝛼𝑛+1−𝛼𝑛||‖‖𝑥𝛾𝑓𝑛‖‖+‖‖𝐴𝑇𝑥𝑛‖‖.(3.6) It follows that ‖‖𝑆𝑦𝑛+1−𝑆𝑦𝑛‖‖−‖‖𝑥𝑛+1−𝑥𝑛‖‖≤‖‖𝑦𝑛+1−𝑦𝑛‖‖−‖‖𝑥𝑛+1−𝑥𝑛‖‖=𝛼𝛾−𝛾𝛼𝑛+1‖‖𝑥𝑛+1−𝑥𝑛‖‖+||𝛼𝑛+1−𝛼𝑛||‖‖𝑥𝛾𝑓𝑛‖‖+‖‖𝐴𝑇𝑥𝑛‖‖,(3.7) which implies, from (C1) and the boundedness of {𝑥𝑛}, {𝑓(𝑥𝑛)}, and {𝐴𝑇𝑥𝑛}, that limsup𝑛→∞‖‖𝑆𝑦𝑛+1−𝑆𝑦𝑛‖‖−‖‖𝑥𝑛+1−𝑥𝑛‖‖≤0.(3.8) Hence, by Lemma 2.2, we have ‖‖𝑆𝑦𝑛−𝑥𝑛‖‖⟶0as𝑛⟶∞.(3.9) Consequently, it follows from (3.1) that lim𝑛→∞‖‖𝑥𝑛+1−𝑥𝑛‖‖=lim𝑛→∞1−𝛽𝑛‖‖𝑆𝑦𝑛−𝑥𝑛‖‖=0.(3.10) This completes the proof.

Remark 3.2. The conclusion ‖𝑥𝑛+1−𝑥𝑛‖→0 is important to prove the strong convergence of the iterative algorithms which have been extensively studied by many authors, see, for example, [3, 6, 7].

If we take 𝑆=𝐼 in (3.1), we have the following iterative algorithm:

𝑥𝑛+1=𝛽𝑛𝑥𝑛+1−𝛽𝑛𝑦𝑛,𝑦𝑛=𝛼𝑛𝑥𝛾𝑓𝑛+𝐼−𝛼𝑛𝐴𝑇𝑥𝑛,𝑛≥0.(3.11) Now we state and prove the strong convergence of iterative scheme (3.11).

Theorem 3.3. Let {𝑥𝑛} be a sequence in 𝐻 generated by the algorithm (3.11) with the sequences {𝛼𝑛} and {𝛽𝑛} satisfying the following control conditions: (C1)lim𝑛→∞𝛼𝑛=0,(C2)lim𝑛→∞𝛼𝑛=∞,(C3)0<liminf𝑛→∞𝛽𝑛≤limsup𝑛→∞𝛽𝑛<1.Then {𝑥𝑛} converges strongly to a fixed point 𝑥∗ of 𝑇 which solves the variational inequality ⟨(𝐴−𝛾𝑓)𝑥∗,𝑥∗−𝑥⟩≤0,𝑥∈𝐹(𝑇).(3.12)

Proof. From Lemma 3.1, we have ‖‖𝑥𝑛+1−𝑥𝑛‖‖⟶0.(3.13)
On the other hand, we have ‖‖𝑥𝑛−𝑇𝑥𝑛‖‖≤‖‖𝑥𝑛+1−𝑥𝑛‖‖+‖‖𝑥𝑛+1−𝑇𝑥𝑛‖‖=‖‖𝑥𝑛+1−𝑥𝑛‖‖+‖‖𝛽𝑥𝑛−𝑇𝑥𝑛+1−𝛽𝑛𝑦𝑛−𝑇𝑥𝑛‖‖≤‖‖𝑥𝑛+1−𝑥𝑛‖‖+𝛽𝑛‖‖𝑥𝑛−𝑇𝑥𝑛‖‖+1−𝛽𝑛‖‖𝑦𝑛−𝑇𝑥𝑛‖‖≤‖‖𝑥𝑛+1−𝑥𝑛‖‖+𝛽𝑛‖‖𝑥𝑛−𝑇𝑥𝑛‖‖+1−𝛽𝑛𝛼𝑛‖‖𝑥𝛾𝑓𝑛‖‖+‖‖𝐴𝑇𝑥𝑛‖‖,(3.14) that is, ‖‖𝑥𝑛−𝑇𝑥𝑛‖‖≤11−𝛽𝑛‖‖𝑥𝑛+1−𝑥𝑛‖‖+𝛼𝑛‖‖𝑥𝛾𝑓𝑛‖‖+‖‖𝐴𝑇𝑥𝑛‖‖,(3.15) this together with (C1), (C3), and (3.13), we obtain lim𝑛→∞‖‖𝑥𝑛−𝑇𝑥𝑛‖‖=0.(3.16)
Next, we show that, for any 𝑥∗∈𝐹(𝑇), limsup𝑛→∞⟨𝑦𝑛−𝑥∗𝑥,𝛾𝑓∗−𝐴𝑥∗⟩≤0.(3.17)
In fact, we take a subsequence {𝑥𝑛𝑘} of {𝑥𝑛} such that limsup𝑛→∞⟨𝑥𝑛−𝑥∗𝑥,𝛾𝑓∗−𝐴𝑥∗⟩=lim𝑘→∞⟨𝑥𝑛𝑘−𝑥∗𝑥,𝛾𝑓∗−𝐴𝑥∗⟩.(3.18) Since {𝑥𝑛} is bounded, we may assume that 𝑥𝑛𝑘0𝑥00086⇀𝑧, where “⇀” denotes the weak convergence. Note that 𝑧∈𝐹(𝑇) by virtue of Lemma 2.3 and (3.16). It follows from the variational inequality (2.3) in Lemma 2.4 that limsup𝑛→∞⟨𝑥𝑛−𝑥∗𝑥,𝛾𝑓∗−𝐴𝑥∗⟩=⟨𝑧−𝑥∗𝑥,𝛾𝑓∗−𝐴𝑥∗⟩≤0.(3.19) By Lemma 3.1 (noting 𝑆=𝐼), we have ‖‖𝑦𝑛−𝑥𝑛‖‖⟶0.(3.20) Hence, we get limsup𝑛→∞⟨𝑦𝑛−𝑥∗𝑥,𝛾𝑓∗−𝐴𝑥∗⟩≤0.(3.21)
Finally, we prove that {𝑥𝑛} converges to the point 𝑥∗. In fact, from (3.2) we have ‖‖𝑦𝑛−𝑥∗‖‖≤‖‖𝑥𝑛−𝑥∗‖‖+𝛼𝑛‖‖𝑥𝛾𝑓∗−𝐴𝑥∗‖‖.(3.22) Therefore, from (3.16), we have ‖‖𝑥𝑛+1−𝑥∗‖‖2=‖‖𝛽𝑛𝑥𝑛−𝑥∗)+(1−𝛽𝑛𝑦𝑛−𝑥∗)‖‖2≤𝛽𝑛‖‖𝑥𝑛−𝑥∗‖‖2+1−𝛽𝑛‖‖𝑦𝑛−𝑥∗‖‖2=𝛽𝑛‖‖𝑥𝑛−𝑥∗‖‖2+1−𝛽𝑛‖‖𝛼𝑛𝑥(𝛾𝑓𝑛−𝐴𝑥∗)+(𝐼−𝛼𝑛𝐴)(𝑇𝑥𝑛−𝑥∗)‖‖2≤𝛽𝑛‖‖𝑥𝑛−𝑥∗‖‖2+1−𝛽𝑛(1−𝛼𝑛𝛾)2‖‖𝑥𝑛−𝑥∗‖‖2+2𝛼𝑛𝑥⟨𝛾𝑓𝑛−𝐴𝑥∗,𝑦𝑛−𝑥∗⟩=1−2𝛼𝑛𝛾+1−𝛽𝑛𝛼2𝑛𝛾2‖‖𝑥𝑛−𝑥∗‖‖2+2𝛼𝑛𝑥⟨𝛾𝑓𝑛𝑥−𝛾𝑓∗,𝑦𝑛−𝑥∗⟩+2𝛼𝑛𝑥⟨𝛾𝑓∗−𝐴𝑥∗,𝑦𝑛−𝑥∗⟩≤1−2𝛼𝑛𝛾+1−𝛽𝑛𝛼2𝑛𝛾2‖‖𝑥𝑛−𝑥∗‖‖2+2𝛼𝑛‖‖𝑥𝛾𝛼𝑛−𝑥∗‖‖‖‖𝑦𝑛−𝑥∗‖‖+2𝛼𝑛𝑥⟨𝛾𝑓∗−𝐴𝑥∗,𝑦𝑛−𝑥∗⟩≤1−2𝛼𝑛‖‖𝑥𝛾−𝛾𝛼𝑛−𝑥∗‖‖2+1−𝛽𝑛𝛼2𝑛𝛾2‖‖𝑥𝑛−𝑥∗‖‖2+2𝛼2𝑛‖‖𝑥𝛾𝛼𝑛−𝑥∗‖‖‖‖𝑥𝛾𝑓∗−𝐴𝑥∗‖‖+2𝛼𝑛𝑥⟨𝛾𝑓∗−𝐴𝑥∗,𝑦𝑛−𝑥∗⟩.(3.23) Since {𝑥𝑛}, 𝑓(𝑥∗) and 𝐴𝑥∗ are all bounded, we can choose a constant 𝑀>0 such that 1𝛾−𝛾𝛼1−𝛽𝑛𝛾22‖‖𝑥𝑛−𝑥∗‖‖2‖‖𝑥+𝛾𝛼𝑛−𝑥∗‖‖‖‖𝑥𝛾𝑓∗−𝐴𝑥∗‖‖≤𝑀,𝑛≥0.(3.24) It follows from (3.23) that ‖‖𝑥𝑛+1−𝑥∗‖‖2≤1−2𝛼𝛾−𝛼𝛾𝑛‖‖𝑥𝑛−𝑥∗‖‖2+2𝛼𝛾−𝛼𝛾𝑛𝛿𝑛,(3.25) where 𝛿𝑛=𝛼𝑛1𝑀+𝑥𝛾−𝛾𝛼⟨𝛾𝑓∗−𝐴𝑥∗,𝑦𝑛−𝑥∗⟩.(3.26) By (C1) and (3.17), we get limsup𝑛→∞𝛽𝑛≤0.(3.27) Now, applying Lemma 2.1 and (3.25), we conclude that 𝑥𝑛→𝑥∗. This completes the proof.

Taking 𝑇=𝐼 in (3.1), we have the following iterative algorithm:

𝑥𝑛+1=𝛽𝑛𝑥𝑛+1−𝛽𝑛𝑆𝑦𝑛,𝑦𝑛=𝛼𝑛𝑥𝛾𝑓𝑛+𝐼−𝛼𝑛𝐴𝑥𝑛,𝑛≥0.(3.28)

Now we state and prove the strong convergence of iterative scheme (3.28).

Theorem 3.4. Let {𝑥𝑛} be a sequence in 𝐻 generated by the algorithm (3.28) with the sequences {𝛼𝑛} and {𝛽𝑛} satisfying the following control conditions: (C1)lim𝑛→∞𝛼𝑛=0,(C2)lim𝑛→∞𝛼𝑛=∞,(C3)0<liminf𝑛→∞𝛽𝑛≤limsup𝑛→∞𝛽𝑛<1.Then {𝑥𝑛} converges strongly to a fixed point 𝑥∗ of 𝑆 which solves the variational inequality ⟨(𝐴−𝛾𝑓)𝑥∗,𝑥∗−𝑥⟩≤0,𝑥∈𝐹(𝑆).(3.29)

Proof. From Lemma 3.1, we have ‖‖𝑥𝑛−𝑆𝑦𝑛‖‖⟶0.(3.30) Thus, we have ‖‖𝑥𝑛−𝑆𝑥𝑛‖‖≤‖‖𝑥𝑛−𝑆𝑦𝑛‖‖+‖‖𝑆𝑦𝑛−𝑆𝑥𝑛‖‖≤‖‖𝑥𝑛−𝑆𝑦𝑛‖‖+‖‖𝑦𝑛−𝑥𝑛‖‖≤‖‖𝑥𝑛−𝑆𝑦𝑛‖‖+𝛼𝑛‖‖𝑥𝛾𝑓𝑛‖‖+‖‖𝐴𝑥𝑛‖‖⟶0.(3.31) By the similar argument as (3.17), we also can prove that limsup𝑛→∞⟨𝑦𝑛−𝑥∗𝑥,𝛾𝑓∗−𝐴𝑥∗⟩≤0.(3.32) From (3.28), we obtain ‖‖𝑥𝑛+1−𝑥∗‖‖2=‖‖𝛽𝑛𝑥𝑛−𝑥∗)+(1−𝛽𝑛(𝑆𝑦𝑛−𝑥∗)‖‖2≤𝛽𝑛‖‖𝑥𝑛−𝑥∗‖‖2+1−𝛽𝑛‖‖𝑆𝑦𝑛−𝑥∗‖‖2≤𝛽𝑛‖‖𝑥𝑛−𝑥∗‖‖2+1−𝛽𝑛‖‖𝑦𝑛−𝑥∗‖‖2=𝛽𝑛‖‖𝑥𝑛−𝑥∗‖‖2+1−𝛽𝑛‖‖𝛼𝑛𝑥(𝛾𝑓𝑛−𝐴𝑥∗)+(𝐼−𝛼𝑛𝑥𝐴)𝑛−𝑥∗)‖‖2≤𝛽𝑛‖‖𝑥𝑛−𝑥∗‖‖2+1−𝛽𝑛1−𝛼𝑛𝛾2‖‖𝑥𝑛−𝑥∗‖‖2𝑥+2⟨𝛾𝑓𝑛−𝐴𝑥∗,𝑦𝑛−𝑥∗⟩.(3.33) The remainder of proof follows from the similar argument of Theorem 3.3. This completes the proof.

From the above results, we have the following corollaries.

Corollary 3.5. Let {𝑥𝑛} be a sequence in 𝐻 generated by the following algorithm 𝑥𝑛+1=𝛽𝑛𝑥𝑛+1−𝛽𝑛𝑦𝑛,𝑦𝑛=𝛼𝑛𝑓𝑥𝑛+1−𝛼𝑛𝑇𝑥𝑛,𝑛≥0,(3.34) where the sequences {𝛼𝑛} and {𝛽𝑛} satisfy the following control conditions: (C1)lim𝑛→∞𝛼𝑛=0, (C2)lim𝑛→∞𝛼𝑛=∞, (C3)0<liminf𝑛→∞𝛽𝑛≤limsup𝑛→∞𝛽𝑛<1.Then {𝑥𝑛} converges strongly to a fixed point 𝑥∗ of 𝑇 which solves the variational inequality ⟨(𝐼−𝑓)𝑥∗,𝑥∗−𝑥⟩≤0,𝑥∈𝐹(𝑇).(3.35)

Corollary 3.6. Let {𝑥𝑛} be a sequence in 𝐻 generated by the following algorithm 𝑥𝑛+1=𝛽𝑛𝑥𝑛+1−𝛽𝑛𝑆𝑦𝑛,𝑦𝑛=𝛼𝑛𝑓𝑥𝑛+1−𝛼𝑛𝑥𝑛,𝑛≥0,(3.36) where the sequences {𝛼𝑛} and {𝛽𝑛} satisfy the following control conditions: (C1)lim𝑛→∞𝛼𝑛=0, (C2)lim𝑛→∞𝛼𝑛=∞, (C3)0<liminf𝑛→∞𝛽𝑛≤limsup𝑛→∞𝛽𝑛<1.Then {𝑥𝑛} converges strongly to a fixed point 𝑥∗ of 𝑆 which solves the variational inequality ⟨(𝐼−𝑓)𝑥∗,𝑥∗−𝑥⟩≤0,𝑥∈𝐹(𝑆).(3.37)

Remark 3.7. Theorems 3.3 and 3.4 provide the strong convergence results of the algorithms (3.11) and (3.28) by using the control conditions (C1) and (C2), which are weaker conditions than the previous known ones. In this respect, our results can be considered as an improvement of the many known results.

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Copyright © 2009 Yonghong Yao 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.

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