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 𝑘-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 ⟨⋅,⋅⟩ and norm ‖⋅‖, 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 𝑥∈𝐾 such that 𝐹(𝑥,𝑦)≥0,∀𝑦∈𝐾,(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 𝑇∶𝐾→𝐻, let 𝐹(𝑥,𝑦)=⟨𝑇𝑥,𝑦−𝑥⟩ for all 𝑥,𝑦∈𝐾. Then, 𝑧∈EP(𝐹) if and only if ⟨𝑇𝑧,𝑦−𝑧⟩≥0 for all 𝑦∈𝐾; 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 𝑇∶𝐾→𝐻 is called a 𝑘-strict pseudocontraction if there exists a constant 𝑘∈[0,1) such that ‖𝑇𝑥−𝑇𝑦‖2≤‖𝑥−𝑦‖2+𝑘‖(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦‖2,∀𝑥,𝑦∈𝐾.(1.2) We use 𝐹(𝑇) to denote the fixed point set of the mapping 𝑇, that is, 𝐹(𝑇)∶={𝑥∈𝐾∶𝑇𝑥=𝑥}. As 𝑘=0, 𝑇 is said to be nonexpansive, that is, ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖,forall𝑥,𝑦∈𝐾. 𝑇 is said to be pseudocontractive if 𝑘=1 and is also said to be strongly pseudocontractive if there exists a positive constant 𝜆∈(0,1) such that 𝑇+𝜆𝐼 is pseudocontractive. Clearly, the class of 𝑘-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 𝑘-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 𝑘𝑖-strict pseudocontractions {𝑇𝑖}𝑁𝑖=1 in Hilbert space 𝐻: ∀𝑥0∈𝐾,𝑥𝑛+1=1−𝛼𝑛𝑥𝑛+𝛼𝑛𝑁𝑖=1𝜆𝑖𝑇𝑖𝑥𝑛,(1.3) where {𝛼𝑛}⊂(0,1), and {𝜆𝑖}𝑁𝑖=1 is a finite sequence of positive numbers such that ∑𝑁𝑖=1𝜆𝑖=1. They proved that the sequence {𝑥𝑛} 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 ⋂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 ⋂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𝑢𝑛+1,𝑦𝑟⟨𝑦−𝑢𝑛,𝑢𝑛−𝑥𝑛𝑦⟩≥0,∀𝑦∈𝐾,𝑛=𝛼𝑛𝑢𝑛+1−𝛼𝑛𝑁𝑖=1𝜂𝑖(𝑛)𝑇𝑖𝑢𝑛,𝑥𝑛+1=𝛽𝑛𝑢+𝛾𝑛𝑥𝑛+1−𝛽𝑛−𝛾𝑛𝑦𝑛,𝑛≥0,(1.4) where 𝑢∈𝐾 is a given point, {𝑇𝑖}𝑁𝑖=𝑖∶𝐾→𝐻 is a finite family of 𝑘𝑖-strictly pseudocontractive nonself-mappings, {𝜂𝑖(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 𝑘𝑖-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 𝑥∈𝐻, 𝑃𝐾𝑥 is the only point in 𝐾 such that ‖𝑥−𝑃𝐾𝑥‖=inf{‖𝑥−𝑧‖∶𝑧∈𝐾}. we write 𝑥𝑛⇀𝑥 and 𝑥𝑛→𝑥 indicate that the sequence {𝑥𝑛} convergence weakly and strongly to 𝑥, respectively.

It is well known that Hilbert space 𝐻 satisfies Opial's condition [8], that is, for any sequence {𝑥𝑛} with 𝑥𝑛⇀𝑥 and every 𝑦∈𝐻 with 𝑦≠𝑥, we have liminfğ‘›â†’âˆžâ€–â€–ğ‘¥ğ‘›â€–â€–âˆ’ğ‘¥<liminfğ‘›â†’âˆžâ€–â€–ğ‘¥ğ‘›â€–â€–.−𝑦(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(𝑥,𝑥)=0 for all 𝑥∈𝐾. (A2)𝐹 is monotone, that is, 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)≤0 for all 𝑥,𝑦∈𝐾. (A3) For each 𝑥,𝑦,𝑧∈𝐾, lim𝑡→0𝐹(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝐹(𝑥,𝑦).(A4) For each 𝑥∈𝐾,𝑦↦𝐹(𝑥,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 𝑟>0 and 𝑥∈𝐻, there exists 𝑧∈𝐾 such that 1𝐹(𝑧,𝑦)+𝑟⟨𝑦−𝑧,𝑧−𝑥⟩≥0,∀𝑦∈𝐾.(2.2) Further, if 𝑇𝑟𝑥={𝑧∈𝐾∶𝐹(𝑧,𝑦)+(1/𝑟)⟨𝑦−𝑧,𝑧−𝑥⟩≥0,forall𝑦∈𝐾}, then the following holds. (1)𝑇𝑟 is single-valued. (2)𝑇𝑟 is firmly nonexpansive, that is, ‖𝑇𝑟𝑥−𝑇𝑟𝑦‖2≤⟨𝑇𝑟𝑥−𝑇𝑟𝑦,𝑥−𝑦⟩, forall𝑥,𝑦∈𝐻.(3)𝐹(𝑇𝑟)=EP(𝐹). (4)EP(𝐹) is closed and convex.

Lemma 2.2 (see [7]). Let (𝐸,⟨⋅,⋅⟩) be an inner product space. Then, for all 𝑥,𝑦,𝑧∈𝐸 and 𝛼,𝛽,𝛾∈[0,1] with 𝛼+𝛽+𝛾=1, we have ‖𝛼𝑥+𝛽𝑦+𝛾𝑧‖2=𝛼‖𝑥‖2+𝛽‖𝑦‖2+𝛾‖𝑧‖2−𝛼𝛽‖𝑥−𝑦‖2−𝛼𝛾‖𝑥−𝑧‖2−𝛽𝛾‖𝑦−𝑧‖2.(2.3)

Lemma 2.3 (see [19]). Let {𝑥𝑛} and {𝑧𝑛} be bounded sequence in Banach space 𝐸, and let {𝜆𝑛} be a sequence in [0,1] such that 0<liminfğ‘›â†’âˆžğœ†ğ‘›â‰¤limsupğ‘›â†’âˆžğœ†ğ‘›<1. Suppose 𝑥𝑛+1=𝜆𝑛𝑥𝑛+(1−𝜆𝑛)𝑧𝑛 and limsupğ‘›â†’âˆžî€·â€–â€–ğ‘§ğ‘›+1−𝑧𝑛‖‖−‖‖𝑥𝑛+1−𝑥𝑛‖‖≤0,∀𝑛≥0.(2.4) Then limğ‘›â†’âˆžâ€–ğ‘§ğ‘›âˆ’ğ‘¥ğ‘›â€–=0.

Lemma 2.4 (see [2, 10]). Let 𝑇∶𝐾→𝐻 be a 𝑘-strict pseudocontraction. For 𝜆∈[𝑘,1), define 𝑆∶𝐾→𝐻 by 𝑆𝑥=𝜆𝑥+(1−𝜆)𝑇𝑥 for each 𝑥∈𝐾. Then, as 𝜆∈[𝑘,1), 𝑆 is a nonexpansive mapping such that 𝐹(𝑆)=𝐹(𝑇).

Lemma 2.5 (see [10]). If 𝑇∶𝐾→𝐻 is a 𝑘-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 𝑥∈𝐻 and 𝑧∈𝐾, then 𝑧=𝑃𝐾𝑥 if and only if there holds the relation: ⟨𝑥−𝑧,𝑧−𝑦⟩≥0,∀𝑦∈𝐾.(2.5)

Lemma 2.7 (see [20]). Assume {ğ‘Žğ‘›} is a sequence of nonnegative real numbers such that ğ‘Žğ‘›+1≤1âˆ’ğ›¾ğ‘›î€¸ğ‘Žğ‘›+𝛾𝑛𝛿𝑛,𝑛≥0,(2.6) where {𝛾n} is a sequence in (0,1) and {𝛿𝑛} is a real sequence such that (i)âˆ‘âˆžğ‘›=0𝛾𝑛=∞,(ii)limsupğ‘›â†’âˆžğ›¿ğ‘›â‰¤0 or âˆ‘âˆžğ‘›=0|𝛾𝑛𝛿𝑛|<∞. Then limğ‘›â†’âˆžğ‘Žğ‘›=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 𝑁≥1, assume that {𝑇𝑖}𝑁𝑖=1∶𝐾→𝐻 is a finite family of 𝑘𝑖-strict pseudocontractions. Suppose that {𝜆𝑖}𝑁𝑖=1 is a positive sequence such that ∑𝑁𝑖=1𝜆𝑖=1. Then ∑𝑁𝑖=1𝜆𝑖𝑇𝑖 is a 𝑘-strict pseudocontraction with 𝑘=max{𝑘𝑖∶1≤𝑖≤𝑁}.

Proposition 2.9 (see, e.g., Acedo and Xu [12]). Let {𝑇𝑖}𝑁𝑖=1 and {𝜆𝑖}𝑁𝑖=1 be given as in Proposition 2.8 above. Then ∑𝐹(𝑁i=1𝜆𝑖𝑇𝑖⋂)=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=1∶𝐾→𝐻 be a finite family of 𝑘𝑖-strict pseudocontractions such that 𝑘=max{𝑘𝑖∶1≤𝑖≤𝑁} and ⋂ℱ=𝑁𝑖=1𝐹(𝑇𝑖)⋂EP(𝐹)≠𝜙. Assume {𝜂𝑖(𝑛)}𝑁𝑖=1 is a finite sequences of positive numbers such that ∑𝑁𝑖=1𝜂𝑖(𝑛)=1 for all 𝑛≥0. Given 𝑢∈𝐾 and 𝑥0∈𝐾, {𝛼𝑛},{𝛽𝑛}, and {𝛾𝑛} are some sequences in (0,1); the following control conditions are satisfied. (i)𝑘≤𝛼𝑛≤𝜆<1 for all 𝑛≥0 and limğ‘›â†’âˆžğ›¼ğ‘›=𝛼,(ii)limğ‘›â†’âˆžğ›½ğ‘›=0 and âˆ‘âˆžğ‘›=0𝛽n=∞,(iii)0<liminfğ‘›â†’âˆžğ›¾ğ‘›â‰¤limsupğ‘›â†’âˆžğ›¾ğ‘›<1,(iv)limğ‘›â†’âˆž|𝜂𝑖(𝑛+1)−𝜂𝑖(𝑛)|=0. Then the sequence {𝑥𝑛} generated by (1.4) converges strongly to ğ‘žâˆˆâ„±, where ğ‘ž=𝑃ℱ𝑢.

Proof. From Lemma 2.1, we see that EP(𝐹)=𝐹(𝑇𝑟), and note that 𝑢𝑛 can be rewritten as 𝑢𝑛=𝑇𝑟𝑥𝑛. Putting 𝐴𝑛=∑𝑁𝑖=1𝜂𝑖(𝑛)𝑇𝑖, we have 𝐴𝑛∶𝐾→𝐻 is a 𝑘-strict pseudocontraction and 𝐹(𝐴𝑛⋂)=𝑁𝑖=1𝐹(𝑇𝑖) by Propositions 2.8 and 2.9, where 𝑘=max{𝑘𝑖∶1≤𝑖≤𝑁}.
From (1.4), condition (i), and Lemma 2.2, taking a point 𝑝∈ℱ, we have ‖‖𝑦𝑛‖‖−𝑝2=‖‖𝛼𝑛(𝑢𝑛−𝑝)+(1−𝛼𝑛)(𝐴𝑛𝑢𝑛‖‖−𝑝)2=𝛼𝑛‖‖𝑢𝑛‖‖−𝑝2+1−𝛼𝑛‖‖𝐴𝑛𝑢𝑛‖‖−𝑝2−𝛼𝑛1−𝛼𝑛‖‖𝑢𝑛−𝐴𝑛𝑢𝑛‖‖2≤𝛼𝑛‖‖𝑢𝑛‖‖−𝑝2+1−𝛼𝑛‖‖𝑢𝑛‖‖−𝑝2‖‖𝑢+𝑘𝑛−𝐴𝑛𝑢𝑛‖‖2−𝛼𝑛1−𝛼𝑛‖‖𝑢𝑛−𝐴𝑛𝑢𝑛‖‖2=‖‖𝑢𝑛‖‖−𝑝2−1−𝛼𝑛𝛼𝑛‖‖𝑢−𝑘𝑛−𝐴𝑛𝑢𝑛‖‖2≤‖‖𝑇𝑟𝑥𝑛−𝑇𝑟𝑝‖‖2≤‖‖𝑥𝑛‖‖−𝑝2.(3.1) Furthermore, we have ‖‖𝑦𝑛‖‖≤‖‖𝑢−𝑝𝑛‖‖≤‖‖𝑥−𝑝𝑛‖‖.−𝑝(3.2) It follows from (1.4) and (3.2) that ‖‖𝑥𝑛+1‖‖=‖‖𝛽−𝑝𝑛𝑢+𝛾𝑛𝑥𝑛+1−𝛽𝑛−𝛾𝑛𝑦𝑛‖‖−𝑝≤𝛽𝑛‖𝑢−𝑝‖+𝛾𝑛‖‖𝑥𝑛‖‖+−𝑝1−𝛽𝑛−𝛾𝑛‖‖𝑦𝑛‖‖−𝑝≤𝛽𝑛‖𝑢−𝑝‖+1−𝛽𝑛‖‖𝑥𝑛‖‖‖‖𝑥−𝑝≤max‖𝑢−𝑝‖,0‖‖.−𝑝(3.3) Consequently, sequence {𝑥𝑛} is bounded and so are {𝑢𝑛} and {𝑦𝑛}.
Define a mapping 𝑇𝑛𝑥∶=𝛼𝑛𝑥+(1−𝛼𝑛)𝐴𝑛𝑥 for each 𝑥∈𝐾. Then 𝑇𝑛∶𝐾→𝐻 is nonexpansive. Indeed, by using (1.1), condition (i), and Lemma 2.2, we have for all 𝑥,𝑦∈𝐾 that ‖‖𝑇𝑛𝑥−𝑇𝑛𝑦‖‖2=‖‖𝛼𝑛(𝑥−𝑦)+1−𝛼𝑛𝐴𝑛𝑥−𝐴𝑛𝑦‖‖2=𝛼𝑛‖𝑥−𝑦‖2+1−𝛼𝑛‖‖𝐴𝑛𝑥−𝐴𝑛𝑦‖‖2−𝛼𝑛1−𝛼𝑛‖‖𝑥−𝐴𝑛𝑥−𝑦−𝐴𝑛𝑦‖‖2≤𝛼𝑛‖𝑥−𝑦‖2+1−𝛼𝑛‖𝑥−𝑦‖2‖‖+𝑘𝑥−𝐴𝑛𝑥−𝑦−𝐴𝑛𝑦‖‖2−𝛼𝑛1−𝛼𝑛‖‖𝑥−𝐴𝑛𝑥−𝑦−𝐴𝑛𝑦‖‖2=‖𝑥−𝑦‖2−1−𝛼𝑛𝛼𝑛‖‖−𝑘𝑥−𝐴𝑛𝑥−𝑦−𝐴𝑛𝑦‖‖2≤‖𝑥−𝑦‖2,(3.4) which shows that 𝑇𝑛∶𝐾→𝐻 is nonexpansive.
Next we show that limğ‘›â†’âˆžâ€–ğ‘¥ğ‘›+1−𝑥𝑛‖=0. Setting 𝑥𝑛+1=𝛾𝑛𝑥𝑛+(1−𝛾𝑛)𝑧𝑛, we have 𝑧𝑛+1−𝑧𝑛=𝑥𝑛+2−𝛾𝑛+1𝑥𝑛+11−𝛾𝑛+1−𝑥𝑛+1−𝛾𝑛𝑥𝑛1−𝛾𝑛=𝛽𝑛+1𝑢+1−𝛽𝑛+1−𝛾𝑛+1𝑦𝑛+11−𝛾𝑛+1−𝛽𝑛𝑢+1−𝛽𝑛−𝛾𝑛𝑦𝑛1−𝛾𝑛=𝛽𝑛+11−𝛾𝑛+1𝑢−𝑦𝑛+1+𝑦𝑛+1−𝑦𝑛−𝛽𝑛1−𝛾𝑛𝑢−𝑦𝑛.(3.5) It follows that ‖‖𝑧𝑛+1−𝑧𝑛‖‖≤𝛽𝑛+11−𝛾𝑛+1‖‖𝑢−𝑦𝑛+1‖‖+‖‖𝑦𝑛+1−𝑦𝑛‖‖+𝛽𝑛1−𝛾𝑛‖‖𝑢−𝑦𝑛‖‖.(3.6) From (1.4), we have 𝑦𝑛=𝑇𝑛𝑢𝑛 and ‖‖𝑦𝑛+1−𝑦𝑛‖‖≤‖‖𝑇𝑛+1𝑢𝑛+1−𝑇𝑛+1𝑢𝑛‖‖+‖‖𝑇𝑛+1𝑢𝑛−𝑇𝑛𝑢𝑛‖‖≤‖‖𝑢𝑛+1−𝑢𝑛‖‖+‖‖𝛼𝑛+1𝑢𝑛+1−𝛼𝑛+1𝐴𝑛+1𝑢𝑛−𝛼𝑛𝑢𝑛+1−𝛼𝑛𝐴𝑛𝑢𝑛‖‖≤‖‖𝑢𝑛+1−𝑢𝑛‖‖+||𝛼𝑛+1−𝛼𝑛||‖‖𝑢𝑛−𝐴𝑛𝑢𝑛‖‖+1−𝛼𝑛+1‖‖𝐴𝑛+1𝑢𝑛−𝐴𝑛𝑢𝑛‖‖≤‖‖𝑢𝑛+1−𝑢𝑛‖‖+||𝛼𝑛+1−𝛼𝑛||‖‖𝑢𝑛−𝐴𝑛𝑢𝑛‖‖+1−𝛼𝑛+1𝑁𝑖=1||𝜂𝑖(𝑛+1)−𝜂𝑖(𝑛)||‖‖𝑇𝑖𝑢𝑛‖‖.(3.7) By Lemma 2.1, 𝑢𝑛=𝑇𝑟𝑥𝑛 and 𝑢𝑛+1=𝑇𝑟𝑥𝑛+1, we have 𝐹𝑢𝑛+1,𝑦𝑟⟨𝑦−𝑢𝑛,𝑢𝑛−𝑥𝑛,⟩≥0,∀𝑦∈𝐾(3.8)𝐹𝑢𝑛+1+1,𝑦𝑟𝑦−𝑢𝑛+1,𝑢𝑛+1−𝑥𝑛+1≥0,∀𝑦∈𝐾.(3.9) Putting 𝑦=𝑢𝑛+1 in (3.8) and 𝑦=𝑢𝑛 in (3.9), we obtain 𝐹𝑢𝑛,𝑢𝑛+1+1𝑟𝑢𝑛+1−𝑢𝑛,𝑢𝑛−𝑥𝑛,𝐹𝑢≥0𝑛+1,𝑢𝑛+1𝑟𝑢𝑛−𝑢𝑛+1,𝑢𝑛+1−𝑥𝑛+1≥0.(3.10) So, from (A2) and 𝑟>0, we have 𝑢𝑛+1−𝑢𝑛,𝑢𝑛−𝑢𝑛+1+𝑢𝑛+1−𝑥𝑛−𝑢𝑛+1−𝑥𝑛+1≥0,(3.11) and hence ‖‖𝑢𝑛+1−𝑢𝑛‖‖2≤𝑢𝑛+1−𝑢𝑛,𝑥𝑛+1−𝑥𝑛,(3.12) which implies that ‖‖𝑢𝑛+1−𝑢𝑛‖‖≤‖‖𝑥𝑛+1−𝑥𝑛‖‖.(3.13) Combining (3.6), (3.7), and (3.13), we have ‖‖𝑧𝑛+1−𝑧𝑛‖‖≤𝛽𝑛+11−𝛾𝑛+1‖‖𝑢−𝑦𝑛+1‖‖+‖‖𝑥𝑛+1−𝑥𝑛‖‖+𝛽𝑛1−𝛾𝑛‖‖𝑢−𝑦𝑛‖‖+||𝛼𝑛+1−𝛼𝑛||‖‖𝑢𝑛−𝐴𝑛𝑢𝑛‖‖+1−𝛼𝑛+1𝑁𝑖=1||𝜂𝑖(𝑛+1)−𝜂𝑖(𝑛)||‖‖𝑇𝑖𝑢𝑛‖‖.(3.14) This together with (i), (ii) and (iv) imply that limsupğ‘›â†’âˆžî€·â€–â€–ğ‘§ğ‘›+1−𝑧𝑛‖‖−‖‖𝑥𝑛+1−𝑥𝑛‖‖≤0.(3.15) Hence, by Lemma 2.3, we obtain limğ‘›â†’âˆžâ€–â€–ğ‘§ğ‘›âˆ’ğ‘¥ğ‘›â€–â€–=0.(3.16) Consequently, limğ‘›â†’âˆžâ€–â€–ğ‘¥ğ‘›+1−𝑥𝑛‖‖=limğ‘›â†’âˆžî€·1−𝛾𝑛‖‖𝑧𝑛−𝑥𝑛‖‖=0.(3.17)
On the other hand, by (1.4) and (iii), we have ‖‖𝑥𝑛+1−𝑦𝑛‖‖≤𝛽𝑛‖‖𝑢−𝑦𝑛‖‖+𝛾𝑛‖‖𝑥𝑛−𝑥𝑛+1‖‖+𝛾𝑛‖‖𝑥𝑛+1−𝑦𝑛‖‖,(3.18) which implies that ‖‖𝑥𝑛+1−𝑦𝑛‖‖≤𝛽𝑛1−𝛾𝑛‖‖𝑢−𝑦𝑛‖‖+𝛾𝑛1−𝛾𝑛‖‖𝑥𝑛−𝑥𝑛+1‖‖.(3.19) Combining (ii), (3.17), and (3.19), we have limğ‘›â†’âˆžâ€–â€–ğ‘¥ğ‘›+1−𝑦𝑛‖‖=0.(3.20) Note that ‖‖𝑥𝑛−𝑦𝑛‖‖≤‖‖𝑥𝑛−𝑥𝑛+1‖‖+‖‖𝑥𝑛+1−𝑦𝑛‖‖,(3.21) which together with (3.17) and (3.20) implies limğ‘›â†’âˆžâ€–â€–ğ‘¥ğ‘›âˆ’ğ‘¦ğ‘›â€–â€–=0.(3.22) Moreover, for ⋂𝑝∈ℱ=𝑁𝑖=1𝐹(𝑇𝑖)⋂EP(𝐹), we have ‖‖𝑢𝑛‖‖−𝑝2=‖‖𝑇𝑟𝑥𝑛−𝑇𝑟𝑝‖‖2≤⟨𝑥𝑛−𝑝,𝑢𝑛=1−𝑝⟩2‖‖𝑥𝑛‖‖−𝑝2+‖‖𝑢𝑛‖‖−𝑝2−‖‖𝑥𝑛−𝑢𝑛‖‖2,(3.23) and hence ‖‖𝑢𝑛‖‖−𝑝2≤‖‖𝑥𝑛‖‖−𝑝2−‖‖𝑥𝑛−𝑢𝑛‖‖2.(3.24) From Lemma 2.2, (3.2) and (3.24), we have ‖‖𝑥𝑛+1‖‖−𝑝2=‖‖𝛽𝑛𝑢+𝛾𝑛𝑥𝑛+(1−𝛽𝑛−𝛾𝑛)𝑦𝑛‖‖−𝑝2≤𝛽𝑛‖𝑢−𝑝‖2+𝛾𝑛‖‖𝑥𝑛‖‖−𝑝2+1−𝛽𝑛−𝛾𝑛‖‖𝑦𝑛‖‖−𝑝2≤𝛽𝑛‖𝑢−𝑝‖2+𝛾𝑛‖‖𝑥𝑛‖‖−𝑝2+1−𝛽𝑛−𝛾𝑛‖‖𝑥𝑛‖‖−𝑝2−‖‖𝑥𝑛−𝑢𝑛‖‖2≤𝛽𝑛‖𝑢−𝑝‖2+‖‖𝑥𝑛‖‖−𝑝2−1−𝛽𝑛−𝛾𝑛‖‖𝑥𝑛−𝑢𝑛‖‖2,(3.25) and hence 1−𝛽𝑛−𝛾𝑛‖‖𝑥𝑛−𝑢𝑛‖‖2≤𝛽𝑛‖𝑢−𝑝‖2+‖‖𝑥𝑛‖‖−𝑝2−‖‖𝑥𝑛+1‖‖−𝑝2≤𝛽𝑛‖𝑢−𝑝‖2+‖‖𝑥𝑛−𝑥𝑛+1‖‖‖‖𝑥𝑛‖‖+‖‖𝑥−𝑝𝑛+1‖‖.−𝑝(3.26) By (ii) and (3.17), we obtain limğ‘›â†’âˆžâ€–â€–ğ‘¥ğ‘›âˆ’ğ‘¢ğ‘›â€–â€–=0.(3.27) It follows from (3.22) and (3.27) that limğ‘›â†’âˆžâ€–â€–ğ‘¦ğ‘›âˆ’ğ‘¢ğ‘›â€–â€–=0.(3.28) Define 𝑆𝑛∶𝐾→𝐻 by 𝑆𝑛𝑥=𝛼𝑥+(1−𝛼)𝐴𝑛𝑥. Then, 𝑆𝑛 is a nonexpansive with 𝐹(𝑆𝑛)=𝐹(𝐴𝑛) by Lemma 2.4. Note that limğ‘›â†’âˆžğ›¼ğ‘›=𝛼∈[𝑘,1) by condition (i) and ‖‖𝑢𝑛−𝑆𝑛𝑢𝑛‖‖≤‖‖𝑢𝑛−𝑦𝑛‖‖+‖‖𝑦𝑛−𝑆𝑛𝑢𝑛‖‖≤‖‖𝑢𝑛−𝑦𝑛‖‖+‖‖𝛼𝑛𝑢𝑛+1−𝛼𝑛𝐴𝑛𝑢𝑛−𝛼𝑢𝑛+(1−𝛼)𝐴𝑛𝑢𝑛‖‖≤‖‖𝑢𝑛−𝑦𝑛‖‖+||𝛼𝑛||‖‖𝑢−𝛼𝑛−𝐴𝑛𝑢𝑛‖‖,(3.29) which combines with condition (i) and (3.28) yielding that limğ‘›â†’âˆžâ€–â€–ğ‘¢ğ‘›âˆ’ğ‘†ğ‘›ğ‘¢ğ‘›â€–â€–=0.(3.30)
We now show that limsupğ‘›â†’âˆžâŸ¨ğ‘¢âˆ’ğ‘ž,ğ‘¥ğ‘›âˆ’ğ‘žâŸ©â‰¤0, where ğ‘ž=𝑃ℱ𝑢. To see this, we choose a subsequence {𝑥𝑛𝑖} of {𝑥𝑛} such that limsupğ‘›â†’âˆžâŸ¨ğ‘¢âˆ’ğ‘ž,ğ‘¥ğ‘›âˆ’ğ‘žâŸ©=limğ‘–â†’âˆžî«ğ‘¢âˆ’ğ‘ž,𝑥𝑛𝑖.âˆ’ğ‘ž(3.31) Since {𝑢𝑛𝑖} is bounded, there exists a subsequence {𝑢𝑛𝑖𝑗} of {𝑢𝑛𝑖} converging weakly to 𝑢∗. Without loss of generality, we assume that 𝑢𝑛𝑖⇀𝑢∗ as ğ‘–â†’âˆž. Form (3.27), we obtain 𝑥𝑛𝑖⇀𝑢∗ as ğ‘–â†’âˆž. Since 𝐾 is closed and convex, 𝐾 is weakly closed. So, we have 𝑢∗∈𝐾 and 𝑢∗∈𝐹(𝑆𝑛). Otherwise, from 𝑢∗≠𝑆𝑛𝑢∗ and Opial's condition, we obtain liminfğ‘–â†’âˆžâ€–â€–ğ‘¢ğ‘›ğ‘–âˆ’ğ‘¢âˆ—â€–â€–<liminfğ‘–â†’âˆžâ€–â€–ğ‘¢ğ‘›ğ‘–âˆ’ğ‘†ğ‘›ğ‘–ğ‘¢âˆ—â€–â€–â‰¤liminfğ‘–â†’âˆžî€·â€–â€–ğ‘¢ğ‘›ğ‘–âˆ’ğ‘†ğ‘›ğ‘–ğ‘¢ğ‘›ğ‘–â€–â€–+‖‖𝑆𝑛𝑖𝑢𝑛𝑖−𝑆𝑛𝑖𝑢∗‖‖≤liminfğ‘–â†’âˆžâ€–â€–ğ‘¢ğ‘›ğ‘–âˆ’ğ‘¢âˆ—â€–â€–.(3.32) This is a contradiction. Hence, we get 𝑢∗∈𝐹(𝑆𝑛)=𝐹(𝐴𝑛). Moreover, by 𝑢𝑛=𝑇𝑟𝑥𝑛, we have 𝐹𝑢𝑛+1,𝑦𝑟⟨𝑦−𝑢𝑛,𝑢𝑛−𝑥𝑛⟩≥0,∀𝑦∈𝐾.(3.33) It follows from (A2) that 1𝑟⟨𝑦−𝑢𝑛,𝑢𝑛−𝑥𝑛⟩≥𝐹𝑦,𝑢𝑛.(3.34) Replacing 𝑛 by 𝑛𝑖, we have 1𝑟𝑦−𝑢𝑛𝑖,𝑢𝑛𝑖−𝑥𝑛𝑖≥𝐹𝑦,𝑢𝑛𝑖.(3.35) Since 𝑢𝑛𝑖−𝑥𝑛𝑖→0 and 𝑢𝑛𝑖⇀𝑢∗, it follows from (A4) that 𝐹(𝑦,𝑢∗)≤0 for all 𝑦∈𝐾. Put 𝑧𝑡=𝑡𝑦+(1−𝑡)𝑢∗ for all 𝑡∈(0,1] and 𝑦∈𝐾. Then, we have 𝑧𝑡∈𝐾, and hence, 𝐹(𝑧𝑡,𝑢∗)≤0. By (A1) and (A4), we have 𝑧0=𝐹𝑡,𝑧𝑡𝑧≤𝑡𝐹𝑡+𝑧,𝑦(1−𝑡)𝐹𝑡,𝑢∗𝑧≤𝑡𝐹𝑡,,𝑦(3.36) which implies 𝐹(𝑧𝑡,𝑦)≥0. From (A3), we have 𝐹(𝑢∗,𝑦)≥0 for all 𝑦∈𝐾, and hence, 𝑢∗∈EP(𝐹). Therefore, 𝑢∗∈𝐹(𝑆𝑛)∩EP(𝐹). From Lemma 2.6, we know that ⟨𝑢−𝑃ℱ𝑢,𝑢∗−𝑃ℱ𝑢⟩≤0.(3.37) It follows from (3.31) and (3.37) that limsupğ‘›â†’âˆžâŸ¨ğ‘¢âˆ’ğ‘ž,ğ‘¥ğ‘›âˆ’ğ‘žâŸ©=limğ‘–â†’âˆžî«ğ‘¢âˆ’ğ‘ž,ğ‘¥ğ‘›ğ‘–î¬âˆ’ğ‘ž=âŸ¨ğ‘¢âˆ’ğ‘ž,ğ‘¢âˆ—âˆ’ğ‘žâŸ©â‰¤0.(3.38)
Finally, we prove that ğ‘¥ğ‘›â†’ğ‘ž=𝑃ℱ𝑢 as ğ‘›â†’âˆž. From (1.4) again, we have ‖‖𝑥𝑛+1â€–â€–âˆ’ğ‘ž2=𝛽𝑛𝑢+𝛾𝑛𝑥𝑛+1âˆ’ğ›½ğ‘›âˆ’ğ›¾ğ‘›î€¸ğ‘¦ğ‘›âˆ’ğ‘ž,𝑥𝑛+1î¬âˆ’ğ‘ž=ğ›½ğ‘›î«ğ‘¢âˆ’ğ‘ž,𝑥𝑛+1î¬âˆ’ğ‘ž+ğ›¾ğ‘›î«ğ‘¥ğ‘›âˆ’ğ‘ž,𝑥𝑛+1+î€·âˆ’ğ‘ž1âˆ’ğ›½ğ‘›âˆ’ğ›¾ğ‘›ğ‘¦î€¸î«ğ‘›âˆ’ğ‘ž,𝑥𝑛+1î¬âˆ’ğ‘žâ‰¤ğ›½ğ‘›î«ğ‘¢âˆ’ğ‘ž,𝑥𝑛+1î¬âˆ’ğ‘ž+ğ›¾ğ‘›â€–â€–ğ‘¥ğ‘›â€–â€–â€–â€–ğ‘¥âˆ’ğ‘žğ‘›+1‖‖+î€·âˆ’ğ‘ž1âˆ’ğ›½ğ‘›âˆ’ğ›¾ğ‘›î€¸â€–â€–ğ‘¦ğ‘›â€–â€–â€–â€–ğ‘¥âˆ’ğ‘žğ‘›+1â€–â€–â‰¤î€·âˆ’ğ‘ž1âˆ’ğ›½ğ‘›î€¸â€–â€–ğ‘¥ğ‘›â€–â€–â€–â€–ğ‘¥âˆ’ğ‘žğ‘›+1â€–â€–âˆ’ğ‘ž+ğ›½ğ‘›î«ğ‘¢âˆ’ğ‘ž,𝑥𝑛+1î¬â‰¤âˆ’ğ‘ž1−𝛽𝑛2î‚€â€–â€–ğ‘¥ğ‘›â€–â€–âˆ’ğ‘ž2+‖‖𝑥𝑛+1â€–â€–âˆ’ğ‘ž2+ğ›½ğ‘›î«ğ‘¢âˆ’ğ‘ž,𝑥𝑛+1î¬â‰¤âˆ’ğ‘ž1−𝛽𝑛2â€–â€–ğ‘¥ğ‘›â€–â€–âˆ’ğ‘ž2+12‖‖𝑥𝑛+1â€–â€–âˆ’ğ‘ž2+ğ›½ğ‘›âŸ¨ğ‘¢âˆ’ğ‘ž,𝑥𝑛+1âˆ’ğ‘žâŸ©,(3.39) which implies that ‖‖𝑥𝑛+1â€–â€–âˆ’ğ‘ž2≤1âˆ’ğ›½ğ‘›î€¸â€–â€–ğ‘¥ğ‘›â€–â€–âˆ’ğ‘ž2+2ğ›½ğ‘›âŸ¨ğ‘¢âˆ’ğ‘ž,𝑥𝑛+1âˆ’ğ‘žâŸ©.(3.40) It follows from (3.38), (3.40), and Lemma 2.7 that limğ‘›â†’âˆžâ€–ğ‘¥ğ‘›âˆ’ğ‘žâ€–=0. This completes the proof.

As 𝑁=1, that is, 𝐴𝑛=𝑇 and 𝜂𝑖(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 𝑇∶𝐾→𝐻 be a 𝑘-strict pseudocontractions such that ⋂ℱ=𝐹(𝑇)EP(𝐹)≠𝜙. Let {𝑥𝑛} be a sequence generated in the following manner: 𝐹𝑢𝑛+1,𝑦𝑟⟨𝑦−𝑢𝑛,𝑢𝑛−𝑥𝑛𝑦⟩≥0,∀𝑦∈𝐾,𝑛=𝛼𝑛𝑢𝑛+1−𝛼𝑛𝑇𝑢𝑛,𝑥𝑛+1=𝛽𝑛𝑢+𝛾𝑛𝑥𝑛+1−𝛽𝑛−𝛾𝑛𝑦𝑛,𝑛≥0,(3.41) where 𝑢∈𝐾 and 𝑥0∈𝐾, {𝛼𝑛},{𝛽𝑛}, and {𝛾𝑛} are some sequences in (0,1). If the following control conditions are satisfied: (i)𝑘≤𝛼𝑛≤𝜆<1 for all 𝑛≥0 and limğ‘›â†’âˆžğ›¼ğ‘›=𝛼,(ii)limğ‘›â†’âˆžğ›½ğ‘›=0 and âˆ‘âˆžğ‘›=0𝛽𝑛=∞,(iii)0<liminfğ‘›â†’âˆžğ›¾ğ‘›â‰¤limsupğ‘›â†’âˆžğ›¾ğ‘›<1,then {𝑥𝑛} converges strongly to ğ‘žâˆˆâ„±, where ğ‘ž=𝑃ℱ𝑢.

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).