Convergence Theorems for Equilibrium Problems and Fixed-Point Problems of an Infinite Family of 𝑘_𝑖-Strictly Pseudocontractive Mapping in Hilbert Spaces

Abstract

We first extend the definition of W_n from an infinite family of nonexpansive mappings to an infinite family of strictly pseudocontractive mappings, and then propose an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of an infinite family of 𝑘_𝑖-strictly pseudocontractive mappings in Hilbert spaces. The results obtained in this paper extend and improve the recent ones announced by many others. Furthermore, a numerical example is presented to illustrate the effectiveness of the proposed scheme.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and induced norm ‖⋅‖. Let 𝐶 be a nonempty closed convex subset of 𝐻 and let 𝐹∶𝐶×𝐶→𝑅 be a bifunction. We consider the following equilibrium problem (EP) which is to find 𝑧∈𝐶 such that EP∶𝐹(𝑧,𝑦)≥0,∀𝑦∈𝐶.(1.1) Denote the set of solutions of EP by EP(𝐹). Given a mapping 𝑇∶𝐶→𝐻, let 𝐹(𝑥,𝑦)=⟨𝑇𝑥,𝑦−𝑥⟩ for all 𝑥,𝑦∈𝐶. Then, 𝑧∈EP(𝐹) if and only if ⟨𝑇𝑥,𝑦−𝑥⟩≥0 for all 𝑦∈𝐶, 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 [1–13].

A mapping 𝐵∶𝐶→𝐶 is called 𝜃-Lipschitzian if there exists a positive constant 𝜃 such that ‖𝐵𝑥−𝐵𝑦‖≤𝜃‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(1.2)𝐵 is said to be 𝜂-strongly monotone if there exists a positive constant 𝜂 such that ⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥𝜂‖𝑥−𝑦‖²,∀𝑥,𝑦∈𝐶.(1.3) A mapping 𝑆∶𝐶→𝐶 is said to be 𝑘-strictly pseudocontractive mapping if there exists a constant 0≤𝑘<1 such that ‖S𝑥−𝑆𝑦‖²≤‖𝑥−𝑦‖²+𝑘‖(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦‖²,(1.4) for all 𝑥,𝑦∈𝐶 and 𝐹(𝑆) denotes the set of fixed point of the mapping 𝑆, that is 𝐹(𝑆)={𝑥∈𝐶∶𝑆𝑥=𝑥}.

If 𝑘=1, then 𝑆 is said to a pseudocontractive mapping, that is, ‖𝑆𝑥−𝑆𝑦‖²≤‖𝑥−𝑦‖²+‖(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦‖²,(1.5) is equivalent to ⟨(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦,𝑥−𝑦⟩≥0,(1.6) for all 𝑥,𝑦∈𝐶.

The class of 𝑘-strict pseudo-contractive mappings extends the class of nonexpansive mappings (A mapping 𝑇 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖, for all 𝑥,𝑦∈𝐶). That is, 𝑆 is nonexpansive if and only if 𝑆 is a 0-strict pseudocontractive mapping. Clearly, the class of 𝑘-strictly pseudocontractive mappings falls into the one between classes of nonexpansive mappings and pseudo-contractive mapping.

In 2006, Marino and Xu [14] introduced the general iterative method and proved that for a given 𝑥₀∈𝐻, the sequence {𝑥_𝑛} generated by the algorithm 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐵𝑇𝑥_𝑛,𝑛∈𝑁,(1.7) where 𝑇 is a self-nonexpansive mapping on 𝐻, 𝑓 is an 𝛼-contraction of 𝐻 into itself (i.e., ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝛼‖𝑥−𝑦‖, for all 𝑥,𝑦∈𝐻 and 𝛼∈(0,1)), {𝛼_𝑛}⊂(0,1) satisfies certain conditions, 𝐵 is strongly positive bounded linear operator on 𝐻, and converges strongly to fixed point 𝑥^∗ of 𝑇 which is the unique solution to the following variational inequality: ⟨(𝛾𝑓−𝐵)𝑥^∗,𝑥^∗−𝑥⟩≤0,∀𝑥∈𝐹(𝑇).(1.8)

Tian [15] considered the following iterative method, for a nonexpansive mapping 𝑇:𝐻→𝐻 with 𝐹(𝑇)≠∅, 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐹𝑇𝑥_𝑛,𝑛∈𝑁,(1.9) where 𝐹 is 𝑘-Lipschitzian and 𝜂-strongly monotone operator. The sequence {𝑥_𝑛} converges strongly to fixed-point 𝑞 in 𝐹(𝑇) which is the unique solution to the following variational inequality: ⟨(𝛾𝑓−𝜇𝐹)𝑞,𝑝−𝑞⟩≤0,𝑝∈𝐹(𝑇).(1.10)

For finding a common element of EP(𝐹)∩𝐹(𝑆), S. Takahashi and W. Takahashi [16] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solution (1.1) and the set of fixed points of a nonexpansive mapping in a Hilbert space. Let 𝑆∶𝐶→𝐻 be a nonexpansive mapping. Starting with arbitrary initial point 𝑥₁∈𝐻, define sequences {𝑥_𝑛} and {𝑢_𝑛} recursively by 𝐹𝑢_𝑛,𝑦+1𝑟_𝑛𝑥⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑆𝑢_𝑛,∀𝑛∈𝑁.(1.11) They proved that under certain appropriate conditions imposed on {𝛼_𝑛} and {𝑟_𝑛}, the sequences {𝑥_𝑛} and {𝑢_𝑛} converge strongly to 𝑧∈𝐹(𝑆)∩EP(𝐹), where 𝑧=𝑃_{𝐹(𝑆)∩EP(𝐹)}𝑓(𝑧).

Liu [17] introduced the following scheme: 𝑥₁∈𝐻 and 𝐹𝑢_𝑛,𝑦+1𝑟_𝑛𝑦⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛𝑥_𝑛+1=𝛽_𝑛𝑢_𝑛+1−𝛽_𝑛𝑆𝑢_𝑛,=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐵𝑦_𝑛,∀𝑛∈𝑁,(1.12) where 𝑆 is a 𝑘-strict pseudo-contractive mapping and 𝐵 is a strongly positive bounded linear operator. They proved that under certain appropriate conditions imposed on {𝛼_𝑛},{𝛽_𝑛}, and {𝑟_𝑛}, the sequence {𝑥_𝑛} converges strongly to 𝑧∈𝐹(𝑆)∩EP(𝐹), where 𝑧=𝑃_{𝐹(𝑆)∩EP(𝐹)}(𝐼−𝐵+𝛾𝑓)(𝑧).

In [18], the concept of 𝑊 mapping had been modified for a countable family {𝑇_𝑛}_𝑛∈𝑁 of nonexpansive mappings by defining the sequence {𝑊_𝑛}_𝑛∈𝑁 of 𝑊-mappings generated by {𝑇_𝑛}_𝑛∈𝑁 and {𝜆_𝑛}⊂(0,1), proceeding backward 𝑈𝑈_𝑛,𝑛+1∶=𝐼,_𝑛,𝑛𝑈∶=𝜆_𝑛𝑇_𝑛𝑈⋅⋅⋅_{𝑛,𝑛+1𝑛,𝑘}+1−𝜆_𝑛𝐼,𝑈∶=𝜆_𝑘𝑇_𝑘𝑈⋅⋅⋅_{𝑛,𝑘+1𝑛,2}+1−𝜆_𝑘𝐼,𝑊∶=𝜆₂𝑇₂𝑈_𝑛,3+1−𝜆₂𝐼,_𝑛=𝑈_𝑛,1∶=𝜆₁𝑇₁𝑈_𝑛,2+1−𝜆₁𝐼.(1.13) Yao et al. [19] using this concept, introduced the following algorithm: 𝑥₁∈𝐻 and 𝐹𝑢_𝑛,𝑦+1𝑟_𝑛𝑥⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛼_𝑛−𝛽_𝑛𝑊_𝑛𝑢_𝑛,∀𝑛∈𝑁.(1.14) They proved that under certain appropriate conditions imposed on {𝛼_𝑛} and {𝑟_𝑛}, the sequences {𝑥_𝑛} and {𝑢_𝑛} converge strongly to 𝑧∈⋂^∞𝑖=1𝐹(𝑇_𝑖)∩EP(𝐹).

Colao and Marino [20] considered the following explicit viscosity scheme 𝐹𝑢_𝑛,𝑦+1𝑟_𝑛𝑥⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝑟𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑊_𝑛𝑢_𝑛,∀𝑛∈𝑁,(1.15) where 𝐴 is a strongly positive operator on 𝐻. Under certain appropriate conditions, the sequences {𝑥_𝑛} and {𝑢_𝑛} converge strongly to 𝑧∈⋂^∞𝑖=1𝐹(𝑇_𝑖)∩EP(𝐹).

Motivated and inspired by these facts, in this paper, we first extend the definition of 𝑊_𝑛 from an infinite family of nonexpansive mappings to an infinite family of strictly pseudo-contractive mappings, and then propose the iteration scheme (3.2) for finding an element of EP(𝐹)⋂^∞𝑖=1𝐹(𝑆_𝑖), where {𝑆_𝑖} is an infinite family of 𝑘_𝑖-strictly pseudo-contractive mappings of 𝐶 into itself. Finally, the convergence theorem of the iteration scheme is obtained. Our results include Yao et al. [19], Colao and Marino [20] as some special cases.

2. Preliminaries

Throughout this paper, we always assume that 𝐶 is a nonempty closed convex subset of a Hilbert space 𝐻. We write 𝑥_𝑛⇀𝑥 to indicate that the sequence {𝑥_𝑛} converges weakly to 𝑥. 𝑥_𝑛→𝑥 implies that {𝑥_𝑛} converges strongly to 𝑥. We denote by 𝑁 and 𝑅 the sets of positive integers and real numbers, respectively. For any 𝑥∈𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃_𝐶𝑥, such that ‖‖𝑥−𝑃_𝐶𝑥‖‖≤‖𝑥−𝑦‖,∀𝑦∈𝐶.(2.1) Such a 𝑃_𝐶 is called the metric projection of 𝐻 onto 𝐶. It is known that 𝑃_𝐶 is nonexpansive. Furthermore, for 𝑥∈𝐻 and 𝑢∈𝐶, 𝑢=𝑃_𝐶𝑥⟺⟨𝑥−𝑢,𝑢−𝑦⟩≥0,∀𝑦∈𝐶.(2.2) It is widely known that 𝐻 satisfies Opials condition [21], that is, for any sequence {𝑥_𝑛} with 𝑥_𝑛⇀𝑥, the inequality lim_𝑛→∞inf‖‖𝑥_𝑛−𝑥‖‖<lim_𝑛→∞inf‖‖𝑥_𝑛−𝑦‖‖(2.3) holds for every 𝑦∈𝐻 with 𝑦≠𝑥.

In order to solve the equilibrium problem for a bifunction 𝐹∶𝐶×𝐶→𝑅, we assume that 𝐹 satisfies the following conditions:(A1)𝐹(𝑥,𝑥)=0,for all 𝑥∈𝐶.(A2)𝐹 is monotone, that is, 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)≤0, for all 𝑥,𝑦∈𝐶.(A3)lim_𝑡↓0𝐹(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝐹(𝑥,𝑦), for all 𝑥,𝑦,𝑧∈𝐶.(A4) For each 𝑥∈𝐶,𝑦↦𝐹(𝑥,𝑦) is convex and lower semicontinuous.

Let us recall the following lemmas which will be useful for our paper.

Lemma 2.1 2.1 (see [22]). Let 𝐹 be a bifunction from 𝐶×𝐶 into 𝑅 satisfying (A1), (A2), (A3), and (A4). Then, for any 𝑟>0 and 𝑥∈𝐻, there exists 𝑧∈𝐶 such that 𝐹(𝑧,𝑦)+1𝑟(𝑦−𝑧,𝑧−𝑥)≥0,∀𝑦∈𝐶.(2.4) Furthermore, if 𝑇_𝑟𝑥={𝑧∈𝐶∶𝐹(𝑧,𝑦)+(1/𝑟)(𝑦−𝑧,𝑧−𝑥)≥0,∀𝑦∈𝐶}, then the following hold:(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⟩,∀𝑥,𝑦∈𝐻.(2.5)(3)𝐹(𝑇_𝑟)=EP(𝐹). (4)EP(𝐹) is closed and convex.

Lemma 2.2 2.2 (see [23]). Let 𝑆∶𝐶→𝐻 be a k-strictly pseudo-contractive mapping. Define 𝑇∶𝐶→𝐻 by 𝑇𝑥=𝜆𝑥+(1−𝜆)𝑆𝑥 for each 𝑥∈𝐶. Then, as 𝜆∈[𝑘,1), 𝑇 is nonexpansive mapping such that 𝐹(𝑇)=𝐹(𝑆).

Lemma 2.3 2.3 (see [24]). In a Hilbert space 𝐻, there holds the inequality ‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩,∀𝑥,𝑦∈𝐻.(2.6)

Lemma 2.4 (see [25]). Let 𝐻 be a Hilbert space and 𝐶 be a closed convex subset of 𝐻, and 𝑇∶𝐶→𝐶 a nonexpansive mapping with 𝐹(𝑇)≠∅. If {𝑥_𝑛} is a sequence in 𝐶 weakly converging to 𝑥 and if {(𝐼−𝑇)𝑥_𝑛} converges strongly to 𝑦, then (𝐼−𝑇)𝑥=𝑦.

Lemma 2.5 (see [26]). Let {𝑥_𝑛} and {𝑧_𝑛} be bounded sequences in a Banach space E and {𝛾_𝑛} be a sequence in [0,1] satisfying the following condition 0<lim_𝑛→∞inf𝛾_𝑛≤lim_𝑛→∞sup𝛾_𝑛<1.(2.7) Suppose that 𝑥_𝑛+1=𝛾_𝑛𝑥_𝑛+(1−𝛾_𝑛)𝑧_𝑛,𝑛≥0 and lim_𝑛→∞sup(‖𝑧_𝑛+1−𝑧_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. Then lim_𝑛→∞‖𝑧_𝑛−𝑥_𝑛‖=0.

Lemma 2.6 (see [27]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝑏_𝑛𝑎_𝑛+𝑏_𝑛𝛿_𝑛,𝑛≥0,(2.8) where {𝑏_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence in 𝑅, such that(i)∑^∞𝑖=1𝑏_𝑖=∞.(ii)lim_𝑛→∞sup𝛿_𝑛≤0 or ∑^∞𝑖=1|𝑏_𝑛𝛿_𝑛|<∞.
Then, lim_𝑛→∞𝑎_𝑛=0.

Let {𝑆_𝑖} be an infinite family of 𝑘_𝑖-strictly pseudo-contractive mappings of 𝐶 into itself, we define a mapping 𝑊_𝑛 of 𝐶 into itself as follows, 𝑈𝑈_𝑛,𝑛+1∶=𝐼,_𝑛,𝑛𝑈∶=𝜏_𝑛𝑆^𝑛𝑈⋅⋅⋅_{𝑛,𝑛+1𝑛,𝑘}+1−𝜏_𝑛𝐼,𝑈∶=𝜏_𝑘𝑆^𝑘𝑈⋅⋅⋅_{𝑛,𝑘+1𝑛,2}+1−𝜏_𝑘𝐼,𝑊∶=𝜏₂𝑆^2𝑈_𝑛,3+1−𝜏₂𝐼,_𝑛=𝑈_𝑛,1∶=𝜏₁𝑆^1𝑈_𝑛,2+1−𝜏₁𝐼,(2.9) where 0≤𝜏_𝑖≤1, 𝑆^𝑖=𝜎_𝑖𝐼+(1−𝜎_𝑖)𝑆_𝑖 and 𝜎_𝑖∈[𝑘_𝑖,1) for 𝑖∈𝑁. We can obtain 𝑆^𝑖 is a nonexpansive mapping and 𝐹(𝑆_𝑖)=𝐹(𝑆^𝑖) by Lemma 2.2. Furthermore, we obtain that 𝑊_𝑛 is a nonexpansive mapping.

Remark 2.7. If 𝑘_𝑖=0, and 𝜎_𝑖=0 for 𝑖∈𝑁, then the definition of 𝑊_𝑛 in (2.9) reduces to the definition of 𝑊_𝑛 in (1.13).

To establish our results, we need the following technical lemmas.

Lemma 2.8 (see [18]). Let C be a nonempty closed convex subset of a strictly convex Banach space. Let {𝑆^𝑖} be an infinite family of nonexpansive mappings of 𝐶 into itself and let {𝜏_𝑖} be a real sequence such that 0<𝜏_𝑖≤𝑏<1 for every 𝑖∈𝑁. Then, for every 𝑥∈𝐶 and 𝑘∈𝑁, the limit lim_𝑛→∞𝑈_𝑛,𝑘𝑥 exists.

In view of the previous lemma, we will define 𝑊𝑥∶=lim_𝑛→∞𝑊_𝑛𝑥=lim_𝑛→∞𝑈_𝑛,1𝑥,𝑥∈𝐶.(2.10)

Lemma 2.9 (see [18]). Let 𝐶 be a nonempty closed convex subset of a strictly convex Banach space. Let {𝑆^𝑖} be an infinite family of nonexpansive mappings of 𝐶 into itself such that ⋂^∞𝑖=1𝐹(𝑆^𝑖)≠∅ and let {𝜏_𝑖} be a real sequence such that 0<𝜏_𝑖≤𝑏<1 for every 𝑖∈𝑁. Then, 𝐹(𝑊)=⋂^∞𝑖=1𝐹(𝑆^𝑖)≠∅.

The following lemmas follow from Lemmas 2.2, 2.8, and 2.9.

Lemma 2.10. Let 𝐶 be a nonempty closed convex subset of a strictly convex Banach space. Let {𝑆_𝑖} be an infinite family of 𝑘_𝑖-strictly pseudo-contractive mappings of 𝐶 into itself such that ⋂^∞𝑖=1𝐹(𝑆_𝑖)≠∅. Define 𝑆^𝑖=𝜎_𝑖𝐼+(1−𝜎_𝑖)𝑆_𝑖 and 𝜎_𝑖∈[𝑘_𝑖,1) and let {𝜏_𝑖} be a real sequence such that 0<𝜏_𝑖≤𝑏<1 for every 𝑖∈𝑁. Then, 𝐹(𝑊)=⋂^∞𝑖=1𝐹(𝑆_𝑖)=⋂^∞𝑖=1𝐹(𝑆^𝑖)≠∅.

Lemma 2.11 (see [28]). Let 𝐶 be a nonempty closed convex subset of a Hilbert space. Let {𝑆^𝑖} be an infinite family of nonexpansive mappings of 𝐶 into itself such that ⋂^∞𝑖=1𝐹(𝑆^𝑖)≠∅ and let {𝜏_𝑖} be a real sequence such that 0<𝜏_𝑖≤𝑏<1 for every 𝑖∈𝑁. If 𝐾 is any bounded subset of 𝐶, then lim_𝑛→∞sup_𝑥∈𝐾‖‖𝑊𝑥−𝑊_𝑛𝑥‖‖=0.(2.11)

3. Main Results

Let 𝐻 be a real Hilbert space and 𝐹 be a 𝑘-Lipschitzian and 𝜂-strongly monotone operator with 𝑘>0, 𝜂>0, 0<𝜇<2𝜂/𝑘² and 0<𝑡<1. Then, for 𝑡∈min{0,{1,1/𝜏}}, 𝑆=(𝐼−𝑡𝜇𝐹)∶𝐻→𝐻 is a contraction with contractive coefficient 1−𝑡𝜏 and 𝜏=(1/2)𝜇(2𝜂−𝜇𝑘²).

In fact, from (1.2) and (1.3), we obtain ‖𝑆𝑥−𝑆𝑦‖²=‖𝑥−𝑦−𝑡𝜇(𝐹𝑥−𝐹𝑦)‖²≤=‖‖𝑥𝑥−−𝑦𝑦‖‖²₂+𝑡²𝜇²‖𝐹𝑥−𝐹𝑦‖²−2𝑡𝜇⟨𝐹𝑥−𝐹𝑦,𝑥−𝑦⟩+𝑘²𝑡²𝜇²‖𝑥−𝑦‖²−2𝑡𝜂𝜇‖𝑥−𝑦‖²≤1−𝑡𝜇2𝜂−𝜇𝑘²‖𝑥−𝑦‖²≤(1−𝑡𝜏)²‖𝑥−𝑦‖².(3.1) Thus, 𝑆=(1−𝑡𝜇𝐹) is a contraction with contractive coefficient 1−𝑡𝜏∈(0,1).

Now, we show the strong convergence results for an infinite family 𝑘_𝑖-strictly pseudo-contractive mappings in Hilbert space.

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝐹 be a bifunction from 𝐶×𝐶→𝑅 satisfying (A1)–(A4). Let 𝑆_𝑖∶𝐶→𝐶 be a 𝑘_𝑖-strictly pseudo-contractive mapping with ⋂^∞𝑖=1𝐹(𝑆_𝑖)∩𝐸𝑃≠∅ and {𝜏_𝑖} be a real sequence such that 0<𝜏_𝑖≤𝑏<1, 𝑖∈𝑁. Let 𝑓 be a contraction of 𝐻 into itself with 𝛽∈(0,1) and 𝐵 be 𝑘-Lipschitzian and 𝜂-strongly monotone operator on H with coefficients 𝑘,𝜂>0, 0<𝜇<2𝜂/𝑘², 0<𝑟<(1/2)𝜇(2𝜂−𝜇𝑘²)/𝛽=(𝜏/𝛽) and 𝜏<1. Let {𝑥_𝑛} be a sequence generated by 𝐹𝑢_𝑛,𝑦+1𝜆_𝑛𝑦⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛𝑥_𝑛+1=𝛿_𝑛𝑢_𝑛+1−𝛿_𝑛𝑊_𝑛𝑢_𝑛,=𝛼_𝑛𝑟𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜇𝛼_𝑛𝐵𝑦_𝑛,∀𝑛∈𝑁,(3.2) where 𝑢_𝑛=𝑇_𝜆𝑛𝑥_𝑛 and {𝑊_𝑛∶𝐶→𝐶} is the sequence defined by (2.9). If {𝛼_𝑛}, {𝛽_𝑛}, {𝛿_𝑛}, and {𝜆_𝑛} satisfy the following conditions:(i){𝛼_𝑛}⊂(0,1), lim_𝑛→∞𝛼_𝑛=0,   ∑^∞𝑖=1𝛼_𝑛=∞,(ii)0<lim_𝑛→∞inf𝛽_𝑛≤lim_𝑛→∞sup𝛽_𝑛<1, (iii)0<lim_𝑛→∞inf𝛿_𝑛≤lim_𝑛→∞sup𝛿_𝑛<1, lim_𝑛→∞|𝛿_𝑛+1−𝛿_𝑛|=0, (iv){𝜆_𝑛}⊂(0,∞), lim_𝑛→∞𝜆_𝑛>0, lim_𝑛→∞|𝜆_𝑛+1−𝜆_𝑛|=0.
Then {𝑥_𝑛} converges strongly to 𝑧∈⋂^∞𝑖=1𝐹(𝑆_𝑖)∩EP≠∅, where 𝑧 is the unique solution of variational inequality lim_𝑛→∞sup⟨(𝑟𝑓−𝜇𝐵)𝑧,𝑝−𝑧⟩≤0,∀𝑝∈^∞_𝑖=1𝐹𝑆_𝑖∩EP≠∅,(3.3) that is, 𝑧=𝑃_{𝐹(𝑊)∩𝐸𝑃(𝐹)}(𝐼−𝜇𝐵+𝑟𝑓)𝑧, which is the optimality condition for the minimization problem min_{⋂𝑧∈∞𝑖=1𝐹(𝑆𝑖)∩𝐸𝑃}12⟨𝜇𝐵𝑧,𝑧⟩−ℎ(𝑧),(3.4) where ℎ is a potential function for 𝑟𝑓 (i.e., ℎ^(𝑧)=𝑟𝑓(𝑧) for 𝑧∈𝐻).

Proof . We divide the proof into five steps.
Step 1. We prove that {𝑥_𝑛} is bounded.
Noting the conditions (i) and (ii), we may assume, without loss of generality, that 𝛼_𝑛/(1−𝛽_𝑛)≤min{1,1/𝜏}. For 𝑥,𝑦∈𝐶, we obtain ‖‖1≤−𝛽1_𝑛−𝐼𝛽_𝑛−𝛼_𝑛𝜇𝐵𝑥−1−𝛽_𝑛𝐼−𝛼_𝑛𝜇𝐵𝑦‖‖≤1−𝛽_𝑛‖‖‖𝐼−1−𝛼^𝑛𝛽_𝑛𝜇𝐵𝑥−𝐼−1−𝛼^𝑛𝛽_𝑛𝜇𝐵𝑦‖‖‖=1−𝛽_𝑛1−1−𝛼^𝑛𝛽_𝑛𝜏‖𝑥−𝑦‖−𝛼_𝑛𝜏‖𝑥−𝑦‖.(3.5) Take 𝑝∈⋂^∞𝑖=1𝐹(𝑆_𝑖)∩EP≠∅. Since 𝑢_𝑛=𝑇_𝜆𝑛𝑥_𝑛 and 𝑝=𝑇_𝜆𝑛𝑝, then from Lemma 2.1, we know that, for any 𝑛∈𝑁, ‖‖𝑢_𝑛−𝑝‖‖=‖‖𝑇_𝜆𝑛𝑥_𝑛−𝑇_𝜆𝑛𝑝‖‖≤‖‖𝑥_𝑛−𝑝‖‖.(3.6) Furthermore, since 𝑊_𝑛𝑝=𝑝 and (3.6), we have ‖‖𝑦_𝑛−𝑝‖‖==‖‖‖‖𝛿𝛿_𝑛𝑢_𝑛+1−𝛿_𝑛𝑊_𝑛𝑢_𝑛−𝑝_𝑛‖‖≤𝛿_𝑛𝑢_𝑛−𝑝+1−𝛿_𝑛𝑊_𝑛𝑢_𝑛−𝑝‖‖≤‖‖𝑢‖‖𝑢_𝑛−𝑝‖‖+1−𝛿_𝑛‖‖𝑊_𝑛𝑢_𝑛−𝑝_𝑛‖‖−𝑝‖‖≤‖‖𝑥_𝑛−𝑝‖‖.(3.7) Thus, it follows from (3.7) that ‖‖𝑥_𝑛+1−𝑝‖‖==‖‖‖‖𝛼𝛼_𝑛𝑟𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜇𝛼_𝑛𝐵𝑦_𝑛−𝑝_𝑛‖‖+𝑟𝛽_𝑛𝑓𝑥_𝑛−𝑓(𝑝)+𝛼_𝑛(𝑟𝑓(𝑝)−𝜇𝐵𝑝)≤𝛼_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛𝐼−𝜇𝛼_𝑛𝐵𝑦_𝑛−𝑝‖‖+𝑟𝛽1‖‖−𝑥_𝑛𝛽_𝑛−𝑝‖‖+𝛼_𝑛‖𝑟𝑓(𝑝)−𝜇𝐵𝑝‖+𝛽_𝑛‖‖𝑥_𝑛−𝑝‖‖≤1−𝛼_𝑛−𝜏𝛼_𝑛‖‖𝑦_𝑛−𝑝‖‖≤max‖‖(𝑥𝜏_𝑛−𝑟𝛽)‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛‖𝑟𝑓(𝑝)−𝜇𝐵𝑝‖−𝑝‖‖,‖𝑟𝑓(𝑝)−𝜇𝐵𝑝‖𝜏−𝑟𝛽.(3.8) By induction, we have ‖‖𝑥_𝑛−𝑝‖‖≤max‖‖𝑥₁−𝑝‖‖,‖𝑟𝑓(𝑝)−𝜇𝐵𝑝‖𝜏−𝑟𝛽,𝑛≥1.(3.9) Hence, {𝑥_𝑛} is bounded and we also obtain that {𝑢_𝑛}, {𝑊_𝑛𝑢_𝑛}, {𝑦_𝑛}, {𝐵𝑦_𝑛}, and {𝑓(𝑥_𝑛)} are all bounded. Without loss of generality, we can assume that there exists a bounded set 𝐾⊂𝐶 such that {𝑢_𝑛}, {𝑊_𝑛𝑢_𝑛}, {𝑦_𝑛}, {𝐵𝑦_𝑛}, {𝑓(𝑥_𝑛)}∈𝐾, for all 𝑛∈𝑁.
Step 2. We show that lim_𝑛→∞‖𝑥_𝑛−𝑥_𝑛+1‖=0.
Let 𝑥_𝑛+1=(1−𝛽_𝑛)𝑧_𝑛+𝛽_𝑛𝑥_𝑛. We note that 𝑧_𝑛=𝑥^𝑛+1−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛=𝛼^𝑛𝑟𝑓𝑥_𝑛+1−𝛽_𝑛𝐼−𝜇𝛼_𝑛𝐵𝑦_𝑛1−𝛽_𝑛,(3.10) and then 𝑧_𝑛+1−𝑧_𝑛=𝛼^𝑛+1𝑟𝑓𝑥_𝑛+1+1−𝛽_𝑛+1𝐼−𝜇𝛼_𝑛+1𝐵𝑦_𝑛+11−𝛽_𝑛+1−𝛼^𝑛𝑟𝑓𝑥_𝑛+1−𝛽_𝑛𝐼−𝜇𝛼_𝑛𝐵𝑦_𝑛1−𝛽_𝑛=𝛼^𝑛+11−𝛽_𝑛+1𝑟𝑓𝑥_𝑛+1−𝜇𝐵𝑦_𝑛+1−1−𝛼^𝑛𝛽_𝑛𝑟𝑓𝑥_𝑛−𝜇𝐵𝑦_𝑛+𝑦_𝑛+1−𝑦_𝑛.(3.11) Therefore, ‖‖𝑧_𝑛+1−𝑧_𝑛‖‖≤𝛼^𝑛+11−𝛽_𝑛+1‖‖𝑟𝑓𝑥_𝑛+1‖‖+‖‖𝜇𝐵𝑦_𝑛+1+𝛼_𝑛‖‖1−𝛽_𝑛‖‖𝑟𝑓𝑥_𝑛‖‖+‖‖𝜇𝐵𝑦_𝑛‖‖+‖‖𝑦_𝑛+1−𝑦_𝑛‖‖.(3.12) It follows from (3.2) that ‖‖𝑦_𝑛+1−𝑦_𝑛‖‖≤=‖‖||𝛿𝛿_𝑛+1𝑛+1𝑢_𝑛+1+1−𝛿_𝑛+1𝑊_𝑛+1𝑢_𝑛+1−𝛿_𝑛𝑢_𝑛+1−𝛿_𝑛𝑊_𝑛𝑢_𝑛‖‖+||𝛿−_𝑛+1𝛿_𝑛||‖‖𝑢_𝑛‖‖+𝛿_𝑛+1‖‖𝑢_𝑛+1−𝑢_𝑛‖‖+1−𝛿_𝑛+1‖‖𝑊_𝑛+1𝑢_𝑛+1−𝑊_𝑛𝑢_𝑛‖‖−𝛿_𝑛||‖‖𝑊_𝑛𝑢_𝑛‖‖.(3.13)
We will estimate ‖𝑢_𝑛+1−𝑢_𝑛‖. From 𝑢_𝑛+1=𝑇_𝜆𝑛+1𝑥_𝑛+1 and 𝑢_𝑛=𝑇_𝜆𝑛𝑥_𝑛, we obtain 𝐹𝑢_𝑛+1,𝑦+1𝜆_𝑛+1𝑦−𝑢_𝑛+1𝐹𝑢,𝑢_𝑛+1−𝑦_𝑛+1≥0,∀𝑦∈𝐶,(3.14)_𝑛,𝑦+1𝜆_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑦_𝑛⟩≥0,∀𝑦∈𝐶.(3.15)
Taking 𝑦=𝑢_𝑛 in (3.14) and 𝑦=𝑢_𝑛+1 in (3.15), we have 𝐹𝑢_𝑛+1,𝑢_𝑛+1𝜆_𝑛+1𝑢_𝑛𝐹𝑢−𝑢_𝑛+1,𝑢_𝑛+1−𝑥_𝑛+1≥0,_𝑛,𝑢_𝑛+1+1𝜆_𝑛𝑢_𝑛+1−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥0.(3.16)
So, from (A2), one has 𝑢^𝑛+1−𝑢_𝑛,𝑢^𝑛−𝑥_n𝜆_𝑛−𝑢^𝑛+1−𝑥_𝑛+1𝜆_𝑛+1≥0,(3.17) furthermore, 𝑢^𝑛+1−𝑢_𝑛,𝑢_𝑛−𝑢_𝑛+1−𝑥_𝑛−𝜆^𝑛𝜆_𝑛+1𝑢_𝑛+1−𝑥_𝑛+1≥0.(3.18) Since lim_𝑛→∞𝜆_𝑛>0, we assume that there exists a real number such that 𝜆_𝑛>𝑎>0 for all 𝑛∈𝑁. Thus, we obtain ‖‖𝑢_𝑛+1−𝑢_𝑛‖‖²≤𝑢_𝑛+1−𝑢_𝑛,𝑥_𝑛+1−𝑥_𝑛+1−𝜆^𝑛𝜆_𝑛+1𝑢^𝑛+1≤‖‖𝑢_𝑛+1−𝑥_𝑛+1−𝑢_𝑛‖‖‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+||||1−𝜆^𝑛𝜆_𝑛+1||||‖‖𝑢_𝑛+1−𝑥_𝑛+1‖‖,(3.19) which means ‖‖𝑢_𝑛+1−𝑢_𝑛‖‖≤‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+||||1−𝜆^𝑛𝜆_𝑛+1||||‖‖𝑢_𝑛+1≤‖‖𝑥_𝑛+1−𝑥_𝑛+1‖‖−𝑥_𝑛‖‖+1𝑎||𝜆^𝑛+1≤‖‖𝑥_𝑛+1−𝜆_𝑛||‖‖𝑢_𝑛+1−𝑥_𝑛+1‖‖−𝑥_𝑛‖‖+𝐿₁||𝜆_𝑛+1−𝜆_𝑛||,(3.20) where 𝐿₁=sup{‖𝑢_𝑛+1−𝑥_𝑛+1‖∶𝑛∈𝑁}.
Next, we estimate ‖𝑊_𝑛+1𝑢_𝑛+1−𝑊_𝑛𝑢_𝑛‖. Notice that ‖‖𝑊_𝑛+1𝑢_𝑛+1−𝑊_𝑛𝑢_𝑛‖‖≤=‖‖‖‖𝑊𝑢_𝑛+1𝑛+1𝑢_𝑛+1−𝑊_𝑛+1𝑢_𝑛+𝑊_𝑛+1𝑢_𝑛−𝑊_𝑛𝑢_𝑛‖‖−𝑢_𝑛‖‖+‖‖𝑊_𝑛+1𝑢_𝑛−𝑊_𝑛𝑢_𝑛‖‖.(3.21) From (2.9), we obtain ‖‖𝑊_𝑛+1𝑢_𝑛−𝑊_𝑛𝑢_𝑛‖‖≤=𝜏‖‖₁𝜏₁𝑆^1𝑈_𝑛+1,2𝑢_𝑛−𝜏₁𝑆^1𝑈_𝑛,2𝑢_𝑛‖‖=𝜏₁‖‖𝑈_𝑛+1,2𝑢_𝑛−𝑈_𝑛,2𝑢_𝑛‖‖≤𝜏₁‖‖𝜏₂𝑆^2𝑈_𝑛+1,3𝑢_𝑛−𝜏₂𝑆^2𝑈_𝑛,3𝑢_𝑛‖‖≤≤𝜏⋯₁𝜏₂‖‖𝑈_𝑛+1,3𝑢_𝑛−𝑈_𝑛,3𝑢_𝑛‖‖≤𝐿𝜏_22𝑛⋅⋅⋅𝜏_𝑛‖‖𝑈_{𝑛+1,𝑛+1}𝑢_𝑛−𝑈_𝑛,𝑛+1𝑢_𝑛‖‖_𝑖=1𝜏_𝑖,(3.22) where 𝐿₂≥0 is a constant such that ‖𝑈_{𝑛+1,𝑛+1}𝑢_𝑛−𝑈_𝑛,𝑛+1𝑢_𝑛‖≤𝐿₂, for all 𝑛∈𝑁.
Substituting (3.20) and (3.22) into (3.21), we obtain ‖‖𝑊_𝑛+1𝑢_𝑛+1−𝑊_𝑛𝑢_𝑛‖‖≤‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+𝐿₁||𝜆_𝑛+1−𝜆_𝑛||+𝐿_2𝑛_𝑖=1𝜏_𝑖.(3.23) Hence, we have ‖‖𝑦_𝑛+1−𝑦_𝑛‖‖≤||+𝛿1_𝑛+1−−𝛿𝛿_𝑛||_𝑛+1‖‖𝑢_𝑛‖‖+‖‖𝑊_𝑛𝑢_𝑛‖‖+‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤𝐿₃𝐿^2𝑛_𝑖=1𝜏_𝑖+𝐿₁||𝜆_𝑛+1−𝜆_𝑛||||𝛿_𝑛+1−𝛿_𝑛||+‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+1−𝛿_𝑛+1𝐿_2𝑛_𝑖=1𝜏_𝑖+𝐿₁||𝜆_𝑛+1−𝜆_𝑛||,(3.24) where 𝐿₃=sup{‖𝑢_𝑛‖+‖𝑊_𝑛𝑢_𝑛‖∶𝑛∈𝑁}.
Furthermore, ‖‖𝑧_𝑛+1−𝑧_𝑛‖‖≤𝛼^𝑛+11−𝛽_𝑛+1‖‖𝑟𝑓𝑥_𝑛+1‖‖+‖‖𝜇𝐵𝑦_𝑛+1+𝛼_𝑛‖‖1−𝛽_𝑛‖‖𝑟𝑓𝑥_𝑛‖‖+‖‖𝜇𝐵𝑦_𝑛+‖‖𝑥_𝑛+1‖‖−𝑥_𝑛‖‖+𝐿₁||𝜆_𝑛+1−𝜆_𝑛||+𝐿₂1−𝛿_𝑛+1^𝑛_𝑖=1𝜏_𝑖+𝐿₃||𝛿_𝑛+1−𝛿_𝑛||.(3.25) It follows from (3.25) that ‖‖𝑧≤_𝑛+1−𝑧𝛼_𝑛‖‖_𝑛+1−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖1−𝛽_𝑛+1‖‖𝑟𝑓𝑥_𝑛+1‖‖+‖‖𝜇𝐵𝑦_𝑛+1‖‖+𝛼^𝑛1−𝛽_𝑛‖‖𝑟𝑓𝑥_𝑛‖‖+‖‖𝜇𝐵𝑦_𝑛+𝐿₁‖‖||𝜆_𝑛+1−𝜆_𝑛||+𝐿₂1−𝛿_𝑛+1^𝑛_𝑖=1𝜏_𝑖+𝐿₃||𝛿_𝑛+1−𝛿_𝑛||.(3.26) By the conditions (i), (iii), and (iv), we obtain lim_𝑛→∞sup‖‖𝑧_𝑛+1−𝑧_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤0.(3.27) Hence, by Lemma 2.5, one has lim_𝑛→∞‖‖𝑧_𝑛−𝑥_𝑛‖‖=0,(3.28) which implies lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=lim_𝑛→∞1−𝛽_𝑛‖‖𝑧_𝑛−𝑥_𝑛‖‖=0.(3.29)Step 3. We claim that lim_𝑛→∞‖𝑊𝑢_𝑛−𝑢_𝑛‖=0.
Notice that ‖‖𝑊𝑢_𝑛−𝑢_𝑛‖‖≤=‖‖‖‖𝑊𝑊𝑢𝑢_𝑛𝑛−𝑊_𝑛𝑢_𝑛+𝑊_𝑛𝑢_𝑛−𝑢_𝑛‖‖≤sup_𝑢∈𝐾−𝑊_𝑛𝑢_𝑛‖‖+‖‖𝑊_𝑛𝑢_𝑛−𝑢_𝑛‖‖‖‖𝑊𝑢−𝑊_𝑛𝑢‖‖+‖‖𝑊_𝑛𝑢_𝑛−𝑢_𝑛‖‖.(3.30) It follows from (3.2) that ‖‖𝑊_𝑛𝑢_𝑛−𝑢_𝑛‖‖≤=‖‖‖‖𝑦𝑊_𝑛𝑛𝑢_𝑛−𝑦_𝑛+𝑦_𝑛−𝑢_𝑛‖‖=‖‖𝑦_𝑛−𝑢_𝑛‖‖+‖‖𝑊_𝑛𝑢_𝑛−𝑦_𝑛‖‖≤‖‖𝑥_𝑛−𝑢_𝑛‖‖+𝛿_𝑛‖‖𝑊_𝑛𝑢_𝑛−𝑢_𝑛‖‖−𝑢_𝑛‖‖+‖‖𝑦_𝑛−𝑥_𝑛‖‖+𝛿_𝑛‖‖𝑊_𝑛𝑢_𝑛−𝑢_𝑛‖‖.(3.31) By the condition (iii), we obtain ‖‖𝑊_𝑛𝑢_𝑛−𝑢_𝑛‖‖≤11−𝛿_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖+‖‖𝑦_𝑛−𝑥_𝑛‖‖.(3.32)
First, we show lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛‖=0. From (3.2), for all 𝑝∈⋂^∞𝑖=1𝐹(𝑆_𝑖)∩EP(𝐹), applying Lemma 2.3 and noting that ‖⋅‖ is convex, we obtain ‖‖𝑥_𝑛+1−𝑝‖‖²=‖‖𝛼_𝑛𝑟𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜇𝛼_𝑛𝐵𝑦_𝑛−𝑝‖‖²=‖‖𝛼_𝑛𝑟𝑓𝑥_𝑛+𝜇𝐵𝑦_𝑛+𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛𝑦_𝑛−𝑝‖‖²≤≤𝛽‖‖_𝑛𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛𝑦_𝑛−𝑝‖‖²+2𝛼_𝑛𝑟𝑓𝑥_𝑛+𝜇𝐵𝑦_𝑛,𝑥_𝑛+1−𝑝≤𝛽_𝑛‖‖𝑥_𝑛−𝑝‖‖²+1−𝛽_𝑛‖‖𝑦_𝑛−𝑝‖‖²+2𝛼_𝑛‖‖𝑟𝑓𝑥_𝑛+𝜇𝐵𝑦_𝑛‖‖‖‖𝑥_𝑛+1−𝑝‖‖‖‖𝑥_𝑛−𝑝‖‖²+1−𝛽_𝑛‖‖𝑢_𝑛−𝑝‖‖²+2𝛼_𝑛‖‖𝑟𝑓𝑥_𝑛+𝜇𝐵𝑦_𝑛‖‖‖‖𝑥_𝑛+1−𝑝‖‖.(3.33) Since 𝑢_𝑛=𝑇_𝜆𝑛𝑥_𝑛, 𝑝=𝑇_𝜆𝑛𝑝, we have ‖‖𝑢_𝑛−𝑝‖‖²==‖‖1𝑇_𝜆𝑛𝑥_𝑛−𝑇_𝜆𝑛p‖‖²≤⟨𝑥_𝑛−𝑝,𝑢_𝑛−𝑝⟩2‖‖𝑥^𝑛−𝑝‖‖²+‖‖𝑢_𝑛−𝑝‖‖²−‖‖𝑥_𝑛−𝑢_𝑛‖‖²,(3.34) which implies ‖‖𝑢_𝑛−𝑝‖‖²≤‖‖𝑥_𝑛−𝑝‖‖²−‖‖𝑥_𝑛−𝑢_𝑛‖‖².(3.35) Substituting (3.35) into (3.33), we have ‖‖𝑥_𝑛+1−𝑝‖‖²≤‖‖𝑥_𝑛−𝑝‖‖²−1−𝛽_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖²+2𝛼_𝑛‖‖𝑟𝑓𝑥_𝑛+𝜇𝐵𝑦_𝑛‖‖‖‖𝑥_𝑛+1−𝑝‖‖,(3.36) which means 1−𝛽_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖²≤≤‖‖‖‖𝑥𝑥_𝑛−𝑝‖‖²−‖‖𝑥_𝑛+1−𝑝‖‖²+2𝛼_𝑛‖‖𝑟𝑓𝑥_𝑛+𝜇𝐵𝑦_𝑛‖‖‖‖𝑥_𝑛+1−𝑝_𝑛+1‖‖−𝑥_𝑛‖‖‖‖𝑥_𝑛−𝑝‖‖+‖‖𝑥_𝑛+1−𝑝‖‖+2𝛼_𝑛‖‖𝑟𝑓𝑥_𝑛+𝜇𝐵𝑦_𝑛‖‖‖‖𝑥_𝑛+1−𝑝‖‖.(3.37) Noticing lim_𝑛→∞𝛼_𝑛=0 and lim_𝑛→∞inf(1−𝛽_𝑛)>0, we have lim_𝑛→∞‖‖𝑥_𝑛−𝑢_𝑛‖‖=0.(3.38)
Second, we show lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0. It follows from (3.2) that ‖‖𝑦_𝑛−𝑥_𝑛‖‖≤=‖‖‖‖𝑦𝛼_𝑛𝑛−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤𝛼_𝑛𝑟𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜇𝛼_𝑛𝐵𝑦_𝑛−𝑦_𝑛‖‖+‖‖𝑥_𝑛+1−𝑥_𝑛‖‖‖‖𝑟𝑓𝑥_𝑛+𝜇𝐵𝑦_𝑛‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑦_𝑛‖‖+‖‖𝑥_𝑛+1−𝑥_𝑛‖‖.(3.39) This implies that 1−𝛽_𝑛‖‖𝑦_𝑛−𝑥_𝑛‖‖≤𝛼_𝑛‖‖𝑟𝑓𝑥_𝑛+𝜇𝐵𝑦_𝑛‖‖+‖‖𝑥_𝑛+1−𝑥_𝑛‖‖.(3.40) Noticing lim_𝑛→∞𝛼_𝑛=0, lim_𝑛→∞inf(1−𝛽_𝑛)>0 and (3.30), we have lim_𝑛→∞‖‖𝑦_𝑛−𝑥_𝑛‖‖=0.(3.41) Thus, substituting (3.41) and (3.38) into (3.32), we obtain lim_𝑛→∞‖‖𝑊_𝑛𝑢_𝑛−𝑢_𝑛‖‖=0.(3.42) Furthermore, (3.42), (3.30), and Lemma 2.11 lead to lim_𝑛→∞‖‖𝑊𝑢_𝑛−𝑢_𝑛‖‖=0.(3.43)
Step 4. Letting 𝑧=𝑃_{𝐹(𝑊)∩EP(𝐹)}(𝐼−𝜇𝐵+𝑟𝑓)𝑧, we show lim_𝑛→∞sup⟨(𝑟𝑓−𝜇𝐵)𝑧,𝑥_𝑛−𝑧⟩≤0.(3.44) We know that 𝑃_{𝐹(𝑊)∩EP(𝐹)}(𝐼−𝜇𝐵+𝑟𝑓) is a contraction. Indeed, for any 𝑥,𝑦∈𝐻, we have ‖‖𝑃≤≤‖(1(𝐼−−𝜏_{𝐹(𝑊)∩EP(𝐹)}(𝜇+𝐼𝐵−𝑟𝛽+𝜇)𝐵𝑟‖𝑓𝑥+)−𝑥𝑟𝑓𝑦−‖)(𝑥,𝐼−−𝑃𝜇𝐵+𝑟𝑓)𝑦_{𝐹(𝑊)∩EP(𝐹)}(‖𝐼−𝜇𝐵+𝑟𝑓)𝑦‖‖(3.45) and hence 𝑃_{𝐹(𝑊)∩EP(𝐹)}(𝐼−𝜇𝐵+𝑟𝑓) is a contraction due to (1−𝜏+𝑟𝛽)∈(0,1). Thus, Banach’s Contraction Mapping Principle guarantees that 𝑃_{𝐹(𝑊)∩EP(𝐹)}(𝐼−𝜇𝐵+𝑟𝑓) has a unique fixed point, which implies 𝑧=𝑃_{𝐹(𝑊)∩EP(𝐹)}(𝐼−𝜇𝐵+𝑟𝑓)𝑧.
Since {𝑢_𝑛𝑖}⊂{𝑢_𝑛} is bounded in 𝐶, without loss of generality, we can assume that {𝑢_𝑛𝑖}⇀𝜔, it follows from (3.43) that 𝑊𝑢_𝑛𝑖⇀𝜔. Since 𝐶 is closed and convex, 𝐶 is weakly closed. Thus we have 𝜔∈𝐶.
Let us show 𝜔∈𝐹(𝑊). For the sake of contradiction, suppose that 𝜔∉𝐹(𝑊), that is, 𝑊𝜔≠𝜔. Since {𝑢_𝑛𝑖}⇀𝜔, by the Opial condition, we have lim_𝑛→∞inf‖‖𝑢_𝑛𝑖−𝜔‖‖≤<lliimm_{𝑛→∞𝑛→∞}inf‖‖𝑢_𝑛𝑖−𝑊𝜔‖‖≤lim_𝑛→∞inf‖‖𝑢_𝑛𝑖−𝑊𝑢_𝑛𝑖‖‖+‖‖𝑊𝑢_𝑛𝑖−𝑊𝜔‖‖inf‖‖𝑢_𝑛𝑖−𝑊𝑢_𝑛𝑖‖‖+‖‖𝑢_𝑛𝑖−𝜔‖‖.(3.46) It follows (3.43) that lim_𝑛→∞inf‖‖𝑢_𝑛𝑖−𝜔‖‖<lim_𝑛→∞inf‖‖𝑢_𝑛𝑖−𝜔‖‖.(3.47) This is a contradiction, which shows that 𝜔∈𝐹(𝑊).
Next, we prove that 𝜔∈EP(𝐹). By (3.2), we obtain 𝐹𝑢_𝑛,𝑦+1𝜆_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0.(3.48) It follows from (A2) that 1𝜆_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥𝐹𝑦,𝑢_𝑛.(3.49) Replacing 𝑛 by 𝑛_𝑖, we have 𝑦−𝑢^𝑛𝑖,1𝜆_𝑛𝑖𝑢_𝑛𝑖−𝑥_𝑛𝑖≥𝐹𝑦,𝑢_𝑛𝑖.(3.50) Since (1/𝜆_𝑛𝑖)(𝑢_𝑛𝑖−𝑥_𝑛𝑖)→0 and {𝑢_𝑛𝑖}⇀𝜔, it follows from (A4) that 𝐹(𝑦,𝜔)≥0 for all 𝑦∈𝐶. Put 𝑧_𝑡=𝑡𝑦+(1−𝑡)𝜔 for all 𝑡∈(0,1] and 𝑦∈𝐶. Then, we have 𝑧_𝑡∈𝐶 and then 𝐹(𝑧_𝑡,𝜔)≥0. Hence, from (A1) and (A4), we have 0=𝐹𝑧_𝑡,𝑧_𝑡≤𝑡𝐹𝑧_𝑡,𝑦+(1−𝑡)𝐹𝑧_𝑡,𝑦≤𝑡𝐹𝑧_𝑡,𝑦,(3.51) which means 𝐹(𝑧_𝑡,𝑦)≥0. From (A3), we obtain 𝐹(𝜔,𝑦)≥0 for 𝑦∈𝐶 and then 𝜔∈EP(𝐹). Therefore, 𝜔∈𝐹(𝑊)∩EP(𝐹).
Since 𝑧=𝑃_{𝐹(𝑊)∩EP(𝐹)}(𝐼−𝜇𝐵+𝑟𝑓)𝑧, it follows from (3.38), (3.42), and Lemma 2.11 that lim_𝑖→∞sup⟨(𝑟𝑓−𝜇𝐵)𝑧,𝑥_𝑛−𝑧⟩≤=lliimm_{𝑛→∞𝑖→∞}(𝑟𝑓−𝜇𝐵)𝑧,𝑥_𝑛𝑖−𝑧+lim_𝑖→∞(𝑟𝑓−𝜇𝐵)𝑧,𝑥_𝑛𝑖−𝑢_𝑛𝑖+lim_𝑖→∞(𝑟𝑓−𝜇𝐵)𝑧,𝑢_𝑛𝑖−𝑊_𝑛𝑖𝑢_𝑛𝑖+lim_𝑖→∞(𝑟𝑓−𝜇𝐵)𝑧,𝑊_𝑛𝑖𝑢_𝑛𝑖−𝑊𝑢_𝑛𝑖=⟨(𝑟𝑓−𝜇𝐵(𝑟)𝑓𝑧−,𝜔𝜇−𝐵)𝑧𝑧⟩,𝑊≤𝑢0._𝑛𝑖−𝑧(3.52)
Step 5. Finally we prove that 𝑥_𝑛→𝜔 as 𝑛→∞. In fact, from (3.2) and (3.7), we obtain ‖‖𝑥_𝑛+1−𝜔‖‖²=‖‖𝛼_𝑛𝑟𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜇𝛼_𝑛𝐵𝑦_𝑛−𝜔‖‖²=‖‖𝛼+_𝑛𝛽𝑟_𝑛𝑓𝑥_𝑛−𝑓(𝜔)+𝛼_𝑛(𝑟𝑓(𝜔)−𝜇𝐵𝜔)𝑥_𝑛−𝜔+1−𝛽_𝑛𝐼−𝜇𝛼_𝑛𝐵𝑦_𝑛−𝜔‖‖²=𝛼+_𝑛𝑟𝛽_𝑛𝑓𝑥_𝑛−𝑓(𝜔),𝑥_𝑛+1−𝜔+𝛼_𝑛𝑟𝑓(𝜔)−𝜇𝐵𝜔,𝑥_𝑛+1−𝜔≤𝛼_𝑛𝑥_𝑛−𝜔,𝑥_𝑛+1−𝜔+1−𝛽_𝑛𝐼−𝜇𝛼_𝑛𝐵𝑦_𝑛−𝜔,𝑥_𝑛+1−𝜔𝑟𝛽‖‖𝑥^𝑛−𝜔‖‖²+‖‖𝑥_𝑛+1−𝜔‖‖²2+𝛼^𝑛𝑟𝑓(𝜔)−𝜇𝐵𝜔,𝑥_𝑛+1+𝛽_𝑛−𝜔‖‖𝑥_𝑛−𝜔‖‖²+‖‖𝑥_𝑛+1−𝜔‖‖²2+1−𝛽^𝑛−𝛼_𝑛𝜏‖‖𝑦^𝑛−𝜔‖‖²+‖‖𝑥_𝑛+1−𝜔‖‖²≤1−𝛼_𝑛2(𝜏−𝑟𝛽)2‖‖𝑥^𝑛−𝜔‖‖²+‖‖𝑥_𝑛+1−𝜔‖‖²+𝛼^𝑛𝑟𝑓(𝜔)−𝜇𝐵𝜔,𝑥_𝑛+1−𝜔,(3.53) which implies ‖‖𝑥_𝑛+1−𝜔‖‖²≤1−𝛼^𝑛(𝜏−𝑟𝛽)1+𝛼_𝑛(𝜏−𝑟𝛽)‖‖𝑥^𝑛−𝜔‖‖²+2𝛼^𝑛(𝜏−𝑟𝛽)1+𝛼_𝑛(𝜏−𝑟𝛽)(𝜏−𝑟𝛽)𝑟𝑓(𝜔)−𝜇𝐵𝜔,𝑥^𝑛+1≤1−𝛼_𝑛−𝜔(𝜏−𝑟𝛽)‖‖𝑥_𝑛−𝜔‖‖²+2𝛼^𝑛(𝜏−𝑟𝛽)1+𝛼_𝑛(𝜏−𝑟𝛽)(𝜏−𝑟𝛽)⟨𝑟𝑓(𝜔)−𝜇𝐵𝜔,𝑥^𝑛+1−𝜔⟩.(3.54) From condition (i) and (3.7), we know that ∑^𝑛𝑖=1𝛼_𝑛(𝜏−𝑟𝛽)=∞ and lim_𝑖→∞sup(2/(1+𝛼_𝑛(𝜏−𝑟𝛽))(𝜏−𝑟𝛽))⟨𝑟𝑓(𝜔)−𝜇𝐵𝜔,𝑥_𝑛+1−𝜔⟩≤0. we can conclude from Lemma 2.6 that 𝑥_𝑛→𝜔 as 𝑛→∞. This completes the proof of Theorem 3.1.

Remark 3.2. If 𝑟=1, 𝜇=1, 𝐵=𝐼 and 𝛿_𝑖=0, 𝑘_𝑖=0, 𝜎_𝑖=0 for 𝑖∈𝑁, then Theorem 3.1 reduces to Theorem 3.5 of Yao et al. [19]. Furthermore, we extend the corresponding results of Yao et al. [19] from one infinite family of nonexpansive mapping to an infinite family of strictly pseudo-contractive mappings.

Remark 3.3. If 𝜇=1 and 𝛿_𝑖=0, 𝑘_𝑖=0, 𝜎_𝑖=0 for 𝑖∈𝑁, then Theorem 3.1 reduces to Theorem 10 of Colao and Marino [20]. Furthermore, we extend the corresponding results of Colao and Marino [20] from one infinite family of nonexpansive mapping to an infinite family of strictly pseudo-contractive mappings, and from a strongly positive bounded linear operator 𝐴 to a 𝑘-Lipschitzian and 𝜂-strongly monotone operator 𝐵.

Theorem 3.4. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝐹 be a bifunction from 𝐶×𝐶→𝑅 satisfying (A1)–(A4). Let 𝑆∶𝐶→𝐶 be a nonexpansive mapping with 𝐹(𝑆)∩𝐸𝑃≠∅. Let 𝑓 be a contraction of 𝐻 into itself with 𝛽∈(0,1) and 𝐵 be 𝑘-Lipschitzian and 𝜂-strongly monotone operator on 𝐻 with coefficients 𝑘,𝜂>0, 0<𝜇<2𝜂/𝑘², 0<𝑟<(1/2)𝜇(2𝜂−𝜇𝑘²)/𝛽=𝜏/𝛽 and 𝜏<1. Let {𝑥_𝑛} be sequence generated by 𝐹𝑢_𝑛,𝑦+1𝜆_𝑛𝑦⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛𝑥_𝑛+1=𝛿_𝑛𝑢_𝑛+1−𝛿_𝑛𝑆_𝑛𝑢_𝑛,=𝛼_𝑛𝑟𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜇𝛼_𝑛𝐵𝑦_𝑛,∀𝑛∈𝑁,(3.55) where 𝑢_𝑛=𝑇_𝜆𝑛𝑥_𝑛. If {𝛼_𝑛},{𝛽_𝑛},{𝛿_𝑛}, and {𝜆_𝑛} satisfy the following conditions:(i){𝛼_𝑛}⊂(0,1), lim_𝑛→∞𝛼_𝑛=0,   ∑^∞𝑖=1𝛼_𝑛=∞,(ii)0<lim_𝑛→∞inf𝛽_𝑛≤lim_𝑛→∞sup𝛽_𝑛<1, (iii)0<lim_𝑛→∞inf𝛿_𝑛≤lim_𝑛→∞sup𝛿_𝑛<1, lim_𝑛→∞|𝛿_𝑛+1−𝛿_𝑛|=0, (iv){𝜆_𝑛}⊂(0,∞), lim_𝑛→∞𝜆_𝑛>0, lim_𝑛→∞|𝜆_𝑛+1−𝜆_𝑛|=0.
Then {𝑥_𝑛} converges strongly to 𝑧∈𝐹(𝑆)∩𝐸𝑃≠∅, where 𝑧 is the unique solution of variational inequality lim_𝑛→∞sup⟨(𝑟𝑓−𝜇𝐵)𝑧,𝑝−𝑧⟩≤0,∀𝑝∈𝐹(𝑆)∩𝐸𝑃≠∅,(3.56) that is, 𝑧=𝑃_{𝐹(𝑆)∩𝐸𝑃(𝐹)}(𝐼−𝜇𝐵+𝑟𝑓)𝑧.

Proof . By Theorem 3.1, letting 𝑘_𝑖=0, 𝜎_𝑖=0, 𝜏_𝑖=1 and 𝑆_𝑖=𝑆 for 𝑖∈𝑁, we can obtain Theorem 3.4.

4. Numerical Example

Now, we present a numerical example to illustrate our theoretical analysis results obtained in Section 3.

Example 4.1. Let 𝐻=𝑅, 𝐶=[−1,1],   𝑆_𝑛=𝐼, 𝜏_𝑛=𝜏∈(0,1),  𝜆_𝑛=1, 𝑛∈𝑁, 𝐹(𝑥,𝑦)=0, for all 𝑥,𝑦∈𝐶, 𝐵=𝐼, 𝑟=𝜇=1, 𝑓(𝑥)=(1/10)𝑥, for all 𝑥, with contraction coefficient 𝛽=1/5, 𝛿_𝑛=1/2, 𝛼_𝑛=1/𝑛, 𝛽_𝑛=1/4+1/2𝑛 for every 𝑛∈𝑁. Then {𝑥_𝑛} is the sequence generated by 𝑥_𝑛+1=1−910𝑛𝑥^𝑛,(4.1) and {𝑥_𝑛}→0, as 𝑛→∞, where 0 is the unique solution of the minimization problem min_𝑥∈𝐶920𝑥²+𝑐.(4.2)

Proof. We divide the proof into four steps.
Step 1. We show 𝑇_𝜆𝑛𝑥=𝑃_𝐶𝑥,∀𝑥∈𝐻,(4.3) where 𝑃_𝐶𝑥=𝑥|𝑥𝑥,|,𝑥𝑥∈∉𝐶𝐶.,(4.4)
Since 𝐹(𝑥,𝑦)=0, for all 𝑥,𝑦∈𝐶, due to the definition of 𝑇_𝜆𝑛(𝑥), for all 𝑥∈𝐻, by Lemma 2.1, we obtain 𝑇_𝜆𝑛𝑥={𝑧∈𝐶∶(𝑦−𝑧,𝑧−𝑥)≥0,∀𝑦∈𝐶}.(4.5)
By the property of 𝑃_𝐶, for 𝑥∈𝐶, we have 𝑇_𝜆𝑛𝑥=𝑃_𝐶𝑥=𝐼𝑥. Furthermore, it follows from (3) in Lemma 2.1 that EP(𝐹)=𝐶.(4.6)
Step 2. We show that 𝑊_𝑛=𝐼.(4.7)
It follows from (2.9) that 𝑊𝑊₁=𝑈_1,1=𝜏₁𝑆^1𝑈_1,2+1−𝜏₁𝐼=𝜏₁𝑆^1+1−𝜏₁𝐼,₂==𝜏𝑈_2,11=𝜏₁𝑆^1𝑈_2,2+1−𝜏₁𝐼=𝜏₁𝑆^1𝜏₂𝑆^2𝑈_2,3+1−𝜏₂𝐼+1−𝜏₁𝐼𝑊𝜏₂𝑆^1𝑆^2+𝜏₁1−𝜏₂𝑆^1+1−𝜏₁𝐼,₃==𝜏𝑈_3,11=𝜏₁𝑆^1𝑈_3,2+1−𝜏₁𝐼=𝜏₁𝑆^1𝜏₂𝑆^2𝑈_3,3+1−𝜏₂𝐼+1−𝜏₁𝐼=𝜏₁𝜏₂𝑆^1𝑆^2𝑈_3,3+𝜏₁1−𝜏₂𝑆^1+1−𝜏₁𝐼=𝜏₁𝜏₂𝑆^1𝑆^2𝜏₃𝑆^3𝑈_3,4+1−𝜏₃𝐼+𝜏₁1−𝜏₂𝑆^1+1−𝜏₁𝐼𝜏₂𝜏₃𝑆^1𝑆^2𝑆^3+𝜏₁𝜏₂1−𝜏₃𝑆^1𝑆^2+𝜏₁1−𝜏₂𝑆^1+1−𝜏₁𝐼.(4.8) Furthermore, we obtain 𝑊_𝑛=𝑈_𝑛,1=𝜏₁𝜏₂𝜏₃⋅⋅⋅𝜏_𝑛𝑆^1𝑆^2𝑆^3⋅⋅⋅𝑆^𝑛+𝜏₁𝜏₂⋅⋅⋅𝜏_𝑛−11−𝜏_𝑛𝑆^1𝑆^2⋅⋅⋅𝑆^𝑛−1+𝜏₁𝜏₂⋅⋅⋅𝜏_𝑛−21−𝜏_𝑛−1𝑆^1𝑆^2⋅⋅⋅𝑆^𝑛−2+⋅⋅⋅+𝜏₁1−𝜏₂𝑆^1+1−𝜏₁𝐼.(4.9) Since 𝑆^𝑖=𝐼, 𝜏_𝑖=𝜏 for 𝑖∈𝑁, one has 𝑊_𝑛=𝜏^𝑛+𝜏^𝑛−1(1−𝜏)+⋅⋅⋅+𝜏(1−𝜏)+(1−𝜏)𝐼=𝐼.(4.10)
Step 3. We show that 𝑥_𝑛+1=1−910𝑛𝑥^𝑛,(4.11){𝑥_𝑛}→0, as 𝑛→∞, where 0 is the unique solution of the minimization problem min_𝑥∈𝐶920𝑥²+𝑐.(4.12)
In fact, we can see that 𝐵=𝐼 is 𝑘-Lipschitzian and 𝜂-strongly monotone operator on 𝐻 with coefficient 𝑘=1, 𝜂=3/4 such that 0<𝜇<2𝜂/𝑘², 0<𝑟<(1/2)𝜇(2𝜂−𝜇𝑘²)/𝛽=𝜏/𝛽, so we take 𝑟=𝜇=1. Since 𝑆^𝑛=𝐼, 𝑛∈𝑁, we have ^∞_𝑖=1𝐹𝑆_𝑖=𝐻.(4.13) Furthermore, we obtain ^∞_𝑖=1𝐹𝑆_𝑖∩EP(𝐹)=𝐶=[−1,1].(4.14)
Next, we need prove {𝑥_𝑛}→0, as 𝑛→∞. Since 𝑦_𝑛=𝑢_𝑛 for all 𝑛∈𝑁, we have 𝑥_𝑛+1=𝛼_𝑛𝑟𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜇𝛼_𝑛𝐵𝑦_𝑛=1−910𝑛𝑥^𝑛,(4.15) for all 𝑛∈𝑁.
Thus, we obtain a special sequence {𝑥_𝑛} of (3.2) in Theorem 3.1 as follows 𝑥_𝑛+1=1−910𝑛𝑥^𝑛.(4.16) By Lemma 2.6, it is obviously that 𝑥_𝑛→0, 0 is the unique solution of the minimization problem min_𝑥∈𝐶920𝑥²+𝑐,(4.17) where 𝑐 is a constant number.
Step 4. Finally, we use software Matlab 7.0 to give the numerical experiment results and then obtain Table 1 which show that the iteration process of the sequence {𝑥_𝑛} is a monotonedecreasing sequence and converges to 0. From Table 1 and the corresponding graph Figure 1, we show that the more the iteration steps are, the more slowly the sequence {𝑥_𝑛} converges to 0.

Table 1 

This table shows the value of sequence {𝑥_𝑛} on each iteration step (initial value 𝑥₁=0.2).

Figure 1 

The corresponding graph at 𝑥=0.2.

Acknowledgments

The authors thank the anonymous referees and the editor for their constructive comments and suggestions, which greatly improved this paper. This project is supported by the Natural Science Foundation of China (Grants nos. 11171180, 11171193, 11126233, and 10901096) and Shandong Provincial Natural Science Foundation (Grants no. ZR2009AL019 and ZR2011AM016) and the Project of Shandong Province Higher Educational Science and Technology Program (Grant no. J09LA53).