Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 632857 | https://doi.org/10.1155/2011/632857

Utith Inprasit, Weerayuth Nilsrakoo, "A Halpern-Mann Type Iteration for Fixed Point Problems of a Relatively Nonexpansive Mapping and a System of Equilibrium Problems", Abstract and Applied Analysis, vol. 2011, Article ID 632857, 22 pages, 2011. https://doi.org/10.1155/2011/632857

A Halpern-Mann Type Iteration for Fixed Point Problems of a Relatively Nonexpansive Mapping and a System of Equilibrium Problems

Academic Editor: Norimichi Hirano
Received17 Mar 2011
Revised03 Jun 2011
Accepted13 Jun 2011
Published21 Sep 2011

Abstract

A new modified Halpern-Mann type iterative method is constructed. Strong convergence of the scheme to a common element of the set of fixed points of a relatively nonexpansive mapping and the set of common solutions to a system of equilibrium problems in a uniformly convex real Banach space which is also uniformly smooth is proved. The results presented in this work improve on the corresponding ones announced by many others.

1. Introduction

Throughout this paper, we denote by โ„• and โ„ the sets of positive integers and real numbers, respectively. Let ๐ธ be a Banach space, ๐ธโˆ— the dual space of ๐ธ, and ๐ถ a nonempty closed convex subset of ๐ธ. Let ๐นโˆถ๐ถร—๐ถโ†’โ„ be a bifunction. The equilibrium problem is to find ๐‘ฅโˆˆ๐ถ such that๐น(๐‘ฅ,๐‘ฆ)โ‰ฅ0โˆ€๐‘ฆโˆˆ๐ถ.(1.1) The set of solutions of (1.1) is denoted by EP(๐น). The equilibrium problems include fixed point problems, optimization problems, variational inequality problems, and Nash equilibrium problems as special cases. Some methods have been proposed to solve the equilibrium problems (see, e.g., [1, 2]). In 2005, Combettes and Hirstoaga [3] introduced an iterative scheme of finding the best approximation to the initial data when EP(๐น) is nonempty, and they also proved a strong convergence theorem.

Let ๐ธ be a smooth Banach space and ๐ฝ the normalized duality mapping from ๐ธ to ๐ธโˆ—. Alber [4] considered the following functional ๐œ‘โˆถ๐ธร—๐ธโ†’[0,โˆž) defined by ๐œ‘(๐‘ฅ,๐‘ฆ)=โ€–๐‘ฅโ€–2โˆ’2โŸจ๐‘ฅ,๐ฝ๐‘ฆโŸฉ+โ€–๐‘ฆโ€–2(๐‘ฅ,๐‘ฆโˆˆ๐ธ).(1.2) Using this functional, Matsushita and Takahashi [5, 6] studied and investigated the following mappings in Banach spaces. A mapping ๐‘†โˆถ๐ถโ†’๐ธ is relatively nonexpansive if the following properties are satisfied: (R1)๐น(๐‘†)โ‰ โˆ…, (R2)๐œ‘(๐‘,๐‘†๐‘ฅ)โ‰ค๐œ‘(๐‘,๐‘ฅ) for all ๐‘โˆˆ๐น(๐‘†) and ๐‘ฅโˆˆ๐ถ,(R3)๎๐น(๐‘†)=๐น(๐‘†),

where ๐น(๐‘†) and ๎๐น(๐‘†) denote the set of fixed points of ๐‘† and the set of asymptotic fixed points of ๐‘†, respectively. It is known that ๐‘† satisfies condition (R3) if and only if ๐ผโˆ’๐‘† is demiclosed at zero, where ๐ผ is the identity mapping; that is, whenever a sequence {๐‘ฅ๐‘›} in ๐ถ converges weakly to ๐‘ and {๐‘ฅ๐‘›โˆ’๐‘†๐‘ฅ๐‘›} converges strongly to 0, it follows that ๐‘โˆˆ๐น(๐‘†). In a Hilbert space ๐ป, the duality mapping ๐ฝ is an identity mapping and ๐œ‘(๐‘ฅ,๐‘ฆ)=โ€–๐‘ฅโˆ’๐‘ฆโ€–2 for all ๐‘ฅ,๐‘ฆโˆˆ๐ป. Hence, if ๐‘†โˆถ๐ถโ†’๐ป is nonexpansive (i.e., โ€–๐‘†๐‘ฅโˆ’๐‘†๐‘ฆโ€–โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€– for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ), then it is relatively nonexpansive. Several articles have appeared providing methods for approximating fixed points of relatively nonexpansive mappings (see, e.g., [5โ€“19] and the references therein). Matsushita and Takahashi [5] introduced the following iteration: a sequence {๐‘ฅ๐‘›} defined by๐‘ฅ๐‘›+1=ฮ ๐ถ๐ฝโˆ’1๎€ท๐›ผ๐‘›๐ฝ๐‘ฅ๐‘›+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐ฝ๐‘†๐‘ฅ๐‘›๎€ธ๐‘›=1,2,โ€ฆ,(1.3) where ๐‘ฅ1โˆˆ๐ถ is arbitrary, {๐›ผ๐‘›} is an appropriate sequence in [0,1], ๐‘† is a relatively nonexpansive mapping, and ฮ ๐ถ denotes the generalized projection from ๐ธ onto a closed convex subset ๐ถ of ๐ธ. They proved that the sequence {๐‘ฅ๐‘›} converges weakly to a fixed point of ๐‘‡. Moreover, Matsushita and Takahashi [6] proposed the following modification of iteration (1.3):๐‘ฅ1๐‘ฆโˆˆ๐ถisarbitrary,๐‘›=๐ฝโˆ’1๎€ท๐›ผ๐‘›๐ฝ๐‘ฅ๐‘›+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐ฝ๐‘†๐‘ฅ๐‘›๎€ธ,๐ถ๐‘›=๎€ฝ๎€ท๐‘งโˆˆ๐ถโˆถ๐œ‘๐‘ง,๐‘ฆ๐‘›๎€ธ๎€ทโ‰ค๐œ‘๐‘ง,๐‘ฅ๐‘›,๐‘„๎€ธ๎€พ๐‘›=๎€ฝ๐‘งโˆˆ๐ถโˆถโŸจ๐‘ฅ๐‘›โˆ’๐‘ง,๐ฝ๐‘ฅ1โˆ’๐ฝ๐‘ฅ๐‘›๎€พ,๐‘ฅโŸฉโ‰ฅ0๐‘›+1=ฮ ๐ถ๐‘›โˆฉ๐‘„๐‘›๐‘ฅ1,๐‘›=1,2,โ€ฆ,(1.4) and proved that the sequence {๐‘ฅ๐‘›} converges strongly to ฮ ๐น(๐‘†)๐‘ฅ1. The iteration (1.4) is called the hybrid method. To generate the iterative sequence, we use the generalized metric projection onto ๐ถ๐‘›โˆฉ๐‘„๐‘› for ๐‘›โˆˆโ„•. It always exists, because each ๐ถ๐‘›โˆฉ๐‘„๐‘› is nonempty, closed, and convex. However, in a practical case, it is not easy to be calculated. In particular, as ๐‘› becomes larger, the shape of ๐ถ๐‘›โˆฉ๐‘„๐‘› becomes more complicate, and the projection will take much more time to be calculated.

In order to overcome this difficulty, Nilsrakoo and Saejung [15] modified Halpern and Mann's iterations for finding a fixed point of a relatively nonexpansive mapping in a Banach space as follows: ๐‘ฅโˆˆ๐ธ,๐‘ฅ1โˆˆ๐ถ and๐‘ฅ๐‘›+1=ฮ ๐ถ๐ฝโˆ’1๎€ท๐›ผ๐‘›๐ฝ๐‘ฅ+๐›ฝ๐‘›๐ฝ๐‘ฅ๐‘›+๐›พ๐‘›๐ฝ๐‘†๐‘ฅ๐‘›๎€ธ,๐‘›=1,2,โ€ฆ,(1.5) where {๐›ผ๐‘›}, {๐›ฝ๐‘›}, and {๐›พ๐‘›} are appropriate sequences in [0,1] with ๐›ผ๐‘›+๐›ฝ๐‘›+๐›พ๐‘›โ‰ก1, and they proved that {๐‘ฅ๐‘›} converges strongly to ฮ ๐น(๐‘†)๐‘ฅ.

Many authors studied the problems of finding a common element of the set of fixed points for a mapping and the set of common solutions to a system of equilibrium problems in the setting of Hilbert space and uniformly smooth and uniformly convex Banach space, respectively (see, e.g., [20โ€“33] and the references therein). In a Hilbert space ๐ป, S. Takahashi and W. Takahashi [34] introduced the iteration as follows: sequence {๐‘ฅ๐‘›} generated by ๐‘ฅ,๐‘ฅ1โˆˆ๐ถ, ๐‘ข๐‘›๎€ท๐‘ขโˆˆ๐ถsuchthat๐น๐‘›๎€ธ+1,๐‘ฆ๐‘Ÿ๐‘›โŸจ๐‘ฆโˆ’๐‘ข๐‘›,๐‘ข๐‘›โˆ’๐‘ฅ๐‘›๐‘ฅโŸฉโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ,๐‘›+1=๐›ผ๐‘›๎€ท๐‘ฅ+1โˆ’๐›ผ๐‘›๎€ธ๐‘†๐‘ข๐‘›,๐‘›=1,2,โ€ฆ,(1.6) where {๐›ผ๐‘›} is an appropriate sequence in [0,1], ๐‘† is nonexpansive, and {๐‘Ÿ๐‘›} is an appropriate positive real sequence. They proved that {๐‘ฅ๐‘›} converges strongly to an element in ๐น(๐‘†)โˆฉEP(๐น). In 2009, Takahashi and Zembayashi [30] proposed the iteration in a uniformly smooth and uniformly convex Banach space as follows: a sequence {๐‘ฅ๐‘›} generated by ๐‘ข1โˆˆ๐ธ, ๐‘ฅ๐‘›๎€ท๐‘ฅโˆˆ๐ถsuchthat๐น๐‘›๎€ธ+1,๐‘ฆ๐‘Ÿ๐‘›โŸจ๐‘ฆโˆ’๐‘ฅ๐‘›,๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘ข๐‘›๐‘ขโŸฉโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ,๐‘›+1=๐ฝโˆ’1๎€ท๐›ผ๐‘›๐ฝ๐‘ฅ๐‘›+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐ฝ๐‘†๐‘ฅ๐‘›๎€ธ,๐‘›=1,2,โ€ฆ,(1.7) where ๐‘† is relatively nonexpansive, {๐›ผ๐‘›} is an appropriate sequence in [0,1], and {๐‘Ÿ๐‘›} is an appropriate positive real sequence. They proved that if ๐ฝ is weakly sequentially continuous, then {๐‘ฅ๐‘›} converges weakly to an element in ๐น(๐‘†)โˆฉEP(๐น). Consequently, there are many results presented strong convergence theorems for finding a common element of the set of fixed points for a mapping and the set of common solutions to a system of equilibrium problems by using the hybrid method. However, Nilsrakoo [35] introduced the Halpern-Mann iteration guaranteeing the strong convergence as follows: ๐‘ฅโˆˆ๐ถ,๐‘ข1โˆˆ๐ธ and๐‘ฅ๐‘›๎€ท๐‘ฅโˆˆ๐ถsuchthat๐น๐‘›๎€ธ+1,๐‘ฆ๐‘Ÿ๐‘›โŸจ๐‘ฆโˆ’๐‘ฅ๐‘›,๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘ข๐‘›๐‘ฆโŸฉโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ,๐‘›=ฮ ๐ถ๐ฝโˆ’1๎€ท๐›ผ๐‘›๎€ท๐ฝ๐‘ฅ+1โˆ’๐›ผ๐‘›๎€ธ๐ฝ๐‘ฅ๐‘›๎€ธ,๐‘ข๐‘›+1=๐ฝโˆ’1๎€ท๐›ฝ๐‘›๐ฝ๐‘ฅ๐‘›+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐ฝ๐‘†๐‘ฆ๐‘›๎€ธ,๐‘›=1,2,โ€ฆ,(1.8) and proved that {๐‘ข๐‘›} and {๐‘ฅ๐‘›} converge strongly to ฮ ๐น(๐‘†)โˆฉEP(๐น)๐‘ฅ.

Motivated by Nilsrakoo and Saejung [15] and Nilsrakoo [35], we present a strong convergence theorem of a new modified Halpern-Mann iterative scheme to find a common element of the set of fixed points of a relatively nonexpansive mapping and the set of common solutions to a system of equilibrium problems in a uniformly convex real Banach space which is also uniformly smooth. The results in this work improve on the corresponding ones announced by many others.

2. Preliminaries

We collect together some definitions and preliminaries which are needed in this paper. We say that a Banach space ๐ธ is strictly convex if the following implication holds for ๐‘ฅ,๐‘ฆโˆˆ๐ธ: โ€–โ€–โ€–โ€–๐‘ฅโ€–=โ€–๐‘ฆโ€–=1,๐‘ฅโ‰ ๐‘ฆimply๐‘ฅ+๐‘ฆ2โ€–โ€–โ€–<1.(2.1) It is also said to be uniformly convex if for any ๐œ€>0, there exists ๐›ฟ>0 such that โ€–โ€–โ€–โ€–๐‘ฅโ€–=โ€–๐‘ฆโ€–=1,โ€–๐‘ฅโˆ’๐‘ฆโ€–โ‰ฅ๐œ€imply๐‘ฅ+๐‘ฆ2โ€–โ€–โ€–โ‰ค1โˆ’๐›ฟ.(2.2) It is known that if ๐ธ is a uniformly convex Banach space, then ๐ธ is reflexive and strictly convex. We say that ๐ธ is uniformly smooth if the dual space ๐ธโˆ— of ๐ธ is uniformly convex. A Banach space ๐ธ is smooth if the limit lim๐‘กโ†’0((โ€–๐‘ฅ+๐‘ก๐‘ฆโ€–โˆ’โ€–๐‘ฅโ€–)/๐‘ก) exists for all norm one elements ๐‘ฅ and ๐‘ฆ in ๐ธ. It is not hard to show that if ๐ธ is reflexive, then ๐ธ is smooth if and only if ๐ธโˆ— is strictly convex.

Let ๐ธ be a smooth Banach space. The function ๐œ‘โˆถ๐ธร—๐ธโ†’โ„ (see [4]) is defined by ๐œ‘(๐‘ฅ,๐‘ฆ)=โ€–๐‘ฅโ€–2โˆ’2โŸจ๐‘ฅ,๐ฝ๐‘ฆโŸฉ+โ€–๐‘ฆโ€–2(๐‘ฅ,๐‘ฆโˆˆ๐ธ),(2.3) where the duality mapping ๐ฝโˆถ๐ธโ†’๐ธโˆ— is given by โŸจ๐‘ฅ,๐ฝ๐‘ฅโŸฉ=โ€–๐‘ฅโ€–2=โ€–๐ฝ๐‘ฅโ€–2(๐‘ฅโˆˆ๐ธ).(2.4) It is obvious from the definition of the function ๐œ‘ that ()โ€–๐‘ฅโ€–โˆ’โ€–๐‘ฆโ€–2)โ‰ค๐œ‘(๐‘ฅ,๐‘ฆ)โ‰ค(โ€–๐‘ฅโ€–+โ€–๐‘ฆโ€–2,(2.5)๐œ‘(๐‘ฅ,๐‘ฆ)=๐œ‘(๐‘ฅ,๐‘ง)+๐œ‘(๐‘ง,๐‘ฆ)+2โŸจ๐‘ฅโˆ’๐‘ง,๐ฝ๐‘งโˆ’๐ฝ๐‘ฆโŸฉ,(2.6) for all ๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐ธ. Moreover,๐œ‘๎ƒฉ๐‘ฅ,๐ฝโˆ’1๎ƒฉ๐‘›๎“๐‘–=1๐œ†๐‘–๐ฝ๐‘ฆ๐‘–โ‰ค๎ƒช๎ƒช๐‘›๎“๐‘–=1๐œ†๐‘–๐œ‘๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธ,(2.7) for all ๐œ†๐‘–โˆˆ[0,1] with โˆ‘๐‘›๐‘–=1๐œ†๐‘–=1 and ๐‘ฅ,๐‘ฆ๐‘–โˆˆ๐ธ.

The following lemma is an analogue of Xu's inequality [36, Theoremโ€‰โ€‰2] with respect to ๐œ‘.

Lemma 2.1 (see [15, Lemmaโ€‰โ€‰2.2]). Let ๐ธ be a uniformly smooth Banach space and ๐‘Ÿ>0. Then, there exists a continuous, strictly increasing, and convex function ๐‘”โˆถ[0,2๐‘Ÿ]โ†’[0,โˆž) such that ๐‘”(0)=0 and ๐œ‘๎€ท๐‘ฅ,๐ฝโˆ’1๎€ธ(๐œ†๐ฝ๐‘ฆ+(1โˆ’๐œ†)๐ฝ๐‘ง)โ‰ค๐œ†๐œ‘(๐‘ฅ,๐‘ฆ)+(1โˆ’๐œ†)๐œ‘(๐‘ฅ,๐‘ง)โˆ’๐œ†(1โˆ’๐œ†)๐‘”(โ€–๐ฝ๐‘ฆโˆ’๐ฝ๐‘งโ€–),(2.8) for all ๐œ†โˆˆ[0,1], ๐‘ฅโˆˆ๐ธ and ๐‘ฆ,๐‘งโˆˆ๐ต๐‘Ÿโˆถ={๐‘งโˆˆ๐ธโˆถโ€–๐‘งโ€–โ‰ค๐‘Ÿ}.

It is also easy to see that if {๐‘ฅ๐‘›} and {๐‘ฆ๐‘›} are bounded sequences of a smooth Banach space ๐ธ, then ๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›โ†’0 implies that ๐œ‘(๐‘ฅ๐‘›,๐‘ฆ๐‘›)โ†’0.

Lemma 2.2 (see [37, Propositionโ€‰โ€‰2]). Let ๐ธ be a uniformly convex and smooth Banach space, and let {๐‘ฅ๐‘›} and {๐‘ฆ๐‘›} be two sequences of ๐ธ such that {๐‘ฅ๐‘›} or {๐‘ฆ๐‘›} is bounded. If ๐œ‘(๐‘ฅ๐‘›,๐‘ฆ๐‘›)โ†’0, then ๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›โ†’0.

Remark 2.3. For any bounded sequences {๐‘ฅ๐‘›} and {๐‘ฆ๐‘›} in a uniformly convex and uniformly smooth Banach space ๐ธ, we have ๐œ‘๎€ท๐‘ฅ๐‘›,๐‘ฆ๐‘›๎€ธโŸถ0โŸบ๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›โŸถ0โŸบ๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘ฆ๐‘›โŸถ0.(2.9)

Let ๐ถ be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space ๐ธ. It is known that [4, 37] for any ๐‘ฅโˆˆ๐ธ, there exists a unique point ฬ‚๐‘ฅโˆˆ๐ถ such that ๐œ‘(ฬ‚๐‘ฅ,๐‘ฅ)=min๐‘ฆโˆˆ๐ถ๐œ‘(๐‘ฆ,๐‘ฅ).(2.10) Following Alber [4], we denote such an element ฬ‚๐‘ฅ by ฮ ๐ถ๐‘ฅ. The mapping ฮ ๐ถ is called the generalized projection from ๐ธ onto ๐ถ. It is easy to see that in a Hilbert space, the mapping ฮ ๐ถ coincides with the metric projection ๐‘ƒ๐ถ. Concerning the generalized projection, the followings are well known.

Lemma 2.4 (see [37, Propositionsโ€‰โ€‰4 andโ€‰โ€‰5]). Let ๐ถ be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space ๐ธ, ๐‘ฅโˆˆ๐ธ and ฬ‚๐‘ฅโˆˆ๐ถ. Then, (a)ฬ‚๐‘ฅ=ฮ ๐ถ๐‘ฅ if and only if โŸจ๐‘ฆโˆ’ฬ‚๐‘ฅ,๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅโŸฉโ‰ค0 for all ๐‘ฆโˆˆ๐ถ,(b)๐œ‘(๐‘ฆ,ฮ ๐ถ๐‘ฅ)+๐œ‘(ฮ ๐ถ๐‘ฅ,๐‘ฅ)โ‰ค๐œ‘(๐‘ฆ,๐‘ฅ) for all ๐‘ฆโˆˆ๐ถ.

Remark 2.5. The generalized projection mapping ฮ ๐ถ above is relatively nonexpansive and ๐น(ฮ ๐ถ)=๐ถ.

Let ๐ธ be a reflexive, strictly convex, and smooth Banach space. The duality mapping ๐ฝโˆ— from ๐ธโˆ— onto ๐ธโˆ—โˆ—=๐ธ coincides with the inverse of the duality mapping ๐ฝ from ๐ธ onto ๐ธโˆ—; that is, ๐ฝโˆ—=๐ฝโˆ’1. We make use of the following mapping ๐‘‰โˆถ๐ธร—๐ธโˆ—โ†’โ„ studied in Alber [4]:๐‘‰๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธ=โ€–๐‘ฅโ€–2โˆ’2โŸจ๐‘ฅ,๐‘ฅโˆ—โŸฉ+โ€–๐‘ฅโˆ—โ€–2,(2.11) for all ๐‘ฅโˆˆ๐ธ and ๐‘ฅโˆ—โˆˆ๐ธโˆ—. Obviously, ๐‘‰(๐‘ฅ,๐‘ฅโˆ—)=๐œ‘(๐‘ฅ,๐ฝโˆ’1(๐‘ฅโˆ—)) for all ๐‘ฅโˆˆ๐ธ and ๐‘ฅโˆ—โˆˆ๐ธโˆ—. We know the following lemma (see [4] and [38, Lemmaโ€‰โ€‰3.2]).

Lemma 2.6. Let ๐ธ be a reflexive, strictly convex, and smooth Banach space, and let ๐‘‰ be as in (2.11). Then ๐‘‰๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธ๎ซ๐ฝ+2โˆ’1๎€ท๐‘ฅโˆ—๎€ธโˆ’๐‘ฅ,๐‘ฆโˆ—๎ฌ๎€ทโ‰ค๐‘‰๐‘ฅ,๐‘ฅโˆ—+๐‘ฆโˆ—๎€ธ,(2.12) for all ๐‘ฅโˆˆ๐ธ and ๐‘ฅโˆ—,๐‘ฆโˆ—โˆˆ๐ธโˆ—.

Lemma 2.7 (see [39, Lemmaโ€‰โ€‰2.1]). Let {๐‘Ž๐‘›} be a sequence of nonnegative real numbers. Suppose that ๐‘Ž๐‘›+1โ‰ค๎€ท1โˆ’๐›พ๐‘›๎€ธ๐‘Ž๐‘›+๐›พ๐‘›๐›ฟ๐‘›(2.13) for all ๐‘›โˆˆโ„•, where the sequences {๐›พ๐‘›} in (0,1) and {๐›ฟ๐‘›} in โ„ satisfy conditions: lim๐‘›โ†’โˆž๐›พ๐‘›=0, โˆ‘โˆž๐‘›=1๐›พ๐‘›=โˆž, and limsup๐‘›โ†’โˆž๐›ฟ๐‘›โ‰ค0. Then lim๐‘›โ†’โˆž๐‘Ž๐‘›=0.

Lemma 2.8 (see [40, Lemmaโ€‰โ€‰3.1]). Let {๐‘Ž๐‘›} be a sequence of real numbers such that there exists a subsequence {๐‘›๐‘–} of {๐‘›} such that ๐‘Ž๐‘›๐‘–<๐‘Ž๐‘›๐‘–+1 for all ๐‘–โˆˆโ„•. Then, there exists a nondecreasing sequence {๐‘š๐‘˜}โŠ‚โ„• such that ๐‘š๐‘˜โ†’โˆž and the following properties are satisfied by all (sufficiently large) numbers ๐‘˜โˆˆโ„•: ๐‘Ž๐‘š๐‘˜โ‰ค๐‘Ž๐‘š๐‘˜+1,๐‘Ž๐‘˜โ‰ค๐‘Ž๐‘š๐‘˜+1.(2.14) In fact, ๐‘š๐‘˜=max{๐‘—โ‰ค๐‘˜โˆถ๐‘Ž๐‘—<๐‘Ž๐‘—+1}.

For solving the equilibrium problem, we usually assume that a bifunction ๐นโˆถ๐ถร—๐ถโ†’โ„ satisfies the following conditions (see, e.g., [1, 3, 30]): (A1)๐น(๐‘ฅ,๐‘ฅ)=0 for all ๐‘ฅโˆˆ๐ถ,(A2)๐น is monotone, that is, ๐น(๐‘ฅ,๐‘ฆ)+๐น(๐‘ฆ,๐‘ฅ)โ‰ค0, for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ, (A3)for all ๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐ถ, limsup๐‘กโ†’0๐น(๐‘ก๐‘ง+(1โˆ’๐‘ก)๐‘ฅ,๐‘ฆ)โ‰ค๐น(๐‘ฅ,๐‘ฆ), (A4)for all ๐‘ฅโˆˆ๐ถ, ๐น(๐‘ฅ,โ‹…) is convex and lower semicontinuous.

The following lemma is a result which appeared in Blum and Oettli [1, Corollaryโ€‰โ€‰1].

Lemma 2.9 (see [1, Corollaryโ€‰โ€‰1]). Let ๐ถ be a closed convex subset of a smooth, strictly convex, and reflexive Banach space ๐ธ. Let ๐นโˆถ๐ถร—๐ถโ†’โ„ be a bifunction satisfying conditions (A1)โ€“(A4), and let ๐‘Ÿ>0 and ๐‘ฅโˆˆ๐ธ. Then, there exists ๐‘งโˆˆ๐ถ such that 1๐น(๐‘ง,๐‘ฆ)+๐‘ŸโŸจ๐‘ฆโˆ’๐‘ง,๐ฝ๐‘งโˆ’๐ฝ๐‘ฅโŸฉโ‰ฅ0โˆ€๐‘ฆโˆˆ๐ถ.(2.15)

The following lemma gives a characterization of a solution of an equilibrium problem.

Lemma 2.10 (see [30, Lemmaโ€‰โ€‰2.8]). Let ๐ถ be a nonempty closed convex subset of a reflexive, strictly convex, and uniformly smooth Banach space ๐ธ. Let ๐นโˆถ๐ถร—๐ถโ†’โ„ be a bifunction satisfying conditions (A1)โ€“(A4). For ๐‘Ÿ>0, define a mapping ๐‘‡๐น๐‘Ÿโˆถ๐ธโ†’๐ถ so-called the resolvent of ๐น as follows: ๐‘‡๐น๐‘Ÿ๎‚†1(๐‘ฅ)=๐‘งโˆˆ๐ถโˆถ๐น(๐‘ง,๐‘ฆ)+๐‘Ÿ๎‚‡,โŸจ๐‘ฆโˆ’๐‘ง,๐ฝ๐‘งโˆ’๐ฝ๐‘ฅโŸฉโ‰ฅ0โˆ€๐‘ฆโˆˆ๐ถ(2.16) for all ๐‘ฅโˆˆ๐ธ. Then, the followings hold: (i)๐‘‡๐‘Ÿ is single-valued, (ii)๐‘‡๐‘Ÿ is a firmly nonexpansive-type mapping [11], that is, for all ๐‘ฅ,๐‘ฆโˆˆ๐ธ๎ซ๐‘‡๐น๐‘Ÿ๐‘ฅโˆ’๐‘‡๐น๐‘Ÿ๐‘ฆ,๐ฝ๐‘‡๐น๐‘Ÿ๐‘ฅโˆ’๐ฝ๐‘‡๐น๐‘Ÿ๐‘ฆ๎ฌโ‰ค๎ซ๐‘‡๐น๐‘Ÿ๐‘ฅโˆ’๐‘‡๐น๐‘Ÿ๎ฌ๐‘ฆ,๐ฝ๐‘ฅโˆ’๐ฝ๐‘ฆ,(2.17)(iii)for all ๐‘ฅโˆˆ๐ธ and ๐‘โˆˆEP(๐น), ๐œ‘๎€ท๐‘,๐‘‡๐น๐‘Ÿ๐‘ฅ๎€ธ๎€ทโ‰ค๐œ‘๐‘ง,๐‘‡๐น๐‘Ÿ๐‘ฅ๎€ธ๎€ท๐‘‡+๐œ‘๐น๐‘Ÿ๎€ธ๐‘ฅ,๐‘ฅโ‰ค๐œ‘(๐‘,๐‘ฅ),(2.18)(iv)๐น(๐‘‡๐น๐‘Ÿ)=EP(๐น), (v)EP(๐น) is closed and convex.

Remark 2.11. Some well-known examples of resolvents of bifunctions satisfying conditions (A1)โ€“(A4) are presented in [3, Lemmaโ€‰โ€‰2.15].

Lemma 2.12 โ€‰2.12 (see [8, Lemmaโ€‰โ€‰2.3]). Let ๐ถ be a nonempty closed convex subset of a Banach space ๐ธ, ๐น a bifunction from ๐ถร—๐ถโ†’โ„ satisfying conditions (A1)โ€“(A4), and ๐‘งโˆˆ๐ถ. Then, ๐‘งโˆˆEP(๐น) if and only if ๐น(๐‘ฆ,๐‘ง)โ‰ค0 for all ๐‘ฆโˆˆ๐ถ.

Lemma 2.13 โ€‰2.13 (see [6], Propositionโ€‰โ€‰2.4). Let ๐ถ be a nonempty closed convex subset of a strictly convex and smooth Banach space ๐ธ and ๐‘†โˆถ๐ถโ†’๐ธ a relatively nonexpansive mapping. Then ๐น(๐‘†) is closed and convex.

3. Main Results

In this section, we introduce a modified Halpern-Mann type iteration without using the generalized metric projection and prove a strong convergence theorem for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions to a system of equilibrium problems in a uniformly convex and uniformly smooth Banach space.

Theorem 3.1. Let ๐ธ a uniformly convex and uniformly smooth Banach space, ๐ถ a nonempty closed convex subset of ๐ธ, {๐น๐‘–}๐‘š๐‘–=1 be a finite family of a bifunction of ๐ถร—๐ถ into โ„ satisfying conditions (A1)โ€“(A4), and ๐‘†โˆถ๐ถโ†’๐ธ a relatively nonexpansive mapping such that ฮฉโˆถ=๐น(๐‘†)โˆฉ(โˆฉ๐‘š๐‘–=1EP(๐น๐‘–))โ‰ โˆ…. Let {๐‘‡๐น๐‘–๐‘Ÿ๐‘–,๐‘›}๐‘š๐‘–=1 be a finite family of the resolvents of ๐น๐‘– with positive real sequences {๐‘Ÿ๐‘–,๐‘›} such that liminf๐‘›โ†’โˆž๐‘Ÿ๐‘–,๐‘›>0 for all ๐‘–=1,2,โ€ฆ,๐‘š. Let {๐‘ฅ๐‘›} be a sequence generated by ๐‘ฅ,๐‘ฅ1โˆˆ๐ธ and ๐‘ฅ๐‘›+1=๐ฝโˆ’1๎‚€๐›ผ๐‘›๐ฝ๐‘ฅ+๐›ฝ๐‘›๐ฝ๐‘ฅ๐‘›+๐›พ๐‘›๐ฝ๐‘†๐‘‡๐น๐‘š๐‘Ÿ๐‘š,๐‘›๐‘‡๐น๐‘šโˆ’1๐‘Ÿ๐‘šโˆ’1,๐‘›,โ‹ฏ๐‘‡๐น1๐‘Ÿ1,๐‘›๐‘ฅ๐‘›๎‚(๐‘›โ‰ฅ1),(3.1) where {๐›ผ๐‘›}, {๐›ฝ๐‘›}, and {๐›พ๐‘›} are sequences in [0,1] satisfying the following conditions: (i)๐›ผ๐‘›+๐›ฝ๐‘›+๐›พ๐‘›โ‰ก1, (ii)lim๐‘›โ†’โˆž๐›ผ๐‘›=0, (iii)โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž,(iv)liminf๐‘›โ†’โˆž๐›ฝ๐‘›(1โˆ’๐›ฝ๐‘›)>0. Then, {๐‘ฅ๐‘›} converges strongly to ฮ ฮฉ๐‘ฅ.

Proof. For each ๐‘›โ‰ฅ1, setting ๐‘ง๐‘˜๐‘›=๐‘‡๐น๐‘˜๐‘Ÿ๐‘˜,๐‘›๐‘‡๐น๐‘˜โˆ’1๐‘Ÿ๐‘˜1,๐‘›โ‹ฏ๐‘‡๐น1๐‘Ÿ1,๐‘›๐‘ฅ๐‘›๐‘ฆ,(๐‘˜=1,2,โ€ฆ,๐‘š),๐‘›=๐ฝโˆ’1๎‚ต๐›ฝ๐‘›1โˆ’๐›ผ๐‘›๐ฝ๐‘ฅ๐‘›+๐›พ๐‘›1โˆ’๐›ผ๐‘›๐ฝ๐‘†๐‘ง๐‘š๐‘›๎‚ถ.(3.2) We can see that ๐‘ง๐‘˜๐‘›=๐‘‡๐น๐‘˜๐‘˜,๐‘›๐‘ง๐‘›๐‘˜โˆ’1. Since ฮฉ is nonempty, closed, and convex, we put ฬ‚๐‘ฅ=ฮ ฮฉ๐‘ฅ. By Lemma 2.10(iii), we get ๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ง๐‘š๐‘›๎€ธ๎€ทโ‰ค๐œ‘ฬ‚๐‘ฅ,๐‘ง๐‘›๐‘šโˆ’1๎€ธ๎€ท๐‘งโˆ’๐œ‘๐‘š๐‘›,๐‘ง๐‘›๐‘šโˆ’1๎€ธ๎€ทโ‰ค๐œ‘ฬ‚๐‘ฅ,๐‘ง๐‘›๐‘šโˆ’2๎€ธ๎€ท๐‘งโˆ’๐œ‘๐‘›๐‘šโˆ’1,๐‘ง๐‘›๐‘šโˆ’2๎€ธ๎€ท๐‘งโˆ’๐œ‘๐‘š๐‘›,๐‘ง๐‘›๐‘šโˆ’1๎€ธโ‹ฎ๎€ทโ‰ค๐œ‘ฬ‚๐‘ฅ,๐‘ฅ๐‘›๎€ธโˆ’๐‘š๎“๐‘˜=1๐œ‘๎€ท๐‘ง๐‘˜๐‘›,๐‘ง๐‘›๐‘˜โˆ’1๎€ธ,(3.3) where ๐‘ง0๐‘›=๐‘ฅ๐‘›. This together with (2.7) gives ๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฆ๐‘›๎€ธโ‰ค๐›ฝ๐‘›1โˆ’๐›ผ๐‘›๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฅ๐‘›๎€ธ+๐›พ๐‘›1โˆ’๐›ผ๐‘›๐œ‘๎€ทฬ‚๐‘ฅ,๐‘†๐‘ง๐‘š๐‘›๎€ธโ‰ค๐›ฝ๐‘›1โˆ’๐›ผ๐‘›๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฅ๐‘›๎€ธ+๐›พ๐‘›1โˆ’๐›ผ๐‘›๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ง๐‘š๐‘›๎€ธ๎€ทโ‰ค๐œ‘ฬ‚๐‘ฅ,๐‘ฅ๐‘›๎€ธ.(3.4) By Lemma 2.6, we obtain ๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฅ๐‘›+1๎€ธ๎€ท=๐‘‰ฬ‚๐‘ฅ,๐ฝ๐‘ฅ๐‘›+1๎€ธ๎€ทโ‰ค๐‘‰ฬ‚๐‘ฅ,๐ฝ๐‘ฅ๐‘›+1โˆ’๐›ผ๐‘›(๎€ธ๎ซ๐‘ฅ๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅ)โˆ’2๐‘›+1โˆ’ฬ‚๐‘ฅ,โˆ’๐›ผ๐‘›(๎ฌ๎€ท๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅ)=๐œ‘ฬ‚๐‘ฅ,๐ฝโˆ’1๎€ท๐›ผ๐‘›๎€ท๐ฝฬ‚๐‘ฅ+1โˆ’๐›ผ๐‘›๎€ธ๐ฝ๐‘ฆ๐‘›๎€ธ๎€ธ+2๐›ผ๐‘›๎ซ๐‘ฅ๐‘›+1๎ฌโˆ’ฬ‚๐‘ฅ,๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅโ‰ค๐›ผ๐‘›๎€ท๐œ‘(ฬ‚๐‘ฅ,ฬ‚๐‘ฅ)+1โˆ’๐›ผ๐‘›๎€ธ๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฆ๐‘›๎€ธ+2๐›ผ๐‘›๎ซ๐‘ฅ๐‘›+1๎ฌโ‰ค๎€ทโˆ’ฬ‚๐‘ฅ,๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅ1โˆ’๐›ผ๐‘›๎€ธ๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฅ๐‘›๎€ธ+2๐›ผ๐‘›๎ซ๐‘ฅ๐‘›+1๎ฌ.โˆ’ฬ‚๐‘ฅ,๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅ(3.5) Next, we show that {๐‘ฅ๐‘›} is bounded. We consider ๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฅ๐‘›+1๎€ธ๎€ทโ‰ค๐œ‘ฬ‚๐‘ฅ,๐ฝโˆ’1๎€ท๐›ผ๐‘›๐ฝ๐‘ฅ+๐›ฝ๐‘›๐ฝ๐‘ฅ๐‘›+๐›พ๐‘›๐ฝ๐‘†๐‘ง๐‘š๐‘›๎€ท๎€ธ๎€ธ=๐œ‘ฬ‚๐‘ฅ,๐ฝโˆ’1๎€ท๐›ผ๐‘›๎€ท๐ฝ๐‘ฅ+1โˆ’๐›ผ๐‘›๎€ธ๐ฝ๐‘ฆ๐‘›๎€ธโ‰ค๐›ผ๐‘›๎€ท๐œ‘(ฬ‚๐‘ฅ,๐‘ฅ)+1โˆ’๐›ผ๐‘›๎€ธ๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฆ๐‘›๎€ธโ‰ค๐›ผ๐‘›๐œ‘๎€ท(ฬ‚๐‘ฅ,๐‘ฅ)+1โˆ’๐›ผ๐‘›๎€ธ๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฅ๐‘›๎€ธ๎€ฝ๎€ทโ‰คmax๐œ‘(ฬ‚๐‘ฅ,๐‘ฅ),๐œ‘ฬ‚๐‘ฅ,๐‘ฅ๐‘›.๎€ธ๎€พ(3.6) By induction, we have ๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฅ๐‘›+1๎€ธ๎€ฝ๐œ‘๎€ทโ‰คmax(ฬ‚๐‘ฅ,๐‘ฅ),๐œ‘ฬ‚๐‘ฅ,๐‘ฅ1๎€ธ๎€พ,(3.7) for all ๐‘›โ‰ฅ1. This implies that {๐‘ฅ๐‘›} is bounded, and so are {๐‘ฅ๐‘›}, {๐‘ข๐‘›}, {๐‘ฆ๐‘›}, {๐‘ง๐‘š๐‘›}, and {๐‘†๐‘ง๐‘š๐‘›}. Let ๐‘”โˆถ[0,2๐‘Ÿ]โ†’[0,โˆž) be a function satisfying the properties of Lemma 2.1, where ๐‘Ÿ=sup{โ€–๐‘ฅ๐‘›โ€–,โ€–๐‘†๐‘ง๐‘š๐‘›โ€–โˆถ๐‘›โ‰ฅ1}. It follows from (3.3) that ๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฆ๐‘›๎€ธโ‰ค๐›ฝ๐‘›1โˆ’๐›ผ๐‘›๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฅ๐‘›๎€ธ+๐›พ๐‘›1โˆ’๐›ผ๐‘›๐œ‘๎€ทฬ‚๐‘ฅ,๐‘†๐‘ง๐‘š๐‘›๎€ธโˆ’๐›ฝ๐‘›๐›พ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ2๐‘”๎€ทโ€–โ€–๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘†๐‘ง๐‘š๐‘›โ€–โ€–๎€ธโ‰ค๐›ฝ๐‘›1โˆ’๐›ผ๐‘›๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฅ๐‘›๎€ธ+๐›พ๐‘›1โˆ’๐›ผ๐‘›๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ง๐‘š๐‘›๎€ธโˆ’๐›ฝ๐‘›๐›พ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ2๐‘”๎€ทโ€–โ€–๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘†๐‘ง๐‘š๐‘›โ€–โ€–๎€ธ๎€ทโ‰ค๐œ‘ฬ‚๐‘ฅ,๐‘ฅ๐‘›๎€ธโˆ’๐›พ๐‘›1โˆ’๐›ผ๐‘›๐‘š๎“๐‘˜=1๐œ‘๎€ท๐‘ง๐‘˜๐‘›,๐‘ง๐‘›๐‘˜โˆ’1๎€ธโˆ’๐›ฝ๐‘›๐›พ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ2๐‘”๎€ทโ€–โ€–๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘†๐‘ง๐‘š๐‘›โ€–โ€–๎€ธ.(3.8)
The rest of the proof will be divided into two cases.Case 1. Suppose that there exists ๐‘›0โˆˆโ„• such that {๐œ‘(ฬ‚๐‘ฅ,๐‘ฅ๐‘›)}โˆž๐‘›=๐‘›0 is nonincreasing. In this situation, {๐œ‘(ฬ‚๐‘ฅ,๐‘ฅ๐‘›)} is then convergent. Then, ๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฅ๐‘›๎€ธ๎€ทโˆ’๐œ‘ฬ‚๐‘ฅ,๐‘ฅ๐‘›+1๎€ธโŸถ0.(3.9) Notice that ๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฅ๐‘›+1๎€ธโ‰ค๐›ผ๐‘›๐œ‘๎€ท(ฬ‚๐‘ฅ,๐‘ฅ)+1โˆ’๐›ผ๐‘›๎€ธ๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฆ๐‘›๎€ธ.(3.10) From condition (ii), ๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฅ๐‘›๎€ธ๎€ทโˆ’๐œ‘ฬ‚๐‘ฅ,๐‘ฆ๐‘›๎€ธ๎€ท=๐œ‘ฬ‚๐‘ฅ,๐‘ฅ๐‘›๎€ธ๎€ทโˆ’๐œ‘ฬ‚๐‘ฅ,๐‘ฅ๐‘›+1๎€ธ๎€ท+๐œ‘ฬ‚๐‘ฅ,๐‘ฅ๐‘›+1๎€ธ๎€ทโˆ’๐œ‘ฬ‚๐‘ฅ,๐‘ฆ๐‘›๎€ธ๎€ทโ‰ค๐œ‘ฬ‚๐‘ฅ,๐‘ฅ๐‘›๎€ธ๎€ทโˆ’๐œ‘ฬ‚๐‘ฅ,๐‘ฅ๐‘›+1๎€ธ+๐›ผ๐‘›๎€ท๎€ท๐œ‘(ฬ‚๐‘ฅ,๐‘ฅ)โˆ’๐œ‘ฬ‚๐‘ฅ,๐‘ฆ๐‘›๎€ธ๎€ธโŸถ0.(3.11) It follows from (3.8) that ๐›พ๐‘›1โˆ’๐›ผ๐‘›๐‘š๎“๐‘˜=1๐œ‘๎€ท๐‘ง๐‘˜๐‘›,๐‘ง๐‘›๐‘˜โˆ’1๎€ธ+๐›ฝ๐‘›๐›พ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ2๐‘”๎€ทโ€–โ€–๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘†๐‘ง๐‘š๐‘›โ€–โ€–๎€ธโŸถ0.(3.12) By the assumptions (i), (ii), and (iv), ๐œ‘๎€ท๐‘ง๐‘˜๐‘›,๐‘ง๐‘›๐‘˜โˆ’1๎€ธ๎€ทโ€–โ€–โŸถ0(๐‘˜=1,2,โ€ฆ,๐‘š),๐‘”๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘†๐‘ง๐‘š๐‘›โ€–โ€–๎€ธโŸถ0.(3.13) By Remark 2.3, we get ๐‘ง๐‘˜๐‘›โˆ’๐‘ง๐‘›๐‘˜โˆ’1โŸถ0(๐‘˜=1,2,โ€ฆ,๐‘š).(3.14) From ๐‘” is continuous strictly increasing with ๐‘”(0)=0, we have ๐‘ง๐‘š๐‘›โˆ’๐‘†๐‘ง๐‘š๐‘›๎€ท๐‘ฅโŸถ0,๐œ‘๐‘›,๐‘†๐‘ง๐‘š๐‘›๎€ธโŸถ0.(3.15) Consequently, ๐œ‘๎€ท๐‘ฅ๐‘›,๐‘ฆ๐‘›๎€ธโ‰ค๐›ฝ๐‘›1โˆ’๐›ผ๐‘›๐œ‘๎€ท๐‘ฅ๐‘›,๐‘ฅ๐‘›๎€ธ+๐›พ๐‘›1โˆ’๐›ผ๐‘›๐œ‘๎€ท๐‘ฅ๐‘›,๐‘†๐‘ง๐‘š๐‘›๎€ธ=๐›พ๐‘›1โˆ’๐›ผ๐‘›๐œ‘๎€ท๐‘ฅ๐‘›,๐‘†๐‘ง๐‘š๐‘›๎€ธ๐œ‘๎€ท๐‘ฆโŸถ0,๐‘›,๐‘ฅ๐‘›+1๎€ธโ‰ค๐›ผ๐‘›๐œ‘๎€ท๐‘ฆ๐‘›๎€ธ+๎€ท,๐‘ฅ1โˆ’๐›ผ๐‘›๎€ธ๐œ‘๎€ท๐‘ฆ๐‘›,๐‘ฆ๐‘›๎€ธ=๐›ผ๐‘›๐œ‘๎€ท๐‘ฆ๐‘›๎€ธ,๐‘ฅโŸถ0.(3.16) This implies that ๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โŸถ0.(3.17) Since {๐‘ฅ๐‘›} is bounded and ๐ธ is reflexive, we choose a subsequence {๐‘ฅ๐‘›๐‘—} of {๐‘ฅ๐‘›} such that ๐‘ฅ๐‘›๐‘—โ‡€๐‘ค and limsup๐‘›โ†’โˆžโŸจ๐‘ฅ๐‘›โˆ’ฬ‚๐‘ฅ,๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅโŸฉ=lim๐‘—โ†’โˆž๎‚ฌ๐‘ฅ๐‘›๐‘—๎‚ญโˆ’ฬ‚๐‘ฅ,๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅ=โŸจ๐‘คโˆ’ฬ‚๐‘ฅ,๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅโŸฉ.(3.18) Let ๐‘˜=1,2,โ€ฆ,๐‘š be fixed. Then, ๐‘ง๐‘˜๐‘›๐‘—โ‡€๐‘ค as ๐‘—โ†’โˆž. From liminf๐‘›โ†’โˆž๐‘Ÿ๐‘˜,๐‘›>0 and (3.14), we have lim๐‘›โ†’โˆž1๐‘Ÿ๐‘˜,๐‘›โ€–โ€–๐ฝ๐‘ง๐‘˜๐‘›โˆ’๐ฝ๐‘ง๐‘›๐‘˜โˆ’1โ€–โ€–=0.(3.19) Then, ๐น๐‘˜๎€ท๐‘ง๐‘˜๐‘›๎€ธ+1,๐‘ฆ๐‘Ÿ๐‘˜,๐‘›๎ซ๐‘ฆโˆ’๐‘ง๐‘˜๐‘›,๐ฝ๐‘ง๐‘˜๐‘›โˆ’๐ฝ๐‘ง๐‘›๐‘˜โˆ’1๎ฌโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ.(3.20) Replacing ๐‘› by ๐‘›๐‘—, we have from (A2) that 1๐‘Ÿ๐‘˜,๐‘›๐‘—๎‚ฌ๐‘ฆโˆ’๐‘ง๐‘˜๐‘›๐‘—,๐ฝ๐‘ง๐‘˜๐‘›๐‘—โˆ’๐ฝ๐‘ง๐‘›๐‘˜โˆ’1๐‘—๎‚ญโ‰ฅโˆ’๐น๐‘˜๎‚€๐‘ง๐‘˜๐‘›๐‘—๎‚,๐‘ฆโ‰ฅ๐น๐‘˜๎‚€๐‘ฆ,๐‘ง๐‘˜๐‘›๐‘—๎‚,โˆ€๐‘ฆโˆˆ๐ถ.(3.21) Letting ๐‘—โ†’โˆž, we have from (3.19) and (A4) that ๐น๐‘˜(๐‘ฆ,๐‘ค)โ‰ค0,โˆ€๐‘ฆโˆˆ๐ถ.(3.22)From Lemma 2.12, we have ๐‘คโˆˆEP(๐น๐‘˜). Since ๐‘† satisfies condition (R3) and ๐‘ง๐‘š๐‘›โˆ’๐‘†๐‘ง๐‘š๐‘›โ†’0, we have ๐‘คโˆˆ๐น(๐‘†). It follows that ๐‘คโˆˆฮฉ. By Lemma 2.4(a), we immediately obtain that limsup๐‘›โ†’โˆž๎ซ๐‘ฅ๐‘›+1๎ฌโˆ’ฬ‚๐‘ฅ,๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅ=limsup๐‘›โ†’โˆžโŸจ๐‘ฅ๐‘›โˆ’ฬ‚๐‘ฅ,๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅโŸฉ=โŸจ๐‘คโˆ’ฬ‚๐‘ฅ,๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅโŸฉโ‰ค0.(3.23) It follows from Lemma 2.7 and (3.5) that ๐œ‘(ฬ‚๐‘ฅ,๐‘ฅ๐‘›)โ†’0. Then, ๐‘ฅ๐‘›โ†’ฬ‚๐‘ฅ.Case 2. Suppose that there exists a subsequence {๐‘›๐‘–} of {๐‘›} such that ๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฅ๐‘›๐‘–๎€ธ๎€ท<๐œ‘ฬ‚๐‘ฅ,๐‘ฅ๐‘›๐‘–+1๎€ธ,(3.24) for all ๐‘–โˆˆโ„•. Then, by Lemma 2.8, there exists a nondecreasing sequence of positive integer numbers {โ„“๐‘—} such that โ„“๐‘—โ†’โˆž, ๐œ‘๎‚€ฬ‚๐‘ฅ,๐‘ฅโ„“๐‘—๎‚๎‚€โ‰ค๐œ‘ฬ‚๐‘ฅ,๐‘ฅโ„“๐‘—+1๎‚๎€ท,๐œ‘ฬ‚๐‘ฅ,๐‘ฅ๐‘—๎€ธ๎‚€โ‰ค๐œ‘ฬ‚๐‘ฅ,๐‘ฅโ„“๐‘—+1๎‚,(3.25) for all sufficiently large numbers ๐‘—. We may assume without loss of generality that ๐›ผโ„“๐‘—>0 for all sufficiently large numbers ๐‘—. Since ๐œ‘๎‚€ฬ‚๐‘ฅ,๐‘ฅโ„“๐‘—+1๎‚โ‰ค๐›ผโ„“๐‘—๎‚€๐œ‘(ฬ‚๐‘ฅ,๐‘ฅ)+1โˆ’๐›ผโ„“๐‘—๎‚๐œ‘๎‚€ฬ‚๐‘ฅ,๐‘ฆโ„“๐‘—๎‚,(3.26) we obtain ๐œ‘๎‚€ฬ‚๐‘ฅ,๐‘ฅโ„“๐‘—๎‚๎‚€โˆ’๐œ‘ฬ‚๐‘ฅ,๐‘ฆโ„“๐‘—๎‚๎‚€=๐œ‘ฬ‚๐‘ฅ,๐‘ฅโ„“๐‘—๎‚๎‚€โˆ’๐œ‘ฬ‚๐‘ฅ,๐‘ฅโ„“๐‘—+1๎‚๎‚€+๐œ‘ฬ‚๐‘ฅ,๐‘ฅโ„“๐‘—+1๎‚๎‚€โˆ’๐œ‘ฬ‚๐‘ฅ,๐‘ฆโ„“๐‘—๎‚โ‰ค๐›ผโ„“๐‘—๎‚€๎‚€๐œ‘(ฬ‚๐‘ฅ,๐‘ฅ)โˆ’๐œ‘ฬ‚๐‘ฅ,๐‘ฆโ„“๐‘—๎‚๎‚โŸถ0.(3.27) It follows from (3.8) that ๐›พโ„“๐‘—1โˆ’๐›ผโ„“๐‘—๐‘š๎“๐‘˜=1๐œ‘๎‚€๐‘ง๐‘˜โ„“๐‘—,๐‘งโ„“๐‘˜โˆ’1๐‘—๎‚+๐›ฝโ„“๐‘—๐›พโ„“๐‘—๎‚€1โˆ’๐›ผโ„“๐‘—๎‚2๐‘”๎‚€โ€–โ€–๐ฝ๐‘ฅโ„“๐‘—โˆ’๐ฝ๐‘†๐‘ง๐‘šโ„“๐‘—โ€–โ€–๎‚โŸถ0.(3.28) Using the same proof of Case 1, we also obtain limsup๐‘—โ†’โˆž๎‚ฌ๐‘ฅโ„“๐‘—+1๎‚ญโˆ’ฬ‚๐‘ฅ,๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅโ‰ค0.(3.29) From (3.5), we have ๐œ‘๎‚€ฬ‚๐‘ฅ,๐‘ฅโ„“๐‘—+1๎‚โ‰ค๎‚€1โˆ’๐›ผโ„“๐‘—๎‚๐œ‘๎‚€ฬ‚๐‘ฅ,๐‘ฅโ„“๐‘—๎‚+2๐›ผโ„“๐‘—๎‚ฌ๐‘ฅโ„“๐‘—+1๎‚ญโˆ’ฬ‚๐‘ฅ,๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅ.(3.30) Since ๐œ‘(ฬ‚๐‘ฅ,๐‘ฅโ„“๐‘—)โ‰ค๐œ‘(ฬ‚๐‘ฅ,๐‘ฅโ„“๐‘—+1), we have ๐›ผโ„“๐‘—๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฅ๐›ผ๎€ธ๎‚€โ‰ค๐œ‘ฬ‚๐‘ฅ,๐‘ฅโ„“๐‘—๎‚๎‚€โˆ’๐œ‘ฬ‚๐‘ฅ,๐‘ฅโ„“๐‘—+1๎‚+2๐›ผโ„“๐‘—๎‚ฌ๐‘ฅโ„“๐‘—+1๎‚ญโˆ’ฬ‚๐‘ฅ,๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅโ‰ค2๐›ผโ„“๐‘—๎‚ฌ๐‘ฅโ„“๐‘—+1๎‚ญ.โˆ’ฬ‚๐‘ฅ,๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅ(3.31) In particular, since ๐›ผโ„“๐‘—>0, we get ๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฅ๐‘š๐‘˜๎€ธ๎‚ฌ๐‘ฅโ‰ค2โ„“๐‘—+1๎‚ญโˆ’ฬ‚๐‘ฅ,๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅ.(3.32) It follows from (3.29) that ๐œ‘(ฬ‚๐‘ฅ,๐‘ฅโ„“๐‘—)โ†’0. This together with (3.30) gives ๐œ‘๎‚€ฬ‚๐‘ฅ,๐‘ฅโ„“๐‘—+1๎‚โŸถ0.(3.33) But ๐œ‘(ฬ‚๐‘ฅ,๐‘ฅ๐‘—)โ‰ค๐œ‘(ฬ‚๐‘ฅ,๐‘ฅโ„“๐‘—+1) for all sufficiently large numbers ๐‘—, we conclude that ๐‘ฅ๐‘—โ†’ฬ‚๐‘ฅ.
From the two cases, we can conclude that {๐‘ฅ๐‘›} converges strongly to ฬ‚๐‘ฅ and the proof is finished.

Setting ๐‘š=1, ๐น1=๐นโ‰ก0, and ๐‘Ÿ1,๐‘›โ‰ก๐‘Ÿ๐‘› in Theorem 3.1, we have the following.

Corollary 3.2. Let ๐ธ be a uniformly convex and uniformly smooth Banach space, ๐ถ a nonempty closed convex subset of ๐ธ, ๐น a bifunction of ๐ถร—๐ถ into โ„ satisfying conditions (A1)โ€“(A4), and ๐‘†โˆถ๐ถโ†’๐ธ be a relatively nonexpansive mapping such that ๐น(๐‘†)โˆฉEP(๐น)โ‰ โˆ…. Let ๐‘‡๐น๐‘Ÿ๐‘› be the resolvent of ๐น with a positive real sequence {๐‘Ÿ๐‘›} such that liminf๐‘›โ†’โˆž๐‘Ÿ๐‘›>0. Let {๐‘ฅ๐‘›} be a sequence generated by ๐‘ฅ,๐‘ฅ1โˆˆ๐ธ and ๐‘ฅ๐‘›+1=๐ฝโˆ’1๎€ท๐›ผ๐‘›๐ฝ๐‘ฅ+๐›ฝ๐‘›๐ฝ๐‘ฅ๐‘›+๐›พ๐‘›๐ฝ๐‘†๐‘‡๐น๐‘Ÿ๐‘›๐‘ฅ๐‘›๎€ธ(๐‘›โ‰ฅ1),(3.34) where {๐›ผ๐‘›}, {๐›ฝ๐‘›}, and {๐›พ๐‘›} are sequences in [0,1] satisfying the following conditions: (i)๐›ผ๐‘›+๐›ฝ๐‘›+๐›พ๐‘›โ‰ก1, (ii)lim๐‘›โ†’โˆž๐›ผ๐‘›=0, (iii)โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž, (iv)liminf๐‘›โ†’โˆž๐›ฝ๐‘›(1โˆ’๐›ฝ๐‘›)>0. Then, {๐‘ฅ๐‘›} converges strongly to ฮ ๐น(๐‘†)โˆฉEP(๐น)๐‘ฅ.

Setting ๐น1โ‰ก0 and ๐‘Ÿ1,๐‘›โ‰ก1 in Corollary 3.2, we have the following result.

Corollary 3.3. Let ๐ธ be a uniformly convex and uniformly smooth Banach space, ๐ถ a nonempty closed convex subset of ๐ธ, and ๐‘†โˆถ๐ถโ†’๐ธ a relatively nonexpansive mapping such that ๐น(๐‘†)โ‰ โˆ…. Let {๐‘ฅ๐‘›} be a sequence generated by ๐‘ฅ,๐‘ฅ1โˆˆ๐ธ and ๐‘ฅ๐‘›+1=๐ฝโˆ’1๎€ท๐›ผ๐‘›๐ฝ๐‘ฅ+๐›ฝ๐‘›๐ฝ๐‘ฅ๐‘›+๐›พ๐‘›๐ฝ๐‘†ฮ ๐ถ๐‘ฅ๐‘›๎€ธ(๐‘›โ‰ฅ1),(3.35) where {๐›ผ๐‘›}, {๐›ฝ๐‘›}, and {๐›พ๐‘›} are sequences in [0,1] satisfying the following conditions: (i)๐›ผ๐‘›+๐›ฝ๐‘›+๐›พ๐‘›โ‰ก1, (ii)lim๐‘›โ†’โˆž๐›ผ๐‘›=0, (iii)โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž, (iv)liminf๐‘›โ†’โˆž๐›ฝ๐‘›(1โˆ’๐›ฝ๐‘›)>0. Then {๐‘ฅ๐‘›} converges strongly to ฮ ๐น(๐‘†)๐‘ฅ.

Next, we prove a strong convergence theorem for finding an element of the set of solutions to a system of equilibrium problems in a uniformly convex and uniformly smooth Banach space.

Theorem 3.4. Let ๐ธ be a uniformly convex and uniformly smooth Banach space, ๐ถ a nonempty closed convex subset of ๐ธ, {๐น๐‘–}๐‘š๐‘–=1 a finite family of a bifunction of ๐ถร—๐ถ into โ„ satisfying conditions (A1)โ€“(A4), and โˆฉ๐‘š๐‘–=1EP(๐น๐‘–)โ‰ โˆ…. Let {๐‘‡๐น๐‘–๐‘Ÿ๐‘–,๐‘›}๐‘š๐‘–=1 be a finite family of the resolvents of ๐น๐‘– with positive real sequences {๐‘Ÿ๐‘–,๐‘›} such that liminf๐‘›โ†’โˆž๐‘Ÿ๐‘–,n>0 for all ๐‘–=1,2,โ€ฆ,๐‘š. Let {๐‘ฅ๐‘›} be a sequence generated by ๐‘ฅ,๐‘ฅ1โˆˆ๐ธ and ๐‘ฅ๐‘›+1=๐ฝโˆ’1๎‚€๐›ผ๐‘›๐ฝ๐‘ฅ+๐›ฝ๐‘›๐ฝ๐‘ฅ๐‘›+๐›พ๐‘›๐ฝ๐‘‡๐น๐‘š๐‘Ÿ๐‘š,๐‘›๐‘‡๐น๐‘šโˆ’1๐‘Ÿ๐‘šโˆ’1,๐‘›,โ€ฆ,๐‘‡๐น1๐‘Ÿ1,๐‘›๐‘ฅ๐‘›๎‚(๐‘›โ‰ฅ1),(3.36) where {๐›ผ๐‘›}, {๐›ฝ๐‘›}, and {๐›พ๐‘›} are sequences in [0,1] satisfying the following conditions: (i)๐›ผ๐‘›+๐›ฝ๐‘›+๐›พ๐‘›โ‰ก1, (ii)lim๐‘›โ†’โˆž๐›ผ๐‘›=0, (iii)โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž, (iv)liminf๐‘›โ†’โˆž๐›ฝ๐‘›(1โˆ’๐›ฝ๐‘›)>0 or liminf๐‘›โ†’โˆž๐›ฝ๐‘›=0. Then, {๐‘ฅ๐‘›} converges strongly to ฮ โˆฉ๐‘š๐‘–=1EP(๐น๐‘–)๐‘ฅ.

Proof. For each ๐‘›โ‰ฅ1, setting ๐‘ง๐‘˜๐‘›=๐‘‡๐น๐‘˜๐‘Ÿ๐‘˜,๐‘›๐‘‡๐น๐‘˜โˆ’1๐‘Ÿ๐‘˜1,๐‘›โ‹ฏ๐‘‡๐น1๐‘Ÿ1,๐‘›๐‘ฅ๐‘›๐‘ฆ,(๐‘˜=1,2,โ€ฆ,๐‘š),๐‘›=๐ฝโˆ’1๎‚ต๐›ฝ๐‘›1โˆ’๐›ผ๐‘›๐ฝ๐‘ฅ๐‘›+๐›พ๐‘›1โˆ’๐›ผ๐‘›๐ฝ๐‘ง๐‘š๐‘›๎‚ถ.(3.37) Since โˆฉ๐‘š๐‘–=1EP(๐น๐‘–) is nonempty, closed, and convex, we put ฬ‚๐‘ฅ=ฮ โˆฉ๐‘š๐‘–=1EP(๐น๐‘–)๐‘ฅ. Using the same proof of Theorem 3.1 when ๐‘† is the identity operator, we can see that ๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฆ๐‘›๎€ธ๎€ทโ‰ค๐œ‘ฬ‚๐‘ฅ,๐‘ฅ๐‘›๎€ธโˆ’๐›พ๐‘›1โˆ’๐›ผ๐‘›๐‘š๎“๐‘˜=1๐œ‘๎€ท๐‘ง๐‘˜๐‘›,๐‘ง๐‘›๐‘˜โˆ’1๎€ธ,๐œ‘๎€ท(3.38)ฬ‚๐‘ฅ,๐‘ฅ๐‘›+1๎€ธโ‰ค๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฅ๐‘›๎€ธ+2๐›ผ๐‘›๎ซ๐‘ฅ๐‘›+1๎ฌโˆ’ฬ‚๐‘ฅ,๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅ.(3.39)
The rest of the proof will be divided into two cases.Case 1. Suppose that there exists ๐‘›0โˆˆโ„• such that {๐œ‘(ฬ‚๐‘ฅ,๐‘ฅ๐‘›)}โˆž๐‘›=๐‘›0 is non-increasing. In this situation, {๐œ‘(ฬ‚๐‘ฅ,๐‘ฅ๐‘›)} is then convergent. Then, ๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฅ๐‘›๎€ธ๎€ทโˆ’๐œ‘ฬ‚๐‘ฅ,๐‘ฅ๐‘›+1๎€ธโŸถ0.(3.40) Notice that ๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฅ๐‘›+1๎€ธโ‰ค๐›ผ๐‘›๐œ‘๎€ท(ฬ‚๐‘ฅ,๐‘ฅ)+1โˆ’๐›ผ๐‘›๎€ธ๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฆ๐‘›๎€ธ.(3.41) From condition (ii), ๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฅ๐‘›๎€ธ๎€ทโˆ’๐œ‘ฬ‚๐‘ฅ,๐‘ฆ๐‘›๎€ธ๎€ท=๐œ‘ฬ‚๐‘ฅ,๐‘ฅ๐‘›๎€ธ๎€ทโˆ’๐œ‘ฬ‚๐‘ฅ,๐‘ฅ๐‘›+1๎€ธ๎€ท+๐œ‘ฬ‚๐‘ฅ,๐‘ฅ๐‘›+1๎€ธ๎€ทโˆ’๐œ‘ฬ‚๐‘ฅ,๐‘ฆ๐‘›๎€ธ๎€ทโ‰ค๐œ‘ฬ‚๐‘ฅ,๐‘ฅ๐‘›๎€ธ๎€ทโˆ’๐œ‘ฬ‚๐‘ฅ,๐‘ฅ๐‘›+1๎€ธ+๐›ผ๐‘›๎€ท๎€ท๐œ‘(ฬ‚๐‘ฅ,๐‘ฅ)โˆ’๐œ‘ฬ‚๐‘ฅ,๐‘ฆ๐‘›๎€ธ๎€ธโŸถ0.(3.42) It follows from (3.38) that ๐›พ๐‘›1โˆ’๐›ผ๐‘›๐‘š๎“๐‘˜=1๐œ‘๎€ท๐‘ง๐‘˜๐‘›,๐‘ง๐‘›๐‘˜โˆ’1๎€ธโŸถ0.(3.43) By the assumptions (i), (ii), and (iv), ๐œ‘๎€ท๐‘ง๐‘˜๐‘›,๐‘ง๐‘›๐‘˜โˆ’1๎€ธโŸถ0(๐‘˜=1,2,โ€ฆ,๐‘š).(3.44) By Remark 2.3, we get ๐‘ง๐‘˜๐‘›โˆ’๐‘ง๐‘›๐‘˜โˆ’1โŸถ0(๐‘˜=1,2,โ€ฆ,๐‘š).(3.45) Consequently, ๐œ‘๎€ท๐‘ฅ๐‘›,๐‘ฆ๐‘›๎€ธโ‰ค๐›ฝ๐‘›1โˆ’๐›ผ๐‘›๐œ‘๎€ท๐‘ฅ๐‘›,๐‘ฅ๐‘›๎€ธ+๐›พ๐‘›1โˆ’๐›ผ๐‘›๐œ‘๎€ท๐‘ฅ๐‘›,๐‘ง๐‘š๐‘›๎€ธ=๐›พ๐‘›1โˆ’๐›ผ๐‘›๐œ‘๎€ท๐‘ง0๐‘›,๐‘ง๐‘š๐‘›๎€ธ๐œ‘๎€ท๐‘ฆโŸถ0,๐‘›,๐‘ฅ๐‘›+1๎€ธโ‰ค๐›ผ๐‘›๐œ‘๎€ท๐‘ฆ๐‘›๎€ธ+๎€ท,๐‘ฅ1โˆ’๐›ผ๐‘›๎€ธ๐œ‘๎€ท๐‘ฆ๐‘›,๐‘ฆ๐‘›๎€ธ=๐›ผ๐‘›๐œ‘๎€ท๐‘ฆ๐‘›๎€ธ,๐‘ฅโŸถ0.(3.46) This implies that ๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โŸถ0.(3.47) Since {๐‘ฅ๐‘›} is bounded and ๐ธ is reflexive, we choose a subsequence {๐‘ฅ๐‘›๐‘—} of {๐‘ฅ๐‘›} such that ๐‘ฅ๐‘›๐‘—โ‡€๐‘ค and limsup๐‘›โ†’โˆžโŸจ๐‘ฅ๐‘›โˆ’ฬ‚๐‘ฅ,๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅโŸฉ=lim๐‘—โ†’โˆž๎‚ฌ๐‘ฅ๐‘›๐‘—๎‚ญโˆ’ฬ‚๐‘ฅ,๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅ=โŸจ๐‘คโˆ’ฬ‚๐‘ฅ,๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅโŸฉ.(3.48) Let ๐‘˜=1,2,โ€ฆ,๐‘š be fixed. Then, ๐‘ง๐‘˜๐‘›๐‘—โ‡€๐‘ค as ๐‘—โ†’โˆž. From liminf๐‘›โ†’โˆž๐‘Ÿ๐‘˜,๐‘›>0 and (3.14), we have lim๐‘›โ†’โˆž1๐‘Ÿ๐‘˜,๐‘›โ€–โ€–๐ฝz๐‘˜๐‘›โˆ’๐ฝ๐‘ง๐‘›๐‘˜โˆ’1โ€–โ€–=0.(3.49) Then, ๐น๐‘˜๎€ท๐‘ง๐‘˜๐‘›๎€ธ+1,๐‘ฆ๐‘Ÿ๐‘˜,๐‘›๎ซ๐‘ฆโˆ’๐‘ง๐‘˜๐‘›,๐ฝ๐‘ง๐‘˜๐‘›โˆ’๐ฝ๐‘ง๐‘›๐‘˜โˆ’1๎ฌโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ.(3.50) Replacing ๐‘› by ๐‘›๐‘—, we have from (A2) that 1๐‘Ÿ๐‘˜,๐‘›๐‘—๎‚ฌ๐‘ฆโˆ’๐‘ง๐‘˜๐‘›๐‘—,๐ฝ๐‘ง๐‘˜๐‘›๐‘—โˆ’๐ฝ๐‘ง๐‘›๐‘˜โˆ’1๐‘—๎‚ญโ‰ฅโˆ’๐น๐‘˜๎‚€๐‘ง๐‘˜๐‘›๐‘—๎‚,๐‘ฆโ‰ฅ๐น๐‘˜๎‚€๐‘ฆ,๐‘ง๐‘˜๐‘›๐‘—๎‚,โˆ€๐‘ฆโˆˆ๐ถ.(3.51) Letting ๐‘—โ†’โˆž, we have from (3.49) and (A4) that ๐น๐‘˜(๐‘ฆ,๐‘ค)โ‰ค0,โˆ€๐‘ฆโˆˆ๐ถ.(3.52) From Lemma 2.12, we have ๐‘คโˆˆEP(๐น๐‘˜). By Lemma 2.4(a), we immediately obtain that limsup๐‘›โ†’โˆž๎ซ๐‘ฅ๐‘›+1๎ฌโˆ’ฬ‚๐‘ฅ,๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅ=limsup๐‘›โ†’โˆžโŸจ๐‘ฅ๐‘›โˆ’ฬ‚๐‘ฅ,๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅโŸฉ=โŸจ๐‘คโˆ’ฬ‚๐‘ฅ,๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅโŸฉโ‰ค0.(3.53) It follows from Lemma 2.7 and (3.39) that ๐œ‘(ฬ‚๐‘ฅ,๐‘ฅ๐‘›)โ†’0. Then, ๐‘ฅ๐‘›โ†’ฬ‚๐‘ฅ.Case 2. Suppose that there exists a subsequence {๐‘›๐‘–} of {๐‘›} such that ๐œ‘๎€ทฬ‚๐‘ฅ,๐‘ฅ๐‘›๐‘–๎€ธ๎€ท<๐œ‘ฬ‚๐‘ฅ,๐‘ฅ๐‘›๐‘–+1๎€ธ,(3.54) for all ๐‘–โˆˆโ„•. Using the same proof of Case 2 in Theorem 3.1, we also conclude that ๐‘ฅ๐‘—โ†’ฬ‚๐‘ฅ.
From the two cases, we can conclude that {๐‘ฅ๐‘›} converges strongly to ฬ‚๐‘ฅ.

Finally, we give two explicit examples validating the assumptions in Theorem 3.1 as follows.

Example 3.5 (Optimization). Let ๐ธ be a uniformly convex and uniformly smooth Banach space, ๐ถ a nonempty bounded closed convex subset of ๐ธ, and ๐‘“โˆถ๐ถโ†’โ„ a lower semicontinuous and convex functional. For instance, let ๐ธ=โ„, ๐ถ=[0,1] and ๐‘“โˆถ[0,1]โ†’โ„ be defined dy ๎‚ป๐‘“(๐‘ฅ)=0,if๐‘ฅ=0,1;๐‘ฅlog๐‘ฅ+(1โˆ’๐‘ฅ)log(1โˆ’๐‘ฅ),if๐‘ฅโˆˆ(0,1).(3.55) Then ๐‘“ is lower semicontinuous and convex. For each ๐‘–=1,2,โ€ฆ,๐‘š, let ๐น๐‘–โˆถ๐ถร—๐ถโ†’โ„ be defined by ๐น๐‘–(๐‘ฅ,๐‘ฆ)โˆถ=๐‘“(๐‘ฆ)โˆ’๐‘“(๐‘ฅ) for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ. It is known [1, 11] that ๐น๐‘– satisfies conditions (A1)โ€“(A4), and EP(๐น๐‘–)โ‰ โˆ…. Let ๐‘†=ฮ ๐ถ. Then, ๐‘† is relatively nonexpansive of ๐ธ into ๐ถ (see [5, 6]) and ๐น(๐‘†)=๐ถ. Then, ฮฉโˆถ=๐น(๐‘†)โˆฉ(โˆฉ๐‘š๐‘–=1EP(๐น๐‘–))=EP(๐น๐‘–)โ‰ โˆ…. Applying Theorem 3.1, we conclude that the sequence defined by (3.1) converges strongly to ฮ ฮฉ๐‘ฅ.

Example 3.6 (The convex feasibility problem). Let ๐ธ be a real Hilbert space, let ๐ถ1,๐ถ2,โ€ฆ,๐ถ๐‘š be nonempty closed convex subsets of ๐ธ satisfying ๐ถโˆถ=โˆฉ๐‘š๐‘–=1๐ถ๐‘–โ‰ โˆ… (e.g., ๐ถ1=๐ถ2=โ‹ฏ=๐ถ๐‘š=๐ถโ‰ โˆ…). Let {๐น๐‘–}๐‘š๐‘–=1 be a finite family of bifunctions of ๐ธร—๐ธ into โ„ defined by ๐น๐‘–1(๐‘ฅ,๐‘ฆ)=2โŸจ๐‘ฆโˆ’๐‘ฅ,๐‘ฅโˆ’๐‘ƒ๐ถ๐‘–๐‘ฅโŸฉโˆ€๐‘ฅ,๐‘ฆโˆˆ๐ธ,(3.56) where ๐‘ƒ๐ถ๐‘– is a metric projection from ๐ธ onto ๐ถ๐‘–. It is known [3, Lemmaโ€‰โ€‰2.15(iv)] that ๐น๐‘– satisfies conditions (A1)โ€“(A4) and EP(๐น๐‘–)=๐ถ๐‘–. Let ๐‘†=๐‘ƒ๐ถ. Then, ๐‘† is relatively nonexpansive of ๐ธ into ๐ถ (see [5, 6]) and then ฮฉโˆถ=๐น(๐‘†)โˆฉ(โˆฉ๐‘š๐‘–=1EP(๐น๐‘–))=๐ถโ‰ โˆ…. Applying Theorem 3.1, we conclude that the sequence defined by (3.1) converges strongly to ฮ ฮฉ๐‘ฅ.

4. Deduced Theorems in Hilbert Spaces

In Hilbert spaces, if ๐‘† is quasi-nonexpansive such that ๐ผโˆ’๐‘† is demiclosed at zero, then ๐‘† is relatively nonexpansive. We obtain the following result.

Theorem 4.1. Let ๐ป be a Hilbert space, ๐ถ a nonempty closed convex subset of ๐ป, {๐น๐‘–}๐‘š๐‘–=1 a finite family of a bifunction of ๐ถร—๐ถ into โ„ satisfying conditions (A1)โ€“(A4), and ๐‘†โˆถ๐ถโ†’๐ธ a quasi-nonexpansive mapping such that ๐ผโˆ’๐‘† is demiclosed at zero and ฮฉโˆถ=๐น(๐‘†)โˆฉ(โˆฉ๐‘š๐‘–=1EP(๐น๐‘–))โ‰ โˆ…. Let {๐‘‡๐น๐‘–๐‘Ÿ๐‘–,๐‘›}๐‘š๐‘–=1 be a finite family of the resolvents of ๐น๐‘– with real sequences {๐‘Ÿ๐‘–,๐‘›} such that liminf๐‘›โ†’โˆž๐‘Ÿ๐‘–,๐‘›>0 for all ๐‘–=1,2,โ€ฆ,๐‘š. Let {๐‘ฅ๐‘›} be a sequence generated by ๐‘ฅ,๐‘ฅ1โˆˆ๐ป and ๐‘ฅ๐‘›+1=๐›ผ๐‘›๐‘ฅ+๐›ฝ๐‘›๐‘ฅ๐‘›+๐›พ๐‘›๐‘†๐‘‡๐น๐‘š๐‘Ÿ๐‘š,๐‘›๐‘‡๐น๐‘šโˆ’1๐‘Ÿ๐‘šโˆ’1,๐‘›,โ€ฆ,๐‘‡๐น1๐‘Ÿ1,๐‘›๐‘ฅ๐‘›(๐‘›โ‰ฅ1),(4.1) where {๐›ผ๐‘›}, {๐›ฝ๐‘›}, and {๐›พ๐‘›} are sequences in [0,1] satisfying the following conditions: (i)๐›ผ๐‘›+๐›ฝ๐‘›+๐›พ๐‘›โ‰ก1, (ii)lim๐‘›โ†’โˆž๐›ผ๐‘›=0, (iii)โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž, (iv)liminf๐‘›โ†’โˆž๐›ฝ๐‘›(1โˆ’๐›ฝ๐‘›)>0. Then {๐‘ฅ๐‘›} converges strongly to ๐‘ƒฮฉ๐‘ฅ.

Applying Theorem 4.1 and using the technique in [41], we have the following result.

Theorem 4.2. Let ๐ป be a Hilbert space, ๐ถ a nonempty closed convex subset of ๐ป, ๐‘“ a contraction of ๐ป into itself (i.e., there is ๐‘Žโˆˆ(0,1) such that โ€–๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฆ)โ€–โ‰ค๐‘Žโ€–๐‘ฅโˆ’๐‘ฆโ€– for all ๐‘ฅ,๐‘ฆโˆˆ๐ป), {๐น๐‘–}๐‘š๐‘–=1 a finite family of a bifunction of ๐ถร—๐ถ into โ„ satisfying conditions (A1)โ€“(A4), and ๐‘†โˆถ๐ถโ†’๐ธ be a nonexpansive mapping such that ฮฉโˆถ=๐น(๐‘†)โˆฉ(โˆฉ๐‘š๐‘–=1EP(๐น๐‘–))โ‰ โˆ…. Let {๐‘‡๐น๐‘–๐‘Ÿ๐‘–,๐‘›}๐‘š๐‘–=1 be a finite family of the resolvents of ๐น๐‘– with real sequences {๐‘Ÿ๐‘–,๐‘›} such that liminf๐‘›โ†’โˆž๐‘Ÿ๐‘–,๐‘›>0 for all ๐‘–=1,2,โ€ฆ,๐‘š. Let {๐‘ฅ๐‘›} be a sequence generated by ๐‘ฅ,๐‘ฅ1โˆˆ๐ป and ๐‘ฅ๐‘›+1=๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๐›ฝ๐‘›๐‘ฅ๐‘›+๐›พ๐‘›๐‘†๐‘‡๐น๐‘š๐‘Ÿ๐‘š,๐‘›๐‘‡๐น๐‘šโˆ’1๐‘Ÿ๐‘šโˆ’1,๐‘›,โ€ฆ,๐‘‡๐น1๐‘Ÿ1,๐‘›๐‘ฅ๐‘›,(๐‘›โ‰ฅ1),(4.2) where {๐›ผ๐‘›}, {๐›ฝ๐‘›}, and {๐›พ๐‘›} are sequences in [0,1] satisfying the following conditions: (i)๐›ผ๐‘›+๐›ฝ๐‘›+๐›พ๐‘›โ‰ก1, (ii)lim๐‘›โ†’โˆž๐›ผ๐‘›=0, (iii)โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž, (iv)liminf๐‘›โ†’โˆž๐›ฝ๐‘›(1โˆ’๐›ฝ๐‘›)>0. Then, {๐‘ฅ๐‘›} converges strongly to ๐‘ง such that ๐‘ง=๐‘ƒฮฉ๐‘“(๐‘ง).

Proof. We note that ๐‘ƒฮฉ๐‘“ is contraction. By Banach contraction principle, let ๐‘ง be the fixed point of ๐‘ƒฮฉ๐‘“ and {๐‘ฆ๐‘›} a sequence generated by ๐‘ฆ1=๐‘ฅ1โˆˆ๐ป and ๐‘ฆ๐‘›+1=๐›ผ๐‘›๐‘“(๐‘ง)+๐›ฝ๐‘›๐‘ฆ๐‘›+๐›พ๐‘›๐‘†๐‘‡๐น๐‘š๐‘Ÿ๐‘š,๐‘›๐‘‡๐น๐‘šโˆ’1๐‘Ÿ๐‘šโˆ’1,๐‘›,โ€ฆ,๐‘‡๐น1๐‘Ÿ1,๐‘›๐‘ฆ๐‘›,(๐‘›โ‰ฅ1).(4.3) Using Theorem 4.1, we have ๐‘ฆ๐‘›โ†’๐‘ง=๐‘ƒฮฉ๐‘“(๐‘ง). Since ๐‘† and ๐‘‡๐น๐‘˜๐‘Ÿ๐‘˜,๐‘›(๐‘˜=1,2,โ€ฆ,๐‘š) are nonexpansive, โ€–โ€–๐‘ฆ๐‘›+1โˆ’๐‘ฅ๐‘›+1โ€–โ€–โ‰ค๐›ผ๐‘›โ€–โ€–๐‘“๎€ท๐‘ฅ๐‘›๎€ธโ€–โ€–โˆ’๐‘“(๐‘ง)+๐›ฝ๐‘›โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–+๐›พ๐‘›โ€–โ€–๐‘†๐‘‡๐น๐‘š๐‘Ÿ๐‘š,๐‘›๐‘‡๐น๐‘šโˆ’1๐‘Ÿ๐‘šโˆ’1,๐‘›,โ€ฆ,๐‘‡๐น1๐‘Ÿ1,๐‘›๐‘ฆ๐‘›โˆ’๐‘†๐‘‡๐น๐‘š๐‘Ÿ๐‘š,๐‘›๐‘‡๐น๐‘šโˆ’1๐‘Ÿ๐‘šโˆ’1,๐‘›,โ€ฆ,๐‘‡๐น1๐‘Ÿ1,๐‘›๐‘ฅ๐‘›โ€–โ€–โ‰ค๐›ผ๐‘›๐‘Žโ€–โ€–๐‘ฅ๐‘›โ€–โ€–+๎€ท๐›ฝโˆ’๐‘ง๐‘›+๐›พ๐‘›๎€ธโ€–โ€–๐‘ฆ๐‘›โˆ’๐‘›๐‘›โ€–โ€–โ‰ค๐›ผ๐‘›๐‘Ž๎€ทโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›โ€–โ€–+โ€–โ€–๐‘ฆ๐‘›โ€–โ€–๎€ธ+๎€ท๐›ฝโˆ’๐‘ง๐‘›+๐›พ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›โ€–โ€–=๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฆ(1โˆ’๐‘Ž)๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–+๐›ผ๐‘›๎‚€๐‘Ž(1โˆ’๐‘Ž)โ€–โ€–๐‘ฆ1โˆ’๐‘Ž๐‘›โ€–โ€–๎‚.โˆ’๐‘ง(4.4) Applying Lemma 2.7, ๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›โ†’0 and so ๐‘ฅ๐‘›โ†’๐‘ง=๐‘ƒฮฉ๐‘“(๐‘ง).

Setting ๐‘š=1, ๐น1=๐นโ‰ก0, and ๐‘Ÿ1,๐‘›โ‰ก๐‘Ÿ๐‘› in Theorem 4.1, we have the following.

Corollary 4.3. Let ๐ป be a Hilbert space, ๐ถ a nonempty closed convex subset of ๐ป, ๐น a bifunction of ๐ถร—๐ถ into โ„ satisfying conditions (A1)โ€“(A4), and ๐‘†โˆถ๐ถโ†’๐ธ a quasi-nonexpansive mapping such that ๐ผโˆ’๐‘† is demiclosed at zero and ๐น(๐‘†)โˆฉEP(๐น)โ‰ โˆ…. Let ๐‘‡๐น๐‘Ÿ๐‘› be the resolvent of ๐น with a positive real sequence {๐‘Ÿ๐‘›} such that liminf๐‘›โ†’โˆž๐‘Ÿ๐‘›>0. Let {๐‘ฅ๐‘›} be a sequence generated by ๐‘ฅ,๐‘ฅ1โˆˆ๐ป and ๐‘ฅ๐‘›+1=๐›ผ๐‘›๐‘ฅ+๐›ฝ๐‘›๐‘ฅ๐‘›+๐›พ๐‘›๐‘†๐‘‡๐น๐‘Ÿ๐‘›๐‘ฅ๐‘›(๐‘›โ‰ฅ1),(4.5) where {๐›ผ๐‘›}, {๐›ฝ๐‘›}, and {๐›พ๐‘›} are sequences in [0,1] satisfying the following conditions: (i)๐›ผ๐‘›+๐›ฝ๐‘›+๐›พ๐‘›โ‰ก1, (ii)lim๐‘›โ†’โˆž๐›ผ๐‘›=0, (iii)โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž, (iv)liminf๐‘›โ†’โˆž๐›ฝ๐‘›(1โˆ’๐›ฝ๐‘›)>0. Then, {๐‘ฅ๐‘›} converges strongly to ๐‘ƒ๐น(๐‘†)โˆฉEP(๐น)๐‘ฅ.

Corollary 4.4. Let ๐ป be a Hilbert space, ๐ถ a nonempty closed convex subset of ๐ป, ๐‘“ a contraction of ๐ป into itself, ๐น a bifunction of ๐ถร—๐ถ into โ„ satisfying conditions (A1)โ€“(A4), and ๐‘†โˆถ๐ถโ†’๐ธ a nonexpansive mapping such that ๐น(๐‘†)โˆฉEP(๐น)โ‰ โˆ…. Let ๐‘‡๐น๐‘Ÿ๐‘› be the resolvent of ๐น with a positive real sequence {๐‘Ÿ๐‘›} such that liminf๐‘›โ†’โˆž๐‘Ÿ๐‘›>0. Let {๐‘ฅ๐‘›} be a sequence generated by ๐‘ฅ,๐‘ฅ1โˆˆ๐ป and ๐‘ฅ๐‘›+1=๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๐›ฝ๐‘›๐‘ฅ๐‘›+๐›พ๐‘›๐‘†๐‘‡๐น๐‘Ÿ๐‘›๐‘ฅ๐‘›(๐‘›โ‰ฅ1),(4.6) where {๐›ผ๐‘›}, {๐›ฝ๐‘›}, and {๐›พ๐‘›} are sequences in [0,1] satisfying the following conditions: (i)๐›ผ๐‘›+๐›ฝ๐‘›+๐›พ๐‘›โ‰ก1, (ii)lim๐‘›โ†’โˆž๐›ผ๐‘›=0, (iii)โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž, (iv)liminf๐‘›โ†’โˆž๐›ฝ๐‘›(1โˆ’๐›ฝ๐‘›)>0. Then, {๐‘ฅ๐‘›} converges strongly to ๐‘ง such that ๐‘ง=๐‘ƒ๐น(๐‘†)โˆฉEP(๐น)๐‘“(๐‘ง).

Remark 4.5. Corollary 4.4 improves and extends [42, Theoremโ€‰โ€‰5]. More precisely, the conditions lim๐‘›โ†’โˆž(๐‘Ÿ๐‘›+1โˆ’๐‘Ÿ๐‘›)=โˆž are removed.

Setting ๐นโ‰ก0 and ๐‘Ÿ๐‘›โ‰ก1 in Corollary 4.3, we have the following.

Corollary 4.6. Let ๐ป be a Hilbert space, ๐ถ a nonempty closed convex subset of ๐ป, and ๐‘†โˆถ๐ถโ†’๐ธ a quasi-nonexpansive mapping such that ๐ผโˆ’๐‘† is demiclosed at zero and ๐น(๐‘†)โ‰ โˆ…. Let {๐‘ฅ๐‘›} be a sequence generated by ๐‘ฅ,๐‘ฅ1โˆˆ๐ป and ๐‘ฅ๐‘›+1=๐›ผ๐‘›๐‘ฅ+๐›ฝ๐‘›๐‘ฅ๐‘›+๐›พ๐‘›๐‘†๐‘ƒ๐ถ๐‘ฅ๐‘›(๐‘›โ‰ฅ1),(4.7) where {๐›ผ๐‘›}, {๐›ฝ๐‘›}, and {๐›พ๐‘›} are sequences in [0,1] satisfying the following conditions: (i)๐›ผ๐‘›+๐›ฝ๐‘›+๐›พ๐‘›โ‰ก1, (ii)lim๐‘›โ†’โˆž๐›ผ๐‘›=0, (iii)โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž, (iv)liminf๐‘›โ†’โˆž๐›ฝ๐‘›(1โˆ’๐›ฝ๐‘›)>0. Then, {๐‘ฅ๐‘›} converges strongly to ๐‘ƒ๐น(๐‘†)๐‘ฅ.

Applying Theorem 3.4, we have the following result.

Theorem 4.7. Let ๐ป be a Hilbert space, ๐ถ a nonempty closed convex subset of ๐ป, {๐น๐‘–