Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 416476 | 23 pages | https://doi.org/10.1155/2012/416476

Convergence Theorems for Equilibrium Problems and Fixed-Point Problems of an Infinite Family of π‘˜π‘–-Strictly Pseudocontractive Mapping in Hilbert Spaces

Academic Editor: Jong Hae Kim
Received28 Feb 2012
Revised08 Jun 2012
Accepted24 Jun 2012
Published31 Jul 2012

Abstract

We first extend the definition of Wn 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 π‘˜π‘–-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 βŸ¨β‹…,β‹…βŸ© and induced norm β€–β‹…β€–. Let 𝐢 be a nonempty closed convex subset of 𝐻 and let πΉβˆΆπΆΓ—πΆβ†’π‘… be a bifunction. We consider the following equilibrium problem (EP) which is to find π‘§βˆˆπΆ such that EP∢𝐹(𝑧,𝑦)β‰₯0,βˆ€π‘¦βˆˆπΆ.(1.1) Denote the set of solutions of EP by EP(𝐹). Given a mapping π‘‡βˆΆπΆβ†’π», let 𝐹(π‘₯,𝑦)=βŸ¨π‘‡π‘₯,π‘¦βˆ’π‘₯⟩ for all π‘₯,π‘¦βˆˆπΆ. Then, π‘§βˆˆEP(𝐹) if and only if βŸ¨π‘‡π‘₯,π‘¦βˆ’π‘₯⟩β‰₯0 for all π‘¦βˆˆπΆ, 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 π΅βˆΆπΆβ†’πΆ is called πœƒ-Lipschitzian if there exists a positive constant πœƒ such that ‖𝐡π‘₯βˆ’π΅π‘¦β€–β‰€πœƒβ€–π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπΆ.(1.2)𝐡 is said to be πœ‚-strongly monotone if there exists a positive constant πœ‚ such that ⟨𝐡π‘₯βˆ’π΅π‘¦,π‘₯βˆ’π‘¦βŸ©β‰₯πœ‚β€–π‘₯βˆ’π‘¦β€–2,βˆ€π‘₯,π‘¦βˆˆπΆ.(1.3) A mapping π‘†βˆΆπΆβ†’πΆ is said to be π‘˜-strictly pseudocontractive mapping if there exists a constant 0β‰€π‘˜<1 such that β€–Sπ‘₯βˆ’π‘†π‘¦β€–2≀‖π‘₯βˆ’π‘¦β€–2β€–+π‘˜β€–(πΌβˆ’π‘†)π‘₯βˆ’(πΌβˆ’π‘†)𝑦2,(1.4) for all π‘₯,π‘¦βˆˆπΆ and 𝐹(𝑆) denotes the set of fixed point of the mapping 𝑆, that is 𝐹(𝑆)={π‘₯βˆˆπΆβˆΆπ‘†π‘₯=π‘₯}.

If π‘˜=1, then 𝑆 is said to a pseudocontractive mapping, that is, ‖𝑆π‘₯βˆ’π‘†π‘¦β€–2≀‖π‘₯βˆ’π‘¦β€–2β€–+β€–(πΌβˆ’π‘†)π‘₯βˆ’(πΌβˆ’π‘†)𝑦2,(1.5) is equivalent to ⟨(πΌβˆ’π‘†)π‘₯βˆ’(πΌβˆ’π‘†)𝑦,π‘₯βˆ’π‘¦βŸ©β‰₯0,(1.6) for all π‘₯,π‘¦βˆˆπΆ.

The class of π‘˜-strict pseudo-contractive mappings extends the class of nonexpansive mappings (A mapping 𝑇 is said to be nonexpansive if ‖𝑇π‘₯βˆ’π‘‡π‘¦β€–β‰€β€–π‘₯βˆ’π‘¦β€–, for all π‘₯,π‘¦βˆˆπΆ). That is, 𝑆 is nonexpansive if and only if 𝑆 is a 0-strict pseudocontractive mapping. Clearly, the class of π‘˜-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 π‘₯0∈𝐻, the sequence {π‘₯𝑛} generated by the algorithm π‘₯𝑛+1=𝛼𝑛π‘₯𝛾𝑓𝑛+ξ€·πΌβˆ’π›Όπ‘›π΅ξ€Έπ‘‡π‘₯𝑛,π‘›βˆˆπ‘,(1.7) where 𝑇 is a self-nonexpansive mapping on 𝐻, 𝑓 is an 𝛼-contraction of 𝐻 into itself (i.e., ‖𝑓(π‘₯)βˆ’π‘“(𝑦)‖≀𝛼‖π‘₯βˆ’π‘¦β€–, for all π‘₯,π‘¦βˆˆπ» and π›Όβˆˆ(0,1)), {𝛼𝑛}βŠ‚(0,1) satisfies certain conditions, 𝐡 is strongly positive bounded linear operator on 𝐻, and converges strongly to fixed point π‘₯βˆ— of 𝑇 which is the unique solution to the following variational inequality: ⟨(π›Ύπ‘“βˆ’π΅)π‘₯βˆ—,π‘₯βˆ—βˆ’π‘₯βŸ©β‰€0,βˆ€π‘₯∈𝐹(𝑇).(1.8)

Tian [15] considered the following iterative method, for a nonexpansive mapping 𝑇:𝐻→𝐻 with 𝐹(𝑇)β‰ βˆ…, π‘₯𝑛+1=𝛼𝑛π‘₯𝛾𝑓𝑛+ξ€·πΌβˆ’πœ‡π›Όπ‘›πΉξ€Έπ‘‡π‘₯𝑛,π‘›βˆˆπ‘,(1.9) where 𝐹 is π‘˜-Lipschitzian and πœ‚-strongly monotone operator. The sequence {π‘₯𝑛} converges strongly to fixed-point π‘ž in 𝐹(𝑇) which is the unique solution to the following variational inequality: ⟨(π›Ύπ‘“βˆ’πœ‡πΉ)π‘ž,π‘βˆ’π‘žβŸ©β‰€0,π‘βˆˆπΉ(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 π‘†βˆΆπΆβ†’π» be a nonexpansive mapping. Starting with arbitrary initial point π‘₯1∈𝐻, define sequences {π‘₯𝑛} and {𝑒𝑛} recursively by 𝐹𝑒𝑛+1,π‘¦π‘Ÿπ‘›βŸ¨π‘¦βˆ’π‘’π‘›,π‘’π‘›βˆ’π‘₯𝑛π‘₯⟩β‰₯0,βˆ€π‘¦βˆˆπΆ,𝑛+1=𝛼𝑛𝑓π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘†π‘’π‘›,βˆ€π‘›βˆˆπ‘.(1.11) They proved that under certain appropriate conditions imposed on {𝛼𝑛} and {π‘Ÿπ‘›}, the sequences {π‘₯𝑛} and {𝑒𝑛} converge strongly to π‘§βˆˆπΉ(𝑆)∩EP(𝐹), where 𝑧=𝑃𝐹(𝑆)∩EP(𝐹)𝑓(𝑧).

Liu [17] introduced the following scheme: π‘₯1∈𝐻 and 𝐹𝑒𝑛+1,π‘¦π‘Ÿπ‘›βŸ¨π‘¦βˆ’π‘’π‘›,π‘’π‘›βˆ’π‘₯π‘›π‘¦βŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ,𝑛=𝛽𝑛𝑒𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘†π‘’π‘›,π‘₯𝑛+1=𝛼𝑛π‘₯𝛾𝑓𝑛+ξ€·πΌβˆ’π›Όπ‘›π΅ξ€Έπ‘¦π‘›,βˆ€π‘›βˆˆπ‘,(1.12) where 𝑆 is a π‘˜-strict pseudo-contractive mapping and 𝐡 is a strongly positive bounded linear operator. They proved that under certain appropriate conditions imposed on {𝛼𝑛},{𝛽𝑛}, and {π‘Ÿπ‘›}, the sequence {π‘₯𝑛} converges strongly to π‘§βˆˆπΉ(𝑆)∩EP(𝐹), where 𝑧=𝑃𝐹(𝑆)∩EP(𝐹)(πΌβˆ’π΅+𝛾𝑓)(𝑧).

In [18], the concept of π‘Š mapping had been modified for a countable family {𝑇𝑛}π‘›βˆˆπ‘ of nonexpansive mappings by defining the sequence {π‘Šπ‘›}π‘›βˆˆπ‘ of π‘Š-mappings generated by {𝑇𝑛}π‘›βˆˆπ‘ and {πœ†π‘›}βŠ‚(0,1), proceeding backward π‘ˆπ‘›,𝑛+1π‘ˆβˆΆ=𝐼,𝑛,π‘›βˆΆ=πœ†π‘›π‘‡π‘›π‘ˆπ‘›,𝑛+1+ξ€·1βˆ’πœ†π‘›ξ€Έπ‘ˆπΌ,⋅⋅⋅𝑛,π‘˜βˆΆ=πœ†π‘˜π‘‡π‘˜π‘ˆπ‘›,π‘˜+1+ξ€·1βˆ’πœ†π‘˜ξ€Έπ‘ˆπΌ,⋅⋅⋅𝑛,2∢=πœ†2𝑇2π‘ˆπ‘›,3+ξ€·1βˆ’πœ†2ξ€Έπ‘ŠπΌ,𝑛=π‘ˆπ‘›,1∢=πœ†1𝑇1π‘ˆπ‘›,2+ξ€·1βˆ’πœ†1𝐼.(1.13) Yao et al. [19] using this concept, introduced the following algorithm: π‘₯1∈𝐻 and 𝐹𝑒𝑛+1,π‘¦π‘Ÿπ‘›βŸ¨π‘¦βˆ’π‘’π‘›,π‘’π‘›βˆ’π‘₯𝑛π‘₯⟩β‰₯0,βˆ€π‘¦βˆˆπΆ,𝑛+1=𝛼𝑛𝑓π‘₯𝑛+𝛽𝑛π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›βˆ’π›½π‘›ξ€Έπ‘Šπ‘›π‘’π‘›,βˆ€π‘›βˆˆπ‘.(1.14) They proved that under certain appropriate conditions imposed on {𝛼𝑛} and {π‘Ÿπ‘›}, the sequences {π‘₯𝑛} and {𝑒𝑛} converge strongly to β‹‚π‘§βˆˆβˆžπ‘–=1𝐹(𝑇𝑖)∩EP(𝐹).

Colao and Marino [20] considered the following explicit viscosity scheme 𝐹𝑒𝑛+1,π‘¦π‘Ÿπ‘›βŸ¨π‘¦βˆ’π‘’π‘›,π‘’π‘›βˆ’π‘₯𝑛π‘₯⟩β‰₯0,βˆ€π‘¦βˆˆπΆ,𝑛+1=𝛼𝑛π‘₯π‘Ÿπ‘“π‘›ξ€Έ+𝛽𝑛π‘₯𝑛+ξ€·ξ€·1βˆ’π›½π‘›ξ€ΈπΌβˆ’π›Όπ‘›π΄ξ€Έπ‘Šπ‘›π‘’π‘›,βˆ€π‘›βˆˆπ‘,(1.15) where 𝐴 is a strongly positive operator on 𝐻. Under certain appropriate conditions, the sequences {π‘₯𝑛} and {𝑒𝑛} converge strongly to β‹‚π‘§βˆˆβˆžπ‘–=1𝐹(𝑇𝑖)∩EP(𝐹).

Motivated and inspired by these facts, in this paper, we first extend the definition of π‘Šπ‘› 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 β‹‚EP(𝐹)βˆžπ‘–=1𝐹(𝑆𝑖), where {𝑆𝑖} is an infinite family of π‘˜π‘–-strictly pseudo-contractive mappings of 𝐢 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 𝐢 is a nonempty closed convex subset of a Hilbert space 𝐻. We write π‘₯𝑛⇀π‘₯ to indicate that the sequence {π‘₯𝑛} converges weakly to π‘₯. π‘₯𝑛→π‘₯ implies that {π‘₯𝑛} converges strongly to π‘₯. We denote by 𝑁 and 𝑅 the sets of positive integers and real numbers, respectively. For any π‘₯∈𝐻, there exists a unique nearest point in 𝐢, denoted by 𝑃𝐢π‘₯, such that β€–β€–π‘₯βˆ’π‘ƒπΆπ‘₯‖‖≀‖π‘₯βˆ’π‘¦β€–,βˆ€π‘¦βˆˆπΆ.(2.1) Such a 𝑃𝐢 is called the metric projection of 𝐻 onto 𝐢. It is known that 𝑃𝐢 is nonexpansive. Furthermore, for π‘₯∈𝐻 and π‘’βˆˆπΆ, 𝑒=𝑃𝐢π‘₯⟺⟨π‘₯βˆ’π‘’,π‘’βˆ’π‘¦βŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ.(2.2) It is widely known that 𝐻 satisfies Opialξ…žs condition [21], that is, for any sequence {π‘₯𝑛} with π‘₯𝑛⇀π‘₯, the inequality limπ‘›β†’βˆžβ€–β€–π‘₯infπ‘›β€–β€–βˆ’π‘₯<limπ‘›β†’βˆžβ€–β€–π‘₯infπ‘›β€–β€–βˆ’π‘¦(2.3) holds for every π‘¦βˆˆπ» with 𝑦≠π‘₯.

In order to solve the equilibrium problem for a bifunction πΉβˆΆπΆΓ—πΆβ†’π‘…, we assume that 𝐹 satisfies the following conditions:(A1)𝐹(π‘₯,π‘₯)=0,for all π‘₯∈𝐢.(A2)𝐹 is monotone, that is, 𝐹(π‘₯,𝑦)+𝐹(𝑦,π‘₯)≀0, for all π‘₯,π‘¦βˆˆπΆ.(A3)lim𝑑↓0𝐹(𝑑𝑧+(1βˆ’π‘‘)π‘₯,𝑦)≀𝐹(π‘₯,𝑦), for all π‘₯,𝑦,π‘§βˆˆπΆ.(A4) For each π‘₯∈𝐢,𝑦↦𝐹(π‘₯,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 𝐢×𝐢 into 𝑅 satisfying (A1), (A2), (A3), and (A4). Then, for any π‘Ÿ>0 and π‘₯∈𝐻, there exists π‘§βˆˆπΆ such that 1𝐹(𝑧,𝑦)+π‘Ÿ(π‘¦βˆ’π‘§,π‘§βˆ’π‘₯)β‰₯0,βˆ€π‘¦βˆˆπΆ.(2.4) Furthermore, if π‘‡π‘Ÿπ‘₯={π‘§βˆˆπΆβˆΆπΉ(𝑧,𝑦)+(1/π‘Ÿ)(π‘¦βˆ’π‘§,π‘§βˆ’π‘₯)β‰₯0,βˆ€π‘¦βˆˆπΆ}, then the following hold:(1)π‘‡π‘Ÿ is single-valued.(2)π‘‡π‘Ÿ is firmly nonexpansive, that is, β€–β€–π‘‡π‘Ÿπ‘₯βˆ’π‘‡π‘Ÿπ‘¦β€–β€–2β‰€βŸ¨π‘‡π‘Ÿπ‘₯βˆ’π‘‡π‘Ÿπ‘¦,π‘₯βˆ’π‘¦βŸ©,βˆ€π‘₯,π‘¦βˆˆπ».(2.5)(3)𝐹(π‘‡π‘Ÿ)=EP(𝐹). (4)EP(𝐹) is closed and convex.

Lemma 2.2 2.2 (see [23]). Let π‘†βˆΆπΆβ†’π» be a k-strictly pseudo-contractive mapping. Define π‘‡βˆΆπΆβ†’π» by 𝑇π‘₯=πœ†π‘₯+(1βˆ’πœ†)𝑆π‘₯ for each π‘₯∈𝐢. Then, as πœ†βˆˆ[π‘˜,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 β€–π‘₯+𝑦‖2≀‖π‘₯β€–2+2βŸ¨π‘¦,π‘₯+π‘¦βŸ©,βˆ€π‘₯,π‘¦βˆˆπ».(2.6)

Lemma 2.4 (see [25]). Let 𝐻 be a Hilbert space and 𝐢 be a closed convex subset of 𝐻, and π‘‡βˆΆπΆβ†’πΆ a nonexpansive mapping with 𝐹(𝑇)β‰ βˆ…. If {π‘₯𝑛} is a sequence in 𝐢 weakly converging to π‘₯ and if {(πΌβˆ’π‘‡)π‘₯𝑛} converges strongly to 𝑦, then (πΌβˆ’π‘‡)π‘₯=𝑦.

Lemma 2.5 (see [26]). Let {π‘₯𝑛} and {𝑧𝑛} be bounded sequences in a Banach space E and {𝛾𝑛} be a sequence in [0,1] satisfying the following condition 0<limπ‘›β†’βˆžinf𝛾𝑛≀limπ‘›β†’βˆžsup𝛾𝑛<1.(2.7) Suppose that π‘₯𝑛+1=𝛾𝑛π‘₯𝑛+(1βˆ’π›Ύπ‘›)𝑧𝑛,𝑛β‰₯0 and limπ‘›β†’βˆžsup(‖𝑧𝑛+1βˆ’π‘§π‘›β€–βˆ’β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖)≀0. Then limπ‘›β†’βˆžβ€–π‘§π‘›βˆ’π‘₯𝑛‖=0.

Lemma 2.6 (see [27]). Assume that {π‘Žπ‘›} is a sequence of nonnegative real numbers such that π‘Žπ‘›+1≀1βˆ’π‘π‘›ξ€Έπ‘Žπ‘›+𝑏𝑛𝛿𝑛,𝑛β‰₯0,(2.8) where {𝑏𝑛} is a sequence in (0,1) and {𝛿𝑛} is a sequence in 𝑅, such that(i)βˆ‘βˆžπ‘–=1𝑏𝑖=∞.(ii)limπ‘›β†’βˆžsup𝛿𝑛≀0 or βˆ‘βˆžπ‘–=1|𝑏𝑛𝛿𝑛|<∞.
Then, limπ‘›β†’βˆžπ‘Žπ‘›=0.

Let {𝑆𝑖} be an infinite family of π‘˜π‘–-strictly pseudo-contractive mappings of 𝐢 into itself, we define a mapping π‘Šπ‘› of 𝐢 into itself as follows, π‘ˆπ‘›,𝑛+1π‘ˆβˆΆ=𝐼,𝑛,π‘›βˆΆ=πœπ‘›π‘†ξ…žπ‘›π‘ˆπ‘›,𝑛+1+ξ€·1βˆ’πœπ‘›ξ€Έπ‘ˆπΌ,⋅⋅⋅𝑛,π‘˜βˆΆ=πœπ‘˜π‘†ξ…žπ‘˜π‘ˆπ‘›,π‘˜+1+ξ€·1βˆ’πœπ‘˜ξ€Έπ‘ˆπΌ,⋅⋅⋅𝑛,2∢=𝜏2π‘†ξ…ž2π‘ˆπ‘›,3+ξ€·1βˆ’πœ2ξ€Έπ‘ŠπΌ,𝑛=π‘ˆπ‘›,1∢=𝜏1π‘†ξ…ž1π‘ˆπ‘›,2+ξ€·1βˆ’πœ1𝐼,(2.9) where 0β‰€πœπ‘–β‰€1, π‘†ξ…žπ‘–=πœŽπ‘–πΌ+(1βˆ’πœŽπ‘–)𝑆𝑖 and πœŽπ‘–βˆˆ[π‘˜π‘–,1) for π‘–βˆˆπ‘. We can obtain π‘†ξ…žπ‘– is a nonexpansive mapping and 𝐹(𝑆𝑖)=𝐹(π‘†ξ…žπ‘–) by Lemma 2.2. Furthermore, we obtain that π‘Šπ‘› is a nonexpansive mapping.

Remark 2.7. If π‘˜π‘–=0, and πœŽπ‘–=0 for π‘–βˆˆπ‘, then the definition of π‘Šπ‘› in (2.9) reduces to the definition of π‘Šπ‘› 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 {π‘†ξ…žπ‘–} be an infinite family of nonexpansive mappings of 𝐢 into itself and let {πœπ‘–} be a real sequence such that 0<πœπ‘–β‰€π‘<1 for every π‘–βˆˆπ‘. Then, for every π‘₯∈𝐢 and π‘˜βˆˆπ‘, the limit limπ‘›β†’βˆžπ‘ˆπ‘›,π‘˜π‘₯ exists.

In view of the previous lemma, we will define π‘Šπ‘₯∢=limπ‘›β†’βˆžπ‘Šπ‘›π‘₯=limπ‘›β†’βˆžπ‘ˆπ‘›,1π‘₯,π‘₯∈𝐢.(2.10)

Lemma 2.9 (see [18]). Let 𝐢 be a nonempty closed convex subset of a strictly convex Banach space. Let {π‘†ξ…žπ‘–} be an infinite family of nonexpansive mappings of 𝐢 into itself such that β‹‚βˆžπ‘–=1𝐹(π‘†ξ…žπ‘–)β‰ βˆ… and let {πœπ‘–} be a real sequence such that 0<πœπ‘–β‰€π‘<1 for every π‘–βˆˆπ‘. Then, ⋂𝐹(π‘Š)=βˆžπ‘–=1𝐹(π‘†ξ…žπ‘–)β‰ βˆ….

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

Lemma 2.10. Let 𝐢 be a nonempty closed convex subset of a strictly convex Banach space. Let {𝑆𝑖} be an infinite family of π‘˜π‘–-strictly pseudo-contractive mappings of 𝐢 into itself such that β‹‚βˆžπ‘–=1𝐹(𝑆𝑖)β‰ βˆ…. Define π‘†ξ…žπ‘–=πœŽπ‘–πΌ+(1βˆ’πœŽπ‘–)𝑆𝑖 and πœŽπ‘–βˆˆ[π‘˜π‘–,1) and let {πœπ‘–} be a real sequence such that 0<πœπ‘–β‰€π‘<1 for every π‘–βˆˆπ‘. Then, ⋂𝐹(π‘Š)=βˆžπ‘–=1𝐹(𝑆𝑖⋂)=βˆžπ‘–=1𝐹(π‘†ξ…žπ‘–)β‰ βˆ….

Lemma 2.11 (see [28]). Let 𝐢 be a nonempty closed convex subset of a Hilbert space. Let {π‘†ξ…žπ‘–} be an infinite family of nonexpansive mappings of 𝐢 into itself such that β‹‚βˆžπ‘–=1𝐹(π‘†ξ…žπ‘–)β‰ βˆ… and let {πœπ‘–} be a real sequence such that 0<πœπ‘–β‰€π‘<1 for every π‘–βˆˆπ‘. If 𝐾 is any bounded subset of 𝐢, then limπ‘›β†’βˆžsupπ‘₯βˆˆπΎβ€–β€–π‘Šπ‘₯βˆ’π‘Šπ‘›π‘₯β€–β€–=0.(2.11)

3. Main Results

Let 𝐻 be a real Hilbert space and 𝐹 be a π‘˜-Lipschitzian and πœ‚-strongly monotone operator with π‘˜>0, πœ‚>0, 0<πœ‡<2πœ‚/π‘˜2 and 0<𝑑<1. Then, for π‘‘βˆˆmin{0,{1,1/𝜏}}, 𝑆=(πΌβˆ’π‘‘πœ‡πΉ)βˆΆπ»β†’π» is a contraction with contractive coefficient 1βˆ’π‘‘πœ and 𝜏=(1/2)πœ‡(2πœ‚βˆ’πœ‡π‘˜2).

In fact, from (1.2) and (1.3), we obtain ‖𝑆π‘₯βˆ’π‘†π‘¦β€–2=β€–π‘₯βˆ’π‘¦βˆ’π‘‘πœ‡(𝐹π‘₯βˆ’πΉπ‘¦)β€–2=β€–π‘₯βˆ’π‘¦β€–2+𝑑2πœ‡2‖𝐹π‘₯βˆ’πΉπ‘¦β€–2βˆ’2π‘‘πœ‡βŸ¨πΉπ‘₯βˆ’πΉπ‘¦,π‘₯βˆ’π‘¦βŸ©β‰€β€–π‘₯βˆ’π‘¦β€–2+π‘˜2𝑑2πœ‡2β€–π‘₯βˆ’π‘¦β€–2βˆ’2π‘‘πœ‚πœ‡β€–π‘₯βˆ’π‘¦β€–2≀1βˆ’π‘‘πœ‡2πœ‚βˆ’πœ‡π‘˜2ξ€Έξ€Έβ€–π‘₯βˆ’π‘¦β€–2≀(1βˆ’π‘‘πœ)2β€–π‘₯βˆ’π‘¦β€–2.(3.1) Thus, 𝑆=(1βˆ’π‘‘πœ‡πΉ) is a contraction with contractive coefficient 1βˆ’π‘‘πœβˆˆ(0,1).

Now, we show the strong convergence results for an infinite family π‘˜π‘–-strictly pseudo-contractive mappings in Hilbert space.

Theorem 3.1. Let 𝐢 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝐹 be a bifunction from 𝐢×𝐢→𝑅 satisfying (A1)–(A4). Let π‘†π‘–βˆΆπΆβ†’πΆ be a π‘˜π‘–-strictly pseudo-contractive mapping with β‹‚βˆžπ‘–=1𝐹(𝑆𝑖)βˆ©πΈπ‘ƒβ‰ βˆ… and {πœπ‘–} be a real sequence such that 0<πœπ‘–β‰€π‘<1, π‘–βˆˆπ‘. Let 𝑓 be a contraction of 𝐻 into itself with π›½βˆˆ(0,1) and 𝐡 be π‘˜-Lipschitzian and πœ‚-strongly monotone operator on H with coefficients π‘˜,πœ‚>0, 0<πœ‡<2πœ‚/π‘˜2, 0<π‘Ÿ<(1/2)πœ‡(2πœ‚βˆ’πœ‡π‘˜2)/𝛽=(𝜏/𝛽) and 𝜏<1. Let {π‘₯𝑛} be a sequence generated by 𝐹𝑒𝑛+1,π‘¦πœ†π‘›βŸ¨π‘¦βˆ’π‘’π‘›,π‘’π‘›βˆ’π‘₯π‘›π‘¦βŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ,𝑛=𝛿𝑛𝑒𝑛+ξ€·1βˆ’π›Ώπ‘›ξ€Έπ‘Šπ‘›π‘’π‘›,π‘₯𝑛+1=𝛼𝑛π‘₯π‘Ÿπ‘“π‘›ξ€Έ+𝛽𝑛π‘₯𝑛+ξ€·ξ€·1βˆ’π›½π‘›ξ€ΈπΌβˆ’πœ‡π›Όπ‘›π΅ξ€Έπ‘¦π‘›,βˆ€π‘›βˆˆπ‘,(3.2) where 𝑒𝑛=π‘‡πœ†π‘›π‘₯𝑛 and {π‘Šπ‘›βˆΆπΆβ†’πΆ} is the sequence defined by (2.9). If {𝛼𝑛}, {𝛽𝑛}, {𝛿𝑛}, and {πœ†π‘›} satisfy the following conditions:(i){𝛼𝑛}βŠ‚(0,1), limπ‘›β†’βˆžπ›Όπ‘›=0, β€‰β€‰βˆ‘βˆžπ‘–=1𝛼𝑛=∞,(ii)0<limπ‘›β†’βˆžinf𝛽𝑛≀limπ‘›β†’βˆžsup𝛽𝑛<1, (iii)0<limπ‘›β†’βˆžinf𝛿𝑛≀limπ‘›β†’βˆžsup𝛿𝑛<1, limπ‘›β†’βˆž|𝛿𝑛+1βˆ’π›Ώπ‘›|=0, (iv){πœ†π‘›}βŠ‚(0,∞), limπ‘›β†’βˆžπœ†π‘›>0, limπ‘›β†’βˆž|πœ†π‘›+1βˆ’πœ†π‘›|=0.
Then {π‘₯𝑛} converges strongly to β‹‚π‘§βˆˆβˆžπ‘–=1𝐹(𝑆𝑖)∩EPβ‰ βˆ…, where 𝑧 is the unique solution of variational inequality limπ‘›β†’βˆžsup⟨(π‘Ÿπ‘“βˆ’πœ‡π΅)𝑧,π‘βˆ’π‘§βŸ©β‰€0,βˆ€π‘βˆˆβˆžξ™π‘–=1πΉξ€·π‘†π‘–ξ€Έβˆ©EPβ‰ βˆ…,(3.3) that is, 𝑧=𝑃𝐹(π‘Š)βˆ©πΈπ‘ƒ(𝐹)(πΌβˆ’πœ‡π΅+π‘Ÿπ‘“)𝑧, which is the optimality condition for the minimization problem minβ‹‚π‘§βˆˆβˆžπ‘–=1𝐹(𝑆𝑖)βˆ©πΈπ‘ƒ12βŸ¨πœ‡π΅π‘§,π‘§βŸ©βˆ’β„Ž(𝑧),(3.4) where β„Ž is a potential function for π‘Ÿπ‘“ (i.e., β„Žξ…ž(𝑧)=π‘Ÿπ‘“(𝑧) for π‘§βˆˆπ»).

Proof . We divide the proof into five steps.
Step 1. We prove that {π‘₯𝑛} is bounded.
Noting the conditions (i) and (ii), we may assume, without loss of generality, that 𝛼𝑛/(1βˆ’π›½π‘›)≀min{1,1/𝜏}. For π‘₯,π‘¦βˆˆπΆ, we obtain β€–β€–ξ€·ξ€·1βˆ’π›½π‘›ξ€ΈπΌβˆ’π›Όπ‘›ξ€Έπœ‡π΅π‘₯βˆ’ξ€·ξ€·1βˆ’π›½π‘›ξ€ΈπΌβˆ’π›Όπ‘›ξ€Έπ‘¦β€–β€–β‰€ξ€·πœ‡π΅1βˆ’π›½π‘›ξ€Έβ€–β€–β€–ξ‚΅π›ΌπΌβˆ’π‘›1βˆ’π›½π‘›ξ‚Άξ‚΅π›Όπœ‡π΅π‘₯βˆ’πΌβˆ’π‘›1βˆ’π›½π‘›ξ‚Άπ‘¦β€–β€–β€–β‰€ξ€·πœ‡π΅1βˆ’π›½π‘›ξ€Έξ‚΅π›Ό1βˆ’π‘›1βˆ’π›½π‘›πœξ‚Ά=ξ€·β€–π‘₯βˆ’π‘¦β€–1βˆ’π›½π‘›βˆ’π›Όπ‘›πœξ€Έβ€–π‘₯βˆ’π‘¦β€–.(3.5) Take β‹‚π‘βˆˆβˆžπ‘–=1𝐹(𝑆𝑖)∩EPβ‰ βˆ…. Since 𝑒𝑛=π‘‡πœ†π‘›π‘₯𝑛 and 𝑝=π‘‡πœ†π‘›π‘, then from Lemma 2.1, we know that, for any π‘›βˆˆπ‘, ‖‖𝑒𝑛‖‖=β€–β€–π‘‡βˆ’π‘πœ†π‘›π‘₯π‘›βˆ’π‘‡πœ†π‘›π‘β€–β€–β‰€β€–β€–π‘₯𝑛‖‖.βˆ’π‘(3.6) Furthermore, since π‘Šπ‘›π‘=𝑝 and (3.6), we have ‖‖𝑦𝑛‖‖=β€–β€–π›Ώβˆ’π‘π‘›π‘’π‘›+ξ€·1βˆ’π›Ώπ‘›ξ€Έπ‘Šπ‘›π‘’π‘›β€–β€–=β€–β€–π›Ώβˆ’π‘π‘›ξ€·π‘’π‘›ξ€Έ+ξ€·βˆ’π‘1βˆ’π›Ώπ‘›π‘Šξ€Έξ€·π‘›π‘’π‘›ξ€Έβ€–β€–βˆ’π‘β‰€π›Ώπ‘›β€–β€–π‘’π‘›β€–β€–+ξ€·βˆ’π‘1βˆ’π›Ώπ‘›ξ€Έβ€–β€–π‘Šπ‘›π‘’π‘›β€–β€–β‰€β€–β€–π‘’βˆ’π‘π‘›β€–β€–β‰€β€–β€–π‘₯βˆ’π‘π‘›β€–β€–.βˆ’π‘(3.7) Thus, it follows from (3.7) that β€–β€–π‘₯𝑛+1β€–β€–=β€–β€–π›Όβˆ’π‘π‘›ξ€·π‘₯π‘Ÿπ‘“π‘›ξ€Έ+𝛽𝑛π‘₯𝑛+ξ€·ξ€·1βˆ’π›½π‘›ξ€ΈπΌβˆ’πœ‡π›Όπ‘›π΅ξ€Έπ‘¦π‘›β€–β€–=β€–β€–π›Όβˆ’π‘π‘›π‘Ÿξ€·π‘“ξ€·π‘₯π‘›ξ€Έξ€Έβˆ’π‘“(𝑝)+𝛼𝑛(π‘Ÿπ‘“(𝑝)βˆ’πœ‡π΅π‘)+𝛽𝑛π‘₯𝑛+βˆ’π‘ξ€·ξ€·1βˆ’π›½π‘›ξ€ΈπΌβˆ’πœ‡π›Όπ‘›π΅π‘¦ξ€Έξ€·π‘›ξ€Έβ€–β€–βˆ’π‘β‰€π›Όπ‘›β€–β€–π‘₯π‘Ÿπ›½π‘›β€–β€–βˆ’π‘+𝛼𝑛(β€–π‘Ÿπ‘“π‘)βˆ’πœ‡π΅π‘β€–+𝛽𝑛‖‖π‘₯𝑛‖‖+ξ€·βˆ’π‘1βˆ’π›½π‘›βˆ’πœπ›Όπ‘›ξ€Έβ€–β€–π‘¦π‘›β€–β€–β‰€ξ€·βˆ’π‘1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘₯(πœβˆ’π‘Ÿπ›½)π‘›β€–β€–βˆ’π‘+𝛼𝑛‖‖π‘₯β€–π‘Ÿπ‘“(𝑝)βˆ’πœ‡π΅π‘β€–β‰€max𝑛‖‖,β€–βˆ’π‘π‘Ÿπ‘“(𝑝)βˆ’πœ‡π΅π‘β€–ξ‚Ό.πœβˆ’π‘Ÿπ›½(3.8) By induction, we have β€–β€–π‘₯𝑛‖‖‖‖π‘₯βˆ’π‘β‰€max1β€–β€–,(βˆ’π‘β€–π‘Ÿπ‘“π‘)βˆ’πœ‡π΅π‘β€–ξ‚Όπœβˆ’π‘Ÿπ›½,𝑛β‰₯1.(3.9) Hence, {π‘₯𝑛} is bounded and we also obtain that {𝑒𝑛}, {π‘Šπ‘›π‘’π‘›}, {𝑦𝑛}, {𝐡𝑦𝑛}, and {𝑓(π‘₯𝑛)} are all bounded. Without loss of generality, we can assume that there exists a bounded set πΎβŠ‚πΆ such that {𝑒𝑛}, {π‘Šπ‘›π‘’π‘›}, {𝑦𝑛}, {𝐡𝑦𝑛}, {𝑓(π‘₯𝑛)}∈𝐾, for all π‘›βˆˆπ‘.
Step 2. We show that limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘₯𝑛+1β€–=0.
Let π‘₯𝑛+1=(1βˆ’π›½π‘›)𝑧𝑛+𝛽𝑛π‘₯𝑛. We note that 𝑧𝑛=π‘₯𝑛+1βˆ’π›½π‘›π‘₯𝑛1βˆ’π›½π‘›=𝛼𝑛π‘₯π‘Ÿπ‘“π‘›ξ€Έ+ξ€·ξ€·1βˆ’π›½π‘›ξ€ΈπΌβˆ’πœ‡π›Όπ‘›π΅ξ€Έπ‘¦π‘›1βˆ’π›½π‘›,(3.10) and then 𝑧𝑛+1βˆ’π‘§π‘›=𝛼𝑛+1ξ€·π‘₯π‘Ÿπ‘“π‘›+1ξ€Έ+ξ€·ξ€·1βˆ’π›½π‘›+1ξ€ΈπΌβˆ’πœ‡π›Όπ‘›+1𝐡𝑦𝑛+11βˆ’π›½π‘›+1βˆ’π›Όπ‘›ξ€·π‘₯π‘Ÿπ‘“π‘›ξ€Έ+ξ€·ξ€·1βˆ’π›½π‘›ξ€ΈπΌβˆ’πœ‡π›Όπ‘›π΅ξ€Έπ‘¦π‘›1βˆ’π›½π‘›=𝛼𝑛+11βˆ’π›½π‘›+1ξ€·ξ€·π‘₯π‘Ÿπ‘“π‘›+1ξ€Έβˆ’πœ‡π΅π‘¦π‘›+1ξ€Έβˆ’π›Όπ‘›1βˆ’π›½π‘›ξ€·ξ€·π‘₯π‘Ÿπ‘“π‘›ξ€Έβˆ’πœ‡π΅π‘¦π‘›ξ€Έ+𝑦𝑛+1βˆ’π‘¦π‘›.(3.11) Therefore, ‖‖𝑧𝑛+1βˆ’π‘§π‘›β€–β€–β‰€π›Όπ‘›+11βˆ’π›½π‘›+1ξ€·β€–β€–ξ€·π‘₯π‘Ÿπ‘“π‘›+1ξ€Έβ€–β€–+β€–β€–πœ‡π΅π‘¦π‘›+1β€–β€–ξ€Έ+𝛼𝑛1βˆ’π›½π‘›ξ€·β€–β€–ξ€·π‘₯π‘Ÿπ‘“π‘›ξ€Έβ€–β€–+β€–β€–πœ‡π΅π‘¦π‘›β€–β€–ξ€Έ+‖‖𝑦𝑛+1βˆ’π‘¦π‘›β€–β€–.(3.12) It follows from (3.2) that ‖‖𝑦𝑛+1βˆ’π‘¦π‘›β€–β€–=‖‖𝛿𝑛+1𝑒𝑛+1+ξ€·1βˆ’π›Ώπ‘›+1ξ€Έπ‘Šπ‘›+1𝑒𝑛+1βˆ’ξ€·π›Ώπ‘›π‘’π‘›+ξ€·1βˆ’π›Ώπ‘›ξ€Έπ‘Šπ‘›π‘’π‘›ξ€Έβ€–β€–β‰€||𝛿𝑛+1βˆ’π›Ώπ‘›||‖‖𝑒𝑛‖‖+𝛿𝑛+1‖‖𝑒𝑛+1βˆ’π‘’π‘›β€–β€–+ξ€·1βˆ’π›Ώπ‘›+1ξ€Έβ€–β€–π‘Šπ‘›+1𝑒𝑛+1βˆ’π‘Šπ‘›π‘’π‘›β€–β€–+||𝛿𝑛+1βˆ’π›Ώπ‘›||β€–β€–π‘Šπ‘›π‘’π‘›β€–β€–.(3.13)
We will estimate ‖𝑒𝑛+1βˆ’π‘’π‘›β€–. From 𝑒𝑛+1=π‘‡πœ†π‘›+1π‘₯𝑛+1 and 𝑒𝑛=π‘‡πœ†π‘›π‘₯𝑛, we obtain 𝐹𝑒𝑛+1ξ€Έ+1,π‘¦πœ†π‘›+1ξ«π‘¦βˆ’π‘’π‘›+1,𝑒𝑛+1βˆ’π‘¦π‘›+1𝐹𝑒β‰₯0,βˆ€π‘¦βˆˆπΆ,(3.14)𝑛+1,π‘¦πœ†π‘›βŸ¨π‘¦βˆ’π‘’π‘›,π‘’π‘›βˆ’π‘¦π‘›βŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ.(3.15)
Taking 𝑦=𝑒𝑛 in (3.14) and 𝑦=𝑒𝑛+1 in (3.15), we have 𝐹𝑒𝑛+1,𝑒𝑛+1πœ†π‘›+1ξ«π‘’π‘›βˆ’π‘’π‘›+1,𝑒𝑛+1βˆ’π‘₯𝑛+1𝐹𝑒β‰₯0,𝑛,𝑒𝑛+1ξ€Έ+1πœ†π‘›ξ«π‘’π‘›+1βˆ’π‘’π‘›,π‘’π‘›βˆ’π‘₯𝑛β‰₯0.(3.16)
So, from (A2), one has 𝑒𝑛+1βˆ’π‘’π‘›,π‘’π‘›βˆ’π‘₯nπœ†π‘›βˆ’π‘’π‘›+1βˆ’π‘₯𝑛+1πœ†π‘›+1ξƒ’β‰₯0,(3.17) furthermore, 𝑒𝑛+1βˆ’π‘’π‘›,π‘’π‘›βˆ’π‘’π‘›+1βˆ’π‘₯π‘›βˆ’πœ†π‘›πœ†π‘›+1𝑒𝑛+1βˆ’π‘₯𝑛+1ξ€Έξƒ’β‰₯0.(3.18) Since limπ‘›β†’βˆžπœ†π‘›>0, we assume that there exists a real number such that πœ†π‘›>π‘Ž>0 for all π‘›βˆˆπ‘. Thus, we obtain ‖‖𝑒𝑛+1βˆ’π‘’π‘›β€–β€–2≀𝑒𝑛+1βˆ’π‘’π‘›,π‘₯𝑛+1βˆ’π‘₯𝑛+ξ‚΅πœ†1βˆ’π‘›πœ†π‘›+1𝑒𝑛+1βˆ’π‘₯𝑛+1≀‖‖𝑒𝑛+1βˆ’π‘’π‘›β€–β€–ξƒ―β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+||||πœ†1βˆ’π‘›πœ†π‘›+1||||‖‖𝑒𝑛+1βˆ’π‘₯𝑛+1β€–β€–ξƒ°,(3.19) which means ‖‖𝑒𝑛+1βˆ’π‘’π‘›β€–β€–β‰€β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+||||πœ†1βˆ’π‘›πœ†π‘›+1||||‖‖𝑒𝑛+1βˆ’π‘₯𝑛+1‖‖≀‖‖π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+1π‘Ž||πœ†π‘›+1βˆ’πœ†π‘›||‖‖𝑒𝑛+1βˆ’π‘₯𝑛+1‖‖≀‖‖π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+𝐿1||πœ†π‘›+1βˆ’πœ†π‘›||,(3.20) where 𝐿1=sup{‖𝑒𝑛+1βˆ’π‘₯𝑛+1β€–βˆΆπ‘›βˆˆπ‘}.
Next, we estimate β€–π‘Šπ‘›+1𝑒𝑛+1βˆ’π‘Šπ‘›π‘’π‘›β€–. Notice that β€–β€–π‘Šπ‘›+1𝑒𝑛+1βˆ’π‘Šπ‘›π‘’π‘›β€–β€–=β€–β€–π‘Šπ‘›+1𝑒𝑛+1βˆ’π‘Šπ‘›+1𝑒𝑛+π‘Šπ‘›+1π‘’π‘›βˆ’π‘Šπ‘›π‘’π‘›β€–β€–β‰€β€–β€–π‘’π‘›+1βˆ’π‘’π‘›β€–β€–+β€–β€–π‘Šπ‘›+1π‘’π‘›βˆ’π‘Šπ‘›π‘’π‘›β€–β€–.(3.21) From (2.9), we obtain β€–β€–π‘Šπ‘›+1π‘’π‘›βˆ’π‘Šπ‘›π‘’π‘›β€–β€–=β€–β€–πœ1π‘†ξ…ž1π‘ˆπ‘›+1,2π‘’π‘›βˆ’πœ1π‘†ξ…ž1π‘ˆπ‘›,2π‘’π‘›β€–β€–β‰€πœ1β€–β€–π‘ˆπ‘›+1,2π‘’π‘›βˆ’π‘ˆπ‘›,2𝑒𝑛‖‖=𝜏1β€–β€–πœ2π‘†ξ…ž2π‘ˆπ‘›+1,3π‘’π‘›βˆ’πœ2π‘†ξ…ž2π‘ˆπ‘›,3π‘’π‘›β€–β€–β‰€πœ1𝜏2β€–β€–π‘ˆπ‘›+1,3π‘’π‘›βˆ’π‘ˆπ‘›,3π‘’π‘›β€–β€–β‰€β‹―β‰€πœ1𝜏2β‹…β‹…β‹…πœπ‘›β€–β€–π‘ˆπ‘›+1,𝑛+1π‘’π‘›βˆ’π‘ˆπ‘›,𝑛+1𝑒𝑛‖‖≀𝐿2𝑛𝑖=1πœπ‘–,(3.22) where 𝐿2β‰₯0 is a constant such that β€–π‘ˆπ‘›+1,𝑛+1π‘’π‘›βˆ’π‘ˆπ‘›,𝑛+1𝑒𝑛‖≀𝐿2, for all π‘›βˆˆπ‘.
Substituting (3.20) and (3.22) into (3.21), we obtain β€–β€–π‘Šπ‘›+1𝑒𝑛+1βˆ’π‘Šπ‘›π‘’π‘›β€–β€–β‰€β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+𝐿1||πœ†π‘›+1βˆ’πœ†π‘›||+𝐿2𝑛𝑖=1πœπ‘–.(3.23) Hence, we have ‖‖𝑦𝑛+1βˆ’π‘¦π‘›β€–β€–β‰€||𝛿𝑛+1βˆ’π›Ώπ‘›||‖‖𝑒𝑛‖‖+β€–β€–π‘Šπ‘›π‘’π‘›β€–β€–ξ€Έ+β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+ξ€·1βˆ’π›Ώπ‘›+1𝐿2𝑛𝑖=1πœπ‘–+𝐿1||πœ†π‘›+1βˆ’πœ†π‘›||≀𝐿3||𝛿𝑛+1βˆ’π›Ώπ‘›||+β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+ξ€·1βˆ’π›Ώπ‘›+1𝐿2𝑛𝑖=1πœπ‘–+𝐿1||πœ†π‘›+1βˆ’πœ†π‘›||,(3.24) where 𝐿3=sup{‖𝑒𝑛‖+β€–π‘Šπ‘›π‘’π‘›β€–βˆΆπ‘›βˆˆπ‘}.
Furthermore, ‖‖𝑧𝑛+1βˆ’π‘§π‘›β€–β€–β‰€π›Όπ‘›+11βˆ’π›½π‘›+1ξ€·β€–β€–ξ€·π‘₯π‘Ÿπ‘“π‘›+1ξ€Έβ€–β€–+β€–β€–πœ‡π΅π‘¦π‘›+1β€–β€–ξ€Έ+𝛼𝑛1βˆ’π›½π‘›ξ€·β€–β€–ξ€·π‘₯π‘Ÿπ‘“π‘›ξ€Έβ€–β€–+β€–β€–πœ‡π΅π‘¦π‘›β€–β€–ξ€Έ+β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+𝐿1||πœ†π‘›+1βˆ’πœ†π‘›||+𝐿2ξ€·1βˆ’π›Ώπ‘›+1𝑛𝑖=1πœπ‘–+𝐿3||𝛿𝑛+1βˆ’π›Ώπ‘›||.(3.25) It follows from (3.25) that ‖‖𝑧𝑛+1βˆ’π‘§π‘›β€–β€–βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀𝛼𝑛+11βˆ’π›½π‘›+1ξ€·β€–β€–ξ€·π‘₯π‘Ÿπ‘“π‘›+1ξ€Έβ€–β€–+β€–β€–πœ‡π΅π‘¦π‘›+1β€–β€–ξ€Έ+𝛼𝑛1βˆ’π›½π‘›ξ€·β€–β€–ξ€·π‘₯π‘Ÿπ‘“π‘›ξ€Έβ€–β€–+β€–β€–πœ‡π΅π‘¦π‘›β€–β€–ξ€Έ+𝐿1||πœ†π‘›+1βˆ’πœ†π‘›||+𝐿2ξ€·1βˆ’π›Ώπ‘›+1𝑛𝑖=1πœπ‘–+𝐿3||𝛿𝑛+1βˆ’π›Ώπ‘›||.(3.26) By the conditions (i), (iii), and (iv), we obtain limπ‘›β†’βˆžξ€·β€–β€–π‘§sup𝑛+1βˆ’π‘§π‘›β€–β€–βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀0.(3.27) Hence, by Lemma 2.5, one has limπ‘›β†’βˆžβ€–β€–π‘§π‘›βˆ’π‘₯𝑛‖‖=0,(3.28) which implies limπ‘›β†’βˆžβ€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖=limπ‘›β†’βˆžξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π‘§π‘›βˆ’π‘₯𝑛‖‖=0.(3.29)Step 3. We claim that limπ‘›β†’βˆžβ€–π‘Šπ‘’π‘›βˆ’π‘’π‘›β€–=0.
Notice that β€–β€–π‘Šπ‘’π‘›βˆ’π‘’π‘›β€–β€–=β€–β€–π‘Šπ‘’π‘›βˆ’π‘Šπ‘›π‘’π‘›+π‘Šπ‘›π‘’π‘›βˆ’π‘’π‘›β€–β€–β‰€β€–β€–π‘Šπ‘’π‘›βˆ’π‘Šπ‘›π‘’π‘›β€–β€–+β€–β€–π‘Šπ‘›π‘’π‘›βˆ’π‘’π‘›β€–β€–β‰€supπ‘’βˆˆπΎβ€–β€–π‘Šπ‘’βˆ’π‘Šπ‘›π‘’β€–β€–+β€–β€–π‘Šπ‘›π‘’π‘›βˆ’π‘’π‘›β€–β€–.(3.30) It follows from (3.2) that β€–β€–π‘Šπ‘›π‘’π‘›βˆ’π‘’π‘›β€–β€–=β€–β€–π‘Šπ‘›π‘’π‘›βˆ’π‘¦π‘›+π‘¦π‘›βˆ’π‘’π‘›β€–β€–β‰€β€–β€–π‘¦π‘›βˆ’π‘’π‘›β€–β€–+β€–β€–π‘Šπ‘›π‘’π‘›βˆ’π‘¦π‘›β€–β€–=β€–β€–π‘¦π‘›βˆ’π‘’π‘›β€–β€–+π›Ώπ‘›β€–β€–π‘Šπ‘›π‘’π‘›βˆ’π‘’π‘›β€–β€–β‰€β€–β€–π‘₯π‘›βˆ’π‘’π‘›β€–β€–+β€–β€–π‘¦π‘›βˆ’π‘₯𝑛‖‖+π›Ώπ‘›β€–β€–π‘Šπ‘›π‘’π‘›βˆ’π‘’π‘›β€–β€–.(3.31) By the condition (iii), we obtain β€–β€–π‘Šπ‘›π‘’π‘›βˆ’π‘’π‘›β€–β€–β‰€11βˆ’π›Ώπ‘›ξ€·β€–β€–π‘₯π‘›βˆ’π‘’π‘›β€–β€–+β€–β€–π‘¦π‘›βˆ’π‘₯𝑛‖‖.(3.32)
First, we show limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘’π‘›β€–=0. From (3.2), for all β‹‚π‘βˆˆβˆžπ‘–=1𝐹(𝑆𝑖)∩EP(𝐹), applying Lemma 2.3 and noting that β€–β‹…β€– is convex, we obtain β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘2=‖‖𝛼𝑛π‘₯π‘Ÿπ‘“π‘›ξ€Έ+𝛽𝑛π‘₯𝑛+ξ€·ξ€·1βˆ’π›½π‘›ξ€ΈπΌβˆ’πœ‡π›Όπ‘›π΅ξ€Έπ‘¦π‘›β€–β€–βˆ’π‘2=‖‖𝛼𝑛π‘₯π‘Ÿπ‘“π‘›ξ€Έ+πœ‡π΅π‘¦π‘›ξ€Έ+𝛽𝑛π‘₯𝑛+ξ€·βˆ’π‘1βˆ’π›½π‘›π‘¦ξ€Έξ€·π‘›ξ€Έβ€–β€–βˆ’π‘2≀‖‖𝛽𝑛π‘₯𝑛+ξ€·βˆ’π‘1βˆ’π›½π‘›π‘¦ξ€Έξ€·π‘›ξ€Έβ€–β€–βˆ’π‘2+2𝛼𝑛π‘₯π‘Ÿπ‘“π‘›ξ€Έ+πœ‡π΅π‘¦π‘›,π‘₯𝑛+1ξ¬βˆ’π‘β‰€π›½π‘›β€–β€–π‘₯π‘›β€–β€–βˆ’π‘2+ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π‘¦π‘›β€–β€–βˆ’π‘2+2𝛼𝑛‖‖π‘₯π‘Ÿπ‘“π‘›ξ€Έ+πœ‡π΅π‘¦π‘›β€–β€–β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘β‰€π›½π‘›β€–β€–π‘₯π‘›β€–β€–βˆ’π‘2+ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π‘’π‘›β€–β€–βˆ’π‘2+2𝛼𝑛‖‖π‘₯π‘Ÿπ‘“π‘›ξ€Έ+πœ‡π΅π‘¦π‘›β€–β€–β€–β€–π‘₯𝑛+1β€–β€–.βˆ’π‘(3.33) Since 𝑒𝑛=π‘‡πœ†π‘›π‘₯𝑛, 𝑝=π‘‡πœ†π‘›π‘, we have β€–β€–π‘’π‘›β€–β€–βˆ’π‘2=β€–β€–π‘‡πœ†π‘›π‘₯π‘›βˆ’π‘‡πœ†π‘›pβ€–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,(3.34) which implies 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.(3.35) Substituting (3.35) into (3.33), we have β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘2≀‖‖π‘₯π‘›β€–β€–βˆ’π‘2βˆ’ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π‘₯π‘›βˆ’π‘’π‘›β€–β€–2+2𝛼𝑛‖‖π‘₯π‘Ÿπ‘“π‘›ξ€Έ+πœ‡π΅π‘¦π‘›β€–β€–β€–β€–π‘₯𝑛+1β€–β€–,βˆ’π‘(3.36) which means ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π‘₯π‘›βˆ’π‘’π‘›β€–β€–2≀‖‖π‘₯π‘›β€–β€–βˆ’π‘2βˆ’β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘2+2𝛼𝑛‖‖π‘₯π‘Ÿπ‘“π‘›ξ€Έ+πœ‡π΅π‘¦π‘›β€–β€–β€–β€–π‘₯𝑛+1‖‖≀‖‖π‘₯βˆ’π‘π‘›+1βˆ’π‘₯𝑛‖‖‖‖π‘₯𝑛‖‖+β€–β€–π‘₯βˆ’π‘π‘›+1β€–β€–ξ€Έβˆ’π‘+2𝛼𝑛‖‖π‘₯π‘Ÿπ‘“π‘›ξ€Έ+πœ‡π΅π‘¦π‘›β€–β€–β€–β€–π‘₯𝑛+1β€–β€–.βˆ’π‘(3.37) Noticing limπ‘›β†’βˆžπ›Όπ‘›=0 and limπ‘›β†’βˆžinf(1βˆ’π›½π‘›)>0, we have limπ‘›β†’βˆžβ€–β€–π‘₯π‘›βˆ’π‘’π‘›β€–β€–=0.(3.38)
Second, we show limπ‘›β†’βˆžβ€–π‘¦π‘›βˆ’π‘₯𝑛‖=0. It follows from (3.2) 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‘₯𝑛‖‖=‖‖𝛼𝑛π‘₯π‘Ÿπ‘“π‘›ξ€Έ+𝛽𝑛π‘₯𝑛+ξ€·ξ€·1βˆ’π›½π‘›ξ€ΈπΌβˆ’πœ‡π›Όπ‘›π΅ξ€Έπ‘¦π‘›βˆ’π‘¦π‘›β€–β€–+β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀𝛼𝑛‖‖π‘₯π‘Ÿπ‘“π‘›ξ€Έ+πœ‡π΅π‘¦π‘›β€–β€–+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘¦π‘›β€–β€–+β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖.(3.39) This implies that ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π‘¦π‘›βˆ’π‘₯𝑛‖‖≀𝛼𝑛‖‖π‘₯π‘Ÿπ‘“π‘›ξ€Έ+πœ‡π΅π‘¦π‘›β€–β€–+β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖.(3.40) Noticing limπ‘›β†’βˆžπ›Όπ‘›=0, limπ‘›β†’βˆžinf(1βˆ’π›½π‘›)>0 and (3.30), we have limπ‘›β†’βˆžβ€–β€–π‘¦π‘›βˆ’π‘₯𝑛‖‖=0.(3.41) Thus, substituting (3.41) and (3.38) into (3.32), we obtain limπ‘›β†’βˆžβ€–β€–π‘Šπ‘›π‘’π‘›βˆ’π‘’π‘›β€–β€–=0.(3.42) Furthermore, (3.42), (3.30), and Lemma 2.11 lead to limπ‘›β†’βˆžβ€–β€–π‘Šπ‘’π‘›βˆ’π‘’π‘›β€–β€–=0.(3.43)
Step 4. Letting 𝑧=𝑃𝐹(π‘Š)∩EP(𝐹)(πΌβˆ’πœ‡π΅+π‘Ÿπ‘“)𝑧, we show limπ‘›β†’βˆžsup⟨(π‘Ÿπ‘“βˆ’πœ‡π΅)𝑧,π‘₯π‘›βˆ’π‘§βŸ©β‰€0.(3.44) We know that 𝑃𝐹(π‘Š)∩EP(𝐹)(πΌβˆ’πœ‡π΅+π‘Ÿπ‘“) is a contraction. Indeed, for any π‘₯,π‘¦βˆˆπ», we have ‖‖𝑃𝐹(π‘Š)∩EP(𝐹)(πΌβˆ’πœ‡π΅+π‘Ÿπ‘“)π‘₯βˆ’π‘ƒπΉ(π‘Š)∩EP(𝐹)β€–β€–β€–(πΌβˆ’πœ‡π΅+π‘Ÿπ‘“)𝑦≀‖(πΌβˆ’πœ‡π΅+π‘Ÿπ‘“)π‘₯βˆ’(πΌβˆ’πœ‡π΅+π‘Ÿπ‘“)𝑦≀(1βˆ’πœ+π‘Ÿπ›½)β€–π‘₯βˆ’π‘¦β€–,(3.45) and hence 𝑃𝐹(π‘Š)∩EP(𝐹)(πΌβˆ’πœ‡π΅+π‘Ÿπ‘“) is a contraction due to (1βˆ’πœ+π‘Ÿπ›½)∈(0,1). Thus, Banach’s Contraction Mapping Principle guarantees that 𝑃𝐹(π‘Š)∩EP(𝐹)(πΌβˆ’πœ‡π΅+π‘Ÿπ‘“) has a unique fixed point, which implies 𝑧=𝑃𝐹(π‘Š)∩EP(𝐹)(πΌβˆ’πœ‡π΅+π‘Ÿπ‘“)𝑧.
Since {𝑒𝑛𝑖}βŠ‚{𝑒𝑛} is bounded in 𝐢, without loss of generality, we can assume that {𝑒𝑛𝑖}β‡€πœ”, it follows from (3.43) that π‘Šπ‘’π‘›π‘–β‡€πœ”. Since 𝐢 is closed and convex, 𝐢 is weakly closed. Thus we have πœ”βˆˆπΆ.
Let us show πœ”βˆˆπΉ(π‘Š). For the sake of contradiction, suppose that πœ”βˆ‰πΉ(π‘Š), that is, π‘Šπœ”β‰ πœ”. Since {𝑒𝑛𝑖}β‡€πœ”, by the Opial condition, we have limπ‘›β†’βˆžβ€–β€–π‘’infπ‘›π‘–β€–β€–βˆ’πœ”<limπ‘›β†’βˆžβ€–β€–π‘’infπ‘›π‘–β€–β€–βˆ’π‘Šπœ”β‰€limπ‘›β†’βˆžξ€½β€–β€–π‘’infπ‘›π‘–βˆ’π‘Šπ‘’π‘›π‘–β€–β€–+β€–β€–π‘Šπ‘’π‘›π‘–β€–β€–ξ€Ύβˆ’π‘Šπœ”β‰€limπ‘›β†’βˆžξ€½β€–β€–π‘’infπ‘›π‘–βˆ’π‘Šπ‘’π‘›π‘–β€–β€–+‖‖𝑒𝑛𝑖‖‖.βˆ’πœ”(3.46) It follows (3.43) that limπ‘›β†’βˆžβ€–β€–π‘’infπ‘›π‘–β€–β€–βˆ’πœ”<limπ‘›β†’βˆžβ€–β€–π‘’inf𝑛𝑖‖‖.βˆ’πœ”(3.47) This is a contradiction, which shows that πœ”βˆˆπΉ(π‘Š).
Next, we prove that πœ”βˆˆEP(𝐹). By (3.2), we obtain 𝐹𝑒𝑛+1,π‘¦πœ†π‘›βŸ¨π‘¦βˆ’π‘’π‘›,π‘’π‘›βˆ’π‘₯π‘›βŸ©β‰₯0.(3.48) It follows from (A2) that 1πœ†π‘›βŸ¨π‘¦βˆ’π‘’π‘›,π‘’π‘›βˆ’π‘₯π‘›ξ€·βŸ©β‰₯𝐹𝑦,𝑒𝑛.(3.49) Replacing 𝑛 by 𝑛𝑖, we have ξ„”π‘¦βˆ’π‘’π‘›π‘–,1πœ†π‘›π‘–ξ€·π‘’π‘›π‘–βˆ’π‘₯𝑛𝑖β‰₯𝐹𝑦,𝑒𝑛𝑖.(3.50) Since (1/πœ†π‘›π‘–)(π‘’π‘›π‘–βˆ’π‘₯𝑛𝑖)β†’0 and {𝑒𝑛𝑖}β‡€πœ”, it follows from (A4) that 𝐹(𝑦,πœ”)β‰₯0 for all π‘¦βˆˆπΆ. Put 𝑧𝑑=𝑑𝑦+(1βˆ’π‘‘)πœ” for all π‘‘βˆˆ(0,1] and π‘¦βˆˆπΆ. Then, we have π‘§π‘‘βˆˆπΆ and then 𝐹(𝑧𝑑,πœ”)β‰₯0. Hence, from (A1) and (A4), we have 𝑧0=𝐹𝑑,𝑧𝑑𝑧≀𝑑𝐹𝑑+𝑧,𝑦(1βˆ’π‘‘)𝐹𝑑𝑧,𝑦≀𝑑𝐹𝑑,𝑦,(3.51) which means 𝐹(𝑧𝑑,𝑦)β‰₯0. From (A3), we obtain 𝐹(πœ”,𝑦)β‰₯0 for π‘¦βˆˆπΆ and then πœ”βˆˆEP(𝐹). Therefore, πœ”βˆˆπΉ(π‘Š)∩EP(𝐹).
Since 𝑧=𝑃𝐹(π‘Š)∩EP(𝐹)(πΌβˆ’πœ‡π΅+π‘Ÿπ‘“)𝑧, it follows from (3.38), (3.42), and Lemma 2.11 that limπ‘–β†’βˆžsup⟨(π‘Ÿπ‘“βˆ’πœ‡π΅)𝑧,π‘₯π‘›βˆ’π‘§βŸ©β‰€limπ‘›β†’βˆžξ«(π‘Ÿπ‘“βˆ’πœ‡π΅)𝑧,π‘₯π‘›π‘–ξ¬βˆ’π‘§=limπ‘–β†’βˆžξ«(π‘Ÿπ‘“βˆ’πœ‡π΅)𝑧,π‘₯π‘›π‘–βˆ’π‘’π‘›π‘–ξ¬+limπ‘–β†’βˆžξ«(π‘Ÿπ‘“βˆ’πœ‡π΅)𝑧,π‘’π‘›π‘–βˆ’π‘Šπ‘›π‘–π‘’π‘›π‘–ξ¬+limπ‘–β†’βˆžξ«(π‘Ÿπ‘“βˆ’πœ‡π΅)𝑧,π‘Šπ‘›π‘–π‘’π‘›π‘–βˆ’π‘Šπ‘’π‘›π‘–ξ¬+limπ‘–β†’βˆžξ«(π‘Ÿπ‘“βˆ’πœ‡π΅)𝑧,π‘Šπ‘’π‘›π‘–ξ¬βˆ’π‘§=⟨(π‘Ÿπ‘“βˆ’πœ‡π΅)𝑧,πœ”βˆ’π‘§βŸ©β‰€0.(3.52)
Step 5. Finally we prove that π‘₯π‘›β†’πœ” as π‘›β†’βˆž. In fact, from (3.2) and (3.7), we obtain β€–β€–π‘₯𝑛+1β€–β€–βˆ’πœ”2=‖‖𝛼𝑛π‘₯π‘Ÿπ‘“π‘›ξ€Έ+𝛽𝑛π‘₯𝑛+ξ€·ξ€·1βˆ’π›½π‘›ξ€ΈπΌβˆ’πœ‡π›Όπ‘›π΅ξ€Έπ‘¦π‘›β€–β€–βˆ’πœ”2=β€–β€–π›Όπ‘›π‘Ÿξ€·π‘“ξ€·π‘₯π‘›ξ€Έξ€Έβˆ’π‘“(πœ”)+𝛼𝑛(π‘Ÿπ‘“(πœ”)βˆ’πœ‡π΅πœ”)+𝛽𝑛π‘₯𝑛+βˆ’πœ”ξ€·ξ€·1βˆ’π›½π‘›ξ€ΈπΌβˆ’πœ‡π›Όπ‘›π΅π‘¦ξ€Έξ€·π‘›ξ€Έβ€–β€–βˆ’πœ”2=π›Όπ‘›π‘Ÿξ«π‘“ξ€·π‘₯π‘›ξ€Έβˆ’π‘“(πœ”),π‘₯𝑛+1ξ¬βˆ’πœ”+π›Όπ‘›ξ«π‘Ÿπ‘“(πœ”)βˆ’πœ‡π΅πœ”,π‘₯𝑛+1ξ¬βˆ’πœ”+𝛽𝑛π‘₯π‘›βˆ’πœ”,π‘₯𝑛+1+βˆ’πœ”ξ«ξ€·ξ€·1βˆ’π›½π‘›ξ€ΈπΌβˆ’πœ‡π›Όπ‘›π΅π‘¦ξ€Έξ€·π‘›ξ€Έβˆ’πœ”,π‘₯𝑛+1ξ¬βˆ’πœ”β‰€π›Όπ‘›β€–β€–π‘₯π‘Ÿπ›½π‘›β€–β€–βˆ’πœ”2+β€–β€–π‘₯𝑛+1β€–β€–βˆ’πœ”22+π›Όπ‘›ξ«π‘Ÿπ‘“(πœ”)βˆ’πœ‡π΅πœ”,π‘₯𝑛+1ξ¬βˆ’πœ”+𝛽𝑛‖‖π‘₯π‘›β€–β€–βˆ’πœ”2+β€–β€–π‘₯𝑛+1β€–β€–βˆ’πœ”22+ξ€·1βˆ’π›½π‘›βˆ’π›Όπ‘›πœξ€Έβ€–β€–π‘¦π‘›β€–β€–βˆ’πœ”2+β€–β€–π‘₯𝑛+1β€–β€–βˆ’πœ”22≀1βˆ’π›Όπ‘›(πœβˆ’π‘Ÿπ›½)2ξ‚€β€–β€–π‘₯π‘›β€–β€–βˆ’πœ”2+β€–β€–π‘₯𝑛+1β€–β€–βˆ’πœ”2+π›Όπ‘›ξ«π‘Ÿπ‘“(πœ”)βˆ’πœ‡π΅πœ”,π‘₯𝑛+1,βˆ’πœ”(3.53) which implies β€–β€–π‘₯𝑛+1β€–β€–βˆ’πœ”2≀1βˆ’π›Όπ‘›(πœβˆ’π‘Ÿπ›½)1+𝛼𝑛(β€–β€–π‘₯πœβˆ’π‘Ÿπ›½)π‘›β€–β€–βˆ’πœ”2+2𝛼𝑛(πœβˆ’π‘Ÿπ›½)ξ€·1+𝛼𝑛(πœβˆ’π‘Ÿπ›½)(πœβˆ’π‘Ÿπ›½)π‘Ÿπ‘“(πœ”)βˆ’πœ‡π΅πœ”,π‘₯𝑛+1ξ¬β‰€ξ€·βˆ’πœ”1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘₯(πœβˆ’π‘Ÿπ›½)π‘›β€–β€–βˆ’πœ”2+2𝛼𝑛(πœβˆ’π‘Ÿπ›½)ξ€·1+𝛼𝑛(πœβˆ’π‘Ÿπ›½)(πœβˆ’π‘Ÿπ›½)βŸ¨π‘Ÿπ‘“(πœ”)βˆ’πœ‡π΅πœ”,π‘₯𝑛+1βˆ’πœ”βŸ©.(3.54) From condition (i) and (3.7), we know that βˆ‘π‘›π‘–=1𝛼𝑛(πœβˆ’π‘Ÿπ›½)=∞ and limπ‘–β†’βˆžsup(2/(1+𝛼𝑛(πœβˆ’π‘Ÿπ›½))(πœβˆ’π‘Ÿπ›½))βŸ¨π‘Ÿπ‘“(πœ”)βˆ’πœ‡π΅πœ”,π‘₯𝑛+1βˆ’πœ”βŸ©β‰€0. we can conclude from Lemma 2.6 that π‘₯π‘›β†’πœ” as π‘›β†’βˆž. This completes the proof of Theorem 3.1.

Remark 3.2. If π‘Ÿ=1, πœ‡=1, 𝐡=𝐼 and 𝛿𝑖=0, π‘˜π‘–=0, πœŽπ‘–=0 for π‘–βˆˆπ‘, 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 πœ‡=1 and 𝛿𝑖=0, π‘˜π‘–=0, πœŽπ‘–=0 for π‘–βˆˆπ‘, 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 π‘˜-Lipschitzian and πœ‚-strongly monotone operator 𝐡.

Theorem 3.4. Let 𝐢 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝐹 be a bifunction from 𝐢×𝐢→𝑅 satisfying (A1)–(A4). Let π‘†βˆΆπΆβ†’πΆ be a nonexpansive mapping with 𝐹(𝑆)βˆ©πΈπ‘ƒβ‰ βˆ…. Let 𝑓 be a contraction of 𝐻 into itself with π›½βˆˆ(0,1) and 𝐡 be π‘˜-Lipschitzian and πœ‚-strongly monotone operator on 𝐻 with coefficients π‘˜,πœ‚>0, 0<πœ‡<2πœ‚/π‘˜2, 0<π‘Ÿ<(1/2)πœ‡(2πœ‚βˆ’πœ‡π‘˜2)/𝛽=𝜏/𝛽 and 𝜏<1. Let {π‘₯𝑛} be sequence generated by 𝐹𝑒𝑛+1,π‘¦πœ†π‘›βŸ¨π‘¦βˆ’π‘’π‘›,π‘’π‘›βˆ’π‘₯π‘›π‘¦βŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ,𝑛=𝛿𝑛𝑒𝑛+ξ€·1βˆ’π›Ώπ‘›ξ€Έπ‘†π‘›π‘’π‘›,π‘₯𝑛+1=𝛼𝑛π‘₯π‘Ÿπ‘“π‘›ξ€Έ+𝛽𝑛π‘₯𝑛+ξ€·ξ€·1βˆ’π›½π‘›ξ€ΈπΌβˆ’πœ‡π›Όπ‘›π΅ξ€Έπ‘¦π‘›,βˆ€π‘›βˆˆπ‘,(3.55) where 𝑒𝑛=π‘‡πœ†π‘›π‘₯𝑛. If {𝛼𝑛},{𝛽𝑛},{𝛿𝑛}, and {πœ†π‘›} satisfy the following conditions:(i){𝛼𝑛}βŠ‚(0,1), limπ‘›β†’βˆžπ›Όπ‘›=0, β€‰β€‰βˆ‘βˆžπ‘–=1𝛼𝑛=∞,(ii)0<limπ‘›β†’βˆžinf𝛽𝑛≀limπ‘›β†’βˆžsup𝛽𝑛<1, (iii)0<limπ‘›β†’βˆžinf𝛿𝑛≀limπ‘›β†’βˆžsup𝛿𝑛<1, limπ‘›β†’βˆž|𝛿𝑛+1βˆ’π›Ώπ‘›|=0, (iv){πœ†π‘›}βŠ‚(0,∞), limπ‘›β†’βˆžπœ†π‘›>0, limπ‘›β†’βˆž|πœ†π‘›+1βˆ’πœ†π‘›|=0.
Then {π‘₯𝑛} converges strongly to π‘§βˆˆπΉ(𝑆)βˆ©πΈπ‘ƒβ‰ βˆ…, where 𝑧 is the unique solution of variational inequality limπ‘›β†’βˆžsup⟨(π‘Ÿπ‘“βˆ’πœ‡π΅)𝑧,π‘βˆ’π‘§βŸ©β‰€0,βˆ€π‘βˆˆπΉ(𝑆)βˆ©πΈπ‘ƒβ‰ βˆ…,(3.56) that is, 𝑧=𝑃𝐹(𝑆)βˆ©πΈπ‘ƒ(𝐹)(πΌβˆ’πœ‡π΅+π‘Ÿπ‘“)𝑧.

Proof . By Theorem 3.1, letting π‘˜π‘–=0, πœŽπ‘–=0, πœπ‘–=1 and 𝑆𝑖=𝑆 for π‘–βˆˆπ‘, 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, 𝐢=[βˆ’1,1],   𝑆𝑛=𝐼, πœπ‘›=𝜏∈(0,1), β€‰πœ†π‘›=1, π‘›βˆˆπ‘, 𝐹(π‘₯,𝑦)=0, for all π‘₯,π‘¦βˆˆπΆ, 𝐡=𝐼, π‘Ÿ=πœ‡=1, 𝑓(π‘₯)=(1/10)π‘₯, for all π‘₯, with contraction coefficient 𝛽=1/5, 𝛿𝑛=1/2, 𝛼𝑛=1/𝑛, 𝛽𝑛=1/4+1/2𝑛 for every π‘›βˆˆπ‘. Then {π‘₯𝑛} is the sequence generated by π‘₯𝑛+1=ξ‚€91βˆ’ξ‚π‘₯10𝑛𝑛,(4.1) and {π‘₯𝑛}β†’0, as π‘›β†’βˆž, where 0 is the unique solution of the minimization problem minπ‘₯∈𝐢9π‘₯20