About this Journal Submit a Manuscript Table of Contents
Journal of Applied Mathematics
Volume 2012 (2012), Article ID 417234, 13 pages
http://dx.doi.org/10.1155/2012/417234
Research Article

Approximation of Common Fixed Points of Nonexpansive Semigroups in Hilbert Spaces

Department of Mathematics, Institute of Applied Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China

Received 14 October 2011; Accepted 11 December 2011

Academic Editor: Rudong Chen

Copyright © 2012 Dan Zhang 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

Let 𝐻 be a real Hilbert space. Consider on 𝐻 a nonexpansive semigroup 𝑆={𝑇(𝑠)0𝑠<} with a common fixed point, a contraction 𝑓 with the coefficient 0<𝛼<1, and a strongly positive linear bounded self-adjoint operator 𝐴 with the coefficient 𝛾>  0. Let 0<𝛾<𝛾/𝛼. It is proved that the sequence {𝑥𝑛} generated by the iterative method 𝑥0𝐻,𝑥𝑛+1=𝛼𝑛𝛾𝑓(𝑥𝑛)+𝛽𝑛𝑥𝑛+((1𝛽𝑛)𝐼𝛼𝑛𝐴)(1/𝑠𝑛)𝑠𝑛0𝑇(𝑠)𝑥𝑛𝑑𝑠,𝑛0 converges strongly to a common fixed point 𝑥𝐹(𝑆), where 𝐹(𝑆) denotes the common fixed point of the nonexpansive semigroup. The point 𝑥 solves the variational inequality (𝛾𝑓𝐴)𝑥,𝑥𝑥0 for all 𝑥𝐹(𝑆).

1. Introduction and Preliminaries

Let 𝐻 be a real Hilbert space and 𝑇 be a nonlinear mapping with the domain 𝐷(𝑇). A point 𝑥𝐷(𝑇) is a fixed point of 𝑇 provided 𝑇𝑥=𝑥. Denote by 𝐹(𝑇) the set of fixed points of 𝑇; that is, 𝐹(𝑇)={𝑥𝐷(𝑇)𝑇𝑥=𝑥}. Recall that 𝑇 is said to be nonexpansive if𝑇𝑥𝑇𝑦𝑥𝑦,𝑥,𝑦𝐷(𝐴).(1.1)

Recall that a family 𝑆={𝑇(𝑠)𝑠0} of mappings from 𝐻 into itself is called a one-parameter nonexpansive semigroup if it satisfies the following conditions:(i)𝑇(0)𝑥=𝑥, forall𝑥𝐻; (ii)𝑇(𝑠+𝑡)𝑥=𝑇(𝑠)𝑇(𝑡)𝑥, forall𝑠,𝑡0 and forall𝑥𝐻;(iii)𝑇(𝑠)𝑥𝑇(𝑠)𝑦𝑥𝑦, forall𝑠0 and forall𝑥,𝑦𝐻;(iv)for all 𝑥𝐶, 𝑠𝑇(𝑠)𝑥 is continuous.

We denote by 𝐹(𝑆) the set of common fixed points of 𝑆, that is, 𝐹(𝑆)=0𝑠<𝐹(𝑇(𝑠)). It is known that 𝐹(𝑆) is closed and convex; see [1]. Let 𝐶 be a nonempty closed and convex subset of 𝐻. One classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping; see [2, 3]. More precisely, take 𝑡(0,1) and define a contraction 𝑇𝑡𝐶𝐶 by 𝑇𝑡𝑥=𝑡𝑢+(1𝑡)𝑇𝑥,𝑥𝐶,(1.2) where 𝑢𝐶 is a fixed point. Banach’s contraction mapping principle guarantees that 𝑇𝑡 has a unique fixed point 𝑥𝑡 in 𝐶. If 𝑇 enjoys a nonempty fixed point set, Browder [2] proved the following well-known strong convergence theorem.

Theorem B. Let 𝐶 be a bounded closed convex subset of a Hilbert space 𝐻 and let 𝑇 be a nonexpansive mapping on 𝐶. Fix 𝑢𝐶 and define 𝑧𝑡𝐶𝑎𝑠𝑧𝑡=𝑡𝑢+(1𝑡)𝑇𝑧𝑡 for 𝑡(0,1). Then as 𝑡0, {𝑧𝑡} converges strongly to a element of 𝐹(𝑇) nearest to 𝑢.

As motivated by Theorem B, Halpern [4] considered the following explicit iteration: 𝑥0𝐶,𝑥𝑛+1=𝛼𝑛𝑢+1𝛼𝑛𝑇𝑥𝑛,𝑛0,(1.3) and proved the following theorem.

Theorem H. Let 𝐶 be a bounded closed convex subset of a Hilbert space 𝐻 and let 𝑇 be a nonexpansive mapping on 𝐶. Define a real sequence {𝛼𝑛} in [0,1] by 𝛼𝑛=𝑛𝜃,0<𝜃<1. Define a sequence {𝑥𝑛} by (1.3). Then {𝑥𝑛} converges strongly to the element of 𝐹(𝑇) nearest to 𝑢.

In 1977, Lions [5] improved the result of Halpern [4], still in Hilbert spaces, by proving the strong convergence of {𝑥𝑛} to a fixed point of 𝑇 where the real sequence {𝛼𝑛} satisfies the following conditions:(C1)lim𝑛𝛼𝑛=0; (C2)𝑛=1𝛼n=; (C3)lim𝑛(𝛼𝑛+1𝛼𝑛)/𝛼2𝑛+1=0.

It was observed that both Halpern’s and Lions’s conditions on the real sequence {𝛼𝑛} excluded the canonical choice 𝛼𝑛=1/(𝑛+1). This was overcome in 1992 by Wittmann [6], who proved, still in Hilbert spaces, the strong convergence of {𝑥𝑛} to a fixed point of 𝑇 if {𝛼𝑛} satisfies the following conditions:(C1)lim𝑛𝛼𝑛=0; (C2)𝑛=1𝛼𝑛=; (C3)𝑛=1|𝛼𝑛+1𝛼𝑛|<.

Recall that a mapping 𝑓𝐻𝐻 is an 𝛼-contraction if there exists a constant 𝛼(0,1) such that𝑓(𝑥)𝑓(𝑦)𝛼𝑥𝑦,𝑥,𝑦𝐻.(1.4)

Recall that an operator 𝐴 is strongly positive on 𝐻 if there exists a constant 𝛾>0 such that𝐴𝑥,𝑥𝛾𝑥2,𝑥𝐻.(1.5)

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [713] and the references therein. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping 𝑇 on a real Hilbert space 𝐻: min𝑥𝐹(𝑇)12𝐴𝑥,𝑥𝑥,𝑏,(1.6) where 𝐴 is a linear bounded operator on 𝐻 and 𝑏 is a given point in 𝐻. In [11], it is proved that the sequence {𝑥𝑛} defined by the iterative method below, with the initial guess 𝑥0H chosen arbitrarily, 𝑥𝑛+1=𝐼𝛼𝑛𝐴𝑇𝑥𝑛+𝛼𝑛𝑏,𝑛0,(1.7) strongly converges to the unique solution of the minimization problem (1.6) provided that the sequence {𝛼𝑛} satisfies certain conditions.

Recently, Marino and Xu [9] studied the following continuous scheme: 𝑥𝑡𝑥=𝑡𝛾𝑓𝑡+(𝐼𝑡𝐴)𝑇𝑥𝑡,(1.8) where 𝑓 is an 𝛼-contraction on a real Hilbert space 𝐻, 𝐴 is a bounded linear strongly positive operator and 𝛾>0 is a constant. They showed that {𝑥𝑡} strongly converges to a fixed point 𝑥 of 𝑇. Also in [9], they introduced a general explicit iterative scheme by the viscosity approximation method:𝑥𝑛𝐻,𝑥𝑛+1=𝛼𝑛𝑥𝛾𝑓𝑛+𝐼𝛼𝑛𝐴𝑇𝑥𝑛,𝑛0(1.9) and proved that the sequence {𝑥𝑛} generated by (1.9) converges strongly to a unique solution of the variational inequality(𝐴𝛾𝑓)𝑥,𝑥𝑥0,𝑥𝐹(𝑇),(1.10)

which is the optimality condition for the minimization problem min𝑥𝐹(𝑇)12𝐴𝑥,𝑥(𝑥),(1.11)

where is a potential function for 𝛾𝑓 (i.e., (𝑥)=𝛾𝑓(𝑥) for 𝑥𝐻).

In this paper, motivated by Li et al. [8], Marino and Xu [9], Plubtieng and Punpaeng [14], Shioji and Takahashi [15], and Shimizu and Takahashi [16], we consider the mapping 𝑇𝑡 defined as follows: 𝑇𝑡1𝑥=𝑡𝛾𝑓(𝑥)+(𝐼𝑡𝐴)𝜆𝑡𝜆𝑡0𝑇(𝑠)𝑥𝑑𝑠,(1.12) where 𝛾>0 is a constant, 𝑓 is an 𝛼-contraction, 𝐴 is a bounded linear strongly positive self-adjoint operator and {𝜆𝑡} is a positive real divergent net. If 𝛾𝛼<𝛾 for each 0<𝑡<𝐴1, one can see that 𝑇𝑡 is a (1𝑡(𝛾𝛾𝛼))-contraction. So, by Banach’s contraction mapping principle, there exists an unique solution 𝑥𝑡 of the fixed point equation𝑥𝑡𝑥=𝑡𝛾𝑓𝑡1+(𝐼𝑡𝐴)𝜆𝑡𝜆𝑡0𝑇(𝑠)𝑥𝑡𝑑𝑠.(1.13) We show that the sequence {𝑥𝑡} generated by above continuous scheme strongly converges to a common fixed point 𝑥𝐹(𝑆), which is the unique point in 𝐹(𝑆) solving the variational inequality (𝛾𝑓𝐴)𝑥,𝑥𝑥0 for all 𝑥𝐹(𝑆). Furthermore, we also study the following explicit iterative scheme: 𝑥0𝐻,𝑥𝑛+1=𝛼𝑛𝑥𝛾𝑓𝑛+𝛽𝑛𝑥𝑛+1𝛽𝑛𝐼𝛼𝑛𝐴1𝑠𝑛𝑠𝑛0𝑇(𝑠)𝑥𝑛𝑑𝑠,𝑛0.(1.14) We prove that the sequence {𝑥𝑛} generated by (1.14) converges strongly to the same 𝑥.

The results presented in this paper improve and extend the corresponding results announced by Marino and Xu [9], Plubtieng and Punpaeng [14], Shioji and Takahashi [15], and Shimizu and Takahashi [16].

In order to prove our main result, we need the following lemmas.

Lemma 1.1 (see [16]). Let 𝐷 be a nonempty bounded closed convex subset of a Hilbert space 𝐻 and let 𝑆={𝑇(𝑡)0𝑡<} be a nonexpansive semigroup on 𝐷. Then, for any 0<, lim𝑡sup𝑥𝐷1𝑡𝑡01𝑇(𝑠)𝑥𝑑𝑠𝑇()𝑡𝑡0𝑇(𝑠)𝑥𝑑𝑠=0.(1.15)

Lemma 1.2 (see [17]). Let 𝐻 be a Hilbert space, 𝐶 a closed convex subset of 𝐻, and 𝑇𝐶𝐶 a nonexpansive mapping with 𝐹(𝑇). Then 𝐼𝑇 is demiclosed, that is, if {𝑥𝑛} is a sequence in 𝐶 weakly converging to 𝑥 and if {(𝐼𝑇)𝑥𝑛} strongly converges to 𝑦, then (𝐼𝑇)𝑥=𝑦.

Lemma 1.3 (see [18]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and let 𝑃𝐶 be the metric projection from 𝐻 onto 𝐶( i.e., for 𝑥𝐻,𝑃𝐶𝑥 is the only point in 𝐶 such that 𝑥𝑃𝐶𝑥=inf{𝑥𝑧𝑧𝐶}). Given 𝑥𝐻 and 𝑧𝐶. Then 𝑧=𝑃𝐶𝑥 if and only if there holds the relations 𝑥𝑧,𝑦𝑧0,𝑦𝐶.(1.16)

Lemma 1.4. Let 𝐻 be a Hilbert space, 𝑓 a 𝛼-contraction, and 𝐴 a strongly positive linear bounded self-adjoint operator with the coefficient 𝛾>0. Then, for 0<𝛾<𝛾/𝛼, 𝑥𝑦,(𝐴𝛾𝑓)𝑥(𝐴𝛾𝑓)𝑦𝛾𝛾𝛼𝑥𝑦2,𝑥,𝑦𝐻.(1.17) That is, 𝐴𝛾𝑓 is strongly monotone with coefficient 𝛾𝛼𝛾.

Proof. From the definition of strongly positive linear bounded operator, we have 𝑥𝑦,𝐴(𝑥𝑦)𝛾𝑥𝑦2.(1.18) On the other hand, it is easy to see 𝑥𝑦,𝛾𝑓𝑥𝛾𝑓𝑦𝛾𝛼𝑥𝑦2.(1.19) Therefore, we have 𝑥𝑦,(𝐴𝛾𝑓)𝑥(𝐴𝛾𝑓)𝑦=𝑥𝑦,𝐴(𝑥𝑦)𝑥𝑦,𝛾𝑓𝑥𝛾𝑓𝑦𝛾𝛾𝛼𝑥𝑦2(1.20) for all 𝑥,𝑦𝐻. This completes the proof.

Remark 1.5. Taking 𝛾=1 and 𝐴=𝐼, the identity mapping, we have the following inequality: 𝑥𝑦,(𝐼𝑓)𝑥(𝐼𝑓)𝑦(1𝛼)𝑥𝑦2,𝑥,𝑦𝐻.(1.21) Furthermore, if 𝑓 is a nonexpansive mapping in Remark 1.5, we have 𝑥𝑦,(𝐼𝑓)𝑥(𝐼𝑓)𝑦0,𝑥,𝑦𝐻.(1.22)

Lemma 1.6 (see [9]). Assume 𝐴 is a strongly positive linear bounded self-adjoint operator on a Hilbert space 𝐻 with coefficient 𝛾>0 and 0<𝜌𝐴1. Then 𝐼𝜌𝐴1𝜌𝛾.

Lemma 1.7 (see [12]). Let {𝛼𝑛} be a sequence of nonnegative real numbers satisfying the following condition: 𝛼𝑛+11𝛾𝑛𝛼𝑛+𝛾𝑛𝜎𝑛,𝑛0,(1.23) where {𝛾𝑛} is a sequence in (0,1) and {𝜎𝑛} is a sequence of real numbers such that(i)lim𝑛𝛾𝑛=0 and 𝑛=0𝛾𝑛=,(ii)either limsup𝑛𝜎𝑛0 or 𝑛=0|𝛾𝑛𝜎𝑛|<. Then {𝛼𝑛}𝑛=0 converges to zero.

2. Main Results

Lemma 2.1. Let 𝐻 a real Hilbert space and 𝑆={𝑇(𝑠)0𝑠<} a nonexpansive semigroup on 𝐻 such that 𝐹(𝑆). Let {𝜆𝑡}0<𝑡<1 be a continuous net of positive real numbers such that lim𝑡0𝜆𝑡=. Let 𝑓𝐻𝐻 be an 𝛼-contraction, 𝐴 a strongly positive linear bounded self-adjoint operator of 𝐻 into itself with coefficient 𝛾>0. Assume that 0<𝛾<𝛾/𝛼. Let {𝑥𝑡} be a sequence defined by (1.13). Then (i){𝑥𝑡} is bounded for all 𝑡(0,𝐴1);(ii)lim𝑡0𝑇(𝜏)𝑥𝑡𝑥𝑡=0 for all 0𝜏<;(iii)𝑥𝑡 defines a continuous curve from (0,𝐴1) into 𝐻.

Proof. (i) Taking 𝑝𝐹(𝑆), we have 𝑥𝑡𝑥𝑝𝑡𝛾𝑓𝑡1+(𝐼𝑡𝐴)𝜆𝑡𝜆𝑡0𝑇(𝑠)𝑥𝑡𝑥𝑑𝑠𝑝𝑡𝛾𝑓𝑡+𝐴𝑝1𝑡𝛾1𝜆𝑡𝜆𝑡0𝑇(𝑠)𝑥𝑡𝑥𝑝𝑑𝑠𝑡𝛾𝑓𝑡+𝐴𝑝1𝑡𝛾𝑥𝑡𝑓𝑥𝑝𝑡𝛾𝑡𝑓(𝑝)+𝑡𝛾𝑓(𝑝)𝐴𝑝+1𝑡𝛾𝑥𝑡𝑝1𝑡𝑥𝛾𝛾𝛼𝑡𝑝+𝑡𝛾𝑓(𝑝)𝐴𝑝.(2.1) It follows that 𝑥𝑡1𝑝𝛾𝛼𝛾𝛾𝑓(𝑝)𝐴𝑝.(2.2) This implies that {𝑥𝑡} is not only bounded, but also that {𝑥𝑡} is contained in 𝐵(𝑝,1/(𝛾𝛾𝛼)𝛾𝑓(𝑝)𝐴𝑝) of center 𝑝 and radius 1/(𝛾𝛾𝛼)𝛾𝑓(𝑝)𝐴𝑝, for all fixed 𝑝𝐹(𝑆). Moreover for 𝑝𝐹(𝑆) and 𝑡(0,𝐴1), 1𝜆𝑡𝜆𝑡0𝑇(𝑠)𝑥𝑡=1𝑑𝑠𝑝𝜆𝑡𝜆𝑡0𝑇(𝑠)𝑥𝑡𝑥𝑇(𝑠)𝑝𝑑𝑠𝑡1𝑝𝛾𝛾𝛼𝛾𝑓(𝑝)𝐴𝑝.(2.3)
(ii) Observe that 𝑇(𝜏)𝑥𝑡𝑥𝑡𝑇(𝜏)𝑥𝑡1𝑇(𝜏)𝜆𝑡𝜆𝑡0𝑇(𝑠)𝑥𝑡+1𝑑𝑠𝑇(𝜏)𝜆𝑡𝜆𝑡0𝑇(𝑠)𝑥𝑡1𝑑𝑠𝜆𝑡𝜆𝑡0𝑇(𝑠)𝑥𝑡+1𝑑𝑠𝜆𝑡𝜆𝑡0𝑇(𝑠)𝑥𝑡𝑑𝑠𝑥𝑡𝑥2𝑡1𝜆𝑡𝜆𝑡0𝑇(𝑠)𝑥𝑡+1𝑑𝑠𝑇(𝜏)𝜆𝑡𝜆𝑡0𝑇(𝑠)𝑥𝑡1𝑑𝑠𝜆𝑡𝜆𝑡0𝑇(𝑠)𝑥𝑡𝑥𝑑𝑠=2𝑡𝛾𝑓𝑡1𝐴𝜆𝑡𝜆𝑡0𝑇(𝑠)𝑥𝑡+𝑇1𝑑𝑠(𝜏)𝜆𝑡𝜆𝑡0𝑇(𝑠)𝑥𝑡1𝑑𝑠𝜆𝑡𝜆𝑡0𝑇(𝑠)𝑥𝑡.𝑑𝑠(2.4) Taking 𝐵(𝑝,1/(𝛾𝛾𝛼)𝛾𝑓(𝑝)𝐴𝑝) as 𝐷 in Lemma 1.1 and passing to lim𝑡0 in (2.4), we can obtain (ii) immediately.
(iii) Taking 𝑡1,𝑡2(0,𝐴1) and fixing 𝑝𝐹(𝑆), we see that 𝑥𝑡1𝑥𝑡2𝑡1𝑡2𝑥𝛾𝑓𝑡1+𝑡2𝛾𝑓𝑥𝑡1𝑥𝑓𝑡2𝑡1𝑡2𝐴1𝜆𝑡1𝜆𝑡10𝑇(𝑠)𝑥𝑡1+𝑑𝑠𝐼𝑡2𝐴1𝜆𝑡1𝜆𝑡10𝑇(𝑠)𝑥𝑡11𝑑𝑠𝜆𝑡2𝜆𝑡20𝑇(𝑠)𝑥𝑡2||𝑡𝑑𝑠1𝑡2||𝛾𝑓𝑥𝑡1+𝑡2𝑥𝛾𝛼𝑡1𝑥𝑡2+||𝑡1𝑡2||1𝐴𝜆𝑡1𝜆𝑡10𝑇(𝑠)𝑥𝑡1+𝑑𝑠1𝑡2𝛾1𝜆𝑡1𝜆𝑡10𝑇(𝑠)𝑥𝑡11𝑑𝑠𝜆𝑡2𝜆𝑡10𝑇(𝑠)𝑥𝑡21𝑑𝑠𝜆𝑡2𝜆𝑡2𝜆𝑡1𝑇(𝑠)𝑥𝑡2||𝑡𝑑𝑠1t2||𝛾𝑓𝑥𝑡1+𝑡2𝑥𝛾𝛼𝑡1𝑥𝑡2+||𝑡1𝑡2||1𝐴𝜆𝑡1𝜆𝑡10𝑇(𝑠)𝑥𝑡1+𝑑𝑠1𝑡2𝛾𝑥𝑡1𝑥𝑡2+||||1𝜆𝑡11𝜆𝑡2||||𝜆𝑡10𝑇(𝑠)𝑥𝑡2+1𝑑𝑠𝜆𝑡2𝜆𝑡2𝜆𝑡1𝑇(𝑠)𝑥𝑡2.𝑑𝑠(2.5) Thus applying (2.3), we arrive at 𝑥𝑡1𝑥𝑡2||𝑡1𝑡2||𝛾𝑓𝑥𝑡1+𝑡2𝑥𝛾𝛼𝑡1𝑥𝑡2+||𝑡1𝑡2||1𝐴+𝛾𝛾𝛼𝛾𝑓(𝑝)𝐴𝑝+𝑝1𝑡2𝛾𝑥𝑡1𝑥𝑡2+2𝜆𝑡2||𝜆𝑡2𝜆𝑡1||1(||𝑡𝛾𝛾𝛼𝛾𝑓𝑝)𝐴𝑝+𝑝1𝑡2||𝛾𝑓𝑥𝑡11+𝐴+𝛾𝛾𝛼𝛾𝑓(𝑝)𝐴𝑝+𝑝1𝑡2𝑥𝛾𝛾𝛼𝑡1𝑥𝑡2+2𝜆𝑡2||𝜆𝑡2𝜆𝑡1||1.𝛾𝛾𝛼𝛾𝑓(𝑝)𝐴𝑝+𝑝(2.6) It follows that 𝑥𝑡1𝑥𝑡2𝑀1||𝑡1𝑡2||+𝑀2||𝜆𝑡2𝜆𝑡1||,(2.7) where 𝑀1=𝛾𝑓𝑥𝛾𝛾𝛼𝑡1+𝐴𝛾𝑓(𝑝)𝐴𝑝+𝛾𝛾𝛼𝐴𝑝𝑡2𝛾𝛾𝛼2(2.8) and 𝑀2=2𝛾𝑓(𝑝)𝐴𝑝+𝛾𝛾𝛼𝑝𝜆𝑡2𝑡2𝛾𝛾𝛼2.(2.9) This inequality, together with the continuity of the net {𝜆𝑡}, gives the continuity of the curve {𝑥𝑡}.

Theorem 2.2. Let 𝐻 be a real Hilbert space 𝐻 and 𝑆={𝑇(𝑠)0𝑠<} a nonexpansive semigroup such that 𝐹(𝑆). Let {𝜆𝑡}0<𝑡<1 be a net of positive real numbers such that 𝑙𝑖𝑚𝑡0𝜆𝑡=. Let 𝑓 be an 𝛼-contraction and let 𝐴 be a strongly positive linear bounded self-adjoint operator on 𝐻 with the coefficient 𝛾>0. Assume that 0<𝛾<𝛾/𝛼. Then sequence {𝑥𝑡} defined by (1.13) strongly converges as 𝑡0 to 𝑥𝐹(𝑆), which solves the following variational inequality: (𝛾𝑓𝐴)𝑥,𝑝𝑥0,𝑝𝐹(𝑆).(2.10) Equivalently, one has 𝑃𝐹(𝑆)(𝐼𝐴+𝛾𝑓)𝑥=𝑥.(2.11)

Proof. The uniqueness of the solution of the variational inequality (2.10) is a consequence of the strong monotonicity of 𝐴𝛾𝑓 (Lemma 1.4) and it was proved in [9]. Next, we will use 𝑥𝐹(𝑆) to denote the unique solution of (2.10). To prove that 𝑥𝑡𝑥(𝑡0), we write, for a given 𝑝𝐹(𝑆), 𝑥𝑡𝑥𝑝=𝑡𝛾𝑓𝑡1𝐴𝑝+(𝐼𝑡𝐴)𝜆𝑡𝜆𝑡0𝑇(𝑠)𝑥𝑡𝑑𝑠𝑝.(2.12) Using 𝑥𝑡𝑝 to make inner product, we obtain that 𝑥𝑡𝑝2=1(𝐼𝑡𝐴)𝜆𝑡𝜆𝑡0𝑇(𝑠)𝑥𝑡𝑑𝑠𝑝,𝑥𝑡𝑥𝑝+𝑡𝛾𝑓𝑡𝐴𝑝,𝑥𝑡𝑝1𝑡𝛾𝑥𝑡𝑝2𝑥+𝑡𝛾𝑓𝑡𝐴𝑝,𝑥𝑡.𝑝(2.13) It follows that 𝑥𝑡𝑝21𝛾𝛾𝑓𝑥𝑡𝑓(𝑝),𝑥t𝑝+𝛾𝑓(𝑝)𝐴𝑝,𝑥𝑡𝑝𝛾𝛼𝛾𝑥𝑡𝑝2+1𝛾𝛾𝑓(𝑝)𝐴𝑝,𝑥𝑡𝑝,(2.14) which yields that 𝑥𝑡𝑝21𝛾𝛼𝛾𝛾𝑓(𝑝)𝐴𝑝,𝑥𝑡𝑝.(2.15) Since 𝐻 is a Hilbert space and {𝑥𝑡} is bounded as 𝑡0, we have that if {𝑡𝑛} is a sequence in (0,1) such that 𝑡𝑛0 and 𝑥𝑡𝑛𝑥. By (2.15), we see 𝑥𝑡𝑛𝑥. Moreover, by (ii) of Lemma 2.1 we have 𝑥𝐹(𝑆). We next prove that 𝑥 solves the variational inequality (2.10). From (1.13), we arrive at (𝐴𝛾𝑓)𝑥𝑡1=𝑡𝑥(𝐼𝑡𝐴)𝑡1𝜆𝑡𝜆𝑡0𝑇(𝑠)𝑥𝑡𝑑𝑠.(2.16) For 𝑝𝐹(𝑆), it follows from (1.22) that (𝐴𝛾𝑓)𝑥𝑡,𝑥𝑡1𝑝=𝑡𝑥(𝐼𝑡𝐴)𝑡1𝜆𝑡𝜆𝑡0𝑇(𝑠)𝑥𝑡𝑑𝑠,𝑥𝑡1𝑝=𝑡1𝜆𝑡𝜆𝑡0(𝐼𝑇(𝑠))𝑥𝑡(𝐼𝑇(𝑠))𝑝𝑑𝑠,𝑥𝑡+𝐴1𝑝𝜆𝑡𝜆𝑡0(𝐼𝑇(𝑠))𝑥𝑡𝑑𝑠,𝑥𝑡1𝑝=𝑡𝜆𝑡𝜆𝑡0(𝐼𝑇(𝑠))𝑥𝑡(𝐼𝑇(𝑠))𝑝,𝑥𝑡+𝐴1𝑝𝑑𝑠𝜆𝑡𝜆𝑡0(𝐼𝑇(𝑠))𝑥𝑡𝑑𝑠,𝑥𝑡𝐴1𝑝𝜆𝑡𝜆𝑡0(𝐼𝑇(𝑠))𝑥𝑡𝑑𝑠,𝑥𝑡=𝐴𝑥𝑝𝑡𝛾𝑓𝑡1𝑡𝐴𝜆𝑡𝜆𝑡0𝑇(𝑠)𝑥𝑡𝑑𝑠,𝑥𝑡𝐴𝑥𝑝=𝑡𝛾𝑓𝑡1𝐴𝜆𝑡𝜆𝑡0𝑇(𝑠)𝑥𝑡𝑑𝑠,𝑥𝑡.𝑝(2.17) Passing to lim𝑡0, since {𝑥𝑡} is a bounded sequence, we obtain (𝐴𝛾𝑓)𝑥,𝑥𝑝0,(2.18) that is, 𝑥 satisfies the variational inequality (2.10). By the uniqueness it follows 𝑥=𝑥. In a summary, we have shown that each cluster point of {𝑥𝑡} (as 𝑡0) equals 𝑥. Therefore, 𝑥𝑡𝑥 as 𝑡0. The variational inequality (2.10) can be rewritten as (𝐼𝐴+𝛾𝑓)𝑥𝑥,𝑥𝑝,𝑝𝐹(𝑆).(2.19) This, by Lemma 1.3, is equivalent to 𝑃𝐹(𝑆)(𝐼𝐴+𝛾𝑓)𝑥=𝑥.(2.20) This completes the proof.

Remark 2.3. Theorem 2.2 which include the corresponding results of Shioji and Takahashi [15] as a special case is reduced to Theorem 3.1 of Plubtieng and Punpaeng [14] when 𝐴=𝐼, the identity mapping and 𝛾=1.

Theorem 2.4. Let 𝐻 be a real Hilbert space 𝐻 and 𝑆={𝑇(𝑠)0𝑠<} a nonexpansive semigroup such that 𝐹(𝑆). Let {𝑠𝑛} be a positive real divergent sequence and let {𝛼𝑛} and {𝛽𝑛} be sequences in (0,1) satisfying the following conditions lim𝑛𝛼𝑛=lim𝑛𝛽𝑛=0 and 𝑛=0𝛼𝑛=. Let 𝑓 be an 𝛼-contraction and let 𝐴 be a strongly positive linear bounded self-adjoint operator with the coefficient 𝛾>0. Assume that 0<𝛾<𝛾/𝛼. Then sequence {𝑥𝑛} defined by (1.14) strongly converges to 𝑥𝐹(𝑆), which solves the variational inequality (2.10).

Proof. We divide the proof into three parts.Step 1. Show the sequence {𝑥𝑛} is bounded.
Noticing that lim𝑛𝛼𝑛=lim𝑛𝛽𝑛=0, we may assume, with no loss of generality, that 𝛼𝑛/(1𝛽𝑛)<𝐴1 for all 𝑛0. From Lemma 1.6, we know that (1𝛽𝑛)𝐼𝛼𝑛𝐴(1𝛽𝑛𝛼𝑛𝛾). Picking 𝑝𝐹(𝑆), we have 𝑥𝑛+1=𝛼𝑝𝑛𝑥𝛾𝑓𝑛𝐴𝑝+𝛽𝑛𝑥𝑛+𝑝1𝛽𝑛𝐼𝛼𝑛𝐴1𝑠𝑛𝑠𝑛0𝑇(𝑠)𝑥𝑛𝑑𝑠𝑝𝛼𝑛𝑥𝛾𝑓𝑛𝐴𝑝+𝛽𝑛𝑥𝑛+𝑝1𝛽𝑛𝛼𝑛𝛾1𝑠𝑛𝑠𝑛0𝑇(𝑠)𝑥𝑛𝑑𝑠𝑝𝛼𝑛𝛾𝑓𝑥𝑛𝑓(𝑝)+𝛼𝑛𝛾𝑓(𝑝)𝐴𝑝+𝛽𝑛𝑥𝑛+𝑝1𝛽𝑛𝛼𝑛𝛾𝑥𝑛𝑝1𝛼𝑛𝑥𝛾𝛾𝛼𝑛𝑝+𝛼𝑛𝛾𝑓(𝑝)𝐴𝑝.(2.21) By simple inductions, we see that𝑥𝑛𝑥𝑝max0,(𝑝𝐴𝑝𝛾𝑓𝑝)𝛾𝛾𝛼,(2.22) which yields that the sequence {𝑥𝑛} is bounded.
Step 2. Show that limsup𝑛(𝛾𝑓𝐴)𝑥,𝑦𝑛𝑥0,(2.23) where 𝑥 is obtained in Theorem 2.2 and 𝑦𝑛=(1/𝑠𝑛)𝑠𝑛0𝑇(𝑠)𝑥𝑛𝑑𝑠.
Putting 𝑧0=𝑃𝐹(𝑆)𝑥0, from (2.22) we see that the closed ball 𝑀 of center 𝑧0 and radius max{𝑧0𝑝,𝐴𝑧0𝛾𝑓(𝑧0)/(𝛾𝛾𝛼)} is 𝑇(𝑠)-invariant for each 𝑠[0,) and contain {𝑥𝑛}. Therefore, we assume, without loss of generality, 𝑆={𝑇(𝑠)0𝑠<} is a nonexpansive semigroup on 𝑀. It follows from Lemma 1.1 that lim𝑛𝑦𝑛𝑇()𝑦𝑛=0(2.24) for all 0<. Taking a suitable subsequence {𝑦𝑛𝑖} of {𝑦𝑛}, we see that limsup𝑛(𝛾𝑓𝐴)𝑥,𝑦𝑛𝑥=lim𝑖(𝛾𝑓𝐴)𝑥,𝑦𝑛𝑖𝑥.(2.25) Since the sequence {𝑦𝑛} is also bounded, we may assume that 𝑦𝑛𝑖𝑥. From the demiclosedness principle, we have 𝑥𝐹(𝑆). Therefore, we have limsup𝑛(𝛾𝑓𝐴)𝑥,𝑦𝑛𝑥=(𝛾𝑓𝐴)𝑥,𝑥𝑥0.(2.26) On the other hand, we have 𝑥𝑛+1𝑦𝑛𝛼𝑛𝑥𝛾𝑓𝑛𝐴𝑥𝑛+𝛽𝑛𝑥𝑛𝑦𝑛.(2.27) From the assumption lim𝑛𝛼𝑛=lim𝑛𝛽𝑛=0, we see that lim𝑛𝑥𝑛+1𝑦𝑛=0,(2.28) which combines with (2.26) gives that limsup𝑛(𝛾𝑓𝐴)𝑥,𝑥𝑛+1𝑥0.(2.29)
Step 3. Show 𝑥𝑛𝑥 as 𝑛.
Note that𝑥𝑛+1𝑥2=𝛼𝑛𝑥𝛾𝑓𝑛𝐴𝑥+𝛽𝑛𝑥𝑛𝑥+1𝛽𝑛𝐼𝛼𝑛𝐴𝑦𝑛𝑥,𝑥𝑛+1𝑥=𝛼𝑛𝑥𝛾𝑓𝑛𝐴𝑥,𝑥𝑛+1𝑥+𝛽𝑛𝑥𝑛𝑥,𝑥𝑛+1𝑥+1𝛽𝑛𝐼𝛼𝑛𝐴𝑦𝑛𝑥,𝑥𝑛+1𝑥𝛼𝑛𝛾𝑓𝑥𝑛𝑥𝑓,𝑥𝑛+1𝑥+𝑥𝛾𝑓𝐴𝑥,𝑥𝑛+1𝑥+𝛽𝑛𝑥𝑛𝑥𝑥𝑛+1𝑥+1𝛽𝑛𝐼𝛼𝑛𝐴𝑦𝑛𝑥𝑥𝑛+1𝑥𝛼𝑛𝑥𝛼𝛾𝑛𝑥𝑥𝑛+1𝑥+𝛼𝑛𝑥𝛾𝑓𝐴𝑥,𝑥𝑛+1𝑥+𝛽𝑛𝑥𝑛𝑥𝑥𝑛+1𝑥+1𝛽𝑛𝛼𝑛𝛾𝑥𝑛𝑥𝑥𝑛+1𝑥=1𝛼𝑛𝑥𝛾𝛾𝛼𝑛𝑥𝑥𝑛+1𝑥+𝛼𝑛𝑥𝛾𝑓𝐴𝑥,𝑥𝑛+1𝑥1𝛼𝑛𝛾𝛾𝛼2𝑥𝑛𝑥2+𝑥𝑛+1𝑥2+𝛼𝑛𝑥𝛾𝑓𝐴𝑥,𝑥𝑛+1𝑥.1𝛼𝑛𝛾𝛾𝛼2𝑥𝑛𝑥2+12𝑥𝑛+1𝑥2+𝛼𝑛𝑥𝛾𝑓𝐴𝑥,𝑥𝑛+1𝑥.(2.30) It follows that 𝑥𝑛+1𝑥21𝛼𝑛𝑥𝛾𝛾𝛼𝑛𝑥2+2𝛼𝑛𝑥𝛾𝑓𝐴𝑥,𝑥𝑛+1𝑥.(2.31)
By using Lemma 1.7, we can obtain the desired conclusion easily.

Remark 2.5. If 𝛾=1 and 𝐴=𝐼, the identity mapping, then Theorem 2.4 is reduced to Theorem 3.3 of Plubtieng and Punpaeng [14].
If the sequence {𝛽𝑛}0, then Theorem 2.4 is reduced to the following.

Corollary 2.6. Let 𝐻 be a real Hilbert space 𝐻 and 𝑆={𝑇(𝑠)0𝑠<} a nonexpansive semigroup such that 𝐹(𝑆). Let {𝑠𝑛} be a positive real divergent sequence and let {𝛼𝑛} be a sequence in (0,1) satisfying the following conditions lim𝑛𝛼𝑛=0 and 𝑛=0𝛼𝑛=. Let 𝑓 be a 𝛼-contraction and let 𝐴 be a strongly positive linear bounded self-adjoint operator with the coefficient 𝛾>0. Assume that 0<𝛾<𝛾/𝛼. Let {𝑥𝑛} be a sequence generated by the following manner: 𝑥0𝐻,𝑥𝑛+1=𝛼𝑛𝑥𝛾𝑓𝑛+𝐼𝛼𝑛𝐴1𝑠𝑛𝑠𝑛0𝑇(𝑠)𝑥𝑛𝑑𝑠,𝑛0.(2.32) Then the sequence {𝑥𝑛} defined by above iterative algorithm converges strongly to 𝑥𝐹(𝑆), which solves the variational inequality (2.10).

Remark 2.7. Corollary 2.6 includes Theorem 2 of Shioji and Takahashi [15] as a special case.

Remark 2.8. Theorem 2.2 and Corollary 2.6 improve Theorem 3.2 and Theorem 3.4 of Marino and Xu [9] from a single nonexpansive mapping to a nonexpansive semigroup, respectively.

Acknowledgment

The present studies were supported by the National Natural Science Foundation of China (11071169), (11126334) and the Natural Science Foundation of Zhejiang Province (Y6110287).

References

  1. F. E. Browder, “Nonexpansive nonlinear operators in a Banach space,” Proceedings of the National Academy of Sciences of the United States of America, vol. 54, pp. 1041–1044, 1965. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. F. E. Browder, “Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces,” Archive for Rational Mechanics and Analysis, vol. 24, pp. 82–90, 1967.
  3. S. Reich, “Strong convergence theorems for resolvents of accretive operators in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 75, no. 1, pp. 287–292, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. B. Halpern, “Fixed points of nonexpanding maps,” Bulletin of the American Mathematical Society, vol. 73, pp. 957–961, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. P.-L. Lions, “Approximation de points fixes de contractions,” Comptes Rendus de l'Académie des Sciences, vol. 284, no. 21, pp. A1357–A1359, 1977. View at Zentralblatt MATH
  6. R. Wittmann, “Approximation of fixed points of nonexpansive mappings,” Archiv der Mathematik, vol. 58, no. 5, pp. 486–491, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. F. Deutsch and I. Yamada, “Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 19, no. 1-2, pp. 33–56, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. S. Li, L. Li, and Y. Su, “General iterative methods for a one-parameter nonexpansive semigroup in Hilbert space,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 9, pp. 3065–3071, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. 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 Zentralblatt MATH
  10. G. Marino, V. Colao, X. Qin, and S. M. Kang, “Strong convergence of the modified Mann iterative method for strict pseudo-contractions,” Computers & Mathematics with Applications, vol. 57, no. 3, pp. 455–465, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659–678, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. I. Yamada, N. Ogura, Y. Yamashita, and K. Sakaniwa, “Quadratic optimization of fixed points of nonexpansive mappings in Hilbert space,” Numerical Functional Analysis and Optimization, vol. 19, no. 1-2, pp. 165–190, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. S. Plubtieng and R. Punpaeng, “Fixed-point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces,” Mathematical and Computer Modelling, vol. 48, no. 1-2, pp. 279–286, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. N. Shioji and W. Takahashi, “Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 34, no. 1, pp. 87–99, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. T. Shimizu and W. Takahashi, “Strong convergence to common fixed points of families of nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 211, no. 1, pp. 71–83, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28, Cambridge University Press, Cambridge, UK, 1990. View at Publisher · View at Google Scholar
  18. S.-S. Chang, Y. J. Cho, and H. Zhou, Iterative Methods for Nonlinear Operator Equations in Banach Spaces, Nova Science Publishers, Huntington, NY, USA, 2002.