Strong Convergence Theorems for Equilibrium Problems and 𝑘-Strict Pseudocontractions in Hilbert Spaces

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 𝑘_𝑖-strict pseudocontractions {𝑇_𝑖}^𝑁𝑖=1 in Hilbert space 𝐻: ∀𝑥₀∈𝐾,𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑁_𝑖=1𝜆_𝑖𝑇_𝑖𝑥_𝑛,(1.3) where {𝛼_𝑛}⊂(0,1), and {𝜆_𝑖}^𝑁𝑖=1 is a finite sequence of positive numbers such that ∑^𝑁𝑖=1𝜆_𝑖=1. They proved that the sequence {𝑥_𝑛} defined by (1.3) converges weakly to a common fixed point of {𝑇_𝑖}^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𝑢_𝑛,𝑦+1𝑟⟨𝑦−𝑢^𝑛𝑦,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐾,_𝑛=𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛^𝑁_𝑖=1𝑥_𝑛+1𝜂^𝑖(𝑛)𝑇_𝑖𝑢_𝑛,=𝛽_𝑛𝑢+𝛾_𝑛𝑥_𝑛+1−𝛽_𝑛−𝛾_𝑛𝑦_𝑛,𝑛≥0,(1.4) where 𝑢∈𝐾 is a given point, {𝑇_𝑖}^{𝑁𝑖=𝑖}∶𝐾→𝐻 is a finite family of 𝑘_𝑖-strictly pseudocontractive nonself-mappings, {𝜂^𝑖(𝑛)}^𝑁𝑖=1 is a finite sequences of positive numbers, {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} 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 𝑘_𝑖-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 𝑃_𝐾 to denote the metric or nearest point projection of 𝐻 onto 𝐾; that is, for 𝑥∈𝐻, 𝑃_𝐾𝑥 is the only point in 𝐾 such that ‖𝑥−𝑃_𝐾𝑥‖=inf{‖𝑥−𝑧‖∶𝑧∈𝐾}. we write 𝑥_𝑛⇀𝑥 and 𝑥_𝑛→𝑥 indicate that the sequence {𝑥_𝑛} 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 {𝑥_𝑛} with 𝑥_𝑛⇀𝑥 and every 𝑦∈𝐻 with 𝑦≠𝑥, we have liminf_𝑛→∞‖‖𝑥_𝑛−𝑥‖‖<liminf_𝑛→∞‖‖𝑥_𝑛−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, lim_𝑡→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 𝐹(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝑧−𝑥⟩≥0,∀𝑦∈𝐾.(2.2) Further, if 𝑇_𝑟𝑥={𝑧∈𝐾∶𝐹(𝑧,𝑦)+(1/𝑟)⟨𝑦−𝑧,𝑧−𝑥⟩≥0,forall𝑦∈𝐾}, then the following holds. (1)𝑇_𝑟 is single-valued. (2)𝑇_𝑟 is firmly nonexpansive, that is, ‖𝑇_𝑟𝑥−𝑇_𝑟𝑦‖²≤⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝑥−𝑦⟩, forall𝑥,𝑦∈𝐻.(3)𝐹(𝑇_𝑟)=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 {𝑥_𝑛} and {𝑧_𝑛} be bounded sequence in Banach space 𝐸, and let {𝜆_𝑛} be a sequence in [0,1] such that 0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<1. Suppose 𝑥_𝑛+1=𝜆_𝑛𝑥_𝑛+(1−𝜆_𝑛)𝑧_𝑛 and limsup_𝑛→∞‖‖𝑧_𝑛+1−𝑧_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤0,∀𝑛≥0.(2.4) Then lim_𝑛→∞‖𝑧_𝑛−𝑥_𝑛‖=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 𝑃_𝐹(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=𝑃_𝐾𝑥 if and only if there holds the relation: ⟨𝑥−𝑧,𝑧−𝑦⟩≥0,∀𝑦∈𝐾.(2.5)

Lemma 2.7 (see [20]). Assume {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛾_𝑛𝛿_𝑛,𝑛≥0,(2.6) where {𝛾_n} is a sequence in (0,1) and {𝛿_𝑛} is a real sequence such that (i)∑^∞𝑛=0𝛾_𝑛=∞,(ii)limsup_𝑛→∞𝛿_𝑛≤0 or ∑^∞𝑛=0|𝛾_𝑛𝛿_𝑛|<∞. Then lim_𝑛→∞𝑎_𝑛=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 {𝑇_𝑖}^𝑁𝑖=1∶𝐾→𝐻 is a finite family of 𝑘_𝑖-strict pseudocontractions. Suppose that {𝜆_𝑖}^𝑁𝑖=1 is a positive sequence such that ∑^𝑁𝑖=1𝜆_𝑖=1. Then ∑^𝑁𝑖=1𝜆_𝑖𝑇_𝑖 is a 𝑘-strict pseudocontraction with 𝑘=max{𝑘_𝑖∶1≤𝑖≤𝑁}.

Proposition 2.9 (see, e.g., Acedo and Xu [12]). Let {𝑇_𝑖}^𝑁𝑖=1 and {𝜆_𝑖}^𝑁𝑖=1 be given as in Proposition 2.8 above. Then 𝐹(∑^𝑁_i=1𝜆_𝑖𝑇_𝑖)=⋂^𝑁𝑖=1𝐹(𝑇_𝑖).

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 {𝑇_𝑖}^𝑁𝑖=1∶𝐾→𝐻 be a finite family of 𝑘_𝑖-strict pseudocontractions such that 𝑘=max{𝑘_𝑖∶1≤𝑖≤𝑁} and ℱ=⋂^𝑁𝑖=1𝐹(𝑇_𝑖)⋂EP(𝐹)≠𝜙. Assume {𝜂^𝑖(𝑛)}^𝑁𝑖=1 is a finite sequences of positive numbers such that ∑^𝑁𝑖=1𝜂^𝑖(𝑛)=1 for all 𝑛≥0. Given 𝑢∈𝐾 and 𝑥₀∈𝐾, {𝛼_𝑛},{𝛽_𝑛}, and {𝛾_𝑛} are some sequences in (0,1); the following control conditions are satisfied. (i)𝑘≤𝛼_𝑛≤𝜆<1 for all 𝑛≥0 and lim_𝑛→∞𝛼_𝑛=𝛼,(ii)lim_𝑛→∞𝛽_𝑛=0 and ∑^∞𝑛=0𝛽_n=∞,(iii)0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<1,(iv)lim_𝑛→∞|𝜂^𝑖(𝑛+1)−𝜂^𝑖(𝑛)|=0. Then the sequence {𝑥_𝑛} generated by (1.4) converges strongly to 𝑞∈ℱ, where 𝑞=𝑃_ℱ𝑢.

Proof. From Lemma 2.1, we see that EP(𝐹)=𝐹(𝑇_𝑟), and note that 𝑢_𝑛 can be rewritten as 𝑢_𝑛=𝑇_𝑟𝑥_𝑛. Putting 𝐴_𝑛=∑^𝑁𝑖=1𝜂^𝑖(𝑛)𝑇_𝑖, we have 𝐴_𝑛∶𝐾→𝐻 is a 𝑘-strict pseudocontraction and 𝐹(𝐴_𝑛)=⋂^𝑁𝑖=1𝐹(𝑇_𝑖) by Propositions 2.8 and 2.9, where 𝑘=max{𝑘_𝑖∶1≤𝑖≤𝑁}.
From (1.4), condition (i), and Lemma 2.2, taking a point 𝑝∈ℱ, we have ‖‖𝑦_𝑛−𝑝‖‖²=‖‖𝛼_𝑛(𝑢_𝑛−𝑝)+(1−𝛼_𝑛)(𝐴_𝑛𝑢_𝑛−𝑝)‖‖²=𝛼_𝑛‖‖𝑢_𝑛−𝑝‖‖²+1−𝛼_𝑛‖‖𝐴_𝑛𝑢_𝑛−𝑝‖‖²−𝛼_𝑛1−𝛼_𝑛‖‖𝑢_𝑛−𝐴_𝑛𝑢_𝑛‖‖²≤−𝛼_𝑛𝛼‖‖_𝑛𝑢_𝑛−𝑝‖‖²+1−𝛼_𝑛‖‖𝑢_𝑛−𝑝‖‖²+𝑘‖‖𝑢_𝑛−𝐴_𝑛𝑢_𝑛‖‖²1−𝛼_𝑛‖‖𝑢_𝑛−𝐴_𝑛𝑢_𝑛‖‖²=‖‖𝑢_𝑛−𝑝‖‖²−1−𝛼_𝑛𝛼_𝑛−𝑘‖‖𝑢_𝑛−𝐴_𝑛𝑢_𝑛‖‖²≤‖‖𝑇_𝑟𝑥_𝑛−𝑇_𝑟𝑝‖‖²≤‖‖𝑥_𝑛−𝑝‖‖².(3.1) Furthermore, we have ‖‖𝑦_𝑛−𝑝‖‖≤‖‖𝑢_𝑛−𝑝‖‖≤‖‖𝑥_𝑛−𝑝‖‖.(3.2) It follows from (1.4) and (3.2) that ‖‖𝑥_𝑛+1−𝑝‖‖=≤‖‖𝛽𝛽_𝑛𝑛𝑢+𝛾_𝑛𝑥_𝑛+1−𝛽_𝑛−𝛾_𝑛𝑦_𝑛−𝑝‖‖≤𝛽_𝑛‖𝑢−𝑝‖+𝛾_𝑛‖‖𝑥_𝑛−𝑝‖‖+1−𝛽_𝑛−𝛾_𝑛‖‖𝑦_𝑛−𝑝‖‖≤ma‖x𝑢−‖𝑝𝑢‖−+𝑝‖1,‖‖−𝑥₀𝛽_𝑛‖‖𝑥_𝑛−𝑝‖‖−𝑝‖‖.(3.3) Consequently, sequence {𝑥_𝑛} is bounded and so are {𝑢_𝑛} and {𝑦_𝑛}.
Define a mapping 𝑇_𝑛𝑥∶=𝛼_𝑛𝑥+(1−𝛼_𝑛)𝐴_𝑛𝑥 for each 𝑥∈𝐾. Then 𝑇_𝑛∶𝐾→𝐻 is nonexpansive. Indeed, by using (1.1), condition (i), and Lemma 2.2, we have for all 𝑥,𝑦∈𝐾 that ‖‖𝑇_𝑛𝑥−𝑇_𝑛𝑦‖‖²=‖‖𝛼_𝑛(𝑥−𝑦)+1−𝛼_𝑛𝐴_𝑛𝑥−𝐴_𝑛𝑦‖‖²=𝛼_𝑛‖𝑥−𝑦‖²+1−𝛼_𝑛‖‖𝐴_𝑛𝑥−𝐴_𝑛𝑦‖‖²−𝛼_𝑛1−𝛼_𝑛‖‖𝑥−𝐴_𝑛𝑥−𝑦−𝐴_𝑛𝑦‖‖²≤−𝛼_𝑛𝛼‖_𝑛𝑥−𝑦‖²+1−𝛼_𝑛‖𝑥−𝑦‖²+𝑘‖‖𝑥−𝐴_𝑛𝑥−𝑦−𝐴_𝑛𝑦‖‖²1−𝛼_𝑛‖‖𝑥−𝐴_𝑛𝑥−𝑦−𝐴_𝑛𝑦‖‖²=‖𝑥−𝑦‖²−1−𝛼_𝑛𝛼_𝑛−𝑘‖‖𝑥−𝐴_𝑛𝑥−𝑦−𝐴_𝑛𝑦‖‖²≤‖𝑥−𝑦‖²,(3.4) which shows that 𝑇_𝑛∶𝐾→𝐻 is nonexpansive.
Next we show that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. Setting 𝑥_𝑛+1=𝛾_𝑛𝑥_𝑛+(1−𝛾_𝑛)𝑧_𝑛, we have 𝑧_𝑛+1−𝑧_𝑛=𝑥^𝑛+2−𝛾_𝑛+1𝑥_𝑛+11−𝛾_𝑛+1−𝑥^𝑛+1−𝛾_𝑛𝑥_𝑛1−𝛾_𝑛=𝛽^𝑛+1𝑢+1−𝛽_𝑛+1−𝛾_𝑛+1𝑦_𝑛+11−𝛾_𝑛+1−𝛽^𝑛𝑢+1−𝛽_𝑛−𝛾_𝑛𝑦_𝑛1−𝛾_𝑛=𝛽^𝑛+11−𝛾_𝑛+1𝑢−𝑦_𝑛+1+𝑦_𝑛+1−𝑦_𝑛−𝛽^𝑛1−𝛾_𝑛𝑢−𝑦_𝑛.(3.5) It follows that ‖‖𝑧_𝑛+1−𝑧_𝑛‖‖≤𝛽^𝑛+11−𝛾_𝑛+1‖‖𝑢−𝑦_𝑛+1‖‖+‖‖𝑦_𝑛+1−𝑦_𝑛‖‖+𝛽^𝑛1−𝛾_𝑛‖‖𝑢−𝑦_𝑛‖‖.(3.6) From (1.4), we have 𝑦_𝑛=𝑇_𝑛𝑢_𝑛 and ‖‖𝑦_𝑛+1−𝑦_𝑛‖‖≤≤‖‖‖‖𝑇𝑢_𝑛+1𝑛+1𝑢_𝑛+1−𝑇_𝑛+1𝑢_𝑛‖‖+‖‖𝑇_𝑛+1𝑢_𝑛−𝑇_𝑛𝑢_𝑛‖‖≤‖‖𝑢_𝑛+1−𝑢_𝑛‖‖+‖‖𝛼_𝑛+1𝑢_𝑛+1−𝛼_𝑛+1𝐴_𝑛+1𝑢_𝑛−𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝐴_𝑛𝑢_𝑛‖‖≤‖‖𝑢_𝑛+1−𝑢_𝑛‖‖+||𝛼_𝑛+1−𝛼_𝑛||‖‖𝑢_𝑛−𝐴_𝑛𝑢_𝑛‖‖+1−𝛼_𝑛+1‖‖𝐴_𝑛+1𝑢_𝑛−𝐴_𝑛𝑢_𝑛‖‖−𝑢_𝑛‖‖+||𝛼_𝑛+1−𝛼_𝑛||‖‖𝑢_𝑛−𝐴_𝑛𝑢_𝑛‖‖+1−𝛼_𝑛+1^𝑁_𝑖=1||𝜂^𝑖(𝑛+1)−𝜂^𝑖(𝑛)||‖‖𝑇_𝑖𝑢_𝑛‖‖.(3.7) By Lemma 2.1, 𝑢_𝑛=𝑇_𝑟𝑥_𝑛 and 𝑢_𝑛+1=𝑇_𝑟𝑥_𝑛+1, we have 𝐹𝑢_𝑛,𝑦+1𝑟⟨𝑦−𝑢^𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐾,(3.8)𝐹𝑢_𝑛+1,𝑦+1𝑟𝑦−𝑢^𝑛+1,𝑢_𝑛+1−𝑥_𝑛+1≥0,∀𝑦∈𝐾.(3.9) Putting 𝑦=𝑢_𝑛+1 in (3.8) and 𝑦=𝑢_𝑛 in (3.9), we obtain 𝐹𝑢_𝑛,𝑢_𝑛+1+1𝑟𝑢^𝑛+1𝐹𝑢−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥0_𝑛+1,,𝑢_𝑛+1𝑟𝑢^𝑛−𝑢_𝑛+1,𝑢_𝑛+1−𝑥_𝑛+1≥0.(3.10) So, from (A2) and 𝑟>0, we have 𝑢_𝑛+1−𝑢_𝑛,𝑢_𝑛−𝑢_𝑛+1+𝑢_𝑛+1−𝑥_𝑛−𝑢_𝑛+1−𝑥_𝑛+1≥0,(3.11) and hence ‖‖𝑢_𝑛+1−𝑢_𝑛‖‖²≤𝑢_𝑛+1−𝑢_𝑛,𝑥_𝑛+1−𝑥_𝑛,(3.12) which implies that ‖‖𝑢_𝑛+1−𝑢_𝑛‖‖≤‖‖𝑥_𝑛+1−𝑥_𝑛‖‖.(3.13) Combining (3.6), (3.7), and (3.13), we have ‖‖𝑧_𝑛+1−𝑧_𝑛‖‖≤𝛽^𝑛+11−𝛾_𝑛+1‖‖𝑢−𝑦_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+𝛽^𝑛1−𝛾_𝑛+||𝛼_𝑛+1‖‖𝑢−𝑦_𝑛‖‖−𝛼_𝑛||‖‖𝑢_𝑛−𝐴_𝑛𝑢_𝑛‖‖+1−𝛼_𝑛+1^𝑁_𝑖=1||𝜂^𝑖(𝑛+1)−𝜂^𝑖(𝑛)||‖‖𝑇_𝑖𝑢_𝑛‖‖.(3.14) This together with (i), (ii) and (iv) imply that limsup_𝑛→∞‖‖𝑧_𝑛+1−𝑧_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤0.(3.15) Hence, by Lemma 2.3, we obtain lim_𝑛→∞‖‖𝑧_𝑛−𝑥_𝑛‖‖=0.(3.16) Consequently, lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=lim_𝑛→∞1−𝛾_𝑛‖‖𝑧_𝑛−𝑥_𝑛‖‖=0.(3.17)
On the other hand, by (1.4) and (iii), we have ‖‖𝑥_𝑛+1−𝑦_𝑛‖‖≤𝛽_𝑛‖‖𝑢−𝑦_𝑛‖‖+𝛾_𝑛‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+𝛾_𝑛‖‖𝑥_𝑛+1−𝑦_𝑛‖‖,(3.18) which implies that ‖‖𝑥_𝑛+1−𝑦_𝑛‖‖≤𝛽^𝑛1−𝛾_𝑛‖‖𝑢−𝑦_𝑛‖‖+𝛾^𝑛1−𝛾_𝑛‖‖𝑥_𝑛−𝑥_𝑛+1‖‖.(3.19) Combining (ii), (3.17), and (3.19), we have lim_𝑛→∞‖‖𝑥_𝑛+1−𝑦_𝑛‖‖=0.(3.20) Note that ‖‖𝑥_𝑛−𝑦_𝑛‖‖≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑦_𝑛‖‖,(3.21) which together with (3.17) and (3.20) implies lim_𝑛→∞‖‖𝑥_𝑛−𝑦_𝑛‖‖=0.(3.22) Moreover, for 𝑝∈ℱ=⋂^𝑁𝑖=1𝐹(𝑇_𝑖)⋂EP(𝐹), we have ‖‖𝑢_𝑛−𝑝‖‖²==‖‖1𝑇_𝑟𝑥_𝑛−𝑇_𝑟𝑝‖‖²≤⟨𝑥_𝑛−𝑝,𝑢_𝑛−𝑝⟩2‖‖𝑥^𝑛−𝑝‖‖²+‖‖𝑢_𝑛−𝑝‖‖²−‖‖𝑥_𝑛−𝑢_𝑛‖‖²,(3.23) and hence ‖‖𝑢_𝑛−𝑝‖‖²≤‖‖𝑥_𝑛−𝑝‖‖²−‖‖𝑥_𝑛−𝑢_𝑛‖‖².(3.24) From Lemma 2.2, (3.2) and (3.24), we have ‖‖𝑥_𝑛+1−𝑝‖‖²=‖‖𝛽_𝑛𝑢+𝛾_𝑛𝑥_𝑛+(1−𝛽_𝑛−𝛾_𝑛)𝑦_𝑛−𝑝‖‖²≤𝛽_𝑛‖𝑢−𝑝‖²+𝛾_𝑛‖‖𝑥_𝑛−𝑝‖‖²+1−𝛽_𝑛−𝛾_𝑛‖‖𝑦_𝑛−𝑝‖‖²≤≤𝛽𝛽_𝑛𝑛‖𝑢−𝑝‖²+𝛾_𝑛‖‖𝑥_𝑛−𝑝‖‖²+1−𝛽_𝑛−𝛾_𝑛‖‖𝑥_𝑛−𝑝‖‖²−‖‖𝑥_𝑛−𝑢_𝑛‖‖²‖𝑢−𝑝‖²+‖‖𝑥_𝑛−𝑝‖‖²−1−𝛽_𝑛−𝛾_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖²,(3.25) and hence 1−𝛽_𝑛−𝛾_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖²≤𝛽_𝑛‖𝑢−𝑝‖²+‖‖𝑥_𝑛−𝑝‖‖²−‖‖𝑥_𝑛+1−𝑝‖‖²≤𝛽_𝑛‖𝑢−𝑝‖²+‖‖𝑥_𝑛−𝑥_𝑛+1‖‖‖‖𝑥_𝑛−𝑝‖‖+‖‖𝑥_𝑛+1−𝑝‖‖.(3.26) By (ii) and (3.17), we obtain lim_𝑛→∞‖‖𝑥_𝑛−𝑢_𝑛‖‖=0.(3.27) It follows from (3.22) and (3.27) that lim_𝑛→∞‖‖𝑦_𝑛−𝑢_𝑛‖‖=0.(3.28) Define 𝑆_𝑛∶𝐾→𝐻 by 𝑆_𝑛𝑥=𝛼𝑥+(1−𝛼)𝐴_𝑛𝑥. Then, 𝑆_𝑛 is a nonexpansive with 𝐹(𝑆_𝑛)=𝐹(𝐴_𝑛) by Lemma 2.4. Note that lim_𝑛→∞𝛼_𝑛=𝛼∈[𝑘,1) by condition (i) and ‖‖𝑢_𝑛−𝑆_𝑛𝑢_𝑛‖‖≤≤‖‖‖‖𝑢𝑢_𝑛𝑛−𝑦_𝑛‖‖+‖‖𝑦_𝑛−𝑆_𝑛𝑢_𝑛‖‖≤‖‖𝑢_𝑛−𝑦_𝑛‖‖+‖‖𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝐴_𝑛𝑢_𝑛−𝛼𝑢_𝑛+(1−𝛼)𝐴_𝑛𝑢_𝑛‖‖−𝑦_𝑛‖‖+||𝛼_𝑛−𝛼||‖‖𝑢_𝑛−𝐴_𝑛𝑢_𝑛‖‖,(3.29) which combines with condition (i) and (3.28) yielding that lim_𝑛→∞‖‖𝑢_𝑛−𝑆_𝑛𝑢_𝑛‖‖=0.(3.30)
We now show that limsup_𝑛→∞⟨𝑢−𝑞,𝑥_𝑛−𝑞⟩≤0, where 𝑞=𝑃_ℱ𝑢. To see this, we choose a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that limsup_𝑛→∞⟨𝑢−𝑞,𝑥_𝑛−𝑞⟩=lim_𝑖→∞𝑢−𝑞,𝑥_𝑛𝑖−𝑞.(3.31) Since {𝑢_𝑛𝑖} is bounded, there exists a subsequence {𝑢_𝑛𝑖𝑗} of {𝑢_𝑛𝑖} converging weakly to 𝑢^∗. Without loss of generality, we assume that 𝑢_𝑛𝑖⇀𝑢^∗ as 𝑖→∞. Form (3.27), we obtain 𝑥_𝑛𝑖⇀𝑢^∗ as 𝑖→∞. Since 𝐾 is closed and convex, 𝐾 is weakly closed. So, we have 𝑢^∗∈𝐾 and 𝑢^∗∈𝐹(𝑆_𝑛). Otherwise, from 𝑢^∗≠𝑆_𝑛𝑢^∗ and Opial's condition, we obtain liminf_𝑖→∞‖‖𝑢_𝑛𝑖−𝑢^∗‖‖<≤lliimmiinnf_𝑖→∞f_𝑖→∞‖‖𝑢_𝑛𝑖−𝑆_𝑛𝑖𝑢^∗‖‖≤liminf_𝑖→∞‖‖𝑢_𝑛𝑖−𝑆_𝑛𝑖𝑢_𝑛𝑖‖‖+‖‖𝑆_𝑛𝑖𝑢_𝑛𝑖−𝑆_𝑛𝑖𝑢^∗‖‖‖‖𝑢_𝑛𝑖−𝑢^∗‖‖.(3.32) This is a contradiction. Hence, we get 𝑢^∗∈𝐹(𝑆_𝑛)=𝐹(𝐴_𝑛). Moreover, by 𝑢_𝑛=𝑇_𝑟𝑥_𝑛, we have 𝐹𝑢_𝑛,𝑦+1𝑟⟨𝑦−𝑢^𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐾.(3.33) It follows from (A2) that 1𝑟⟨𝑦−𝑢^𝑛,𝑢_𝑛−𝑥_𝑛⟩≥𝐹𝑦,𝑢_𝑛.(3.34) Replacing 𝑛 by 𝑛_𝑖, we have 1𝑟𝑦−𝑢^𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖≥𝐹𝑦,𝑢_𝑛𝑖.(3.35) Since 𝑢_𝑛𝑖−𝑥_𝑛𝑖→0 and 𝑢_𝑛𝑖⇀𝑢^∗, it follows from (A4) that 𝐹(𝑦,𝑢^∗)≤0 for all 𝑦∈𝐾. Put 𝑧_𝑡=𝑡𝑦+(1−𝑡)𝑢^∗ for all 𝑡∈(0,1] and 𝑦∈𝐾. Then, we have 𝑧_𝑡∈𝐾, and hence, 𝐹(𝑧_𝑡,𝑢^∗)≤0. By (A1) and (A4), we have 0=𝐹𝑧_𝑡,𝑧_𝑡≤𝑡𝐹𝑧_𝑡,𝑦+(1−𝑡)𝐹𝑧_𝑡,𝑢^∗≤𝑡𝐹𝑧_𝑡,𝑦,(3.36) which implies 𝐹(𝑧_𝑡,𝑦)≥0. From (A3), we have 𝐹(𝑢^∗,𝑦)≥0 for all 𝑦∈𝐾, and hence, 𝑢^∗∈EP(𝐹). Therefore, 𝑢^∗∈𝐹(𝑆_𝑛)∩EP(𝐹). From Lemma 2.6, we know that ⟨𝑢−𝑃_ℱ𝑢,𝑢^∗−𝑃_ℱ𝑢⟩≤0.(3.37) It follows from (3.31) and (3.37) that limsup_𝑛→∞⟨𝑢−𝑞,𝑥_𝑛−𝑞⟩=lim_𝑖→∞𝑢−𝑞,𝑥_𝑛𝑖−𝑞=⟨𝑢−𝑞,𝑢^∗−𝑞⟩≤0.(3.38)
Finally, we prove that 𝑥_𝑛→𝑞=𝑃_ℱ𝑢 as 𝑛→∞. From (1.4) again, we have ‖‖𝑥_𝑛+1−𝑞‖‖²==𝛽_𝑛𝛽_𝑛𝑢+𝛾_𝑛𝑥_𝑛+1−𝛽_𝑛−𝛾_𝑛𝑦_𝑛−𝑞,𝑥_𝑛+1−𝑞≤𝛽_𝑛𝑢−𝑞,𝑥_𝑛+1−𝑞+𝛾_𝑛𝑥_𝑛−𝑞,𝑥_𝑛+1−𝑞+1−𝛽_𝑛−𝛾_𝑛𝑦_𝑛−𝑞,𝑥_𝑛+1−𝑞≤1−𝑢𝛽−_𝑛𝑞,𝑥_𝑛+1−𝑞+𝛾_𝑛‖‖𝑥_𝑛−𝑞‖‖‖‖𝑥_𝑛+1−𝑞‖‖+1−𝛽_𝑛−𝛾_𝑛‖‖𝑦_𝑛−𝑞‖‖‖‖𝑥_𝑛+1−𝑞‖‖≤1−𝛽_𝑛‖‖𝑥_𝑛−𝑞‖‖‖‖𝑥_𝑛+1−𝑞‖‖+𝛽_𝑛𝑢−𝑞,𝑥_𝑛+1−𝑞2‖‖𝑥^𝑛≤1−𝛽_𝑛−𝑞‖‖²+‖‖𝑥_𝑛+1−𝑞‖‖²+𝛽^𝑛𝑢−𝑞,𝑥_𝑛+1−𝑞2‖‖𝑥^𝑛−𝑞‖‖²+12‖‖𝑥^𝑛+1−𝑞‖‖²+𝛽_𝑛⟨𝑢−𝑞,𝑥_𝑛+1−𝑞⟩,(3.39) which implies that ‖‖𝑥_𝑛+1−𝑞‖‖²≤1−𝛽_𝑛‖‖𝑥_𝑛−𝑞‖‖²+2𝛽_𝑛⟨𝑢−𝑞,𝑥_𝑛+1−𝑞⟩.(3.40) It follows from (3.38), (3.40), and Lemma 2.7 that lim_𝑛→∞‖𝑥_𝑛−𝑞‖=0. This completes the proof.

As 𝑁=1, that is, 𝐴_𝑛=𝑇 and 𝜂^𝑖(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 {𝑥_𝑛} be a sequence generated in the following manner: 𝐹𝑢_𝑛,𝑦+1𝑟⟨𝑦−𝑢^𝑛𝑦,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐾,_𝑛𝑥_𝑛+1=𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝑇𝑢_𝑛,=𝛽_𝑛𝑢+𝛾_𝑛𝑥_𝑛+1−𝛽_𝑛−𝛾_𝑛𝑦_𝑛,𝑛≥0,(3.41) where 𝑢∈𝐾 and 𝑥₀∈𝐾, {𝛼_𝑛},{𝛽_𝑛}, and {𝛾_𝑛} are some sequences in (0,1). If the following control conditions are satisfied: (i)𝑘≤𝛼_𝑛≤𝜆<1 for all 𝑛≥0 and lim_𝑛→∞𝛼_𝑛=𝛼,(ii)lim_𝑛→∞𝛽_𝑛=0 and ∑^∞𝑛=0𝛽_𝑛=∞,(iii)0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<1,then {𝑥_𝑛} converges strongly to 𝑞∈ℱ, where 𝑞=𝑃_ℱ𝑢.

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).