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Abstract and Applied Analysis
VolumeΒ 2011, Article IDΒ 276874, 13 pages
http://dx.doi.org/10.1155/2011/276874
Research Article

Strong Convergence Theorems for Equilibrium Problems and π‘˜-Strict Pseudocontractions in Hilbert Spaces

School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China

Received 23 March 2011; Revised 31 May 2011; Accepted 23 June 2011

Academic Editor: LjubisaΒ Kocinac

Copyright Β© 2011 Dao-Jun Wen. 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 a new iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed point of a finite family of π‘˜-strictly pseudo-contractive nonself-mappings. Strong convergence theorems are established in a real Hilbert space under some suitable conditions. Our theorems presented in this paper improve and extend the corresponding results announced by many others.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product βŸ¨β‹…,β‹…βŸ© and norm β€–β‹…β€–, respectively. Let 𝐾 be a nonempty closed convex subset of 𝐻. Let 𝐹 be a bifunction from 𝐾×𝐾 into ℝ, where ℝ denotes the set of real numbers. We consider the following problem: Find π‘₯∈𝐾 such that 𝐹(π‘₯,𝑦)β‰₯0,βˆ€π‘¦βˆˆπΎ,(1.1) which is called equilibrium problem. We use EP(𝐹) to denote the set of solution of the problem (1.1). Given a mapping π‘‡βˆΆπΎβ†’π», let 𝐹(π‘₯,𝑦)=βŸ¨π‘‡π‘₯,π‘¦βˆ’π‘₯⟩ for all π‘₯,π‘¦βˆˆπΎ. Then, π‘§βˆˆEP(𝐹) if and only if βŸ¨π‘‡π‘§,π‘¦βˆ’π‘§βŸ©β‰₯0 for all π‘¦βˆˆπΎ; that is, 𝑧 is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.1). Some methods have been proposed to solve the equilibrium problem (see, e.g., [1–3]).

Recall that a nonself-mapping π‘‡βˆΆπΎβ†’π» is called a π‘˜-strict pseudocontraction if there exists a constant π‘˜βˆˆ[0,1) such that ‖𝑇π‘₯βˆ’π‘‡π‘¦β€–2≀‖π‘₯βˆ’π‘¦β€–2+π‘˜β€–(πΌβˆ’π‘‡)π‘₯βˆ’(πΌβˆ’π‘‡)𝑦‖2,βˆ€π‘₯,π‘¦βˆˆπΎ.(1.2) We use 𝐹(𝑇) to denote the fixed point set of the mapping 𝑇, that is, 𝐹(𝑇)∢={π‘₯βˆˆπΎβˆΆπ‘‡π‘₯=π‘₯}. As π‘˜=0, 𝑇 is said to be nonexpansive, that is, ‖𝑇π‘₯βˆ’π‘‡π‘¦β€–β‰€β€–π‘₯βˆ’π‘¦β€–,forallπ‘₯,π‘¦βˆˆπΎ. 𝑇 is said to be pseudocontractive if π‘˜=1 and is also said to be strongly pseudocontractive if there exists a positive constant πœ†βˆˆ(0,1) such that 𝑇+πœ†πΌ is pseudocontractive. Clearly, the class of π‘˜-strict pseudocontractions falls into the one between classes of nonexpansive mappings and pseudocontractions. We remark also that the class of strongly pseudocontractive mappings is independent of the class of π‘˜-strict pseudocontractions (see, e.g., [4, 5]).

Iterative methods for equilibrium problem and nonexpansive mappings have been extensively investigated; see, for example, [1–18] and the references therein. However, iterative methods for strict pseudocontractions are far less developed than those for nonexpansive mappings though Browder and Petryshyn [5] initiated their work in 1967; the reason is probably that the second term appearing in the right-hand side of (1.2) impedes the convergence analysis for iterative algorithms used to find a fixed point of the strict pseudocontraction 𝑇. On the other hand, strict pseudocontractions have more powerful applications than nonexpansive mappings do in solving inverse problems (see, e.g., [6]). Therefore, it is interesting to develop the theory of iterative methods for equilibrium problem and strict pseudocontractions.

In 2007, Acedo and Xu [12] proposed the following parallel algorithm for a finite family of π‘˜π‘–-strict pseudocontractions {𝑇𝑖}𝑁𝑖=1 in Hilbert space 𝐻: βˆ€π‘₯0∈𝐾,π‘₯𝑛+1=ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘₯𝑛+𝛼𝑛𝑁𝑖=1πœ†π‘–π‘‡π‘–π‘₯𝑛,(1.3) where {𝛼𝑛}βŠ‚(0,1), and {πœ†π‘–}𝑁𝑖=1 is a finite sequence of positive numbers such that βˆ‘π‘π‘–=1πœ†π‘–=1. They proved that the sequence {π‘₯𝑛} defined by (1.3) converges weakly to a common fixed point of {𝑇𝑖}𝑁𝑖=1 under some appropriate conditions. Moreover, by applying additional projections, they further proved that algorithm can be modified to have strong convergence.

Recently, S. Takahashi and W. Takahashi [13] studied the equilibrium problem and fixed point of nonexpansive self-mappings 𝑇 in Hilbert spaces by a viscosity approximation methods for finding an element of β‹‚EP(𝐹)𝐹(𝑇). Very recently, by using the general approximation method, Qin et al. [14] obtained a strong convergence theorem for finding an element of 𝐹(𝑇). On the other hand, Ceng et al. [16] proposed an iterative scheme for finding an element of β‹‚EP(𝐹)𝐹(𝑇) and then obtained some weak and strong convergence theorems.

In this paper, inspired and motivated by research going in this area, we introduce a modified parallel iteration, which is defined in the following way:𝐹𝑒𝑛+1,π‘¦π‘ŸβŸ¨π‘¦βˆ’π‘’π‘›,π‘’π‘›βˆ’π‘₯π‘›π‘¦βŸ©β‰₯0,βˆ€π‘¦βˆˆπΎ,𝑛=𝛼𝑛𝑒𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘ξ“π‘–=1πœ‚π‘–(𝑛)𝑇𝑖𝑒𝑛,π‘₯𝑛+1=𝛽𝑛𝑒+𝛾𝑛π‘₯𝑛+ξ€·1βˆ’π›½π‘›βˆ’π›Ύπ‘›ξ€Έπ‘¦π‘›,𝑛β‰₯0,(1.4) where π‘’βˆˆπΎ is a given point, {𝑇𝑖}𝑁𝑖=π‘–βˆΆπΎβ†’π» is a finite family of π‘˜π‘–-strictly pseudocontractive nonself-mappings, {πœ‚π‘–(𝑛)}𝑁𝑖=1 is a finite sequences of positive numbers, {𝛼𝑛}, {𝛽𝑛}, and {𝛾𝑛} are some sequences in (0,1).

Our purpose is not only to modify the parallel algorithm (1.3) to the case of equilibrium problems and common fixed point for a finite family of π‘˜π‘–-strictly pseudocontractive nonself-mappings, but also to establish strong convergence theorems in a real Hilbert space under some different conditions. Our theorems presented in this paper improve and extend the main results of [9, 12–14, 16].

2. Preliminaries

Let 𝐾 be a nonempty closed and convex subset of a Hilbert space 𝐻. We use 𝑃𝐾 to denote the metric or nearest point projection of 𝐻 onto 𝐾; that is, for π‘₯∈𝐻, 𝑃𝐾π‘₯ is the only point in 𝐾 such that β€–π‘₯βˆ’π‘ƒπΎπ‘₯β€–=inf{β€–π‘₯βˆ’π‘§β€–βˆΆπ‘§βˆˆπΎ}. we write π‘₯𝑛⇀π‘₯ and π‘₯𝑛→π‘₯ indicate that the sequence {π‘₯𝑛} convergence weakly and strongly to π‘₯, respectively.

It is well known that Hilbert space 𝐻 satisfies Opial's condition [8], that is, for any sequence {π‘₯𝑛} with π‘₯𝑛⇀π‘₯ and every π‘¦βˆˆπ» with 𝑦≠π‘₯, we have liminfπ‘›β†’βˆžβ€–β€–π‘₯π‘›β€–β€–βˆ’π‘₯<liminfπ‘›β†’βˆžβ€–β€–π‘₯𝑛‖‖.βˆ’π‘¦(2.1)

To study the equilibrium problem (1.1), we may assume that the bifunction 𝐹 of 𝐾×𝐾 into ℝ satisfies the following conditions. (A1)𝐹(π‘₯,π‘₯)=0 for all π‘₯∈𝐾. (A2)𝐹 is monotone, that is, 𝐹(π‘₯,𝑦)+𝐹(𝑦,π‘₯)≀0 for all π‘₯,π‘¦βˆˆπΎ. (A3) For each π‘₯,𝑦,π‘§βˆˆπΎ, lim𝑑→0𝐹(𝑑𝑧+(1βˆ’π‘‘)π‘₯,𝑦)≀𝐹(π‘₯,𝑦).(A4) For each π‘₯∈𝐾,𝑦↦𝐹(π‘₯,𝑦) is convex and lower semi-continuous.

In order to prove our main results, we need the following Lemmas and Propositions.

Lemma 2.1 (see [1, 3]). Let 𝐹 be a bifunction from 𝐾×𝐾 into ℝ satisfying (A1)–(A4). Then, for any π‘Ÿ>0 and π‘₯∈𝐻, there exists π‘§βˆˆπΎ such that 1𝐹(𝑧,𝑦)+π‘ŸβŸ¨π‘¦βˆ’π‘§,π‘§βˆ’π‘₯⟩β‰₯0,βˆ€π‘¦βˆˆπΎ.(2.2) Further, if π‘‡π‘Ÿπ‘₯={π‘§βˆˆπΎβˆΆπΉ(𝑧,𝑦)+(1/π‘Ÿ)βŸ¨π‘¦βˆ’π‘§,π‘§βˆ’π‘₯⟩β‰₯0,forallπ‘¦βˆˆπΎ}, then the following holds. (1)π‘‡π‘Ÿ is single-valued. (2)π‘‡π‘Ÿ is firmly nonexpansive, that is, β€–π‘‡π‘Ÿπ‘₯βˆ’π‘‡π‘Ÿπ‘¦β€–2β‰€βŸ¨π‘‡π‘Ÿπ‘₯βˆ’π‘‡π‘Ÿπ‘¦,π‘₯βˆ’π‘¦βŸ©, forallπ‘₯,π‘¦βˆˆπ».(3)𝐹(π‘‡π‘Ÿ)=EP(𝐹). (4)EP(𝐹) is closed and convex.

Lemma 2.2 (see [7]). Let (𝐸,βŸ¨β‹…,β‹…βŸ©) be an inner product space. Then, for all π‘₯,𝑦,π‘§βˆˆπΈ and 𝛼,𝛽,π›Ύβˆˆ[0,1] with 𝛼+𝛽+𝛾=1, we have ‖𝛼π‘₯+𝛽𝑦+𝛾𝑧‖2=𝛼‖π‘₯β€–2+𝛽‖𝑦‖2+𝛾‖𝑧‖2βˆ’π›Όπ›½β€–π‘₯βˆ’π‘¦β€–2βˆ’π›Όπ›Ύβ€–π‘₯βˆ’π‘§β€–2βˆ’π›½π›Ύβ€–π‘¦βˆ’π‘§β€–2.(2.3)

Lemma 2.3 (see [19]). Let {π‘₯𝑛} and {𝑧𝑛} be bounded sequence in Banach space 𝐸, and let {πœ†π‘›} be a sequence in [0,1] such that 0<liminfπ‘›β†’βˆžπœ†π‘›β‰€limsupπ‘›β†’βˆžπœ†π‘›<1. Suppose π‘₯𝑛+1=πœ†π‘›π‘₯𝑛+(1βˆ’πœ†π‘›)𝑧𝑛 and limsupπ‘›β†’βˆžξ€·β€–β€–π‘§π‘›+1βˆ’π‘§π‘›β€–β€–βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀0,βˆ€π‘›β‰₯0.(2.4) Then limπ‘›β†’βˆžβ€–π‘§π‘›βˆ’π‘₯𝑛‖=0.

Lemma 2.4 (see [2, 10]). Let π‘‡βˆΆπΎβ†’π» be a π‘˜-strict pseudocontraction. For πœ†βˆˆ[π‘˜,1), define π‘†βˆΆπΎβ†’π» by 𝑆π‘₯=πœ†π‘₯+(1βˆ’πœ†)𝑇π‘₯ for each π‘₯∈𝐾. Then, as πœ†βˆˆ[π‘˜,1), 𝑆 is a nonexpansive mapping such that 𝐹(𝑆)=𝐹(𝑇).

Lemma 2.5 (see [10]). If π‘‡βˆΆπΎβ†’π» is a π‘˜-strict pseudocontraction, then the fixed point set 𝐹(𝑇) is closed convex so that the projection 𝑃𝐹(𝑇) is well defined.

Lemma 2.6 (see [9]). Let 𝐾 be a nonempty bounded closed convex subset of 𝐻. Given π‘₯∈𝐻 and π‘§βˆˆπΎ, then 𝑧=𝑃𝐾π‘₯ if and only if there holds the relation: ⟨π‘₯βˆ’π‘§,π‘§βˆ’π‘¦βŸ©β‰₯0,βˆ€π‘¦βˆˆπΎ.(2.5)

Lemma 2.7 (see [20]). Assume {π‘Žπ‘›} is a sequence of nonnegative real numbers such that π‘Žπ‘›+1≀1βˆ’π›Ύπ‘›ξ€Έπ‘Žπ‘›+𝛾𝑛𝛿𝑛,𝑛β‰₯0,(2.6) where {𝛾n} is a sequence in (0,1) and {𝛿𝑛} is a real sequence such that (i)βˆ‘βˆžπ‘›=0𝛾𝑛=∞,(ii)limsupπ‘›β†’βˆžπ›Ώπ‘›β‰€0 or βˆ‘βˆžπ‘›=0|𝛾𝑛𝛿𝑛|<∞. Then limπ‘›β†’βˆžπ‘Žπ‘›=0.

Proposition 2.8 (see, e.g., Acedo and Xu [12]). Let 𝐾 be a nonempty closed convex subset of Hilbert space 𝐻. Given an integer 𝑁β‰₯1, assume that {𝑇𝑖}𝑁𝑖=1βˆΆπΎβ†’π» is a finite family of π‘˜π‘–-strict pseudocontractions. Suppose that {πœ†π‘–}𝑁𝑖=1 is a positive sequence such that βˆ‘π‘π‘–=1πœ†π‘–=1. Then βˆ‘π‘π‘–=1πœ†π‘–π‘‡π‘– is a π‘˜-strict pseudocontraction with π‘˜=max{π‘˜π‘–βˆΆ1≀𝑖≀𝑁}.

Proposition 2.9 (see, e.g., Acedo and Xu [12]). Let {𝑇𝑖}𝑁𝑖=1 and {πœ†π‘–}𝑁𝑖=1 be given as in Proposition 2.8 above. Then βˆ‘πΉ(𝑁i=1πœ†π‘–π‘‡π‘–β‹‚)=𝑁𝑖=1𝐹(𝑇𝑖).

3. Main Results

Theorem 3.1. Let 𝐾 be a nonempty closed convex subset of Hilbert space 𝐻, and let 𝐹 be a bifunction from 𝐾×𝐾 into ℝ satisfying (A1)–(A4). Let {𝑇𝑖}𝑁𝑖=1βˆΆπΎβ†’π» be a finite family of π‘˜π‘–-strict pseudocontractions such that π‘˜=max{π‘˜π‘–βˆΆ1≀𝑖≀𝑁} and β‹‚β„±=𝑁𝑖=1𝐹(𝑇𝑖)β‹‚EP(𝐹)β‰ πœ™. Assume {πœ‚π‘–(𝑛)}𝑁𝑖=1 is a finite sequences of positive numbers such that βˆ‘π‘π‘–=1πœ‚π‘–(𝑛)=1 for all 𝑛β‰₯0. Given π‘’βˆˆπΎ and π‘₯0∈𝐾, {𝛼𝑛},{𝛽𝑛}, and {𝛾𝑛} are some sequences in (0,1); the following control conditions are satisfied. (i)π‘˜β‰€π›Όπ‘›β‰€πœ†<1 for all 𝑛β‰₯0 and limπ‘›β†’βˆžπ›Όπ‘›=𝛼,(ii)limπ‘›β†’βˆžπ›½π‘›=0 and βˆ‘βˆžπ‘›=0𝛽n=∞,(iii)0<liminfπ‘›β†’βˆžπ›Ύπ‘›β‰€limsupπ‘›β†’βˆžπ›Ύπ‘›<1,(iv)limπ‘›β†’βˆž|πœ‚π‘–(𝑛+1)βˆ’πœ‚π‘–(𝑛)|=0. Then the sequence {π‘₯𝑛} generated by (1.4) converges strongly to π‘žβˆˆβ„±, where π‘ž=𝑃ℱ𝑒.

Proof. From Lemma 2.1, we see that EP(𝐹)=𝐹(π‘‡π‘Ÿ), and note that 𝑒𝑛 can be rewritten as 𝑒𝑛=π‘‡π‘Ÿπ‘₯𝑛. Putting 𝐴𝑛=βˆ‘π‘π‘–=1πœ‚π‘–(𝑛)𝑇𝑖, we have π΄π‘›βˆΆπΎβ†’π» is a π‘˜-strict pseudocontraction and 𝐹(𝐴𝑛⋂)=𝑁𝑖=1𝐹(𝑇𝑖) by Propositions 2.8 and 2.9, where π‘˜=max{π‘˜π‘–βˆΆ1≀𝑖≀𝑁}.
From (1.4), condition (i), and Lemma 2.2, taking a point π‘βˆˆβ„±, we have β€–β€–π‘¦π‘›β€–β€–βˆ’π‘2=‖‖𝛼𝑛(π‘’π‘›βˆ’π‘)+(1βˆ’π›Όπ‘›)(π΄π‘›π‘’π‘›β€–β€–βˆ’π‘)2=π›Όπ‘›β€–β€–π‘’π‘›β€–β€–βˆ’π‘2+ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π΄π‘›π‘’π‘›β€–β€–βˆ’π‘2βˆ’π›Όπ‘›ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘’π‘›βˆ’π΄π‘›π‘’π‘›β€–β€–2β‰€π›Όπ‘›β€–β€–π‘’π‘›β€–β€–βˆ’π‘2+ξ€·1βˆ’π›Όπ‘›ξ€Έξ‚ƒβ€–β€–π‘’π‘›β€–β€–βˆ’π‘2‖‖𝑒+π‘˜π‘›βˆ’π΄π‘›π‘’π‘›β€–β€–2ξ‚„βˆ’π›Όπ‘›ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘’π‘›βˆ’π΄π‘›π‘’π‘›β€–β€–2=β€–β€–π‘’π‘›β€–β€–βˆ’π‘2βˆ’ξ€·1βˆ’π›Όπ‘›π›Όξ€Έξ€·π‘›ξ€Έβ€–β€–π‘’βˆ’π‘˜π‘›βˆ’π΄π‘›π‘’π‘›β€–β€–2β‰€β€–β€–π‘‡π‘Ÿπ‘₯π‘›βˆ’π‘‡π‘Ÿπ‘β€–β€–2≀‖‖π‘₯π‘›β€–β€–βˆ’π‘2.(3.1) Furthermore, we have β€–β€–π‘¦π‘›β€–β€–β‰€β€–β€–π‘’βˆ’π‘π‘›β€–β€–β‰€β€–β€–π‘₯βˆ’π‘π‘›β€–β€–.βˆ’π‘(3.2) It follows from (1.4) and (3.2) that β€–β€–π‘₯𝑛+1β€–β€–=β€–β€–π›½βˆ’π‘π‘›π‘’+𝛾𝑛π‘₯𝑛+ξ€·1βˆ’π›½π‘›βˆ’π›Ύπ‘›ξ€Έπ‘¦π‘›β€–β€–βˆ’π‘β‰€π›½π‘›β€–π‘’βˆ’π‘β€–+𝛾𝑛‖‖π‘₯𝑛‖‖+ξ€·βˆ’π‘1βˆ’π›½π‘›βˆ’π›Ύπ‘›ξ€Έβ€–β€–π‘¦π‘›β€–β€–βˆ’π‘β‰€π›½π‘›ξ€·β€–π‘’βˆ’π‘β€–+1βˆ’π›½π‘›ξ€Έβ€–β€–π‘₯𝑛‖‖‖‖π‘₯βˆ’π‘β‰€maxβ€–π‘’βˆ’π‘β€–,0β€–β€–ξ€Ύ.βˆ’π‘(3.3) Consequently, sequence {π‘₯𝑛} is bounded and so are {𝑒𝑛} and {𝑦𝑛}.
Define a mapping 𝑇𝑛π‘₯∢=𝛼𝑛π‘₯+(1βˆ’π›Όπ‘›)𝐴𝑛π‘₯ for each π‘₯∈𝐾. Then π‘‡π‘›βˆΆπΎβ†’π» is nonexpansive. Indeed, by using (1.1), condition (i), and Lemma 2.2, we have for all π‘₯,π‘¦βˆˆπΎ that ‖‖𝑇𝑛π‘₯βˆ’π‘‡π‘›π‘¦β€–β€–2=‖‖𝛼𝑛(π‘₯βˆ’π‘¦)+1βˆ’π›Όπ‘›π΄ξ€Έξ€·π‘›π‘₯βˆ’π΄π‘›π‘¦ξ€Έβ€–β€–2=𝛼𝑛‖π‘₯βˆ’π‘¦β€–2+ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π΄π‘›π‘₯βˆ’π΄π‘›π‘¦β€–β€–2βˆ’π›Όπ‘›ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘₯βˆ’π΄π‘›ξ€·π‘₯βˆ’π‘¦βˆ’π΄π‘›π‘¦ξ€Έβ€–β€–2≀𝛼𝑛‖π‘₯βˆ’π‘¦β€–2+ξ€·1βˆ’π›Όπ‘›ξ€Έξ‚ƒβ€–π‘₯βˆ’π‘¦β€–2β€–β€–+π‘˜π‘₯βˆ’π΄π‘›ξ€·π‘₯βˆ’π‘¦βˆ’π΄π‘›π‘¦ξ€Έβ€–β€–2ξ‚„βˆ’π›Όπ‘›ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘₯βˆ’π΄π‘›ξ€·π‘₯βˆ’π‘¦βˆ’π΄π‘›π‘¦ξ€Έβ€–β€–2=β€–π‘₯βˆ’π‘¦β€–2βˆ’ξ€·1βˆ’π›Όπ‘›π›Όξ€Έξ€·π‘›ξ€Έβ€–β€–βˆ’π‘˜π‘₯βˆ’π΄π‘›ξ€·π‘₯βˆ’π‘¦βˆ’π΄π‘›π‘¦ξ€Έβ€–β€–2≀‖π‘₯βˆ’π‘¦β€–2,(3.4) which shows that π‘‡π‘›βˆΆπΎβ†’π» is nonexpansive.
Next we show that limπ‘›β†’βˆžβ€–π‘₯𝑛+1βˆ’π‘₯𝑛‖=0. Setting π‘₯𝑛+1=𝛾𝑛π‘₯𝑛+(1βˆ’π›Ύπ‘›)𝑧𝑛, we have 𝑧𝑛+1βˆ’π‘§π‘›=π‘₯𝑛+2βˆ’π›Ύπ‘›+1π‘₯𝑛+11βˆ’π›Ύπ‘›+1βˆ’π‘₯𝑛+1βˆ’π›Ύπ‘›π‘₯𝑛1βˆ’π›Ύπ‘›=𝛽𝑛+1𝑒+1βˆ’π›½π‘›+1βˆ’π›Ύπ‘›+1𝑦𝑛+11βˆ’π›Ύπ‘›+1βˆ’π›½π‘›ξ€·π‘’+1βˆ’π›½π‘›βˆ’π›Ύπ‘›ξ€Έπ‘¦π‘›1βˆ’π›Ύπ‘›=𝛽𝑛+11βˆ’π›Ύπ‘›+1ξ€·π‘’βˆ’π‘¦π‘›+1ξ€Έ+𝑦𝑛+1βˆ’π‘¦π‘›ξ€Έβˆ’π›½π‘›1βˆ’π›Ύπ‘›ξ€·π‘’βˆ’π‘¦π‘›ξ€Έ.(3.5) It follows that ‖‖𝑧𝑛+1βˆ’π‘§π‘›β€–β€–β‰€π›½π‘›+11βˆ’π›Ύπ‘›+1β€–β€–π‘’βˆ’π‘¦π‘›+1β€–β€–+‖‖𝑦𝑛+1βˆ’π‘¦π‘›β€–β€–+𝛽𝑛1βˆ’π›Ύπ‘›β€–β€–π‘’βˆ’π‘¦π‘›β€–β€–.(3.6) From (1.4), we have 𝑦𝑛=𝑇𝑛𝑒𝑛 and ‖‖𝑦𝑛+1βˆ’π‘¦π‘›β€–β€–β‰€β€–β€–π‘‡π‘›+1𝑒𝑛+1βˆ’π‘‡π‘›+1𝑒𝑛‖‖+‖‖𝑇𝑛+1π‘’π‘›βˆ’π‘‡π‘›π‘’π‘›β€–β€–β‰€β€–β€–π‘’π‘›+1βˆ’π‘’π‘›β€–β€–+‖‖𝛼𝑛+1𝑒𝑛+ξ€·1βˆ’π›Όπ‘›+1𝐴𝑛+1π‘’π‘›βˆ’ξ€Ίπ›Όπ‘›π‘’π‘›+ξ€·1βˆ’π›Όπ‘›ξ€Έπ΄π‘›π‘’π‘›ξ€»β€–β€–β‰€β€–β€–π‘’π‘›+1βˆ’π‘’π‘›β€–β€–+||𝛼𝑛+1βˆ’π›Όπ‘›||β€–β€–π‘’π‘›βˆ’π΄π‘›π‘’π‘›β€–β€–+ξ€·1βˆ’π›Όπ‘›+1‖‖𝐴𝑛+1π‘’π‘›βˆ’π΄π‘›π‘’π‘›β€–β€–β‰€β€–β€–π‘’π‘›+1βˆ’π‘’π‘›β€–β€–+||𝛼𝑛+1βˆ’π›Όπ‘›||β€–β€–π‘’π‘›βˆ’π΄π‘›π‘’π‘›β€–β€–+ξ€·1βˆ’π›Όπ‘›+1𝑁𝑖=1||πœ‚π‘–(𝑛+1)βˆ’πœ‚π‘–(𝑛)||‖‖𝑇𝑖𝑒𝑛‖‖.(3.7) By Lemma 2.1, 𝑒𝑛=π‘‡π‘Ÿπ‘₯𝑛 and 𝑒𝑛+1=π‘‡π‘Ÿπ‘₯𝑛+1, we have 𝐹𝑒𝑛+1,π‘¦π‘ŸβŸ¨π‘¦βˆ’π‘’π‘›,π‘’π‘›βˆ’π‘₯𝑛,⟩β‰₯0,βˆ€π‘¦βˆˆπΎ(3.8)𝐹𝑒𝑛+1ξ€Έ+1,π‘¦π‘Ÿξ«π‘¦βˆ’π‘’π‘›+1,𝑒𝑛+1βˆ’π‘₯𝑛+1β‰₯0,βˆ€π‘¦βˆˆπΎ.(3.9) Putting 𝑦=𝑒𝑛+1 in (3.8) and 𝑦=𝑒𝑛 in (3.9), we obtain 𝐹𝑒𝑛,𝑒𝑛+1ξ€Έ+1π‘Ÿξ«π‘’π‘›+1βˆ’π‘’π‘›,π‘’π‘›βˆ’π‘₯𝑛,𝐹𝑒β‰₯0𝑛+1,𝑒𝑛+1π‘Ÿξ«π‘’π‘›βˆ’π‘’π‘›+1,𝑒𝑛+1βˆ’π‘₯𝑛+1β‰₯0.(3.10) So, from (A2) and π‘Ÿ>0, we have 𝑒𝑛+1βˆ’π‘’π‘›,π‘’π‘›βˆ’π‘’π‘›+1+𝑒𝑛+1βˆ’π‘₯π‘›βˆ’ξ€·π‘’π‘›+1βˆ’π‘₯𝑛+1β‰₯0,(3.11) and hence ‖‖𝑒𝑛+1βˆ’π‘’π‘›β€–β€–2≀𝑒𝑛+1βˆ’π‘’π‘›,π‘₯𝑛+1βˆ’π‘₯𝑛,(3.12) which implies that ‖‖𝑒𝑛+1βˆ’π‘’π‘›β€–β€–β‰€β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖.(3.13) Combining (3.6), (3.7), and (3.13), we have ‖‖𝑧𝑛+1βˆ’π‘§π‘›β€–β€–β‰€π›½π‘›+11βˆ’π›Ύπ‘›+1β€–β€–π‘’βˆ’π‘¦π‘›+1β€–β€–+β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+𝛽𝑛1βˆ’π›Ύπ‘›β€–β€–π‘’βˆ’π‘¦π‘›β€–β€–+||𝛼𝑛+1βˆ’π›Όπ‘›||β€–β€–π‘’π‘›βˆ’π΄π‘›π‘’π‘›β€–β€–+ξ€·1βˆ’π›Όπ‘›+1𝑁𝑖=1||πœ‚π‘–(𝑛+1)βˆ’πœ‚π‘–(𝑛)||‖‖𝑇𝑖𝑒𝑛‖‖.(3.14) This together with (i), (ii) and (iv) imply that limsupπ‘›β†’βˆžξ€·β€–β€–π‘§π‘›+1βˆ’π‘§π‘›β€–β€–βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀0.(3.15) Hence, by Lemma 2.3, we obtain limπ‘›β†’βˆžβ€–β€–π‘§π‘›βˆ’π‘₯𝑛‖‖=0.(3.16) Consequently, limπ‘›β†’βˆžβ€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖=limπ‘›β†’βˆžξ€·1βˆ’π›Ύπ‘›ξ€Έβ€–β€–π‘§π‘›βˆ’π‘₯𝑛‖‖=0.(3.17)
On the other hand, by (1.4) and (iii), we have β€–β€–π‘₯𝑛+1βˆ’π‘¦π‘›β€–β€–β‰€π›½π‘›β€–β€–π‘’βˆ’π‘¦π‘›β€–β€–+𝛾𝑛‖‖π‘₯π‘›βˆ’π‘₯𝑛+1β€–β€–+𝛾𝑛‖‖π‘₯𝑛+1βˆ’π‘¦π‘›β€–β€–,(3.18) which implies that β€–β€–π‘₯𝑛+1βˆ’π‘¦π‘›β€–β€–β‰€π›½π‘›1βˆ’π›Ύπ‘›β€–β€–π‘’βˆ’π‘¦π‘›β€–β€–+𝛾𝑛1βˆ’π›Ύπ‘›β€–β€–π‘₯π‘›βˆ’π‘₯𝑛+1β€–β€–.(3.19) Combining (ii), (3.17), and (3.19), we have limπ‘›β†’βˆžβ€–β€–π‘₯𝑛+1βˆ’π‘¦π‘›β€–β€–=0.(3.20) Note that β€–β€–π‘₯π‘›βˆ’π‘¦π‘›β€–β€–β‰€β€–β€–π‘₯π‘›βˆ’π‘₯𝑛+1β€–β€–+β€–β€–π‘₯𝑛+1βˆ’π‘¦π‘›β€–β€–,(3.21) which together with (3.17) and (3.20) implies limπ‘›β†’βˆžβ€–β€–π‘₯π‘›βˆ’π‘¦π‘›β€–β€–=0.(3.22) Moreover, for β‹‚π‘βˆˆβ„±=𝑁𝑖=1𝐹(𝑇𝑖)β‹‚EP(𝐹), we have β€–β€–π‘’π‘›β€–β€–βˆ’π‘2=β€–β€–π‘‡π‘Ÿπ‘₯π‘›βˆ’π‘‡π‘Ÿπ‘β€–β€–2β‰€βŸ¨π‘₯π‘›βˆ’π‘,𝑒𝑛=1βˆ’π‘βŸ©2ξ‚€β€–β€–π‘₯π‘›β€–β€–βˆ’π‘2+β€–β€–π‘’π‘›β€–β€–βˆ’π‘2βˆ’β€–β€–π‘₯π‘›βˆ’π‘’π‘›β€–β€–2,(3.23) and hence β€–β€–π‘’π‘›β€–β€–βˆ’π‘2≀‖‖π‘₯π‘›β€–β€–βˆ’π‘2βˆ’β€–β€–π‘₯π‘›βˆ’π‘’π‘›β€–β€–2.(3.24) From Lemma 2.2, (3.2) and (3.24), we have β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘2=‖‖𝛽𝑛𝑒+𝛾𝑛π‘₯𝑛+(1βˆ’π›½π‘›βˆ’π›Ύπ‘›)π‘¦π‘›β€–β€–βˆ’π‘2β‰€π›½π‘›β€–π‘’βˆ’π‘β€–2+𝛾𝑛‖‖π‘₯π‘›β€–β€–βˆ’π‘2+ξ€·1βˆ’π›½π‘›βˆ’π›Ύπ‘›ξ€Έβ€–β€–π‘¦π‘›β€–β€–βˆ’π‘2β‰€π›½π‘›β€–π‘’βˆ’π‘β€–2+𝛾𝑛‖‖π‘₯π‘›β€–β€–βˆ’π‘2+ξ€·1βˆ’π›½π‘›βˆ’π›Ύπ‘›ξ€Έξ‚€β€–β€–π‘₯π‘›β€–β€–βˆ’π‘2βˆ’β€–β€–π‘₯π‘›βˆ’π‘’π‘›β€–β€–2ξ‚β‰€π›½π‘›β€–π‘’βˆ’π‘β€–2+β€–β€–π‘₯π‘›β€–β€–βˆ’π‘2βˆ’ξ€·1βˆ’π›½π‘›βˆ’π›Ύπ‘›ξ€Έβ€–β€–π‘₯π‘›βˆ’π‘’π‘›β€–β€–2,(3.25) and hence ξ€·1βˆ’π›½π‘›βˆ’π›Ύπ‘›ξ€Έβ€–β€–π‘₯π‘›βˆ’π‘’π‘›β€–β€–2β‰€π›½π‘›β€–π‘’βˆ’π‘β€–2+β€–β€–π‘₯π‘›β€–β€–βˆ’π‘2βˆ’β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘2β‰€π›½π‘›β€–π‘’βˆ’π‘β€–2+β€–β€–π‘₯π‘›βˆ’π‘₯𝑛+1β€–β€–ξ€·β€–β€–π‘₯𝑛‖‖+β€–β€–π‘₯βˆ’π‘π‘›+1β€–β€–ξ€Έ.βˆ’π‘(3.26) By (ii) and (3.17), we obtain limπ‘›β†’βˆžβ€–β€–π‘₯π‘›βˆ’π‘’π‘›β€–β€–=0.(3.27) It follows from (3.22) and (3.27) that limπ‘›β†’βˆžβ€–β€–π‘¦π‘›βˆ’π‘’π‘›β€–β€–=0.(3.28) Define π‘†π‘›βˆΆπΎβ†’π» by 𝑆𝑛π‘₯=𝛼π‘₯+(1βˆ’π›Ό)𝐴𝑛π‘₯. Then, 𝑆𝑛 is a nonexpansive with 𝐹(𝑆𝑛)=𝐹(𝐴𝑛) by Lemma 2.4. Note that limπ‘›β†’βˆžπ›Όπ‘›=π›Όβˆˆ[π‘˜,1) by condition (i) and β€–β€–π‘’π‘›βˆ’π‘†π‘›π‘’π‘›β€–β€–β‰€β€–β€–π‘’π‘›βˆ’π‘¦π‘›β€–β€–+β€–β€–π‘¦π‘›βˆ’π‘†π‘›π‘’π‘›β€–β€–β‰€β€–β€–π‘’π‘›βˆ’π‘¦π‘›β€–β€–+‖‖𝛼𝑛𝑒𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ΄π‘›π‘’π‘›βˆ’ξ€Ίπ›Όπ‘’π‘›+(1βˆ’π›Ό)π΄π‘›π‘’π‘›ξ€»β€–β€–β‰€β€–β€–π‘’π‘›βˆ’π‘¦π‘›β€–β€–+||𝛼𝑛||β€–β€–π‘’βˆ’π›Όπ‘›βˆ’π΄π‘›π‘’π‘›β€–β€–,(3.29) which combines with condition (i) and (3.28) yielding that limπ‘›β†’βˆžβ€–β€–π‘’π‘›βˆ’π‘†π‘›π‘’π‘›β€–β€–=0.(3.30)
We now show that limsupπ‘›β†’βˆžβŸ¨π‘’βˆ’π‘ž,π‘₯π‘›βˆ’π‘žβŸ©β‰€0, where π‘ž=𝑃ℱ𝑒. To see this, we choose a subsequence {π‘₯𝑛𝑖} of {π‘₯𝑛} such that limsupπ‘›β†’βˆžβŸ¨π‘’βˆ’π‘ž,π‘₯π‘›βˆ’π‘žβŸ©=limπ‘–β†’βˆžξ«π‘’βˆ’π‘ž,π‘₯𝑛𝑖.βˆ’π‘ž(3.31) Since {𝑒𝑛𝑖} is bounded, there exists a subsequence {𝑒𝑛𝑖𝑗} of {𝑒𝑛𝑖} converging weakly to π‘’βˆ—. Without loss of generality, we assume that π‘’π‘›π‘–β‡€π‘’βˆ— as π‘–β†’βˆž. Form (3.27), we obtain π‘₯π‘›π‘–β‡€π‘’βˆ— as π‘–β†’βˆž. Since 𝐾 is closed and convex, 𝐾 is weakly closed. So, we have π‘’βˆ—βˆˆπΎ and π‘’βˆ—βˆˆπΉ(𝑆𝑛). Otherwise, from π‘’βˆ—β‰ π‘†π‘›π‘’βˆ— and Opial's condition, we obtain liminfπ‘–β†’βˆžβ€–β€–π‘’π‘›π‘–βˆ’π‘’βˆ—β€–β€–<liminfπ‘–β†’βˆžβ€–β€–π‘’π‘›π‘–βˆ’π‘†π‘›π‘–π‘’βˆ—β€–β€–β‰€liminfπ‘–β†’βˆžξ€·β€–β€–π‘’π‘›π‘–βˆ’π‘†π‘›π‘–π‘’π‘›π‘–β€–β€–+β€–β€–π‘†π‘›π‘–π‘’π‘›π‘–βˆ’π‘†π‘›π‘–π‘’βˆ—β€–β€–ξ€Έβ‰€liminfπ‘–β†’βˆžβ€–β€–π‘’π‘›π‘–βˆ’π‘’βˆ—β€–β€–.(3.32) This is a contradiction. Hence, we get π‘’βˆ—βˆˆπΉ(𝑆𝑛)=𝐹(𝐴𝑛). Moreover, by 𝑒𝑛=π‘‡π‘Ÿπ‘₯𝑛, we have 𝐹𝑒𝑛+1,π‘¦π‘ŸβŸ¨π‘¦βˆ’π‘’π‘›,π‘’π‘›βˆ’π‘₯π‘›βŸ©β‰₯0,βˆ€π‘¦βˆˆπΎ.(3.33) It follows from (A2) that 1π‘ŸβŸ¨π‘¦βˆ’π‘’π‘›,π‘’π‘›βˆ’π‘₯π‘›ξ€·βŸ©β‰₯𝐹𝑦,𝑒𝑛.(3.34) Replacing 𝑛 by 𝑛𝑖, we have 1π‘Ÿξ«π‘¦βˆ’π‘’π‘›π‘–,π‘’π‘›π‘–βˆ’π‘₯𝑛𝑖β‰₯𝐹𝑦,𝑒𝑛𝑖.(3.35) Since π‘’π‘›π‘–βˆ’π‘₯𝑛𝑖→0 and π‘’π‘›π‘–β‡€π‘’βˆ—, it follows from (A4) that 𝐹(𝑦,π‘’βˆ—)≀0 for all π‘¦βˆˆπΎ. Put 𝑧𝑑=𝑑𝑦+(1βˆ’π‘‘)π‘’βˆ— for all π‘‘βˆˆ(0,1] and π‘¦βˆˆπΎ. Then, we have π‘§π‘‘βˆˆπΎ, and hence, 𝐹(𝑧𝑑,π‘’βˆ—)≀0. By (A1) and (A4), we have 𝑧0=𝐹𝑑,𝑧𝑑𝑧≀𝑑𝐹𝑑+𝑧,𝑦(1βˆ’π‘‘)𝐹𝑑,π‘’βˆ—ξ€Έξ€·π‘§β‰€π‘‘πΉπ‘‘ξ€Έ,,𝑦(3.36) which implies 𝐹(𝑧𝑑,𝑦)β‰₯0. From (A3), we have 𝐹(π‘’βˆ—,𝑦)β‰₯0 for all π‘¦βˆˆπΎ, and hence, π‘’βˆ—βˆˆEP(𝐹). Therefore, π‘’βˆ—βˆˆπΉ(𝑆𝑛)∩EP(𝐹). From Lemma 2.6, we know that βŸ¨π‘’βˆ’π‘ƒβ„±π‘’,π‘’βˆ—βˆ’π‘ƒβ„±π‘’βŸ©β‰€0.(3.37) It follows from (3.31) and (3.37) that limsupπ‘›β†’βˆžβŸ¨π‘’βˆ’π‘ž,π‘₯π‘›βˆ’π‘žβŸ©=limπ‘–β†’βˆžξ«π‘’βˆ’π‘ž,π‘₯π‘›π‘–ξ¬βˆ’π‘ž=βŸ¨π‘’βˆ’π‘ž,π‘’βˆ—βˆ’π‘žβŸ©β‰€0.(3.38)
Finally, we prove that π‘₯π‘›β†’π‘ž=𝑃ℱ𝑒 as π‘›β†’βˆž. From (1.4) again, we have β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘ž2=𝛽𝑛𝑒+𝛾𝑛π‘₯𝑛+ξ€·1βˆ’π›½π‘›βˆ’π›Ύπ‘›ξ€Έπ‘¦π‘›βˆ’π‘ž,π‘₯𝑛+1ξ¬βˆ’π‘ž=π›½π‘›ξ«π‘’βˆ’π‘ž,π‘₯𝑛+1ξ¬βˆ’π‘ž+𝛾𝑛π‘₯π‘›βˆ’π‘ž,π‘₯𝑛+1+ξ€·βˆ’π‘ž1βˆ’π›½π‘›βˆ’π›Ύπ‘›π‘¦ξ€Έξ«π‘›βˆ’π‘ž,π‘₯𝑛+1ξ¬βˆ’π‘žβ‰€π›½π‘›ξ«π‘’βˆ’π‘ž,π‘₯𝑛+1ξ¬βˆ’π‘ž+𝛾𝑛‖‖π‘₯𝑛‖‖‖‖π‘₯βˆ’π‘žπ‘›+1β€–β€–+ξ€·βˆ’π‘ž1βˆ’π›½π‘›βˆ’π›Ύπ‘›ξ€Έβ€–β€–π‘¦π‘›β€–β€–β€–β€–π‘₯βˆ’π‘žπ‘›+1β€–β€–β‰€ξ€·βˆ’π‘ž1βˆ’π›½π‘›ξ€Έβ€–β€–π‘₯𝑛‖‖‖‖π‘₯βˆ’π‘žπ‘›+1β€–β€–βˆ’π‘ž+π›½π‘›ξ«π‘’βˆ’π‘ž,π‘₯𝑛+1ξ¬β‰€βˆ’π‘ž1βˆ’π›½π‘›2ξ‚€β€–β€–π‘₯π‘›β€–β€–βˆ’π‘ž2+β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘ž2+π›½π‘›ξ«π‘’βˆ’π‘ž,π‘₯𝑛+1ξ¬β‰€βˆ’π‘ž1βˆ’π›½π‘›2β€–β€–π‘₯π‘›β€–β€–βˆ’π‘ž2+12β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘ž2+π›½π‘›βŸ¨π‘’βˆ’π‘ž,π‘₯𝑛+1βˆ’π‘žβŸ©,(3.39) which implies that β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘ž2≀1βˆ’π›½π‘›ξ€Έβ€–β€–π‘₯π‘›β€–β€–βˆ’π‘ž2+2π›½π‘›βŸ¨π‘’βˆ’π‘ž,π‘₯𝑛+1βˆ’π‘žβŸ©.(3.40) It follows from (3.38), (3.40), and Lemma 2.7 that limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘žβ€–=0. This completes the proof.

As 𝑁=1, that is, 𝐴𝑛=𝑇 and πœ‚π‘–(𝑛)≑1 in Theorem 3.1, we have the following results immediately.

Theorem 3.2. Let 𝐾 be a nonempty closed convex subset of Hilbert space 𝐻, and let 𝐹 be a bifunction from 𝐾×𝐾 into ℝ satisfying (A1)–(A4). Let π‘‡βˆΆπΎβ†’π» be a π‘˜-strict pseudocontractions such that β‹‚β„±=𝐹(𝑇)EP(𝐹)β‰ πœ™. Let {π‘₯𝑛} be a sequence generated in the following manner: 𝐹𝑒𝑛+1,π‘¦π‘ŸβŸ¨π‘¦βˆ’π‘’π‘›,π‘’π‘›βˆ’π‘₯π‘›π‘¦βŸ©β‰₯0,βˆ€π‘¦βˆˆπΎ,𝑛=𝛼𝑛𝑒𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘’π‘›,π‘₯𝑛+1=𝛽𝑛𝑒+𝛾𝑛π‘₯𝑛+ξ€·1βˆ’π›½π‘›βˆ’π›Ύπ‘›ξ€Έπ‘¦π‘›,𝑛β‰₯0,(3.41) where π‘’βˆˆπΎ and π‘₯0∈𝐾, {𝛼𝑛},{𝛽𝑛}, and {𝛾𝑛} are some sequences in (0,1). If the following control conditions are satisfied: (i)π‘˜β‰€π›Όπ‘›β‰€πœ†<1 for all 𝑛β‰₯0 and limπ‘›β†’βˆžπ›Όπ‘›=𝛼,(ii)limπ‘›β†’βˆžπ›½π‘›=0 and βˆ‘βˆžπ‘›=0𝛽𝑛=∞,(iii)0<liminfπ‘›β†’βˆžπ›Ύπ‘›β‰€limsupπ‘›β†’βˆžπ›Ύπ‘›<1,then {π‘₯𝑛} converges strongly to π‘žβˆˆβ„±, where π‘ž=𝑃ℱ𝑒.

Acknowledgment

This work was supported by the National Science Foundation of China (11001287), the Natural Science Foundation Project of Chongqing (CSTC 2010BB9254), and the Science and Technology Research Project of Chongqing Municipal Education Commission (KJ 110701).

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