Table of Contents
ISRN Mathematical Analysis
Volumeย 2011, Article IDย 595263, 18 pages
http://dx.doi.org/10.5402/2011/595263
Research Article

A New Hybrid Algorithm for a System of Generalized Mixed Equilibrium Problems and Fixed Point of Quasi-๐œ™-Asymptotically Nonexpansive Mappings

Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China

Received 6 March 2011; Accepted 11 April 2011

Academic Editors: S.ย Deng and K. A.ย Lurie

Copyright ยฉ 2011 Jinhua Zhu and Shih-Sen Chang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The purpose of this paper is to use a new hybrid algorithm for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems and the set of common fixed points for an infinite family of quasi-๐œ™-asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the paper improve and extend some recent results.

1. Introduction

Throughout this paper, we denote by โ„• and โ„ the sets of positive integers and real numbers, respectively. We also assume that ๐ธ is a real Banach space, ๐ธโˆ— is the dual space of ๐ธ, ๐ถ is a nonempty closed convex subset of ๐ธ, and โŸจโ‹…,โ‹…โŸฉ is the pairing between ๐ธ and ๐ธโˆ—. In the sequel, we denote the strong convergence and weak convergence of a sequence {๐‘ฅ๐‘›} by ๐‘ฅ๐‘›โ†’๐‘ฅ and ๐‘ฅ๐‘›โ‡€๐‘ฅ, respectively.

Let ๐œ“โˆถ๐ถโ†’โ„ be a real-valued function, ๐ดโˆถ๐ถโ†’๐ธโˆ— a nonlinear mapping, and {ฮ˜๐‘–โˆถ๐ถร—๐ถโ†’โ„,๐‘–=1,2,โ€ฆ,๐‘} a finite family of equilibrium bifunctions, that is, ฮ˜๐‘–(๐‘ข,๐‘ข)=0 for each ๐‘ขโˆˆ๐ถ. The โ€œso-calledโ€ system of generalized mixed equilibrium problems (SGMEP) for functions (ฮ˜1,ฮ˜2,โ€ฆ,ฮ˜๐‘,๐ด,๐œ“) is to find a common element ๐‘ฅโˆ—โˆˆ๐ถ such that ฮ˜1๎€ท๐‘ฅโˆ—๎€ธ,๐‘ฆ+โŸจ๐ด๐‘ฅโˆ—,๐‘ฆโˆ’๐‘ฅโˆ—๎€ท๐‘ฅโŸฉ+๐œ“(๐‘ฆ)โˆ’๐œ“โˆ—๎€ธฮ˜โ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ,2๎€ท๐‘ฅโˆ—๎€ธ,๐‘ฆ+โŸจ๐ด๐‘ฅโˆ—,๐‘ฆโˆ’๐‘ฅโˆ—๎€ท๐‘ฅโŸฉ+๐œ“(๐‘ฆ)โˆ’๐œ“โˆ—๎€ธโ‹ฎฮ˜โ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ,๐‘๎€ท๐‘ฅโˆ—๎€ธ,๐‘ฆ+โŸจ๐ด๐‘ฅโˆ—,๐‘ฆโˆ’๐‘ฅโˆ—๎€ท๐‘ฅโŸฉ+๐œ“(๐‘ฆ)โˆ’๐œ“โˆ—๎€ธโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ.(1.1)

We denote the set of solutions of (1.1) by โ‹‚ฮฉโˆถ=๐‘๐‘–=1ฮฉ(ฮ˜๐‘–,๐œ“), where ฮฉ(ฮ˜๐‘–,๐œ“) is the set of solutions to the generalized mixed equilibrium problem: ฮ˜๐‘–๎€ท๐‘ฅโˆ—๎€ธ,๐‘ฆ+โŸจ๐ด๐‘ฅโˆ—,๐‘ฆโˆ’๐‘ฅโˆ—๎€ท๐‘ฅโŸฉ+๐œ“(๐‘ฆ)โˆ’๐œ“โˆ—๎€ธโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ.(1.2)

Special examples are as follows.(i)If ๐ด=0 and ๐‘=1, the problem (1.1) is equivalent to finding ๐‘ฅโˆ—โˆˆ๐ถ such that ฮ˜๎€ท๐‘ฅโˆ—๎€ธ๎€ท๐‘ฅ,๐‘ฆ+๐œ“(๐‘ฆ)โˆ’๐œ“โˆ—๎€ธโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ,(1.3)

which is called the mixed equilibrium problem (MEP) [1]. The set of solutions to (1.3) is denoted by MEP.(ii)If ฮ˜=0 and ๐‘=1, the problem (1.1) is equivalent to finding ๐‘ฅโˆ—โˆˆ๐ถ such that โŸจ๐ด๐‘ฅโˆ—,๐‘ฆโˆ’๐‘ฅโˆ—๎€ท๐‘ฅโŸฉ+๐œ“(๐‘ฆ)โˆ’๐œ“โˆ—๎€ธโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ,(1.4)

which is called the mixed variational inequality of Browder type (VI) [2]. The set of solutions to (1.4) is denoted by VI(๐ถ,๐ด,๐œ“).

Recently, many authors studied the problems of finding a common element of the set of fixed point for a nonexpansive mapping and the set of solutions for an equilibrium problem in the setting of Hilbert space and uniformly smooth and uniformly convex Banach space, respectively (see, e.g., [3โ€“5] and the references therein).

Motivated and inspired by the researches going on in this direction, the purpose of this paper is using a hybrid algorithm for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems and the set of common fixed points for an infinite family of quasi-๐œ™-asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results improve and extend the corresponding results in [6โ€“12].

2. Preliminaries

First, we recall some definitions and conclusions.

The mapping ๐ฝโˆถ๐ธโ†’2๐ธโˆ— defined by ๐ฝ๎€ฝ๐‘ฅ(๐‘ฅ)=โˆ—โˆˆ๐ธโˆ—โˆถโŸจ๐‘ฅ,๐‘ฅโˆ—โŸฉ=โ€–๐‘ฅโ€–=โ€–๐‘ฅโˆ—โ€–๎€พ,๐‘ฅโˆˆ๐ธ,(2.1) is called the normalized duality mapping. By the Hahn-Banach theorem, ๐ฝ(๐‘ฅ)โ‰ โˆ… for each ๐‘ฅโˆˆ๐ธ.

A Banach space ๐ธ is said to be strictly convex if โ€–๐‘ฅ+๐‘ฆโ€–/2<1 for all ๐‘ฅ,๐‘ฆโˆˆ๐‘ˆ={๐‘งโˆˆ๐ธโˆถโ€–๐‘งโ€–=1} with ๐‘ฅโ‰ ๐‘ฆ. ๐ธ is said to be uniformly convex if, for each ๐œ–โˆˆ(0,2], there exists ๐›ฟ>0 such that โ€–๐‘ฅ+๐‘ฆโ€–/2<1โˆ’๐›ฟ for all ๐‘ฅ,๐‘ฆโˆˆ๐‘ˆ with โ€–๐‘ฅโˆ’๐‘ฆโ€–โ‰ฅ๐œ–. ๐ธ is said to be smooth if the limit lim๐‘กโ†’0โ€–๐‘ฅ+๐‘ก๐‘ฆโ€–โˆ’โ€–๐‘ฅโ€–๐‘ก(2.2) exists for all ๐‘ฅ,๐‘ฆโˆˆ๐‘ˆ. ๐ธ is said to be uniformly smooth if the above limit exists uniformly in ๐‘ฅ,๐‘ฆโˆˆ๐‘ˆ.

Remark 2.1. The following basic properties can be found in Cioranescu [13]. (i)If ๐ธ is a uniformly smooth Banach space, then ๐ฝ is uniformly continuous on each bounded subset of ๐ธ.(ii)If ๐ธ is a reflexive and strictly convex Banach space, then ๐ฝโˆ’1 is norm-weak continuous.(iii)If ๐ธ is a smooth, strictly convex, and reflexive Banach space, then ๐ฝ is single-valued, one-to-one, and onto.(iv)A Banach space ๐ธ is uniformly smooth if and only if ๐ธโˆ— is uniformly convex.(v)Each uniformly convex Banach space ๐ธ has the Kadec-Klee property, that is, for any sequence {๐‘ฅ๐‘›}โŠ‚๐ธ, if ๐‘ฅ๐‘›โ‡€๐‘ฅโˆˆ๐ธ and โ€–๐‘ฅ๐‘›โ€–โ†’โ€–๐‘ฅโ€–, then ๐‘ฅ๐‘›โ†’๐‘ฅ.
Next we assume that ๐ธ is a smooth, strictly convex, and reflexive Banach space and ๐ถ is a nonempty closed convex subset of ๐ธ. In the sequel, we always use ๐œ™โˆถ๐ธร—๐ธโ†’โ„+ to denote the Lyapunov functional defined by ๐œ™(๐‘ฅ,๐‘ฆ)=โ€–๐‘ฅโ€–2โˆ’2โŸจ๐‘ฅ,๐ฝ๐‘ฆโŸฉ+โ€–๐‘ฆโ€–2,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ธ.(2.3)
It is obvious from the definition of 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,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ธ.(2.4)
Following Alber [14], the generalized projection ฮ ๐ถโˆถ๐ธโ†’๐ถ is defined by ฮ ๐ถ(๐‘ฅ)=arginf๐‘ฆโˆˆ๐ถ๐œ™(๐‘ฆ,๐‘ฅ),โˆ€๐‘ฅโˆˆ๐ธ.(2.5)

Lemma 2.2 (see [14, 15]). Let ๐ธ be a smooth, strictly convex, and reflexive Banach space and ๐ถ a nonempty closed convex subset of ๐ธ. Then the following conclusions hold: (a)๐œ™(๐‘ฅ,ฮ ๐ถ๐‘ฆ)+๐œ™(ฮ ๐ถ๐‘ฆ,๐‘ฆ)โ‰ค๐œ™(๐‘ฅ,๐‘ฆ), for all๐‘ฅโˆˆ๐ถand๐‘ฆโˆˆ๐ธ;(b)if ๐‘ฅโˆˆ๐ธ and ๐‘งโˆˆ๐ถ, then ๐‘ง=ฮ ๐ถ๐‘ฅโŸบโŸจ๐‘งโˆ’๐‘ฆ,๐ฝ๐‘ฅโˆ’๐ฝ๐‘งโŸฉโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ;(2.6)(c)for ๐‘ฅ,๐‘ฆโˆˆ๐ธ, ๐œ™(๐‘ฅ,๐‘ฆ)=0 if and only ๐‘ฅ=๐‘ฆ.

Remark 2.3. If ๐ธ is a real Hilbert space ๐ป, then ๐œ™(๐‘ฅ,๐‘ฆ)=โ€–๐‘ฅโˆ’๐‘ฆโ€–2 and ฮ ๐ถ is the metric projection ๐‘ƒ๐ถ of ๐ป onto ๐ถ.
Let ๐ธ be a smooth, strictly convex, and reflexive Banach space, ๐ถ a nonempty closed convex subset of ๐ธ,๐‘‡โˆถ๐ถโ†’๐ถ a mapping, and ๐น(๐‘‡) the set of fixed points of ๐‘‡. A point ๐‘โˆˆ๐ถ is said to be an asymptotic fixed point of ๐‘‡ if there exists a sequence {๐‘ฅ๐‘›}โŠ‚๐ถ such that ๐‘ฅ๐‘›โ‡€๐‘ and โ€–๐‘ฅ๐‘›โˆ’๐‘‡๐‘ฅ๐‘›โ€–โ†’0. We denoted the set of all asymptotic fixed points of ๐‘‡ by ๎‚๐น(๐‘‡).

Definition 2.4 (see [16]). (1) A mapping ๐‘‡โˆถ๐ถโ†’๐ถ is said to be relatively nonexpansive if ๎‚๐น(๐‘‡)โ‰ โˆ…,๐น(๐‘‡)=๐น(๐‘‡), and ๐œ™(๐‘,๐‘‡๐‘ฅ)โ‰ค๐œ™(๐‘,๐‘ฅ),โˆ€๐‘ฅโˆˆ๐ถ,๐‘โˆˆ๐น(๐‘‡).(2.7)
(2) A mapping๐‘‡โˆถ๐ถโ†’๐ถis said to beclosed if, for any sequence {๐‘ฅ๐‘›}โŠ‚๐ถwith๐‘ฅ๐‘›โ†’๐‘ฅ and๐‘‡๐‘ฅ๐‘›โ†’๐‘ฆ, then๐‘‡๐‘ฅ=๐‘ฆ.

Definition 2.5 (see [9]). (1) A mapping ๐‘‡โˆถ๐ถโ†’๐ถ is said to be quasi-๐œ™-nonexpansive if ๐น(๐‘‡)โ‰ โˆ… and ๐œ™(๐‘,๐‘‡๐‘ฅ)โ‰ค๐œ™(๐‘,๐‘ฅ),โˆ€๐‘ฅโˆˆ๐ถ,๐‘โˆˆ๐น(๐‘‡).(2.8)
(2) A mapping ๐‘‡โˆถ๐ถโ†’๐ถ is said to be quasi-๐œ™-asymptotically nonexpansive if ๐น(๐‘‡)โ‰ โˆ… and there exists a real sequence {๐‘˜๐‘›}โŠ‚[1,โˆž) with ๐‘˜๐‘›โ†’1 such that ๐œ™(๐‘,๐‘‡๐‘›๐‘ฅ)โ‰ค๐‘˜๐‘›๐œ™(๐‘,๐‘ฅ),โˆ€๐‘›โ‰ฅ1,๐‘ฅโˆˆ๐ถ,๐‘โˆˆ๐น(๐‘‡).(2.9)
(3) A mapping ๐‘‡โˆถ๐ถโ†’๐ถ is said to be uniformly ๐ฟ-Lipschitz continuous if there exists a constant ๐ฟ>0 such that โ€–๐‘‡๐‘›๐‘ฅโˆ’๐‘‡๐‘›๐‘ฆโ€–โ‰ค๐ฟโ€–๐‘ฅโˆ’๐‘ฆโ€–,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ,โˆ€๐‘›โ‰ฅ1.(2.10)

Remark 2.6. (1) From the definition, it is easy to know that each relatively nonexpansive mapping is closed.
(2) The class of quasi-๐œ™-asymptotically nonexpansive mappings contains properly the class of quasi-๐œ™-nonexpansive mappings as a subclass, and the class of quasi-๐œ™-nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true.

Lemma 2.7 (see [6]). Let ๐ธ be a uniformly convex Banach space, ๐‘Ÿ>0 a positive number, and ๐ต๐‘Ÿ(0) a closed ball of ๐ธ. Then, for any given sequence {๐‘ฅ๐‘–}โˆž๐‘–=1โŠ‚๐ต๐‘Ÿ(0) and for any given sequence {๐œ†๐‘–}โˆž๐‘–=1 of positive numbers with โˆ‘โˆž๐‘–=1๐œ†๐‘–=1, then there exists a continuous, strictly increasing, and convex function ๐‘”โˆถ[0,2๐‘Ÿ)โ†’[0,โˆž) with ๐‘”(0)=0 such that for any positive integers ๐‘–,๐‘— with ๐‘–<๐‘—, โ€–โ€–โ€–โ€–โˆž๎“๐‘›=1๐œ†๐‘›๐‘ฅ๐‘›โ€–โ€–โ€–โ€–2โ‰คโˆž๎“๐‘›=1๐œ†๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–2โˆ’๐œ†๐‘–๐œ†๐‘—๐‘”๎€ทโ€–โ€–๐‘ฅ๐‘–โˆ’๐‘ฅ๐‘—โ€–โ€–๎€ธ.(2.11)

Lemma 2.8 (see [6]). Let ๐ธ be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property and ๐ถ a nonempty closed convex subset of ๐ธ. Let Tโˆถ๐ถโ†’๐ถ be a closed and quasi-๐œ™-asymptotically nonexpansive mapping with a sequence {๐‘˜๐‘›}โŠ‚[1,โˆž),๐‘˜๐‘›โ†’1. Then ๐น(๐‘‡) is a closed convex subset of ๐ถ.

For solving the system of generalized mixed equilibrium problems (1.1), let us assume that the function ๐œ“โˆถ๐ถโ†’โ„ is convex and lower semicontinuous, the nonlinear mapping ๐ดโˆถ๐ถโ†’๐ธโˆ— is continuous and monotone, and the bifunction ฮ˜๐‘–โˆถ๐ถร—๐ถโ†’โ„,๐‘–=1,2,โ€ฆ,๐‘ satisfies the following conditions:(๐ด1)ฮ˜๐‘–(๐‘ฅ,๐‘ฅ)=0,forall๐‘ฅโˆˆ๐ถ;(๐ด2)ฮ˜๐‘– is monotone, that is, ฮ˜๐‘–(๐‘ฅ,๐‘ฆ)+ฮ˜๐‘–(๐‘ฆ,๐‘ฅ)โ‰ค0,forall๐‘ฅ,๐‘ฆโˆˆ๐ถ;(๐ด3)limsup๐‘กโ†“0ฮ˜๐‘–(๐‘ฅ+๐‘ก(๐‘งโˆ’๐‘ฅ),๐‘ฆ)โ‰คฮ˜๐‘–(๐‘ฅ,๐‘ฆ)forall๐‘ฅ,๐‘ง,๐‘ฆโˆˆ๐ถ;(๐ด4)the function ๐‘ฆโ†ฆฮ˜๐‘–(๐‘ฅ,๐‘ฆ) is convex and lower semicontinuous.

Lemma 2.9. Let ๐ธ be a smooth, strictly convex, and reflexive Banach space and ๐ถ a nonempty closed convex subset of ๐ธ. Let ฮ˜โˆถ๐ถร—๐ถโ†’โ„ be a bifunction satisfying conditions (๐ด1)-(๐ด4). Let ๐‘Ÿ>0 and ๐‘ฅโˆˆ๐ธ. Then, the following hold. (i)(Blum and Oettli [17]) There exists ๐‘งโˆˆ๐ถ such that 1ฮ˜(๐‘ง,๐‘ฆ)+๐‘ŸโŸจ๐‘ฆโˆ’๐‘ง,๐ฝ๐‘งโˆ’๐ฝ๐‘ฅโŸฉโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ.(2.12)(ii)(Takahashi and Zembayashi [3]) Define a mapping ๐‘‡๐‘Ÿโˆถ๐ธโ†’๐ถ by ๐‘‡๐‘Ÿ๎‚†1(๐‘ฅ)=๐‘งโˆˆ๐ถโˆถฮ˜(๐‘ง,๐‘ฆ)+๐‘Ÿ๎‚‡โŸจ๐‘ฆโˆ’๐‘ง,๐ฝ๐‘งโˆ’๐ฝ๐‘ฅโŸฉโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ,๐‘ฅโˆˆ๐ธ.(2.13)Then, the following conclusions hold: (a)๐‘‡๐‘Ÿ is single valued;(b)๐‘‡๐‘Ÿ is firmly nonexpansive-type mapping, that is, for all ๐‘ง,๐‘ฆโˆˆ๐ธ, โŸจ๐‘‡๐‘Ÿ๐‘งโˆ’๐‘‡๐‘Ÿ๐‘ฆ,๐ฝ๐‘‡๐‘Ÿ๐‘งโˆ’๐ฝ๐‘‡๐‘Ÿ๐‘ฆโŸฉโ‰คโŸจ๐‘‡๐‘Ÿ๐‘งโˆ’๐‘‡๐‘Ÿ๐‘ฆ,๐ฝ๐‘งโˆ’๐ฝ๐‘ฆโŸฉ;(2.14)(c)๐น(๐‘‡๐‘Ÿฬƒ)=EP(ฮ˜)=F(T๐‘Ÿ); (d)EP(ฮ˜) is closed and convex;(e)๐œ™(๐‘ž,๐‘‡๐‘Ÿ๐‘ฅ)+๐œ™(๐‘‡๐‘Ÿ๐‘ฅ,๐‘ฅ)โ‰ค๐œ™(๐‘ž,๐‘ฅ),forall๐‘žโˆˆ๐น(T๐‘Ÿ).

Lemma 2.10 (see [7]). Let ๐ธ be a smooth, strictly convex, and reflexive Banach space and ๐ถ a nonempty closed convex subset of ๐ธ. Let ๐ดโˆถ๐ถโ†’๐ธโˆ— be a continuous and monotone mapping, ๐œ“โˆถ๐ถโ†’โ„ a lower semicontinuous and convex function, and ฮ˜โˆถ๐ถร—๐ถโ†’โ„ a bifunction satisfying the conditions (๐ด1)-(๐ด4). Let ๐‘Ÿ>0 be any given number and ๐‘ฅโˆˆ๐ธ be any given point. Then, the following hold. (i)There exists ๐‘ขโˆˆ๐ถ such that 1ฮ˜(๐‘ข,๐‘ฆ)+โŸจ๐ด๐‘ข,๐‘ฆโˆ’๐‘ขโŸฉ+๐œ“(๐‘ฆ)โˆ’๐œ“(๐‘ข)+๐‘ŸโŸจ๐‘ฆโˆ’๐‘ข,๐ฝ๐‘ขโˆ’๐ฝ๐‘ฅโŸฉโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ.(2.15)(ii)If we define a mapping ๐พฮ˜๐‘Ÿโˆถ๐ถโ†’๐ถ by ๐พฮ˜๐‘Ÿ๎‚†1(๐‘ฅ)=๐‘ขโˆˆ๐ถโˆถฮ˜(๐‘ข,๐‘ฆ)+โŸจ๐ด๐‘ข,๐‘ฆโˆ’๐‘ขโŸฉ+๐œ“(๐‘ฆ)โˆ’๐œ“(๐‘ข)+๐‘Ÿ๎‚‡โŸจ๐‘ฆโˆ’๐‘ข,๐ฝ๐‘ขโˆ’๐ฝ๐‘ฅโŸฉโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ,โˆ€๐‘ฅโˆˆ๐ถ.(2.16)then, the mapping ๐พฮ˜๐‘Ÿ has the following properties: (a)๐พฮ˜๐‘Ÿ is single valued;(b)๐พฮ˜๐‘Ÿ is a firmly nonexpansive-type mapping, that is, ๎ซ๐พฮ˜๐‘Ÿ๐‘งโˆ’๐พฮ˜๐‘Ÿ๐‘ฆ,๐ฝ๐พฮ˜๐‘Ÿ๐‘งโˆ’๐ฝ๐พฮ˜๐‘Ÿ๐‘ฆ๎ฌโ‰ค๎ซ๐พฮ˜๐‘Ÿ๐‘งโˆ’๐พฮ˜๐‘Ÿ๎ฌ๐‘ฆ,๐ฝ๐‘งโˆ’๐ฝ๐‘ฆ,โˆ€๐‘ง,๐‘ฆโˆˆ๐ธ;(2.17)(c)๐น(๐พฮ˜๐‘Ÿฬƒ)=ฮฉ(ฮ˜,๐œ“)=F(๐พฮ˜๐‘Ÿ); (d)ฮฉ(ฮ˜,๐œ“) is closed and convex;(e)๐œ™๎€ท๐‘ž,๐พฮ˜๐‘Ÿ๐‘ฅ๎€ธ๎€ท๐พ+๐œ™ฮ˜๐‘Ÿ๎€ธ๎€ท๐พ๐‘ฅ,๐‘ฅโ‰ค๐œ™(๐‘ž,๐‘ฅ),โˆ€๐‘žโˆˆ๐นฮ˜๐‘Ÿ๎€ธ.(2.18)

Remark 2.11. It follows from Lemma 2.9 that the mapping ๐พฮ˜๐‘Ÿ is a relatively nonexpansive mapping. Thus, it is quasi-๐œ™-asymptotically nonexpansive.

3. Main Results

In this section, we will use the hybrid method to prove some strong convergence theorems for finding a common element of the set of solutions for a system of the generalized mixed equilibrium problems (1.1) and the set of common fixed points for an infinite family of quasi-๐œ™-asymptotically nonexpansive mappings in Banach spaces.

Theorem 3.1. Let ๐ธ be a uniformly smooth and strictly convex Banach space with Kleac-Klee property and ๐ถ a nonempty closed convex subset of ๐ธ. Let ๐ดโˆถ๐ถโ†’๐ธโˆ— be a continuous and monotone mapping, ๐œ“โˆถ๐ถโ†’โ„ a lower semicontinuous and convex function, and {ฮ˜๐‘–โˆถ๐ถร—๐ถโ†’โ„,๐‘–=1,2,โ€ฆ,๐‘} be a finite family of bifunction satisfying conditions (๐ด1)-(๐ด4). Let {๐‘†๐‘–}โˆž๐‘–=1โˆถ๐ถโ†’๐ถ an infinite family of closed and uniformly quasi-๐œ™-asymptotically nonexpansive mappings with a sequence {๐‘˜๐‘›}โŠ‚[1,โˆž) and ๐‘˜๐‘›โ†’1. Suppose that for each ๐‘–โ‰ฅ1,๐‘†๐‘– is uniformly ๐ฟ๐‘–-Lipschitz continuous and that โ‹‚๐บโˆถ=โˆž๐‘–=1๐น(๐‘†๐‘–)โ‹‚ฮฉ is a nonempty and bounded subset in ๐ถ, where โ‹‚ฮฉ=๐‘๐‘–=1ฮฉ(ฮ˜๐‘–,๐œ“). Let {๐‘ฅ๐‘›},{๐‘ฆ๐‘›},{๐‘ง๐‘›}, and {๐‘ข๐‘›(๐‘˜)},๐‘˜=1,2,โ€ฆ,๐‘, be the sequences generated by ๐‘ฅ0โˆˆ๐ถ,๐ถ0๐‘ง=๐ถ,๐‘›=๐ฝโˆ’1๎ƒฉ๐›ผ๐‘›,0๐ฝ๐‘ฅ๐‘›+โˆž๎“๐‘–=1๐›ผ๐‘›,๐‘–๐ฝ๐‘†๐‘›๐‘–๐‘ฅ๐‘›๎ƒช,๐‘ฆ๐‘›=๐ฝโˆ’1๎€ท๐›ผ๐‘›๐ฝ๐‘ง๐‘›+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐ฝ๐‘ฅ๐‘›๎€ธ,๐‘ข๐‘›(๐‘)ฮ˜โˆˆ๐ถ๐‘ ๐‘ข๐‘โ„Ž๐‘กโ„Ž๐‘Ž๐‘ก,โˆ€๐‘ฆโˆˆ๐ถ,๐‘๎‚€๐‘ข๐‘›(๐‘)๎‚+๎‚ฌ,๐‘ฆ๐ด๐‘ข๐‘›(๐‘),๐‘ฆโˆ’๐‘ข๐‘›(๐‘)๎‚ญ๎‚€๐‘ข+๐œ“(๐‘ฆ)โˆ’๐œ“๐‘›(๐‘)๎‚+1๐‘Ÿ๐‘›๎‚ฌ๐‘ฆโˆ’๐‘ข๐‘›(๐‘),๐ฝ๐‘ข๐‘›(๐‘)โˆ’๐ฝ๐‘ข๐‘›(๐‘โˆ’1)๎‚ญฮ˜โ‰ฅ0,๐‘โˆ’1๎‚€๐‘ข๐‘›(๐‘โˆ’1)๎‚+๎‚ฌ,๐‘ฆ๐ด๐‘ข๐‘›(๐‘โˆ’1),๐‘ฆโˆ’๐‘ข๐‘›(๐‘โˆ’1)๎‚ญ๎‚€๐‘ข+๐œ“(๐‘ฆ)โˆ’๐œ“๐‘›(๐‘โˆ’1)๎‚+1๐‘Ÿ๐‘›๎‚ฌ๐‘ฆโˆ’๐‘ข๐‘›(๐‘โˆ’1),๐ฝ๐‘ข๐‘›(๐‘โˆ’1)โˆ’๐ฝ๐‘ข๐‘›(๐‘โˆ’2)๎‚ญโ‹ฎฮ˜โ‰ฅ0,1๎‚€๐‘ข๐‘›(1)๎‚+๎‚ฌ,๐‘ฆ๐ด๐‘ข๐‘›(1),๐‘ฆโˆ’๐‘ข๐‘›(1)๎‚ญ๎‚€๐‘ข+๐œ“(๐‘ฆ)โˆ’๐œ“๐‘›(1)๎‚+1๐‘Ÿ๐‘›๎‚ฌ๐‘ฆโˆ’๐‘ข๐‘›(1),๐ฝ๐‘ข๐‘›(1)โˆ’๐ฝ๐‘ฆ๐‘›๎‚ญ๐ถโ‰ฅ0,๐‘›+1=๎‚†๐‘ฃโˆˆ๐ถ๐‘›๎‚€โˆถ๐œ™๐‘ฃ,๐‘ข๐‘›(๐‘)๎‚๎€ทโ‰ค๐œ™๐‘ฃ,๐‘ฅ๐‘›๎€ธ+๐œ‰๐‘›๎‚‡๐‘ฅ,โˆ€๐‘›โ‰ฅ0,๐‘›+1=ฮ ๐ถ๐‘›+1๐‘ฅ0,(3.1) where ๐‘ข๐‘›(1)=๐พฮ˜1๐‘Ÿ๐‘›๐‘ฆ๐‘›,๐‘ข๐‘›(๐‘–)=๐พฮ˜๐‘–๐‘Ÿ๐‘›๐‘ข๐‘›(๐‘–โˆ’1)=๐พฮ˜๐‘–๐‘Ÿ๐‘›๐พฮ˜๐‘–โˆ’1๐‘Ÿ๐‘›๐‘ข๐‘›(๐‘–โˆ’2)=๐พฮ˜๐‘–๐‘Ÿ๐‘›โ‹ฏ๐พฮ˜2๐‘Ÿ๐‘›๐‘ข๐‘›(1)=๐พฮ˜๐‘–๐‘Ÿ๐‘›โ‹ฏ๐พฮ˜2๐‘Ÿ๐‘›๐พฮ˜1๐‘Ÿ๐‘›๐‘ฆ๐‘›,๐‘–=1,2,โ€ฆ,๐‘,(3.2) and ๐พฮ˜๐‘–๐‘Ÿ๐‘›โˆถ๐ถโ†’๐ถ, ๐‘–=1,2,โ€ฆ,๐‘ is the mapping defined by (2.16),๐‘Ÿ๐‘›โˆˆ[๐‘‘,โˆž) for some ๐‘‘>0, ๐œ‰๐‘›=sup๐‘ขโˆˆ๐บ(๐‘˜๐‘›โˆ’1)๐œ™(๐‘ข,๐‘ฅ๐‘›), ฮ ๐ถ๐‘›+1 is the generalized projection of ๐ธ onto the set ๐ถ๐‘›+1, and for each ๐‘›โ‰ฅ0, {๐›ผ๐‘›,๐‘–}, {๐›ผ๐‘›} are sequences in [0,1] satisfying the following conditions: (a)โˆ‘โˆž๐‘–=0๐›ผ๐‘›,๐‘–=1 for all ๐‘›โ‰ฅ0;(b)liminf๐‘›โ†’โˆž๐›ผ๐‘›,0โ‹…๐›ผ๐‘›,๐‘–>0 for all ๐‘–โ‰ฅ1;(c)0<๐›ผโ‰ค๐›ผ๐‘›<1 for some ๐›ผโˆˆ(0,1).
Then {๐‘ฅ๐‘›} converges strongly to ฮ ๐บ๐‘ฅ0, where ฮ ๐บ is the generalized projection from ๐ธ onto G.

Proof. We divide the proof of Theorem 3.1 into five steps.(I)We first prove that ๐บ and ๐ถ๐‘› both are closed and convex subset of ๐ถ for all ๐‘›โ‰ฅ0.
In fact, it follows from Lemmas 2.8 and 2.10 that ๐น(๐‘†๐‘–), ๐‘–โ‰ฅ1, and ฮฉ both are closed and convex. Therefore ๐บ is a closed and convex subset in ๐ถ. Furthermore, it is obvious that ๐ถ0=๐ถ is closed and convex. Suppose that ๐ถ๐‘› is closed and convex for some ๐‘›โ‰ฅ1. Since the inequality ๐œ™(๐‘ฃ,๐‘ข๐‘›(๐‘))โ‰ค๐œ™(๐‘ฃ,๐‘ฅ๐‘›)+๐œ‰๐‘› is equivalent to 2๎‚ฌ๐‘ฃ,๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘ข๐‘›(๐‘)๎‚ญโ‰คโ€–โ€–๐‘ฅ๐‘›โ€–โ€–2โˆ’โ€–โ€–๐‘ข๐‘›(๐‘)โ€–โ€–2+๐œ‰๐‘›,(3.3) therefore, we have๐ถ๐‘›+1=๎‚†๐‘ฃโˆˆ๐ถ๐‘›๎‚ฌโˆถ2๐‘ฃ,๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘ข๐‘›(๐‘)๎‚ญโ‰คโ€–โ€–๐‘ฅ๐‘›โ€–โ€–2โˆ’โ€–โ€–๐‘ข๐‘›(๐‘)โ€–โ€–2+๐œ‰๐‘›๎‚‡.(3.4)
This implies that ๐ถ๐‘›+1 is closed and convex. The desired conclusions are proved. These in turn show that ฮ ๐บ๐‘ฅ0 and ฮ ๐ถ๐‘›๐‘ฅ0 are well defined.
(II)We prove that {๐‘ฅ๐‘›} is a bounded sequence in ๐ถ.By the definition of ๐ถ๐‘›, we have ๐‘ฅ๐‘›=ฮ ๐ถ๐‘›๐‘ฅ0 for all ๐‘›โ‰ฅ0. It follows from Lemma 2.2(a) that ๐œ™๎€ท๐‘ฅ๐‘›,๐‘ฅ0๎€ธ๎€ทฮ =๐œ™๐ถ๐‘›๐‘ฅ0,๐‘ฅ0๎€ธ๎€ทโ‰ค๐œ™๐‘ข,๐‘ฅ0๎€ธ๎€ทโˆ’๐œ™๐‘ข,ฮ ๐ถ๐‘›๐‘ฅ0๎€ธ๎€ทโ‰ค๐œ™๐‘ข,๐‘ฅ0๎€ธ,โˆ€๐‘›โ‰ฅ0,๐‘ขโˆˆ๐บ.(3.5)
This implies that {๐œ™(๐‘ฅ๐‘›,๐‘ฅ0)} is bounded. By virtue of (2.4), {๐‘ฅ๐‘›} is bounded. Denote by ๐‘€=sup๐‘›โ‰ฅ0๎€ฝโ€–โ€–๐‘ฅ๐‘›โ€–โ€–๎€พ<โˆž.(3.6)
In view of the structure of {๐ถ๐‘›}, we have ๐ถ๐‘›+1โŠ‚๐ถ๐‘›, ๐‘ฅ๐‘›=ฮ ๐ถ๐‘›๐‘ฅ0, and ๐‘ฅ๐‘›+1=ฮ ๐ถ๐‘›+1๐‘ฅ0. This implies that ๐‘ฅ๐‘›+1โˆˆ๐ถ๐‘› and ๐œ™๎€ท๐‘ฅ๐‘›,๐‘ฅ0๎€ธ๎€ท๐‘ฅโ‰ค๐œ™๐‘›+1,๐‘ฅ0๎€ธ,โˆ€๐‘›โ‰ฅ1.(3.7) Therefore, {๐œ™(๐‘ฅ๐‘›,๐‘ฅ0)} is convergent. Without loss of generality, we can assume that lim๐‘›โ†’โˆž๐œ™๎€ท๐‘ฅ๐‘›,๐‘ฅ0๎€ธ=๐‘Ÿโ‰ฅ0.(3.8)
(III)Next, we prove that โ‹‚๐บโˆถ=โˆž๐‘–=1๐น(๐‘†๐‘–)โ‹‚(โ‹‚๐‘๐‘–=1ฮฉ(ฮ˜๐‘–,๐œ“))โŠ‚๐ถ๐‘› for all ๐‘›โ‰ฅ0.
Indeed, it is obvious that ๐บโŠ‚๐ถ0=๐ถ. Suppose that ๐บโŠ‚๐ถ๐‘› for some ๐‘›โ‰ฅ0. Since ๐‘ข๐‘›(๐‘)=๐พฮ˜๐‘๐‘Ÿ๐‘›๐พฮ˜๐‘โˆ’1๐‘Ÿ๐‘›โ‹ฏ๐พฮ˜2๐‘Ÿ๐‘›๐พฮ˜1๐‘Ÿ๐‘›๐‘ฆ๐‘›, by Lemma 2.10 and Remark 2.6, ๐พฮ˜๐‘–๐‘Ÿ๐‘› is quasi-๐œ™-asymptotically nonexpansive. Since ๐ธ is uniformly smooth, ๐ธโˆ— is uniformly convex. For any given ๐‘ขโˆˆ๐บโŠ‚๐ถ๐‘› and for any positive integer ๐‘—>0, from Lemma 2.7 we have ๐œ™๎‚€๐‘ข,๐‘ข๐‘›(๐‘)๎‚๎‚€=๐œ™๐‘ข,๐พฮ˜๐‘๐‘Ÿ๐‘›๐พฮ˜๐‘โˆ’1๐‘Ÿ๐‘›โ‹ฏ๐พฮ˜2๐‘Ÿ๐‘›๐พฮ˜1๐‘Ÿ๐‘›๐‘ฆ๐‘›๎‚๎€ทโ‰ค๐œ™๐‘ข,๐‘ฆ๐‘›๎€ธ๎€ท=๐œ™๐‘ข,๐ฝโˆ’1๎€ท๐›ผ๐‘›๐ฝ๐‘ง๐‘›+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐ฝ๐‘ฅ๐‘›๎€ธ๎€ธโ‰คโ€–๐‘ขโ€–2๎ซโˆ’2๐‘ข,๐›ผ๐‘›๐ฝ๐‘ง๐‘›+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐ฝ๐‘ฅ๐‘›๎ฌ+โ€–โ€–๐›ผ๐‘›๐ฝ๐‘ง๐‘›+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐ฝ๐‘ฅ๐‘›โ€–โ€–2โ‰คโ€–๐‘ขโ€–2๎ซโˆ’2๐‘ข,๐›ผ๐‘›๐ฝ๐‘ง๐‘›+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐ฝ๐‘ฅ๐‘›๎ฌ+๐›ผ๐‘›โ€–โ€–๐ฝ๐‘ง๐‘›โ€–โ€–2+๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐ฝ๐‘ฅ๐‘›โ€–โ€–2=โ€–๐‘ขโ€–2๎ซโˆ’2๐‘ข,๐›ผ๐‘›๐ฝ๐‘ง๐‘›+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐ฝ๐‘ฅ๐‘›๎ฌ+๐›ผ๐‘›โ€–โ€–๐‘ง๐‘›โ€–โ€–2+๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–2=๐›ผ๐‘›๐œ™๎€ท๐‘ข,๐‘ง๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐œ™๎€ท๐‘ข,๐‘ฅ๐‘›๎€ธ=๐›ผ๐‘›๐œ™๎ƒฉ๐‘ข,๐ฝโˆ’1๎ƒฉ๐›ผ๐‘›,0๐ฝ๐‘ฅ๐‘›+โˆž๎“๐‘–=1๐›ผ๐‘›,๐‘–๐ฝ๐‘†๐‘›๐‘–๐‘ฅ๐‘›+๎€ท๎ƒช๎ƒช1โˆ’๐›ผ๐‘›๎€ธ๐œ™๎€ท๐‘ข,๐‘ฅ๐‘›๎€ธ=๐›ผ๐‘›โŽ›โŽœโŽœโŽโ€–๐‘ขโ€–2โˆ’2๐›ผ๐‘›,0โŸจ๐‘ข,๐ฝ๐‘ฅ๐‘›โŸฉโˆ’2โˆž๎“๐‘–=1๐›ผ๐‘›,๐‘–๎ซ๐‘ข,๐ฝ๐‘†๐‘›๐‘–๐‘ฅ๐‘›๎ฌ+โ€–โ€–โ€–โ€–๐›ผ๐‘›,0๐ฝ๐‘ฅ๐‘›+โˆž๎“๐‘–=1๐›ผ๐‘›,๐‘–๐ฝ๐‘†๐‘›๐‘–๐‘ฅ๐‘›โ€–โ€–โ€–โ€–2โŽžโŽŸโŽŸโŽ +๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐œ™๎€ท๐‘ข,๐‘ฅ๐‘›๎€ธโ‰ค๐›ผ๐‘›๎ƒฉโ€–๐‘ขโ€–2โˆ’2๐›ผ๐‘›,0โŸจ๐‘ข,๐ฝ๐‘ฅ๐‘›โŸฉโˆ’2โˆž๎“๐‘–=1๐›ผ๐‘›,๐‘–๎ซ๐‘ข,๐ฝ๐‘†๐‘›๐‘–๐‘ฅ๐‘›๎ฌ+๐›ผ๐‘›,0โ€–โ€–๐ฝ๐‘ฅ๐‘›โ€–โ€–2+โˆž๎“๐‘–=1๐›ผ๐‘›,๐‘–โ€–โ€–๐ฝ๐‘†๐‘›๐‘–๐‘ฅ๐‘›โ€–โ€–2โˆ’๐›ผ๐‘›,0๐›ผ๐‘›,๐‘—๐‘”๎€ทโ€–โ€–๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘†๐‘›๐‘—๐‘ฅ๐‘›โ€–โ€–๎€ธ๎ƒช+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐œ™๎€ท๐‘ข,๐‘ฅ๐‘›๎€ธโ‰ค๐›ผ๐‘›๎ƒฉโ€–๐‘ขโ€–2โˆ’2๐›ผ๐‘›,0โŸจ๐‘ข,๐ฝ๐‘ฅ๐‘›โŸฉโˆ’2โˆž๎“๐‘–=1๐›ผ๐‘›,๐‘–๎ซ๐‘ข,๐ฝ๐‘†๐‘›๐‘–๐‘ฅ๐‘›๎ฌ+๐›ผ๐‘›,0โ€–โ€–๐‘ฅ๐‘›โ€–โ€–2+โˆž๎“๐‘–=1๐›ผ๐‘›,๐‘–โ€–๐‘†๐‘›๐‘–๐‘ฅ๐‘›โ€–2โˆ’๐›ผ๐‘›,0๐›ผ๐‘›,๐‘—๐‘”๎€ทโ€–โ€–๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘†๐‘›๐‘—๐‘ฅ๐‘›โ€–โ€–๎€ธ๎ƒช+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐œ™๎€ท๐‘ข,๐‘ฅ๐‘›๎€ธ=๐›ผ๐‘›๎ƒฉ๐›ผ๐‘›,0๐œ™๎€ท๐‘ข,๐‘ฅ๐‘›๎€ธ+โˆž๎“๐‘–=1๐›ผ๐‘›,๐‘–๐œ™๎€ท๐‘ข,๐‘†๐‘›๐‘–๐‘ฅ๐‘›๎€ธโˆ’๐›ผ๐‘›,0๐›ผ๐‘›,๐‘—๐‘”๎€ทโ€–โ€–๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘†๐‘›๐‘—๐‘ฅ๐‘›โ€–โ€–๎€ธ๎ƒช+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐œ™๎€ท๐‘ข,๐‘ฅ๐‘›๎€ธโ‰ค๐›ผ๐‘›๎ƒฉ๐›ผ๐‘›,0๐œ™๎€ท๐‘ข,๐‘ฅ๐‘›๎€ธ+โˆž๎“๐‘–=1๐›ผ๐‘›,๐‘–๐‘˜๐‘›๐œ™๎€ท๐‘ข,๐‘ฅ๐‘›๎€ธโˆ’๐›ผ๐‘›,0๐›ผ๐‘›,๐‘—๐‘”๎€ทโ€–โ€–๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘†๐‘›๐‘—๐‘ฅ๐‘›โ€–โ€–๎€ธ๎ƒช+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐œ™๎€ท๐‘ข,๐‘ฅ๐‘›๎€ธโ‰ค๐›ผ๐‘›๎€ท๐‘˜๐‘›๐œ™๎€ท๐‘ข,๐‘ฅ๐‘›๎€ธโˆ’๐›ผ๐‘›,0๐›ผ๐‘›,๐‘—๐‘”๎€ทโ€–โ€–๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘†๐‘›๐‘—๐‘ฅ๐‘›โ€–โ€–+๎€ท๎€ธ๎€ธ1โˆ’๐›ผ๐‘›๎€ธ๐œ™๎€ท๐‘ข,๐‘ฅ๐‘›๎€ธโ‰ค๐›ผ๐‘›๎‚ต๐œ™๎€ท๐‘ข,๐‘ฅ๐‘›๎€ธ+sup๐‘ขโˆˆ๐บ๎€ท๐‘˜๐‘›๎€ธ๐œ™๎€ทโˆ’1๐‘ข,๐‘ฅ๐‘›๎€ธโˆ’๐›ผ๐‘›,0๐›ผ๐‘›,๐‘—๐‘”๎€ทโ€–โ€–๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘†๐‘›๐‘—๐‘ฅ๐‘›โ€–โ€–๎€ธ๎‚ถ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐œ™๎€ท๐‘ข,๐‘ฅ๐‘›๎€ธ๎€ท=๐œ™๐‘ข,๐‘ฅ๐‘›๎€ธ+๐›ผ๐‘›๐œ‰๐‘›โˆ’๐›ผ๐‘›๐›ผ๐‘›,0๐›ผ๐‘›,๐‘—๐‘”๎€ทโ€–โ€–๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘†๐‘›๐‘–๐‘ฅ๐‘›โ€–โ€–๎€ธ๎€ทโ‰ค๐œ™๐‘ข,๐‘ฅ๐‘›๎€ธ+๐œ‰๐‘›.(3.9) Hence ๐‘ขโˆˆ๐ถ๐‘›+1, and so ๐บโŠ‚๐ถ๐‘› for all ๐‘›โ‰ฅ0. By the way, from the definition of {๐œ‰๐‘›}, (2.4), and (3.6), it is easy to see that ๐œ‰๐‘›=sup๐‘ขโˆˆ๐บ๎€ท๐‘˜๐‘›๎€ธ๐œ™๎€ทโˆ’1๐‘ข,๐‘ฅ๐‘›๎€ธโ‰คsup๐‘ขโˆˆ๐บ๎€ท๐‘˜๐‘›๎€ธ()โˆ’1โ€–๐‘ขโ€–+๐‘€2(โŸถ0as๐‘›โŸถโˆž).(3.10)
(IV)Now, we prove that {๐‘ฅ๐‘›} converges strongly to some point โ‹‚๐‘โˆˆ๐บโˆถ=โˆž๐‘–=1๐น(๐‘†๐‘–)โ‹‚ฮฉ.
First, we prove that {๐‘ฅ๐‘›} converges strongly to some point โ‹‚๐‘โˆˆโˆž๐‘–=1๐น(๐‘†๐‘–).
In fact, since {๐‘ฅ๐‘›} is bounded in ๐ถ and ๐ธ is reflexive, there exists a subsequence {๐‘ฅ๐‘›๐‘–}โŠ‚{๐‘ฅ๐‘›} such that ๐‘ฅ๐‘›๐‘–โ‡€๐‘. Again since ๐ถ๐‘› is closed and convex for each ๐‘›โ‰ฅ1, it is weakly closed, and so ๐‘โˆˆ๐ถ๐‘› for each ๐‘›โ‰ฅ0. Since ๐‘ฅ๐‘›=ฮ ๐ถ๐‘›๐‘ฅ0, from the definition of ฮ ๐ถ๐‘›, we have ๐œ™๎€ท๐‘ฅ๐‘›๐‘–,๐‘ฅ0๎€ธ๎€ทโ‰ค๐œ™๐‘,๐‘ฅ0๎€ธ.,๐‘›โ‰ฅ0(3.11) Since liminf๐‘›๐‘–โ†’โˆž๐œ™๎€ท๐‘ฅ๐‘›๐‘–,๐‘ฅ0๎€ธ=liminf๐‘›๐‘–โ†’โˆž๎‚†โ€–โ€–๐‘ฅ๐‘›๐‘–โ€–โ€–2๎ซ๐‘ฅโˆ’2๐‘›๐‘–,๐ฝ๐‘ฅ0๎ฌ+โ€–โ€–๐‘ฅ0โ€–โ€–2๎‚‡โ‰ฅโ€–๐‘โ€–2โˆ’2โŸจ๐‘,๐ฝ๐‘ฅ0โ€–โ€–๐‘ฅโŸฉ+0โ€–โ€–2๎€ท=๐œ™๐‘,๐‘ฅ0๎€ธ,(3.12) we have ๐œ™๎€ท๐‘,๐‘ฅ0๎€ธโ‰คliminf๐‘›๐‘–โ†’โˆž๐œ™๎€ท๐‘ฅ๐‘›๐‘–,๐‘ฅ0๎€ธโ‰คlimsup๐‘›๐‘–โ†’โˆž๐œ™๎€ท๐‘ฅ๐‘›๐‘–,๐‘ฅ0๎€ธ๎€ทโ‰ค๐œ™๐‘,๐‘ฅ0๎€ธ.(3.13) This implies that lim๐‘›๐‘–โ†’โˆž๐œ™(๐‘ฅ๐‘›๐‘–,๐‘ฅ0)=๐œ™(๐‘,๐‘ฅ0), that is, โ€–๐‘ฅ๐‘›๐‘–โ€–โ†’โ€–๐‘โ€–. In view of the Kadec-Klee property of ๐ธ, we obtain that lim๐‘›โ†’โˆž๐‘ฅ๐‘›๐‘–=๐‘.
Now we prove that ๐‘ฅ๐‘›โ†’๐‘(๐‘›โ†’โˆž). In fact, if there exists a subsequence {๐‘ฅ๐‘›๐‘—}โŠ‚{๐‘ฅ๐‘›} such that ๐‘ฅ๐‘›๐‘—โ†’๐‘ž, then we have ๐œ™(๐‘,๐‘ž)=lim๐‘›๐‘–โ†’โˆž,๐‘›๐‘—โ†’โˆž๐œ™๎‚€๐‘ฅ๐‘›๐‘–,๐‘ฅ๐‘›๐‘—๎‚โ‰คlim๐‘›๐‘–โ†’โˆž,๐‘›๐‘—โ†’โˆž๐œ™๎€ท๐‘ฅ๐‘›๐‘–๐‘ฅ0๎€ธ๎‚€ฮ โˆ’๐œ™๐ถ๐‘›๐‘—๐‘ฅ0,๐‘ฅ0๎‚=lim๐‘›๐‘–โ†’โˆž,๐‘›๐‘—โ†’โˆž๐œ™๎€ท๐‘ฅ๐‘›๐‘–,๐‘ฅ0๎€ธ๎‚€๐‘ฅโˆ’๐œ™๐‘›๐‘—,๐‘ฅ0๎‚=0(by(3.8)).(3.14) Therefore we have ๐‘=๐‘ž. This implies that lim๐‘›โ†’โˆž๐‘ฅ๐‘›=๐‘.(3.15)
Now we prove that โ‹‚๐‘โˆˆโˆž๐‘–=1๐น(๐‘†๐‘–). In fact, by the construction of ๐ถ๐‘›, we have that ๐ถ๐‘›+1โŠ‚๐ถ๐‘› and ๐‘ฅ๐‘›+1=ฮ ๐ถ๐‘›+1๐‘ฅ0. Therefore, by Lemma 2.2(a) we have ๐œ™๎€ท๐‘ฅ๐‘›+1,๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅ=๐œ™๐‘›+1,ฮ ๐ถ๐‘›๐‘ฅ0๎€ธ๎€ท๐‘ฅโ‰ค๐œ™๐‘›+1,๐‘ฅ0๎€ธ๎€ทฮ โˆ’๐œ™๐ถ๐‘›๐‘ฅ0,๐‘ฅ0๎€ธ๎€ท๐‘ฅ=๐œ™๐‘›+1,๐‘ฅ0๎€ธ๎€ท๐‘ฅโˆ’๐œ™๐‘›,๐‘ฅ0๎€ธ(โŸถ0as๐‘›โŸถโˆž).(3.16) In view of ๐‘ฅ๐‘›+1โˆˆ๐ถ๐‘› and noting the construction of ๐ถ๐‘›+1 we obtain that ๐œ™๎‚€๐‘ฅ๐‘›+1,๐‘ข๐‘›(๐‘)๎‚๎€ท๐‘ฅโ‰ค๐œ™๐‘›+1,๐‘ฅ๐‘›๎€ธ+๐œ‰๐‘›โŸถ0(as๐‘›โŸถโˆž).(3.17) From (2.4) it yields (โ€–๐‘ฅ๐‘›+1โ€–โˆ’โ€–๐‘ข๐‘›(๐‘)โ€–)2โ†’0. Since โ€–๐‘ฅ๐‘›+1โ€–โ†’โ€–๐‘โ€–, we have โ€–โ€–๐‘ข๐‘›(๐‘)โ€–โ€–โŸถโ€–๐‘โ€–(as๐‘›โŸถโˆž).(3.18)
Hence we have โ€–โ€–๐ฝ๐‘ข๐‘›(๐‘)โ€–โ€–โŸถโ€–๐ฝ๐‘โ€–(as๐‘›โŸถโˆž).(3.19)
This implies that {๐ฝ๐‘ข๐‘›(๐‘)} is bounded in ๐ธโˆ—. Since ๐ธ is reflexive, and so ๐ธโˆ— is reflexive, there exists a subsequence {๐ฝ๐‘ข๐‘›(๐‘)๐‘–}โŠ‚{๐ฝ๐‘ข๐‘›(๐‘)} such that ๐ฝ๐‘ข๐‘›(๐‘)๐‘–โ‡€๐‘0โˆˆ๐ธโˆ—. In view of the reflexive of ๐ธ, we see that ๐ฝ(๐ธ)=๐ธโˆ—. Hence there exists ๐‘ฅโˆˆ๐ธ such that ๐ฝ๐‘ฅ=๐‘0. Since ๐œ™๎‚€๐‘ฅ๐‘›๐‘–,๐‘ข๐‘›(๐‘)๐‘–๎‚=โ€–โ€–๐‘ฅ๐‘›๐‘–โ€–โ€–2๎‚ฌ๐‘ฅโˆ’2๐‘›๐‘–,๐ฝ๐‘ข๐‘›(๐‘)๐‘–๎‚ญ+โ€–โ€–๐‘ข๐‘›(๐‘)๐‘–โ€–โ€–2=โ€–โ€–๐‘ฅ๐‘›๐‘–โ€–โ€–2๎‚ฌ๐‘ฅโˆ’2๐‘›๐‘–,๐ฝ๐‘ข๐‘›(๐‘)๐‘–๎‚ญ+โ€–โ€–๐ฝ๐‘ข๐‘›(๐‘)๐‘–โ€–โ€–2,(3.20) taking liminf๐‘›โ†’โˆž on both sides of above equality and in view of the weak lower semicontinuity of norm โ€–โ‹…โ€–, it yields that 0โ‰ฅโ€–๐‘โ€–2โˆ’2โŸจ๐‘,๐‘0โ€–โ€–๐‘โŸฉ+0โ€–โ€–2=โ€–๐‘โ€–2โˆ’2โŸจ๐‘,๐ฝ๐‘ฅโŸฉ+โ€–๐ฝ๐‘ฅโ€–2=โ€–๐‘โ€–2โˆ’2โŸจ๐‘,๐ฝ๐‘ฅโŸฉ+โ€–๐‘ฅโ€–2=๐œ™(๐‘,๐‘ฅ).(3.21) That is, ๐‘=๐‘ฅ. This implies that ๐‘0=๐ฝ๐‘, and so ๐ฝ๐‘ข๐‘›(๐‘)โ‡€๐ฝ๐‘. It follows from (3.19) and the Kadec-Klee property of ๐ธโˆ— that ๐ฝ๐‘ข๐‘›(๐‘)๐‘–โ†’๐ฝ๐‘ (as ๐‘›โ†’โˆž). Noting that ๐ฝโˆ’1โˆถ๐ธโˆ—โ†’๐ธ is norm-weak continuous, it yields that ๐‘ข๐‘›(๐‘)๐‘–โ‡€๐‘. It follows from (3.18) and the Kadec-Klee property of ๐ธ that lim๐‘›๐‘–โ†’โˆž๐‘ข๐‘›(๐‘)๐‘–=๐‘.
By the similar way as given in the proof of (3.15), we can also prove that lim๐‘›โ†’โˆž๐‘ข๐‘›(๐‘)=๐‘.(3.22)
From (3.15) and (3.22) we have that โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ข๐‘›(๐‘)โ€–โ€–โŸถ0(as๐‘›โŸถโˆž).(3.23)
Since ๐ฝ is uniformly continuous on any bounded subset of ๐ธ, we have โ€–โ€–๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘ข๐‘›(๐‘)โ€–โ€–โŸถ0(as๐‘›โŸถโˆž).(3.24)
For any ๐‘—โ‰ฅ1 and any ๐‘ขโˆˆ๐บ, it follows from (3.9), (3.15), and (3.22) that ๐›ผ๐‘›๐›ผ๐‘›,0๐›ผ๐‘›,๐‘—๐‘”๎€ทโ€–โ€–๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘†๐‘›๐‘–๐‘ฅ๐‘›โ€–โ€–๎€ธ๎€ทโ‰ค๐œ™๐‘ข,๐‘ฅ๐‘›๎€ธ๎‚€โˆ’๐œ™๐‘ข,๐‘ข๐‘›(๐‘)๎‚+๐›ผ๐‘›๐œ‰๐‘›.(3.25)
Since ๐œ™๎€ท๐‘ข,๐‘ฅ๐‘›๎€ธ๎‚€โˆ’๐œ™๐‘ข,๐‘ข๐‘›(๐‘)๎‚=โ€–โ€–๐‘ฅ๐‘›โ€–โ€–2โˆ’โ€–โ€–๐‘ข๐‘›(๐‘)โ€–โ€–2๎‚ฌโˆ’2๐‘ข,๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘ข๐‘›(๐‘)๎‚ญโ‰ค|||โ€–โ€–๐‘ฅ๐‘›โ€–โ€–2โˆ’โ€–โ€–๐‘ข๐‘›(๐‘)โ€–โ€–2|||โ€–โ€–+2โ€–๐‘ขโ€–โ‹…๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘ข๐‘›(๐‘)โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ข๐‘›(๐‘)โ€–โ€–๎‚€โ€–โ€–๐‘ฅ๐‘›โ€–โ€–+โ€–โ€–๐‘ข๐‘›(๐‘)โ€–โ€–๎‚โ€–โ€–+2โ€–๐‘ขโ€–โ‹…๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘ข๐‘›(๐‘)โ€–โ€–,(3.26)
From (3.23) and (3.24), it follows that ๐œ™๎€ท๐‘ข,๐‘ฅ๐‘›๎€ธ๎‚€โˆ’๐œ™๐‘ข,๐‘ข๐‘›(๐‘)๎‚โŸถ0(as๐‘›โŸถโˆž).(3.27)
In view of conditions (b) liminf๐‘›โ†’โˆž๐›ผ๐‘›,0๐›ผ๐‘›,๐‘—>0 and (c) ๐›ผ๐‘›โ‰ฅ๐›ผ>0, we see that ๐‘”๎€ทโ€–โ€–๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘†๐‘›๐‘–๐‘ฅ๐‘›โ€–โ€–๎€ธ(โŸถ0as๐‘›โŸถโˆž).(3.28)
It follows from the property of ๐‘” that โ€–โ€–๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘†๐‘›๐‘–๐‘ฅ๐‘›โ€–โ€–โŸถ0(as๐‘›โŸถโˆž).(3.29)
Since ๐‘ฅ๐‘›โ†’๐‘ and ๐ฝ is uniformly continuous, it yields ๐ฝ๐‘ฅ๐‘›โ†’๐ฝ๐‘. Hence from (3.29) we have ๐ฝ๐‘†๐‘›๐‘—๐‘ฅ๐‘›โŸถ๐ฝ๐‘(as๐‘›โŸถโˆž).(3.30) Since ๐ฝโˆ’1โˆถ๐ธโˆ—โ†’๐ธ is norm-weak continuous, it follows that ๐‘†๐‘›๐‘—๐‘ฅ๐‘›โ‡€๐‘(foreach๐‘—โ‰ฅ1).(3.31) On the other hand, for each ๐‘—โ‰ฅ1 we have ||โ€–โ€–๐‘†๐‘›๐‘—๐‘ฅ๐‘›โ€–โ€–||=||โ€–โ€–๐ฝ๎€ท๐‘†โˆ’โ€–๐‘โ€–๐‘›๐‘—๐‘ฅ๐‘›๎€ธโ€–โ€–||โ‰คโ€–โ€–โˆ’โ€–๐ฝ๐‘โ€–๐ฝ๐‘†๐‘›๐‘—๐‘ฅ๐‘›โ€–โ€–(โˆ’๐ฝ๐‘โŸถ0as๐‘›โŸถโˆž).(3.32) This together with (3.31) shows that ๐‘†๐‘›๐‘—๐‘ฅ๐‘›โŸถ๐‘(foreach๐‘—โ‰ฅ1).(3.33)
Furthermore, by the assumption that for each ๐‘—โ‰ฅ1, ๐‘†๐‘— is uniformly ๐ฟ๐‘–-Lipschitz continuous, hence we have โ€–โ€–๐‘†๐‘—๐‘›+1๐‘ฅ๐‘›โˆ’๐‘†๐‘›๐‘—๐‘ฅ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘†๐‘—๐‘›+1๐‘ฅ๐‘›โˆ’๐‘†๐‘—๐‘›+1๐‘ฅ๐‘›+1โ€–โ€–+โ€–โ€–๐‘†๐‘—๐‘›+1๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›+1โ€–โ€–+โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–+โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘†๐‘›๐‘—๐‘ฅ๐‘›โ€–โ€–โ‰ค๎€ท๐ฟ๐‘—๎€ธโ€–โ€–๐‘ฅ+1๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–+โ€–โ€–๐‘†๐‘—๐‘›+1๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›+1โ€–โ€–+โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘†๐‘›๐‘—๐‘ฅ๐‘›โ€–โ€–.(3.34)
This together with (3.15) and (3.33) yields โ€–๐‘†๐‘—๐‘›+1๐‘ฅ๐‘›โˆ’๐‘†๐‘›๐‘—๐‘ฅ๐‘›โ€–โ†’0 (as ๐‘›โ†’โˆž). Hence from (3.33) we have ๐‘†๐‘—๐‘›+1๐‘ฅ๐‘›โ†’๐‘, that is, ๐‘†๐‘—๐‘†๐‘›๐‘—๐‘ฅ๐‘›โ†’๐‘. In view of (3.33) and the closeness of ๐‘†๐‘—, it yields that ๐‘†๐‘—๐‘=๐‘,for all๐‘—โ‰ฅ1. This implies that โ‹‚๐‘โˆˆโˆž๐‘–=1๐น(๐‘†๐‘–).
Next, we prove that โ‹‚๐‘โˆˆ๐‘๐‘–=1ฮฉ(ฮ˜๐‘–,๐œ“). Since ๐‘ข๐‘›(๐‘)=๐พฮ˜๐‘๐‘Ÿ๐‘›๐‘ข๐‘›(๐‘โˆ’1), from (2.18), (3.9), and (3.27), we have ๐œ™๎‚€๐‘ข๐‘›(๐‘),๐‘ข๐‘›(๐‘โˆ’1)๎‚๎‚€๐พ=๐œ™ฮ˜๐‘๐‘Ÿ๐‘›๐‘ข๐‘›(๐‘โˆ’1),๐‘ข๐‘›(๐‘โˆ’1)๎‚๎‚€โ‰ค๐œ™๐‘ข,๐‘ข๐‘›(๐‘โˆ’1)๎‚๎‚€โˆ’๐œ™๐‘ข,๐พฮ˜๐‘๐‘Ÿ๐‘›๐‘ข๐‘›(๐‘โˆ’1)๎‚๎€ทโ‰ค๐œ™๐‘ข,๐‘ฅ๐‘›๎€ธ+๐œ‰๐‘›๎‚€โˆ’๐œ™๐‘ข,๐พฮ˜๐‘๐‘Ÿ๐‘›๐‘ข๐‘›(๐‘โˆ’1)๎‚๎€ท=๐œ™๐‘ข,๐‘ฅ๐‘›๎€ธ+๐œ‰๐‘›๎‚€โˆ’๐œ™๐‘ข,๐‘ข๐‘›(๐‘)๎‚โŸถ0(as๐‘›โŸถโˆž).(3.35) From (2.4) it yields (โ€–๐‘ข๐‘›(๐‘)โ€–โˆ’โ€–๐‘ข๐‘›(๐‘โˆ’1)โ€–)2โ†’0. Since โ€–๐‘ข๐‘›(๐‘)โ€–โ†’โ€–๐‘โ€–, we have โ€–โ€–๐‘ข๐‘›(๐‘โˆ’1)โ€–โ€–โŸถโ€–๐‘โ€–(as.๐‘›โŸถโˆž)(3.36)
Hence we have โ€–โ€–๐ฝ๐‘ข๐‘›(๐‘โˆ’1)โ€–โ€–โŸถโ€–๐ฝ๐‘โ€–(as๐‘›โŸถโˆž).(3.37)
This implies that {๐ฝ๐‘ข๐‘›(๐‘โˆ’1)} is bounded in ๐ธโˆ—. Since ๐ธ is reflexive, and so ๐ธโˆ— is reflexive, there exists a subsequence {๐ฝ๐‘ข๐‘›(๐‘โˆ’1)๐‘–}โŠ‚{๐ฝ๐‘ข๐‘›(๐‘โˆ’1)} such that ๐ฝ๐‘ข๐‘›(๐‘โˆ’1)๐‘–โ‡€๐‘0โˆˆ๐ธโˆ—. In view of the reflexive of ๐ธ, we see that ๐ฝ(๐ธ)=๐ธโˆ—. Hence there exists ๐‘ฅโˆˆ๐ธ such that ๐ฝ๐‘ฅ=๐‘0. Since ๐œ™๎‚€๐‘ข๐‘›(๐‘)๐‘–,๐‘ข๐‘›(๐‘โˆ’1)๐‘–๎‚=โ€–โ€–๐‘ข๐‘›(๐‘)๐‘–โ€–โ€–2๎‚ฌ๐‘ขโˆ’2๐‘›(๐‘)๐‘–,๐ฝ๐‘ข๐‘›(๐‘โˆ’1)๐‘–๎‚ญ+โ€–โ€–๐‘ข๐‘›(๐‘โˆ’1)๐‘–โ€–โ€–2=โ€–โ€–๐‘ข๐‘›(๐‘)๐‘–โ€–โ€–2๎‚ฌ๐‘ขโˆ’2๐‘›(๐‘)๐‘–,๐ฝ๐‘ข๐‘›(๐‘โˆ’1)๐‘–๎‚ญ+โ€–โ€–๐ฝ๐‘ข๐‘›(๐‘โˆ’1)๐‘–โ€–โ€–2,(3.38) taking liminf๐‘›โ†’โˆž on both sides of equality above and in view of the weak lower semicontinuity of norm โ€–โ‹…โ€–, it yields that 0โ‰ฅโ€–๐‘โ€–2โˆ’2โŸจ๐‘,๐‘0โ€–โ€–๐‘โŸฉ+0โ€–โ€–2=โ€–๐‘โ€–2โˆ’2โŸจ๐‘,๐ฝ๐‘ฅโŸฉ+โ€–๐ฝ๐‘ฅโ€–2=โ€–๐‘โ€–2โˆ’2โŸจ๐‘,๐ฝ๐‘ฅโŸฉ+โ€–๐‘ฅโ€–2=๐œ™(๐‘,๐‘ฅ).(3.39) That is, ๐‘=๐‘ฅ. This implies that ๐‘0=๐ฝ๐‘, and so ๐ฝ๐‘ข๐‘›(๐‘โˆ’1)๐‘–โ‡€๐ฝ๐‘. It follows from (3.37) and the Kadec-Klee property of ๐ธโˆ— that ๐ฝ๐‘ข๐‘›(๐‘โˆ’1)โ†’๐ฝ๐‘ (as ๐‘›โ†’โˆž). Note that ๐ฝโˆ’1โˆถ๐ธโˆ—โ†’๐ธ is hemi-continuous, it yields that ๐‘ข๐‘›(๐‘โˆ’1)๐‘–โ‡€๐‘. It follows from (3.36) and the Kadec-Klee property of ๐ธ that lim๐‘›๐‘–โ†’โˆž๐‘ข๐‘›(๐‘โˆ’1)๐‘–=๐‘.
By the similar way as given in the proof of (3.15), we can also prove that lim๐‘›โ†’โˆž๐‘ข๐‘›(๐‘โˆ’1).=๐‘(3.40) From (3.22) and (3.40) we have that โ€–๐‘ข๐‘›(๐‘)โˆ’๐‘ข๐‘›(๐‘โˆ’1)โ€–โŸถ0(as๐‘›โŸถโˆž).(3.41) Since ๐ฝ is uniformly continuous on any bounded subset of ๐ธ, we have โ€–๐ฝ๐‘ข๐‘›(๐‘)โˆ’๐ฝ๐‘ข๐‘›(๐‘โˆ’1)โ€–โŸถ0(as๐‘›โŸถโˆž).(3.42)
Since ๐‘ข๐‘›(๐‘–)=๐พฮ˜๐‘–๐‘Ÿ๐‘›๐‘ข๐‘›(๐‘–โˆ’1),๐‘–=2,3,โ€ฆ,๐‘,๐‘ข๐‘›(1)=๐พฮ˜1๐‘Ÿ๐‘›๐‘ฆ๐‘›,(3.43)
by the similar way as above, we can also prove that โ€–๐‘ข๐‘›(๐‘–)โˆ’๐‘ข๐‘›(๐‘–โˆ’1)โ€–โŸถ0,โ€–๐ฝ๐‘ข๐‘›(๐‘–)โˆ’๐ฝ๐‘ข๐‘›(๐‘–โˆ’1)โ€–โŸถ0,โ€–๐‘ฆ๐‘›โˆ’๐‘ข๐‘›(1)โ€–โŸถ0,โ€–๐ฝ๐‘ฆ๐‘›โˆ’๐ฝ๐‘ข๐‘›(1)โ€–โŸถ0(as๐‘›โŸถโˆž),๐‘–=2,3,โ€ฆ,๐‘.(3.44) From (3.44) and the assumption that ๐‘Ÿ๐‘›โ‰ฅ๐‘‘,forall๐‘›โ‰ฅ0, we have lim๐‘›โ†’โˆžโ€–โ€–๐ฝ๐‘ข๐‘›(๐‘–)โˆ’๐ฝ๐‘ข๐‘›(๐‘–โˆ’1)โ€–โ€–๐‘Ÿ๐‘›=0,lim๐‘›โ†’โˆžโ€–โ€–๐ฝ๐‘ฆ๐‘›โˆ’๐ฝ๐‘ข๐‘›(1)โ€–โ€–๐‘Ÿ๐‘›=0,๐‘–=2,3,โ€ฆ,๐‘.(3.45) Now, we define functions ๐ป๐‘–โˆถ๐ถร—๐ถโ†’โ„,๐‘–=1,2,โ€ฆ,๐‘ by ๐ป๐‘–(๐‘ฅ,๐‘ฆ)=ฮ˜๐‘–.(๐‘ฅ,๐‘ฆ)+โŸจ๐ด๐‘ฅ,๐‘ฆโˆ’๐‘ฅโŸฉ+๐œ“(๐‘ฆ)โˆ’๐œ“(๐‘ฅ),โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ(3.46) By Lemma 2.9, we know that the function ๐ป๐‘– satisfies conditions (๐ด1)-(๐ด4). Since ๐ป๐‘–๎‚€๐‘ข๐‘›(๐‘–)๎‚+1,๐‘ฆ๐‘Ÿ๐‘›๎‚ฌ๐‘ฆโˆ’๐‘ข๐‘›(๐‘–),๐ฝ๐‘ข๐‘›(๐‘–)โˆ’๐ฝ๐‘ข๐‘›(๐‘–โˆ’1)๎‚ญ๐ปโ‰ฅ0,1๎‚€๐‘ข๐‘›(1)๎‚+1,๐‘ฆ๐‘Ÿ๐‘›๎‚ฌ๐‘ฆโˆ’๐‘ข๐‘›(1),๐ฝ๐‘ข๐‘›(1)โˆ’๐ฝ๐‘ฆ๐‘›๎‚ญโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ,(3.47) by condition (๐ด2), we have 1๐‘Ÿ๐‘›๎‚ฌ๐‘ฆโˆ’๐‘ข๐‘›(๐‘–),๐ฝ๐‘ข๐‘›(๐‘–)โˆ’๐ฝ๐‘ข๐‘›(๐‘–โˆ’1)๎‚ญโ‰ฅโˆ’๐ป๐‘–๎‚€๐‘ข๐‘›(๐‘–)๎‚,๐‘ฆโ‰ฅ๐ป๐‘–๎‚€๐‘ฆ,๐‘ข๐‘›(๐‘–)๎‚,1๐‘Ÿ๐‘›๎‚ฌ๐‘ฆโˆ’๐‘ข๐‘›(1),๐ฝ๐‘ข๐‘›(1)โˆ’๐ฝ๐‘ฆ๐‘›๎‚ญโ‰ฅโˆ’๐ป1๎‚€๐‘ข๐‘›(1)๎‚,๐‘ฆโ‰ฅ๐ป1๎‚€๐‘ฆ,๐‘ข๐‘›(1)๎‚,โˆ€๐‘ฆโˆˆ๐ถ.(3.48) By the assumption that ๐‘ฆโ†ฆ๐ป๐‘–(๐‘ฅ,๐‘ฆ) is convex and lower semicontinuous, letting ๐‘›โ†’โˆž in (3.48), from (3.45) and (3.48), for each ๐‘–, we have ๐ปi(๐‘ฆ,๐‘)โ‰ค0, for all ๐‘ฆโˆˆ๐ถ.
For ๐‘กโˆˆ(0,1] and ๐‘ฆโˆˆ๐ถ, letting ๐‘ฆ๐‘ก=๐‘ก๐‘ฆ+(1โˆ’๐‘ก)๐‘, there are ๐‘ฆ๐‘กโˆˆ๐ถ and ๐ป๐‘–(๐‘ฆ๐‘ก,๐‘)โ‰ค0. By conditions (๐ด1) and (๐ด4), we have 0=๐ป๐‘–๎€ท๐‘ฆ๐‘ก,๐‘ฆ๐‘ก๎€ธโ‰ค๐‘ก๐ป๐‘–๎€ท๐‘ฆ๐‘ก๎€ธ+,๐‘ฆ(1โˆ’๐‘ก)๐ป๐‘–๎€ท๐‘ฆ๐‘ก๎€ธ,๐‘โ‰ค๐‘ก๐ป๐‘–๎€ท๐‘ฆ๐‘ก๎€ธ.,๐‘ฆ(3.49) Dividing both sides of the above equation by ๐‘ก, we have ๐ป๐‘–(๐‘ฆ๐‘ก,๐‘ฆ)โ‰ฅ0,for all ๐‘ฆโˆˆ๐ถ. Letting ๐‘กโ†“0, from condition (๐ด3), we have ๐ป๐‘–(๐‘,๐‘ฆ)โ‰ฅ0, for all ๐‘ฆโˆˆ๐ถ, that is, โ‹‚๐‘โˆˆ๐‘๐‘–=1ฮฉ(ฮ˜๐‘–,๐œ“), that is, ๐‘โˆˆฮฉ, and โ‹‚๐‘โˆˆโˆž๐‘–=1๐น(๐‘†๐‘–)โ‹‚ฮฉ.
(V)Now, we prove ๐‘ฅ๐‘›โ†’ฮ โ‹‚โˆž๐‘–=1๐น(๐‘†๐‘–)โ‹‚ฮฉ๐‘ฅ0.
Let ๐‘ค=ฮ โ‹‚โˆž๐‘–=1๐น(๐‘†๐‘–)โ‹‚ฮฉ๐‘ฅ0. From โ‹‚๐‘คโˆˆโˆž๐‘–=1๐น(๐‘†๐‘–)โ‹‚ฮฉโŠ‚๐ถ๐‘›+1 and ๐‘ฅ๐‘›+1=ฮ ๐ถ๐‘›+1๐‘ฅ0, we have ๐œ™(๐‘ฅ๐‘›+1,๐‘ฅ0)โ‰ค๐œ™(๐‘ค,๐‘ฅ0), for all ๐‘›โ‰ฅ0. This implies that ๐œ™๎€ท๐‘,๐‘ฅ0๎€ธ=lim๐‘›โ†’โˆž๐œ™๎€ท๐‘ฅ๐‘›,๐‘ฅ0๎€ธ๎€ทโ‰ค๐œ™๐‘ค,๐‘ฅ0๎€ธ.(3.50)
By the definition of ฮ โ‹‚โˆž๐‘–=1๐น(๐‘†๐‘–)โ‹‚ฮฉ๐‘ฅ0 and (3.50), we have ๐‘=๐‘ค. Therefore, ๐‘ฅ๐‘›โ†’ฮ โ‹‚โˆž๐‘–=1๐น(๐‘†๐‘–)โ‹‚ฮฉ๐‘ฅ0. This completes the proof of Theorem 3.1.

Theorem 3.2. Let ๐ธ be a uniformly smooth and strictly convex Banach space with Kleac-Klee property and ๐ถ a nonempty closed convex subset of ๐ธ. Let ๐ดโˆถ๐ถโ†’๐ธโˆ— be a continuous and monotone mapping, ๐œ“โˆถ๐ถโ†’โ„ a lower semicontinuous and convex function, and {ฮ˜๐‘–โˆถ๐ถร—๐ถโ†’โ„,๐‘–=1,2,โ€ฆ,๐‘} a finite family of bifunction satisfying conditions (๐ด1)-(๐ด4). Let {๐‘†๐‘–}โˆž๐‘–=1โˆถ๐ถโ†’๐ถ be an infinite family of closed and quasi-๐œ™-nonexpansive mappings. Suppose that โ‹‚๐บโˆถ=โˆž๐‘–=1๐น(๐‘†๐‘–)โ‹‚ฮฉโ‰ โˆ…, where โ‹‚ฮฉ=๐‘๐‘–=1ฮฉ(ฮ˜๐‘–,๐œ“). Let {๐‘ฅ๐‘›}, {๐‘ฆ๐‘›}, {๐‘ง๐‘›}, and {๐‘ข๐‘›(๐‘˜)}, ๐‘˜=1,2,โ€ฆ,๐‘, be the sequences generated by ๐‘ฅ0โˆˆ๐ถ,๐ถ0๐‘ง=๐ถ,๐‘›=๐ฝโˆ’1๎ƒฉ๐›ผ๐‘›,0๐ฝ๐‘ฅ๐‘›+โˆž๎“๐‘–=1๐›ผ๐‘›,๐‘–๐ฝ๐‘†๐‘›๐‘–๐‘ฅ๐‘›๎ƒช,๐‘ฆ๐‘›=๐ฝโˆ’1๎€ท๐›ผ๐‘›+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐ฝ๐‘ง๐‘›๎€ธ,๐‘ข๐‘›(๐‘)ฮ˜โˆˆ๐ถ๐‘ ๐‘ข๐‘โ„Ž๐‘กโ„Ž๐‘Ž๐‘ก,โˆ€๐‘ฆโˆˆ๐ถ,๐‘๎‚€๐‘ข๐‘›(๐‘)๎‚+๎‚ฌ,๐‘ฆ๐ด๐‘ข๐‘›(๐‘),๐‘ฆโˆ’๐‘ข๐‘›(๐‘)๎‚ญ๎‚€๐‘ข+๐œ“(๐‘ฆ)โˆ’๐œ“๐‘›(๐‘)๎‚+1๐‘Ÿ๐‘›๎‚ฌ๐‘ฆโˆ’๐‘ข๐‘›(๐‘),๐ฝ๐‘ข๐‘›(๐‘)โˆ’๐ฝ๐‘ข๐‘›(๐‘โˆ’1)๎‚ญฮ˜โ‰ฅ0,๐‘โˆ’1๎‚€๐‘ข๐‘›(๐‘โˆ’1)๎‚+๎‚ฌ,๐‘ฆ๐ด๐‘ข๐‘›(๐‘โˆ’1),๐‘ฆโˆ’๐‘ข๐‘›(๐‘โˆ’1)๎‚ญ๎‚€๐‘ข+๐œ“(๐‘ฆ)โˆ’๐œ“๐‘›(๐‘โˆ’1)๎‚+1๐‘Ÿ๐‘›๎‚ฌ๐‘ฆโˆ’๐‘ข๐‘›(๐‘โˆ’1),๐ฝ๐‘ข๐‘›(๐‘โˆ’1)โˆ’๐ฝ๐‘ข๐‘›(๐‘โˆ’2)๎‚ญโ‹ฎฮ˜โ‰ฅ0,1๎‚€๐‘ข๐‘›(1)๎‚+๎‚ฌ,๐‘ฆ๐ด๐‘ข๐‘›(1),๐‘ฆโˆ’๐‘ข๐‘›(1)๎‚ญ๎‚€๐‘ข+๐œ“(๐‘ฆ)โˆ’๐œ“๐‘›(1)๎‚+1๐‘Ÿ๐‘›๎‚ฌ๐‘ฆโˆ’๐‘ข๐‘›(1),๐ฝ๐‘ข๐‘›(1)โˆ’๐ฝ๐‘ฆ๐‘›๎‚ญ๐ถโ‰ฅ0,๐‘›+1=๎‚†๐‘ฃโˆˆ๐ถ๐‘›๎‚€โˆถ๐œ™๐‘ฃ,๐‘ข๐‘›(๐‘)๎‚๎€ทโ‰ค๐œ™๐‘ฃ,๐‘ฅ๐‘›๎€ธ๎‚‡๐‘ฅ,โˆ€๐‘›โ‰ฅ0,๐‘›+1=ฮ ๐ถ๐‘›+1๐‘ฅ0,(3.51) where ๐‘Ÿ๐‘›โˆˆ[๐‘‘,โˆž) for some ๐‘‘>0 and for ๐‘–โ‰ฅ0,{๐›ผ๐‘›,๐‘–},{๐›ผ๐‘›} are sequences in [0,1] satisfying the following conditions: (a)โˆ‘โˆž๐‘–=0๐›ผ๐‘›,๐‘–=1, for all ๐‘›โ‰ฅ0;(b)liminf๐‘›โ†’โˆž๐›ผ๐‘›,0โ‹…๐›ผ๐‘›,๐‘–>0 for all ๐‘–โ‰ฅ1;(c)0<๐›ผโ‰ค๐›ผ๐‘›<1 for some ๐›ผโˆˆ(0,1).Then {๐‘ฅ๐‘›} converges strongly to ฮ ๐บ๐‘ฅ0.

Proof. Since {๐‘†๐‘–}โˆž๐‘–=1โˆถ๐ถโ†’๐ถ is an infinite family of closed quasi-๐œ™-nonexpansive mappings, it is an infinite family of closed and uniformly quasi-๐œ™-asymptotically nonexpansive mappings with sequence {๐‘˜๐‘›=1}. Hence ๐œ‰๐‘›=sup๐‘ขโˆˆ๐บ(๐‘˜๐‘›โˆ’1)๐œ™(๐‘ข,๐‘ฅ๐‘›)=0. Therefore, the conditions appearing in Theorem 3.1โ€”โ€œ๐บ is bounded subset in ๐ถโ€ and for each ๐‘–โ‰ฅ1,๐‘†๐‘– is uniformly ๐ฟ๐‘–-Lipschitz continuousโ€” are of no use here. In fact, by the same methods as given in the proofs of (3.15), (3.22), (3.31), (3.37), and (3.42), we can prove that ๐‘ฅ๐‘›โ†’๐‘, ๐‘ข๐‘›(๐‘–)โ†’๐‘, ๐‘ฆ๐‘›โ†’๐‘, and ๐‘†๐‘—๐‘ฅ๐‘›โ†’๐‘ (as ๐‘›โ†’โˆž), for each ๐‘—โ‰ฅ1, that is, โ‹‚๐‘โˆˆโˆž๐‘–=1๐น(๐‘†๐‘–)โ‹‚ฮฉ. Therefore, all conditions in Theorem 3.1 are satisfied. The conclusion of Theorem 3.2 is obtained from Theorem 3.1 immediately.

Remark 3.3. Theorems 3.1 and 3.2 improve and extend the corresponding results in [10โ€“12, 18].(a)For the framework of spaces, we extend the space from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space with the Kadec-Klee property (note that each uniformly convex Banach space must have Kadec-Klee property).(b)For the mappings, we extend the mappings from nonexpansive mappings, relatively nonexpansive mappings, or quasi-๐œ™-nonexpansive mapping to an infinite family of quasi-๐œ™-asymptotically nonexpansive mappings.(c)We extend a single generalized mixed equilibrium problem to a system of generalized mixed equilibrium problems.

References

  1. L.-C. Ceng and J.-C. Yao, โ€œA hybrid iterative scheme for mixed equilibrium problems and fixed point problems,โ€ Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 186โ€“201, 2008. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet
  2. F. E. Browder, โ€œExistence and approximation of solutions of nonlinear variational inequalities,โ€ Proceedings of the National Academy of Sciences of the United States of America, vol. 56, pp. 1080โ€“1086, 1966. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet
  3. W. Takahashi and K. Zembayashi, โ€œStrong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces,โ€ Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 1, pp. 45โ€“57, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet
  4. S. Takahashi and W. Takahashi, โ€œViscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces,โ€ Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 506โ€“515, 2007. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet
  5. X. L. Qin, M. Shang, and Y. F. Su, โ€œA general iterative method for equilibrium problems and fixed point problems in Hilbert spaces,โ€ Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 11, pp. 3897โ€“3909, 2008. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet
  6. S.-S. Chang, J. K. Kim, and X. R. Wang, โ€œModified block iterative algorithm for solving convex feasibility problems in Banach spaces,โ€ Journal of Inequalities and Applications, Article ID 869684, 14 pages, 2010. View at Google Scholar ยท View at Zentralblatt MATH
  7. S. S. Zhan, โ€œExistence and approximation of solutions of set-valued variational inclusions in Banach spaces,โ€ Applied Mathematics and Mechanics, vol. 30, no. 9, pp. 1105โ€“1112, 2009. View at Google Scholar
  8. Y. Su, H.-K. Xu, and X. Zhang, โ€œStrong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications,โ€ Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 12, pp. 3890โ€“3906, 2010. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  9. H. Y. Zhou, G. Gao, and B. Tan, โ€œConvergence theorems of a modified hybrid algorithm for a family of quasi-ϕ-asymptotically nonexpansive mappings,โ€ Journal of Applied Mathematics and Computing, vol. 32, no. 2, pp. 453โ€“464, 2010. View at Publisher ยท View at Google Scholar
  10. S. Plubtieng and K. Ungchittrakool, โ€œHybrid iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces,โ€ Fixed Point Theory and Applications, Article ID 583082, 19 pages, 2008. View at Google Scholar ยท View at Zentralblatt MATH
  11. S. Plubtieng and K. Ungchittrakool, โ€œStrong convergence theorems of block iterative methods for a finite family of relatively nonexpansive mappings in Banach spaces,โ€ Journal of Nonlinear and Convex Analysis, vol. 8, no. 3, pp. 431โ€“450, 2007. View at Google Scholar ยท View at Zentralblatt MATH
  12. S.-S. Chang, H. W. J. Lee, and C. K. Chan, โ€œA new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization,โ€ Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3307โ€“3319, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet
  13. I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht,The Netherlands, 1990.
  14. Y. I. Alber, โ€œMetric and generalized projection operators in Banach spaces: properties and applications,โ€ in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. Kartsatos, Ed., vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15โ€“50, Marcel Dekker, New York, NY, USA, 1996. View at Google Scholar ยท View at Zentralblatt MATH
  15. S. Kamimura and W. Takahashi, โ€œStrong convergence of a proximal-type algorithm in a Banach space,โ€ SIAM Journal on Optimization, vol. 13, no. 3, pp. 938โ€“945, 2002. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet
  16. S.-Y. Matsushita and W. Takahashi, โ€œA strong convergence theorem for relatively nonexpansive mappings in a Banach space,โ€ Journal of Approximation Theory, vol. 134, no. 2, pp. 257โ€“266, 2005. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet
  17. E. Blum and W. Oettli, โ€œFrom optimization and variational inequalities to equilibrium problems,โ€ The Mathematics Student, vol. 63, no. 1–4, pp. 123โ€“145, 1994. View at Google Scholar ยท View at Zentralblatt MATH
  18. A. Aleyner and S. Reich, โ€œBlock-iterative algorithms for solving convex feasibility problems in Hilbert and in Banach spaces,โ€ Journal of Mathematical Analysis and Applications, vol. 343, no. 1, pp. 427โ€“435, 2008. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet