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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 658905, 11 pages
http://dx.doi.org/10.1155/2012/658905
Research Article

Two General Algorithms for Computing Fixed Points of Nonexpansive Mappings in Banach Spaces

School of Mathematical Sciences, Yancheng Teachers University, Jiangsu, Yancheng 224051, China

Received 16 January 2012; Accepted 23 February 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Shuang Wang. 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

Recently, Yao et al. (2011) introduced two algorithms for solving a system of nonlinear variational inequalities. In this paper, we consider two general algorithms and obtain the extension results for computing fixed points of nonexpansive mappings in Banach spaces. Moreover, the fixed points solve the same system of nonlinear variational inequalities.

1. Introduction

Let 𝑋 be a real Banach space and let 𝐶 be a nonempty closed convex subset of 𝑋. Recall that a mapping 𝑇𝐶𝐶 is said to be nonexpansive if 𝑇𝑥𝑇𝑦𝑥𝑦, for all 𝑥,𝑦𝐶. We denote by Fix(𝑇) the set of fixed points of 𝑇.

Recently, Yao et al. [1] considered the following algorithms:𝑥𝑡=Π𝐶(𝐼𝑡𝐹)Π𝐶(𝐼𝜆𝐴)Π𝐶(𝐼𝜇𝐵)𝑥𝑡,(1.1) and for an arbitrary point 𝑥0𝐶,𝑥𝑛+1=𝛽𝑛𝑥𝑛+1𝛽𝑛Π𝐶𝐼𝛼𝑛𝐹Π𝐶(𝐼𝜆𝐴)Π𝐶(𝐼𝜇𝐵)𝑥𝑛,𝑛0,(1.2) where Π𝐶𝑋𝐶 is a sunny nonexpansive retraction, 𝐹𝐶𝑋 is a strongly positive bounded linear operator and 𝐴,𝐵𝐶𝑋 are 𝛼-inverse-strongly accretive and 𝛽-inverse-strongly accretive operators, respectively. They proved that the {𝑥𝑡} defined by (1.1) and {𝑥𝑛} defined by (1.2) converge strongly to a unique solution 𝑥 of the variational inequality 𝐹(𝑥),𝑗(𝑥𝑧)0. Furthermore, they proved that the above algorithms converge strongly to some solutions of a system of nonlinear inequalities, which involves finding (𝑥,𝑦)𝐶×𝐶 such that𝜆𝐴𝑦+𝑥𝑦,𝑗𝑥𝑥0,𝑥𝐶,𝜇𝐵𝑥+𝑦𝑥,𝑗𝑥𝑦0,𝑥𝐶.(1.3) For related works, please see [25] and the references therein.

In this paper, we introduce two general algorithms (3.3) and (3.22) (defined below) and prove that the proposed algorithms strongly converge to 𝑥Fix(𝑇) which solves the variational inequality 𝐹𝑥,𝑗(𝑥𝑢)0,𝑢Fix(𝑇), where 𝐹𝐶𝑋 is a 𝛽-Lipschitzian and 𝜂-strongly accretive operator. It is worth pointing out that our proofs contain some new techniques.

2. Preliminaries

Let 𝑋 be a real Banach space with norm and let 𝑋 be its dual space. The value of 𝑓𝑋 and 𝑥𝑋 will be denoted by 𝑥,𝑓. For the sequence {𝑥𝑛} in 𝑋, we write 𝑥𝑛𝑥 to indicate that the sequence {𝑥𝑛} converges weakly to 𝑥. 𝑥𝑛𝑥 means that {𝑥𝑛} converges strongly to 𝑥.

Let 𝜂>0, a mapping 𝐹 of 𝐶 into 𝑋 is said to be 𝜂-strongly accretive if there exists 𝑗(𝑥𝑦)𝐽(𝑥𝑦) such that𝐹𝑥𝐹𝑦,𝑗(𝑥𝑦)𝜂𝑥𝑦2,(2.1) for all 𝑥,𝑦𝐶. A mapping 𝐹 from 𝐶 into 𝑋 is said to be 𝛽-Lipschitzian if, for 𝛽>0,𝐹𝑥𝐹𝑦𝛽𝑥𝑦,(2.2) for all 𝑥,𝑦𝐶. From the definition of 𝐹 (see [1]), we note that a strongly positive bounded linear operator 𝐹 is a 𝐹-Lipschitzian and 𝛾-strongly accretive operator.

Let 𝑈={𝑥𝑋𝑥=1}. A Banach space 𝑋 is said to be uniformly convex if for each 𝜖(0,2], there exists 𝛿>0 such that for any 𝑥,𝑦𝑈,𝑥𝑦𝜖implies𝑥+𝑦21𝛿.(2.3) It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space 𝑋 is said to be smooth if the limitlim𝑡0𝑥+𝑡𝑦𝑥𝑡,(2.4) exists for all 𝑥,𝑦𝑈. It is said to be uniformly smooth if the limit (2.4) is attained uniformly for 𝑥,𝑦𝑈. Also, we define a function 𝜌[0,)[0,) called the modulus of smoothness of 𝑋 as follows:1𝜌(𝜏)=sup2(.𝑥+𝑦+𝑥𝑦)1𝑥,𝑦𝑋,𝑥=1,𝑦=𝜏(2.5) It is known that 𝑋 is uniformly smooth if and only if lim𝜏0𝜌(𝜏)/𝜏=0. Let 𝑞 be a fixed real number with 1<𝑞2. Then a Banach space 𝑋 is said to be 𝑞-uniformly smooth if there exists a constant 𝑐>0 such that 𝜌(𝜏)𝑐𝜏𝑞 for all 𝜏>0.

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

Lemma 2.1 (see [6]). Let 𝑞 be a given real number with 1<𝑞2 and let 𝑋 be a 𝑞-uniformly smooth Banach space. Then 𝑥+𝑦𝑞𝑥𝑞+𝑞𝑦,𝐽𝑞(𝑥)+2𝐾𝑦𝑞,(2.6) for all 𝑥,𝑦𝑋, where 𝐾 is the 𝑞-uniformly smooth constant of 𝑋 and 𝐽𝑞 is the generalized duality mapping from 𝑋 into 2𝑋 defined by 𝐽𝑞(𝑥)=𝑓𝑋𝑥,𝑓=𝑥𝑞,𝑓=𝑥𝑞1,(2.7) for all 𝑥𝑋.

Lemma 2.2 (see [7]). Let 𝐶 be a closed convex subset of a smooth Banach space 𝑋, let 𝐷 be a nonempty subset of 𝐶 and Π be a retraction from 𝐶 onto 𝐷. Then Π is sunny and nonexpansive if and only if 𝑢Π(𝑢),𝑗(𝑦Π(𝑢))0,(2.8) for all 𝑢𝐶 and 𝑦𝐷.

Lemma 2.3 (see [8]). Let 𝐶 be a nonempty bounded closed convex subset of a uniformly convex Banach space 𝑋 and let 𝑇 be a nonexpansive mapping of 𝐶 into itself. If {𝑥𝑛} is a sequence of 𝐶 such that 𝑥𝑛𝑥 and 𝑥𝑛𝑇𝑥𝑛0, then 𝑥 is a fixed point of 𝑇.

Lemma 2.4 (see [9, 10]). Let {𝑠𝑛} be a sequence of nonnegative real numbers satisfying 𝑠𝑛+11𝜆𝑛𝑠𝑛+𝜆𝑛𝛿𝑛+𝛾𝑛,𝑛0,(2.9) where {𝜆𝑛}, {𝛿𝑛} and {𝛾𝑛} satisfy the following conditions: (i) {𝜆𝑛}[0,1] and 𝑛=0𝜆𝑛=, (ii) limsup𝑛𝛿𝑛0 or 𝑛=0𝜆𝑛𝛿𝑛<, and (iii) 𝛾𝑛0(𝑛0), 𝑛=0𝛾𝑛<. Then lim𝑛𝑠𝑛=0.

Lemma 2.5 (see [11]). Let {𝑥𝑛} and {𝑧𝑛} be bounded sequences in Banach space 𝐸 and {𝛾𝑛} be a sequence in [0,1] which satisfies the following condition: 0<liminf𝑛𝛾𝑛limsup𝑛𝛾𝑛<1.(2.10) Suppose that 𝑥𝑛+1=𝛾𝑛𝑥𝑛+(1𝛾𝑛)𝑧𝑛,𝑛0, and limsup𝑛(𝑧𝑛+1𝑧𝑛𝑥𝑛+1𝑥𝑛)0. Then lim𝑛𝑧𝑛𝑥𝑛=0.

In addition, we need the following extension of Lemma 2.5 in Wang and Hu [2] in a 2-uniformly smooth Banach space.

Lemma 2.6. Let 𝐶 be a nonempty closed convex subset of a real 2-uniformly smooth Banach space 𝑋. Let 𝐹𝐶𝑋 be a 𝛽-Lipschitzian and 𝜂-strongly accretive operator with 0<𝜂2𝛽𝐾 and 0<𝑡<𝜂/2𝛽2𝐾2. Then 𝑆=(𝐼𝑡𝐹)𝐶𝑋 is a contraction with contraction coefficient 𝜏𝑡=12𝑡(𝜂𝑡𝛽2𝐾2).

Proof. By Lemma 2.1, we have 𝑆𝑥𝑆𝑦2=(𝑥𝑦)𝑡𝐹𝑥𝐹𝑦2=𝑥𝑦22𝑡𝐹𝑥𝐹𝑦,𝑗(𝑥𝑦)+2𝑡2𝐾2𝐹𝑥𝐹𝑦2𝑥𝑦22𝑡𝜂𝑥𝑦2+2𝑡2𝛽2𝐾2𝑥𝑦2=12𝑡𝜂𝑡𝛽2𝐾2𝑥𝑦2,(2.11) for all 𝑥,𝑦𝐶. From 0<𝜂2𝛽𝐾 and 0<𝑡<𝜂/2𝛽2𝐾2, we have 𝑆𝑥𝑆𝑦𝜏𝑡𝑥𝑦,(2.12) where 𝜏𝑡=12𝑡(𝜂𝑡𝛽2𝐾2)(0,1). Hence 𝑆 is a contraction with contraction coefficient 𝜏𝑡.

3. Main Results

Let 𝐶 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝑋. Let 𝑇𝐶𝐶 be a nonexpansive mapping with Fix(𝑇). Let 𝐹𝐶𝑋 be a 𝛽-Lipschitzian and 𝜂-strongly accretive operator with 0<𝜂2𝛽𝐾. Let 𝑡(0,𝜂/2𝛽2𝐾2) and 𝜏𝑡=12𝑡(𝜂𝑡𝛽2𝐾2), consider a mapping 𝑆𝑡 on 𝐶 defined by𝑆𝑡𝑥=Π𝐶𝐼𝑡𝐹𝑇𝑥,𝑥𝐶,(3.1) where Π𝐶 is a sunny nonexpansive retraction from 𝑋 onto 𝐶. It is easy to see that 𝑆𝑡 is a contraction. Indeed, from Lemma 2.6, we have𝑆𝑡𝑥𝑆𝑡𝑦Π𝐶𝐼𝑡𝐹𝑇𝑥Π𝐶𝐼𝑡𝐹𝑇𝑦𝐼𝑡𝐹𝑇𝑥𝐼𝑡𝐹𝑇𝑦𝜏𝑡𝑇𝑥𝑇𝑦𝜏𝑡𝑥𝑦,(3.2) for all 𝑥,𝑦𝐶. Therefore, the following implicit method is well defined:𝑥𝑡=Π𝐶𝐼𝑡𝐹𝑇𝑥𝑡,𝑥𝑡𝐶.(3.3)

Theorem 3.1. The net {𝑥𝑡} generated by the implicit method (3.3) converges in norm, as 𝑡0+ to the unique solution 𝑥Fix(𝑇) of the variational inequality: 𝐹𝑥𝑥,𝑗𝑢0,𝑢Fix(𝑇).(3.4)

Proof. We first show that the solution set of (3.4) is singleton. As a matter of fact, we assume that 𝑥Fix(𝑇) and ̃𝑥Fix(𝑇) both are solutions to (3.4), then 𝐹𝑥𝑥,𝑗̃𝑥0,(3.5)𝐹̃𝑥,𝑗̃𝑥𝑥0.(3.6) Adding (3.5) to (3.6), we get 𝐹𝑥𝑥𝐹̃𝑥,𝑗̃𝑥0.(3.7) The strong accretive of 𝐹 implies that 𝑥=̃𝑥, and the uniqueness is proved. Below we use 𝑥Fix(𝑇) to denote the unique solution of (3.4).
Next, we prove that {𝑥𝑡} is bounded. Taking 𝑢Fix(𝑇), from (3.3) and using Lemma 2.6, we have𝑥𝑡=Π𝑢𝐶𝐼𝑡𝐹𝑇𝑥𝑡Π𝐶𝑢𝐼𝑡𝐹𝑇𝑥𝑡𝐼𝑡𝐹𝑇𝑢𝑡𝐹𝑇𝑢𝐼𝑡𝐹𝑇𝑥𝑡𝐼𝑡𝐹𝑇𝑢+𝑡𝐹𝑢𝜏𝑡𝑥𝑡𝑢+𝑡,𝐹𝑢(3.8) that is, 𝑥𝑡𝑡𝑢1𝜏𝑡.𝐹𝑢(3.9) Observe that lim𝑡0+𝑡1𝜏𝑡=1𝜂.(3.10) From 𝑡0+, we may assume, without loss of generality, that 𝑡𝜂/2𝛽2𝐾2𝜖, where 𝜖 is an arbitrarily small positive number. Thus, we have 𝑡/(1𝜏𝑡) to be continuous, for all 𝑡[0,𝜂/2𝛽2𝐾2𝜖]. Therefore, we obtain 𝑀1𝑡=sup1𝜏𝑡𝜂𝑡0,2𝛽2𝐾2𝜖<+.(3.11) From (3.9) and (3.11), we have {𝑥𝑡} bounded and so is {𝐹𝑇𝑥𝑡}.
On the other hand, from (3.3), we obtain𝑥𝑡𝑇𝑥𝑡=Π𝐶𝐼𝑡𝐹𝑇𝑥𝑡Π𝐶𝑇𝑥𝑡𝐼𝑡𝐹𝑇𝑥𝑡𝑇𝑥𝑡=𝑡𝐹𝑇𝑥𝑡0𝑡0+.(3.12)
Next, we show that {𝑥𝑡} is relatively norm-compact as 𝑡0+. Assume that {𝑡𝑛}(0,𝜂/2𝛽2𝐾2) such that 𝑡𝑛0+ as 𝑛. Put 𝑥𝑛=𝑥𝑡𝑛. It follows from (3.12) that𝑥𝑛𝑇𝑥𝑛0(𝑛).(3.13)
For a given 𝑢Fix(𝑇), by (3.3) and using Lemma 2.2, we have𝑥𝑡𝐼𝑡𝐹𝑇𝑥𝑡𝑥,𝑗𝑡𝑢0.(3.14) By (3.14) and using Lemma 2.6, we have 𝑥𝑡𝑢2=𝑥𝑡𝑥𝑢,𝑗𝑡=𝑥𝑢𝑡𝐼𝑡𝐹𝑇𝑥𝑡𝑥,𝑗𝑡+𝑢𝐼𝑡𝐹𝑇𝑥𝑡𝑥𝑢,𝑗𝑡𝑢𝐼𝑡𝐹𝑇𝑥𝑡𝑥𝑢,𝑗𝑡𝑢𝐼𝑡𝐹𝑇𝑥𝑡𝐼𝑡𝐹𝑥𝑇𝑢,𝑗𝑡𝑢+𝑡𝐹𝑢,𝑗𝑢𝑥𝑡𝜏𝑡𝑥𝑡𝑢2+𝑡𝐹𝑢,𝑗𝑢𝑥𝑡,(3.15) that is, 𝑥𝑡𝑢2𝑡1𝜏𝑡𝐹𝑢,𝑗𝑢𝑥𝑡𝑀1𝐹𝑢,𝑗𝑢𝑥𝑡.(3.16) In particular, 𝑥𝑛𝑢2𝑀1𝐹𝑢,𝑗𝑢𝑥𝑛.(3.17)
Since {𝑥𝑡} is bounded, without loss of generality, we may assume that {𝑥𝑛} converges weakly to a point ̃𝑥. Noticing (3.13) we can use Lemma 2.3 to get ̃𝑥Fix(𝑇). Therefore we can substitute ̃𝑥 for 𝑢 in (3.17) to get𝑥𝑛̃𝑥𝑀1𝐹̃𝑥,𝑗̃𝑥𝑥𝑛.(3.18) Consequently, the weak convergence of {𝑥𝑛} to ̃𝑥 actually implies that 𝑥𝑛̃𝑥. This has proved the relative norm compactness of the net {𝑥𝑡} as 𝑡0+.
We next show that ̃𝑥 solves the variational inequality (3.4). Observe that𝑥𝑡=Π𝐶𝐼𝑡𝐹𝑇𝑥𝑡𝐼𝑡𝐹𝑇𝑥𝑡𝐼𝑡𝐹𝑥𝑡+𝐼𝑡𝐹𝑇𝑥𝑡+𝑥𝑡𝑡𝐹𝑥𝑡𝐹𝑥𝑡=1𝑡Π𝐶𝐼𝑡𝐹𝑇𝑥𝑡𝐼𝑡𝐹𝑇𝑥𝑡𝐼𝑡𝐹𝑥𝑡+𝐼𝑡𝐹𝑇𝑥𝑡.(3.19) For any 𝑢Fix(𝑇), we have 𝐹𝑥𝑡𝑥,𝑗𝑡1𝑢=𝑡Π𝐶𝐼𝑡𝐹𝑇𝑥𝑡𝐼𝑡𝐹𝑇𝑥𝑡𝑥,𝑗𝑡1𝑢𝑡𝐼𝑡𝐹𝑥𝑡𝐼𝑡𝐹𝑇𝑥𝑡𝑥,𝑗𝑡1𝑢𝑡𝑥𝑡𝑇𝑥𝑡𝑥,𝑗𝑡+𝑢𝐹𝑥𝑡𝐹𝑇𝑥𝑡𝑥,𝑗𝑡1𝑢𝑡(𝐼𝑇)𝑥𝑡𝑥(𝐼𝑇)𝑢,𝑗𝑡𝑥𝑢+𝛽𝑡𝑇𝑥𝑡𝑥𝑡𝑢𝛽𝑀2𝑥𝑡𝑇𝑥𝑡,(3.20) where 𝑀2=sup{𝑥𝑡𝑢,𝑡(0,𝜂/2𝛽2𝐾2)}.
Now replacing 𝑡 in (3.20) with 𝑡𝑛 and letting 𝑛, we have𝐹̃𝑥,𝑗(̃𝑥𝑢)0.(3.21) That is, ̃𝑥Fix(𝑇) is a solution of (3.4), hence ̃𝑥=𝑥 by uniqueness. In summary, we have shown that each cluster point of {𝑥𝑡} (at 𝑡0) equals 𝑥. Therefore, 𝑥𝑡𝑥 as 𝑡0.

Theorem 3.2. Let 𝐶 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝑋 with a weakly sequentially continuous duality mapping 𝑗. Let 𝐹𝐶𝑋 be a 𝛽-Lipschitzian and 𝜂-strongly accretive operator with 0<𝜂2𝛽𝐾. Suppose that 𝑇𝐶𝐶 is a nonexpansive mapping with Fix(𝑇). Let Π𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let {𝛼𝑛} and {𝛽𝑛} be two real sequences in (0,1) and satisfy the conditions:(A1)lim𝑛𝛼𝑛=0 and 𝑛=0𝛼𝑛=,(A2)0<liminf𝑛𝛽𝑛limsup𝑛𝛽𝑛<1.
For given 𝑥1𝐶 arbitrarily, let the sequence {𝑥𝑛} be generated by 𝑦𝑛=Π𝐶𝐼𝛼𝑛𝐹𝑇𝑥𝑛,𝑥𝑛+1=𝛽𝑛𝑥𝑛+1𝛽𝑛𝑦𝑛,𝑛0.(3.22) Then the sequence {𝑥𝑛} strongly converges to 𝑥Fix(𝑇) which solves the variational inequality (3.4).

Proof. We proceed with the following steps.
Step 1. We claim that {𝑥𝑛} is bounded. From lim𝑛𝛼𝑛=0, we may assume, without loss of generality, that 0<𝛼𝑛𝜂/2𝛽2𝐾2𝜖 for all 𝑛. In fact, let 𝑢Fix(𝑇), from (3.22) and using Lemma 2.6, we have 𝑦𝑛=Π𝑢𝐶𝐼𝛼𝑛𝐹𝑇𝑥𝑛Π𝐶𝑢𝐼𝛼𝑛𝐹𝑇𝑥𝑛𝐼𝛼𝑛𝐹𝑇𝑢𝛼𝑛𝐹𝑢𝜏𝛼𝑛𝑥𝑛𝑢+𝛼𝑛,𝐹𝑢(3.23) where 𝜏𝛼𝑛=12𝛼𝑛(𝜂𝛼𝑛𝛽2𝐾2)(0,1). Then from (3.22) and (3.23), we obtain 𝑥𝑛+1𝑢𝛽𝑛𝑥𝑛+𝑢1𝛽𝑛𝑦𝑛𝑢𝛽𝑛𝑥𝑛+𝑢1𝛽𝑛𝜏𝛼𝑛𝑥𝑛𝑢+𝛼𝑛𝐹𝑢11𝛽𝑛1𝜏𝛼𝑛𝑥𝑛+𝑢1𝛽𝑛𝛼𝑛𝑥𝐹𝑢max𝑛,𝛼𝑢𝑛𝐹𝑢1𝜏𝛼𝑛.(3.24) By induction, we have 𝑥𝑛𝑥𝑢max1𝑢,𝑀3𝐹𝑢,(3.25) where 𝑀3=sup{𝛼𝑛/(1𝜏𝛼𝑛)0<𝛼𝑛𝜂/2𝛽2𝐾2𝜖}<+. Therefore, {𝑥𝑛} is bounded. We also obtain that {𝑦𝑛} and {𝐹𝑇𝑥𝑛} are bounded.Step 2. We claim that lim𝑛𝑥𝑛𝑦𝑛=0. Observe that 𝑦𝑛+1𝑦𝑛=Π𝐶𝐼𝛼𝑛+1𝐹𝑇𝑥𝑛+1Π𝐶𝐼𝛼𝑛𝐹𝑇𝑥𝑛𝐼𝛼𝑛+1𝐹𝑇𝑥𝑛+1𝐼𝛼𝑛𝐹𝑇𝑥𝑛𝑇𝑥𝑛+1𝑇𝑥𝑛+𝛼𝑛+1𝐹𝑇𝑥𝑛+1+𝛼𝑛𝐹𝑇𝑥𝑛𝑥𝑛+1𝑥𝑛+𝛼𝑛+1𝐹𝑇𝑥𝑛+1+𝛼𝑛𝐹𝑇𝑥𝑛.(3.26) Therefore, we have limsup𝑛𝑦𝑛+1𝑦𝑛𝑥𝑛+1𝑥𝑛0.(3.27) From (3.22), (3.27), and using Lemma 2.5, we have lim𝑛𝑥𝑛𝑦𝑛=0.Step 3. We claim that lim𝑛𝑦𝑛𝑇𝑦𝑛=0. Observe that 𝑦𝑛𝑇𝑦𝑛=Π𝐶𝐼𝛼𝑛𝐹𝑇𝑥𝑛Π𝐶𝑇𝑦𝑛𝑇𝑥𝑛𝑇𝑦𝑛+𝛼𝑛𝐹𝑇𝑥𝑛𝑥𝑛𝑦𝑛+𝛼𝑛𝐹𝑇𝑥𝑛.(3.28) Hence, from Step 2 and lim𝑛𝛼𝑛=0, we have lim𝑛𝑦𝑛𝑇𝑦𝑛=0.(3.29)Step 4. We claim that limsup𝑛𝐹𝑥,𝑗(𝑥𝑦𝑛)0, where 𝑥=lim𝑡0𝑥𝑡 and 𝑥𝑡 is defined by (3.3). Since 𝑦𝑛 is bounded, there exists a subsequence {𝑦𝑛𝑘} of {𝑦𝑛} which converges weakly to 𝜔. From Step 3, we obtain 𝑇𝑦𝑛𝑘𝜔. From Lemma 2.3, we have 𝜔Fix(𝑇). Hence, using Theorem 3.1, we have 𝑥Fix(𝑇) and limsup𝑛𝐹𝑥𝑥,𝑗𝑦𝑛=lim𝑘𝐹𝑥𝑥,𝑗𝑦𝑛𝑘=𝐹𝑥𝑥,𝑗𝜔0.(3.30)Step 5. We claim that {𝑥𝑛} converges strongly to 𝑥Fix(𝑇). From (3.22) and using Lemma 2.2, we have Π𝐶𝐼𝛼𝑛𝐹𝑇𝑥𝑛𝐼𝛼𝑛𝐹𝑇𝑥𝑛𝑦,𝑗𝑛𝑥0.(3.31) Observe that 𝑦𝑛𝑥2=Π𝐶𝐼𝛼𝑛𝐹𝑇𝑥𝑛𝑥𝑦,𝑗𝑛𝑥=Π𝐶𝐼𝛼𝑛𝐹𝑇𝑥𝑛𝐼𝛼𝑛𝐹𝑇𝑥𝑛𝑦,𝑗𝑛𝑥+𝐼𝛼𝑛𝐹𝑇𝑥𝑛𝑥𝑦,𝑗𝑛𝑥𝐼𝛼𝑛𝐹𝑇𝑥𝑛𝑥𝑦,𝑗𝑛𝑥𝐼𝛼𝑛𝐹𝑇𝑥𝑛𝐼𝛼𝑛𝐹𝑇𝑥𝑦,𝑗𝑛𝑥+𝛼𝑛𝐹𝑥𝑥,𝑗𝑦𝑛𝐼𝛼𝑛𝐹𝑇𝑥𝑛𝐼𝛼𝑛𝐹𝑇𝑥𝑦𝑛𝑥+𝛼𝑛𝐹𝑥𝑥,𝑗𝑦𝑛𝜏𝛼𝑛𝑥𝑛𝑥𝑦𝑛𝑥+𝛼𝑛𝐹𝑥𝑥,𝑗𝑦𝑛𝜏2𝛼𝑛2𝑥𝑛𝑥2+12𝑦𝑛𝑥2+𝛼𝑛𝐹𝑥𝑥,𝑗𝑦𝑛,(3.32) that is, 𝑦𝑛𝑥2𝜏𝛼𝑛𝑥𝑛𝑥2+2𝛼𝑛𝐹𝑥𝑥,𝑗𝑦𝑛.(3.33) By (3.22) and (3.33), we have 𝑥𝑛+1𝑥2𝛽𝑛𝑥𝑛𝑥2+1𝛽𝑛𝑦𝑛𝑥2𝛽𝑛𝑥𝑛𝑥2+1𝛽𝑛𝜏𝛼𝑛𝑥𝑛𝑥2+2𝛼𝑛𝐹𝑥𝑥,𝑗𝑦𝑛11𝛽𝑛1𝜏𝛼𝑛𝑥𝑛𝑥2+2𝑀31𝛽𝑛1𝜏𝛼𝑛𝐹𝑥𝑥,𝑗𝑦𝑛=1𝜆𝑛𝑥𝑛𝑥2+𝜆𝑛𝛿𝑛,(3.34) where 𝜆𝑛=(1𝛽𝑛)(1𝜏𝛼𝑛), 𝛿𝑛=2𝑀3𝐹𝑥,𝑗(𝑥𝑦𝑛). It is easy to see that 𝜆𝑛0, 𝑛=1𝜆𝑛= and limsup𝑛𝛿𝑛0. Hence, by Lemma 2.4, the sequence {𝑥𝑛} converges strongly to 𝑥Fix(𝑇). From 𝑥=lim𝑡0𝑥𝑡 and Theorem 3.1, we have 𝑥 to be the unique solution of the variational inequality (3.4).

Taking 𝑇=Π𝐶(𝐼𝜆𝐴)Π𝐶(𝐼𝜇𝐵) and 𝐹=𝐹, where 0<𝜆𝛼/𝐾2 and 0<𝜇𝜂/𝐾2, we obtain the following theorems immediately.

Corollary 3.3 (see [1, Theorem 3.5]). The net {𝑥𝑡} generated by the implicit method (1.1) converges in norm, as 𝑡0+, to the unique solution ̃𝑥 of variational inequalitỹ𝑥Ω,𝐹(̃𝑥),𝑗(̃𝑥𝑧)0,𝑧Ω.(3.35)

Corollary 3.4 (see [1, Theorem 3.7]). Let 𝐶 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝑋 and let Π𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let the mappings 𝐴,𝐵𝐶𝑋 be 𝛼-inverse-strongly accretive and 𝛽-inverse-strongly accretive operators, respectively. Let 𝐹𝐶𝐻 be a strongly positive linear bounded operator with coefficient 𝛾>0. For given 𝑥0𝐶, let the sequence {𝑥𝑛} be generated iteratively by (1.2). Suppose that the sequences {𝛼𝑛} and {𝛽𝑛} satisfy the conditions (A1) and (A2), then {𝑥𝑛} converges strongly to ̃𝑥Ω which solves the variational inequality (3.35).

Acknowledgments

Supported by the Natural Science Foundation of Yancheng Teachers University under Grant (11YCKL009) and Professor and Doctor Foundation of Yancheng Teachers University under Grant (11YSYJB0202).

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