Abstract and Applied Analysis

Abstract and Applied Analysis / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 818970 | 14 pages | https://doi.org/10.1155/2012/818970

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

Academic Editor: Yonghong Yao
Received15 May 2012
Accepted08 Jun 2012
Published05 Jul 2012

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 š» be a real Hilbert space with inner product āŸØā‹…,ā‹…āŸ© and norm ||ā‹…||, and š¶ be a nonempty closed convex subset of š». Let š‘ƒš¶ be the projection of š» onto š¶, it is known that projection operator š‘ƒš¶ is nonexpansive and satisfies the following: āŸØš‘„āˆ’š‘¦,š‘ƒš¶š‘„āˆ’š‘ƒš¶ā€–ā€–š‘ƒš‘¦āŸ©ā‰„š¶š‘„āˆ’š‘ƒš¶š‘¦ā€–ā€–2,āˆ€š‘„,š‘¦āˆˆš».(1.1) Moreover, š‘ƒš¶š‘„ is characterized by the properties š‘ƒš¶š‘„āˆˆš¶ and āŸØš‘„āˆ’š‘ƒš¶š‘„,š‘ƒš¶š‘„āˆ’š‘¦āŸ©ā‰„0 for all š‘¦āˆˆš¶.

Let š“āˆ¶š¶ā†’š» be a mapping. Recall that the classical variational inequality, denoted by VI(š¶,š“), is to find š‘¢āˆˆš‘‰ such that āŸØš“š‘¢,š‘£āˆ’š‘¢āŸ©ā‰„0,āˆ€š‘£āˆˆš¶.(1.2)

One can see that the variational inequality (1.2) is equivalent to a fixed point problem. š“ element š‘„āˆ—āˆˆš¶ is a solution of the variational inequality (1.2) if and only if š‘„āˆ—āˆˆš¶ is a fixed point of the mapping š‘ƒš¶(š¼āˆ’šœ†š“), where š¼ is the identity mapping and šœ†>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, [1ā€“18] and the references therein.

For a monotone mapping š“āˆ¶š¶ā†’š», Noor [2] studied the following problem of finding (š‘„āˆ—;š‘¦āˆ—)āˆˆš¶Ć—š¶ such that: āŸØšœ†š“š‘¦āˆ—+š‘„āˆ—āˆ’š‘¦āˆ—,š‘„āˆ’š‘„āˆ—āŸ©ā‰„0,āˆ€š‘„āˆˆš¶,āŸØšœ‡š“š‘„āˆ—+š‘¦āˆ—āˆ’š‘„āˆ—,š‘„āˆ’š‘¦āˆ—āŸ©ā‰„0,āˆ€š‘„āˆˆš¶,(1.3) where šœ†,šœ‡>0 are constants. If we add up the requirement that š‘„āˆ—=š‘¦āˆ—, 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 [2ā€“9] 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 š›¼-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 š‘„āˆ—,š‘¦āˆ—āˆˆš¶ such that āŸØšœ†š“š‘¦āˆ—+š‘„āˆ—āˆ’š‘¦āˆ—,š‘„āˆ’š‘„āˆ—āŸ©ā‰„0,āˆ€š‘„āˆˆš¶,āŸØšœ‡šµš‘„āˆ—+š‘¦āˆ—āˆ’š‘„āˆ—,š‘„āˆ’š‘¦āˆ—āŸ©ā‰„0,āˆ€š‘„āˆˆš¶,(1.4) where šœ†,šœ‡>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 š¶ be a nonempty closed convex subset of a real smooth Banach space. Let š“,šµāˆ¶š¶ā†’šø be š›¼-inversestrongly accretive mapping and š›½-inverse-strongly accretive mapping. Find (š‘„āˆ—,š‘¦āˆ—)āˆˆš¶Ć—š¶ such that ī«šœ†š“š‘¦āˆ—+š‘„āˆ—āˆ’š‘¦āˆ—ī€·,š‘—š‘„āˆ’š‘„āˆ—ī«ī€øī¬ā‰„0,āˆ€š‘„āˆˆš¶,šœ‡šµš‘„āˆ—+š‘¦āˆ—āˆ’š‘„āˆ—ī€·,š‘—š‘„āˆ’š‘¦āˆ—ī€øī¬ā‰„0,āˆ€š‘„āˆˆš¶,(1.5) where šœ†,šœ‡>0 are constants, and š‘— is the normalized duality mapping. For more details of š‘—, one may see Li [15]. In a real Hilbert space, š‘—=š¼ is the identity mapping, and the system (1.5) is reduced to (1.4). If we add up the requirement that š“=šµ, 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 š›¼-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 [2ā€“4, 6, 8, 10ā€“12, 14].

2. Preliminaries

Let š¶ be a nonempty closed convex subset of a Banach space of šø. Let šøāˆ— be the dual space of šø, and let āŸØā‹…,ā‹…āŸ© denote the pairing between šø and šøāˆ—. For š‘ž>1, the generalized duality mapping š½š‘žāˆ¶šøā†’2šøāˆ— is defined by š½š‘žī€½(š‘„)=š‘“āˆˆšøāˆ—āˆ¶āŸØš‘„,š‘“āŸ©=ā€–š‘„ā€–š‘ž,ā€–š‘“ā€–=ā€–š‘„ā€–š‘žī€¾,(2.1) for all š‘„āˆˆšø. In particular, š½=š½2 is called the normalized duality mapping. It is known that š½š‘ž(š‘„)=||š‘„||š‘žāˆ’2š½(š‘„) for all š‘„āˆˆšø. If šø is a Hilbert space, then š½=š¼ is the identity mapping. Further, we have the following properties of the generalized duality mapping š½š‘ž: (1)š½š‘ž(š‘„)=||š‘„||š‘žāˆ’2š½2(š‘„) for all š‘„āˆˆšø with š‘„ā‰ 0,(2)š½š‘ž(š‘”š‘„)=š‘”š‘žāˆ’1š½š‘ž(š‘„) for all š‘„āˆˆšø and š‘”āˆˆ[0,āˆž),(3)š½š‘ž(āˆ’š‘„)=āˆ’š½š‘ž(š‘„) for all š‘„āˆˆšø.

Let š‘ˆ={š‘„āˆˆšøāˆ¶||š‘„||=1}. šø is said to be uniformly convex if, for any šœ–āˆˆ(0,2], there exists š›æ>0 such that for any š‘„,š‘¦āˆˆš‘ˆ. ||š‘„āˆ’š‘¦||ā‰„šœ– implies ||(š‘„+š‘¦)/2||ā‰¤(1āˆ’š›æ).

It is known that a uniformly convex Banach space is reflexive and strictly convex, šø is said to be GĢ‚ateaux differentiable if the limit Limš‘”ā†’0ā€–š‘„+š‘”š‘¦ā€–āˆ’ā€–š‘„ā€–š‘”(2.2) exists for each š‘„,š‘¦āˆˆš‘ˆ. In this case, šø is said to be smooth. The norm of šø is said to be uniformly GĢ‚ateaux differentiable if, for each š‘¦āˆˆš‘ˆ, the limit (2.2) is attained uniformly for š‘„āˆˆš‘ˆ. The norm of šø is said to be FrĢ‚echet differentiable, if, for each š‘„āˆˆš‘ˆ, the limit (2.2) is attained uniformly for š‘¦āˆˆš‘ˆ. The norm of šø is said to be uniformly FrĢ‚echet differentiable if the limit (2.2) is attained uniformly for š‘„,š‘¦āˆˆš‘ˆ. It is well-known that (uniform) FrĢ‚echet differentiable of the norm of šø implies (uniform) GĢ‚ateaux differentiability of the norm of šø.

The modulus of smoothness of šø is defined by ī‚†1šœŒ(šœ)=sup2ī€·||||||||+||||||||ī€ø||||||||š‘¦||||ī‚‡,š‘„+š‘¦š‘„āˆ’š‘¦āˆ’1āˆ¶š‘„,š‘¦āˆˆšø,|š‘„|=1,ā‰¤š‘”(2.3) where šœŒāˆ¶[0,āˆž)ā†’[0,āˆž) is a function. It is known that a Banach space šø is uniformly smooth if and only if lim(š‘›ā†’āˆž)(šœŒ(š‘”)/š‘”)=0. Let š‘ž be a fixed real number with 1<š‘žā‰¤2. A Banach space šø is said to be š‘ž uniformly smooth if there exists a fixed constant š‘>0 such that šœŒ(š‘”)ā‰¤š‘š‘”š‘ž, for all š‘”>0.

Next, we always assume that šø is a smooth Banach space. Let š¶ be a nonempty closed convex subsets of šø. Recall that an operator š“ of š¶ into šø is said to be accretive if āŸØš“š‘„āˆ’š“š‘¦,š‘—(š‘„āˆ’š‘¦)āŸ©ā‰„0,āˆ€š‘„,š‘¦āˆˆš¶.(2.4)š“ is said to be š›¼-inversestrongly accretive if there exists a constant š›¼>0 such that āŸØš“š‘„āˆ’š“š‘¦,š‘—(š‘„āˆ’š‘¦)āŸ©ā‰„š›¼ā€–š“š‘„āˆ’š“š‘¦ā€–2,āˆ€š‘„,š‘¦āˆˆš¶.(2.5)

Let š· be a subset of š¶ and š‘„ be a mapping of š¶ into š·. Then š‘„ is said to be sunny if š‘„(š‘„š‘„+š‘”(š‘„āˆ’š‘„š‘„))=š‘„š‘„,(2.6) whenever š‘„š‘„+š‘”(š‘„āˆ’š‘„š‘„)āˆˆš¶ for š‘„āˆˆš¶ and š‘”ā‰„0. A subset š· of š¶ is called a sunny nonexpansive retract of š¶ if there exists a sunny nonexpansive retraction from š¶ onto š·.

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 šø be a real 2 uniformly smooth Banach space with the best smooth constant š¾. Then the following inequality holds: ā€–š‘„+š‘¦ā€–2ā‰¤ā€–š‘„ā€–2+2āŸØš‘¦,š‘—š‘„āŸ©+2ā€–š¾š‘¦ā€–2,āˆ€š‘„,š‘¦āˆˆšø.(2.7)

Lemma 2.2 (see [17]). Let š¶ be a closed convex subset of a smooth Banach space šø, let š· be a nonempty subset of š¶, and let š‘„ be a retraction form š¶ onto š·. Then š‘„ is sunny and nonexpansive if and only if āŸØš‘¢āˆ’š‘„š‘¢,š‘—(š‘¦āˆ’š‘„š‘¢)āŸ©ā‰¤0,(2.8) for all š‘¢āˆˆš¶ and š‘¦āˆˆš·.

Lemma 2.3 (see [18]). Let {š‘„š‘›} and {š‘¦š‘›} be bounded sequences in a Banach space šø and a sequence {š›½š‘›} in [0,1] with 0<liminfš‘›ā†’āˆžš›½š‘›ā‰¤limsupš‘›ā†’āˆžš›½š‘›<1.(2.9) Suppose that š‘„š‘›+1=(1āˆ’š›½š‘›)š‘¦š‘›+š›½š‘›š‘„š‘› for all integers š‘›ā‰„0 and limsupš‘›ā†’āˆžī€·||||š‘¦š‘›+1āˆ’š‘¦š‘›||||āˆ’||||š‘„š‘›+1āˆ’š‘„š‘›||||ī€øā‰¤0.(2.10) Then, lim(š‘›ā†’āˆž)||š‘¦š‘›āˆ’š‘„š‘›||=0.

Lemma 2.4 (see [19]). Let š¶ be a nonempty closed convex subset of a real uniformly smooth Banach space. Let š‘†1 and š‘†2 be two nonexpansive mappings from š¶ into itself with a common fixed point. Define a mapping š‘†āˆ¶š¶ā†’š¶ by š‘†š‘„=š›æš‘†1š‘„+(1āˆ’š›æ)š‘†2š‘„,āˆ€š‘„āˆˆš¶,(2.11) where š›æ is a constant in (0,1). Then š‘† is nonexpansive and š¹(š‘†)=š¹(š‘†1)ā‹‚š¹(š‘†2).

Lemma 2.5 (see [20]). Assume that {š›¼š‘›} is a sequence of nonnegative real numbers such that š›¼š‘›+1ā‰¤ī€·1āˆ’š›¾š‘›ī€øš›¼š‘›+š›æš‘›,(2.12) where {š›¾š‘›} is a sequence in (0,1) and {š›æš‘›} is a sequence such that (a)āˆ‘āˆžš‘›=1š›¾š‘›=āˆž, (b)lim(š‘›ā†’āˆž)(š›æš‘›/š›¾š‘›āˆ‘)ā‰¤0,orāˆžš‘›=1|š›æš‘›|<āˆž. Then lim(š‘›ā†’āˆž)š›¼š‘›=0.

Lemma 2.6. Let š¶ be a nonempty closed convex subset of a real 2 uniformly smooth Banach space šø with the best smooth constant š¾. Let the mappings š“,šµāˆ¶š¶ā†’šø be š›¼-inverse strongly accretive and š›½-inverse strongly accretive, respectively, and then one has š¼āˆ’šœ†š“ and š¼āˆ’šœ‡šµ are nonexpansive, where šœ†āˆˆ(0,š›¼/š¾2), šœ‡āˆˆ(0,š›½/š¾2).

Proof. Indeed, for all š‘„,š‘¦āˆˆš¶, from Lemma 2.1, we have ā€–ā€–(š¼āˆ’šœ†š“)š‘„āˆ’(š¼āˆ’šœ†š“)š‘¦2ā€–=ā€–(š‘„āˆ’š‘¦)āˆ’šœ†(š“š‘„āˆ’š“š‘¦)2ā‰¤ā€–š‘„āˆ’š‘¦ā€–2āˆ’2šœ†āŸØš“š‘„āˆ’š“š‘¦,š‘—(š‘„āˆ’š‘¦)āŸ©+2š¾2šœ†2ā€–š“š‘„āˆ’š“š‘¦ā€–2ā‰¤ā€–š‘„āˆ’š‘¦ā€–2āˆ’2šœ†š›¼ā€–š“š‘„āˆ’š“š‘¦ā€–2+2š¾2šœ†2ā€–š“š‘„āˆ’š“š‘¦ā€–2ā‰¤ā€–š‘„āˆ’š‘¦ā€–2ī€·š¾+2šœ†2ī€øšœ†āˆ’š›¼ā€–š“š‘„āˆ’š“š‘¦ā€–2ā‰¤ā€–š‘„āˆ’š‘¦ā€–2.(2.13) This shows that š¼āˆ’šœ†š“ is nonexpansive mapping, so is š¼āˆ’šœ‡šµ.

Lemma 2.7. Let š¶ be a nonempty closed convex subset of a real 2 uniformly smooth Banach space šø. Let š‘„š¶ be the sunny nonexpansive retraction from šø onto š¶. Let š“,šµāˆ¶š¶ā†’šø be š›¼-inverse strongly accretive mapping and š›½-inverse strongly accretive mapping, respectively. Let šŗāˆ¶š¶ā†’š¶ be a mapping defined by šŗ(š‘„)=š‘„š¶ī€ŗš‘„š¶(š‘„āˆ’šœ‡šµš‘„)āˆ’šœ†š“š‘„š¶ī€»(š‘„āˆ’šœ‡šµš‘„),āˆ€š‘„āˆˆš¶.(2.14) If šœ†āˆˆ(0,š›¼/š¾2), šœ‡āˆˆ(0,š›½/š¾2), then šŗāˆ¶š¶ā†’š¶ is nonexpansive.

Proof. For all š‘„,š‘¦āˆˆš¶, from Lemma 2.6, we have ||||šŗ||||=ā€–ā€–š‘„(š‘„)āˆ’šŗ(š‘¦)š¶ī€ŗš‘„š¶(š‘„āˆ’šœ‡šµš‘„)āˆ’šœ†š“š‘„š¶ī€»(š‘„āˆ’šœ‡šµš‘„)āˆ’š‘„š¶ī€ŗš‘„š¶(š‘¦āˆ’šœ‡šµš‘¦)āˆ’šœ†š“š‘„š¶(ī€»ā€–ā€–ā‰¤ā€–ā€–š‘„š‘¦āˆ’šœ‡šµš‘¦)š¶(š‘„āˆ’šœ‡šµš‘„)āˆ’šœ†š“š‘„š¶ī€ŗš‘„(š‘„āˆ’šœ‡šµš‘„)āˆ’š¶(š‘¦āˆ’šœ‡šµš‘¦)āˆ’šœ†š“š‘„š¶ī€»ā€–ā€–=ā€–ā€–((š‘¦āˆ’šœ‡šµš‘¦)š¼āˆ’šœ†š“)š‘„š¶(š¼āˆ’šœ‡šµ)š‘„āˆ’(š¼āˆ’šœ†š“)š‘„š¶(ā€–ā€–ā‰¤ā€–ā€–š‘„š¼āˆ’šœ‡šµ)š‘¦š¶(š¼āˆ’šœ‡šµ)š‘„āˆ’š‘„š¶ā€–ā€–ā€–(š¼āˆ’šœ‡šµ)š‘¦ā‰¤ā€–(š¼āˆ’šœ‡šµ)š‘„āˆ’(š¼āˆ’šœ‡šµ)š‘¦ā‰¤ā€–š‘„āˆ’š‘¦ā€–.(2.15) Therefore, from (2.15), we obtain immediately that the mapping šŗ is nonexpansive.

Lemma 2.8. Let š¶ be a nonempty closed convex subset of a real smooth Banach space šø. Let š‘„š¶ be the sunny nonexpansive retraction from šø onto š¶. Let š“,šµāˆ¶š¶ā†’šø be two possibly nonlinear mappings. For given š‘„āˆ—,š‘¦āˆ—āˆˆš¶, (š‘„āˆ—,š‘¦āˆ—) is a solution of problem (1.5) if and only if š‘„āˆ—=š‘„š¶(š‘¦āˆ—āˆ’šœ†š“š‘¦āˆ—), where š‘¦āˆ—=š‘„š¶(š‘„āˆ—āˆ’šœ‡šµš‘„āˆ—).

Proof. We note that we can rewrite (1.5) as ī«š‘„āˆ—āˆ’ī€·š‘¦āˆ—āˆ’šœ†š“š‘¦āˆ—ī€øī€·,š‘—š‘„āˆ’š‘„āˆ—ī«š‘¦ī€øī¬ā‰„0,āˆ€š‘„āˆˆš¶,āˆ—āˆ’ī€·š‘„āˆ—āˆ’šœ†šµš‘„āˆ—ī€øī€·,š‘—š‘„āˆ’š‘¦āˆ—ī€øī¬ā‰„0,āˆ€š‘„āˆˆš¶.(2.16) From Lemma 2.2, we can deduce that (2.16) is equivalent to š‘„āˆ—=š‘„š¶ī€·š‘¦āˆ—āˆ’šœ†š“š‘¦āˆ—ī€ø,š‘¦āˆ—=š‘„š¶ī€·š‘„āˆ—āˆ’šœ‡šµš‘„āˆ—ī€ø.(2.17) This completes the proof.

Remark 2.9. From Lemma 2.8, we note that š‘„āˆ—=š‘„š¶[š‘„š¶(š‘„āˆ—āˆ’šœ‡šµš‘„āˆ—)āˆ’šœ†š“š‘„š¶(š‘„āˆ—āˆ’šœ‡šµš‘„āˆ—)], which implies that š‘„āˆ— is a fixed point of the mappings šŗ.

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 šø be a uniformly convex and 2 uniformly smooth Banach space with the best smooth constant š¾, š¶ a nonempty closed convex subset of šø, and š‘„š¶ be the sunny nonexpansive retraction from šø onto š¶. Let š“,šµāˆ¶š¶ā†’šø be š›¼-inverse strongly accretive mapping and š›½-inverse strongly accretive mapping, respectively, and š‘†āˆ¶š¶ā†’š¶ a nonexpansive mapping with a fixed point. Assume that ā‹‚š¹=š¹(š‘†)š¹(šŗ)ā‰ āˆ…, where šŗ is defined as Lemma 2.7. Let {š‘„š‘›} be a sequence generated in the following manner: š‘„1š‘¦=š‘¢āˆˆš¶,š‘›=š‘„š¶ī€·š‘„š‘›āˆ’šœ‡šµš‘„š‘›ī€ø,š‘„š‘›+1=š›¼š‘›š‘¢+š›½š‘›š‘„š‘›+š›¾š‘›ī€ŗš›æš‘›š‘†š‘„š‘›+ī€·1āˆ’š›æš‘›ī€øš‘„š¶ī€·š‘¦š‘›āˆ’šœ†š“š‘¦š‘›ī€øī€»,š‘›ā‰„1,(3.1) where š›æš‘›āŠ‚[0,1], šœ†āˆˆ(0,š›¼/š¾2), šœ‡āˆˆ(0,š›½/š¾2), and {š›¼š‘›},{š›½š‘›}, and {š›¾š‘›} are three sequences in (0,1), and the following conditions are satisfied (1)š›¼š‘›+š›½š‘›+š›¾š‘›=1,forallš‘›ā‰„1, (2)lim(š‘›ā†’āˆž)š›¼š‘›āˆ‘=0,āˆžš‘›=0š›¼š‘›=āˆž, (3)0<liminf(š‘›ā†’āˆž)š›½š‘›ā‰¤limsup(š‘›ā†’āˆž)š›½š‘›<1, (4)lim(š‘›ā†’āˆž)š›æš‘›=š›æāˆˆ(0,1).
Then the sequence {š‘„š‘›} defined by (3.1) converges strongly to āˆ’š‘„=š‘„š¹š‘¢, and (āˆ’š‘„,āˆ’š‘¦) is a solution of the problem (1.5), where āˆ’š‘¦=š‘„š¶(āˆ’š‘„āˆ’šœ‡šµāˆ’š‘„), š‘„š¹ is a sunny nonexpansive retraction of š¶ onto š¹.

Proof. We divide the proof of Theorem 3.1 into six steps.
Step 1. First, we prove that š¹ is closed and convex. We know that š¹(š‘†) is closed and convex. Next, we show that š¹(šŗ) is closed and convex. From Lemma 3.1 and 3.2, we can see that š¼āˆ’šœ†š“,š¼āˆ’šœ‡šµ,šŗ are nonexpansive. This shows that ā‹‚š¹=š¹(š‘†)š¹(šŗ) is closed and convex.
Step 2. Now we prove that the sequences {š‘„š‘›},{š‘¦š‘›},{š‘”š‘›},{š“š‘¦š‘›}, and {šµš‘„š‘›} are bounded. Let š‘„āˆ—ā‹‚āˆˆš¹(š‘†)š¹(šŗ), and from Remark 2.9, we obtain that š‘„āˆ—=š‘„š¶ī€ŗš‘„š¶ī€·š‘„āˆ—āˆ’šœ‡šµš‘„āˆ—ī€øāˆ’šœ†š“š‘„š¶ī€·š‘„āˆ—āˆ’šœ‡šµš‘„āˆ—ī€øī€».(3.2) Putting š‘¦āˆ—=š‘„š¶(š‘„āˆ—āˆ’šœ‡šµš‘„āˆ—), we see that š‘„āˆ—=š‘„š¶ī€·š‘¦āˆ—āˆ’šœ†š“š‘¦āˆ—ī€ø.(3.3) Putting š‘”š‘›=š›æš‘›š‘†š‘„š‘›+(1āˆ’š›æš‘›)š‘„š¶(š‘¦š‘›āˆ’šœ†š“š‘¦š‘›) for each š‘›ā‰„1, we arrive at ā€–ā€–š‘”š‘›āˆ’š‘„āˆ—ā€–ā€–=ā€–ā€–š›æš‘›š‘†š‘„š‘›+ī€·1āˆ’š›æš‘›ī€øš‘„š¶ī€·š‘¦š‘›āˆ’šœ†š“š‘¦š‘›ī€øāˆ’š‘„āˆ—ā€–ā€–ā‰¤š›æš‘›ā€–ā€–š‘†š‘„š‘›āˆ’š‘„āˆ—ā€–ā€–+ī€·1āˆ’š›æš‘›ī€øā€–ā€–š‘„š¶ī€·š‘¦š‘›āˆ’šœ†š“š‘¦š‘›ī€øāˆ’š‘„āˆ—ā€–ā€–ā‰¤š›æš‘›ā€–ā€–š‘„š‘›āˆ’š‘„āˆ—ā€–ā€–+ī€·1āˆ’š›æš‘›ī€øā€–ā€–š‘„š¶ī€·š‘¦š‘›āˆ’šœ†š“š‘¦š‘›ī€øāˆ’š‘„š¶ī€·š‘¦āˆ—āˆ’šœ†š“š‘¦āˆ—ī€øā€–ā€–ā‰¤š›æš‘›ā€–ā€–š‘„š‘›āˆ’š‘„āˆ—ā€–ā€–+ī€·1āˆ’š›æš‘›ī€øā€–ā€–š‘¦š‘›āˆ’š‘¦āˆ—ā€–ā€–=š›æš‘›ā€–ā€–š‘„š‘›āˆ’š‘„āˆ—ā€–ā€–+ī€·1āˆ’š›æš‘›ī€øā€–ā€–š‘„š¶ī€·š‘„š‘›āˆ’šœ‡šµš‘„š‘›ī€øāˆ’š‘„š¶ī€·š‘„āˆ—āˆ’šœ‡šµš‘„āˆ—ī€øā€–ā€–ā‰¤š›æš‘›ā€–ā€–š‘„š‘›āˆ’š‘„āˆ—ā€–ā€–+ī€·1āˆ’š›æš‘›ī€øā€–ā€–š‘„š‘›āˆ’š‘„āˆ—ā€–ā€–=ā€–ā€–š‘„š‘›āˆ’š‘„āˆ—ā€–ā€–.(3.4) Hence, it follows that ā€–ā€–š‘„š‘›+1āˆ’š‘„āˆ—ā€–ā€–=ā€–ā€–š›¼š‘›š‘¢+š›½š‘›š‘„š‘›+š›¾š‘›š‘”š‘›āˆ’š‘„āˆ—ā€–ā€–ā‰¤š›¼š‘›ā€–š‘¢āˆ’š‘„āˆ—ā€–+š›½š‘›ā€–ā€–š‘„š‘›āˆ’š‘„āˆ—ā€–ā€–+š›¾š‘›ā€–ā€–š‘”š‘›āˆ’š‘„āˆ—ā€–ā€–ā‰¤š›¼š‘›ā€–š‘¢āˆ’š‘„āˆ—ā€–+š›½š‘›ā€–ā€–š‘„š‘›āˆ’š‘„āˆ—ā€–ā€–+š›¾š‘›ā€–ā€–š‘„š‘›āˆ’š‘„āˆ—ā€–ā€–ā‰¤š›¼š‘›ā€–š‘¢āˆ’š‘„āˆ—ī€·ā€–+1āˆ’š›¼š‘›ī€øā€–ā€–š‘„š‘›āˆ’š‘„āˆ—ā€–ā€–ī€½ā€–ā€–š‘„ā‰¤max1āˆ’š‘„āˆ—ā€–ā€–,ā€–š‘¢āˆ’š‘„āˆ—ā€–ī€¾=ā€–š‘¢āˆ’š‘„āˆ—ā€–.(3.5) Therefore, {š‘„š‘›} is bounded. Hence {š‘„š‘›},{š‘¦š‘›},{š‘”š‘›},{š“š‘¦š‘›}, and {šµš‘„š‘›} are bounded.
On the other hand, we have ā€–ā€–š‘”š‘›+1āˆ’š‘”š‘›ā€–ā€–=ā€–ā€–š›æš‘›+1š‘†š‘„š‘›+1+ī€·1āˆ’š›æš‘›+1ī€øš‘„š¶ī€·š‘¦š‘›+1āˆ’šœ†š“š‘¦š‘›+1ī€øāˆ’ī€·š›æš‘›š‘†š‘„š‘›ī€ø+ī€·1āˆ’š›æš‘›ī€øš‘„š¶ī€·š‘¦š‘›āˆ’šœ†š“š‘¦š‘›ī€øā€–ā€–=ā€–ā€–š›æš‘›+1ī€·š‘†š‘„š‘›+1āˆ’š‘†š‘„š‘›ī€ø+ī€·1āˆ’š›æš‘›+1š‘„ī€øī€·š¶ī€·š‘¦š‘›+1āˆ’šœ†š“š‘¦š‘›+1ī€ø-š‘„š¶ī€·š‘¦š‘›āˆ’šœ†š“š‘¦š‘›+ī€·š›æī€øī€øš‘›+1āˆ’š›æš‘›ī€øī€·š‘†š‘„š‘›āˆ’š‘„š¶ī€·š‘¦š‘›āˆ’šœ†š“š‘¦š‘›ā€–ā€–ī€øī€øā‰¤š›æš‘›+1ā€–ā€–ī€·š‘†š‘„š‘›+1āˆ’š‘†š‘„š‘›ī€øā€–ā€–+ī€·1āˆ’š›æš‘›+1ī€øā€–ā€–š‘„š¶ī€·š‘¦š‘›+1āˆ’šœ†š“š‘¦š‘›+1ī€øāˆ’š‘„š¶ī€·š‘¦š‘›āˆ’šœ†š“š‘¦š‘›ī€øā€–ā€–+||š›æš‘›+1āˆ’š›æš‘›||ā‹…ā€–ā€–š‘†š‘„š‘›āˆ’š‘„š¶ī€·š‘¦š‘›āˆ’šœ†š“š‘¦š‘›ī€øā€–ā€–ā‰¤š›æš‘›+1ā€–ā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–ā€–+ī€·1āˆ’š›æš‘›+1ī€øā€–ā€–š‘¦š‘›+1āˆ’š‘¦š‘›ā€–ā€–+||š›æš‘›+1āˆ’š›æš‘›||ā‹…ā€–ā€–š‘†š‘„š‘›āˆ’š‘„š¶ī€·š‘¦š‘›āˆ’šœ†š“š‘¦š‘›ī€øā€–ā€–=š›æš‘›+1ā€–ā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–ā€–+ī€·1āˆ’š›æš‘›+1ī€ø+ā€–ā€–š‘„š¶ī€·š‘„š‘›+1āˆ’šœ†šµš‘„š‘›+1ī€øāˆ’š‘„š¶ī€·š‘„š‘›āˆ’šœ†šµš‘„š‘›ī€øā€–ā€–+||š›æš‘›+1āˆ’š›æš‘›||ā‹…ā€–ā€–š‘†š‘„š‘›āˆ’š‘„š¶ī€·š‘¦š‘›āˆ’šœ†š“š‘¦š‘›ī€øā€–ā€–ā‰¤š›æš‘›+1ā€–ā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–ā€–+ī€·1āˆ’š›æš‘›+1ī€øā€–ā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–ā€–+||š›æš‘›+1āˆ’š›æš‘›||ā‹…ā€–ā€–š‘†š‘„š‘›āˆ’š‘„š¶ī€·š‘¦š‘›āˆ’šœ†š‘›š“š‘¦š‘›ī€øā€–ā€–ā‰¤ā€–ā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–ā€–+||š›æš‘›+1āˆ’š›æš‘›||š‘€,(3.6) where š‘€ is an appropriate constant such that š‘€ā‰„supš‘›ā‰„1ā€–ā€–š‘†š‘„š‘›āˆ’š‘„š¶ī€·š‘¦š‘›āˆ’šœ†š‘›š“š‘¦š‘›ī€øā€–ā€–.(3.7)
Step 3. We prove that lim(š‘›ā†’āˆž)||š‘„š‘›+1āˆ’š‘„š‘›||=0.
Setting š‘¤š‘›=(š‘„(š‘›+1)āˆ’š›½š‘›š‘„š‘›)/(1āˆ’š›½š‘›) for each š‘›ā‰„1, we see that š‘„š‘›+1=ī€·1āˆ’š›½š‘›ī€øš‘¤š‘›+š›½š‘›š‘„š‘›,āˆ€š‘›ā‰„1.(3.8) Now, we compute ||š‘¤š‘›+1āˆ’š‘¤š‘›|| from ā€–ā€–š‘¤š‘›+1āˆ’š‘¤š‘›ā€–ā€–=ā€–ā€–ā€–š›¼š‘›+1š‘¢+š›¾š‘›+1š‘”š‘›+11āˆ’š›½š‘›+1āˆ’š›¼š‘›š‘¢+š›¾š‘›š‘”š‘›1āˆ’š›½š‘›ā€–ā€–ā€–=ā€–ā€–ā€–š›¼š‘›+11āˆ’š›½š‘›+1š‘¢+1āˆ’š›½š‘›+1āˆ’š›¼š‘›+11āˆ’š›½š‘›+1š‘”š‘›+1āˆ’š›¼š‘›1āˆ’š›½š‘›š‘¢āˆ’1āˆ’š›½š‘›āˆ’š›¼š‘›1āˆ’š›½š‘›š‘”š‘›ā€–ā€–ā€–=ā€–ā€–ā€–š›¼š‘›+11āˆ’š›½š‘›+1ī€·š‘¢āˆ’š‘”š‘›+1ī€ø+š›¼š‘›1āˆ’š›½š‘›ī€·š‘”š‘›ī€øāˆ’š‘¢+š‘”š‘›+1āˆ’š‘”š‘›ā€–ā€–ā€–ā‰¤š›¼š‘›+11āˆ’š›½š‘›+1ā€–ā€–š‘¢āˆ’š‘”š‘›+1ā€–ā€–+š›¼š‘›1āˆ’š›½š‘›ā€–ā€–š‘”š‘›ā€–ā€–+ā€–ā€–š‘”āˆ’š‘¢š‘›+1āˆ’š‘”š‘›ā€–ā€–.(3.9) Combining (3.6) and (3.9), we arrive at ā€–ā€–š‘¤š‘›+1āˆ’š‘¤š‘›ā€–ā€–āˆ’ā€–ā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–ā€–ā‰¤š›¼š‘›+11āˆ’š›½š‘›+1ā€–ā€–š‘¢āˆ’š‘”š‘›+1ā€–ā€–+š›¼š‘›1āˆ’š›½š‘›ā€–ā€–š‘”š‘›ā€–ā€–+||š›æāˆ’š‘¢š‘›+1āˆ’š›æš‘›||š‘€1.(3.10) It follows from the conditions (1.3), (1.4), and (1.5) that limsupš‘›ā†’āˆžī€·ā€–ā€–š‘¤š‘›+1āˆ’š‘¤š‘›ā€–ā€–āˆ’ā€–ā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–ā€–ī€øā‰¤0.(3.11) Hence, by Lemma 2.3, we obtain that lim(š‘›ā†’āˆž)||š‘¤š‘›āˆ’š‘„š‘›||=0. Consequently, limš‘›ā†’āˆžā€–ā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–ā€–=limš‘›ā†’āˆžī€·1āˆ’š›½š‘›ī€øā€–ā€–š‘¤š‘›āˆ’š‘„š‘›ā€–ā€–=0.(3.12) On the other hand, it follows from the algorithm (3.1) that š‘„š‘›+1āˆ’š‘„š‘›=š›¼š‘›ī€·š‘¢āˆ’š‘„š‘›ī€ø+š›¾š‘›ī€·š‘”š‘›āˆ’š‘„š‘›ī€ø.(3.13) From the condition (1.3) and formula (3.12), we see that limš‘›ā†’āˆžā€–ā€–š‘”š‘›āˆ’š‘„š‘›ā€–ā€–=0.(3.14)
Step 4. We prove that lim(š‘›ā†’āˆž)||š‘„š‘›āˆ’š‘‰š‘„š‘›||=0. Define a mapping š‘‰āˆ¶š¶ā†’š¶ by š‘‰š‘„=š›æš‘†š‘„+(1āˆ’š›æ)š‘„š¶ī€·š‘„š¶(š¼āˆ’šœ‡šµ)āˆ’šœ†š“š‘„š¶ī€ø(š¼āˆ’šœ‡šµ)š‘„,āˆ€š‘„āˆˆš¶,(3.15) where lim(š‘›ā†’āˆž)š›æš‘›=š›æāˆˆ(0,1). From Lemma 2.4, we see that š‘‰ is a nonexpansive mapping with ī™š¹ī€·š‘„š¹(š‘‰)=š¹(š‘†)š¶ī€·š‘„š¶(š¼āˆ’šœ‡šµ)āˆ’šœ†š“š‘„š¶ī™š¹ī€·š‘„(š¼āˆ’šœ‡šµ)ī€øī€ø=š¹(š‘†)š¶(š¼āˆ’šœ†š“)š‘„š¶ī€øī™š¹(š¼āˆ’šœ‡šµ)=š¹(š‘†)(šŗ)=š¹.(3.16) On the other hand, we have ā€–ā€–š‘„š‘›āˆ’š‘‰š‘„š‘›ā€–ā€–ā‰¤ā€–ā€–š‘„š‘›āˆ’š‘„š‘›+1ā€–ā€–+ā€–ā€–š‘„š‘›+1āˆ’š‘‰š‘„š‘›ā€–ā€–ā‰¤ā€–ā€–š‘„š‘›āˆ’š‘„š‘›+1ā€–ā€–+š›¼š‘›ā€–ā€–š‘¢š‘›āˆ’š‘‰š‘„š‘›ā€–ā€–+š›½š‘›ā€–ā€–š‘„š‘›āˆ’š‘‰š‘„š‘›ā€–ā€–+š›¾š‘›ā€–ā€–š‘”š‘›āˆ’š‘‰š‘„š‘›ā€–ā€–=ā€–ā€–š‘„š‘›āˆ’š‘„š‘›+1ā€–ā€–+š›¼š‘›ā€–ā€–š‘¢š‘›āˆ’š‘‰š‘„š‘›ā€–ā€–+š›½š‘›ā€–ā€–š‘„š‘›āˆ’š‘‰š‘„š‘›ā€–ā€–+š›¾š‘›ā€–ā€–š›æš‘›š‘†š‘„š‘›+ī€·1āˆ’š›æš‘›ī€øš‘„š¶ī€·š‘¦š‘›āˆ’šœ†š“š‘¦š‘›ī€øāˆ’š›æš‘†š‘„š‘›āˆ’(1āˆ’š›æ)š‘„š¶ī€·š‘¦š‘›āˆ’šœ†š“š‘¦š‘›ī€øā€–ā€–ā‰¤ā€–ā€–š‘„š‘›āˆ’š‘„š‘›+1ā€–ā€–+š›¼š‘›ā€–ā€–š‘¢š‘›āˆ’š‘‰š‘„š‘›ā€–ā€–+š›½š‘›ā€–ā€–š‘„š‘›āˆ’š‘‰š‘„š‘›ā€–ā€–+š›¾š‘›||š›æš‘›||ā‹…ā€–ā€–āˆ’š›æš‘†š‘„š‘›āˆ’š‘„š¶ī€·š‘¦š‘›āˆ’šœ†š‘›š‘¦š‘›ī€øā€–ā€–ā‰¤ā€–ā€–š‘„š‘›āˆ’š‘„š‘›+1ā€–ā€–+š›¼š‘›ā€–ā€–š‘¢š‘›āˆ’š‘‰š‘„š‘›ā€–ā€–+š›½š‘›ā€–ā€–š‘„š‘›āˆ’š‘‰š‘„š‘›ā€–ā€–+š›¾š‘›||š›æš‘›||āˆ’š›æš‘€.(3.17) This implies that ī€·1āˆ’š›½š‘›ī€øā€–ā€–š‘„š‘›āˆ’š‘‰š‘„š‘›ā€–ā€–ā‰¤ā€–ā€–š‘„š‘›āˆ’š‘„š‘›+1ā€–ā€–+š›¼š‘›ā€–ā€–š‘¢š‘›āˆ’š‘‰š‘„š‘›ā€–ā€–+š›¾š‘›||š›æš‘›||āˆ’š›æš‘€.(3.18) It follows from the conditions (1.3), (1.4), (1.5), and (3.18) that limš‘›ā†’āˆžā€–ā€–š‘„š‘›āˆ’š‘‰š‘„š‘›ā€–ā€–=0.(3.19)
Step 5. Next, we show that limsup(š‘›ā†’āˆž)āŸØš‘¢āˆ’āˆ’š‘„,š‘—(š‘„š‘›āˆ’āˆ’š‘„)āŸ©ā‰¤0.
Let š‘§š‘” be the fixed point of the contraction š‘§ā†¦š‘”š‘¢+(1āˆ’š‘”)š‘‰š‘§š‘”, where š‘”āˆˆ(0,1). That is, š‘§š‘”=š‘”š‘¢+(1āˆ’š‘”)š‘‰š‘§š‘”. It follows that ā€–ā€–š‘§š‘”āˆ’š‘„š‘›ā€–ā€–=ā€–ā€–ī€·(1āˆ’š‘”)š‘‰š‘§š‘”āˆ’š‘„š‘›ī€øī€·+š‘”š‘¢āˆ’š‘„š‘›ī€øā€–ā€–.(3.20) On the other hand, for any š‘”āˆˆ(0,1), we see that ā€–ā€–š‘§š‘”āˆ’š‘„š‘›ā€–ā€–2=ī«ī€·(1āˆ’š‘”)š‘‰š‘§š‘”āˆ’š‘„š‘›ī€øī€·+š‘”š‘¢āˆ’š‘„š‘›ī€øī€·š‘§,š‘—š‘”āˆ’š‘„š‘›ī«ī€øī¬=(1āˆ’š‘”)š‘‰š‘§š‘”āˆ’š‘„š‘›ī€·š‘§,š‘—š‘”āˆ’š‘„š‘›ī«ī€øī¬+š‘”š‘¢āˆ’š‘„š‘›ī€·š‘§,š‘—š‘”āˆ’š‘„š‘›ī«ī€øī¬=(1āˆ’š‘”)š‘‰š‘§š‘”āˆ’š‘‰š‘„š‘›ī€·š‘§,š‘—š‘”āˆ’š‘„š‘›ī«ī€øī¬+(1āˆ’š‘”)š‘‰š‘„š‘›āˆ’š‘„š‘›ī€·š‘§,š‘—š‘”āˆ’š‘„š‘›ī«ī€øī¬+š‘”š‘¢āˆ’š‘§š‘”ī€·š‘§,š‘—š‘”āˆ’š‘„š‘›ī«š‘§ī€øī¬+š‘”š‘”āˆ’š‘„š‘›ī€·š‘§,š‘—š‘”āˆ’š‘„š‘›ā€–ā€–š‘§ī€øī¬ā‰¤(1āˆ’š‘”)š‘”āˆ’š‘„š‘›ā€–ā€–2ā€–ā€–+(1āˆ’š‘”)š‘‰š‘„š‘›āˆ’š‘„š‘›ā€–ā€–ā‹…ā€–ā€–š‘§š‘”āˆ’š‘„š‘›ā€–ā€–ī«+š‘”š‘¢āˆ’š‘§š‘”ī€·š‘§,š‘—š‘”āˆ’š‘„š‘›ā€–ā€–š‘§ī€øī¬+š‘”š‘”āˆ’š‘„š‘›ā€–ā€–2ā‰¤ā€–ā€–š‘§š‘”āˆ’š‘„š‘›ā€–ā€–2+ā€–ā€–š‘‰š‘„š‘›āˆ’š‘„š‘›ā€–ā€–ā‹…ā€–ā€–š‘§š‘”āˆ’š‘„š‘›ā€–ā€–ī«+š‘”š‘¢āˆ’š‘§š‘”ī€·š‘§,š‘—š‘”āˆ’š‘„š‘›.ī€øī¬(3.21) It follows that ī«š‘§š‘”ī€·š‘§āˆ’š‘¢,š‘—š‘”āˆ’š‘„š‘›ā‰¤1ī€øī¬š‘”ā€–ā€–š‘‰š‘„š‘›āˆ’š‘„š‘›ā€–ā€–ā‹…ā€–ā€–š‘§š‘”āˆ’š‘„š‘›ā€–ā€–,āˆ€š‘”āˆˆ(0,1).(3.22) In view of (3.19), we see that limsupš‘›ā†’āˆžī«š‘§š‘”ī€·š‘§āˆ’š‘¢,š‘—š‘”āˆ’š‘„š‘›ī€øī¬ā‰¤0.(3.23) On the other hand, we see that š‘„š¹(š‘‰)š‘¢=lim(š‘”ā†’0)š‘§š‘” and š¹(š‘‰)=š¹. It follows that š‘§š‘”ā†’āˆ’š‘„=š‘„š¹š‘¢ as š‘”ā†’0. Owing to the fact that š‘— is strong to weak* uniformly continuous on bounded subsets of šø, we see that limš‘”ā†’0|||ī‚¬š‘¢āˆ’āˆ’š‘„ī‚€š‘„,š‘—š‘›āˆ’āˆ’š‘„āˆ’ī«š‘§ī‚ī‚­š‘”ī€·š‘§āˆ’š‘¢,š‘—š‘”āˆ’š‘„š‘›|||ā‰¤|||ī‚¬ī€øī¬š‘¢āˆ’āˆ’š‘„ī‚€š‘„,š‘—š‘›āˆ’āˆ’š‘„āˆ’ī‚¬ī‚ī‚­š‘¢āˆ’āˆ’š‘„ī€·š‘„,š‘—š‘›āˆ’š‘§š‘”ī€øī‚­|||+|||ī‚¬š‘¢āˆ’āˆ’š‘„ī€·š‘„,š‘—š‘›āˆ’š‘§š‘”ī€øī‚­āˆ’ī‚¬š‘§š‘”āˆ’āˆ’š‘„ī€·š‘§,š‘—š‘”āˆ’š‘„š‘›ī€øī‚­|||ā‰¤|||ī‚¬š‘¢āˆ’āˆ’š‘„ī‚€š‘„,š‘—š‘›āˆ’āˆ’š‘„ī‚ī€·š‘„āˆ’š‘—š‘›āˆ’š‘§š‘”ī€øī‚­|||+|||ī‚¬š‘§š‘”āˆ’āˆ’š‘„ī€·š‘„,š‘—š‘›āˆ’š‘§š‘”ī€øī‚­|||ā‰¤ā€–ā€–š‘¢āˆ’āˆ’š‘„ā€–ā€–ā‹…ā€–ā€–š‘—ī‚€š‘„š‘›āˆ’āˆ’š‘„ī‚ī€·š‘„āˆ’š‘—š‘›āˆ’š‘§š‘”ī€øā€–ā€–+ā€–ā€–š‘§š‘”āˆ’āˆ’š‘„ā€–ā€–ā‹…ā€–ā€–š‘„š‘›āˆ’š‘§š‘”ā€–ā€–=0.(3.24) Hence, for any šœ€>0, there exists š›æ>0 such that forallš‘”āˆˆ(0,š›æ), the following inequality holds: ī‚¬š‘¢āˆ’āˆ’š‘„ī‚€š‘„,š‘—š‘›āˆ’āˆ’š‘„ā‰¤ī«š‘§ī‚ī‚­š‘”ī€·š‘§āˆ’š‘¢,š‘—š‘”āˆ’š‘„š‘›ī€øī¬+šœ€.(3.25) Since šœ€ is arbitrary and (3.23), we see that limsupš‘›ā†’āˆžī‚¬š‘¢āˆ’āˆ’š‘„ī‚€š‘„,š‘—š‘›āˆ’āˆ’š‘„ī‚ī‚­ā‰¤0.(3.26)
Step 6. Finally, we show that š‘„š‘›ā†’āˆ’š‘„ as š‘›ā†’āˆž. Observe that ā€–ā€–š‘„š‘›+1āˆ’āˆ’š‘„ā€–ā€–2=ī‚¬š›¼š‘›š‘¢+š›½š‘›š‘„š‘›+š›¾š‘›š‘”š‘›āˆ’āˆ’š‘„ī‚€š‘„,š‘—š‘›+1āˆ’āˆ’š‘„ī‚ī‚­=š›¼š‘›ī‚¬š‘¢āˆ’āˆ’š‘„ī‚€š‘„,š‘—š‘›+1āˆ’āˆ’š‘„ī‚ī‚­+š›½š‘›ī‚¬š‘„š‘›āˆ’āˆ’š‘„ī‚€š‘„,š‘—š‘›+1āˆ’āˆ’š‘„ī‚ī‚­+š›¾š‘›ī‚¬š‘”š‘›āˆ’āˆ’š‘„ī‚€š‘„,š‘—š‘›+1āˆ’āˆ’š‘„ī‚ī‚­ā‰¤š›¼š‘›ī‚¬š‘¢āˆ’āˆ’š‘„ī‚€š‘„,š‘—š‘›+1āˆ’āˆ’š‘„ī‚ī‚­+š›½š‘›ā€–ā€–š‘„š‘›āˆ’āˆ’š‘„ā€–ā€–ā‹…ā€–ā€–š‘„š‘›+1āˆ’āˆ’š‘„ā€–ā€–+š›¾š‘›ā€–ā€–š‘”š‘›āˆ’āˆ’š‘„ā€–ā€–ā‹…ā€–š‘„ā€–ī‚€š‘›+1āˆ’āˆ’š‘„ī‚ā€–ā€–ā‰¤š›¼š‘›ī‚¬š‘¢āˆ’āˆ’š‘„ī‚€š‘„,š‘—š‘›+1āˆ’āˆ’š‘„ī‚ī‚­+š›½š‘›ā€–ā€–š‘„š‘›āˆ’āˆ’š‘„ā€–ā€–ā‹…ā€–ā€–š‘„š‘›+1āˆ’āˆ’š‘„ā€–ā€–+š›¾š‘›ā€–ā€–š‘„š‘›āˆ’āˆ’š‘„ā€–ā€–ā‹…ā€–ā€–ī‚€š‘„š‘›+1āˆ’āˆ’š‘„ī‚ā€–ā€–=š›¼š‘›ī‚¬š‘¢āˆ’āˆ’š‘„ī‚€š‘„,š‘—š‘›+1āˆ’āˆ’š‘„+ī€·ī‚ī‚­1āˆ’š›¼š‘›ī€øā€–ā€–š‘„š‘›āˆ’āˆ’š‘„ā€–ā€–ā‹…ā€–ā€–š‘„š‘›+1āˆ’āˆ’š‘„ā€–ā€–ā‰¤š›¼š‘›ī‚¬š‘¢āˆ’āˆ’š‘„ī‚€š‘„,š‘—š‘›+1āˆ’āˆ’š‘„+ī‚ī‚­1āˆ’š›¼š‘›2ī‚µā€–ā€–š‘„š‘›āˆ’āˆ’š‘„ā€–ā€–2+ā€–ā€–š‘„š‘›+1āˆ’āˆ’š‘„ā€–ā€–2ī‚¶(3.27) which implies that ā€–ā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–ā€–2ā‰¤ī€·1āˆ’š›¼š‘›ī€øā€–ā€–š‘„š‘›āˆ’āˆ’š‘„ā€–ā€–2+2š›¼š‘›ī‚¬š‘¢āˆ’āˆ’š‘„ī‚€š‘„,š‘—š‘›+1āˆ’āˆ’š‘„.ī‚ī‚­(3.28) From the conditions (1.3) and (3.26) and applying Lemma 2.5 to (3.28), we obtain that limš‘›ā†’āˆžā€–ā€–š‘„š‘›āˆ’āˆ’š‘„ā€–ā€–=0.(3.29) This completes the proof.

Remark 3.2. Since šæš‘ for all š‘ā‰„2 is uniformly convex and 2 uniformly smooth, we see that Theorem 3.1 is applicable to šæš‘ for all š‘ā‰„2. There are a number of sequences satisfying the restrictions (C1)ā€“(C3), for example, š›¼š‘›=1/(š‘›+1),š›½š‘›=š‘›/(2š‘›+1),š›¾š‘›=š‘›2/(2š‘›2+3š‘›+1) for each š‘›ā‰„1.

Corollary 3.3. Let š» be a real Hilbert space and š¶ a nonempty closed convex subset of š». Let š“,šµāˆ¶š¶ā†’š» be š›¼-inverse strongly monotone mapping and š›½-inverse strongly monotone mapping, respectively, and š‘†āˆ¶š¶ā†’š¶ nonexpansive mappings with a fixed point. Assume that ā‹‚š¹=š¹(š‘†)š¹(šŗī…ž)ā‰ āˆ…, where šŗī…ž=š‘ƒš¶[š‘ƒš¶(š‘„āˆ’šœ‡šµš‘„)āˆ’šœ†š“š‘ƒš¶(š‘„āˆ’šœ‡šµš‘„)]. Suppose that {š‘„š‘›} is generated by š‘„1š‘¦=š‘¢āˆˆš¶,š‘›=š‘ƒš¶ī€·š‘„š‘›āˆ’šœ‡šµš‘„š‘›ī€ø,š‘„š‘›+1=š›¼š‘›š‘¢+š›½š‘›š‘„š‘›+š›¾š‘›ī€ŗš›æš‘›š‘†š‘„š‘›+ī€·1āˆ’š›æš‘›ī€øš‘ƒš¶ī€·š‘¦š‘›āˆ’šœ†š“š‘¦š‘›ī€øī€»,š‘›ā‰„1,(3.30) where š›æš‘›āŠ‚[0,1], šœ†āˆˆ(0,2š›¼), šœ‡āˆˆ(0,2š›½) and {š›¼š‘›},{š›½š‘›}, and {š›¾š‘›} are three sequences in (0,1), and the following conditions are satisfied: (1)š›¼š‘›+š›½š‘›+š›¾š‘›=1,forallš‘›ā‰„1, (2)lim(š‘›ā†’āˆž)š›¼š‘›āˆ‘=0,āˆžš‘›=0š›¼š‘›=āˆž, (3)0<liminf(š‘›ā†’āˆž)š›½š‘›ā‰¤limsup(š‘›ā†’āˆž)š›½š‘›<1, (4)lim(š‘›ā†’āˆž)š›æš‘›=š›æāˆˆ(0,1).
Then the sequence {š‘„š‘›} defined by (3.30) converges strongly to āˆ’š‘„=š‘ƒš¹š‘¢, and (āˆ’š‘„,āˆ’š‘¦) is a solution of the problem (1.4), where āˆ’š‘¦=š‘ƒš¶(āˆ’š‘„āˆ’šœ‡šµāˆ’š‘„), š‘ƒš¶ is the projection of š» onto š¶, and š‘ƒš¹ is the projection of š¶ onto š¹.

Remark 3.4. Theorem 3.1 and Corollary 3.3 improve and extend the corresponding results announced by other authors, such as [2ā€“4, 6, 8, 10ā€“12, 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, ā€œFormes bilinéaires coercitives sur les ensembles convexes,ā€ Comptes Rendus de l'Académie des Sciences, vol. 258, pp. 4413ā€“4416, 1964. View at: Google Scholar | Zentralblatt MATH
  2. M. A. Noor, ā€œSome algorithms for general monotone mixed variational inequalities,ā€ Mathematical and Computer Modelling, vol. 29, no. 7, pp. 1ā€“9, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  3. Y. Yao and M. A. Noor, ā€œOn viscosity iterative methods for variational inequalities,ā€ Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 776ā€“787, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  4. M. A. Noor, ā€œNew approximation schemes for general variational inequalities,ā€ Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 217ā€“229, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  5. M. Aslam Noor, ā€œSome developments in general variational inequalities,ā€ Applied Mathematics and Computation, vol. 152, no. 1, pp. 199ā€“277, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  6. R. U. Verma, ā€œGeneralized system for relaxed cocoercive variational inequalities and projection methods,ā€ Journal of Optimization Theory and Applications, vol. 121, no. 1, pp. 203ā€“210, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. S. S. Chang, H. W. Joseph Lee, and C. K. Chan, ā€œGeneralized system for relaxed cocoercive variational inequalities in Hilbert spaces,ā€ Applied Mathematics Letters, vol. 20, no. 3, pp. 329ā€“334, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  8. L.-C. Ceng and J.-C. Yao, ā€œAn extragradient-like approximation method for variational inequality problems and fixed point problems,ā€ Applied Mathematics and Computation, vol. 190, no. 1, pp. 205ā€“215, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  9. X. Qin, S. M. Kang, and M. Shang, ā€œGeneralized system for relaxed cocoercive variational inequalities in Hilbert spaces,ā€ Applicable Analysis, vol. 87, no. 4, pp. 421ā€“430, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  10. H. Iiduka and W. Takahashi, ā€œStrong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings,ā€ Nonlinear Analysis. Theory, Methods & Applications, vol. 61, no. 3, pp. 341ā€“350, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  11. X. Qin, S. Y. Cho, and S. M. Kang, ā€œConvergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications,ā€ Journal of Computational and Applied Mathematics, vol. 233, no. 2, pp. 231ā€“240, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  12. L.-C. Ceng, C.-y. Wang, and J.-C. Yao, ā€œStrong convergence theorems by a relaxed extragradient method for a general system of variational inequalities,ā€ Mathematical Methods of Operations Research, vol. 67, no. 3, pp. 375ā€“390, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  13. D. L. Zhu and P. Marcotte, ā€œCo-coercivity and its role in the convergence of iterative schemes for solving variational inequalities,ā€ SIAM Journal on Optimization, vol. 6, no. 3, pp. 714ā€“726, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  14. Y. Yao, M. Aslam Noor, K. Inayat Noor, Y.-C. Liou, and H. Yaqoob, ā€œModified extragradient methods for a system of variational inequalities in Banach spaces,ā€ Acta Applicandae Mathematicae, vol. 110, no. 3, pp. 1211ā€“1224, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  15. J. Li, ā€œThe generalized projection operator on reflexive Banach spaces and its applications,ā€ Journal of Mathematical Analysis and Applications, vol. 306, no. 1, pp. 55ā€“71, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  16. H. K. Xu, ā€œInequalities in Banach spaces with applications,ā€ Nonlinear Analysis. Theory, Methods & Applications, vol. 16, no. 12, pp. 1127ā€“1138, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  17. R. E. Bruck, Jr., ā€œNonexpansive retracts of Banach spaces,ā€ Bulletin of the American Mathematical Society, vol. 76, pp. 384ā€“386, 1970. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  18. T. Suzuki, ā€œStrong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals,ā€ Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227ā€“239, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  19. R. E. Bruck, Jr., ā€œProperties of fixed-point sets of nonexpansive mappings in Banach spaces,ā€ Transactions of the American Mathematical Society, vol. 179, pp. 251ā€“262, 1973. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  20. S. Reich, ā€œAsymptotic behavior of contractions in Banach spaces,ā€ Journal of Mathematical Analysis and Applications, vol. 44, pp. 57ā€“70, 1973. View at: Publisher Site | Google Scholar | Zentralblatt MATH

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.

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