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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 818970, 14 pages
http://dx.doi.org/10.1155/2012/818970
Research Article

Modified Relaxed Extragradient Method for a General System of Variational Inequalities and Nonexpansive Mappings in Banach Spaces

Department of Mathematics, Zhejiang Normal University, Zhejiang 321004, China

Received 15 May 2012; Accepted 8 June 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Yuanheng Wang and Liu Yang. 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

The purpose of this paper is to introduce a new modified relaxed extragradient method and study for finding some common solutions for a general system of variational inequalities with inversestrongly monotone mappings and nonexpansive mappings in the framework of real Banach spaces. By using the demiclosedness principle, it is proved that the iterative sequence defined by the relaxed extragradient method converges strongly to a common solution for the system of variational inequalities and nonexpansive mappings under quite mild conditions.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product , and norm ||||, and 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝑃𝐶 be the projection of 𝐻 onto 𝐶, it is known that projection operator 𝑃𝐶 is nonexpansive and satisfies the following: 𝑥𝑦,𝑃𝐶𝑥𝑃𝐶𝑃𝑦𝐶𝑥𝑃𝐶𝑦2,𝑥,𝑦𝐻.(1.1) Moreover, 𝑃𝐶𝑥 is characterized by the properties 𝑃𝐶𝑥𝐶 and 𝑥𝑃𝐶𝑥,𝑃𝐶𝑥𝑦0 for all 𝑦𝐶.

Let 𝐴𝐶𝐻 be a mapping. Recall that the classical variational inequality, denoted by VI(𝐶,𝐴), is to find 𝑢𝑉 such that 𝐴𝑢,𝑣𝑢0,𝑣𝐶.(1.2)

One can see that the variational inequality (1.2) is equivalent to a fixed point problem. 𝐴 element 𝑥𝐶 is a solution of the variational inequality (1.2) if and only if 𝑥𝐶 is a fixed point of the mapping 𝑃𝐶(𝐼𝜆𝐴), where 𝐼 is the identity mapping and 𝜆>0 is a constant.

Variational inequalities introduced by Stampacchia [1] in the early sixties have had a great impact and influence in the development of almost all branches of pure and applied sciences and have witnessed an explosive growth in theoretical advances, algorithmic development, and so forth; see, for example, [118] and the references therein.

For a monotone mapping 𝐴𝐶𝐻, Noor [2] studied the following problem of finding (𝑥;𝑦)𝐶×𝐶 such that: 𝜆𝐴𝑦+𝑥𝑦,𝑥𝑥0,𝑥𝐶,𝜇𝐴𝑥+𝑦𝑥,𝑥𝑦0,𝑥𝐶,(1.3) where 𝜆,𝜇>0 are constants. If we add up the requirement that 𝑥=𝑦, then the problem (1.3) is reduced to the classical variational inequality (1.2). The problem of finding solutions of (1.3) by using iterative methods has been studied by many authors; see [29] and the references therein.

Recently, some authors also studied the problem of finding a common element of the fixed point set of nonexpansive mappings and the solution set of variational inequalities for 𝛼-inversestrongly monotone mappings in the framework of real Hilbert spaces [10] and Banach spaces [11].

On the other hands, Ceng et al. [12] introduce the following general system of variational inequalities involving two different operators. For two given operators, consider the problem finding 𝑥,𝑦𝐶 such that 𝜆𝐴𝑦+𝑥𝑦,𝑥𝑥0,𝑥𝐶,𝜇𝐵𝑥+𝑦𝑥,𝑥𝑦0,𝑥𝐶,(1.4) where 𝜆,𝜇>0 are constants. To illustrate the applications of this system, we can refer to an example of related nonlinear optimization problem put forward by Zhu and Marcotte [13]. Very recently, Yao et al. [14] extend the system of variational inequality problems (1.4) to Banach spaces.

In the present paper, motivated and inspired by the methods of Ceng et al. [12], Iiduka and Takahashi [10], Qin et al. [11], and Yao et al. [14], we consider the following general system of variational inequalities in Banach spaces.

Let 𝐶 be a nonempty closed convex subset of a real smooth Banach space. Let 𝐴,𝐵𝐶𝐸 be 𝛼-inversestrongly accretive mapping and 𝛽-inverse-strongly accretive mapping. Find (𝑥,𝑦)𝐶×𝐶 such that 𝜆𝐴𝑦+𝑥𝑦,𝑗𝑥𝑥0,𝑥𝐶,𝜇𝐵𝑥+𝑦𝑥,𝑗𝑥𝑦0,𝑥𝐶,(1.5) where 𝜆,𝜇>0 are constants, and 𝑗 is the normalized duality mapping. For more details of 𝑗, one may see Li [15]. In a real Hilbert space, 𝑗=𝐼 is the identity mapping, and the system (1.5) is reduced to (1.4). If we add up the requirement that 𝐴=𝐵, then the problem (1.4) is reduced to the generalized variational inequality (1.3), in particular.

And we consider the problem of finding a common element of the fixed point set of nonexpansive mappings and the solution set of the general system of variational inequalities for 𝛼-inversestrongly monotone mappings in the framework of real Banach spaces. By using the demiclosedness principle, we prove that under quite mild conditions the iterative sequence defined by the relaxed extragradient method converges strongly to a common solution of this system of variational inequalities and nonexpansive mappings. Our results improve and extend the corresponding results announced by other authors, such as [24, 6, 8, 1012, 14].

2. Preliminaries

Let 𝐶 be a nonempty closed convex subset of a Banach space of 𝐸. Let 𝐸 be the dual space of 𝐸, and let , denote the pairing between 𝐸 and 𝐸. For 𝑞>1, the generalized duality mapping 𝐽𝑞𝐸2𝐸 is defined by 𝐽𝑞(𝑥)=𝑓𝐸𝑥,𝑓=𝑥𝑞,𝑓=𝑥𝑞,(2.1) for all 𝑥𝐸. In particular, 𝐽=𝐽2 is called the normalized duality mapping. It is known that 𝐽𝑞(𝑥)=||𝑥||𝑞2𝐽(𝑥) for all 𝑥𝐸. If 𝐸 is a Hilbert space, then 𝐽=𝐼 is the identity mapping. Further, we have the following properties of the generalized duality mapping 𝐽𝑞: (1)𝐽𝑞(𝑥)=||𝑥||𝑞2𝐽2(𝑥) for all 𝑥𝐸 with 𝑥0,(2)𝐽𝑞(𝑡𝑥)=𝑡𝑞1𝐽𝑞(𝑥) for all 𝑥𝐸 and 𝑡[0,),(3)𝐽𝑞(𝑥)=𝐽𝑞(𝑥) for all 𝑥𝐸.

Let 𝑈={𝑥𝐸||𝑥||=1}. 𝐸 is said to be uniformly convex if, for any 𝜖(0,2], there exists 𝛿>0 such that for any 𝑥,𝑦𝑈. ||𝑥𝑦||𝜖 implies ||(𝑥+𝑦)/2||(1𝛿).

It is known that a uniformly convex Banach space is reflexive and strictly convex, 𝐸 is said to be Ĝateaux differentiable if the limit Lim𝑡0𝑥+𝑡𝑦𝑥𝑡(2.2) exists for each 𝑥,𝑦𝑈. In this case, 𝐸 is said to be smooth. The norm of 𝐸 is said to be uniformly Ĝateaux differentiable if, for each 𝑦𝑈, the limit (2.2) is attained uniformly for 𝑥𝑈. The norm of 𝐸 is said to be Fr̂echet differentiable, if, for each 𝑥𝑈, the limit (2.2) is attained uniformly for 𝑦𝑈. The norm of 𝐸 is said to be uniformly Fr̂echet differentiable if the limit (2.2) is attained uniformly for 𝑥,𝑦𝑈. It is well-known that (uniform) Fr̂echet differentiable of the norm of 𝐸 implies (uniform) Ĝateaux differentiability of the norm of 𝐸.

The modulus of smoothness of 𝐸 is defined by 1𝜌(𝜏)=sup2||||||||+||||||||||||||||𝑦||||,𝑥+𝑦𝑥𝑦1𝑥,𝑦𝐸,|𝑥|=1,𝑡(2.3) where 𝜌[0,)[0,) is a function. It is known that a Banach space 𝐸 is uniformly smooth if and only if lim(𝑛)(𝜌(𝑡)/𝑡)=0. Let 𝑞 be a fixed real number with 1<𝑞2. A Banach space 𝐸 is said to be 𝑞 uniformly smooth if there exists a fixed constant 𝑐>0 such that 𝜌(𝑡)𝑐𝑡𝑞, for all 𝑡>0.

Next, we always assume that 𝐸 is a smooth Banach space. Let 𝐶 be a nonempty closed convex subsets of 𝐸. Recall that an operator 𝐴 of 𝐶 into 𝐸 is said to be accretive if 𝐴𝑥𝐴𝑦,𝑗(𝑥𝑦)0,𝑥,𝑦𝐶.(2.4)𝐴 is said to be 𝛼-inversestrongly accretive if there exists a constant 𝛼>0 such that 𝐴𝑥𝐴𝑦,𝑗(𝑥𝑦)𝛼𝐴𝑥𝐴𝑦2,𝑥,𝑦𝐶.(2.5)

Let 𝐷 be a subset of 𝐶 and 𝑄 be a mapping of 𝐶 into 𝐷. Then 𝑄 is said to be sunny if 𝑄(𝑄𝑥+𝑡(𝑥𝑄𝑥))=𝑄𝑥,(2.6) whenever 𝑄𝑥+𝑡(𝑥𝑄𝑥)𝐶 for 𝑥𝐶 and 𝑡0. A subset 𝐷 of 𝐶 is called a sunny nonexpansive retract of 𝐶 if there exists a sunny nonexpansive retraction from 𝐶 onto 𝐷.

In order to prove the main result, we also need the following lemmas. The following Lemma 2.2 describes characterization of sunny nonexpansive retraction on a smooth Banach space.

Lemma 2.1 (see [16]). Let 𝐸 be a real 2 uniformly smooth Banach space with the best smooth constant 𝐾. Then the following inequality holds: 𝑥+𝑦2𝑥2+2𝑦,𝑗𝑥+2𝐾𝑦2,𝑥,𝑦𝐸.(2.7)

Lemma 2.2 (see [17]). Let 𝐶 be a closed convex subset of a smooth Banach space 𝐸, let 𝐷 be a nonempty subset of 𝐶, and let 𝑄 be a retraction form 𝐶 onto 𝐷. Then 𝑄 is sunny and nonexpansive if and only if 𝑢𝑄𝑢,𝑗(𝑦𝑄𝑢)0,(2.8) for all 𝑢𝐶 and 𝑦𝐷.

Lemma 2.3 (see [18]). Let {𝑥𝑛} and {𝑦𝑛} be bounded sequences in a Banach space 𝐸 and a sequence {𝛽𝑛} in [0,1] with 0<liminf𝑛𝛽𝑛limsup𝑛𝛽𝑛<1.(2.9) Suppose that 𝑥𝑛+1=(1𝛽𝑛)𝑦𝑛+𝛽𝑛𝑥𝑛 for all integers 𝑛0 and limsup𝑛||||𝑦𝑛+1𝑦𝑛||||||||𝑥𝑛+1𝑥𝑛||||0.(2.10) Then, lim(𝑛)||𝑦𝑛𝑥𝑛||=0.

Lemma 2.4 (see [19]). Let 𝐶 be a nonempty closed convex subset of a real uniformly smooth Banach space. Let 𝑆1 and 𝑆2 be two nonexpansive mappings from 𝐶 into itself with a common fixed point. Define a mapping 𝑆𝐶𝐶 by 𝑆𝑥=𝛿𝑆1𝑥+(1𝛿)𝑆2𝑥,𝑥𝐶,(2.11) where 𝛿 is a constant in (0,1). Then 𝑆 is nonexpansive and 𝐹(𝑆)=𝐹(𝑆1)𝐹(𝑆2).

Lemma 2.5 (see [20]). Assume that {𝛼𝑛} is a sequence of nonnegative real numbers such that 𝛼𝑛+11𝛾𝑛𝛼𝑛+𝛿𝑛,(2.12) where {𝛾𝑛} is a sequence in (0,1) and {𝛿𝑛} is a sequence such that (a)𝑛=1𝛾𝑛=, (b)lim(𝑛)(𝛿𝑛/𝛾𝑛)0,or𝑛=1|𝛿𝑛|<. Then lim(𝑛)𝛼𝑛=0.

Lemma 2.6. Let 𝐶 be a nonempty closed convex subset of a real 2 uniformly smooth Banach space 𝐸 with the best smooth constant 𝐾. Let the mappings 𝐴,𝐵𝐶𝐸 be 𝛼-inverse strongly accretive and 𝛽-inverse strongly accretive, respectively, and then one has 𝐼𝜆𝐴 and 𝐼𝜇𝐵 are nonexpansive, where 𝜆(0,𝛼/𝐾2), 𝜇(0,𝛽/𝐾2).

Proof. Indeed, for all 𝑥,𝑦𝐶, from Lemma 2.1, we have (𝐼𝜆𝐴)𝑥(𝐼𝜆𝐴)𝑦2=(𝑥𝑦)𝜆(𝐴𝑥𝐴𝑦)2𝑥𝑦22𝜆𝐴𝑥𝐴𝑦,𝑗(𝑥𝑦)+2𝐾2𝜆2𝐴𝑥𝐴𝑦2𝑥𝑦22𝜆𝛼𝐴𝑥𝐴𝑦2+2𝐾2𝜆2𝐴𝑥𝐴𝑦2𝑥𝑦2𝐾+2𝜆2𝜆𝛼𝐴𝑥𝐴𝑦2𝑥𝑦2.(2.13) This shows that 𝐼𝜆𝐴 is nonexpansive mapping, so is 𝐼𝜇𝐵.

Lemma 2.7. Let 𝐶 be a nonempty closed convex subset of a real 2 uniformly smooth Banach space 𝐸. Let 𝑄𝐶 be the sunny nonexpansive retraction from 𝐸 onto 𝐶. Let 𝐴,𝐵𝐶𝐸 be 𝛼-inverse strongly accretive mapping and 𝛽-inverse strongly accretive mapping, respectively. Let 𝐺𝐶𝐶 be a mapping defined by 𝐺(𝑥)=𝑄𝐶𝑄𝐶(𝑥𝜇𝐵𝑥)𝜆𝐴𝑄𝐶(𝑥𝜇𝐵𝑥),𝑥𝐶.(2.14) If 𝜆(0,𝛼/𝐾2), 𝜇(0,𝛽/𝐾2), then 𝐺𝐶𝐶 is nonexpansive.

Proof. For all 𝑥,𝑦𝐶, from Lemma 2.6, we have ||||𝐺||||=𝑄(𝑥)𝐺(𝑦)𝐶𝑄𝐶(𝑥𝜇𝐵𝑥)𝜆𝐴𝑄𝐶(𝑥𝜇𝐵𝑥)𝑄𝐶𝑄𝐶(𝑦𝜇𝐵𝑦)𝜆𝐴𝑄𝐶(𝑄𝑦𝜇𝐵𝑦)𝐶(𝑥𝜇𝐵𝑥)𝜆𝐴𝑄𝐶𝑄(𝑥𝜇𝐵𝑥)𝐶(𝑦𝜇𝐵𝑦)𝜆𝐴𝑄𝐶=((𝑦𝜇𝐵𝑦)𝐼𝜆𝐴)𝑄𝐶(𝐼𝜇𝐵)𝑥(𝐼𝜆𝐴)𝑄𝐶(𝑄𝐼𝜇𝐵)𝑦𝐶(𝐼𝜇𝐵)𝑥𝑄𝐶(𝐼𝜇𝐵)𝑦(𝐼𝜇𝐵)𝑥(𝐼𝜇𝐵)𝑦𝑥𝑦.(2.15) Therefore, from (2.15), we obtain immediately that the mapping 𝐺 is nonexpansive.

Lemma 2.8. Let 𝐶 be a nonempty closed convex subset of a real smooth Banach space 𝐸. Let 𝑄𝐶 be the sunny nonexpansive retraction from 𝐸 onto 𝐶. Let 𝐴,𝐵𝐶𝐸 be two possibly nonlinear mappings. For given 𝑥,𝑦𝐶, (𝑥,𝑦) is a solution of problem (1.5) if and only if 𝑥=𝑄𝐶(𝑦𝜆𝐴𝑦), where 𝑦=𝑄𝐶(𝑥𝜇𝐵𝑥).

Proof. We note that we can rewrite (1.5) as 𝑥𝑦𝜆𝐴𝑦,𝑗𝑥𝑥𝑦0,𝑥𝐶,𝑥𝜆𝐵𝑥,𝑗𝑥𝑦0,𝑥𝐶.(2.16) From Lemma 2.2, we can deduce that (2.16) is equivalent to 𝑥=𝑄𝐶𝑦𝜆𝐴𝑦,𝑦=𝑄𝐶𝑥𝜇𝐵𝑥.(2.17) This completes the proof.

Remark 2.9. From Lemma 2.8, we note that 𝑥=𝑄𝐶[𝑄𝐶(𝑥𝜇𝐵𝑥)𝜆𝐴𝑄𝐶(𝑥𝜇𝐵𝑥)], which implies that 𝑥 is a fixed point of the mappings 𝐺.

3. Main Result

To solve the general system of variation inequality problem (1.5), now we are in a position to state and prove the main result in this paper.

Theorem 3.1. Let 𝐸 be a uniformly convex and 2 uniformly smooth Banach space with the best smooth constant 𝐾, 𝐶 a nonempty closed convex subset of 𝐸, and 𝑄𝐶 be the sunny nonexpansive retraction from 𝐸 onto 𝐶. Let 𝐴,𝐵𝐶𝐸 be 𝛼-inverse strongly accretive mapping and 𝛽-inverse strongly accretive mapping, respectively, and 𝑆𝐶𝐶 a nonexpansive mapping with a fixed point. Assume that 𝐹=𝐹(𝑆)𝐹(𝐺), where 𝐺 is defined as Lemma 2.7. Let {𝑥𝑛} be a sequence generated in the following manner: 𝑥1𝑦=𝑢𝐶,𝑛=𝑄𝐶𝑥𝑛𝜇𝐵𝑥𝑛,𝑥𝑛+1=𝛼𝑛𝑢+𝛽𝑛𝑥𝑛+𝛾𝑛𝛿𝑛𝑆𝑥𝑛+1𝛿𝑛𝑄𝐶𝑦𝑛𝜆𝐴𝑦𝑛,𝑛1,(3.1) where 𝛿𝑛[0,1], 𝜆(0,𝛼/𝐾2), 𝜇(0,𝛽/𝐾2), and {𝛼𝑛},{𝛽𝑛}, and {𝛾𝑛} are three sequences in (0,1), and the following conditions are satisfied (1)𝛼𝑛+𝛽𝑛+𝛾𝑛=1,forall𝑛1, (2)lim(𝑛)𝛼𝑛=0,𝑛=0𝛼𝑛=, (3)0<liminf(𝑛)𝛽𝑛limsup(𝑛)𝛽𝑛<1, (4)lim(𝑛)𝛿𝑛=𝛿(0,1).
Then the sequence {𝑥𝑛} defined by (3.1) converges strongly to 𝑥=𝑄𝐹𝑢, and (𝑥,𝑦) is a solution of the problem (1.5), where 𝑦=𝑄𝐶(𝑥𝜇𝐵𝑥), 𝑄𝐹 is a sunny nonexpansive retraction of 𝐶 onto 𝐹.

Proof. We divide the proof of Theorem 3.1 into six steps.
Step 1. First, we prove that 𝐹 is closed and convex. We know that 𝐹(𝑆) is closed and convex. Next, we show that 𝐹(𝐺) is closed and convex. From Lemma 3.1 and 3.2, we can see that 𝐼𝜆𝐴,𝐼𝜇𝐵,𝐺 are nonexpansive. This shows that 𝐹=𝐹(𝑆)𝐹(𝐺) is closed and convex.
Step 2. Now we prove that the sequences {𝑥𝑛},{𝑦𝑛},{𝑡𝑛},{𝐴𝑦𝑛}, and {𝐵𝑥𝑛} are bounded. Let 𝑥𝐹(𝑆)𝐹(𝐺), and from Remark 2.9, we obtain that 𝑥=𝑄𝐶𝑄𝐶𝑥𝜇𝐵𝑥𝜆𝐴𝑄𝐶𝑥𝜇𝐵𝑥.(3.2) Putting 𝑦=𝑄𝐶(𝑥𝜇𝐵𝑥), we see that 𝑥=𝑄𝐶𝑦𝜆𝐴𝑦.(3.3) Putting 𝑡𝑛=𝛿𝑛𝑆𝑥𝑛+(1𝛿𝑛)𝑄𝐶(𝑦𝑛𝜆𝐴𝑦𝑛) for each 𝑛1, we arrive at 𝑡𝑛𝑥=𝛿𝑛𝑆𝑥𝑛+1𝛿𝑛𝑄𝐶𝑦𝑛𝜆𝐴𝑦𝑛𝑥𝛿𝑛𝑆𝑥𝑛𝑥+1𝛿𝑛𝑄𝐶𝑦𝑛𝜆𝐴𝑦𝑛𝑥𝛿𝑛𝑥𝑛𝑥+1𝛿𝑛𝑄𝐶𝑦𝑛𝜆𝐴𝑦𝑛𝑄𝐶𝑦𝜆𝐴𝑦𝛿𝑛𝑥𝑛𝑥+1𝛿𝑛𝑦𝑛𝑦=𝛿𝑛𝑥𝑛𝑥+1𝛿𝑛𝑄𝐶𝑥𝑛𝜇𝐵𝑥𝑛𝑄𝐶𝑥𝜇𝐵𝑥𝛿𝑛𝑥𝑛𝑥+1𝛿𝑛𝑥𝑛𝑥=𝑥𝑛𝑥.(3.4) Hence, it follows that 𝑥𝑛+1𝑥=𝛼𝑛𝑢+𝛽𝑛𝑥𝑛+𝛾𝑛𝑡𝑛𝑥𝛼𝑛𝑢𝑥+𝛽𝑛𝑥𝑛𝑥+𝛾𝑛𝑡𝑛𝑥𝛼𝑛𝑢𝑥+𝛽𝑛𝑥𝑛𝑥+𝛾𝑛𝑥𝑛𝑥𝛼𝑛𝑢𝑥+1𝛼𝑛𝑥𝑛𝑥𝑥max1𝑥,𝑢𝑥=𝑢𝑥.(3.5) Therefore, {𝑥𝑛} is bounded. Hence {𝑥𝑛},{𝑦𝑛},{𝑡𝑛},{𝐴𝑦𝑛}, and {𝐵𝑥𝑛} are bounded.
On the other hand, we have 𝑡𝑛+1𝑡𝑛=𝛿𝑛+1𝑆𝑥𝑛+1+1𝛿𝑛+1𝑄𝐶𝑦𝑛+1𝜆𝐴𝑦𝑛+1𝛿𝑛𝑆𝑥𝑛+1𝛿𝑛𝑄𝐶𝑦𝑛𝜆𝐴𝑦𝑛=𝛿𝑛+1𝑆𝑥𝑛+1𝑆𝑥𝑛+1𝛿𝑛+1𝑄𝐶𝑦𝑛+1𝜆𝐴𝑦𝑛+1-𝑄𝐶𝑦𝑛𝜆𝐴𝑦𝑛+𝛿𝑛+1𝛿𝑛𝑆𝑥𝑛𝑄𝐶𝑦𝑛𝜆𝐴𝑦𝑛𝛿𝑛+1𝑆𝑥𝑛+1𝑆𝑥𝑛+1𝛿𝑛+1𝑄𝐶𝑦𝑛+1𝜆𝐴𝑦𝑛+1𝑄𝐶𝑦𝑛𝜆𝐴𝑦𝑛+||𝛿𝑛+1𝛿𝑛||𝑆𝑥𝑛𝑄𝐶𝑦𝑛𝜆𝐴𝑦𝑛𝛿𝑛+1𝑥𝑛+1𝑥𝑛+1𝛿𝑛+1𝑦𝑛+1𝑦𝑛+||𝛿𝑛+1𝛿𝑛||𝑆𝑥𝑛𝑄𝐶𝑦𝑛𝜆𝐴𝑦𝑛=𝛿𝑛+1𝑥𝑛+1𝑥𝑛+1𝛿𝑛+1+𝑄𝐶𝑥𝑛+1𝜆𝐵𝑥𝑛+1𝑄𝐶𝑥𝑛𝜆𝐵𝑥𝑛+||𝛿𝑛+1𝛿𝑛||𝑆𝑥𝑛𝑄𝐶𝑦𝑛𝜆𝐴𝑦𝑛𝛿𝑛+1𝑥𝑛+1𝑥𝑛+1𝛿𝑛+1𝑥𝑛+1𝑥𝑛+||𝛿𝑛+1𝛿𝑛||𝑆𝑥𝑛𝑄𝐶𝑦𝑛𝜆𝑛𝐴𝑦𝑛𝑥𝑛+1𝑥𝑛+||𝛿𝑛+1𝛿𝑛||𝑀,(3.6) where 𝑀 is an appropriate constant such that 𝑀sup𝑛1𝑆𝑥𝑛𝑄𝐶𝑦𝑛𝜆𝑛𝐴𝑦𝑛.(3.7)
Step 3. We prove that lim(𝑛)||𝑥𝑛+1𝑥𝑛||=0.
Setting 𝑤𝑛=(𝑥(𝑛+1)𝛽𝑛𝑥𝑛)/(1𝛽𝑛) for each 𝑛1, we see that 𝑥𝑛+1=1𝛽𝑛𝑤𝑛+𝛽𝑛𝑥𝑛,𝑛1.(3.8) Now, we compute ||𝑤𝑛+1𝑤𝑛|| from 𝑤𝑛+1𝑤𝑛=𝛼𝑛+1𝑢+𝛾𝑛+1𝑡𝑛+11𝛽𝑛+1𝛼𝑛𝑢+𝛾𝑛𝑡𝑛1𝛽𝑛=𝛼𝑛+11𝛽𝑛+1𝑢+1𝛽𝑛+1𝛼𝑛+11𝛽𝑛+1𝑡𝑛+1𝛼𝑛1𝛽𝑛𝑢1𝛽𝑛𝛼𝑛1𝛽𝑛𝑡𝑛=𝛼𝑛+11𝛽𝑛+1𝑢𝑡𝑛+1+𝛼𝑛1𝛽𝑛𝑡𝑛𝑢+𝑡𝑛+1𝑡𝑛𝛼𝑛+11𝛽𝑛+1𝑢𝑡𝑛+1+𝛼𝑛1𝛽𝑛𝑡𝑛+𝑡𝑢𝑛+1𝑡𝑛.(3.9) Combining (3.6) and (3.9), we arrive at 𝑤𝑛+1𝑤𝑛𝑥𝑛+1𝑥𝑛𝛼𝑛+11𝛽𝑛+1𝑢𝑡𝑛+1+𝛼𝑛1𝛽𝑛𝑡𝑛+||𝛿𝑢𝑛+1𝛿𝑛||𝑀1.(3.10) It follows from the conditions (1.3), (1.4), and (1.5) that limsup𝑛𝑤𝑛+1𝑤𝑛𝑥𝑛+1𝑥𝑛0.(3.11) Hence, by Lemma 2.3, we obtain that lim(𝑛)||𝑤𝑛𝑥𝑛||=0. Consequently, lim𝑛𝑥𝑛+1𝑥𝑛=lim𝑛1𝛽𝑛𝑤𝑛𝑥𝑛=0.(3.12) On the other hand, it follows from the algorithm (3.1) that 𝑥𝑛+1𝑥𝑛=𝛼𝑛𝑢𝑥𝑛+𝛾𝑛𝑡𝑛𝑥𝑛.(3.13) From the condition (1.3) and formula (3.12), we see that lim𝑛𝑡𝑛𝑥𝑛=0.(3.14)
Step 4. We prove that lim(𝑛)||𝑥𝑛𝑉𝑥𝑛||=0. Define a mapping 𝑉𝐶𝐶 by 𝑉𝑥=𝛿𝑆𝑥+(1𝛿)𝑄𝐶𝑄𝐶(𝐼𝜇𝐵)𝜆𝐴𝑄𝐶(𝐼𝜇𝐵)𝑥,𝑥𝐶,(3.15) where lim(𝑛)𝛿𝑛=𝛿(0,1). From Lemma 2.4, we see that 𝑉 is a nonexpansive mapping with 𝐹𝑄𝐹(𝑉)=𝐹(𝑆)𝐶𝑄𝐶(𝐼𝜇𝐵)𝜆𝐴𝑄𝐶𝐹𝑄(𝐼𝜇𝐵)=𝐹(𝑆)𝐶(𝐼𝜆𝐴)𝑄𝐶𝐹(𝐼𝜇𝐵)=𝐹(𝑆)(𝐺)=𝐹.(3.16) On the other hand, we have 𝑥𝑛𝑉𝑥𝑛𝑥𝑛𝑥𝑛+1+𝑥𝑛+1𝑉𝑥𝑛𝑥𝑛𝑥𝑛+1+𝛼𝑛𝑢𝑛𝑉𝑥𝑛+𝛽𝑛𝑥𝑛𝑉𝑥𝑛+𝛾𝑛𝑡𝑛𝑉𝑥𝑛=𝑥𝑛𝑥𝑛+1+𝛼𝑛𝑢𝑛𝑉𝑥𝑛+𝛽𝑛𝑥𝑛𝑉𝑥𝑛+𝛾𝑛𝛿𝑛𝑆𝑥𝑛+1𝛿𝑛𝑄𝐶𝑦𝑛𝜆𝐴𝑦𝑛𝛿𝑆𝑥𝑛(1𝛿)𝑄𝐶𝑦𝑛𝜆𝐴𝑦𝑛𝑥𝑛𝑥𝑛+1+𝛼𝑛𝑢𝑛𝑉𝑥𝑛+𝛽𝑛𝑥𝑛𝑉𝑥𝑛+𝛾𝑛||𝛿𝑛||𝛿𝑆𝑥𝑛𝑄𝐶𝑦𝑛𝜆𝑛𝑦𝑛𝑥𝑛𝑥𝑛+1+𝛼𝑛𝑢𝑛𝑉𝑥𝑛+𝛽𝑛𝑥𝑛𝑉𝑥𝑛+𝛾𝑛||𝛿𝑛||𝛿𝑀.(3.17) This implies that 1𝛽𝑛𝑥𝑛𝑉𝑥𝑛𝑥𝑛𝑥𝑛+1+𝛼𝑛𝑢𝑛𝑉𝑥𝑛+𝛾𝑛||𝛿𝑛||𝛿𝑀.(3.18) It follows from the conditions (1.3), (1.4), (1.5), and (3.18) that lim𝑛𝑥𝑛𝑉𝑥𝑛=0.(3.19)
Step 5. Next, we show that limsup(𝑛)𝑢𝑥,𝑗(𝑥𝑛𝑥)0.
Let 𝑧𝑡 be the fixed point of the contraction 𝑧𝑡𝑢+(1𝑡)𝑉𝑧𝑡, where 𝑡(0,1). That is, 𝑧𝑡=𝑡𝑢+(1𝑡)𝑉𝑧𝑡. It follows that 𝑧𝑡𝑥𝑛=(1𝑡)𝑉𝑧𝑡𝑥𝑛+𝑡𝑢𝑥𝑛.(3.20) On the other hand, for any 𝑡(0,1), we see that 𝑧𝑡𝑥𝑛2=(1𝑡)𝑉𝑧𝑡𝑥𝑛+𝑡𝑢𝑥𝑛𝑧,𝑗𝑡𝑥𝑛=(1𝑡)𝑉𝑧𝑡𝑥𝑛𝑧,𝑗𝑡𝑥𝑛+𝑡𝑢𝑥𝑛𝑧,𝑗𝑡𝑥𝑛=(1𝑡)𝑉𝑧𝑡𝑉𝑥𝑛𝑧,𝑗𝑡𝑥𝑛+(1𝑡)𝑉𝑥𝑛𝑥𝑛𝑧,𝑗𝑡𝑥𝑛+𝑡𝑢𝑧𝑡𝑧,𝑗𝑡𝑥𝑛𝑧+𝑡𝑡𝑥𝑛𝑧,𝑗𝑡𝑥𝑛𝑧(1𝑡)𝑡𝑥𝑛2+(1𝑡)𝑉𝑥𝑛𝑥𝑛𝑧𝑡𝑥𝑛+𝑡𝑢𝑧𝑡𝑧,𝑗𝑡𝑥𝑛𝑧+𝑡𝑡𝑥𝑛2𝑧𝑡𝑥𝑛2+𝑉𝑥𝑛𝑥𝑛𝑧𝑡𝑥𝑛+𝑡𝑢𝑧𝑡𝑧,𝑗𝑡𝑥𝑛.(3.21) It follows that 𝑧𝑡𝑧𝑢,𝑗𝑡𝑥𝑛1𝑡𝑉𝑥𝑛𝑥𝑛𝑧𝑡𝑥𝑛,𝑡(0,1).(3.22) In view of (3.19), we see that limsup𝑛𝑧𝑡𝑧𝑢,𝑗𝑡𝑥𝑛0.(3.23) On the other hand, we see that 𝑄𝐹(𝑉)𝑢=lim(𝑡0)𝑧𝑡 and 𝐹(𝑉)=𝐹. It follows that 𝑧𝑡𝑥=𝑄𝐹𝑢 as 𝑡0. Owing to the fact that 𝑗 is strong to weak* uniformly continuous on bounded subsets of 𝐸, we see that lim𝑡0|||𝑢𝑥𝑥,𝑗𝑛𝑥𝑧𝑡𝑧𝑢,𝑗𝑡𝑥𝑛||||||𝑢𝑥𝑥,𝑗𝑛𝑥𝑢𝑥𝑥,𝑗𝑛𝑧𝑡|||+|||𝑢𝑥𝑥,𝑗𝑛𝑧𝑡𝑧𝑡𝑥𝑧,𝑗𝑡𝑥𝑛||||||𝑢𝑥𝑥,𝑗𝑛𝑥𝑥𝑗𝑛𝑧𝑡|||+|||𝑧𝑡𝑥𝑥,𝑗𝑛𝑧𝑡|||𝑢𝑥𝑗𝑥𝑛𝑥𝑥𝑗𝑛𝑧𝑡+𝑧𝑡𝑥𝑥𝑛𝑧𝑡=0.(3.24) Hence, for any 𝜀>0, there exists 𝛿>0 such that forall𝑡(0,𝛿), the following inequality holds: 𝑢𝑥𝑥,𝑗𝑛𝑥𝑧𝑡𝑧𝑢,𝑗𝑡𝑥𝑛+𝜀.(3.25) Since 𝜀 is arbitrary and (3.23), we see that limsup𝑛𝑢𝑥𝑥,𝑗𝑛𝑥0.(3.26)
Step 6. Finally, we show that 𝑥𝑛𝑥 as 𝑛. Observe that 𝑥𝑛+1𝑥2=𝛼𝑛𝑢+𝛽𝑛𝑥𝑛+𝛾𝑛𝑡𝑛𝑥𝑥,𝑗𝑛+1𝑥=𝛼𝑛𝑢𝑥𝑥,𝑗𝑛+1𝑥+𝛽𝑛𝑥𝑛𝑥𝑥,𝑗𝑛+1𝑥+𝛾𝑛𝑡𝑛𝑥𝑥,𝑗𝑛+1𝑥𝛼𝑛𝑢𝑥𝑥,𝑗𝑛+1𝑥+𝛽𝑛𝑥𝑛𝑥𝑥𝑛+1𝑥+𝛾𝑛𝑡𝑛𝑥𝑥𝑛+1𝑥𝛼𝑛𝑢𝑥𝑥,𝑗𝑛+1𝑥+𝛽𝑛𝑥𝑛𝑥𝑥𝑛+1𝑥+𝛾𝑛𝑥𝑛𝑥𝑥𝑛+1𝑥=𝛼𝑛𝑢𝑥𝑥,𝑗𝑛+1𝑥+1𝛼𝑛𝑥𝑛𝑥𝑥𝑛+1𝑥𝛼𝑛𝑢𝑥𝑥,𝑗𝑛+1𝑥+1𝛼𝑛2𝑥𝑛𝑥2+𝑥𝑛+1𝑥2(3.27) which implies that 𝑥𝑛+1𝑥𝑛21𝛼𝑛𝑥𝑛𝑥2+2𝛼𝑛𝑢𝑥𝑥,𝑗𝑛+1𝑥.(3.28) From the conditions (1.3) and (3.26) and applying Lemma 2.5 to (3.28), we obtain that lim𝑛𝑥𝑛𝑥=0.(3.29) This completes the proof.

Remark 3.2. Since 𝐿𝑝 for all 𝑝2 is uniformly convex and 2 uniformly smooth, we see that Theorem 3.1 is applicable to 𝐿𝑝 for all 𝑝2. There are a number of sequences satisfying the restrictions (C1)–(C3), for example, 𝛼𝑛=1/(𝑛+1),𝛽𝑛=𝑛/(2𝑛+1),𝛾𝑛=𝑛2/(2𝑛2+3𝑛+1) for each 𝑛1.

Corollary 3.3. Let 𝐻 be a real Hilbert space and 𝐶 a nonempty closed convex subset of 𝐻. Let 𝐴,𝐵𝐶𝐻 be 𝛼-inverse strongly monotone mapping and 𝛽-inverse strongly monotone mapping, respectively, and 𝑆𝐶𝐶 nonexpansive mappings with a fixed point. Assume that 𝐹=𝐹(𝑆)𝐹(𝐺), where 𝐺=𝑃𝐶[𝑃𝐶(𝑥𝜇𝐵𝑥)𝜆𝐴𝑃𝐶(𝑥𝜇𝐵𝑥)]. Suppose that {𝑥𝑛} is generated by 𝑥1𝑦=𝑢𝐶,𝑛=𝑃𝐶𝑥𝑛𝜇𝐵𝑥𝑛,𝑥𝑛+1=𝛼𝑛𝑢+𝛽𝑛𝑥𝑛+𝛾𝑛𝛿𝑛𝑆𝑥𝑛+1𝛿𝑛𝑃𝐶𝑦𝑛𝜆𝐴𝑦𝑛,𝑛1,(3.30) where 𝛿𝑛[0,1], 𝜆(0,2𝛼), 𝜇(0,2𝛽) and {𝛼𝑛},{𝛽𝑛}, and {𝛾𝑛} are three sequences in (0,1), and the following conditions are satisfied: (1)𝛼𝑛+𝛽𝑛+𝛾𝑛=1,forall𝑛1, (2)lim(𝑛)𝛼𝑛=0,𝑛=0𝛼𝑛=, (3)0<liminf(𝑛)𝛽𝑛limsup(𝑛)𝛽𝑛<1, (4)lim(𝑛)𝛿𝑛=𝛿(0,1).
Then the sequence {𝑥𝑛} defined by (3.30) converges strongly to 𝑥=𝑃𝐹𝑢, and (𝑥,𝑦) is a solution of the problem (1.4), where 𝑦=𝑃𝐶(𝑥𝜇𝐵𝑥), 𝑃𝐶 is the projection of 𝐻 onto 𝐶, and 𝑃𝐹 is the projection of 𝐶 onto 𝐹.

Remark 3.4. Theorem 3.1 and Corollary 3.3 improve and extend the corresponding results announced by other authors, such as [24, 6, 8, 1012, 14].

Acknowledgments

The authors would like to thank editors and referees for many useful comments and suggestions for the improvement of the paper. This work was partially supported by the Natural Science Foundation of Zhejiang Province (Y6100696) and NSFC (11071169).

References

  1. G. Stampacchia, “Formes bilinéaires coercitives sur les ensembles convexes,” Comptes Rendus de l'Académie des Sciences, vol. 258, pp. 4413–4416, 1964. View at Zentralblatt MATH
  2. M. A. Noor, “Some algorithms for general monotone mixed variational inequalities,” Mathematical and Computer Modelling, vol. 29, no. 7, pp. 1–9, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. Y. Yao and M. A. Noor, “On viscosity iterative methods for variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 776–787, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. M. A. Noor, “New approximation schemes for general variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 217–229, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. M. Aslam Noor, “Some developments in general variational inequalities,” Applied Mathematics and Computation, vol. 152, no. 1, pp. 199–277, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. R. U. Verma, “Generalized system for relaxed cocoercive variational inequalities and projection methods,” Journal of Optimization Theory and Applications, vol. 121, no. 1, pp. 203–210, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. S. S. Chang, H. W. Joseph Lee, and C. K. Chan, “Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces,” Applied Mathematics Letters, vol. 20, no. 3, pp. 329–334, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. L.-C. Ceng and J.-C. Yao, “An extragradient-like approximation method for variational inequality problems and fixed point problems,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 205–215, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. X. Qin, S. M. Kang, and M. Shang, “Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces,” Applicable Analysis, vol. 87, no. 4, pp. 421–430, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. H. Iiduka and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol. 61, no. 3, pp. 341–350, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. X. Qin, S. Y. Cho, and S. M. Kang, “Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications,” Journal of Computational and Applied Mathematics, vol. 233, no. 2, pp. 231–240, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. L.-C. Ceng, C.-y. Wang, and J.-C. Yao, “Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities,” Mathematical Methods of Operations Research, vol. 67, no. 3, pp. 375–390, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. D. L. Zhu and P. Marcotte, “Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities,” SIAM Journal on Optimization, vol. 6, no. 3, pp. 714–726, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. Y. Yao, M. Aslam Noor, K. Inayat Noor, Y.-C. Liou, and H. Yaqoob, “Modified extragradient methods for a system of variational inequalities in Banach spaces,” Acta Applicandae Mathematicae, vol. 110, no. 3, pp. 1211–1224, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. J. Li, “The generalized projection operator on reflexive Banach spaces and its applications,” Journal of Mathematical Analysis and Applications, vol. 306, no. 1, pp. 55–71, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis. Theory, Methods & Applications, vol. 16, no. 12, pp. 1127–1138, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. R. E. Bruck, Jr., “Nonexpansive retracts of Banach spaces,” Bulletin of the American Mathematical Society, vol. 76, pp. 384–386, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. T. Suzuki, “Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227–239, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. R. E. Bruck, Jr., “Properties of fixed-point sets of nonexpansive mappings in Banach spaces,” Transactions of the American Mathematical Society, vol. 179, pp. 251–262, 1973. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. S. Reich, “Asymptotic behavior of contractions in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 44, pp. 57–70, 1973. View at Publisher · View at Google Scholar · View at Zentralblatt MATH