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Journal of Applied Mathematics
Volumeย 2012, Article IDย 750732, 19 pages
http://dx.doi.org/10.1155/2012/750732
Research Article

A Modified Halpern-TypeIterative Method of a System of Equilibrium Problems and a Fixed Point for a Totally Quasi-๐“-Asymptotically Nonexpansive Mapping in a Banach Space

1Department of Mathematics and Statistics, Faculty of Science, Thaksin University (TSU), Pa Phayom, Phatthalung 93110, Thailand
2Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand

Received 17 February 2012; Accepted 28 April 2012

Academic Editor: Yeol Jeย Cho

Copyright ยฉ 2012 Preedaporn Kanjanasamranwong et al. 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 introduce the modified Halpern-type iterative method by the generalized f-projection operator for finding a common solution of fixed-point problem of a totally quasi-๐œ™-asymptotically nonexpansive mapping and a system of equilibrium problems in a uniform smooth and strictly convex Banach space with the Kadec-Klee property. Consequently, we prove the strong convergence for a common solution of above two sets. Our result presented in this paper generalize and improve the result of Chang et al., (2012), and some others.

1. Introduction

In 1953, Mann [1] introduced the following iteration process which is now known as Mann's iteration: ๐‘ฅ๐‘›+1=๐›ผ๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‡๐‘ฅ๐‘›,(1.1) where ๐‘‡ is nonexpansive, the initial guess element ๐‘ฅ1โˆˆ๐ถ is arbitrary, and {๐›ผ๐‘›} is a sequence in [0,1]. Mann iteration has been extensively investigated for nonexpansive mappings. In an infinite-dimensional Hilbert space, Mann iteration can conclude only weak conviergence (see [2, 3]).

Later, in 1967, Halpern [4] considered the following algorithm: ๐‘ฅ1โˆˆ๐ถ,๐‘ฅ๐‘›+1=๐›ผ๐‘›๐‘ฅ1+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‡๐‘ฅ๐‘›,โˆ€๐‘›โ‰ฅ0,(1.2) where ๐‘‡ is nonexpansive. He proved the strong convergence theorem of {๐‘ฅ๐‘›} to a fixed point of ๐‘‡ under some control condition {๐›ผ๐‘›}. Many authors improved and studied the result of Halpern [4] such as Qin et al. [5], Wang et al. [6], and reference therein.

In 2008-2009, Takahashi and Zembayashi [7, 8] studied the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in the framework of the Banach spaces.

On the other hand, Li et al. [9] introduced the following hybrid iterative scheme for approximation fixed points of relatively nonexpansive mapping using the generalized ๐‘“-projection operator in a uniformly smooth real Banach space which is also uniformly convex. They obtained strong convergence theorem for finding an element in the fixed point set of ๐‘‡.

Recently, Ofoedu and Shehu [10] extended algorithm of Li et al. [9] to prove a strong convergence theorem for a common solution of a system of equilibrium problem and the set of common fixed points of a pair of relatively quasi-nonexpansive mappings in the Banach spaces by using generalized ๐‘“-projection operator. Chang et al. [11] extended and improved Qin and Su [12] to obtain a strong convergence theorem for finding a common element of the set of solutions for a generalized equilibrium problem, the set of solutions for a variational inequality problem, and the set of common fixed points for a pair of relatively nonexpansive mappings in a Banach space.

Very recently, Chang et al. [13] extended the results of Qin et al. [5] and Wang et al. [6] to consider a modification to the Halpern-type iteration algorithm for a total quasi-๐œ™-asymptotically nonexpansive mapping to have the strong convergence under a limit condition only in the framework of Banach spaces.

The purpose of this paper is to be motivated and inspired by the works mentioned above, we introduce a modified Halpern-type iterative method by using the new hybrid projection algorithm of the generalized ๐‘“-projection operator for solving the common solution of fixed point for totally quasi-๐œ™-asymptoically nonexpansive mappings and the system of equilibrium problems in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. The results presented in this paper improve and extend the corresponding ones announced by many others.

2. Preliminaries and Definitions

Let ๐ธ be a real Banach space with dual ๐ธโˆ—, and let ๐ถ be a nonempty closed and convex subset of ๐ธ. Let {๐œƒ๐‘–}๐‘–โˆˆฮ“โˆถ๐ถร—๐ถโ†’โ„ be a bifunction, where ฮ“ is an arbitrary index set. The system of equilibrium problems is to find ๐‘ฅโˆˆ๐ถ such that ๐œƒ๐‘–(๐‘ฅ,๐‘ฆ)โ‰ฅ0,๐‘–โˆˆฮ“,โˆ€๐‘ฆโˆˆ๐ถ.(2.1) If ฮ“ is a singleton, then problem (2.1) reduces to the equilibrium problem, which is to find ๐‘ฅโˆˆ๐ถ such that ๐œƒ(๐‘ฅ,๐‘ฆ)โ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ.(2.2) A mapping ๐‘‡ from ๐ถ into itself is said to be nonexpansive if โ€–๐‘‡๐‘ฅโˆ’๐‘‡๐‘ฆโ€–โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ.(2.3)๐‘‡ is said to be asymptotically nonexpansive if there exists a sequence {๐‘˜๐‘›}โŠ‚[1,โˆž) with ๐‘˜๐‘›โ†’1 as ๐‘›โ†’โˆž such that โ€–๐‘‡๐‘›๐‘ฅโˆ’๐‘‡๐‘›๐‘ฆโ€–โ‰ค๐‘˜๐‘›โ€–๐‘ฅโˆ’๐‘ฆโ€–,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ.(2.4)๐‘‡ is said to be total asymptotically nonexpansive if there exist nonnegative real sequences ๐œˆ๐‘›, ๐œ‡๐‘› with ๐œˆ๐‘›โ†’0, ๐œ‡๐‘›โ†’0 as ๐‘›โ†’โˆž and a strictly increasing continuous function ๐œ‘โˆถโ„+โ†’โ„+ with ๐œ‘(0)=0 such that โ€–๐‘‡๐‘›๐‘ฅโˆ’๐‘‡๐‘›๐‘ฆโ€–โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–+๐œˆ๐‘›๐œ“(โ€–๐‘ฅโˆ’๐‘ฆโ€–)+๐œ‡๐‘›,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ,โˆ€๐‘›โ‰ฅ1.(2.5) A point ๐‘ฅโˆˆ๐ถ is a fixed point of ๐‘‡ provided ๐‘‡๐‘ฅ=๐‘ฅ. Denote by ๐น(๐‘‡) the fixed point set of ๐‘‡; that is, ๐น(๐‘‡)={๐‘ฅโˆˆ๐ถโˆถ๐‘‡๐‘ฅ=๐‘ฅ}. A point ๐‘ in ๐ถ is called an asymptotic fixed point of ๐‘‡ if ๐ถ contains a sequence {๐‘ฅ๐‘›} which converges weakly to ๐‘ such that lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘‡๐‘ฅ๐‘›โ€–=0. The asymptotic fixed point set of ๐‘‡ is denoted by ๎๐น(๐‘‡).

The normalized duality mapping ๐ฝโˆถ๐ธโ†’2๐ธโˆ— is defined by ๐ฝ(๐‘ฅ)={๐‘ฅโˆ—โˆˆ๐ธโˆ—โˆถโŸจ๐‘ฅ,๐‘ฅโˆ—โŸฉ=โ€–๐‘ฅโ€–2,โ€–๐‘ฅโˆ—โ€–=โ€–๐‘ฅโ€–}. If ๐ธ is a Hilbert space, then ๐ฝ=๐ผ, where ๐ผ is the identity mapping. Consider the functional defined by ๐œ™(๐‘ฅ,๐‘ฆ)=โ€–๐‘ฅโ€–2โˆ’2โŸจ๐‘ฅ,๐ฝ๐‘ฆโŸฉ+โ€–๐‘ฆโ€–2,(2.6) where ๐ฝ is the normalized duality mapping and โŸจโ‹…,โ‹…โŸฉ denote the duality pairing of ๐ธ and ๐ธโˆ—.

If ๐ธ is a Hilbert space, then ๐œ™(๐‘ฆ,๐‘ฅ)=โ€–๐‘ฆโˆ’๐‘ฅโ€–2. It is obvious from the definition of ๐œ™ that ()โ€–๐‘ฆโ€–โˆ’โ€–๐‘ฅโ€–2)โ‰ค๐œ™(๐‘ฆ,๐‘ฅ)โ‰ค(โ€–๐‘ฆโ€–+โ€–๐‘ฅโ€–2,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ธ.(2.7) A mapping ๐‘‡ from ๐ถ into itself is said to be ๐œ™-nonexpansive [14, 15] if ๐œ™(๐‘‡๐‘ฅ,๐‘‡๐‘ฆ)โ‰ค๐œ™(๐‘ฅ,๐‘ฆ),โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ.(2.8)๐‘‡ is said to be quasi-๐œ™-nonexpansive [14, 15] 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‘‡).(2.9)๐‘‡ is said to be asymptotically ๐œ™-nonexpansive [15] if there exists a sequence {๐‘˜๐‘›}โŠ‚[0,โˆž) with ๐‘˜๐‘›โ†’1 as ๐‘›โ†’โˆž such that ๐œ™(๐‘‡๐‘›๐‘ฅ,๐‘‡๐‘›๐‘ฆ)โ‰ค๐‘˜๐‘›๐œ™(๐‘ฅ,๐‘ฆ),โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ.(2.10)๐‘‡ is said to be quasi-๐œ™-asymptotically nonexpansive [15] if ๐น(๐‘‡)โ‰ โˆ… and there exists a sequence {๐‘˜๐‘›}โŠ‚[0,โˆž) with ๐‘˜๐‘›โ†’1 as ๐‘›โ†’โˆž such that ๐œ™(๐‘,๐‘‡๐‘›๐‘ฅ)โ‰ค๐‘˜๐‘›๐œ™(๐‘,๐‘ฅ),โˆ€๐‘ฅโˆˆ๐ถ,๐‘โˆˆ๐น(๐‘‡),โˆ€๐‘›โ‰ฅ1.(2.11)๐‘‡ is said to be totally quasi-๐œ™-asymptotically nonexpansive, if ๐น(๐‘‡)โ‰ โˆ… and there exist nonnegative real sequences ๐œˆ๐‘›, ๐œ‡๐‘› with ๐œˆ๐‘›โ†’0, ๐œ‡๐‘›โ†’0 as ๐‘›โ†’โˆž and a strictly increasing continuous function ๐œ‘โˆถโ„+โ†’โ„+ with ๐œ‘(0)=0 such that ๐œ™(๐‘,๐‘‡๐‘›๐‘ฅ)โ‰ค๐œ™(๐‘,๐‘ฅ)+๐œˆ๐‘›๐œ‘(๐œ™(๐‘,๐‘ฅ))+๐œ‡๐‘›,โˆ€๐‘›โ‰ฅ1,โˆ€๐‘ฅโˆˆ๐ถ,๐‘โˆˆ๐น(๐‘‡).(2.12) A mapping ๐‘‡ from ๐ถ into itself is said to be closed if for any sequence {๐‘ฅ๐‘›}โŠ‚๐ถ such that lim๐‘›โ†’โˆž๐‘ฅ๐‘›=๐‘ฅ0 and lim๐‘›โ†’โˆž๐‘‡๐‘ฅ๐‘›=๐‘ฆ0, then ๐‘‡๐‘ฅ0=๐‘ฆ0.

Alber [16] introduced the generalized projection ฮ ๐ถโˆถ๐ธโ†’๐ถ is a map that assigns to an arbitrary point ๐‘ฅโˆˆ๐ธ the minimum point of the functional ๐œ™(๐‘ฅ,๐‘ฆ); that is, ฮ ๐ถ๐‘ฅ=๐‘ฅ, where ๐‘ฅ is the solution of the minimization problem: ๐œ™๎€ท๎€ธ๐‘ฅ,๐‘ฅ=inf๐‘ฆโˆˆ๐ถ๐œ™(๐‘ฆ,๐‘ฅ).(2.13) The existence and uniqueness of the operator ฮ ๐ถ follows from the properties of the functional ๐œ™(๐‘ฆ,๐‘ฅ) and the strict monotonicity of the mapping ๐ฝ (see, e.g., [16โ€“20]). If ๐ธ is a Hilbert space, then ๐œ™(๐‘ฅ,๐‘ฆ)=โ€–๐‘ฅโˆ’๐‘ฆโ€–2 and ฮ ๐ถ becomes the metric projection ๐‘ƒ๐ถโˆถ๐ปโ†’๐ถ. If ๐ถ is a nonempty, closed, and convex subset of a Hilbert space ๐ป, then ๐‘ƒ๐ถ is nonexpansive. This fact actually characterizes Hilbert spaces, and consequently, it is not available in more general Banach spaces. Later, Wu and Huang [21] introduced a new generalized ๐‘“-projection operator in the Banach space. They extended the definition of the generalized projection operators and proved some properties of the generalized ๐‘“-projection operator. Next, we recall the concept of the generalized ๐‘“-projection operator. Let ๐บโˆถ๐ถร—๐ธโˆ—โ†’โ„โˆช{+โˆž} be a functional defined by ๐บ(๐‘ฆ,๐œ›)=โ€–๐‘ฆโ€–2โˆ’2โŸจ๐‘ฆ,๐œ›โŸฉ+โ€–๐œ›โ€–2+2๐œŒ๐‘“(๐‘ฆ),(2.14) where ๐‘ฆโˆˆ๐ถ,โ€‰โ€‰๐œ›โˆˆ๐ธโˆ—,๐œŒ is positive number, and ๐‘“โˆถ๐ถโ†’โ„โˆช{+โˆž} is proper, convex, and lower semicontinuous. From the definition of ๐บ, Wu and Huang [21] proved the following properties:(1)๐บ(๐‘ฆ,๐œ›) is convex and continuous with respect to ๐œ› when ๐‘ฆ is fixed; (2)๐บ(y,๐œ›) is convex and lower semicontinuous with respect to ๐‘ฆ when ๐œ› is fixed.

Definition 2.1. Let ๐ธ be a real Banach space with its dual ๐ธโˆ—. Let ๐ถ be a nonempty, closed, and convex subset of ๐ธ. We say that ๐œ‹๐‘“๐ถโˆถ๐ธโˆ—โ†’2๐ถ is a generalized ๐‘“-projection operator if ๐œ‹๐‘“๐ถ๎‚ป๐œ›=๐‘ขโˆˆ๐ถโˆถ๐บ(๐‘ข,๐œ›)=inf๐‘ฆโˆˆ๐ถ๐บ(๐‘ฆ,๐œ›),โˆ€๐œ›โˆˆ๐ธโˆ—๎‚ผ.(2.15)

A Banach space ๐ธ with norm โ€–โ‹…โ€– is called strictly convex if โ€–(๐‘ฅ+๐‘ฆ)/2โ€–<1 for all ๐‘ฅ,๐‘ฆโˆˆ๐ธ with โ€–๐‘ฅโ€–=โ€–๐‘ฆโ€–=1 and ๐‘ฅโ‰ ๐‘ฆ. Let ๐‘ˆ={๐‘ฅโˆˆ๐ธโˆถโ€–๐‘ฅโ€–=1} be the unit sphere of ๐ธ. A Banach space ๐ธ is called smooth if the limit lim๐‘กโ†’0((โ€–๐‘ฅ+๐‘ก๐‘ฆโ€–โˆ’โ€–๐‘ฅโ€–)/๐‘ก) exists for each ๐‘ฅ,๐‘ฆโˆˆ๐‘ˆ. It is also called uniformly smooth if the limit exists uniformly for all ๐‘ฅ,๐‘ฆโˆˆ๐‘ˆ. The modulus of smoothness of ๐ธ is the function ๐œŒ๐ธโˆถ[0,โˆž)โ†’[0,โˆž) defined by ๐œŒ๐ธ(๐‘ก)=sup{(โ€–๐‘ฅ+๐‘ฆโ€–+โ€–๐‘ฅโˆ’๐‘ฆโ€–)/2โˆ’1โˆถโ€–๐‘ฅโ€–=1,โ€–๐‘ฆโ€–โ‰ค๐‘ก}. The modulus of convexity of ๐ธ (see [22]) is the function ๐›ฟ๐ธโˆถ[0,2]โ†’[0,1] defined by ๐›ฟ๐ธ(๐œ€)=inf{1โˆ’โ€–(๐‘ฅ+๐‘ฆ)/2โ€–โˆถ๐‘ฅ,๐‘ฆโˆˆ๐ธ,โ€–๐‘ฅโ€–=โ€–๐‘ฆโ€–=1,โ€–๐‘ฅโˆ’๐‘ฆโ€–โ‰ฅ๐œ€}. In this paper 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.

Remark 2.2. The basic properties of ๐ธ, ๐ธโˆ—, ๐ฝ, and ๐ฝโˆ’1 (see [18]) are as follows.(i)If ๐ธ is an arbitrary Banach space, then ๐ฝ is monotone and bounded. (ii)If ๐ธ is a strictly convex, then ๐ฝ is strictly monotone.(iii)If ๐ธ is a smooth, then ๐ฝ is single valued and semicontinuous. (iv)If ๐ธ is uniformly smooth, then ๐ฝ is uniformly norm-to-norm continuous on each bounded subset of ๐ธ. (v)If ๐ธ is reflexive smooth and strictly convex, then the normalized duality mapping ๐ฝ is single valued, one-to-one, and onto. (vi)If ๐ธ is a reflexive strictly convex and smooth Banach space and ๐ฝ is the duality mapping from ๐ธ into ๐ธโˆ—, then ๐ฝโˆ’1 is also single valued, bijective, and is also the duality mapping from ๐ธโˆ— into ๐ธ, and thus ๐ฝ๐ฝโˆ’1=๐ผ๐ธโˆ— and ๐ฝโˆ’1๐ฝ=๐ผ๐ธ. (vii)If ๐ธ is uniformly smooth, then ๐ธ is smooth and reflexive.(viii)๐ธ is uniformly smooth if and only if ๐ธโˆ— is uniformly convex.(ix)If ๐ธ is a reflexive and strictly convex Banach space, then ๐ฝโˆ’1 is norm-weak*-continuous.

Remark 2.3. If ๐ธ is a reflexive, strictly convex, and smooth Banach space, then ๐œ™(๐‘ฅ,๐‘ฆ)=0, if and only if ๐‘ฅ=๐‘ฆ. It is sufficient to show that if ๐œ™(๐‘ฅ,๐‘ฆ)=0 then ๐‘ฅ=๐‘ฆ. From (2.6), we have โ€–๐‘ฅโ€–=โ€–๐‘ฆโ€–. This implies that โŸจ๐‘ฅ,๐ฝ๐‘ฆโŸฉ=โ€–๐‘ฅโ€–2=โ€–๐ฝ๐‘ฆโ€–2. From the definition of ๐ฝ, one has ๐ฝ๐‘ฅ=๐ฝ๐‘ฆ. Therefore, we have ๐‘ฅ=๐‘ฆ (see [18, 20, 23] for more details).

Recall that a Banach space ๐ธ has the Kadec-Klee property [18, 20, 24], if for any sequence {๐‘ฅ๐‘›}โŠ‚๐ธ and ๐‘ฅโˆˆ๐ธ with ๐‘ฅ๐‘›โ‡€๐‘ฅ and โ€–๐‘ฅ๐‘›โ€–โ†’โ€–๐‘ฅโ€–, then โ€–๐‘ฅ๐‘›โˆ’๐‘ฅโ€–โ†’0 as ๐‘›โ†’โˆž. It is well known that if ๐ธ is a uniformly convex Banach space, then ๐ธ has the Kadec-Klee property.

We also need the following lemmas for the proof of our main results.

Lemma 2.4 (see Change et al. [25]). Let ๐ถ be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space ๐ธ with the Kadec-Klee property. Let ๐‘‡โˆถ๐ถโ†’๐ถ be a closed and total quasi-๐œ™-asymptotically nonexpansive mapping with nonnegative real sequence ๐œˆ๐‘› and ๐œ‡๐‘› with ๐œˆ๐‘›โ†’0, ๐œ‡๐‘›โ†’0 as ๐‘›โ†’โˆž and a strictly increasing continuous function ๐œโˆถโ„+โ†’โ„+ with ๐œ(0)=0. If ๐œ‡1=0, then the fixed point set ๐น(๐‘‡) is a closed convex subset of ๐ถ.

Lemma 2.5 (see Wu and Hung [21]). Let ๐ธ be a real reflexive Banach space with its dual ๐ธโˆ— and ๐ถ a nonempty, closed, and convex subset of ๐ธ. The following statement hold:(1)๐œ‹๐‘“๐ถ๐œ› is a nonempty, closed and convex subset of ๐ถ for all ๐œ›โˆˆ๐ธโˆ—; (2)if ๐ธ is smooth, then for all ๐œ›โˆˆ๐ธโˆ—, ๐‘ฅโˆˆ๐œ‹๐‘“๐ถ๐œ› if and only if โŸจ๐‘ฅโˆ’๐‘ฆ,๐œ›โˆ’๐ฝ๐‘ฅโŸฉ+๐œŒ๐‘“(๐‘ฆ)โˆ’๐œŒ๐‘“(๐‘ฅ)โ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ;(2.16)(3)if ๐ธ is strictly convex and ๐‘“โˆถ๐ถโ†’โ„โˆช{+โˆž} is positive homogeneous (i.e., ๐‘“(๐‘ก๐‘ฅ)=๐‘ก๐‘“(๐‘ฅ) for all ๐‘ก>0 such that ๐‘ก๐‘ฅโˆˆ๐ถ where ๐‘ฅโˆˆ๐ถ), then ๐œ‹๐‘“๐ถ๐œ› is single-valued mapping.

Lemma 2.6 (see Fan et al. [26]). Let ๐ธ be a real reflexive Banach space with its dual ๐ธโˆ— and ๐ถ be a nonempty, closed and convex subset of ๐ธ. If ๐ธ is strictly convex, then ๐œ‹๐‘“๐ถ๐œ› is single valued.

Recall that ๐ฝ is single-valued mapping when ๐ธ is a smooth Banach space. There exists a unique element ๐œ›โˆˆ๐ธโˆ— such that ๐œ›=๐ฝ๐‘ฅ where ๐‘ฅโˆˆ๐ธ. This substitution in (2.14) gives ๐บ(๐‘ฆ,๐ฝ๐‘ฅ)=โ€–๐‘ฆโ€–2โˆ’2โŸจ๐‘ฆ,๐ฝ๐‘ฅโŸฉ+โ€–๐‘ฅโ€–2+2๐œŒ๐‘“(๐‘ฆ).(2.17)

Now we consider the second generalized ๐‘“ projection operator in Banach space (see [9]).

Definition 2.7. Let ๐ธ be a real smooth Banach space, and let ๐ถ be a nonempty, closed, and convex subset of ๐ธ. We say that ฮ ๐‘“๐ถโˆถ๐ธโ†’2๐ถ is generalized ๐‘“-projection operator if ฮ ๐‘“๐ถ๎‚ป๐‘ฅ=๐‘ขโˆˆ๐ถโˆถ๐บ(๐‘ข,๐ฝ๐‘ฅ)=inf๐‘ฆโˆˆ๐ถ๎‚ผ๐บ(๐‘ฆ,๐ฝ๐‘ฅ),โˆ€๐‘ฅโˆˆ๐ธ.(2.18)

Lemma 2.8 (see Deimling [27]). Let ๐ธ be a Banach space, and let ๐‘“โˆถ๐ธโ†’โ„โˆช{+โˆž} be a lower semicontinuous convex function. Then there exist ๐‘ฅโˆ—โˆˆ๐ธโˆ— and ๐›ผโˆˆโ„ such that ๐‘“(๐‘ฅ)โ‰ฅโŸจ๐‘ฅ,๐‘ฅโˆ—โŸฉ+๐›ผ,โˆ€๐‘ฅโˆˆ๐ธ.(2.19)

Lemma 2.9 (see Li et al. [9]). Let ๐ธ be a reflexive smooth Banach space, and let ๐ถ be a nonempty, closed, and convex subset of ๐ธ. The following statements hold: (1)ฮ ๐‘“๐ถ๐‘ฅ is nonempty, closed and convex subset of ๐ถ for all ๐‘ฅโˆˆ๐ธ; (2)for all ๐‘ฅโˆˆ๐ธ, ฬ‚๐‘ฅโˆˆฮ ๐‘“๐ถ๐‘ฅ if and only if โŸจฬ‚๐‘ฅโˆ’๐‘ฆ,๐ฝ๐‘ฅโˆ’๐ฝฬ‚๐‘ฅโŸฉ+๐œŒ๐‘“(๐‘ฆ)โˆ’๐œŒ๐‘“(ฬ‚๐‘ฅ)โ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ;(2.20)(3)if ๐ธ is strictly convex, then ฮ ๐‘“๐ถ is single-valued mapping.

Lemma 2.10 (see Li et al. [9]). Let ๐ธ be a real reflexive smooth Banach space, let ๐ถ be a nonempty, closed, and convex subset of ๐ธ, ๐‘ฅโˆˆ๐ธ, and let ฬ‚๐‘ฅโˆˆฮ ๐‘“๐ถ๐‘ฅ. Then ๐œ™(๐‘ฆ,ฬ‚๐‘ฅ)+๐บ(ฬ‚๐‘ฅ,๐ฝ๐‘ฅ)โ‰ค๐บ(๐‘ฆ,๐ฝ๐‘ฅ),โˆ€๐‘ฆโˆˆ๐ถ.(2.21)

Remark 2.11. Let ๐ธ be a uniformly convex and uniformly smooth Banach space and ๐‘“(๐‘ฅ)=0 for all ๐‘ฅโˆˆ๐ธ, then Lemma 2.10 reduces to the property of the generalized projection operator considered by Alber [16].

If ๐‘“(๐‘ฆ)โ‰ฅ0 for all ๐‘ฆโˆˆ๐ถ and ๐‘“(0)=0, then the definition of totally quasi-๐œ™-asymptotically nonexpansive ๐‘‡ is equivalent to if ๐น(๐‘‡)โ‰ โˆ…, and there exist nonnegative real sequences ๐œˆ๐‘›, ๐œ‡๐‘› with ๐œˆ๐‘›โ†’0, ๐œ‡๐‘›โ†’0 as ๐‘›โ†’โˆž and a strictly increasing continuous function ๐œโˆถโ„+โ†’โ„+ with ๐œ(0)=0 such that ๐บ(๐‘,๐‘‡๐‘›๐‘ฅ)โ‰ค๐บ(๐‘,๐‘ฅ)+๐œˆ๐‘›๐œ๐บ(๐‘,๐‘ฅ)+๐œ‡๐‘›,โˆ€๐‘›โ‰ฅ1,โˆ€๐‘ฅโˆˆ๐ถ,๐‘โˆˆ๐น(๐‘‡).(2.22)

For solving the equilibrium problem for a bifunction ๐œƒโˆถ๐ถร—๐ถโ†’โ„, let us assume that ๐œƒ satisfies the following conditions: (A1)๐œƒ(๐‘ฅ,๐‘ฅ)=0 for all ๐‘ฅโˆˆ๐ถ; (A2)๐œƒ is monotone; that is, ๐œƒ(๐‘ฅ,๐‘ฆ)+๐œƒ(๐‘ฆ,๐‘ฅ)โ‰ค0 for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ; (A3)for each ๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐ถ, lim๐‘กโ†“0๐œƒ(๐‘ก๐‘ง+(1โˆ’๐‘ก)๐‘ฅ,๐‘ฆ)โ‰ค๐œƒ(๐‘ฅ,๐‘ฆ);(2.23)(A4)for each ๐‘ฅโˆˆ๐ถ, ๐‘ฆโ†ฆ๐œƒ(๐‘ฅ,๐‘ฆ) is convex and lower semicontinuous.

For example, let ๐ด be a continuous and monotone operator of ๐ถ into ๐ธโˆ— and define ๐œƒ(๐‘ฅ,๐‘ฆ)=โŸจ๐ด๐‘ฅ,๐‘ฆโˆ’๐‘ฅโŸฉ,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ.(2.24) Then, ๐œƒ satisfies (A1)โ€“(A4). The following result is in Blum and Oettli [28].

Lemma 2.12 (see Blum and Oettli [28]). Let ๐ถ be a closed convex subset of a smooth, strictly convex, and reflexive Banach space ๐ธ, let ๐œƒ be a bifunction from ๐ถร—๐ถ to โ„ satisfying (A1)โ€“(A4), and let ๐‘Ÿ>0 and ๐‘ฅโˆˆ๐ธ. Then, there exists ๐‘งโˆˆ๐ถ such that 1๐œƒ(๐‘ง,๐‘ฆ)+๐‘ŸโŸจ๐‘ฆโˆ’๐‘ง,๐ฝ๐‘งโˆ’๐ฝ๐‘ฅโŸฉโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ.(2.25)

Lemma 2.13 (see Takahashi and Zembayashi [8]). Let ๐ถ be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space ๐ธ, and let ๐œƒ be a bifunction from ๐ถร—๐ถ to โ„ satisfying conditions (A1)โ€“(A4). For all ๐‘Ÿ>0 and ๐‘ฅโˆˆ๐ธ, define a mapping ๐‘‡๐œƒ๐‘Ÿโˆถ๐ธโ†’๐ถ as follows: ๐‘‡๐œƒ๐‘Ÿ๎‚†1๐‘ฅ=๐‘งโˆˆ๐ถโˆถ๐œƒ(๐‘ง,๐‘ฆ)+๐‘Ÿ๎‚‡.โŸจ๐‘ฆโˆ’๐‘ง,๐ฝ๐‘งโˆ’๐ฝ๐‘ฅโŸฉโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ(2.26) Then the following hold: (1)๐‘‡๐œƒ๐‘Ÿ is single-valued; (2)๐‘‡๐œƒ๐‘Ÿ is a firmly nonexpansive-type mapping [29]; that is, for all ๐‘ฅ,๐‘ฆโˆˆ๐ธ, โŸจ๐‘‡๐œƒ๐‘Ÿ๐‘ฅโˆ’๐‘‡๐œƒ๐‘Ÿ๐‘ฆ,๐ฝ๐‘‡๐œƒ๐‘Ÿ๐‘ฅโˆ’๐ฝ๐‘‡๐œƒ๐‘Ÿ๐‘ฆโŸฉโ‰คโŸจ๐‘‡๐œƒ๐‘Ÿ๐‘ฅโˆ’๐‘‡๐œƒ๐‘Ÿ๐‘ฆ,๐ฝ๐‘ฅโˆ’๐ฝ๐‘ฆโŸฉ;(2.27)(3)๐น(๐‘‡๐œƒ๐‘Ÿ)=EP(๐œƒ); (4)EP(๐œƒ) is closed and convex.

Lemma 2.14 (see Takahashi and Zembayashi [8]). Let ๐ถ be a closed convex subset of a smooth, strictly convex, and reflexive Banach space ๐ธ, let ๐œƒ be a bifunction from ๐ถร—๐ถ to โ„ satisfying (A1)โ€“(A4), and let ๐‘Ÿ>0. Then, for ๐‘ฅโˆˆ๐ธ and ๐‘žโˆˆ๐น(๐‘‡๐œƒ๐‘Ÿ), ๐œ™๎€ท๐‘ž,๐‘‡๐œƒ๐‘Ÿ๐‘ฅ๎€ธ๎€ท๐‘‡+๐œ™๐œƒ๐‘Ÿ๎€ธ๐‘ฅ,๐‘ฅโ‰ค๐œ™(๐‘ž,๐‘ฅ).(2.28)

3. Main Result

Theorem 3.1. Let ๐ถ be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space ๐ธ with the Kadec-Klee property. For each ๐‘—=1,2,โ€ฆ,๐‘š, let ๐œƒj be a bifunction from ๐ถร—๐ถ to โ„ which satisfies conditions (A1)โ€“(A4). Let ๐‘†โˆถ๐ถโ†’๐ถ be a closed totally quasi-๐œ™-asymptotically nonexpansive mappings with nonnegative real sequences ๐œˆ๐‘›, ๐œ‡๐‘› with ๐œˆ๐‘›โ†’0, ๐œ‡๐‘›โ†’0 as ๐‘›โ†’โˆž, and a strictly increasing continuous function ๐œ“โˆถโ„+โ†’โ„+ with ๐œ“(0)=0. Let ๐‘“โˆถ๐ธโ†’โ„ be a convex and lower semicontinuous function with ๐ถโŠ‚int(๐ท(๐‘“)) such that ๐‘“(๐‘ฅ)โ‰ฅ0 for all ๐‘ฅโˆˆ๐ถ and ๐‘“(0)=0. Assume that โ„ฑโˆถ=๐น(๐‘†)โˆฉ(โˆฉ๐‘š๐‘—=1EP(๐œƒ๐‘—))โ‰ โˆ…. For an initial point ๐‘ฅ1โˆˆ๐ธ and ๐ถ1=๐ถ, one define the sequence {๐‘ฅ๐‘›} by ๐‘ข๐‘›=๐‘‡๐œƒ๐‘š๐‘Ÿ๐‘š,๐‘›๐‘‡๐œƒ๐‘šโˆ’1๐‘Ÿ๐‘šโˆ’1,๐‘›๐‘‡๐œƒ๐‘šโˆ’2๐‘Ÿ๐‘šโˆ’2,๐‘›โ‹ฏ๐‘‡๐œƒ1๐‘Ÿ1,๐‘›๐‘ฅ๐‘›,๐‘ง๐‘›=๐ฝโˆ’1๎€ท๐›ผ๐‘›๐ฝ๐‘ฅ1+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐ฝ๐‘†๐‘›๐‘ข๐‘›๎€ธ,๐ถ๐‘›+1=๎€ฝ๐‘ฃโˆˆ๐ถ๐‘›๎€ทโˆถ๐บ๐‘ฃ,๐ฝ๐‘ง๐‘›๎€ธ๎€ทโ‰ค๐บ๐‘ฃ,๐ฝ๐‘ข๐‘›๎€ธ๎€ทโ‰ค๐บ๐‘ฃ,๐ฝ๐‘ฅ1๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐บ๎€ท๐‘ฃ,๐ฝ๐‘ฅ๐‘›๎€ธ+๐œ๐‘›๎€พ,๐‘ฅ๐‘›+1=ฮ ๐‘“๐ถ๐‘›+1๐‘ฅ1,๐‘›โˆˆโ„•,(3.1) where {๐›ผ๐‘›} is a sequence in [0,1], ๐œ๐‘›=๐œˆ๐‘›sup๐‘žโˆˆโ„ฑ๐œ“(๐บ(๐‘ž,๐‘ฅ๐‘›))+๐œ‡๐‘› and {๐‘Ÿ๐‘—,๐‘›}โŠ‚[๐‘‘,โˆž) for some ๐‘‘>0. If lim๐‘›โ†’โˆž๐›ผ๐‘›=0, then {๐‘ฅ๐‘›} converges strongly to ฮ ๐‘“โ„ฑ๐‘ฅ0.

Proof. We split the proof into four steps.

Step 1. First, we show that ๐ถ๐‘› is closed and convex for all ๐‘›โˆˆโ„•.
Clearly ๐ถ1=๐ถ is closed and convex. Suppose that ๐ถ๐‘› is closed and convex for all ๐‘›โˆˆโ„•. For any ๐œโˆˆ๐ถ๐‘›, we know that ๐บ(๐œ,๐ฝ๐‘ง๐‘›)โ‰ค๐บ(๐œ,๐ฝ๐‘ฅ๐‘›)+๐œ๐‘› is equivalent to 2โŸจ๐œ,๐ฝ๐‘ฅ๐‘›โˆ’๐ฝ๐‘ง๐‘›โ€–โ€–๐‘ฅโŸฉโ‰ค๐‘›โ€–โ€–2โˆ’โ€–โ€–๐‘ง๐‘›โ€–โ€–2+๐œ๐‘›.(3.2) So, ๐ถ๐‘›+1 is closed and convex. Hence by induction ๐ถ๐‘› is closed and convex for all ๐‘›โ‰ฅ1.

Step 2. We will show that the sequence {๐‘ฅ๐‘›} is well defined.
We will show by induction that โ„ฑโŠ‚๐ถ๐‘› for all ๐‘›โˆˆโ„•. It is obvious that โ„ฑโŠ‚๐ถ1=๐ถ. Suppose that โ„ฑโŠ‚๐ถ๐‘› for some ๐‘›โˆˆโ„•. Let ๐‘žโˆˆโ„ฑ, put ๐‘ข๐‘›=๐พ๐‘š๐‘›๐‘ฅ๐‘›,๐พ๐‘—๐‘›=๐‘‡๐œƒ๐‘—๐‘Ÿ๐‘—,๐‘›๐‘‡๐œƒ๐‘—โˆ’1๐‘Ÿ๐‘—โˆ’1,๐‘›โ€ฆ๐‘‡๐œƒ1๐‘Ÿ1,๐‘› for all ๐‘—=1,2,3,โ€ฆ,๐‘š, ๐พ0๐‘›=๐ผ, we have that ๐บ๎€ท๐‘ž,๐ฝ๐‘ข๐‘›๎€ธ๎€ท=&๐บ๐‘ž,๐ฝ๐พ๐‘š๐‘›๐‘ฅ๐‘›๎€ธ๎€ทโ‰ค&๐บ๐‘ž,๐ฝ๐‘ฅ๐‘›๎€ธ.(3.3) From (3.3) and ๐‘† which is a totally quasi-๐œ™ asymptotically nonexpansive mappings, it follows that ๐บ๎€ท๐‘ž,๐ฝ๐‘ง๐‘›๎€ธ๎€ท๎€ท๐›ผ=๐บ๐‘ž,๐‘›๐ฝ๐‘ฅ1+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐ฝ๐‘†๐‘›๐‘ข๐‘›๎€ธ๎€ธ=โ€–๐‘žโ€–2โˆ’2๐›ผ๐‘›โŸจ๐‘ž,๐ฝ๐‘ฅ1๎€ทโŸฉโˆ’21โˆ’๐›ผ๐‘›๎€ธโŸจ๐‘ž,๐ฝ๐‘†๐‘›๐‘ข๐‘›โŸฉ+โ€–โ€–๐›ผ๐‘›๐ฝ๐‘ฅ1+(1โˆ’๐›ผ๐‘›)๐ฝ๐‘†๐‘›๐‘ข๐‘›โ€–โ€–2+2๐œŒ๐‘“(๐‘ž)โ‰คโ€–๐‘žโ€–2โˆ’2๐›ผ๐‘›โŸจ๐‘ž,๐ฝ๐‘ฅ1๎€ทโŸฉโˆ’21โˆ’๐›ผ๐‘›๎€ธโŸจ๐‘ž,๐ฝ๐‘†๐‘›๐‘ข๐‘›โŸฉ+๐›ผ๐‘›โ€–โ€–๐ฝ๐‘ฅ1โ€–โ€–2+๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐ฝ๐‘†๐‘›๐‘ข๐‘›โ€–โ€–2+2๐œŒ๐‘“(๐‘ž)=๐›ผ๐‘›๐บ๎€ท๐‘ž,๐ฝ๐‘ฅ1๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐บ๎€ท๐‘ž,๐ฝ๐‘†๐‘›๐‘ข๐‘›๎€ธโ‰ค๐›ผ๐‘›๐บ๎€ท๐‘ž,๐ฝ๐‘ฅ1๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๐บ๎€ท๎€ธ๎€ท๐‘ž,๐ฝ๐‘ข๐‘›๎€ธ+๐œˆ๐‘›๐œ“๎€ท๐บ๎€ท๐‘ž,๐ฝ๐‘ข๐‘›๎€ธ๎€ธ+๐œ‡๐‘›๎€ธโ‰ค๐›ผ๐‘›๐บ๎€ท๐‘ž,๐ฝ๐‘ฅ1๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐บ๎€ท๐‘ž,๐ฝ๐‘ฅ๐‘›๎€ธ+๐œˆ๐‘›sup๐‘žโˆˆโ„ฑ๐œ“๎€ท๐บ๎€ท๐‘ž,๐ฝ๐‘ฅ๐‘›๎€ธ๎€ธ+๐œ‡๐‘›=๐›ผ๐‘›๐บ๎€ท๐‘ž,๐ฝ๐‘ฅ1๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐บ๎€ท๐‘ž,๐ฝ๐‘ฅ๐‘›๎€ธ+๐œ๐‘›.(3.4)
This shows that ๐‘žโˆˆ๐ถ๐‘›+1 which implies that โ„ฑโŠ‚๐ถ๐‘›+1, and hence, โ„ฑโŠ‚๐ถ๐‘› for all ๐‘›โˆˆโ„•. and the sequence {๐‘ฅ๐‘›} is well defined. From ๐‘ฅ๐‘›=ฮ ๐‘“๐ถ๐‘›๐‘ฅ1, we see that โŸจ๐‘ฅ๐‘›โˆ’๐‘ž,๐ฝ๐‘ฅ1โˆ’๐ฝ๐‘ฅ๐‘›๎€ท๐‘ฅโŸฉ+๐œŒ๐‘“(๐‘ž)โˆ’๐œŒ๐‘“๐‘›๎€ธโ‰ฅ0,โˆ€๐‘žโˆˆ๐ถ๐‘›.(3.5) Since โ„ฑโŠ‚๐ถ๐‘› for each ๐‘›โˆˆโ„•, we arrive at โŸจ๐‘ฅ๐‘›โˆ’๐‘ž,๐ฝ๐‘ฅ1โˆ’๐ฝ๐‘ฅ๐‘›๎€ท๐‘ฅโŸฉ+๐œŒ๐‘“(๐‘ž)โˆ’๐œŒ๐‘“๐‘›๎€ธโ‰ฅ0,โˆ€๐‘žโˆˆโ„ฑ.(3.6) Hence, the sequence {๐‘ฅ๐‘›} is well defined.

Step 3. We will show that ๐‘ฅ๐‘›โ†’๐‘โˆˆโ„ฑโˆถ=๐น(๐‘†)โˆฉ(โˆฉ๐‘š๐‘—=1EP(๐œƒ๐‘—)).
Let ๐‘“โˆถ๐ธโ†’โ„ is convex and lower semicontinuous function, follows from Lemma 2.8, there exist ๐‘ฅโˆ—โˆˆ๐ธโˆ— and ๐›ผโˆˆโ„ such that ๐‘“(๐‘ฆ)โ‰ฅโŸจ๐‘ฆ,๐‘ฅโˆ—โŸฉ+๐›ผ,โˆ€๐‘ฆโˆˆ๐ธ.(3.7) Since ๐‘ฅ๐‘›โˆˆ๐ถ๐‘›โŠ‚๐ธ, it follows that ๐บ๎€ท๐‘ฅ๐‘›,๐ฝ๐‘ฅ1๎€ธ=โ€–โ€–๐‘ฅ๐‘›โ€–โ€–2โˆ’2โŸจ๐‘ฅ๐‘›,๐ฝ๐‘ฅ1โ€–โ€–๐‘ฅโŸฉ+1โ€–โ€–2๎€ท๐‘ฅ+2๐œŒ๐‘“๐‘›๎€ธโ‰ฅโ€–โ€–๐‘ฅ๐‘›โ€–โ€–2โˆ’2โŸจ๐‘ฅ๐‘›,๐ฝ๐‘ฅ1โ€–โ€–๐‘ฅโŸฉ+1โ€–โ€–2+2๐œŒโŸจ๐‘ฅ๐‘›,๐‘ฅโˆ—=โ€–โ€–๐‘ฅโŸฉ+2๐œŒ๐›ผ๐‘›โ€–โ€–2โˆ’2โŸจ๐‘ฅ๐‘›,๐ฝ๐‘ฅ1โˆ’๐œŒ๐‘ฅโˆ—โ€–โ€–๐‘ฅโŸฉ+1โ€–โ€–2โ‰ฅโ€–โ€–๐‘ฅ+2๐œŒ๐›ผ๐‘›โ€–โ€–2โ€–โ€–๐‘ฅโˆ’2๐‘›โ€–โ€–โ€–โ€–๐ฝ๐‘ฅ1โˆ’๐œŒ๐‘ฅโˆ—โ€–โ€–+โ€–โ€–๐‘ฅ1โ€–โ€–2=๎€ทโ€–โ€–๐‘ฅ+2๐œŒ๐›ผ๐‘›โ€–โ€–โˆ’โ€–โ€–๐ฝ๐‘ฅ1โˆ’๐œŒ๐‘ฅโˆ—โ€–โ€–๎€ธ2+โ€–โ€–๐‘ฅ1โ€–โ€–2โˆ’โ€–โ€–๐ฝ๐‘ฅ1โˆ’๐œŒ๐‘ฅโˆ—โ€–โ€–2+2๐œŒ๐›ผ.(3.8) For ๐‘žโˆˆโ„ฑ and ๐‘ฅ๐‘›=ฮ ๐‘“๐ถ๐‘›๐‘ฅ1, we have ๐บ๎€ท๐‘ž,๐ฝ๐‘ฅ1๎€ธ๎€ท๐‘ฅโ‰ฅ๐บ๐‘›,๐ฝ๐‘ฅ1๎€ธโ‰ฅ๎€ทโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’โ€–โ€–๐ฝ๐‘ฅ1โˆ’๐œŒ๐‘ฅโˆ—โ€–โ€–๎€ธ2+โ€–โ€–๐‘ฅ1โ€–โ€–2โˆ’โ€–โ€–๐ฝ๐‘ฅ1โˆ’๐œŒ๐‘ฅโˆ—โ€–โ€–2+2๐œŒ๐›ผ.(3.9) This shows that {๐‘ฅ๐‘›} is bounded and so is {๐บ(๐‘ฅ๐‘›,๐ฝ๐‘ฅ1)}. From the fact that ๐‘ฅ๐‘›+1=ฮ ๐‘“๐ถ๐‘›+1๐‘ฅ1โˆˆ๐ถ๐‘›+1โŠ‚๐ถ๐‘› and ๐‘ฅ๐‘›=ฮ ๐‘“๐ถ๐‘›๐‘ฅ1, it follows from Lemma 2.10 that โ€–โ€–๐‘ฅ0โ‰ค(๐‘›+1โˆ’โ€–โ€–๐‘ฅ๐‘›โ€–)2๎€ท๐‘ฅโ‰ค๐œ™๐‘›+1,๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅโ‰ค๐บ๐‘›+1,๐ฝ๐‘ฅ1๎€ธ๎€ท๐‘ฅโˆ’๐บ๐‘›,๐ฝ๐‘ฅ1๎€ธ.(3.10) That is, {๐บ(๐‘ฅ๐‘›,๐ฝ๐‘ฅ1)} is nondecreasing. Hence, we obtain that lim๐‘›โ†’โˆž๐บ(๐‘ฅ๐‘›,๐ฝ๐‘ฅ1) exists. Taking ๐‘›โ†’โˆž, we obtain lim๐‘›โ†’โˆž๐œ™๎€ท๐‘ฅ๐‘›+1,๐‘ฅ๐‘›๎€ธ=0.(3.11) Since ๐ธ is reflexive, {๐‘ฅ๐‘›} is bounded, and ๐ถ๐‘› is closed and convex for all ๐‘›โˆˆโ„•. Without loss of generality, we can assume that ๐‘ฅ๐‘›โ‡€๐‘โˆˆ๐ถ๐‘›. From the fact that ๐‘ฅ๐‘›=ฮ ๐‘“๐ถ๐‘›๐‘ฅ1, we get that ๐บ๎€ท๐‘ฅ๐‘›,๐ฝ๐‘ฅ1๎€ธ๎€ทโ‰ค๐บ๐‘,๐ฝ๐‘ฅ1๎€ธ,โˆ€๐‘›โˆˆโ„•.(3.12) Since ๐‘“ is convex and lower semicontinuous, we have liminf๐‘›โ†’โˆž๐บ๎€ท๐‘ฅ๐‘›,๐ฝ๐‘ฅ1๎€ธ=liminf๐‘›โ†’โˆž๎‚†โ€–โ€–๐‘ฅ๐‘›โ€–โ€–2โˆ’2โŸจ๐‘ฅ๐‘›,๐ฝ๐‘ฅ1โ€–โ€–๐‘ฅโŸฉ+1โ€–โ€–2๎€ท๐‘ฅ+2๐œŒ๐‘“๐‘›๎€ธ๎‚‡โ‰ฅโ€–๐‘โ€–2โˆ’2โŸจ๐‘,๐ฝ๐‘ฅ1โ€–โ€–๐‘ฅโŸฉ+1โ€–โ€–2๎€ท๐‘ฅ+2๐œŒ๐‘“(๐‘)=๐บ๐‘›,๐ฝ๐‘ฅ1๎€ธ.(3.13) By (3.12) and (3.13), we get ๐บ๎€ท๐‘,๐ฝ๐‘ฅ1๎€ธโ‰คliminf๐‘›โ†’โˆž๐บ๎€ท๐‘ฅ๐‘›,๐ฝ๐‘ฅ1๎€ธโ‰คlimsup๐‘›โ†’โˆž๐บ๎€ท๐‘ฅ๐‘›,๐ฝ๐‘ฅ1๎€ธ๎€ทโ‰ค๐บ๐‘,๐ฝ๐‘ฅ1๎€ธ.(3.14) That is, lim๐‘›โ†’โˆž๐บ(๐‘ฅ๐‘›,๐ฝ๐‘ฅ1)=๐บ(๐‘,๐ฝ๐‘ฅ1); this implies that โ€–๐‘ฅ๐‘›โ€–โ†’โ€–๐‘โ€–; by virtue of the Kadec-Klee property of ๐ธ, we obtain that lim๐‘›โ†’โˆž๐‘ฅ๐‘›=๐‘.(3.15) We also have lim๐‘›โ†’โˆž๐‘ฅ๐‘›+1=๐‘.(3.16) From (3.15), we get that lim๐‘›โ†’โˆž๐œ๐‘›=lim๐‘›โ†’โˆž๎ƒฉ๐œˆ๐‘›sup๐‘žโˆˆโ„ฑ๐œ“๎€ท๐บ๎€ท๐‘ž,๐‘ฅ๐‘›๎€ธ๎€ธ+๐œ‡๐‘›๎ƒช=0.(3.17)
(a) We show that ๐‘โˆˆโˆฉ๐‘š๐‘—=1EP(๐œƒ๐‘—).
Since ๐‘ฅ๐‘›+1=ฮ ๐‘“๐ถ๐‘›+1๐‘ฅ1โˆˆ๐ถ๐‘›+1โŠ‚๐ถ๐‘› and the definition of ๐ถ๐‘›+1, we have ๐บ๎€ท๐‘ฅ๐‘›+1,๐ฝ๐‘ข๐‘›๎€ธโ‰ค๐›ผ๐‘›๐บ๎€ท๐‘ฅ๐‘›+1,๐ฝ๐‘ฅ1๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐บ๎€ท๐‘ฅ๐‘›+1,๐ฝ๐‘ฅ๐‘›๎€ธ+๐œ๐‘›(3.18) is equivalent to ๐œ™๎€ท๐‘ฅ๐‘›+1,๐‘ข๐‘›๎€ธโ‰ค๐›ผ๐‘›๐œ™๎€ท๐‘ฅ๐‘›+1,๐‘ฅ1๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐œ™๎€ท๐‘ฅ๐‘›+1,๐‘ฅ๐‘›๎€ธ+๐œ๐‘›.(3.19) From (3.11), (3.15), and (3.17), it follows that lim๐‘›โ†’โˆž๐œ™๎€ท๐‘ฅ๐‘›+1,๐‘ข๐‘›๎€ธ=0.(3.20) From (2.7), we have ๎€ทโ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–โˆ’โ€–โ€–๐‘ข๐‘›โ€–โ€–๎€ธ2โŸถ0.(3.21) Since โ€–๐‘ฅ๐‘›+1โ€–โ†’โ€–๐‘โ€–, we have โ€–โ€–๐‘ข๐‘›โ€–โ€–โŸถโ€–๐‘โ€–as๐‘›โŸถโˆž.(3.22) It follow that โ€–โ€–๐ฝ๐‘ข๐‘›โ€–โ€–โŸถโ€–๐ฝ๐‘โ€–as๐‘›โŸถโˆž.(3.23) That is, {โ€–๐ฝ๐‘ข๐‘›โ€–} is bounded in ๐ธโˆ— and ๐ธโˆ— is reflexive; we assume that ๐ฝ๐‘ข๐‘›โ‡€๐‘ขโˆ—โˆˆ๐ธโˆ—. In view of ๐ฝ(๐ธ)=๐ธโˆ—, there exists ๐‘ขโˆˆ๐ธ such that ๐ฝ๐‘ข=๐‘ขโˆ—. It follows that ๐œ™๎€ท๐‘ฅ๐‘›+1,๐‘ข๐‘›๎€ธ=โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–2โˆ’2โŸจ๐‘ฅ๐‘›+1,๐ฝ๐‘ฆ๐‘›โ€–โ€–๐‘ขโŸฉ+๐‘›โ€–โ€–2=โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–2โˆ’2โŸจ๐‘ฅ๐‘›+1,๐ฝ๐‘ข๐‘›โ€–โ€–โŸฉ+๐ฝ๐‘ข๐‘›โ€–โ€–2.(3.24) Taking liminf๐‘›โ†’โˆž on both sides of the equality above and โ€–โ‹…โ€– is the weak lower semicontinuous, it yields that 0โ‰ฅโ€–๐‘โ€–2โˆ’2โŸจ๐‘,๐‘ขโˆ—โŸฉ+โ€–๐‘ขโˆ—โ€–2=โ€–๐‘โ€–2โˆ’2โŸจ๐‘,๐ฝ๐‘ขโŸฉ+โ€–๐ฝ๐‘ขโ€–2=โ€–๐‘โ€–2โˆ’2โŸจ๐‘,๐ฝ๐‘ขโŸฉ+โ€–๐‘ขโ€–2=๐œ™(๐‘,๐‘ข).(3.25) That is, ๐‘=๐‘ข, which implies that ๐‘ขโˆ—=๐ฝ๐‘. It follows that ๐ฝ๐‘ข๐‘›โ‡€๐ฝ๐‘โˆˆ๐ธโˆ—. From (3.23) and the Kadec-Klee property of ๐ธโˆ— we have ๐ฝ๐‘ข๐‘›โ†’๐ฝ๐‘ as ๐‘›โ†’โˆž. Note that ๐ฝโˆ’1โˆถ๐ธโˆ—โ†’๐ธ is norm-weakโˆ—-continuous; that is, ๐‘ข๐‘›โ‡€๐‘. From (3.22) and the Kadec-Klee property of ๐ธ, we have lim๐‘›โ†’โˆž๐‘ข๐‘›=๐‘.(3.26) For ๐‘žโˆˆ๐นโŠ‚๐ถ๐‘›, by nonexpansiveness, we observe that ๐œ™๎€ท๐‘ž,๐‘ข๐‘›๎€ธ๎€ท=๐œ™๐‘ž,๐พ๐‘š๐‘›๐‘ฅ๐‘›๎€ธ๎€ทโ‰ค๐œ™๐‘ž,๐พ๐‘›๐‘šโˆ’1๐‘ฅ๐‘›๎€ธ๎€ทโ‰ค๐œ™๐‘ž,๐พ๐‘›๐‘šโˆ’2๐‘ฅ๐‘›๎€ธโ‹ฎ๎€ทโ‰ค๐œ™๐‘ž,๐พ๐‘—๐‘›๐‘ฅ๐‘›๎€ธ.(3.27) By Lemma 2.14, we have for ๐‘—=1,2,3,โ€ฆ,๐‘š๐œ™๎€ท๐พ๐‘—๐‘›๐‘ฅ๐‘›,๐‘ฅ๐‘›๎€ธ๎€ท&โ‰ค๐œ™๐‘ž,๐‘ฅ๐‘›๎€ธ๎€ทโˆ’๐œ™๐‘ž,๐พ๐‘—๐‘›๐‘ฅ๐‘›๎€ธ๎€ทโ‰ค๐œ™๐‘ž,๐‘ฅ๐‘›๎€ธ๎€ทโˆ’๐œ™๐‘ž,๐‘ข๐‘›๎€ธ.(3.28) Since ๐‘ฅ๐‘›,๐‘ข๐‘›โ†’๐‘ as ๐‘›โ†’โˆž, we get ๐œ™(๐พ๐‘—๐‘›๐‘ฅ๐‘›,๐‘ฅ๐‘›)โ†’0 as ๐‘›โ†’โˆž, for ๐‘—=1,2,3,โ€ฆ,๐‘š. From (2.7), it follow that ๎€ทโ€–โ€–๐พ๐‘—๐‘›๐‘ฅ๐‘›โ€–โ€–โˆ’โ€–โ€–๐‘ฅ๐‘›โ€–โ€–๎€ธ2โŸถ0.(3.29) Since โ€–๐‘ฅ๐‘›โ€–โ†’โ€–๐‘โ€–, we also have โ€–โ€–๐พ๐‘—๐‘›๐‘ฅ๐‘›โ€–โ€–โŸถโ€–๐‘โ€–as๐‘›โŸถโˆž.(3.30) Since {๐พ๐‘—๐‘›๐‘ฅ๐‘›} is bounded and ๐ธ is reflexive, without loss of generality we assume that ๐พ๐‘—๐‘›๐‘ฆ๐‘›โ‡€โ„Ž. We know that ๐ถ๐‘› is closed and convex for each ๐‘›โ‰ฅ1 it is obvious that โ„Žโˆˆ๐ถ๐‘›. Again since ๐œ™๎€ท๐พ๐‘—๐‘›๐‘ฅ๐‘›,๐‘ฅ๐‘›๎€ธ=โ€–โ€–๐พ๐‘—๐‘›๐‘ฅ๐‘›โ€–โ€–2๎ซ๐พโˆ’2๐‘—๐‘›๐‘ฅ๐‘›,๐ฝ๐‘ฅ๐‘›๎ฌ+โ€–โ€–๐‘ฅ๐‘›โ€–โ€–2,(3.31) taking liminf๐‘›โ†’โˆž on the both sides of equality above, we have 0&โ‰ฅโ€–โ„Žโ€–2โˆ’2โŸจโ„Ž,๐ฝ๐‘โŸฉ+โ€–๐‘โ€–2=๐œ™(โ„Ž,๐‘).(3.32) That is, โ„Ž=๐‘,for all ๐‘—=1,2,3,โ€ฆ,๐‘š; it follow that ๐พ๐‘—๐‘›๐‘ฅ๐‘›โ‡€๐‘;(3.33) from (3.30), (3.33), and the Kadec-Klee property, it follows that lim๐‘›โ†’โˆž๐พ๐‘—๐‘›๐‘ฅ๐‘›=๐‘,โˆ€๐‘—=1,2,3,โ€ฆ,๐‘š.(3.34) By using triangle inequality, we have โ€–โ€–๐‘ฅ๐‘›โˆ’๐พ๐‘—๐‘›๐‘ฅ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โ€–โ€–+โ€–โ€–โˆ’๐‘๐‘โˆ’๐พ๐‘—๐‘›๐‘ข๐‘›โ€–โ€–.(3.35) Since ๐‘ฅ๐‘›,๐พ๐‘—๐‘›๐‘ฅ๐‘›โ†’๐‘ as ๐‘›โ†’โˆž, we have lim๐‘›โ†’โˆžโ€–โ€–๐‘ฅ๐‘›โˆ’๐พ๐‘—๐‘›๐‘ฅ๐‘›โ€–โ€–=0,โˆ€๐‘—=1,2,3,โ€ฆ,๐‘š.(3.36) Again by using triangle inequality, we have โ€–โ€–๐พ๐‘—๐‘›๐‘ฅ๐‘›โˆ’๐พ๐‘›๐‘—โˆ’1๐‘ฅ๐‘›โ€–โ€–โ‰คโ€–โ€–๐พ๐‘—๐‘›๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–+โ€–โ€–๐‘ฅ๐‘›โˆ’๐พ๐‘›๐‘—โˆ’1๐‘ฅ๐‘›โ€–โ€–.(3.37) From (3.36), we also have lim๐‘›โ†’โˆžโ€–โ€–๐พ๐‘—๐‘›๐‘ฅ๐‘›โˆ’๐พ๐‘›๐‘—โˆ’1๐‘ฅ๐‘›โ€–โ€–=0,โˆ€๐‘—=1,2,3,โ€ฆ,๐‘š.(3.38) Since ๐ฝ is uniformly norm-to-norm continuous, we obtain lim๐‘›โ†’โˆžโ€–โ€–๐ฝ๐พ๐‘—๐‘›๐‘ฅ๐‘›โˆ’๐ฝ๐พ๐‘›๐‘—โˆ’1๐‘ฅ๐‘›โ€–โ€–=0,โˆ€๐‘—=1,2,3,โ€ฆ,๐‘š.(3.39) From ๐‘Ÿ๐‘—,๐‘›>0, we have โ€–๐ฝ๐พ๐‘—๐‘›๐‘ฅ๐‘›โˆ’๐ฝ๐พ๐‘›๐‘—โˆ’1๐‘ฅ๐‘›โ€–/๐‘Ÿ๐‘—,๐‘›โ†’0 as ๐‘›โ†’โˆžfor all ๐‘—=1,2,3,โ€ฆ,๐‘š, and ๐œƒ๐‘—๎€ท๐พ๐‘—๐‘›๐‘ฆ๐‘›๎€ธ+1,๐‘ฆ๐‘Ÿ๐‘—,๐‘›โŸจ๐‘ฆโˆ’๐พ๐‘—๐‘›๐‘ฅ๐‘›,๐ฝ๐พ๐‘—๐‘›๐‘ฅ๐‘›โˆ’๐ฝ๐พ๐‘›๐‘—โˆ’1๐‘ฅ๐‘›โŸฉโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ.(3.40) By (A2), that โ€–โ€–๐‘ฆโˆ’๐พ๐‘—๐‘›๐‘ฆ๐‘›โ€–โ€–โ€–โ€–๐ฝ๐พ๐‘—๐‘›๐‘ฆ๐‘›โˆ’๐ฝ๐พ๐‘›๐‘—โˆ’1๐‘ฅ๐‘›โ€–โ€–๐‘Ÿ๐‘›โ‰ฅ1๐‘Ÿ๐‘—,๐‘›โŸจ๐‘ฆโˆ’๐พ๐‘—๐‘›๐‘ฅ๐‘›,๐ฝ๐พ๐‘—๐‘›๐‘ฆ๐‘›โˆ’๐ฝ๐พ๐‘›๐‘—โˆ’1๐‘ฅ๐‘›โŸฉโ‰ฅโˆ’๐œƒ๐‘—๎€ท๐พ๐‘—๐‘›๐‘ฅ๐‘›๎€ธ,๐‘ฆโ‰ฅ๐œƒ๐‘—๎€ท๐‘ฆ,๐พ๐‘—๐‘›๐‘ฅ๐‘›๎€ธ,โˆ€๐‘ฆโˆˆ๐ถ,(3.41) and ๐พ๐‘—๐‘›๐‘ฅ๐‘›โ†’๐‘ as ๐‘›โ†’โˆž, we get ๐œƒ๐‘—(๐‘ฆ,๐‘)โ‰ค0, for all ๐‘ฆโˆˆ๐ถ. For 0<๐‘ก<1, define ๐‘ฆ๐‘ก=๐‘ก๐‘ฆ+(1โˆ’๐‘ก)๐‘, then ๐‘ฆ๐‘กโˆˆ๐ถ which imply that ๐œƒ๐‘—(๐‘ฆ๐‘ก,๐‘)โ‰ค0. From (A1), we obtain that 0=๐œƒ๐‘—๎€ท๐‘ฆ๐‘ก,๐‘ฆ๐‘ก๎€ธโ‰ค๐‘ก๐œƒ๐‘—๎€ท๐‘ฆ๐‘ก๎€ธ+,๐‘ฆ(1โˆ’๐‘ก)๐œƒ๐‘—๎€ท๐‘ฆ๐‘ก๎€ธ,๐‘โ‰ค๐‘ก๐œƒ๐‘—๎€ท๐‘ฆ๐‘ก๎€ธ.,๐‘ฆ(3.42) We have that ๐œƒ๐‘—(๐‘ฆ๐‘ก,๐‘ฆ)โ‰ฅ0. From (A3), we have ๐œƒ๐‘—(๐‘,๐‘ฆ)โ‰ฅ0, for all ๐‘ฆโˆˆ๐ถ and ๐‘—=1,2,3,โ€ฆ,๐‘š. That is, ๐‘โˆˆEP(๐œƒ๐‘—), for all๐‘—=1,2,3,โ€ฆ,๐‘š. This imply that ๐‘โˆˆโˆฉ๐‘š๐‘—=1EP(๐œƒ๐‘—).
(b) We show that ๐‘โˆˆ๐น(๐‘†).
Since ๐‘ฅ๐‘›+1=ฮ ๐‘“๐ถ๐‘›+1๐‘ฅ1โˆˆ๐ถ๐‘›+1โŠ‚๐ถ๐‘› and the definition of ๐ถ๐‘›+1, we have ๐บ๎€ท๐‘ฅ๐‘›+1,๐ฝ๐‘ง๐‘›๎€ธโ‰ค๐›ผ๐‘›๐บ๎€ท๐‘ฅ๐‘›+1,๐ฝ๐‘ฅ1๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐บ๎€ท๐‘ฅ๐‘›+1,๐ฝ๐‘ฅ๐‘›๎€ธ+๐œ๐‘›(3.43) is equivalent to ๐œ™๎€ท๐‘ฅ๐‘›+1,๐‘ง๐‘›๎€ธโ‰ค๐›ผ๐‘›๐œ™๎€ท๐‘ฅ๐‘›+1,๐‘ฅ1๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐œ™๎€ท๐‘ฅ๐‘›+1,๐‘ฅ๐‘›๎€ธ+๐œ๐‘›.(3.44) Following (3.11), (3.15), and (3.17), we get that lim๐‘›โ†’โˆž๐œ™๎€ท๐‘ฅ๐‘›+1,๐‘ง๐‘›๎€ธ=0.(3.45) From (2.7), we also have โ€–โ€–๐‘ง๐‘›โ€–โ€–โŸถโ€–๐‘โ€–as๐‘›โŸถโˆž.(3.46) It follows that โ€–โ€–๐ฝ๐‘ง๐‘›โ€–โ€–โŸถโ€–๐ฝ๐‘โ€–as๐‘›โŸถโˆž.(3.47) This implies that {โ€–๐ฝ๐‘ง๐‘›โ€–} is bounded in ๐ธโˆ—. Since ๐ธ is reflexive and ๐ธโˆ— is also reflexive, we can assume that ๐ฝ๐‘ง๐‘›โ‡€๐‘งโˆ—โˆˆ๐ธโˆ—. In view of the reflexive of ๐ธ, we see that ๐ฝ(๐ธ)=๐ธโˆ—. There exists ๐‘งโˆˆ๐ธ such that ๐ฝ๐‘ง=๐‘งโˆ—. It follows that ๐œ™๎€ท๐‘ฅ๐‘›+1,๐‘ง๐‘›๎€ธ=โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–2โˆ’2โŸจ๐‘ฅ๐‘›+1,๐ฝ๐‘ง๐‘›โ€–โ€–๐‘งโŸฉ+๐‘›โ€–โ€–2=โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–2โˆ’2โŸจ๐‘ฅ๐‘›+1,๐ฝ๐‘ง๐‘›โ€–โ€–โŸฉ+๐ฝ๐‘ง๐‘›โ€–โ€–2.(3.48) Taking liminf๐‘›โ†’โˆž on both sides of the 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‘งโˆ—โŸฉ+โ€–๐‘งโˆ—โ€–2=โ€–๐‘โ€–2โˆ’2โŸจ๐‘,๐ฝ๐‘งโŸฉ+โ€–๐ฝ๐‘งโ€–2=โ€–๐‘โ€–2โˆ’2โŸจ๐‘,๐ฝ๐‘งโŸฉ+โ€–๐‘งโ€–2=๐œ™(๐‘,๐‘ง);(3.49) That is ๐‘=๐‘ง, which implies that ๐‘งโˆ—=๐ฝ๐‘. It follows that ๐ฝ๐‘ง๐‘›โ‡€๐ฝ๐‘โˆˆ๐ธโˆ—.From (3.47) and the Kadec-Klee property of ๐ธโˆ— we have ๐ฝ๐‘ง๐‘›โ†’๐ฝ๐‘ as ๐‘›โ†’โˆž. Since ๐ฝโˆ’1โˆถ๐ธโˆ—โ†’๐ธ is norm-weakโˆ—-continuous,๐‘ง๐‘›โ‡€๐‘ as ๐‘›โ†’โˆž. From (3.46) and the Kadec-Klee property of ๐ธ, we have lim๐‘›โ†’โˆž๐‘ง๐‘›=๐‘.(3.50) Since {๐‘ฅ๐‘›} is bounded, then a mapping ๐‘† is also bounded. From the condition lim๐‘›โ†’โˆž๐›ผ๐‘›=0, we have that โ€–โ€–๐ฝ๐‘ง๐‘›โˆ’๐ฝ๐‘†๐‘›๐‘ข๐‘›โ€–โ€–=lim๐‘›โ†’โˆž๐›ผ๐‘›โ€–โ€–๐ฝ๐‘ฅ1โˆ’๐ฝ๐‘†๐‘›๐‘ข๐‘›โ€–โ€–=0.(3.51) From (3.47), we get โ€–โ€–๐ฝ๐‘†๐‘›๐‘ข๐‘›โ€–โ€–โŸถโ€–๐ฝ๐‘โ€–as๐‘›โŸถโˆž.(3.52) Since ๐ฝโˆ’1โˆถ๐ธโˆ—โ†’๐ธ is norm-weak*-continuous, ๐‘†๐‘›๐‘ข๐‘›โ‡€๐‘as๐‘›โŸถโˆž.(3.53) On the other hand, we observe that ||โ€–โ€–๐‘†๐‘›๐‘ข๐‘›โ€–โ€–||=โ€–โ€–๐ฝ๎€ท๐‘†โˆ’โ€–๐‘โ€–๐‘›๐‘ข๐‘›๎€ธโ€–โ€–โ€–โ€–๐ฝ๎€ท๐‘†โˆ’โ€–๐ฝ๐‘โ€–โ‰ค๐‘›๐‘ข๐‘›๎€ธโ€–โ€–โˆ’๐ฝ๐‘.(3.54) In view of (3.52), we obtain โ€–๐‘†๐‘›๐‘ข๐‘›โ€–โ†’โ€–๐‘โ€–. Since ๐ธ has the Kadee-Klee property, we get ๐‘†๐‘›๐‘ข๐‘›โŸถ๐‘foreach๐‘›โˆˆโ„•.(3.55) From ๐‘†๐‘›๐‘ข๐‘›โ†’๐‘, we get ๐‘†๐‘›+1๐‘ข๐‘›โ†’๐‘; that is, ๐‘†๐‘†๐‘›๐‘ข๐‘›โ†’๐‘. In view of closeness of ๐‘†, we have ๐‘†๐‘=๐‘. This implies that ๐‘โˆˆ๐น(๐‘†). From (a) and (b), it follows that ๐‘โˆˆโˆฉ๐‘š๐‘—=1EP(๐œƒ๐‘—)โˆฉ๐น(๐‘†).

Step 4. We will show that ๐‘=ฮ ๐‘“โ„ฑ๐‘ฅ1.
Since โ„ฑ is closed and convex set from Lemma 2.9, we have ฮ ๐‘“โ„ฑ๐‘ฅ1 which is single valued, denoted by ๐œ. By definition ๐‘ฅ๐‘›=ฮ ๐‘“๐ถ๐‘›๐‘ฅ1 and ๐‘ฃโˆˆโ„ฑโŠ‚๐ถ๐‘›, we also have ๐บ๎€ท๐‘ฅ๐‘›,๐ฝ๐‘ฅ1๎€ธ๎€ทโ‰คG๐œ,๐ฝ๐‘ฅ1๎€ธ,โˆ€๐‘›โ‰ฅ1.(3.56) By the definition of ๐บ and ๐‘“, we know that, for each given ๐‘ฅ,๐บ(๐œ‰,๐ฝ๐‘ฅ) is convex and lower semicontinuous with respect to ๐œ‰. So ๐บ๎€ท๐‘,๐ฝ๐‘ฅ1๎€ธโ‰คliminf๐‘›โ†’โˆž๐บ๎€ท๐‘ฅ๐‘›,๐ฝ๐‘ฅ1๎€ธโ‰คlimsup๐‘›โ†’โˆž๐บ๎€ท๐‘ฅ๐‘›,๐ฝ๐‘ฅ1๎€ธ๎€ทโ‰ค๐บ๐œ,๐ฝ๐‘ฅ1๎€ธ.(3.57) From the definition of ฮ ๐‘“โ„ฑ๐‘ฅ1 and since ๐‘โˆˆโ„ฑ, we conclude that ๐œ=๐‘=ฮ ๐‘“โ„ฑ๐‘ฅ1 and ๐‘ฅ๐‘›โ†’๐‘ as ๐‘›โ†’โˆž. The proof is completed.

Setting ๐œˆ๐‘›โ‰ก0 and ๐œ‡๐‘›โ‰ก0 in Theorem 3.1, then we have the following corollary.

Corollary 3.2. Let ๐ถ be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space ๐ธ with the Kadec-Klee property. For each ๐‘—=1,2,โ€ฆ,๐‘š, let ๐œƒ๐‘— be a bifunction from ๐ถร—๐ถ to โ„ which satisfies conditions (A1)โ€“(A4). Let ๐‘†โˆถ๐ถโ†’๐ถ be a closed and quasi-๐œ™-asymptotically nonexpansive mappings, and let ๐‘“โˆถ๐ธโ†’โ„ be a convex and lower semicontinuous function with ๐ถโŠ‚int(๐ท(๐‘“)) such that ๐‘“(๐‘ฅ)โ‰ฅ0 for all ๐‘ฅโˆˆ๐ถ and ๐‘“(0)=0. Assume that โ„ฑ=๐น(๐‘†)โˆฉ(โˆฉ๐‘š๐‘—=1EP(๐œƒ๐‘—))โ‰ โˆ…. For an initial point ๐‘ฅ1โˆˆ๐ธ and ๐ถ1=๐ถ, we define the sequence {๐‘ฅ๐‘›} by ๐‘ข๐‘›=๐‘‡๐œƒ๐‘š๐‘Ÿ๐‘š,๐‘›๐‘‡๐œƒ๐‘šโˆ’1๐‘Ÿ๐‘šโˆ’1,๐‘›๐‘‡๐œƒ๐‘šโˆ’2๐‘Ÿ๐‘šโˆ’2,๐‘›โ‹ฏ๐‘‡๐œƒ1๐‘Ÿ1,๐‘›๐‘ฅ๐‘›,๐‘ง๐‘›=๐ฝโˆ’1๎€ท๐›ผ๐‘›๐ฝ๐‘ฅ1+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐ฝ๐‘†๐‘›๐‘ข๐‘›๎€ธ,๐ถ๐‘›+1=๎€ฝ๐‘ฃโˆˆ๐ถ๐‘›๎€ทโˆถ๐บ๐‘ฃ,๐ฝ๐‘ง๐‘›๎€ธ๎€ทโ‰ค๐บ๐‘ฃ,๐ฝ๐‘ข๐‘›๎€ธ๎€ทโ‰ค๐บ๐‘ฃ,๐ฝ๐‘ฅ1๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐บ๎€ท๐‘ฃ,๐ฝ๐‘ฅ๐‘›๎€ธ+๐œ๐‘›๎€พ,๐‘ฅ๐‘›+1=ฮ ๐‘“๐ถ๐‘›+1๐‘ฅ1,๐‘›โˆˆโ„•,(3.58) where {๐›ผ๐‘›} is a sequence in [0,1], ๐œ๐‘›=๐œˆ๐‘›sup๐‘žโˆˆโ„ฑ๐œ“(๐บ(๐‘ž,๐‘ฅ๐‘›))+๐œ‡๐‘›, and {๐‘Ÿ๐‘—,๐‘›}โŠ‚[๐‘‘,โˆž) for some ๐‘‘>0. If lim๐‘›โ†’โˆž๐›ผ๐‘›=0, then {๐‘ฅ๐‘›} converges strongly to ฮ ๐‘“โ„ฑ๐‘ฅ1.

Let ๐ธ be a real Banach space, and let ๐ถ be a nonempty closed convex subset of ๐ธ. Given a mapping ๐ดโˆถ๐ถโ†’๐ธโˆ—, let ๐œƒ(๐‘ฅ,๐‘ฆ)=โŸจ๐ด๐‘ฅ,๐‘ฆโˆ’๐‘ฅโŸฉ for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ. Then ๐‘ฅโˆ—โˆˆEP(๐œƒ) if and only if โŸจ๐ด๐‘ฅโˆ—,๐‘ฆโˆ’๐‘ฅโˆ—โŸฉโ‰ฅ0 for all ๐‘ฆโˆˆ๐ถ; that is, ๐‘ฅโˆ— is a solution of the classical variational inequality problem. The set of this solution is denoted by VI(๐ด,๐ถ). For each ๐‘Ÿ>0 and ๐‘ฅโˆˆ๐ธ, we define the mapping ๐‘‡๐œƒ๐‘Ÿ๐‘ฅ by ๐‘‡๐œƒ๐‘Ÿ๎‚†1๐‘ฅ=๐‘งโˆˆ๐ถโˆถโŸจ๐ด๐‘ง,๐‘ฆโˆ’๐‘งโŸฉ+๐‘Ÿ๎‚‡.โŸจ๐‘ฆโˆ’๐‘ง,๐ฝ๐‘งโˆ’๐ฝ๐‘ฅโŸฉโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ(3.59) Hence, we obtain the following corollary.

Corollary 3.3. Let ๐ถ be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space ๐ธ with the Kadec-Klee property. For each ๐‘—=1,2,โ€ฆ,๐‘š, let{๐ด๐‘—} be a continuous monotone mapping of ๐ถ into ๐ธโˆ—. Let ๐‘†โˆถ๐ถโ†’๐ถ be a closed totally quasi-๐œ™-asymptotically nonexpansive mappings with nonnegative real sequences ๐œˆ๐‘›, ๐œ‡๐‘› with ๐œˆ๐‘›โ†’0, ๐œ‡๐‘›โ†’0 as ๐‘›โ†’โˆž and a strictly increasing continuous function ๐œ“โˆถโ„+โ†’โ„+ with ๐œ“(0)=0, and let ๐‘“โˆถ๐ธโ†’โ„ be a convex and lower semicontinuous function with ๐ถโŠ‚int(๐ท(๐‘“)) such that ๐‘“(๐‘ฅ)โ‰ฅ0 for all ๐‘ฅโˆˆ๐ถ and ๐‘“(0)=0. Assume that โ„ฑ=๐น(๐‘†)โˆฉ(โˆฉ๐‘š๐‘—=1VI(๐ด๐‘—,๐ถ))โ‰ โˆ…. For an initial point ๐‘ฅ1โˆˆ๐ธ and ๐ถ1=๐ถ, one defines the sequence {๐‘ฅ๐‘›} by ๐‘ข๐‘›=๐‘‡๐œƒ๐‘š๐‘Ÿ๐‘š,๐‘›๐‘‡๐œƒ๐‘šโˆ’1๐‘Ÿ๐‘šโˆ’1,๐‘›๐‘‡๐œƒ๐‘šโˆ’2๐‘Ÿ๐‘šโˆ’2,๐‘›โ‹ฏ๐‘‡๐œƒ1๐‘Ÿ1,๐‘›๐‘ฅ๐‘›,๐‘ง๐‘›=๐ฝโˆ’1๎€ท๐›ผ๐‘›๐ฝ๐‘ฅ1+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐ฝ๐‘†๐‘›๐‘ข๐‘›๎€ธ,๐ถ๐‘›+1=๎€ฝ๐‘ฃโˆˆ๐ถ๐‘›๎€ทโˆถ๐บ๐‘ฃ,๐ฝ๐‘ง๐‘›๎€ธ๎€ทโ‰ค๐บ๐‘ฃ,๐ฝ๐‘ข๐‘›๎€ธ๎€ทโ‰ค๐บ๐‘ฃ,๐ฝ๐‘ฅ1๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐บ๎€ท๐‘ฃ,๐ฝ๐‘ฅ๐‘›๎€ธ+๐œ๐‘›๎€พ,๐‘ฅ๐‘›+1=ฮ ๐‘“๐ถ๐‘›+1๐‘ฅ1,๐‘›โˆˆโ„•,(3.60) where ๐œ๐‘›=๐œˆ๐‘›sup๐‘žโˆˆโ„ฑ๐œ“(๐บ(๐‘ž,๐‘ฅ๐‘›))+๐œ‡๐‘›, {๐›ผ๐‘›} is a sequence in [0,1], and {๐‘Ÿ๐‘—,๐‘›}โŠ‚[๐‘‘,โˆž) for some ๐‘‘>0. If lim๐‘›โ†’โˆž๐›ผ๐‘›=0, then {๐‘ฅ๐‘›} converges strongly to ฮ ๐‘“โ„ฑ๐‘ฅ1.

If ๐‘“(๐‘ฅ)=0 for all ๐‘ฅโˆˆ๐ธ, we have ๐บ(๐œ‰,๐ฝ๐‘ฅ)=๐œ™(๐œ‰,๐‘ฅ) and ฮ ๐‘“๐ถ๐‘ฅ=ฮ ๐ถ๐‘ฅ. From Theorem 3.1, we obtain the following corollary.

Corollary 3.4. Let ๐ถ be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space ๐ธ with the Kadec-Klee property. For each ๐‘—=1,2,โ€ฆ,๐‘š, let ๐œƒ๐‘— be a bifunction from ๐ถร—๐ถ to โ„ which satisfies conditions (A1)โ€“(A4). Let ๐‘†โˆถ๐ถโ†’๐ถ be a closed totally quasi-๐œ™-asymptotically nonexpansive mappings with nonnegative real sequences ๐œˆ๐‘›, ๐œ‡๐‘› with ๐œˆ๐‘›โ†’0, ๐œ‡๐‘›โ†’0 as ๐‘›โ†’โˆž and a strictly increasing continuous function ๐œ“โˆถโ„+โ†’โ„+ with ๐œ“(0)=0. Assume that โ„ฑ=๐น(๐‘†)โˆฉ(โˆฉ๐‘š๐‘—=1EP(๐œƒ๐‘—))โ‰ โˆ…. For an initial point ๐‘ฅ1โˆˆ๐ธ and ๐ถ1=๐ถ, we define the sequence {๐‘ฅ๐‘›} by ๐‘ข๐‘›=๐‘‡๐œƒ๐‘š๐‘Ÿ๐‘š,๐‘›๐‘‡๐œƒ๐‘šโˆ’1๐‘Ÿ๐‘šโˆ’1,๐‘›๐‘‡๐œƒ๐‘šโˆ’2๐‘Ÿ๐‘šโˆ’2,๐‘›โ‹ฏ๐‘‡๐œƒ1๐‘Ÿ1,๐‘›๐‘ฅ๐‘›,๐‘ง๐‘›=๐ฝโˆ’1๎€ท๐›ผ๐‘›๐ฝ๐‘ฅ1+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐ฝ๐‘†๐‘›๐‘ข๐‘›๎€ธ,๐ถ๐‘›+1=๎€ฝ๐‘ฃโˆˆ๐ถ๐‘›๎€ทโˆถ๐บ๐‘ฃ,๐ฝ๐‘ง๐‘›๎€ธ๎€ทโ‰ค๐บ๐‘ฃ,๐ฝ๐‘ข๐‘›๎€ธ๎€ทโ‰ค๐บ๐‘ฃ,๐ฝ๐‘ฅ1๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐บ๎€ท๐‘ฃ,๐ฝ๐‘ฅ๐‘›๎€ธ+๐œ๐‘›๎€พ,๐‘ฅ๐‘›+1=ฮ ๐ถ๐‘›+1๐‘ฅ1,๐‘›โˆˆโ„•,(3.61) where {๐›ผ๐‘›} is a sequence in [0,1], ๐œ๐‘›=๐œˆ๐‘›sup๐‘žโˆˆโ„ฑ๐œ“(๐บ(๐‘ž,๐‘ฅ๐‘›))+๐œ‡๐‘›, and {๐‘Ÿ๐‘—,๐‘›}โŠ‚[๐‘‘,โˆž) for some ๐‘‘>0. If lim๐‘›โ†’โˆž๐›ผ๐‘›=0, then {๐‘ฅ๐‘›} converges strongly to ฮ โ„ฑ๐‘ฅ1.

Remark 3.5. Our main result extends and improves the result of Chang et al. [13] in the following sense. (i)From the algorithm we used new method replace by the generalized ๐‘“-projection method which is more general than generalized projection. (ii)For the problem, we extend the result to a common problem of fixed point problems and equilibrium problems.

Acknowledgments

The authors would like to thank The National Research Council of Thailand (NRCT) and Faculty of Science, King Mongkut's University of Technology Thonburi (Grant NRCT-2555). Furthermore, the authors would like to express their thanks to the referees for their helpful comments.

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