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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 720192, 14 pages
http://dx.doi.org/10.1155/2012/720192
Research Article

Implicit and Explicit Iterations with Meir-Keeler-Type Contraction for a Finite Family of Nonexpansive Semigroups in Banach Spaces

1School of Control and Computer Engineering, North China Electric Power University, Baoding 071003, China
2School of Mathematics and Physics, North China Electric Power University, Baoding 071003, China
3Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 31 December 2011; Accepted 28 January 2012

Academic Editor: Rudong Chen

Copyright © 2012 Jiancai Huang 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 implicit and explicit iterative schemes for a finite family of nonexpansive semigroups with the Meir-Keeler-type contraction in a Banach space. Then we prove the strong convergence for the implicit and explicit iterative schemes. Our results extend and improve some recent ones in literatures.

1. Introduction

Let 𝐶 be a nonempty subset of a Banach space 𝐸 and 𝑇𝐶𝐶 be a mapping. We call 𝑇 nonexpansive if 𝑇𝑥𝑇𝑦𝑥𝑦 for all 𝑥,𝑦𝐸. The set of all fixed points of 𝑇 is denoted by Fix(𝑇), that is, Fix(𝑇)={𝑥𝐶𝑥=𝑇𝑥}.

One parameter family 𝒯={𝑇(𝑡)𝑡0} is said to a semigroup of nonexpansive mappings or nonexpansive semigroup on 𝐶 if the following conditions are satisfied: (1)𝑇(0)𝑥=𝑥 for all 𝑥𝐶; (2)𝑇(𝑠+𝑡)=𝑇(𝑠)𝑇(𝑡) for all 𝑠,𝑡0; (3)for each 𝑡0, 𝑇(𝑡)𝑥𝑇(𝑡)𝑦𝑥𝑦 for all 𝑥,𝑦𝐶; (4)for each 𝑥𝐶, the mapping 𝑇()𝑥 from +, where + denotes the set of all nonnegative reals, into 𝐶 is continuous.

We denote by Fix(𝒯) the set of all common fixed points of semigroup 𝒯, that is, Fix(𝒯)={𝑥𝐶𝑇(𝑡)𝑥=𝑥,0𝑡<} and by the set of natural numbers.

Now, we recall some recent work on nonexpansive semigroup in literatures. In [1], Shioji and Takahashi introduced the following implicit iteration for a nonexpansive semigroup in a Hilbert space:𝑥𝑛=𝛼𝑛𝑥+1𝛼𝑛1𝑡𝑛𝑡𝑛0𝑇(𝑠)𝑥𝑛𝑑𝑠,𝑛,(1.1) where {𝛼𝑛}(0,1) and {𝑡𝑛}(0,). Under the certain conditions on {𝛼𝑛} and {𝑡𝑛}, they proved that the sequence {𝑥𝑛} defined by (1.1) converges strongly to an element in Fix(𝒯).

In [2], Suzuki introduced the following implicit iteration for a nonexpansive semigroup in a Hilbert space:𝑥𝑛=𝛼𝑛𝑢+1𝛼𝑛𝑇𝑡𝑛𝑥𝑛,𝑛,(1.2) where {𝛼𝑛}(0,1) and {𝑡𝑛}(0,). Under the conditions that lim𝑛𝑡𝑛=lim𝑛𝛼𝑛/𝑡𝑛=0, he proved that {𝑥𝑛} defined by (1.2) converges strongly to an element of Fix(𝒯). Later on, Xu [3] extended the iteration (1.2) to a uniformly convex Banach space that admits a weakly sequentially continuous duality mapping. Song and Xu [4] also extended the iteration (1.2) to a reflexive and strictly convex Banach space.

In 2007, Chen and He [5] studied the following implicit and explicit viscosity approximation processes for a nonexpansive semigroup in a reflexive Banach space admitting a weakly sequentially continuous duality mapping:𝑥𝑛=𝛼𝑛𝑓𝑥𝑛+1𝛼𝑛𝑇𝑡𝑛𝑥𝑛,𝑦𝑛+1=𝛽𝑛𝑓𝑦𝑛+1𝛽𝑛𝑇𝑡𝑛𝑦𝑛,𝑛,(1.3) where 𝑓 is a contraction, {𝛼𝑛}(0,1) and {𝑡𝑛}(0,). They proved the strong convergence for the above iterations under some certain conditions on the control sequences.

Recently, Chen et al. [6] introduced the following implicit and explicit iterations for nonexpansive semigroups in a reflexive Banach space admitting a weakly sequentially continuous duality mapping:𝑦𝑛=𝛼𝑛𝑥𝑛+1𝛼𝑛𝑇𝑡𝑛𝑥𝑛,𝑥𝑛=𝛽𝑛𝑓𝑥𝑛+1𝛽𝑛𝑦𝑛𝑦,𝑛,(1.4)𝑛=𝛼𝑛𝑥𝑛+1𝛼𝑛𝑇𝑡𝑛𝑥𝑛,𝑥𝑛+1=𝛽𝑛𝑓𝑥𝑛+1𝛽𝑛𝑦𝑛,𝑛,(1.5) where 𝑓 is a contraction, {𝛼𝑛}(0,1) and {𝑡𝑛}(0,). They proved that {𝑥𝑛} defined by (1.4) and (1.5) converges strongly to an element 𝑞 of Fix(𝒯), which is the unique solution of the following variation inequality problem:(𝑓𝐼),𝑗(𝑥𝑞)0,𝑥Fix(𝒯).(1.6)

For more convergence theorems on implicit and explicit iterations for nonexpansive semigroups, refer to [713].

In this paper, we introduce an implicit and explicit iterative process by a generalized contraction for a finite family of nonexpansive semigroups in a Banach space. Then we prove the strong convergence for the iterations and our results extend the corresponding ones of Suzuki [2], Xu [3], Chen and He [5], and Chen et al. [6].

2. Preliminaries

Let 𝐸 be a Banach space and 𝐸 the duality space of 𝐸. We denote the normalized mapping from 𝐸 to 2𝐸 by 𝐽 defined by𝐽(𝑥)=𝑗𝐸𝑥,𝑗𝑥=𝑥2=𝑗,𝑥𝐸,(2.1) where , denotes the generalized duality pairing. For any 𝑥,𝑦𝐸 with 𝑗(𝑥)𝐽(𝑥) and 𝑗(𝑥+𝑦)𝐽(𝑥+𝑦), it is well known that the following inequality holds: 𝑥2+2𝑦,𝑗(𝑥)𝑥+𝑦2𝑥2+2𝑦,𝑗(𝑥+𝑦).(2.2)

The dual mapping 𝐽 is called weakly sequentially continuous if 𝐽 is single valued, and {𝑥𝑛}𝑥𝐸, where denotes the weak convergence, then 𝐽(𝑥𝑛) weakly star converges to 𝐽(𝑥) [1416]. A Banach space 𝐸 is called to satisfy Opial’s condition [17] if for any sequence {𝑥𝑛} in 𝐸, 𝑥𝑛𝑥, limsup𝑛𝑥𝑛𝑥<limsup𝑛𝑥𝑛𝑦,𝑦𝐸with𝑥𝑦.(2.3) It is known that if 𝐸 admits a weakly sequentially continuous duality mapping 𝐽, then 𝐸 is smooth and satisfies Opial’s condition [14].

A function 𝜓++ is said to be an 𝐿-function if 𝜓(0)=0, 𝜓(𝑡)>0 for any 𝑡>0, and for every 𝑡>0 and 𝑠>0, there exists 𝑢>𝑠 such that 𝜓(𝑡)𝑠, for all 𝑡[𝑠,𝑢]. This implies that 𝜓(𝑡)<𝑡 for all 𝑡>0.

Let 𝑓𝐶𝐶 be a mapping. 𝑓 is said to be a (𝜓,𝐿)-contraction if there exists a 𝐿-function 𝜓++ such that 𝑓(𝑥)𝑓(𝑦)<𝜓(𝑥𝑦) for all 𝑥,𝑦𝐶 with 𝑥𝑦. Obviously, if 𝜓(𝑡)=𝑘𝑡 for all 𝑡>0, where 𝑘(0,1), then 𝑓 is a contraction. 𝑓 is called a Meir-Keeler-type mapping if for each 𝜖>0, there exists 𝛿(𝜖)>0 such that for all 𝑥,𝑦𝐶, if 𝜖<𝑥𝑦<𝜖+𝛿, then 𝑓(𝑥)𝑓(𝑦)<𝜖.

In this paper, we always assume that 𝜓(𝑡) is continuous, strictly increasing and lim𝑡𝜂(𝑡)=, where 𝜂(𝑡)=𝑡𝜓(𝑡), is strictly increasing and onto.

The following lemmas will be used in next section.

Lemma 2.1 (see [18]). Let (𝑋,𝑑) be a metric space and 𝑓𝑋𝑋 be a mapping. The following assertions are equivalent: (i)𝑓 is a Meir-Keeler-type mapping;(ii)there exists an 𝐿-function 𝜓++ such that 𝑓 is a (𝜓,𝐿)-contraction.

Lemma 2.2 (see [19]). Let 𝐸 be a Banach space and 𝐶 be a convex subset of 𝐸. Let 𝑇𝐶𝐶 be a nonexpansive mapping and 𝑓 be a (𝜓,𝐿)-contraction. Then the following assertions hold: (i)𝑇𝑓 is a (𝜓,𝐿)-contraction on 𝐶 and has a unique fixed point in 𝐶;(ii)for each 𝛼(0,1), the mapping 𝑥𝛼𝑓(𝑥)+(1𝛼)𝑇𝑥 is of Meir-Keeler-type and it has a unique fixed point in 𝐶.

Lemma 2.3 (see [20]). Let 𝐸 be a Banach space and 𝐶 be a convex subset of 𝐸. Let 𝑓𝐶𝐶 be a Meir-Keeler-type contraction. Then for each 𝜖>0 there exists 𝑟(0,1) such that, for each 𝑥,𝑦𝐶 with 𝑥𝑦𝜖,𝑓(𝑥)𝑓(𝑦)𝑟𝑥𝑦.

Lemma 2.4 (see [21]). Let C be a closed convex subset of a strictly convex Banach space E. Let 𝑇𝑚𝐶𝐶 be a nonexpansive mapping for each 1𝑚𝑟, where 𝑟 is some integer. Suppose that 𝑟𝑚=1Fix(𝑇𝑚) is nonempty. Let {𝜆𝑛} be a sequence of positive numbers with 𝑟𝑛=1𝜆𝑛=1. Then the mapping 𝑆𝐶𝐶 defined by 𝑆𝑥=𝑟𝑚=1𝜆𝑚𝑇𝑚𝑥,𝑥𝐶,(2.4) is well defined, nonexpansive and Fix(𝑆)=𝑟𝑚=1Fix(𝑇𝑚) holds.

Lemma 2.5 (see [22]). Assume that {𝛼𝑛} is a sequence of nonnegative real numbers such that 𝛼𝑛+11𝛾𝑛𝛼𝑛+𝛿𝑛,𝑛,(2.5) where {𝛾𝑛} is a sequence in (0,1) and {𝛿𝑛} is a sequence in such that(i)lim𝑛𝛾𝑛=0;(ii)𝑛=1𝛾𝑛=;(iii)limsup𝑛𝛿𝑛/𝛾𝑛0 or 𝑛=1|𝛿𝑛|<.Then lim𝑛𝛼𝑛=0.

3. Main Results

In this section, by a generalized contraction mapping we mean a Meir-Keeler-type mapping or (𝜓,𝐿)- contraction. In the rest of the paper we suppose that 𝜓 from the definition of the (𝜓,𝐿)-contraction is continuous, strictly increasing and 𝜂(𝑡) is strictly increasing and onto, where 𝜂(𝑡)=𝑡𝜓(𝑡), for all 𝑡+. As a consequence, we have the 𝜂(𝑡) is a bijection on +.

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a reflexive Banach space 𝐸 which admits a weakly sequentially continuous duality mapping 𝐽 from 𝐸 into 𝐸. For every 𝑖=1,,𝑁(𝑁1), let 𝒯𝑖={𝑇𝑖(𝑡)𝑡0} be a semigroup of nonexpansive mappings on 𝐶 such that =𝑁𝑖=1Fix(𝒯𝑖) and 𝑓𝐶𝐶 be a generalized contraction on 𝐶. Let {𝛼𝑛},{𝛽𝑛}[0,1) and {𝑡𝑛}(0,) be the sequences satisfying lim𝑛𝑡𝑛=lim𝑛(𝛼𝑛/𝑡𝑛)=0 and limsup𝑛𝛽𝑛<1. Let {𝑥𝑛} be a sequence generated by 𝑥𝑛=𝛼𝑛𝑓𝑥𝑛+1𝛼𝑛𝑁𝑁𝑖=1𝑦𝑖𝑛,𝑦𝑖𝑛=𝛽𝑛𝑥𝑛+1𝛽𝑛𝑇𝑖𝑡𝑛𝑥𝑛,𝑖=1,,𝑁.(3.1) Then {𝑥𝑛} converges strongly to a point 𝑥, which is the unique solution to the following variational inequality: (𝑓𝐼)𝑥,𝑗𝑥𝑥0,𝑥.(3.2)

Proof. First, we show that the sequence {𝑥𝑛} generated by (3.1) is well defined. For every 𝑛 and 𝑖=1,,𝑁, let 𝑈𝑖𝑛=𝛽𝑛𝐼+(1𝛽𝑛)𝑇𝑖(𝑡𝑛) and define 𝑊𝑛𝐶𝐶 by 𝑊𝑛𝑥=𝛼𝑛𝑓(𝑥)+1𝛼𝑛𝐺𝑛𝑥,𝑥𝐶,(3.3) where 𝐺𝑛𝑥=(1/𝑁)𝑁𝑖=1𝑈𝑖𝑛𝑥. Since 𝑈𝑖𝑛 is nonexpansive, 𝐺𝑛 is nonexpansive. By Lemma 2.2 we see that 𝑊𝑛 is a Meir-Keeler-type contraction for each 𝑛. Hence, each 𝑊𝑛 has a unique fixed point, denoted as 𝑥𝑛, which uniquely solves the fixed point equation (3.3). Hence {𝑥𝑛} generated by (3.1) is well defined.
Now we prove that {𝑥𝑛} generated by (3.1) is bounded. For any 𝑝, we have 𝑦𝑖𝑛𝑝𝛽𝑛𝑥𝑛+𝑝1𝛽𝑛𝑇𝑖𝑡𝑛𝑥𝑛𝑥𝑝𝑛𝑝.(3.4) Using (3.4), we get 𝑥𝑛𝑝2=𝛼𝑛𝑓𝑥𝑛+1𝛼𝑛𝑁𝑁𝑖=1𝑦𝑖𝑛𝑥𝑝,𝑗𝑛𝑝=𝛼𝑛𝑓𝑥𝑛𝑥𝑓(𝑝),𝑗𝑛𝑝+𝛼𝑛𝑥𝑓(𝑝)𝑝,𝑗𝑛+𝑝1𝛼𝑛𝑁𝑁𝑖=1𝑦𝑖𝑛𝑥𝑝,𝑗𝑛𝑝𝛼𝑛𝜓𝑥𝑛𝑥𝑝𝑛𝑝+𝛼𝑛𝑥𝑓(𝑝)𝑝𝑛+𝑝1𝛼𝑛𝑁𝑁𝑖=1𝑦𝑖𝑛𝑥𝑝𝑛𝑝=𝛼𝑛𝜓𝑥𝑛𝑥𝑝𝑛𝑝+𝛼𝑛𝑥𝑓(𝑝)𝑝𝑛+𝑝1𝛼𝑛𝑥𝑛𝑝2(3.5) and hence 𝑥𝑛𝑥𝑝𝜓𝑛𝑝+𝑓(𝑝)𝑝,(3.6) which implies that 𝜂𝑥𝑛=𝑥𝑝𝑛𝑥𝑝𝜓𝑛𝑝𝑓(𝑝)𝑝.(3.7) Hence 𝑥𝑛𝑝𝜂1(𝑓(𝑝)𝑝).(3.8) This shows that {𝑥𝑛} is bounded, and so are {𝑇𝑖(𝑡𝑛)𝑥𝑛}, {𝑓(𝑥𝑛)} and {𝑦𝑖𝑛}.
Since 𝐸 is reflexivity and {𝑥𝑛} is bounded, there exists a subsequence {𝑥𝑛𝑗}{𝑥𝑛} such that 𝑥𝑛𝑗𝑥 for some 𝑥𝐶 as 𝑗. Now we prove that 𝑥. For any fixed 𝑡>0, we have 𝑁𝑖=1𝑥𝑛𝑗𝑇𝑖(𝑡)𝑥𝑁𝑖=1[𝑡/𝑡𝑛𝑖]1𝑘=0𝑇𝑖(𝑘+1)𝑡𝑛𝑗𝑥𝑛𝑗𝑇𝑖𝑘𝑡𝑛𝑗𝑥𝑛𝑗+𝑇𝑖𝑡𝑡𝑛𝑗𝑡𝑛𝑗𝑥𝑛𝑗𝑇𝑖𝑡𝑡𝑛𝑗𝑡𝑛𝑗𝑥+𝑇𝑖𝑡𝑡𝑛𝑗𝑡𝑛𝑗𝑥𝑛𝑗𝑇𝑖(𝑡)𝑥𝑁𝑖=1𝑡𝑡𝑛𝑗𝑇𝑖𝑡𝑛𝑗𝑥𝑛𝑗𝑥𝑛𝑗+𝑥𝑛𝑗𝑥+𝑇𝑖𝑡𝑡𝑡𝑛𝑗𝑡𝑛𝑗𝑥𝑛𝑗𝑥𝑁𝑖=1𝑡𝑡𝑛𝑗𝑇𝑖𝑡𝑛𝑗𝑥𝑛𝑗𝑥𝑛𝑗+𝑥𝑛𝑗𝑥𝑇+max𝑖(𝑠)𝑥𝑥0𝑠𝑡𝑛𝑗𝑁𝛼𝑛𝑗𝑡/𝑡𝑛𝑗1𝛼𝑛𝑗1𝛽𝑛𝑗𝑥𝑛𝑗𝑥𝑓𝑛𝑗𝑥+𝑁𝑛𝑗𝑥+𝑁𝑖=1𝑇max𝑖(𝑠)𝑥𝑥0𝑠𝑡𝑛𝑗𝑁𝑡1𝛼𝑛𝑗1𝛽𝑛𝑗𝛼𝑛𝑗𝑡𝑛𝑗𝑥𝑛𝑗𝑥𝑓𝑛𝑗𝑥+𝑁𝑛𝑗𝑥+𝑁𝑖=1𝑇max𝑖(𝑠)𝑥𝑥0𝑠𝑡𝑛𝑗.(3.9) By hypothesis on {𝑡𝑛},{𝛼𝑛}, {𝛽𝑛}, we have lim𝑗𝑁𝑡1𝛼𝑛𝑗1𝛽𝑛𝑗𝛼𝑛𝑗𝑡𝑛𝑗=0.(3.10) Further, from (3.9) we get limsup𝑁𝑗𝑖=1𝑥𝑛𝑗𝑇𝑖(𝑡)𝑥limsup𝑗𝑁𝑥𝑛𝑗𝑥.(3.11) Since 𝐸 admits a weakly sequentially duality mapping, we see that 𝐸 satisfies Opial’s condition. Thus if 𝑥, we have limsup𝑗𝑁𝑥𝑛𝑗𝑥<limsup𝑁𝑗𝑖=1𝑥𝑛𝑗𝑇𝑖𝑥.(3.12) This contradicts (3.11). So 𝑥.
In (3.5), replacing 𝑝 with 𝑥 and 𝑛 with 𝑛𝑗, we see that 𝑥𝑛𝑗𝑥2=𝛼𝑛𝑗𝑓𝑥𝑛𝑗𝑥𝑓𝑥,𝑗𝑛𝑗𝑥+𝛼𝑛𝑗𝑓𝑥𝑥𝑥,𝑗𝑛𝑗𝑥+1𝛼𝑛𝑗𝑁𝑁𝑖=1𝑦𝑖𝑛𝑗𝑥𝑥,𝑗𝑛𝑗𝑥𝛼𝑛𝑗𝜓𝑥𝑛𝑗𝑥𝑥𝑛𝑗𝑥+𝛼𝑛𝑗𝑓𝑥𝑥𝑥,𝑗𝑛𝑗𝑥+1𝛼𝑛𝑗𝑁𝑁𝑖=1𝑦𝑖𝑛𝑗𝑥𝑥𝑛𝑗𝑥𝛼𝑛𝑗𝜓𝑥𝑛𝑗𝑥𝑥𝑛𝑗𝑥+𝛼𝑛𝑗𝑓𝑥𝑥𝑥,𝑗𝑛𝑗𝑥+1𝛼𝑛𝑗𝑥𝑛𝑝2,(3.13) which implies that 𝑥𝑛𝑗𝑥𝜓𝑥𝑛𝑗𝑥𝑥𝑛𝑗𝑥𝑓𝑥𝑥𝑥,𝑗𝑛𝑗𝑥.(3.14) Now we prove that {𝑥𝑛} is relatively sequentially compact. Since 𝑗 is weakly sequentially continuous, we have lim𝑗𝑥𝑛𝑗𝑥𝜓𝑥𝑛𝑗𝑥𝑥𝑛𝑗𝑥0,(3.15) which implies that lim𝑗𝑥𝑛𝑗𝑥=0,orlim𝑗𝜓𝑥𝑛𝑗𝑥𝑥𝑛𝑗𝑥=0.(3.16) If lim𝑗𝑥𝑛𝑗𝑥=0, then {𝑥𝑛} is relatively sequentially compact. If lim𝑗(𝜓(𝑥𝑛𝑗𝑥)𝑥𝑛𝑗𝑥)=0, we have lim𝑗𝑥𝑛𝑗𝑥=lim𝑗𝜓(𝑥𝑛𝑗𝑥). Since 𝜓 is continuous, lim𝑗𝑥𝑛𝑗𝑥=𝜓(lim𝑗𝑥𝑛𝑗𝑥). By the definition of 𝜓, we conclude that lim𝑗𝑥𝑛𝑗𝑥=0, which implies that {𝑥𝑛} is relatively sequentially compact.
Next, we prove that 𝑥 is the solution to (3.2). Indeed, for any 𝑥, we have 𝑥𝑛𝑥2=𝛼𝑛𝑓𝑥𝑛𝑥𝑛+𝑥𝑛𝑥𝑥,𝑗𝑛+𝑥1𝛼𝑛𝑁𝑁𝑖=1𝑦𝑖𝑛𝑥𝑥,𝑗𝑛𝑥=𝛼𝑛𝑓𝑥𝑛𝑥𝑛𝑥,𝑗𝑛𝑥+𝛼𝑛𝑥𝑛𝑥𝑥,𝑗𝑛+𝑥1𝛼𝑛𝑁𝑁𝑖=1𝛽𝑛𝑥𝑛𝑥𝑥,𝑗𝑛+𝑥1𝛽𝑛𝑇𝑖𝑡𝑛𝑥𝑛𝑥𝑥,𝑗𝑛𝑥𝛼𝑛𝑓𝑥𝑛𝑥𝑛𝑥,𝑗𝑛𝑥+𝛼𝑛𝑥𝑛𝑥2+1𝛼𝑛𝑁𝑁𝑖=1𝛽𝑛𝑥𝑛𝑥2+1𝛽𝑛𝑇𝑖𝑡𝑛𝑥𝑛𝑥𝑥𝑛𝑥𝛼𝑛𝑓𝑥𝑛𝑥𝑛𝑥,𝑗𝑛𝑥+𝛼𝑛𝑥𝑛𝑥2+1𝛼𝑛𝑁𝑁𝑖=1𝛽𝑛𝑥𝑛𝑥2+1𝛽𝑛𝑥𝑛𝑥2=𝛼𝑛𝑓𝑥𝑛𝑥𝑛𝑥,𝑗𝑛+𝑥𝑥𝑛𝑥2.(3.17) Therefore, 𝑓𝑥𝑛𝑥𝑛,𝑗𝑥𝑥𝑛0.(3.18) Since 𝑥𝑛𝑗𝑥 and 𝑗 is weakly sequentially continuous, we have 𝑓𝑥𝑥,𝑗𝑥𝑥=lim𝑗𝑓𝑥𝑛𝑗𝑥𝑛𝑗,𝑗𝑥𝑥𝑛𝑗0.(3.19) This shows that 𝑥 is the solution of the variational inequality (3.2).
Finally, we prove that 𝑥 is the unique solution of the variational inequality (3.2). Assume that ̂𝑥 with ̂𝑥𝑥 is another solution of (3.2). Then there exists 𝜖>0 such that ̂𝑥𝑥>𝜖. By Lemma 2.3 there exists 𝑟(0,1) such that 𝑓(̂𝑥)𝑓(𝑥)𝑟̂𝑥𝑥. Since both ̂𝑥 and 𝑥 are the solution of (3.2), we have 𝑓𝑥𝑥,𝑗̂𝑥𝑥𝑓𝑥0,(̂𝑥)̂𝑥,𝑗̂𝑥0.(3.20) Adding the above inequalities, we get 0<(1𝑟)𝜖2<(1𝑟)̂𝑥𝑥2(𝐼𝑓)𝑥𝑥(𝐼𝑓)̂𝑥,𝑗̂𝑥0,(3.21) which is a contradiction. Therefore, we must have ̂𝑥=𝑥, which implies that 𝑥 is the unique solution of (3.2).
In a similar way it can be shown that each cluster point of sequence {𝑥𝑛} is equal to 𝑥. Therefore, the entire sequence {𝑥𝑛} converges strongly to 𝑥. This completes the proof.

If letting 𝛽𝑛=0 for all 𝑛 in Theorem 3.1, then we get the following.

Corollary 3.2. Let 𝐶 be a nonempty closed convex subset of a reflexive Banach space 𝐸 which admits a weakly sequentially continuous duality mapping 𝐽 from 𝐸 into 𝐸. For every 𝑖=1,,𝑁 ( 𝑁1), let 𝒯𝑖={𝑇𝑖(𝑡)𝑡0} be a semigroup of nonexpansive mappings on 𝐶 such that =𝑁𝑖=1Fix(𝒯𝑖) and 𝑓𝐶𝐶 be a generalized contraction on 𝐶. Let {𝛼𝑛}[0,1) and {𝑡𝑛}(0,) be sequences satisfying lim𝑛𝑡𝑛=lim𝑛(𝛼𝑛/𝑡𝑛)=0. Let {𝑥𝑛} be a sequence generated by 𝑥𝑛=𝛼𝑛𝑓𝑥𝑛+1𝛼𝑛𝑁𝑁𝑖=1𝑇𝑖𝑡𝑛𝑥𝑛.(3.22) Then {𝑥𝑛} converges strongly to a point 𝑥, which is the unique solution to the following variational inequality: (𝑓𝐼)𝑥,𝑗𝑥𝑥0,𝑥.(3.23)

Theorem 3.3. Let 𝐶 be a nonempty closed convex subset of a reflexive and strictly convex Banach space 𝐸 which admits a weakly sequentially continuous duality mapping 𝐽 from 𝐸 into 𝐸. For every 𝑖=1,,𝑁(𝑁1), let 𝒯𝑖={𝑇𝑖(𝑡)𝑡0} be a semigroup of nonexpansive mappings on 𝐶 such that =𝑁𝑖=1Fix(𝒯𝑖) and 𝑓𝐶𝐶 be a generalized contraction on 𝐶. Let {𝛼𝑛},{𝛽𝑛}[0,1) and {𝑡𝑛}(0,) be the sequences satisfying lim𝑛𝑡𝑛=lim𝑛(𝛽𝑛/𝑡𝑛)=0. Let {𝑥𝑛} be a sequence generated 𝑦𝑖𝑛=𝛼𝑛𝑥𝑛+1𝛼𝑛𝑇𝑖𝑡𝑛𝑥𝑛𝑥,𝑖=1,,𝑁,𝑛+1=𝛽𝑛𝑓𝑥𝑛+1𝛽𝑛𝑁𝑁𝑖=1𝑦𝑖𝑛,𝑛.(3.24) Then {𝑥𝑛} converges strongly to a point 𝑥, which is the unique solution of variational inequality (3.2).

Proof. Let 𝑝 and 𝑀=max{𝑥1𝑝,𝜂1(𝑓(𝑝)𝑝}. Now we show by induction that 𝑥𝑛𝑝𝑀,𝑛.(3.25) It is obvious that (3.25) holds for 𝑛=1. Suppose that (3.25) holds for some 𝑛=𝑘, where 𝑘>1. Observe that 𝑦𝑖𝑘=𝛼𝑝𝑘𝑥𝑘+𝑝1𝛼𝑘𝑇𝑖𝑡𝑘𝑥𝑘𝑝𝛼𝑘𝑥𝑘+𝑝1𝛼𝑘𝑇𝑖𝑡𝑘𝑥𝑘𝑥𝑝𝑘.𝑝(3.26) Now, by using (3.24) and (3.26), we have 𝑥𝑘+1𝛽𝑝=𝑘𝑓𝑥𝑘+𝑝1𝛽𝑘𝑁𝑁𝑖=1𝑦𝑖𝑘𝑝𝛽𝑘𝑓𝑥𝑘𝑓(𝑝)+𝛽𝑘𝑓(𝑝)𝑝+1𝛽𝑘𝑁𝑁𝑖=1𝑦𝑖𝑘𝑝𝛽𝑘𝜓𝑥𝑘𝑝+𝛽𝑘𝑓(𝑝)𝑝+1𝛽𝑘𝑁𝑁𝑖=1𝑥𝑘𝑝=𝛽𝑘𝜓𝑥𝑘𝑝+𝛽𝑘𝑓(𝑝)𝑝+1𝛽𝑘𝑥𝑘𝑝=𝛽𝑘𝜓𝑥𝑘𝑝+𝛽𝑘𝜂𝜂1+𝑓(𝑝)𝑝1𝛽𝑘𝑥𝑘𝑝𝛽𝑘𝜓(𝑀)+𝛽𝑘𝜂(𝑀)+1𝛽𝑘𝑀=𝛽𝑘𝜓(𝑀)+𝛽𝑘(𝑀𝜓(𝑀))+1𝛽𝑘𝑀=𝑀.(3.27) By induction we conclude that (3.25) holds for all 𝑛. Therefore, {𝑥𝑛} is bounded and so are {𝑓(𝑥𝑛)}, {𝑦𝑖𝑛}, {𝑇𝑖(𝑡𝑛)𝑥𝑛}.
For each 𝑖=1,,𝑁 and 𝑛, define the mapping 𝑈(𝑡𝑛)=(1/𝑁)𝑁𝑖=1𝑆𝑖(𝑡𝑛), where 𝑆𝑖(𝑡𝑛)=𝛼𝑛𝐼+(1𝛼𝑛)𝑇𝑖(𝑡𝑛). Then we rewrite the sequence (3.24) to 𝑥𝑛+1=𝛽𝑛𝑓𝑥𝑛+1𝛽𝑛𝑈𝑡𝑛𝑥𝑛.(3.28) Obviously, each 𝑈(𝑡𝑛) is nonexpansive. Since {𝑥𝑛} is bounded and 𝐸 is reflexive, we may assume that some subsequence {𝑥𝑛𝑗} of {𝑥𝑛} converges weakly to 𝑝. Next we show that 𝑝. Put 𝑥𝑗=𝑥𝑛𝑗, 𝛽𝑗=𝛽𝑛𝑗, and 𝑡𝑗=𝑡𝑛𝑗 for each 𝑗. Fix 𝑡>0. By (3.28) we have 𝑥𝑗=𝑈(𝑡)𝑝[𝑡/𝑡𝑗]1𝑘=0𝑈(𝑘+1)𝑡𝑗𝑥𝑗𝑈𝑘𝑡𝑗𝑥𝑗+𝑈𝑡𝑡𝑗𝑡𝑗𝑥𝑗𝑡𝑈𝑡𝑗𝑡𝑗𝑝+𝑈𝑡𝑡𝑗𝑡𝑗𝑡𝑝𝑈(𝑡)𝑝𝑡𝑗𝑈𝑡𝑗𝑥𝑗𝑥𝑗+1+𝑥𝑗+1+𝑈𝑡𝑝𝑡𝑡𝑗𝑡𝑗=𝑡𝑝𝑝𝑡𝑗𝛽𝑗𝑈𝑡𝑗𝑥𝑗𝑥𝑓𝑗+𝑥𝑗+1+𝑈𝑡𝑝𝑡𝑡𝑗𝑡𝑗𝑝𝑝𝑡𝛽𝑗𝑡𝑗𝑈𝑡𝑗𝑥𝑗𝑥𝑓𝑗+𝑥𝑗+1𝑝+max𝑈(𝑠)𝑝𝑝0𝑠𝑡𝑗.(3.29) So, for all 𝑗, we have limsup𝑗𝑥𝑗𝑈(𝑡)𝑝limsup𝑗𝑥𝑗+1𝑝=limsup𝑗𝑥𝑗𝑝.(3.30)
Since 𝐸 has a weakly sequentially continuous duality mapping satisfying Opials condition, this implies 𝑝=𝑈(𝑡)𝑝. By Lemma 2.4, we have Fix(𝑈(𝑡))=𝑁𝑖=1Fix(𝑇𝑖(𝑡)) for each 𝑡>0. Therefore, 𝑝. In view of the variational inequality (3.2) and the assumption that duality mapping 𝐽 is weakly sequentially continuous, we conclude that limsup𝑛𝑥(𝑓𝐼)𝑞,𝑗𝑛+1𝑞=lim𝑗𝑥(𝑓𝐼)𝑞,𝑗𝑛𝑗+1𝑞=(𝐼𝑓)𝑞,𝑗(𝑝𝑞)0.(3.31)
Finally, we prove that 𝑥𝑛𝑞 as 𝑛. Suppose that 𝑥𝑛𝑞0. Then there exists 𝜖>0 and subsequence {𝑥𝑛𝑗} of {𝑥𝑛} such that 𝑥𝑛𝑗𝑞𝜖 for all 𝑗. Put 𝑥𝑗=𝑥𝑛𝑗, 𝛽𝑗=𝛽𝑛𝑗 and 𝑡𝑗=𝑡𝑛𝑗. By Lemma 2.3 one has 𝑓(𝑥𝑗)𝑓(𝑞)𝑟𝑥𝑗𝑞 for all 𝑗. Now, from (2.2) and (3.28) we have 𝑥𝑗+1𝑞2=(1𝛽𝑛)(𝑈(𝑡𝑗)𝑥𝑗𝑞)+𝛽𝑛(𝑓(𝑥𝑗)𝑞)21𝛽𝑗2𝑈𝑡𝑗𝑥𝑗𝑞2+2𝛽𝑗𝑓𝑥𝑗𝑥𝑞,𝑗𝑗+1𝑞1𝛽𝑗2𝑥𝑗𝑞2+2𝛽𝑛𝑓𝑥𝑗𝑥𝑓(𝑞),𝑗𝑗+1𝑞+2𝛽𝑗𝑥𝑓(𝑞)𝑞,𝑗𝑗+1𝑞1𝛽𝑗2𝑥𝑗𝑞2+2𝛽𝑗𝑟𝑥𝑗𝑥𝑞𝑗+1𝑞+2𝛽𝑛𝑥𝑓(𝑞)𝑞,𝑗𝑗+1𝑞1𝛽𝑗2𝑥𝑗𝑞2+𝛽𝑗𝑟𝑥𝑗𝑞2+𝑥𝑗+1𝑞2+2𝛽𝑗𝑓𝑥(𝑞)𝑞,𝑗𝑗+1=𝑞1𝛽𝑗2+𝛽𝑗𝑟𝑥𝑗𝑞2+𝛽𝑗𝑟𝑥𝑗+1𝑞2+2𝛽𝑗𝑥𝑓(𝑞)𝑞,𝑗𝑗+1.𝑞(3.32) It follows that 𝑥𝑗+11(2𝑟)𝛽𝑗+𝛽2𝑗1𝛽𝑗𝑟𝑥𝑗𝑞2+2𝛽𝑗1𝛽𝑗𝑟𝑥𝑓(𝑞)𝑞,𝑗𝑗+1𝑞1𝛽𝑗𝑟2(1𝑟)𝛽𝑗1𝛽𝑗𝑟𝑥𝑗𝑞2+2𝛽𝑗1𝛽𝑗𝑟𝑓𝑥(𝑞)𝑞,𝑗𝑗+1𝑞+𝛽2𝑗𝑀=12(1𝑟)𝛽𝑗1𝛽𝑗𝑟𝑥𝑗𝑞2+2𝛽𝑗1𝛽𝑗𝑟𝑥𝑓(𝑞)𝑞,𝑗𝑗+1𝑞+𝛽2𝑗𝑀12(1𝑟)𝛽𝑗𝑥𝑗𝑞2+𝛽𝑗2𝑓𝑥1𝑟(𝑞)𝑞,𝑗𝑗+1𝑞+𝛽𝑗𝑀,(3.33) where 𝑀 is a constant.
Let 𝛾𝑗=2(1𝑟)𝛽𝑗 and 𝛿𝑗=𝛽𝑗((2/(1𝑟))𝑓(𝑞)𝑞,𝑗(𝑥𝑗+1𝑞)+𝛽𝑗𝑀). It follows from (3.33) that 𝑥𝑗+1𝑞1𝛾𝑗𝑥𝑗𝑞+𝛿𝑗.(3.34) It is easy to see that 𝛾𝑗0, 𝑗=1𝛾𝑗= and (noting (3.28)) limsup𝑗𝛿𝑗𝛾𝑗1=limsup(1𝑟)2𝑥𝑓(𝑞)𝑞,𝑗𝑗+1+𝑀𝑞𝛽2(1𝑟)𝑗,limsup𝑛1(1𝑟)2𝑓𝑥(𝑞)𝑞,𝑗𝑗+1𝑞0.(3.35) Using Lemma 2.5, we conclude that 𝑥𝑗𝑞0 as 𝑗. It is a contradiction. Therefore, 𝑥𝑛𝑞 as 𝑛. This completes the proof.

If letting 𝛼𝑛=0 for all 𝑛 in Theorem 3.3, then we get the following.

Corollary 3.4. Let 𝐶 be a nonempty closed convex subset of a reflexive and strictly convex Banach space 𝐸 which admits a weakly sequentially continuous duality mapping 𝐽 from 𝐸 into 𝐸. For every 𝑖=1,,𝑁(𝑁1), let 𝒯𝑖={𝑇𝑖(𝑡)𝑡0} be a semigroup of nonexpansive mappings on 𝐶 such that =𝑁𝑖=1Fix(𝒯𝑖) and 𝑓𝐶𝐶 be a generalized contraction on 𝐶. Let {𝛽𝑛}[0,1) and {𝑡𝑛}(0,) be sequences satisfying lim𝑛𝑡𝑛=lim𝑛(𝛽𝑛/𝑡𝑛)=0. Let {𝑥𝑛} be a sequence generated 𝑥𝑛+1=𝛽𝑛𝑓𝑥𝑛+1𝛽𝑛𝑁𝑁𝑖=1𝑇𝑖𝑡𝑛𝑥𝑛,𝑛.(3.36) Then {𝑥𝑛} converges strongly to a point 𝑥, which is the unique solution of variational inequality (3.2).

Remark 3.5. Theorem 3.1 and Corollary 3.2 extend the corresponding ones of Suzuki [2], Xu [3], and Chen and He [5] from one nonexpansive semigroup to a finite family of nonexpansive semigroups. But Theorem 3.3 and Corollary 3.4 are not the extension of Theorem 3.2 of Chen and He [5] since Banach space in Theorem 3.3 and Corollary 3.4 is required to be strictly convex. But if letting 𝑁=1 in Theorem 3.3 and Corollary 3.4, we can remove the restriction on strict convexity and hence they extend Theorem 3.2 of Chen and He [5] from a contraction to a generalized contraction.

Remark 3.6. Our Theorem 3.1 extends and improves Theorems 3.2 and 4.2 of Song and Xu [4] from a nonexpansive semigroup to a finite family of nonexpansive semigroups and a contraction to a generalized contraction. Our conditions on the control sequences are different with ones of Song and Xu [4].

Acknowledgment

This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (Grant Number: 2011-0021821).

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