About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
VolumeΒ 2012Β (2012), Article IDΒ 215261, 17 pages
http://dx.doi.org/10.1155/2012/215261
Research Article

Modified Hybrid Block Iterative Algorithm for Uniformly Quasi-πœ™-Asymptotically Nonexpansive Mappings

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China

Received 6 June 2012; Accepted 11 July 2012

Academic Editor: YonghongΒ Yao

Copyright Β© 2012 Qiansheng Feng 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

Saewan and Kumam (2010) have proved the convergence theorems for finding the set of solutions of a general equilibrium problems and the common fixed point set of a family of closed and uniformly quasi-πœ™-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space E with Kadec-Klee property. In this paper, authors prove the convergence theorems and do not need the Kadec-Klee property of Banach space and some other conditions used in the paper of S. Saewan and P. Kumam. Therefore, the results presented in this paper improve and extend some recent results.

1. Introduction

Let 𝐢 be a nonempty closed convex subspace of a real Banach space 𝐸. A mapping 𝐴∢𝐷(𝐴)βŠ‚πΈβ†’πΈβˆ— is said to be monotone if for each π‘₯,π‘¦βˆˆπ·(𝐴), the following inequality holds: ⟨π‘₯βˆ’π‘¦,𝐴π‘₯βˆ’π΄π‘¦βŸ©β‰₯0.(1.1)

A mapping π΄βˆΆπΆβ†’πΈβˆ— is called 𝛼-inverse-strongly monotone if there exists 𝛼>0 such that ⟨π‘₯βˆ’π‘¦,𝐴π‘₯βˆ’π΄π‘¦βŸ©β‰₯𝛼‖𝐴π‘₯βˆ’π΄π‘¦β€–2.(1.2)

A monotone mapping 𝑇 is said to be maximal monotone if 𝑅(𝐽+π‘Ÿπ‘‡)=πΈβˆ—, for all π‘Ÿ>0, where 𝐽 is the normalized duality mapping. We denote by π‘‡βˆ’1(0)={π‘₯∈𝐸∢0βˆˆπ‘‡π‘₯} the set of zero points of 𝑇.

Remark 1.1. It is well know that if π΄βˆΆπΆβ†’πΈβˆ— is an 𝛼-inverse-strongly monotone mapping, then it is (1/𝛼)-Lipschitzian, and hence uniformly continuous. Clearly, the class of monotone mappings includes the class of 𝛼-inverse strongly monotone mappings.
Let 𝐢 be a nonempty closed convex subspace of a real Banach space 𝐸 with dual πΈβˆ— and βŸ¨β‹…,β‹…βŸ© is the pairing between 𝐸 and πΈβˆ—. Let π‘“βˆΆπΆΓ—πΆβ†’π‘… be a bifunction and π΄βˆΆπΆβ†’πΈβˆ— be a monotone mapping. The generalized equilibrium problem means that finding a π‘’βˆˆπΆ such that 𝑓(𝑒,𝑦)+βŸ¨π΄π‘’,π‘¦βˆ’π‘’βŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ.(1.3)

The set of solutions of (1.3) is denoted by GEP(𝑓,𝐴), that is, GEP(𝑓,𝐴)={π‘’βˆˆπΆβˆΆπ‘“(𝑒,𝑦)+βŸ¨π΄π‘’,π‘¦βˆ’π‘’βŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ}.(1.4)

If 𝐴=0, then the problem (1.3) is equivalent to that of finding a π‘’βˆˆπΆ such that 𝑓(𝑒,𝑦)β‰₯0,βˆ€π‘¦βˆˆπΆ,(1.5) which is called the equilibrium problem. The solution of (1.5) is denoted by EP(𝑓). If 𝑓=0, then the problem (1.3) is equivalent to that of finding a π‘’βˆˆπΆ such that βŸ¨π΄π‘’,π‘¦βˆ’π‘’βŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ,(1.6) which is called the variational inequality of Browder type. The solution of (1.6) is denoted by VI(𝐢,𝐴).

The problem (1.3) was shown in [1] to cover variational inequality problems, monotone inclusion problems, vector equilibrium problems, numerous problems in physics, minimization problems, saddle point problems, and Nash equilibria in noncooperative games. In addition, there are several other problems, for example, fixed point problem, the complementarity problem, and optimization problem, which can also be written in the form of an EP(𝑓). In other words, the EP(𝑓) is a unifying model for several problems arising in physics, engineering, science, optimization, economics, and so on. In the past two decades, Some methods have been modified for solving the generalized equilibrium problem and the equilibrium problem in Hilbert space and Banach space, see [2–9].

The convex feasibility problem (CFP) is the problem for computing points that lay in the intersection of a finite family of closed convex subsets 𝐢𝑗,𝑗=1,2,…,𝑁, of a Banach space 𝐸. This problem appears in many fields of applied mathematics, such as the theory of optimization [1], Image Reconstruction from projections [10], and Game Theory [11] and plays an important role in these domains. There is a considerable investigation of (CFP) in the framework of Hilbert spaces which captures applications in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [12]. Also the projection methods have dominated in the iterative approaches to (CFP) in Hilbert spaces. In 1993, Kitahara and Takahashi [13] deal with the convex feasibility problem by convex and sunny nonexpansive retractions in a uniformly convex Banach space.

We note that the block iterative method is a common method by many authors to solve (CFP) [14]. In 2008, Plubtieng and Ungchittrakool [15] established block iterative methods for a finite family of relatively nonexpansive mappings and got some strong convergence theorems in a Banach space by using the hybrid method.

In 2009, Takahashi and Zembayashi [16] introduced the following iterative scheme in the case that 𝐸 is uniformly smooth and uniformly convex Banach space: π‘₯0=π‘₯∈𝐢,arbitrarily,𝑦𝑛=π½βˆ’1𝛼𝑛𝐽π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ½π‘₯𝑛;π‘’π‘›βˆˆπΆs.t𝑒.𝑓𝑛+1,π‘¦π‘Ÿπ‘›βŸ¨π‘¦βˆ’π‘’π‘›,π½π‘’π‘›βˆ’π½π‘¦π‘›πΆβŸ©,βˆ€π‘¦βˆˆπΆπ‘›+1=ξ€½π‘§βˆˆπΆπ‘›ξ€·βˆΆπœ™π‘§,π‘’π‘›ξ€Έξ€·β‰€πœ™π‘§,π‘₯𝑛;π‘₯𝑛+1=Π𝐢𝑛+1π‘₯,𝑛=1,2,…,(1.7) where 𝑇 is a relatively nonexpansive mapping and 𝑓 is a bifunction from 𝐢×𝐢 into 𝑅. They prove that the sequence {π‘₯𝑛} converges strongly to π‘ž=Π𝐹(𝑇)∩EP(𝑓) under appropriate conditions.

In the same year, Qin et al. [7] introduced a hybrid projection algorithm to two quasi-πœ™-nonexpansive mappings in Banach spaces as follows: π‘₯0=π‘₯∈𝐢,arbitrarily,𝐢1π‘₯=𝐢;1=Π𝐢1π‘₯0;𝑦𝑛=π½βˆ’1𝛼𝑛𝐽π‘₯𝑛+𝛽𝑛𝐽𝑇π‘₯𝑛+𝛾𝑛𝐽𝑆π‘₯𝑛;π‘’π‘›βˆˆπΆs.t𝑒.𝑓𝑛+1,π‘¦π‘Ÿπ‘›βŸ¨π‘¦βˆ’π‘’π‘›,π½π‘’π‘›βˆ’π½π‘¦π‘›πΆβŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ;𝑛+1=ξ€½π‘§βˆˆπΆπ‘›ξ€·βˆΆπœ™π‘§,π‘’π‘›ξ€Έξ€·β‰€πœ™π‘§,π‘₯𝑛;π‘₯𝑛+1=Π𝐢𝑛+1π‘₯,𝑛=1,2,…,(1.8) where Π𝐢𝑛+1 is the generalized projection from 𝐸 onto 𝐢𝑛+1. They proved that the sequence {π‘₯𝑛} converges strongly to Π𝐹(𝑆)∩𝐹(𝑇)∩EP(𝑓)π‘₯0. Then Petrot et al. [17] improved the notion from a relatively nonexpansive mapping or a quasi-πœ™-nonexpansive mapping to two relatively quasi-nonexpansive mappings; they also proved some strong convergence theorems to find a common element of the set of fixed point of relatively quasi-nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of Banach spaces.

In 2010, Saewan and Kumam [18] introduced the following iterative method to find a common element of the set of solutions of an equilibrium problem and the set of common fixed points of an infinite family of closed and uniformly quasi-πœ™-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. π‘₯0=π‘₯∈𝐸,arbitrarily,𝐢1π‘₯=𝐢;1=Π𝐢1π‘₯0;𝑦𝑛=π½βˆ’1𝛽𝑛𝐽π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ½π‘§π‘›ξ€Έ;𝑧𝑛=π½βˆ’1𝛼𝑛,0𝐽π‘₯𝑛+βˆžξ“π‘–=1𝛼𝑛,𝑖𝐽𝑆𝑛𝑖π‘₯𝑛ξƒͺ;π‘’π‘›βˆˆπΆs.t𝑒.𝑓𝑛,𝑦+βŸ¨π΄π‘¦π‘›,π‘¦βˆ’π‘’π‘›1⟩+π‘Ÿπ‘›βŸ¨π‘¦βˆ’π‘’π‘›,π½π‘’π‘›βˆ’π½π‘¦π‘›πΆβŸ©β‰₯0;𝑛+1=ξ€½π‘§βˆˆπΆπ‘›ξ€·βˆΆπœ™π‘§,π‘’π‘›ξ€Έξ€·β‰€πœ™π‘§,π‘₯𝑛+πœƒπ‘›ξ€Ύ;π‘₯𝑛+1=Π𝐢𝑛+1π‘₯,𝑛=1,2,….(1.9)

They proved that {π‘₯𝑛} converges strongly to Ξ βˆ©βˆžπ‘–=1𝐹(𝑆𝑖)∩GEP(𝑓,𝐴) under the proper conditions. The same year, Chang et al. [19] proposed the modified block iterative algorithm for solving the convex feasibility problems for an infinite family of closed and uniformly quasi-πœ™-asymptotically nonexpansive mapping; they obtain the strong convergence theorems in a Banach space.

Motivated by Saewan and Kumam [18], in this paper we use some new conditions to prove strong convergence theorems for modified block hybrid projection algorithm for finding a common element of the set of solutions of the generalized equilibrium problems and the set of common fixed points of an infinite family of closed and uniformly quasi-πœ™-asymptotically nonexpansive mappings which is more general than closed quasi-πœ™-nonexpansive mappings in a uniformly smooth and strictly convex Banach space 𝐸. In (1.9) we find iterative step 𝑦𝑛 is not essential, so we combine 𝑦𝑛 with 𝑧𝑛 of (1.9), and use an equally continuous mapping that is more weak than uniformly 𝐿-Lipschitz mapping in a uniformly smooth and strictly convex Banach space 𝐸, but the Banach space 𝐸 does not have Kadec-Klee property, under the circumstances we prove strong convergence theorems and get some results same as the results of Saewan and Kumam [18]. The results presented in this paper improve some well-known results in the literature.

2. Preliminaries

The space 𝐸 is said to be smooth if the limit lim𝑑→0β€–π‘₯+π‘‘π‘¦β€–βˆ’β€–π‘₯‖𝑑(2.1) exists for all π‘₯,π‘¦βˆˆπ‘ˆ={π‘§βˆˆπΈβˆΆβ€–π‘§β€–=1}, and 𝐸 is said to be uniformly smooth if the limit (2.1) exists uniformly for all π‘₯,π‘¦βˆˆπ‘ˆ. Then a Banach space 𝐸 is said to be strictly convex if β€–π‘₯+𝑦‖/2≀1 for all π‘₯,π‘¦βˆˆπ‘ˆ and π‘₯≠𝑦. It is said to be uniformly convex if for each πœ€βˆˆ(0,2], there exists 𝛿>0 such that β€–π‘₯+𝑦‖/2≀1βˆ’π›Ώ for all π‘₯,π‘¦βˆˆπ‘ˆ with β€–π‘₯βˆ’π‘¦β€–β‰₯πœ€.

Let 𝐸 be a Banach space and let πΈβˆ— be the topological dual of 𝐸. For all π‘₯∈𝐸 and π‘₯βˆ—βˆˆπΈβˆ—, we denote the value of π‘₯βˆ— at π‘₯ by ⟨π‘₯,π‘₯βˆ—βŸ©. Then, the duality mapping π½βˆΆπΈβ†’2πΈβˆ— is defined by 𝐽(π‘₯)=π‘“βˆˆπΈβˆ—βˆΆβŸ¨π‘₯,π‘“βŸ©=β€–π‘₯β€–2=‖𝑓‖2ξ€Ύ,(2.2) for every π‘₯∈𝐸. By the Hahn-Banach theorem, 𝐽(π‘₯) is nonempty.

The following basic properties can be found in Cioranescu [20].(i)If 𝐸 is a uniformly smooth Banach space, then 𝐽 is uniformly continuous on each bounded subset of 𝐸.(ii)If 𝐸 is a reflexive and strictly convex Banach space, then π½βˆ’1 is norm-weakβˆ—- continuous.(iii)If 𝐸 is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping π½βˆΆπΈβ†’2πΈβˆ— is single-valued, one-to-one, and onto.(iv)A Banach space 𝐸 is uniformly smooth if and only if πΈβˆ— is uniformly convex.

Let 𝐸 be a smooth, strictly convex, and reflexive Banach space and let 𝐢 be a nonempty closed convex subset of 𝐸. Throughout this paper, we denote by πœ™ the function defined by πœ™(π‘₯,𝑦)=β€–π‘₯β€–2βˆ’2⟨π‘₯,π½π‘¦βŸ©+‖𝑦‖2,βˆ€π‘₯,π‘¦βˆˆπΈ.(2.3) Following Alber [21], the generalized projection Π𝐢 from 𝐸 onto 𝐢 is defined by Π𝐢(π‘₯)=argminπ‘’βˆˆπΆπœ™(𝑒,π‘₯),βˆ€π‘₯∈𝐸.(2.4)

If 𝐸 is a Hilbert space, then πœ™(π‘₯,𝑦)=β€–π‘₯βˆ’π‘¦β€–2 and Π𝐢 is the metric projection of 𝐻 onto 𝐢. We know the following lemmas for generalized projections.

Lemma 2.1 (see Alber [21] and Kamimura and Takahashi [22]). Let 𝐢 be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space 𝐸. Then πœ™ξ€·π‘₯,Π𝐢𝑦Π+πœ™πΆξ€Έπ‘¦,π‘¦β‰€πœ™(π‘₯,𝑦),βˆ€π‘₯∈𝐢,π‘¦βˆˆπΈ.(2.5)

Lemma 2.2 (see Alber [21], Kamimura and Takahashi [22]). Let 𝐢 be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space and let π‘₯∈𝐸 and π‘§βˆˆπΆ. Then 𝑧=Π𝐢π‘₯βŸΊβŸ¨π‘¦βˆ’π‘§,𝐽π‘₯βˆ’π½π‘§βŸ©β‰€0,βˆ€π‘¦βˆˆπΆ.(2.6)

Lemma 2.3 (see Kamimura and Takahashi [22]). Let 𝐸 be a smooth and uniformly convex Banach space and let {π‘₯𝑛} and {𝑦𝑛} be sequences in 𝐸 such that either {π‘₯𝑛} or {𝑦𝑛} is bounded. If limπ‘›β†’βˆžπœ™(π‘₯𝑛,𝑦𝑛)=0, then limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘¦π‘›β€–=0.

Let 𝐢 be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸 and let 𝑇 be a mapping from 𝐢 into itself. We denoted by 𝐹(𝑇) the set of fixed points of 𝑇, that is 𝐹(𝑇)={π‘₯βˆΆπ‘‡π‘₯=π‘₯}. A point π‘βˆˆπΆ is said to be an asymptotic fixed point of 𝑇 if there exists {π‘₯𝑛} in 𝐢 which converges weakly to 𝑝 and limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘‡π‘₯𝑛‖=0. We denote the set of all asymptotic fixed points of 𝑇 by 𝐹(𝑇).

A mapping 𝑇 from 𝐢 into itself is said to be relatively nonexpansive [23] if the following conditions are satisfied:(1)𝐹(𝑇) is nonempty,(2)πœ™(𝑒,𝑇π‘₯)β‰€πœ™(𝑒,π‘₯),forallπ‘’βˆˆπΉ(𝑇),π‘₯∈𝐸,(3)𝐹(𝑇)=𝐹(𝑇).

A mapping 𝑇 from 𝐢 into itself is said to be relatively quasi-nonexpansive if the following conditions are satisfied:(1)𝐹(𝑇) is nonempty,(2)πœ™(𝑒,𝑇π‘₯)β‰€πœ™(𝑒,π‘₯),forallπ‘’βˆˆπΉ(𝑇),π‘₯∈𝐸,

The asymptotic behavior of a relatively nonexpansive mapping was studied in [24]. 𝑇 is said to be πœ™-nonexpansive, if πœ™(𝑇π‘₯,𝑇𝑦)β‰€πœ™(π‘₯,𝑦) for π‘₯,π‘¦βˆˆπΆ. 𝑇 is said to be quasi-πœ™-asymptotically nonexpansive if the following conditions are satisfied:(1)𝐹(𝑇) is nonempty,(2)πœ™(𝑒,𝑇𝑛π‘₯)β‰€π‘˜π‘›πœ™(𝑒,π‘₯),forallπ‘’βˆˆπΉ(𝑇),π‘₯∈𝐸 and 𝑛β‰₯1,where {π‘˜π‘›} is a real sequence within [1,∞) and π‘˜π‘›β†’1 as π‘›β†’βˆž.

A mapping 𝑇 is said to be closed if for any sequence {π‘₯𝑛}βŠ‚πΆ with π‘₯𝑛→π‘₯ and 𝑇π‘₯𝑛→𝑦, then 𝑇π‘₯=𝑦. It is easy to know that each relatively nonexpansive mapping is closed. The class of quasi-πœ™-asymptotically nonexpansive mappings contains properly the class of quasi-πœ™-nonexpansive mappings as a subclass and the class of quasi-πœ™-nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true (see more details [24, 25]).

By using the similar method as in Su et al. [26], the following Lemma is not hard to prove.

Lemma 2.4. Let 𝐸 be a strictly convex and uniformly smooth real Banach space, let 𝐢 be a closed convex subset of 𝐸, and let 𝑇 be a closed and quasi-πœ™-asymptotically nonexpansive mapping from 𝐢 into itself with a sequence {π‘˜π‘›}βŠ‚[1,∞) and π‘˜π‘›β†’1 as π‘›β†’βˆž. Then 𝐹(𝑇) is a closed and convex subset of 𝐢.

For solving the equilibrium problem, let us assume that a bifunction 𝑓 satisfies the following conditions:(A1)𝑓(π‘₯,π‘₯)=0,forallπ‘₯∈𝐸,(A2)𝑓 is monotone, that is, 𝑓(π‘₯,𝑦)+𝑓(𝑦,π‘₯)≀0,forallπ‘₯,π‘¦βˆˆπΈ,(A3)for all π‘₯,𝑦,π‘§βˆˆπΈ, limsup𝑑↓0𝑓(𝑑𝑧+(1βˆ’π‘‘)π‘₯,𝑦)≀𝑓(π‘₯,𝑦),(A4)for all π‘₯∈𝐢, 𝑦↦𝑓(π‘₯,𝑦) is convex and lower semicontinuous.

Lemma 2.5 (see Blum and Oettli [1]). Let 𝐢 be a closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸, let 𝑓 be a bifunction from 𝐢×𝐢 to 𝑅=(βˆ’βˆž,+∞) satisfying (A1)–(A4), and let π‘Ÿ>0 and π‘₯∈𝐸. Then, there exists π‘§βˆˆπΆ such that 1𝑓(𝑧,𝑦)+π‘ŸβŸ¨π‘¦βˆ’π‘§,π½π‘§βˆ’π½π‘₯⟩β‰₯0,βˆ€π‘¦βˆˆπΆ.(2.7)

Lemma 2.6 (see Kumam [5]). Let 𝐢 be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space 𝐸. Let 𝑓 be a bifunction from 𝐢×𝐢 to 𝑅=(βˆ’βˆž,+∞) satisfying (A1)–(A4) and let 𝐴 be a monotone mapping from 𝐢 into πΈβˆ—. For π‘Ÿ>0, define a mapping π‘‡π‘ŸβˆΆπΆβ†’πΆ as follows: π‘‡π‘Ÿξ‚†1(π‘₯)=π‘§βˆˆπΆβˆΆπ‘“(𝑧,𝑦)+⟨𝐴π‘₯,π‘¦βˆ’π‘§βŸ©+π‘Ÿξ‚‡,βŸ¨π‘¦βˆ’π‘§,π½π‘§βˆ’π½π‘₯⟩β‰₯0,βˆ€π‘¦βˆˆπΆ(2.8) for all π‘₯∈𝐸. Then, the following hold:(1)π‘‡π‘Ÿ is single-valued,(2)π‘‡π‘Ÿ is a firmly nonexpansive-type mapping [6], that is, for all π‘₯,π‘¦βˆˆπΈ, βŸ¨π‘‡π‘Ÿπ‘₯βˆ’π‘‡π‘Ÿπ‘¦,π½π‘‡π‘Ÿπ‘₯βˆ’π½π‘‡π‘Ÿπ‘¦βŸ©β‰€βŸ¨π‘‡π‘Ÿπ‘₯βˆ’π‘‡π‘Ÿπ‘¦,𝐽π‘₯βˆ’π½π‘¦βŸ©,(2.9)(3)𝐹(π‘‡π‘Ÿ)=GEP(𝑓,𝐴),(4)GEP(𝑓,𝐴) is closed and convex.

Lemma 2.7 (see Kumam [5]). Let 𝐢 be a closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸. Let 𝑓 be a bifunction from 𝐢×𝐢 to 𝑅=(βˆ’βˆž,+∞) satisfying (A1)–(A4) and let 𝐴 be a monotone mapping from 𝐢 into πΈβˆ—. For π‘₯∈𝐸, π‘žβˆˆπΉ(π‘‡π‘Ÿ), then the following holds: πœ™ξ€·π‘ž,π‘‡π‘Ÿπ‘₯𝑇+πœ™π‘Ÿξ€Έπ‘₯,π‘₯β‰€πœ™(π‘ž,π‘₯).(2.10)

Lemma 2.8 (see Chang et al. [19]). Let E be a uniformly convex Banach space, π‘Ÿ>0 a positive number, and π΅π‘Ÿ(0) a closed ball of 𝐸. Then, for any given sequence {π‘₯𝑖}βˆžπ‘–=1βŠ‚π΅π‘Ÿ(0) and for any given sequence {πœ†π‘–}βˆžπ‘–=1 of positive number with βˆ‘βˆžπ‘›=1πœ†π‘›=1, there exists a continuous, strictly increasing, and convex function π‘”βˆΆ[0,2π‘Ÿ)β†’[0,∞) with 𝑔(0)=0 such that, for any positive integer 𝑖,𝑗 with 𝑖≠𝑗, β€–β€–β€–β€–βˆžξ“π‘›=1πœ†π‘›π‘₯𝑛‖‖‖‖2β‰€βˆžξ“π‘›=1πœ†π‘›β€–β€–π‘₯𝑛‖‖2βˆ’πœ†π‘–πœ†π‘–π‘”ξ€·β€–β€–π‘₯π‘–βˆ’π‘₯𝑗‖‖.(2.11)

Definition 2.9. A mapping 𝑆 from 𝐢 into itself is said to be equally continuous if it is follows that limπ‘›β†’βˆžβ€–β€–π‘₯π‘›βˆ’π‘¦π‘›β€–β€–=0β‡’limπ‘›β†’βˆžβ€–β€–π‘†π‘›π‘₯π‘›βˆ’π‘†π‘›π‘¦π‘›β€–β€–=0,βˆ€π‘₯𝑛,π‘¦π‘›βˆˆπΆ.(2.12)
A mapping 𝑆 from 𝐢 into itself is said to be uniformly 𝐿-Lipschitz continuous if there exists a constant 𝐿>0 such that ‖𝑆𝑛π‘₯βˆ’π‘†π‘›π‘¦β€–β‰€πΏβ€–π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπΆ.(2.13)
It is easy to know that each 𝐿-Lipschitz continuous mapping is equally continuous, but the converse may be not true.

Definition 2.10. Let {𝑆𝑖}βˆžπ‘–=1βˆΆπΆβ†’πΆ be a sequence of mapping. {𝑆𝑖}βˆžπ‘–=1 is said to be a family of uniformly quasi-πœ™-asymptotically nonexpansive mappings, if βˆ©βˆžπ‘–=1𝐹(𝑆𝑖)β‰ βˆ…, and there exists a sequence {π‘˜π‘›}βŠ‚[1,∞) with π‘˜π‘›β†’1 such that for each 𝑖β‰₯1, πœ™ξ€·π‘,𝑆𝑛𝑖π‘₯ξ€Έβ‰€π‘˜π‘›πœ™(𝑝,π‘₯),βˆ€π‘βˆˆβˆ©βˆžπ‘–=1𝐹𝑆𝑖,π‘₯∈𝐢,βˆ€π‘›β‰₯1.(2.14)

3. Main Results

Theorem 3.1. Let 𝐸 be a uniformly smooth and uniformly convex Banach space, and let 𝐢 be a nonempty closed convex subset of 𝐸. Let 𝑓 be a bifunction from 𝐢×𝐢 to 𝑅=(βˆ’βˆž,+∞) satisfying (𝐴1)βˆ’(𝐴4) and let 𝐴 be a continuous monotone mapping of 𝐢 into πΈβˆ—. Let {𝑆𝑖}βˆžπ‘–=1βˆΆπΆβ†’πΆ be an infinite family of closed equally continuous and uniformly quasi-πœ™-asymptotically nonexpansive mappings with a sequence {π‘˜π‘›}βŠ‚[1,∞),π‘˜π‘›β†’1 such that 𝐹∢=βˆ©βˆžπ‘–=1𝐹(𝑆𝑖)∩GEP(𝑓,𝐴) is a nonempty and bounded subset in 𝐢. Let {π‘₯𝑛} be a sequence generated by π‘₯0πΆβˆˆπΈπ‘β„Žπ‘œπ‘ π‘’π‘›π‘Žπ‘Ÿπ‘π‘–π‘‘π‘Ÿπ‘Žπ‘Ÿπ‘–π‘™π‘¦,1=𝐢,π‘₯1=Π𝐢1π‘₯0,𝑦𝑛=π½βˆ’1𝛼𝑛,0𝐽π‘₯𝑛+βˆžξ“π‘–=1𝛼𝑛,𝑖𝐽𝑆𝑛𝑖π‘₯𝑛ξƒͺ,𝑒𝑛=π‘‡π‘Ÿπ‘›π‘¦π‘›,𝐢𝑛+1=ξ€½π‘§βˆˆπΆπ‘›ξ€·βˆΆπœ™π‘§,π‘’π‘›ξ€Έξ€·β‰€πœ™π‘§,π‘₯𝑛+πœƒπ‘›ξ€Ύ,π‘₯𝑛+1=Π𝐢𝑛+1π‘₯0,βˆ€π‘›=1,2,3,…,(3.1) where 𝐽 is the duality mapping on 𝐸, {𝛼𝑛,𝑖}βˆžπ‘–=0 are sequences in [0,1] which satisfies βˆ‘βˆžπ‘–=0𝛼𝑛,𝑖=1, πœƒπ‘›=supπ‘βˆˆπΉ(π‘˜π‘›βˆ’1)πœ™(𝑝,π‘₯𝑛), and π‘Ÿπ‘›βˆˆ[π‘Ž,+∞) for some π‘Ž>0. If liminfπ‘›β†’βˆžπ›Όπ‘›,0𝛼𝑛,𝑖>0 for all 𝑛β‰₯0, then {π‘₯𝑛} converges strongly to Π𝐹π‘₯0, where Π𝐹 is the generalized projection from 𝐢 onto 𝐹.

Proof. We first show that 𝐢𝑛 is closed and convex. It is obvious that 𝐢𝑛 is closed. In addition, since πœ™ξ€·π‘§,π‘’π‘›ξ€Έξ€·β‰€πœ™π‘§,π‘₯𝑛+πœƒπ‘›βŸΊβ€–β€–π‘’π‘›β€–β€–2βˆ’β€–β€–π‘₯𝑛‖‖2βˆ’2βŸ¨π‘§,π½π‘’π‘›βˆ’π½π‘₯π‘›βŸ©βˆ’πœƒπ‘›β‰€0,(3.2)
so 𝐢𝑛 is convex, therefore, 𝐢𝑛 is a closed convex subset of 𝐸 for all 𝑛β‰₯0.
Next, we show that πΉβŠ‚πΆπ‘› for all 𝑛β‰₯1. It is clear that πΉβŠ‚πΆ1=𝐢. Suppose πΉβˆˆπΆπ‘› for 𝑛>1, by the property of πœ™, βˆ‘βˆžπ‘–=0𝛼𝑛,𝑖=1, Lemmas 2.6 and 2.8, and uniformly quasi-πœ™-asymptotically nonexpansive of 𝑆𝑛 for each π‘’βˆˆπΉβŠ‚πΆπ‘›, then we haveπœ™ξ€·π‘’,𝑒𝑛=πœ™π‘’,π‘‡π‘Ÿπ‘›π‘¦π‘›ξ€Έξ€·β‰€πœ™π‘’,𝑦𝑛=πœ™π‘’,π½βˆ’1𝛼𝑛,0𝐽π‘₯𝑛+βˆžξ“π‘–=1𝛼𝑛,𝑖𝐽𝑆𝑛𝑖π‘₯𝑛ξƒͺξƒͺ=‖𝑒‖2ξ„”ξƒ©π›Όβˆ’2𝑒,𝑛,0𝐽π‘₯𝑛+βˆžξ“π‘–=1𝛼𝑛,𝑖𝐽𝑆𝑛𝑖π‘₯𝑛+‖‖‖‖𝛼ξƒͺ𝑛,0𝐽π‘₯𝑛+βˆžξ“π‘–=1𝛼𝑛,𝑖𝐽𝑆𝑛𝑖π‘₯𝑛‖‖‖‖2≀‖𝑝‖2βˆ’2𝛼𝑛,0βŸ¨π‘,𝐽π‘₯π‘›βŸ©βˆ’2βˆžξ“π‘–=1𝛼𝑛,𝑖𝑒,𝐽𝑆𝑛𝑖π‘₯𝑛+𝛼𝑛,0β€–β€–π‘₯𝑛‖‖2+βˆžξ“π‘–=1𝛼𝑛,𝑖‖‖𝑆𝑛𝑖π‘₯𝑛‖‖2βˆ’π›Όπ‘›,0𝛼𝑛,𝑗𝑔‖‖𝐽π‘₯π‘›βˆ’π½π‘†π‘›π‘—π‘₯𝑛‖‖≀𝛼𝑛,0πœ™ξ€·π‘’,π‘₯𝑛+βˆžξ“π‘–=1𝛼𝑛,π‘–πœ™ξ€·π‘’,𝑆𝑛𝑖π‘₯𝑛≀𝛼𝑛,0πœ™ξ€·π‘’,π‘₯𝑛+βˆžξ“π‘–=1𝛼𝑛,π‘–π‘˜π‘›πœ™ξ€·π‘’,π‘₯π‘›ξ€Έβ‰€π‘˜π‘›πœ™ξ€·π‘’,π‘₯π‘›ξ€Έξ€·β‰€πœ™π‘’,π‘₯𝑛+πœƒπ‘›.(3.3)
This shows that π‘’βˆˆπΆπ‘›+1 implies that πΉβŠ‚πΆπ‘› for all 𝑛β‰₯1 by induction. On the one hand, since π‘₯𝑛+1=Π𝐢𝑛π‘₯0 and 𝐢𝑛+1βŠ‚πΆπ‘› for all 𝑛β‰₯1, we have πœ™ξ€·π‘₯𝑛,π‘₯0ξ€Έξ€·π‘₯β‰€πœ™π‘›+1,π‘₯0ξ€Έ.(3.4)
Therefore {πœ™(π‘₯𝑛,π‘₯0)} is nondecreasing. In the other hand, by Lemma 2.1, we have πœ™ξ€·π‘₯𝑛,π‘₯0ξ€Έξ€·Ξ =πœ™πΆπ‘›π‘₯0,π‘₯0ξ€Έξ€·β‰€πœ™π‘’,π‘₯0ξ€Έξ€·βˆ’πœ™π‘’,π‘₯π‘›ξ€Έξ€·β‰€πœ™π‘’,π‘₯0ξ€Έ,(3.5) for each π‘’βˆˆπΉ(𝑇)βŠ‚πΆπ‘› for all 𝑛β‰₯0. Therefore, {πœ™(π‘₯𝑛,π‘₯0)} is bounded; this together with (3.4) implies that the limit of {πœ™(π‘₯𝑛,π‘₯0)} exists.
Since {πœ™(π‘₯𝑛,π‘₯0)} is bounded, so {π‘₯𝑛} is bounded by (1.7), together with limπ‘›β†’βˆžπ‘˜π‘›=1, we have that limπ‘›β†’βˆžπœƒπ‘›=0.(3.6)
From Lemma 2.1, we have, for any positive integers 𝑛,π‘š, that πœ™ξ€·π‘₯𝑛+π‘š,π‘₯𝑛π‘₯=πœ™π‘›+π‘š,Π𝐢𝑛π‘₯0ξ€Έξ€·π‘₯β‰€πœ™π‘›+π‘š,π‘₯0ξ€Έξ€·Ξ βˆ’πœ™πΆπ‘›π‘₯0,π‘₯0ξ€Έξ€·π‘₯=πœ™π‘›+π‘š,π‘₯0ξ€Έξ€·π‘₯βˆ’πœ™π‘›,π‘₯0ξ€Έ.(3.7)
Because the limit of {πœ™(π‘₯𝑛,π‘₯0)} exists, then we have limπ‘›β†’βˆžπœ™ξ€·π‘₯𝑛+π‘š,π‘₯𝑛=0(3.8) uniformly for positive integers π‘š>1. Since {π‘₯𝑛} is a bounded sequence, by using Lemma 2.4, we have limπ‘›β†’βˆžβ€–β€–π‘₯𝑛+π‘šβˆ’π‘₯𝑛‖‖=0(3.9) uniformly for positive integers π‘š>1. Hence {π‘₯𝑛} is a Cauchy sequence, therefore, there exists a point π‘βˆˆπΆ such that {π‘₯𝑛} converges strongly to 𝑝.
In addition, from (3.7) we have limπ‘›β†’βˆžπœ™(π‘₯𝑛+1,π‘₯𝑛)=0, this together with the fact π‘₯𝑛+1βˆˆπΆπ‘› implies that πœ™ξ€·π‘₯𝑛+1,𝑒𝑛π‘₯β‰€πœ™π‘›+1,π‘₯𝑛+πœƒπ‘›.(3.10)
Taking limit on both side of (3.10) and from (3.6),we get that limπ‘›β†’βˆžπœ™ξ€·π‘₯𝑛+1,𝑒𝑛=0.(3.11)
By using Lemma 2.4, we have limπ‘›β†’βˆžβ€–β€–π‘₯𝑛+1βˆ’π‘’π‘›β€–β€–=0,(3.12) which implies that {𝑒𝑛} converges strongly to 𝑝.
From (3.3), we have πœ™(𝑒,𝑦𝑛)≀(𝑒,π‘₯𝑛)+πœƒπ‘›, together with 𝑒𝑛=π‘‡π‘Ÿπ‘›π‘¦π‘› and Lemma 2.7, we have πœ™ξ€·π‘’π‘›,𝑦𝑛𝑇=πœ™π‘Ÿπ‘›π‘¦π‘›,π‘¦π‘›ξ€Έξ€·β‰€πœ™π‘’,π‘¦π‘›ξ€Έξ€·βˆ’πœ™π‘’,π‘‡π‘Ÿπ‘›π‘¦π‘›ξ€Έξ€·β‰€πœ™π‘’,π‘₯π‘›ξ€Έξ€·βˆ’πœ™π‘’,π‘‡π‘Ÿπ‘›π‘¦π‘›ξ€Έ+πœƒπ‘›ξ€·=πœ™π‘’,π‘₯π‘›ξ€Έξ€·βˆ’πœ™π‘’,𝑒𝑛+πœƒπ‘›(3.13)
for any π‘’βˆˆπΉ. This implies that limπ‘›β†’βˆžπœ™ξ€·π‘’π‘›,𝑦𝑛=0.(3.14)
Therefore, we have limπ‘›β†’βˆžβ€–β€–π‘’π‘›βˆ’π‘¦π‘›β€–β€–=0,(3.15) which implies that {𝑦𝑛} converges strongly to 𝑝. Thus we have proved that π‘₯𝑛→𝑝,𝑒𝑛→𝑝,𝑦𝑛→𝑝,(3.16) as π‘›β†’βˆž, where π‘βˆˆπΆ. From (3.1) ‖‖𝐽π‘₯π‘›βˆ’π½π‘¦π‘›β€–β€–=‖‖‖‖𝐽π‘₯π‘›βˆ’ξƒ©π›Όπ‘›,0𝐽π‘₯𝑛+βˆžξ“π‘–=1𝛼𝑛,𝑖𝐽𝑆𝑛𝑖π‘₯𝑛ξƒͺβ€–β€–β€–β€–=β€–β€–β€–β€–βˆžξ“π‘–=1𝛼𝑛,𝑖𝐽π‘₯π‘›βˆ’π½π‘†π‘›π‘–π‘₯π‘›ξ€Έβ€–β€–β€–β€–β‰€βˆžξ“π‘–=1𝛼𝑛,𝑖‖‖𝐽π‘₯π‘›βˆ’π½π‘†π‘›π‘–π‘₯𝑛‖‖,(3.17) and hence ‖‖𝐽π‘₯π‘›βˆ’π½π‘†π‘›π‘–π‘₯𝑛‖‖≀1βˆ‘βˆžπ‘–=1𝛼𝑛,𝑖‖‖𝐽π‘₯π‘›βˆ’π½π‘¦π‘›β€–β€–.(3.18)
Taking limit on both side of above inequality, by liminfπ‘›β†’βˆžβˆ‘βˆžπ‘–=1𝛼𝑛,𝑖>0 and from (3.16), we have limπ‘›β†’βˆžβ€–β€–π½π‘₯π‘›βˆ’π½π‘†π‘›π‘–π‘₯𝑛‖‖=0.(3.19)
Since π½βˆ’1 is uniformly norm-to-norm continuous on bounded sets, we have limπ‘›β†’βˆžβ€–β€–π‘₯π‘›βˆ’π‘†π‘›π‘–π‘₯𝑛‖‖=0,(3.20) for each 𝑖β‰₯1, together with (3.16), we get that limπ‘›β†’βˆžπ‘†π‘›π‘–π‘₯𝑛=𝑝,(3.21) for each 𝑖β‰₯1. Since 𝑆𝑖 is equally continuous, we have ‖‖𝑆𝑖𝑛+1π‘₯π‘›βˆ’π‘†π‘›π‘–π‘₯𝑛‖‖=‖‖𝑆𝑖𝑛+1π‘₯π‘›βˆ’π‘†π‘–π‘›+1π‘₯𝑛+1β€–β€–+‖‖𝑆𝑖𝑛+1π‘₯𝑛+1βˆ’π‘₯𝑛+1β€–β€–+β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+β€–β€–π‘₯π‘›βˆ’π‘†π‘–π‘›+1π‘₯𝑛‖‖≀𝐿𝑖‖‖π‘₯+1𝑛+1βˆ’π‘₯𝑛‖‖+‖‖𝑆𝑖𝑛+1π‘₯𝑛+1βˆ’π‘₯𝑛+1β€–β€–+β€–β€–π‘₯π‘›βˆ’π‘†π‘–π‘›+1π‘₯𝑛‖‖.(3.22)
Together with (3.16) and (3.20), we have limπ‘›β†’βˆžβ€–π‘†π‘–π‘›+1π‘₯π‘›βˆ’π‘†π‘›π‘–π‘₯𝑛‖=0. From (3.21), we have 𝑆𝑖𝑛+1π‘₯𝑛→𝑝, that is, 𝑆𝑖𝑆𝑛𝑖π‘₯𝑛→𝑝. In view of closeness of 𝑆𝑖, we have 𝑆𝑖𝑝=𝑝, for all 𝑖β‰₯1. This implies that π‘βˆˆβˆ©βˆžπ‘–=1𝐹(𝑆𝑖).
Next we show π‘βˆˆGEP(𝑓,𝐴). By 𝑒𝑛=π‘‡π‘Ÿπ‘›π‘¦π‘›, we have 𝑓𝑒𝑛,𝑦+βŸ¨π΄π‘¦π‘›,π‘¦βˆ’π‘’π‘›1⟩+π‘Ÿπ‘›βŸ¨π‘¦βˆ’π‘’π‘›,π½π‘’π‘›βˆ’π½π‘¦π‘›βŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ.(3.23)
From (A2), we get that βŸ¨π΄π‘¦π‘›,π‘¦βˆ’π‘’π‘›1⟩+π‘Ÿπ‘›βŸ¨π‘¦βˆ’π‘’π‘›,π½π‘’π‘›βˆ’π½π‘¦π‘›ξ€·π‘’βŸ©β‰₯βˆ’π‘“π‘›ξ€Έξ€·,𝑦β‰₯𝑓𝑦,𝑒𝑛,βˆ€π‘¦βˆˆπΆ,(3.24) and hence 0β‰₯βˆ’βŸ¨π΄π‘¦π‘›,π‘¦βˆ’π‘’π‘›ξƒ‘βŸ©βˆ’π‘¦βˆ’π‘’π‘›,π½π‘’π‘›βˆ’π½π‘¦π‘›π‘Ÿπ‘›ξƒ’ξ€·+𝑓𝑦,𝑒𝑛,βˆ€π‘¦βˆˆπΆ.(3.25)
For 𝑑 with 0<𝑑<1 and π‘¦βˆˆπΆ, let 𝑦𝑑=𝑑𝑦+(1βˆ’π‘‘)𝑝, then π‘¦π‘‘βˆˆπΆ, from (3.25) we have βŸ¨π΄π‘¦π‘‘,π‘¦π‘‘βˆ’π‘’π‘›βŸ©β‰₯βŸ¨π΄π‘¦π‘‘,π‘¦π‘‘βˆ’π‘’π‘›βŸ©βˆ’βŸ¨π΄π‘¦π‘›,π‘¦π‘‘βˆ’π‘’π‘›ξƒ‘π‘¦βŸ©βˆ’π‘‘βˆ’π‘’π‘›,π½π‘’π‘›βˆ’π½π‘¦π‘›π‘Ÿπ‘›ξƒ’ξ€·π‘¦+𝑓𝑑,𝑒𝑛=βŸ¨π΄π‘¦π‘‘βˆ’π΄π‘’π‘›,π‘¦π‘‘βˆ’π‘’π‘›βŸ©+βŸ¨π΄π‘’π‘›βˆ’π΄π‘¦π‘›,π‘¦π‘‘βˆ’π‘’π‘›βŸ©βˆ’ξƒ‘π‘¦π‘‘βˆ’π‘’π‘›,π½π‘’π‘›βˆ’π½π‘¦π‘›π‘Ÿπ‘›ξƒ’ξ€·π‘¦+𝑓𝑑,𝑒𝑛.(3.26) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, 𝐴 is monotone and (3.16), we have βŸ¨π΄π‘¦π‘‘,π‘¦π‘‘βˆ’π‘’π‘›βŸ©β‰₯0.(3.27)
It follows from (A4) that 𝑓𝑦𝑑,𝑝≀liminfπ‘›β†’βˆžπ‘“ξ€·π‘¦π‘‘,𝑒𝑛≀limπ‘›β†’βˆžβŸ¨π΄π‘¦π‘‘,π‘¦π‘‘βˆ’π‘’π‘›βŸ©=βŸ¨π΄π‘¦π‘‘,π‘¦π‘‘βˆ’π‘βŸ©=π‘‘βŸ¨π΄π‘¦π‘‘,π‘¦βˆ’π‘βŸ©.(3.28)
From the conditions (A1) and (A4), we have 𝑦0=𝑓𝑑,𝑦𝑑𝑦≀𝑑𝑓𝑑𝑦,𝑦+(1βˆ’π‘‘)𝑓𝑑𝑦,𝑝≀𝑑𝑓𝑑+,𝑦(1βˆ’π‘‘)π‘‘βŸ¨π΄π‘¦π‘‘ξ€·π‘¦,π‘¦βˆ’π‘βŸ©β‰€π‘“π‘‘ξ€Έ,𝑦+(1βˆ’π‘‘)βŸ¨π΄π‘¦π‘‘,π‘¦βˆ’π‘βŸ©.(3.29)
Letting 𝑑→0, we get 𝑓(𝑝,𝑦)+βŸ¨π΄π‘,π‘¦βˆ’π‘βŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ.(3.30)
This implies that π‘βˆˆGEP(𝑓,𝐴).
Finally, we show that 𝑝=Π𝐹π‘₯0. Let 𝑀=Π𝐹π‘₯0, from π‘βˆˆπΉ, we have πœ™ξ€·π‘,π‘₯0ξ€Έξ€·β‰₯πœ™π‘€,π‘₯0ξ€Έ.(3.31)
Since π‘₯𝑛=Π𝐢𝑛π‘₯0 and π‘€βˆˆπΉβŠ‚πΆπ‘›, πœ™ξ€·π‘₯𝑛,π‘₯0ξ€Έξ€·β‰€πœ™π‘€,π‘₯0ξ€Έ,(3.32)
together with above two hands and limπ‘›β†’βˆžπ‘₯𝑛=𝑝, we obtain πœ™ξ€·π‘,π‘₯0ξ€Έξ€·=πœ™π‘€,π‘₯0ξ€Έ.(3.33)
that is 𝑝=𝑀=Π𝐹π‘₯0. The proof is completed.

By using the similar method of proof as in Theorem 3.1, the following theorem is not hard to prove.

Theorem 3.2. Let 𝐸 be a uniformly smooth and uniformly convex Banach space, and let 𝐢 be a nonempty closed convex subset of 𝐸. Let 𝑓 be a bifunction from 𝐢×𝐢 to 𝑅=(βˆ’βˆž,+∞) satisfying (𝐴1)βˆ’(𝐴4). Let {𝑆𝑖}βˆžπ‘–=1βˆΆπΆβ†’πΆ be an infinite family of closed equally continuous and uniformly quasi-πœ™-asymptotically nonexpansive mappings with a sequence {π‘˜π‘›}βŠ‚[1,∞),π‘˜π‘›β†’1 such that 𝐹∢=βˆ©βˆžπ‘–=1𝐹(𝑆𝑖)∩EP(𝑓) is a nonempty and bounded subset in 𝐢. Let {π‘₯𝑛} be a sequence generated by π‘₯0βˆˆπΈπ‘β„Žπ‘œπ‘ π‘’π‘›π‘Žπ‘Ÿπ‘π‘–π‘‘π‘Ÿπ‘Žπ‘Ÿπ‘–π‘™π‘¦,𝐢1=𝐢,π‘₯1=Π𝐢1π‘₯0,𝑦𝑛=π½βˆ’1𝛼𝑛,0𝐽π‘₯𝑛+βˆžξ“π‘–=1𝛼𝑛,𝑖𝐽𝑆𝑛𝑖π‘₯𝑛ξƒͺ,π‘’π‘›βˆˆπΆs.t𝑒.𝑓𝑛+1,π‘¦π‘Ÿπ‘›βŸ¨π‘¦βˆ’π‘’π‘›,π½π‘’π‘›βˆ’π½π‘¦π‘›πΆβŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ,𝑛+1=ξ€½π‘§βˆˆπΆπ‘›ξ€·βˆΆπœ™π‘§,π‘’π‘›ξ€Έξ€·β‰€πœ™π‘§,π‘₯𝑛+πœƒπ‘›ξ€Ύ,π‘₯𝑛+1=Π𝐢𝑛+1π‘₯0,βˆ€π‘›=1,2,3,…,(3.34) where 𝐽 is the duality mapping on 𝐸, {𝛼𝑛,𝑖}βˆžπ‘–=0 are sequences in [0,1] which satisfies βˆ‘βˆžπ‘–=0𝛼𝑛,𝑖=1, πœƒπ‘›=supπ‘βˆˆπΉ(π‘˜π‘›βˆ’1)πœ™(𝑝,π‘₯𝑛), and π‘Ÿπ‘›βˆˆ[π‘Ž,+∞) for some π‘Ž>0. If liminfπ‘›β†’βˆžπ›Όπ‘›,0𝛼𝑛,𝑖>0 for all 𝑛β‰₯0, then {π‘₯𝑛} converges strongly to Π𝐹π‘₯0, where Π𝐹 is the generalized projection from 𝐢 onto 𝐹.

Proof. In Theorem 3.1, put 𝐴=0 we can obtain the conclusion of Theorem 3.2.

Theorem 3.3. Let 𝐸 be a uniformly smooth and uniformly convex Banach space, and let 𝐢 be a nonempty closed convex subset of 𝐸. Let 𝑓 be a bifunction from 𝐢×𝐢 to 𝑅=(βˆ’βˆž,+∞) satisfying (𝐴1)βˆ’(𝐴4) and let 𝐴 be a continuous monotone mapping of 𝐢 into πΈβˆ—. Let π‘†βˆΆπΆβ†’πΆ be an infinite family of closed equally continuous and quasi-πœ™-asymptotically nonexpansive mappings with a sequence {π‘˜π‘›}βŠ‚[1,∞),π‘˜π‘›β†’1 such that 𝐹∢=𝐹(𝑆)∩GEP(𝑓,𝐴) is a nonempty and bounded subset in 𝐢. Let {π‘₯𝑛} be a sequence generated by π‘₯0πΆβˆˆπΈπ‘β„Žπ‘œπ‘ π‘’π‘›π‘Žπ‘Ÿπ‘π‘–π‘‘π‘Ÿπ‘Žπ‘Ÿπ‘–π‘™π‘¦,1=𝐢,π‘₯1=Π𝐢1π‘₯0,𝑦𝑛=π½βˆ’1𝛼𝑛𝐽π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ½π‘†π‘›π‘₯𝑛,𝑒𝑛=π‘‡π‘Ÿπ‘›π‘¦π‘›,𝐢𝑛+1=ξ€½π‘§βˆˆπΆπ‘›ξ€·βˆΆπœ™π‘§,π‘’π‘›ξ€Έξ€·β‰€πœ™π‘§,π‘₯𝑛+πœƒπ‘›ξ€Ύ,π‘₯𝑛+1=Π𝐢𝑛+1π‘₯0,βˆ€π‘›=1,2,3,…,(3.35) where 𝐽 is the duality mapping on 𝐸, {𝛼𝑛} are sequences in [0,1] which satisfies liminfπ‘›β†’βˆžπ›Όπ‘›(1βˆ’π›Όπ‘›)>0 for all 𝑛β‰₯0, πœƒπ‘›=supπ‘βˆˆπΉ(π‘˜π‘›βˆ’1)πœ™(𝑝,π‘₯𝑛) and π‘Ÿπ‘›βˆˆ[π‘Ž,+∞) for some π‘Ž>0. Then {π‘₯𝑛} converges strongly to Π𝐹π‘₯0, where Π𝐹 is the generalized projection from 𝐢 onto 𝐹.

Proof. In Theorem 3.1, put 𝑆𝑖=𝑆, for 𝑖=1,2,…, we can obtain the conclusion of Theorem 3.3.

4. Application for Optimization Problem

In this section, we study a kind of optimization problem by using the result of this paper. that is, we will give an iterative algorithm of solution for the following optimization problem with nonempty set of solutions: maxβ„Ž(π‘₯),π‘₯∈𝐢,(4.1) where β„Ž(π‘₯) is a convex and lower semicontinuous functional defined on a closed convex subset 𝐢 of a Banach space 𝐻. We denoted by 𝑆 the set of solutions of (4.1). Let 𝐹 be a bifunction from 𝐢×𝐢 to 𝑅 defined by 𝑓(π‘₯,𝑦)=β„Ž(π‘₯)βˆ’β„Ž(𝑦). We consider the following equilibrium problem, that is, to find π‘₯∈𝐢 such that 𝑓(π‘₯,𝑦)β‰₯0,βˆ€π‘¦βˆˆπΆ.(4.2)

It is obvious that EP(𝐹)=𝑆, where EP(𝐹) denote the set of solutions of equilibrium problem (4.2). In addition, it is easy to see that 𝑓(π‘₯,𝑦) satisfies the conditions (A1)–(A4) in the Section 2. Therefore, from the Theorem 3.1, we can obtain the following theorem.

Theorem 4.1. Let 𝐸 be a uniformly smooth and uniformly convex Banach space, and let 𝐢 be a nonempty closed convex subset of 𝐸. Let 𝑓 be a bifunction from 𝐢×𝐢 to 𝑅=(βˆ’βˆž,+∞) satisfying (𝐴1)βˆ’(𝐴4) and let 𝐴 be a continuous monotone mapping of 𝐢 into πΈβˆ—. Let {𝑆𝑖}βˆžπ‘–=1βˆΆπΆβ†’πΆ be an infinite family of closed equally continuous and uniformly quasi-πœ™-asymptotically nonexpansive mappings with a sequence {π‘˜π‘›}βŠ‚[1,∞),π‘˜π‘›β†’1 such that 𝐹∢=βˆ©βˆžπ‘–=1𝐹(𝑆𝑖)βˆ©π‘† is a nonempty and bounded subset in 𝐢. Let {π‘₯𝑛} be a sequence generated by π‘₯0πΆβˆˆπΈπ‘β„Žπ‘œπ‘ π‘’π‘›π‘Žπ‘Ÿπ‘π‘–π‘‘π‘Ÿπ‘Žπ‘Ÿπ‘–π‘™π‘¦,1=𝐢,π‘₯1=Π𝐢1π‘₯0,𝑦𝑛=π½βˆ’1𝛼𝑛,0𝐽π‘₯𝑛+βˆžξ“π‘–=1𝛼𝑛,𝑖𝐽𝑆𝑛𝑖π‘₯𝑛ξƒͺ,π‘’π‘›βˆˆπΆ,s.t.β„Žξ€·π‘’π‘›ξ€Έ1βˆ’β„Ž(𝑦)+π‘Ÿπ‘›βŸ¨π‘¦βˆ’π‘’π‘›,π‘’π‘›βˆ’π‘¦π‘›πΆβŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ,𝑛+1=ξ€½π‘§βˆˆπΆπ‘›ξ€·βˆΆπœ™π‘§,π‘’π‘›ξ€Έξ€·β‰€πœ™π‘§,π‘₯𝑛+πœƒπ‘›ξ€Ύ,π‘₯𝑛+1=Π𝐢𝑛+1π‘₯0βˆ€π‘›=1,2,3,…,(4.3) where 𝐽 is the duality mapping on 𝐸, {𝛼𝑛,𝑖}βˆžπ‘–=0 are sequences in [0,1] which satisfies βˆ‘βˆžπ‘–=0𝛼𝑛,𝑖=1, πœƒπ‘›=supπ‘βˆˆπΉ(π‘˜π‘›βˆ’1)πœ™(𝑝,π‘₯𝑛), and π‘Ÿπ‘›βˆˆ[π‘Ž,+∞) for some π‘Ž>0. If liminfπ‘›β†’βˆžπ›Όπ‘›,0𝛼𝑛,𝑖>0 for all 𝑛β‰₯0, then {π‘₯𝑛} converges strongly to Π𝐹π‘₯0, where Π𝐹 is the generalized projection from 𝐢 onto 𝐹.

Proof. By the proof of Theorem 3.2, we can obtain Theorem 4.1.

It is easy to see that, this paper has some new methods and conditions than the conditions of Takahashi and Zembayashi [16]. In this paper, we prove the convergence theorems for uniformly quasi-πœ™-asymptotically nonexpansive mappings and do not need the Kadec-Klee property of Banach space and use the condition of equally continuous that is more weak different from the condition of uniformly 𝐿-Lipscitz.

Acknowledgment

This project is supported by the National Natural Science Foundation of China under Grant (11071279).

References

  1. E. Blum and W. Oettli, β€œFrom optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994. View at Zentralblatt MATH
  2. P. L. Combettes and S. A. Hirstoaga, β€œEquilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117–136, 2005. View at Zentralblatt MATH
  3. C. Jaiboon, W. Chantarangsi, and P. Kumam, β€œA convergence theorem based on a hybrid relaxed extragradient method for generalized equilibrium problems and fixed point problems of a finite family of nonexpansive mappings,” Nonlinear Analysis: Hybrid Systems, vol. 4, no. 1, pp. 199–215, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  4. A. Kangtunyakarn and S. Suantai, β€œHybrid iterative scheme for generalized equilibrium problems and fixed point problems of finite family of nonexpansive mappings,” Nonlinear Analysis: Hybrid Systems, vol. 3, no. 3, pp. 296–309, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  5. P. Kumam, β€œA hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive mapping,” Nonlinear Analysis: Hybrid Systems, vol. 2, no. 4, pp. 1245–1255, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  6. A. Moudafi, β€œSecond-order differential proximal methods for equilibrium problems,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 1, article 18, 2003. View at Zentralblatt MATH
  7. X. Qin, Y. J. Cho, and S. M. Kang, β€œConvergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 225, no. 1, pp. 20–30, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  8. R. Wangkeeree and R. Wangkeeree, β€œStrong convergence of the iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems of an infinite family of nonexpansive mappings,” Nonlinear Analysis: Hybrid Systems, vol. 3, no. 4, pp. 719–733, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  9. P. Kumam and P. Katchang, β€œA viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings,” Nonlinear Analysis: Hybrid Systems, vol. 3, no. 4, pp. 475–486, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  10. Y. Censor, β€œParallel application of block-iterative methods in medical imaging and radiation therapy,” Mathematical Programming, vol. 42, no. 2, pp. 307–325, 1988. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  11. D. Butnariu and Y. Censor, β€œOn a class of barganing schemes for points in the cores of n-person cooperative games,” In press.
  12. P. L. Combettes, β€œThe convex feasibility problem in inage recovery,” in Advances in Imaging and Electron Physics, P. Hawkes, Ed., vol. 95, pp. 155–270, Academic Press, New York, NY, USA, 1996.
  13. S. Kitahara and W. Takahashi, β€œImage recovery by convex combinations of sunny nonexpansive retractions,” Topological Methods in Nonlinear Analysis, vol. 2, no. 2, pp. 333–342, 1993. View at Zentralblatt MATH
  14. F. Kohsaka and W. Takahashi, β€œBlock iterative methods for a finite family of relatively nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2007, Article ID 21972, 18 pages, 2007. View at Zentralblatt MATH
  15. S. Plubtieng and K. Ungchittrakool, β€œHybrid iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2008, Article ID 583082, 19 pages, 2008. View at Zentralblatt MATH
  16. W. Takahashi and K. Zembayashi, β€œStrong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 1, pp. 45–57, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  17. N. Petrot, K. Wattanawitoon, and P. Kumam, β€œA hybrid projection method for generalized mixed equilibrium problems and fixed point problems in Banach spaces,” Nonlinear Analysis: Hybrid Systems, vol. 4, no. 4, pp. 631–643, 2010. View at Publisher Β· View at Google Scholar Β· View at Scopus
  18. S. Saewan and P. Kumam, β€œModified hybrid block iterative algorithm for convex feasibility problems and generalized equilibrium problems for uniformly quasi-φ-asymptotically nonexpansive mappings,” Abstract and Applied Analysis, vol. 2010, Article ID 357120, 22 pages, 2010. View at Publisher Β· View at Google Scholar
  19. S.-S. Chang, J. K. Kim, and X. R. Wang, β€œModified block iterative algorithm for solving convex feasibility problems in Banach spaces,” Journal of Inequalities and Applications, vol. 2010, Article ID 869684, 14 pages, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  20. I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1990. View at Publisher Β· View at Google Scholar
  21. Y. I. Alber, β€œMetric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G. Kartsatos, Ed., vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15–50, Dekker, New York, NY, USA, 1996. View at Zentralblatt MATH
  22. S. Kamimura and W. Takahashi, β€œStrong convergence of a proximal-type algorithm in a Banach space,” SIAM Journal on Optimization, vol. 13, no. 3, pp. 938–945, 2002. View at Publisher Β· View at Google Scholar
  23. W. Nilsrakoo and S. Saejung, β€œStrong convergence to common fixed points of countable relatively quasi-nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 312454, 19 pages, 2008. View at Zentralblatt MATH
  24. D. Butnariu, S. Reich, and A. J. Zaslavski, β€œWeak convergence of orbits of nonlinear operators in reflexive Banach spaces,” Numerical Functional Analysis and Optimization, vol. 24, no. 5-6, pp. 489–508, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  25. S.-Y. Matsushita and W. Takahashi, β€œA strong convergence theorem for relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  26. Y. Su, D. Wang, and M. Shang, β€œStrong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 284613, 8 pages, 2008. View at Zentralblatt MATH