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Journal of Applied Mathematics
VolumeΒ 2012Β (2012), Article IDΒ 961560, 22 pages
http://dx.doi.org/10.1155/2012/961560
Research Article

Generalized Mixed Equilibrium Problems and Fixed Point Problem for a Countable Family of Total Quasi-πœ™-Asymptotically Nonexpansive Mappings in Banach Spaces

1Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China
2Department of Mathematics, College of Statistics and Mathematics, Yunnan University of Finance and Economics, Yunnan, Kunming 650221, China

Received 4 September 2011; Accepted 25 October 2011

Academic Editor: GiuseppeΒ Marino

Copyright Β© 2012 Jinhua Zhu 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 first to introduce the concept of total quasi-πœ™-asymptotically nonexpansive mapping which contains many kinds of mappings as its special cases and then to use a hybrid algorithm to introduce a new iterative scheme 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 a countable family of total quasi-πœ™-asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in an 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 ℝ and ℝ+ the set of all real numbers and all nonnegative real numbers, respectively. We also assume that 𝐸 is a real Banach space, πΈβˆ— is the dual space of 𝐸, 𝐢 is a nonempty closed convex subset of 𝐸, and βŸ¨β‹…,β‹…βŸ© is the pairing between 𝐸 and πΈβˆ—. In the sequel, we denote the strong convergence and weak convergence of a sequence {π‘₯𝑛} by π‘₯𝑛→π‘₯ and π‘₯𝑛⇀π‘₯, respectively, and π½βˆΆπΈβ†’2πΈβˆ— is the normalized duality mapping defined by 𝐽π‘₯(π‘₯)=βˆ—βˆˆπΈβˆ—βˆΆβŸ¨π‘₯,π‘₯βˆ—βŸ©=β€–π‘₯β€–=β€–π‘₯βˆ—β€–ξ€Ύ,π‘₯∈𝐸.(1.1) Let πœ“βˆΆπΆβ†’β„ be a proper real-valued function, π΄βˆΆπΆβ†’πΈβˆ— a nonlinear mapping, and πΉβˆΆπΆΓ—πΆβ†’β„ a bifunction. The β€œso called” generalized mixed equilibrium problem for 𝐹,𝐴,πœ“ is to find π‘₯βˆ—βˆˆπΆ such that 𝐹π‘₯βˆ—ξ€Έ,𝑦+⟨𝐴π‘₯βˆ—,π‘¦βˆ’π‘₯βˆ—ξ€·π‘₯⟩+πœ“(𝑦)βˆ’πœ“βˆ—ξ€Έβ‰₯0,βˆ€π‘¦βˆˆπΆ.(1.2) We denote the set of solutions of (1.2) by GMEP(𝐹,𝐴,πœ“), that is, ξ€½π‘₯GMEP(𝐹,𝐴,πœ“)=βˆ—ξ€·π‘₯βˆˆπΆβˆΆπΉβˆ—ξ€Έ,𝑦+⟨𝐴π‘₯βˆ—,π‘¦βˆ’π‘₯βˆ—ξ€·π‘₯⟩+πœ“(𝑦)βˆ’πœ“βˆ—ξ€Έξ€Ύ.β‰₯0,βˆ€π‘¦βˆˆπΆ(1.3)

Special Examples
(i)If 𝐴=0, then the problem (1.2) is reduced to the mixed equilibrium problem (MEP), and the set of its solutions is denoted by ξ€½π‘₯MEP(𝐹,πœ“)=βˆ—ξ€·π‘₯βˆˆπΆβˆΆΞ˜βˆ—ξ€Έξ€·π‘₯,𝑦+πœ“(𝑦)βˆ’πœ“βˆ—ξ€Έξ€Ύ.β‰₯0,βˆ€π‘¦βˆˆπΆ(1.4)(ii) If πœ“β‰‘0, then the problem (1.2) is reduced to the generalized equilibrium problem (GEP), and the set of its solutions is denoted by ξ€½π‘₯GEP(𝐹,𝐴)=βˆ—ξ€·π‘₯βˆˆπΆβˆΆπΉβˆ—ξ€Έ,𝑦+⟨𝐴π‘₯βˆ—,π‘¦βˆ’π‘₯βˆ—ξ€Ύ.⟩β‰₯0,βˆ€π‘¦βˆˆπΆ(1.5)(iii)If 𝐴=0,πœ“=0, then the problem (1.2) is reduced to the equilibrium problem (EP), and the set of its solutions is denoted by ξ€½π‘₯EP(𝐹)=βˆ—ξ€·π‘₯βˆˆπΆβˆΆπΉβˆ—ξ€Έξ€Ύ.,𝑦β‰₯0,βˆ€π‘¦βˆˆπΆ(1.6)(iv) If 𝐹=0, then the problem (1.2) is reduced to the mixed variational inequality of Browder type (VI), and the set of its solutions is denoted by ξ€½π‘₯VI(𝐢,𝐴,πœ“)=βˆ—βˆˆπΆβˆΆβŸ¨π΄π‘₯βˆ—,π‘¦βˆ’π‘₯βˆ—ξ€·π‘₯⟩+πœ“(𝑦)βˆ’πœ“βˆ—ξ€Έξ€Ύ.β‰₯0,βˆ€π‘¦βˆˆπΆ(1.7)

These show that the problem (1.2) is very general in the sense that numerous problems in physics, optimization, and economics reduce to finding a solution of (1.2). Recently, some methods have been proposed for the generalized mixed equilibrium problem in Banach space (see, e.g., [1–5]).

A Banach space 𝐸 is said to be strictly convex if β€–π‘₯+𝑦‖/2<1 for all π‘₯,π‘¦βˆˆπ‘ˆ={π‘§βˆˆπΈβˆΆβ€–π‘§β€–=1} with π‘₯≠𝑦. 𝐸 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–. 𝐸 is said to be smooth if the limit lim𝑑→0β€–π‘₯+π‘‘π‘¦β€–βˆ’β€–π‘₯‖𝑑(1.8) exists for all π‘₯,π‘¦βˆˆπ‘ˆ. 𝐸 is said to be uniformly smooth if the above limit exists uniformly in π‘₯,π‘¦βˆˆπ‘ˆ.

Remark 1.1. The following basic properties for Banach space 𝐸 and for the normalized duality mapping 𝐽 can be found in Cioranescu [6].(i)If 𝐸 is an arbitrary Banach space, then 𝐽 is monotone and bounded;(ii)If 𝐸 is a strictly convex Banach space, then 𝐽 is strictly monotone;(iii)If 𝐸 is a a smooth Banach space, then 𝐽 is single-valued, and hemicontinuous; that is, 𝐽 is continuous from the strong topology of 𝐸 to the weak star topology of πΈβˆ—;(iv) If 𝐸 is a uniformly smooth Banach space, then 𝐽 is uniformly continuous on each bounded subset of 𝐸;(v)If 𝐸 is a reflexive and strictly convex Banach space with a strictly convex dual πΈβˆ— and π½βˆ—βˆΆπΈβˆ—β†’πΈ is the normalized duality mapping in πΈβˆ—, then π½βˆ’1=π½βˆ—,π½π½βˆ—=πΌπΈβˆ— and π½βˆ—π½=𝐼𝐸;(vi) If 𝐸 is a smooth, strictly convex and reflexive Banach space, then the normalized duality mapping 𝐽 is single valued, one to one and onto;(vii) A Banach space 𝐸 is uniformly smooth if and only if πΈβˆ— is uniformly convex. If 𝐸 is uniformly smooth, then it is smooth and reflexive.

Recall that a Banach space 𝐸 has the Kadec-Klee property, if for any sequence {π‘₯𝑛}βŠ‚πΈ and π‘₯∈𝐸 with π‘₯𝑛⇀π‘₯∈𝐸 and β€–π‘₯𝑛‖→‖π‘₯β€–, then π‘₯𝑛→π‘₯ (as π‘›β†’βˆž). It is well known that if 𝐸 is a uniformly convex Banach space, then 𝐸 has the Kadec-Klee property.

Next we assume that 𝐸 is a smooth, strictly convex and reflexive Banach space and 𝐢 is a nonempty closed convex subset of 𝐸. In the sequel, we always use πœ™βˆΆπΈΓ—πΈβ†’β„+ to denote the Lyapunov functional defined by πœ™(π‘₯,𝑦)=β€–π‘₯β€–2βˆ’2⟨π‘₯,π½π‘¦βŸ©+‖𝑦‖2,βˆ€π‘₯,π‘¦βˆˆπΈ.(1.9)

It is obvious from the definition of πœ™ that ()β€–π‘₯β€–βˆ’β€–π‘¦β€–2)β‰€πœ™(π‘₯,𝑦)≀(β€–π‘₯β€–+‖𝑦‖2,βˆ€π‘₯,π‘¦βˆˆπΈ.(1.10)

Following Alber [7], the generalized projection Ξ πΆβˆΆπΈβ†’πΆ is defined by Π𝐢(π‘₯)=arginfπ‘¦βˆˆπΆπœ™(𝑦,π‘₯),βˆ€π‘₯∈𝐸.(1.11)

Let π‘‡βˆΆπΆβ†’πΆ be a mapping and 𝐹(𝑇) be the set of fixed points of 𝑇.

Recall that a point π‘βˆˆπΆ is said to be an asymptotic fixed point of 𝑇 if there exists a sequence {π‘₯𝑛}βŠ‚πΆ such that π‘₯𝑛⇀𝑝 and β€–π‘₯π‘›βˆ’π‘‡π‘₯𝑛‖→0. We denoted the set of all asymptotic fixed points of 𝑇 by 𝐹(𝑇). A point π‘βˆˆπΆ is said to be a strong asymptotic fixed point of 𝑇, if there exists a sequence {π‘₯𝑛}βŠ‚πΆ such that π‘₯𝑛→𝑝 and β€–π‘₯π‘›βˆ’π‘‡π‘₯𝑛‖→0. We denoted the set of all strong asymptotic fixed points of 𝑇 by 𝐹(𝑇).

Definition 1.2. (1) A mapping π‘‡βˆΆπΆβ†’πΆ is said to be nonexpansive if ‖𝑇π‘₯βˆ’π‘‡π‘¦β€–β‰€β€–π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπΆ.(1.12)
(2) A mapping π‘‡βˆΆπΆβ†’πΆ is said to be relatively nonexpansive [8] if 𝐹(𝑇)β‰ βˆ…,𝐹(𝑇)=𝐹(𝑇) and πœ™(𝑝,𝑇π‘₯)β‰€πœ™(𝑝,π‘₯),βˆ€π‘₯∈𝐢,π‘βˆˆπΉ(𝑇).(1.13)
(3) A mapping π‘‡βˆΆπΆβ†’πΆ is said to be weak relatively nonexpansive [9] if 𝐹(𝑇)β‰ βˆ…,𝐹(𝑇)=𝐹(𝑇) and πœ™(𝑝,𝑇π‘₯)β‰€πœ™(𝑝,π‘₯),βˆ€π‘₯∈𝐢,π‘βˆˆπΉ(𝑇).(1.14)
(4) A mapping π‘‡βˆΆπΆβ†’πΆ 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.

Definition 1.3. (1) A mapping π‘‡βˆΆπΆβ†’πΆ is said to be quasi-πœ™-nonexpansive [10] if 𝐹(𝑇)β‰ βˆ… and πœ™(𝑝,𝑇π‘₯)β‰€πœ™(𝑝,π‘₯),βˆ€π‘₯∈𝐢,π‘βˆˆπΉ(𝑇).(1.15)
(2) A mapping π‘‡βˆΆπΆβ†’πΆ is said to be quasi-πœ™-asymptotically nonexpansive [11], if 𝐹(𝑇)β‰ βˆ… and there exists a real sequence {π‘˜π‘›}βŠ‚[1,∞) with π‘˜π‘›β†’1 such that πœ™(𝑝,𝑇𝑛π‘₯)β‰€π‘˜π‘›πœ™(𝑝,π‘₯),βˆ€π‘›β‰₯1,π‘₯∈𝐢,π‘βˆˆπΉ(𝑇).(1.16)
(3) A mapping π‘‡βˆΆπΆβ†’πΆ is said to be uniformly 𝐿-Lipschitz continuous, if there exists a constant 𝐿>0 such that ‖𝑇𝑛π‘₯βˆ’π‘‡π‘›π‘¦β€–β‰€πΏβ€–π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπΆ,βˆ€π‘›β‰₯1.(1.17)

Definition 1.4. (1) A mapping π‘‡βˆΆπΆβ†’πΆ is said to be total quasi-πœ™-asymptotically nonexpansive if 𝐹(𝑇)β‰ βˆ… and there exist nonnegative real sequences {πœˆπ‘›},{πœ‡π‘›} with πœˆπ‘›β†’0,πœ‡π‘›β†’0 (as π‘›β†’βˆž) and a strictly increasing continuous function πœβˆΆβ„+→ℝ+ with 𝜁(0)=0 such that for all π‘₯∈𝐢,π‘ƒβˆˆπΉ(𝑇)πœ™(𝑝,𝑇𝑛π‘₯)β‰€πœ™(𝑝,π‘₯)+πœˆπ‘›πœ(πœ™(𝑝,π‘₯))+πœ‡π‘›,βˆ€π‘›β‰₯1.(1.18)
(2) A countable family of mappings {𝑇𝑛}βˆΆπΆβ†’πΆ is said to be uniformly total quasi-πœ™-asymptotically nonexpansive, if β‹‚βˆžπ‘–=1𝐹(𝑇𝑖)β‰ βˆ… and there exist nonnegative real sequences {πœˆπ‘›},{πœ‡π‘›} with πœˆπ‘›β†’0,πœ‡π‘›β†’0 (as π‘›β†’βˆž) and a strictly increasing continuous function πœβˆΆβ„+→ℝ+ with 𝜁(0)=0 such that for all β‹‚π‘₯∈𝐢,π‘βˆˆβˆžπ‘–=1𝐹(𝑇𝑖)πœ™ξ€·π‘,𝑇𝑛𝑖π‘₯ξ€Έβ‰€πœ™(𝑝,π‘₯)+πœˆπ‘›πœ(πœ™(𝑝,π‘₯))+πœ‡π‘›,βˆ€π‘›β‰₯1.(1.19)

Remark 1.5. From the definition, it is easy to know that(1)each relatively nonexpansive mapping is closed;(2) taking 𝜁(𝑑)=𝑑,𝑑β‰₯0,πœˆπ‘›=(π‘˜π‘›βˆ’1) and πœ‡π‘›=0, then (1.16) can be rewritten as πœ™ξ€·π‘,𝑇𝑛𝑖π‘₯ξ€Έβ‰€πœ™(𝑝,π‘₯)+πœˆπ‘›πœ(πœ™(𝑝,π‘₯))+πœ‡π‘›,βˆ€π‘›β‰₯1,π‘₯∈𝐢,π‘βˆˆπΉ(𝑇).(1.20) This implies that each quasi-πœ™-asymptotically nonexpansive mapping must be a total quasi-πœ™-asymptotically nonexpansive mapping, but the converse is not true;(3) the class of quasi-πœ™-asymptotically nonexpansive mappings contains properly the class of quasi-πœ™-nonexpansive mappings as a subclass, but the converse is not true;(4)the class of quasi-πœ™-nonexpansive mappings contains properly the class of weak relatively nonexpansive mappings as a subclass, but the converse is not true;(5)the class of weak relatively nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse is not true.

A mapping π΄βˆΆπΆβ†’πΈβˆ— is said to be 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.(1.21)

Remark 1.6. If 𝐴 is an 𝛼-inverse strongly monotone mapping, then it is 1/𝛼-Lipschitz continuous.
Iterative approximation of fixed points for relatively nonexpansive mappings in the setting of Banach spaces has been studied extensively by many authors. In 2005, Matsushita and Takahashi [12] obtained some weak and strong convergence theorems to approximate a fixed point of a single relatively nonexpansive mapping. Recently, Ofoedu and Malonza [4], Zhang [5], Su et al. [13], Zegeye and Shahzad [14], Wattanawitoon and Kumam [15], Qin et al. [16], Takahashi and Zembayashi [17], Chang et al. [18, 19], Yao et al. [20, 21], Qin et al. [22], and Cho et al. [23, 24] extend the notions from relatively nonexpansive mappings, weakly relatively nonexpansive mappings or quasi-πœ™-nonexpansive mappings to quasi-πœ™-asymptotically nonexpansive mappings and also prove some strongence theorems to approximate a common fixed point of quasi-πœ™-nonexpansive mappings or quasi-πœ™-asymptotically nonexpansive mappings.

The purpose of this paper is first to introduce the concept of total quasi-πœ™-asymptotically nonexpansive mapping which contains many kinds of mappings as its special cases, and then by using a hybrid algorithm to introduce a new iterative scheme 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 a countable family of total quasi-πœ™-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results improve and extend the corresponding results in [8, 11–25].

2. Preliminaries

First, we recall some definitions and conclusions.

Lemma 2.1 (see [7, 26]). Let 𝐸 be a smooth, strictly convex and reflexive Banach space and 𝐢 be a nonempty closed convex subset of 𝐸. Then the following conclusions hold:(a)πœ™(π‘₯,Π𝐢𝑦)+πœ™(Π𝐢𝑦,𝑦)β‰€πœ™(π‘₯,𝑦) for all π‘₯∈𝐢 and π‘¦βˆˆπΈ;(b) if π‘₯∈𝐸 and π‘§βˆˆπΆ, then 𝑧=Π𝐢π‘₯βŸΊβŸ¨π‘§βˆ’π‘¦,𝐽π‘₯βˆ’π½π‘§βŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ;(2.1)(c) for π‘₯,π‘¦βˆˆπΈ,πœ™(π‘₯,𝑦)=0 if and only if π‘₯=𝑦.

Remark 2.2. If 𝐸 is a real Hilbert space 𝐻, then πœ™(π‘₯,𝑦)=β€–π‘₯βˆ’π‘¦β€–2 and Π𝐢 is the metric projection 𝑃𝐢 of 𝐻 onto 𝐢.

Lemma 2.3 (see [18]). 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 numbers with βˆ‘βˆžπ‘–=1πœ†π‘–=1, then 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 integers 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‖‖.(2.2)

Lemma 2.4. Let 𝐸 be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and let 𝐢 be a nonempty closed convex subset of 𝐸. Let π‘‡βˆΆπΆβ†’πΆ be a closed and total quasi-πœ™-asymptotically nonexpansive mapping with nonnegative real sequences {πœˆπ‘›},{πœ‡π‘›} and a strictly increasing continuous functions πœβˆΆβ„+→ℝ+ such that πœ‡1=0,πœˆπ‘›β†’0,πœ‡π‘›β†’0 (as π‘›β†’βˆž) and 𝜁(0)=0. Then 𝐹(𝑇) is a closed convex subset of 𝐢.

Proof. Letting {𝑝𝑛} be a sequence in 𝐹(𝑇) with 𝑝𝑛→𝑝 (as π‘›β†’βˆž), we prove that π‘βˆˆπΉ(𝑇). In fact, from the definition of 𝑇, we have πœ™ξ€·π‘π‘›ξ€Έξ€·π‘,π‘‡π‘β‰€πœ™π‘›ξ€Έ,𝑝+𝜈1πœξ€·πœ™ξ€·π‘π‘›,𝑝+πœ‡1(⟢0asπ‘›βŸΆβˆž).(2.3) Therefore we have limπ‘›β†’βˆžπœ™ξ€·π‘π‘›ξ€Έ,𝑇𝑝=limπ‘›β†’βˆžξ‚€β€–β€–π‘π‘›β€–β€–2βˆ’2βŸ¨π‘π‘›ξ‚,π½π‘‡π‘βŸ©+‖𝑇𝑝‖2=‖𝑝‖2βˆ’2βŸ¨π‘,π½π‘‡π‘βŸ©+‖𝑇𝑝‖2=πœ™(𝑝,𝑇𝑝)=0,(2.4) that is, π‘βˆˆπΉ(𝑇).
Next we prove that 𝐹(𝑇) is convex. For any 𝑝,π‘žβˆˆπΉ(𝑇),π‘‘βˆˆ(0,1), putting 𝑀=𝑑𝑝+(1βˆ’π‘‘)π‘ž, we prove that π‘€βˆˆπΉ(𝑇). Indeed, in view of the definition of πœ™(π‘₯,𝑦), we have πœ™(𝑀,𝑇𝑛𝑀)=‖𝑀‖2βˆ’2βŸ¨π‘€,π½π‘‡π‘›π‘€βŸ©+‖𝑇𝑛𝑀‖2=‖𝑀‖2βˆ’2π‘‘βŸ¨π‘,π½π‘‡π‘›π‘€βŸ©βˆ’2(1βˆ’π‘‘)βŸ¨π‘ž,π½π‘‡π‘›π‘€βŸ©+‖𝑇𝑛𝑀‖2=‖𝑀‖2+π‘‘πœ™(𝑝,𝑇𝑛𝑀)+(1βˆ’π‘‘)πœ™(π‘ž,𝑇𝑛𝑀)βˆ’π‘‘β€–π‘β€–2βˆ’(1βˆ’π‘‘)β€–π‘žβ€–2≀‖𝑀‖2ξ€·+π‘‘πœ™(𝑝,𝑀)+πœˆπ‘›πœ(πœ™(𝑝,𝑀))+πœ‡π‘›ξ€Έξ€·+(1βˆ’π‘‘)πœ™(π‘ž,𝑀)+πœˆπ‘›πœ(πœ™(π‘ž,𝑀))+πœ‡π‘›ξ€Έβˆ’π‘‘β€–π‘β€–2βˆ’(1βˆ’π‘‘)β€–π‘žβ€–2=‖𝑀‖2ξ€·+𝑑‖𝑝‖2βˆ’2βŸ¨π‘,π½π‘€βŸ©+‖𝑀‖2ξ€Έξ€·πœˆ+π‘‘π‘›πœ(πœ™(𝑝,𝑀))+πœ‡π‘›ξ€ΈΓ—ξ€·+(1βˆ’π‘‘)β€–π‘žβ€–2βˆ’2βŸ¨π‘ž,π½π‘€βŸ©+‖𝑀‖2ξ€Έ+ξ€·πœˆ(1βˆ’π‘‘)π‘›πœ(πœ™(π‘ž,𝑀))+πœ‡π‘›ξ€Έβˆ’π‘‘β€–π‘β€–2βˆ’(1βˆ’π‘‘)β€–π‘žβ€–2=‖𝑀‖2βˆ’2βŸ¨π‘€,π½π‘€βŸ©+‖𝑀‖2+π‘‘πœˆπ‘›ξ€·πœˆπœ(πœ™(𝑝,𝑀))+(1βˆ’π‘‘)π‘›πœ(πœ™(π‘ž,𝑀))+πœ‡π‘›=π‘‘πœˆπ‘›ξ€·πœˆπœ(πœ™(𝑝,𝑀))+(1βˆ’π‘‘)π‘›πœ(πœ™(π‘ž,𝑀))+πœ‡π‘›.(2.5) Since πœ‡π‘›β†’0 and πœˆπ‘›β†’0, we have πœ™(𝑀,𝑇𝑛𝑀)β†’0 (as π‘›β†’βˆž). From (1.10) we have ‖𝑇𝑛𝑀‖→‖𝑀‖. Consequently ‖𝐽𝑇𝑛𝑀‖→‖𝐽𝑀‖. This implies that {𝐽𝑇𝑛𝑀} is a bounded sequence. Since 𝐸 is reflexive, πΈβˆ— is also reflexive. So we can assume that 𝐽𝑇𝑛𝑀⇀𝑓0βˆˆπΈβˆ—.(2.6) Again since 𝐸 is reflexive, we have 𝐽(𝐸)=πΈβˆ—. Therefore there exists π‘₯∈𝐸 such that 𝐽π‘₯=𝑓0. By virtue of the weakly lower semicontinuity of norm β€–β‹…β€–, we have 0=liminfπ‘›β†’βˆžπœ™(𝑀,𝑇𝑛𝑀)=liminfπ‘›β†’βˆžξ€·β€–π‘€β€–2βˆ’2βŸ¨π‘€,𝐽(𝑇𝑛𝑀)⟩+‖𝑇𝑛𝑀‖2ξ€Έ=liminfπ‘›β†’βˆžξ€·β€–π‘€β€–2βˆ’2βŸ¨π‘€,𝐽(𝑇𝑛𝑀)⟩+‖𝐽(𝑇𝑛‖𝑀)2ξ€Έβ‰₯‖𝑀‖2βˆ’2βŸ¨π‘€,𝑓0β€–β€–π‘“βŸ©+0β€–β€–2=‖𝑀‖2βˆ’2βŸ¨π‘€,𝐽π‘₯⟩+‖𝐽π‘₯β€–2=‖𝑀‖2βˆ’2βŸ¨π‘€,𝐽π‘₯⟩+β€–π‘₯β€–2=πœ™(𝑀,π‘₯),(2.7) that is, 𝑀=π‘₯ which implies that 𝑓0=𝐽𝑀. Hence from (2.6) we have π½π‘‡π‘›π‘€β‡€π½π‘€βˆˆπΈβˆ—. Since ‖𝐽𝑇𝑛𝑀‖→‖𝑀‖ and πΈβˆ— has the Kadec-Klee property, we have 𝐽𝑇𝑛𝑀→𝐽𝑀. Since 𝐸 is uniformly smooth, πΈβˆ— is uniformly convex, which in turn implies that πΈβˆ— is smooth. From Remark 1.1(iii) it yields that π½βˆ’1βˆΆπΈβˆ—β†’πΈ is hemi-continuous. Therefore we have 𝑇𝑛𝑀⇀𝑀. Again since ‖𝑇𝑛𝑀‖→‖𝑀‖, by using the Kadec-Klee property of 𝐸, we have 𝑇𝑛𝑀→𝑀. This implies that 𝑇𝑇𝑛𝑀=𝑇𝑛+1𝑀→𝑀. Since 𝑇 is closed, we have 𝑀=𝑇𝑀.
This completes the proof of Lemma 2.4.

Lemma 2.5. Let 𝐸 be a smooth, strictly convex and reflexive Banach space and 𝐢 be a nonempty closed convex subset of 𝐸. Let π‘“βˆΆπΆΓ—πΆβ†’β„ be a bifunction satisfying the following conditions:(𝐴1) 𝑓(π‘₯,π‘₯)=0,forallπ‘₯∈𝐢,(𝐴2)𝑓 is monotone, that is, 𝑓(π‘₯,𝑦)+𝑓(𝑦,π‘₯)≀0,forallπ‘₯,π‘¦βˆˆπΆ,(𝐴3)limsup𝑑↓0𝑓(π‘₯+𝑑(π‘§βˆ’π‘₯),𝑦)≀𝑓(π‘₯,𝑦)forallπ‘₯,𝑧,π‘¦βˆˆπΆ,(𝐴4) The function 𝑦↦𝑓(π‘₯,𝑦) is convex and lower semi-continuous.
Then the following conclusions hold:(1) (Blum and Oettli [27]) for any given π‘Ÿ>0 and π‘₯∈𝐸, there exists a unique π‘§βˆˆπΆ such that 1𝑓(𝑧,𝑦)+π‘ŸβŸ¨π‘¦βˆ’π‘§,π½π‘§βˆ’π½π‘₯⟩β‰₯0,βˆ€π‘¦βˆˆπΆ;(2.8)(2) (Takahashi and Zembayashi [28]) for any given π‘Ÿ>0 and π‘₯∈𝐸, define a mapping πΎπ‘“π‘ŸβˆΆπΈβ†’πΆ by πΎπ‘“π‘Ÿξ‚†1(π‘₯)=π‘§βˆˆπΆβˆΆπ‘“(𝑧,𝑦)+π‘Ÿξ‚‡βŸ¨π‘¦βˆ’π‘§,π½π‘§βˆ’π½π‘₯⟩β‰₯0,βˆ€π‘¦βˆˆπΆ,π‘₯∈𝐸.(2.9) Then, the following conclusions hold:(a)πΎπ‘“π‘Ÿ is single-valued;(b)πΎπ‘“π‘Ÿ is firmly nonexpansive-type mapping, that is, for all𝑧,π‘¦βˆˆπΈ, ξ‚¬πΎπ‘“π‘Ÿπ‘§βˆ’πΎπ‘“π‘Ÿπ‘¦,π½πΎπ‘“π‘Ÿπ‘§βˆ’π½πΎπ‘“π‘Ÿπ‘¦ξ‚­β‰€ξ‚¬πΎπ‘“π‘Ÿπ‘§βˆ’πΎπ‘“π‘Ÿξ‚­;𝑦,π½π‘§βˆ’π½π‘¦(2.10)(c)𝐹(πΎπ‘“π‘Ÿ)=EP(𝑓) and πΎπ‘“π‘Ÿ is quasi-πœ™-nonexpansive;(d)EP(𝑓) is closed and convex;(e)πœ™(π‘ž,πΎπ‘“π‘Ÿπ‘₯)+πœ™(πΎπ‘“π‘Ÿπ‘₯,π‘₯)β‰€πœ™(π‘ž,π‘₯),forallπ‘žβˆˆπΉ(πΎπ‘“π‘Ÿ).
For solving the generalized mixed equilibrium problem (1.2), let us assume that the following conditions are satisfied:(1)𝐸 is a smooth, strictly convex, and reflexive Banach space and 𝐢 is a nonempty closed convex subset of 𝐸;(2)π΄βˆΆπΆβ†’πΈβˆ— is 𝛽-inverse strongly monotone mapping;(3)πΉβˆΆπΆΓ—πΆβ†’β„ is bifunction satisfying the conditions (A1), (A3), (A4) in Lemma 2.5 and the following condition (𝐴2)ξ…ž:(𝐴2)ξ…ž for some 𝛾β‰₯0 with 𝛾≀𝛽𝐹(π‘₯,𝑦)+𝐹(𝑦,π‘₯)≀𝛾‖𝐴π‘₯βˆ’π΄π‘¦β€–2,βˆ€π‘₯,π‘¦βˆˆπΆ;(2.11)(4)πœ“βˆΆπΆβ†’β„ is a lower semicontinuous and convex function.
Under the assumptions as above, we have the following results.

Lemma 2.6. Let 𝐸,𝐢,𝐴,𝐹,πœ“ satisfy the above conditions (1)–(4). Denote by Ξ“(π‘₯,𝑦)=𝐹(π‘₯,𝑦)+πœ“(𝑦)βˆ’πœ“(π‘₯)+⟨𝐴π‘₯,π‘¦βˆ’π‘₯⟩,βˆ€π‘₯,π‘¦βˆˆπΆ.(2.12) For any given π‘Ÿ>0 and π‘₯∈𝐸, define a mapping πΎΞ“π‘ŸβˆΆπΈβ†’πΆ by πΎΞ“π‘Ÿξ‚†1(π‘₯)=π‘§βˆˆπΆβˆΆΞ“(𝑧,𝑦)+π‘Ÿξ‚‡.βŸ¨π‘¦βˆ’π‘§,π½π‘§βˆ’π½π‘₯⟩β‰₯0,βˆ€π‘¦βˆˆπΆ(2.13) Then, the following hold:(a)πΎΞ“π‘Ÿ is single-valued;(b)πΎΞ“π‘Ÿ is a firmly nonexpansive-type mapping, that is, for all 𝑧,π‘¦βˆˆπΈ, ξ«πΎΞ“π‘Ÿ(𝑧)βˆ’πΎΞ“π‘Ÿ(𝑦),π½πΎΞ“π‘Ÿ(𝑧)βˆ’π½πΎΞ“r≀𝐾(𝑦)Ξ“π‘Ÿ(𝑧)βˆ’πΎΞ“π‘Ÿξ¬;(𝑦),π½π‘§βˆ’π½π‘¦(2.14)(c)𝐹(πΎΞ“π‘Ÿ)=EP(Ξ“)=GMEP(𝐹,𝐴,πœ“);(d)GMEP(𝐹,𝐴,πœ“) is closed and convex;(e)  πœ™ξ€·π‘ž,πΎΞ“π‘Ÿπ‘₯𝐾+πœ™Ξ“π‘Ÿξ€Έξ€·πΎπ‘₯,π‘₯β‰€πœ™(π‘ž,π‘₯),βˆ€π‘žβˆˆπΉΞ“π‘Ÿξ€Έ.(2.15)

Proof. It follows from Lemma 2.5 that in order to prove the conclusions of Lemma 2.6 it is sufficient to prove that the function Ξ“βˆΆπΆΓ—πΆβ†’β„ satisfies the conditions (A1)–(A4) in Lemma 2.5.
In fact, by the similar method as given in the proof of Lemma 2.4 in [1], we can prove that the function Ξ“ satisfies the conditions (A1), (A3), and (A4). Now we prove that Ξ“ also satisfies the conditions (A2).
Indeed, for any π‘₯,π‘¦βˆˆπΆ, by condition (A2)β€², we have Ξ“(π‘₯,𝑦)+Ξ“(𝑦,π‘₯)=𝐹(π‘₯,𝑦)+πœ“(𝑦)βˆ’πœ“(π‘₯)+⟨𝐴π‘₯,π‘¦βˆ’π‘₯⟩+𝐹(𝑦,π‘₯)+πœ“(π‘₯)βˆ’πœ“(𝑦)+βŸ¨π΄π‘¦,π‘₯βˆ’π‘¦βŸ©=𝐹(π‘₯,𝑦)+𝐹(𝑦,π‘₯)βˆ’βŸ¨π΄π‘₯βˆ’π΄π‘¦,π‘₯βˆ’π‘¦βŸ©β‰€(π›Ύβˆ’π›½)‖𝐴π‘₯βˆ’π΄π‘¦β€–2≀0.(2.16) This implies that the function Ξ“ satisfies the conditions (A2). Therefore the conclusions of Lemma 2.6 can be obtained from Lemma 2.3 immediately.

Remark 2.7. It follows from Lemma 2.5 that the mapping πΎΞ“π‘Ÿ is a relatively nonexpansive mapping. Thus, it is quasi-πœ™-nonexpansive.

3. Main Results

In this section, we shall 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 (1.2) and the set of common fixed points of a countable family of total quasi-πœ™-asymptotically nonexpansive mappings in Banach spaces.

In the sequel, we assume that 𝐸,𝐢,{𝑆𝑖}βˆžπ‘–=1,{𝐴𝑖}𝑀𝑖=1,{𝐹𝑖}𝑀𝑖=1,{πœ“π‘–}𝑀𝑖=1 satisfy the following conditions.(1)Let 𝐸 be a uniformly smooth and strictly convex Banach space with Kleac-Klee property and 𝐢 a nonempty closed convex subset of 𝐸.(2) Let π‘†π‘–βˆΆπΆβ†’πΆ be a countable family of closed and uniformly total quasi-πœ™-asymptotically nonexpansive mappings with nonnegative real sequences {πœˆπ‘›},{πœ‡π‘›} and a strictly increasing continuous functions πœβˆΆβ„+→ℝ+ such that πœˆπ‘›β†’0,πœ‡π‘›β†’0 (as π‘›β†’βˆž) and πœ‡1=0,𝜁(0)=0. Suppose further that for each 𝑖β‰₯1,𝑆𝑖 is a uniformly 𝐿𝑖-Lipschitz mapping, that is, there exists a constant 𝐿𝑖>0 such that ‖‖𝑆𝑛𝑖π‘₯βˆ’π‘†π‘›π‘–π‘¦β€–β€–β‰€πΏπ‘–β€–π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπΆ,βˆ€π‘›β‰₯1.(3.1)(3) Let π΄π‘–βˆΆπΆβ†’πΈβˆ—(𝑖=1,2,…,𝑀) be a finite family of 𝛽𝑖-inverse strongly monotone mappings.(4) Let πΉπ‘–βˆΆπΆβ†’β„(𝑖=1,2,…,𝑀) be a finite family of bifunction satisfying the conditions (A1), (A3), (A4), and the following condition (A2)β€²:(𝐴2)ξ…ž For each 𝑖=1,2,…,𝑀 there exists 𝛾𝑖β‰₯0 with 𝛾𝑖≀𝛽𝑖 such that 𝐹𝑖(π‘₯,𝑦)+𝐹𝑖(𝑦,π‘₯)≀𝛾𝑖‖‖𝐴𝑖π‘₯βˆ’π΄π‘–π‘¦β€–β€–2,βˆ€π‘₯,π‘¦βˆˆπΆ;(3.2)(5) Let πœ“π‘–βˆΆπΆβ†’β„(𝑖=1,2,…,𝑀) be a finite family of lower semicontinuous and convex functions.

Theorem 3.1. Let 𝐸,𝐢,{𝑆𝑖}βˆžπ‘–=1,{𝐴𝑖}𝑀𝑖=1,{𝐹𝑖}𝑀𝑖=1,{πœ“π‘–}𝑀𝑖=1 be the same as above. Suppose that β„±βˆΆ=βˆžξ™π‘–=1𝐹𝑇𝑖𝑀𝑗=1𝐹GMEP𝑗,𝐴𝑗,πœ“π‘—ξ€Έ(3.3) is a nonempty and bounded subset of 𝐢. For any given π‘₯0∈𝐢, let {π‘₯𝑛} be the sequence generated by π‘₯0∈𝐢0𝑧=𝐢,𝑛=π½βˆ’1𝛼𝑛,0𝐽π‘₯𝑛+βˆžξ“π‘–=1𝛼𝑛,𝑖𝐽𝑆𝑛𝑖π‘₯𝑛ξƒͺ,𝑦𝑛=π½βˆ’1𝛼𝑛𝐽𝑧𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ½π‘₯𝑛,𝑒𝑛=πΎΞ“π‘€π‘Ÿπ‘€,π‘›πΎΞ“π‘€βˆ’1π‘ŸMβˆ’1,𝑛⋯𝐾Γ2π‘Ÿ2,𝑛𝐾Γ1π‘Ÿ1,𝑛𝑦𝑛,𝐢𝑛+1=ξ€½π‘£βˆˆπΆπ‘›ξ€·βˆΆπœ™π‘£,π‘’π‘›ξ€Έξ€·β‰€πœ™π‘£,π‘₯𝑛+πœ‚π‘›ξ€Ύ,π‘₯𝑛+1=Π𝐢𝑛+1π‘₯0,βˆ€π‘›β‰₯0,(3.4) where πœ‚π‘›=πœˆπ‘›supπ‘’βˆˆβ„±πœξ€·πœ™ξ€·π‘’,π‘₯𝑛+πœ‡π‘›,βˆ€π‘›β‰₯1,(3.5)πΎΞ“π‘–π‘Ÿπ‘–,π‘›βˆΆπΈβ†’πΆ,𝑖=1,2,…,𝑀 is the mapping defined by (2.13) with Ξ“=Γ𝑖,π‘Ÿ=π‘Ÿπ‘–,𝑛, and Γ𝑖(π‘₯,𝑦)=𝐹𝑖(π‘₯,𝑦)+βŸ¨π΄π‘–π‘₯,π‘¦βˆ’π‘₯⟩+πœ“π‘–(𝑦)βˆ’πœ“π‘–(π‘₯),βˆ€π‘₯,π‘¦βˆˆπΆ.(3.6)π‘Ÿπ‘˜,π‘›βˆˆ[𝑑,∞),π‘˜=1,2,…,𝑀,𝑛β‰₯1 for some 𝑑>0,Π𝐢𝑛+1 is the generalized projection of 𝐸 onto the set 𝐢𝑛+1,and {𝛼𝑛,𝑖},{𝛼𝑛} are sequences in [0,1] satisfying the following conditions:(a)βˆ‘βˆžπ‘–=0𝛼𝑛,𝑖=1 for all 𝑛β‰₯0;(b)liminfπ‘›β†’βˆžπ›Όπ‘›,0⋅𝛼𝑛,𝑖>0 for all 𝑖β‰₯1;(c)0<𝛼≀𝛼𝑛<1 for some π›Όβˆˆ(0,1).Then {π‘₯𝑛} converges strongly to Ξ β„±π‘₯0, where Ξ β„± is the generalized projection from 𝐸 onto β„±.

Proof. We divide the proof of Theorem 3.1 into five steps.
(i) We first prove that β„± and 𝐢𝑛 both are closed and convex subset of 𝐢 for all 𝑛β‰₯0.
In fact, it follows from Lemmas 2.4 and 2.6 that 𝐹(𝑆𝑖),𝑖β‰₯1 and GMEP(𝐹𝑗,𝐴𝑗,πœ“π‘—)(𝑗=1,2,…,𝑀) both are closed and convex. Therefore β„± is a closed and convex subset in 𝐢. Furthermore, it is obvious that 𝐢0=𝐢 is closed and convex. Suppose that 𝐢𝑛 is closed and convex for some 𝑛β‰₯1. Since the inequality πœ™(𝑣,𝑒𝑛)β‰€πœ™(𝑣,π‘₯𝑛)+πœ‚π‘› is equivalent to 2βŸ¨π‘£,𝐽π‘₯π‘›βˆ’π½π‘’π‘›β€–β€–π‘₯βŸ©β‰€π‘›β€–β€–2βˆ’β€–β€–π‘’π‘›β€–β€–2+πœ‚π‘›,(3.7) therefore, we have𝐢𝑛+1=ξ‚†π‘£βˆˆπΆπ‘›βˆΆ2βŸ¨π‘£,𝐽π‘₯π‘›βˆ’π½π‘’π‘›β€–β€–π‘₯βŸ©β‰€π‘›β€–β€–2βˆ’β€–β€–π‘’π‘›β€–β€–2+πœ‚π‘›ξ‚‡.(3.8)
This implies that 𝐢𝑛+1 is closed and convex. The desired conclusions are proved. These in turn show that Ξ β„±π‘₯0 and Π𝐢𝑛π‘₯0 are well defined.
(ii) We prove that {π‘₯𝑛} and {𝑆𝑛𝑖π‘₯𝑛}βˆžπ‘›=0 for all 𝑖β‰₯1 are both bounded sequences in 𝐢.
By the definition of 𝐢𝑛, we have π‘₯𝑛=Π𝐢𝑛π‘₯0 for all 𝑛β‰₯0. It follows from Lemma 2.1 (a) that 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‘’,Π𝐢𝑛π‘₯0ξ€Έξ€·β‰€πœ™π‘’,π‘₯0ξ€Έ,βˆ€π‘›β‰₯0,π‘’βˆˆβ„±.(3.9) This implies that {πœ™(π‘₯𝑛,π‘₯0)} is bounded. By virtue of (1.10), {π‘₯𝑛} is bounded. Since πœ™(𝑒,𝑆𝑛𝑖π‘₯𝑛)β‰€πœ™(𝑒,π‘₯𝑛)+πœˆπ‘›πœ(πœ™(𝑒,π‘₯𝑛))+πœ‡π‘› for all π‘’βˆˆβ„± and 𝑖β‰₯1,{𝑆𝑛𝑖π‘₯𝑛} is bounded for all 𝑖β‰₯1, and so {𝑧𝑛}is bounded in 𝐸. Denote 𝑀 by 𝑀=sup𝑛β‰₯0,𝑖β‰₯1ξ€½β€–β€–π‘₯𝑛‖‖,‖‖𝑆𝑛𝑖π‘₯𝑛‖‖,‖‖𝑧𝑛‖‖<∞.(3.10)
In view of the structure of {𝐢𝑛}, we have 𝐢𝑛+1βŠ‚πΆπ‘›,π‘₯𝑛=Π𝐢𝑛π‘₯0 and π‘₯𝑛+1=Π𝐢𝑛+1π‘₯0. This implies that π‘₯𝑛+1βˆˆπΆπ‘› and πœ™ξ€·π‘₯𝑛,π‘₯0ξ€Έξ€·π‘₯β‰€πœ™π‘›+1,π‘₯0ξ€Έ,βˆ€π‘›β‰₯1.(3.11) Therefore {πœ™(π‘₯𝑛,π‘₯0)} is convergent. Without loss of generality, we can assume that limπ‘›β†’βˆžπœ™ξ€·π‘₯𝑛,π‘₯0ξ€Έ=π‘Ÿβ‰₯0.(3.12)(iii) Next, we prove that β‹‚β„±βˆΆ=βˆžπ‘–=1𝐹(𝑆𝑖)⋂⋂𝑀𝑖=1GMEP(𝐹𝑖,𝐴𝑖,πœ“π‘–)βŠ‚πΆπ‘› for all 𝑛β‰₯0.
Indeed, it is obvious that β„±βŠ‚πΆ0=𝐢. Suppose that β„±βŠ‚πΆπ‘› for some 𝑛β‰₯0. Since 𝑒𝑛=πΎΞ“π‘€π‘Ÿπ‘€,π‘›πΎΞ“π‘€βˆ’1π‘Ÿπ‘€βˆ’1,𝑛⋯𝐾Γ2π‘Ÿ2,𝑛𝐾Γ1π‘Ÿ1,𝑛𝑦𝑛, by Lemma 2.6 and Remark 2.7, πΎΞ“π‘–π‘Ÿπ‘–,𝑛 is quasi-πœ™-nonexpansive. Again since 𝐸 is uniformly smooth, πΈβˆ— is uniformly convex. Hence, For any given π‘’βˆˆβ„±βŠ‚πΆπ‘› and for any positive integer 𝑗>0, from Lemma 2.3 we have πœ™ξ€·π‘’,𝑒𝑛=πœ™π‘’,πΎΞ“π‘€π‘Ÿπ‘€,π‘›πΎΞ“π‘€βˆ’1π‘Ÿπ‘€βˆ’1,𝑛⋯𝐾Γ2π‘Ÿ2,𝑛𝐾Γ1π‘Ÿ1,π‘›π‘¦π‘›ξ‚ξ€·β‰€πœ™π‘’,𝑦𝑛=πœ™π‘’,π½βˆ’1𝛼𝑛𝐽𝑧𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ½π‘₯𝑛≀‖𝑒‖2ξ«βˆ’2𝑒,𝛼𝑛𝐽𝑧𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ½π‘₯𝑛+‖‖𝛼𝑛𝐽𝑧𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ½π‘₯𝑛‖‖2≀‖𝑒‖2ξ«βˆ’2𝑒,𝛼𝑛𝐽𝑧𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ½π‘₯𝑛+𝛼𝑛‖‖𝐽𝑧𝑛‖‖2+ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π½π‘₯𝑛‖‖2=‖𝑒‖2ξ«βˆ’2𝑒,𝛼𝑛𝐽𝑧𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ½π‘₯𝑛+𝛼𝑛‖‖𝑧𝑛‖‖2+ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘₯𝑛‖‖2=π›Όπ‘›πœ™ξ€·π‘’,𝑧𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπœ™ξ€·π‘’,π‘₯𝑛=π›Όπ‘›πœ™ξƒ©π‘’,π½βˆ’1𝛼𝑛,0𝐽π‘₯𝑛+βˆžξ“π‘–=1𝛼𝑛,𝑖𝐽𝑆𝑛𝑖π‘₯𝑛+ξ€·ξƒͺξƒͺ1βˆ’π›Όπ‘›ξ€Έπœ™ξ€·π‘’,π‘₯𝑛=π›Όπ‘›βŽ›βŽœβŽœβŽβ€–π‘’β€–2βˆ’2𝛼𝑛,0βŸ¨π‘’,𝐽π‘₯π‘›βŸ©βˆ’2βˆžξ“π‘–=1𝛼𝑛,𝑖𝑒,𝐽𝑆𝑛𝑖π‘₯𝑛+‖‖‖‖𝛼𝑛,0𝐽π‘₯𝑛+βˆžξ“π‘–=1𝛼𝑛,𝑖𝐽𝑆𝑛𝑖π‘₯𝑛‖‖‖‖2⎞⎟⎟⎠+ξ€·1βˆ’π›Όπ‘›ξ€Έπœ™ξ€·π‘’,π‘₯𝑛≀𝛼𝑛‖𝑒‖2βˆ’2𝛼𝑛,0βŸ¨π‘’,𝐽π‘₯π‘›βŸ©βˆ’2βˆžξ“π‘–=1𝛼𝑛,𝑖𝑒,𝐽𝑆𝑛𝑖π‘₯𝑛+𝛼𝑛,0‖‖𝐽π‘₯𝑛‖‖2+βˆžξ“π‘–=1𝛼𝑛,𝑖‖‖𝐽𝑆𝑛𝑖π‘₯𝑛‖‖2βˆ’π›Όπ‘›,0𝛼𝑛,𝑗𝑔‖‖𝐽π‘₯π‘›βˆ’π½π‘†π‘›π‘—π‘₯𝑛‖‖ξƒͺ+ξ€·1βˆ’π›Όπ‘›ξ€Έπœ™ξ€·π‘’,π‘₯𝑛≀𝛼𝑛‖𝑒‖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‖‖ξƒͺ+ξ€·1βˆ’π›Όπ‘›ξ€Έπœ™ξ€·π‘’,π‘₯𝑛=𝛼𝑛𝛼𝑛,0πœ™ξ€·π‘’,π‘₯𝑛+βˆžξ“π‘–=1𝛼𝑛,π‘–πœ™ξ€·π‘’,𝑆𝑛𝑖π‘₯π‘›ξ€Έβˆ’π›Όπ‘›,0𝛼𝑛,𝑗𝑔‖‖𝐽π‘₯π‘›βˆ’π½π‘†π‘›π‘—π‘₯𝑛‖‖ξƒͺ+ξ€·1βˆ’π›Όπ‘›ξ€Έπœ™ξ€·π‘’,π‘₯𝑛≀𝛼𝑛𝛼𝑛,0πœ™ξ€·π‘’,π‘₯𝑛+βˆžξ“π‘–=1𝛼𝑛,π‘–ξ€½πœ™ξ€·π‘’,π‘₯𝑛+πœˆπ‘›πœξ€·πœ™ξ€·π‘’,π‘₯𝑛+πœ‡π‘›ξ€Ύβˆ’π›Όπ‘›,0𝛼𝑛,𝑗𝑔‖‖𝐽π‘₯π‘›βˆ’π½π‘†π‘›π‘—π‘₯𝑛‖‖ξƒͺ+ξ€·1βˆ’π›Όπ‘›ξ€Έπœ™ξ€·π‘’,π‘₯π‘›ξ€Έβ‰€π›Όπ‘›ξƒ©πœ™ξ€·π‘’,π‘₯𝑛+βˆžξ“π‘–=1𝛼𝑛,π‘–ξ€·πœˆπ‘›πœξ€·πœ™ξ€·π‘’,π‘₯𝑛+πœ‡π‘›ξ€Έβˆ’π›Όπ‘›,0𝛼𝑛,𝑗𝑔‖‖𝐽π‘₯π‘›βˆ’π½π‘†π‘›π‘—π‘₯𝑛‖‖ξƒͺ+ξ€·1βˆ’π›Όπ‘›ξ€Έπœ™ξ€·π‘’,π‘₯π‘›ξ€Έξ€·β‰€πœ™π‘’,π‘₯𝑛+π›Όπ‘›ξ‚΅πœˆπ‘›supπ‘’βˆˆβ„±πœξ€·πœ™ξ€·π‘’,π‘₯𝑛+πœ‡π‘›ξ‚Άβˆ’π›Όπ‘›π›Όπ‘›,0𝛼𝑛,𝑗𝑔‖‖𝐽π‘₯π‘›βˆ’π½π‘†π‘›π‘—π‘₯𝑛‖‖=πœ™π‘’,π‘₯𝑛+π›Όπ‘›πœ‚π‘›βˆ’π›Όπ‘›π›Όπ‘›,0𝛼𝑛,𝑗𝑔‖‖𝐽π‘₯π‘›βˆ’π½π‘†π‘›π‘—π‘₯π‘›β€–β€–ξ€Έξ€·β‰€πœ™π‘’,π‘₯𝑛+πœ‚π‘›.(3.13) Hence π‘’βˆˆπΆπ‘›+1 and so β„±βŠ‚πΆπ‘› for all 𝑛β‰₯0. By the way, from the definition of {πœ‚π‘›} and 𝜁 and (3.10), it is easy to see that πœ‚π‘›=πœˆπ‘›supπ‘’βˆˆβ„±πœξ€·πœ™ξ€·π‘’,π‘₯𝑛+πœ‡π‘›β‰€πœˆπ‘›supπ‘’βˆˆβ„±πœξ€·()‖𝑒‖+𝑀2ξ€Έ+πœ‡π‘›(⟢0asπ‘›βŸΆβˆž).(3.14)(IV) Now, we prove that {π‘₯𝑛} converges strongly to some point π‘βˆˆβ„±βˆΆ=βˆžξ™π‘–=1πΉξ€·π‘†π‘–ξ€Έξ™βˆžξ™π‘—=1𝐹GMEP𝑗,𝐴𝑗,πœ“π‘—ξ€Έ.(3.15)
First, we prove that {π‘₯𝑛} converges strongly to some point β‹‚π‘βˆˆβˆžπ‘–=1𝐹(𝑆𝑖).
In fact, since {π‘₯𝑛} is bounded in 𝐢 and 𝐸 is reflexive, there exists a subsequence {π‘₯𝑛𝑖}βŠ‚{π‘₯𝑛} such that π‘₯𝑛𝑖⇀𝑝. Again since 𝐢𝑛 is closed and convex for each 𝑛β‰₯1, it is weakly closed, and so π‘βˆˆπΆπ‘› for each 𝑛β‰₯0. Since π‘₯𝑛=Π𝐢𝑛π‘₯0, from the definition of Π𝐢𝑛, we have πœ™ξ€·π‘₯𝑛𝑖,π‘₯0ξ€Έξ€·β‰€πœ™π‘,π‘₯0ξ€Έ,𝑛β‰₯0.(3.16) Since liminfπ‘›π‘–β†’βˆžπœ™ξ€·π‘₯𝑛𝑖,π‘₯0ξ€Έ=liminfπ‘›π‘–β†’βˆžξ‚†β€–β€–π‘₯𝑛𝑖‖‖2π‘₯βˆ’2𝑛𝑖,𝐽π‘₯0+β€–β€–π‘₯0β€–β€–2β‰₯‖𝑝‖2βˆ’2βŸ¨π‘,𝐽π‘₯0β€–β€–π‘₯⟩+0β€–β€–2ξ€·=πœ™π‘,π‘₯0ξ€Έ,(3.17) we have πœ™ξ€·π‘,π‘₯0≀liminfπ‘›π‘–β†’βˆžπœ™ξ€·π‘₯𝑛𝑖,π‘₯0≀limsupπ‘›π‘–β†’βˆžπœ™ξ€·π‘₯𝑛𝑖,π‘₯0ξ€Έξ€·β‰€πœ™π‘,π‘₯0ξ€Έ.(3.18) This implies that limπ‘›π‘–β†’βˆžπœ™(π‘₯𝑛𝑖,π‘₯0)=πœ™(𝑝,π‘₯0), that is, β€–π‘₯𝑛𝑖‖→‖𝑝‖. In view of the Kadec-Klee property of 𝐸, we obtain that limπ‘›β†’βˆžπ‘₯𝑛𝑖=𝑝.
Now we prove that π‘₯𝑛→𝑝(π‘›β†’βˆž). In fact, if there exists a subsequence {π‘₯𝑛𝑗}βŠ‚{π‘₯𝑛} such that π‘₯π‘›π‘—β†’π‘ž, then we have πœ™(𝑝,π‘ž)=limπ‘›π‘–β†’βˆž,π‘›π‘—β†’βˆžπœ™ξ‚€π‘₯𝑛𝑖,π‘₯𝑛𝑗≀limπ‘›π‘–β†’βˆž,π‘›π‘—β†’βˆžπœ™ξ€·π‘₯𝑛𝑖π‘₯0ξ€Έξ‚€Ξ βˆ’πœ™πΆπ‘›π‘—π‘₯0,π‘₯0=limπ‘›π‘–β†’βˆž,π‘›π‘—β†’βˆžπœ™ξ€·π‘₯𝑛𝑖,π‘₯0ξ€Έξ‚€π‘₯βˆ’πœ™π‘›π‘—,π‘₯0=0(by(3.12)).(3.19) Therefore we have 𝑝=π‘ž. This implies that limπ‘›β†’βˆžπ‘₯𝑛=𝑝.(3.20)
Now we prove that β‹‚π‘βˆˆβˆžπ‘–=1𝐹(𝑆𝑖). In fact, by the construction of 𝐢𝑛, we have that 𝐢𝑛+1βŠ‚πΆπ‘› and π‘₯𝑛+1=Π𝐢𝑛+1π‘₯0. Therefore by Lemma 2.1(a) we have πœ™ξ€·π‘₯𝑛+1,π‘₯𝑛π‘₯=πœ™π‘›+1,Π𝐢𝑛π‘₯0ξ€Έξ€·π‘₯β‰€πœ™π‘›+1,π‘₯0ξ€Έξ€·Ξ βˆ’πœ™πΆπ‘›π‘₯0,π‘₯0ξ€Έξ€·π‘₯=πœ™π‘›+1,π‘₯0ξ€Έξ€·π‘₯βˆ’πœ™π‘›,π‘₯0ξ€Έ(⟢0asπ‘›βŸΆβˆž).(3.21) In view of π‘₯𝑛+1βˆˆπΆπ‘› and noting the construction of 𝐢𝑛+1 we obtain that πœ™ξ€·π‘₯𝑛+1,𝑒𝑛π‘₯β‰€πœ™π‘›+1,π‘₯𝑛+πœ‚π‘›(⟢0asπ‘›βŸΆβˆž).(3.22) From (1.10) it yields (β€–π‘₯𝑛+1β€–βˆ’β€–π‘’π‘›β€–)2β†’0. Sinceβ€–π‘₯𝑛+1‖→‖𝑝‖, we have β€–β€–π‘’π‘›β€–β€–βŸΆ(‖𝑝‖asπ‘›βŸΆβˆž).(3.23) Hence we have β€–β€–π½π‘’π‘›β€–β€–βŸΆ(‖𝐽𝑝‖asπ‘›βŸΆβˆž).(3.24)
This implies that {𝐽𝑒𝑛} is bounded in πΈβˆ—. Since 𝐸 is reflexive, and so πΈβˆ— is reflexive, there exists a subsequence {𝐽𝑒𝑛𝑖}βŠ‚{𝐽𝑒𝑛} such that 𝐽𝑒𝑛𝑖⇀𝑝0βˆˆπΈβˆ—. In view of the reflexive of 𝐸, we see that 𝐽(𝐸)=πΈβˆ—. Hence there exists π‘₯∈𝐸 such that 𝐽π‘₯=𝑝0. Since πœ™ξ€·π‘₯𝑛𝑖+1,𝑒𝑛𝑖=β€–β€–π‘₯𝑛𝑖+1β€–β€–2π‘₯βˆ’2𝑛𝑖+1,𝐽𝑒𝑛𝑖+‖‖𝑒𝑛𝑖‖‖2=β€–β€–π‘₯𝑛𝑖+1β€–β€–2π‘₯βˆ’2𝑛𝑖+1,𝐽𝑒𝑛𝑖+‖‖𝐽𝑒𝑛𝑖‖‖2(3.25) taking liminfπ‘›β†’βˆž on the both sides of above equality and in view of the weak lower semicontinuity of norm β€–β‹…β€–, then it yields that 0β‰₯‖𝑝‖2βˆ’2βŸ¨π‘,𝑝0β€–β€–π‘βŸ©+0β€–β€–2=‖𝑝‖2βˆ’2βŸ¨π‘,𝐽π‘₯⟩+‖𝐽π‘₯β€–2=‖𝑝‖2βˆ’2βŸ¨π‘,𝐽π‘₯⟩+β€–π‘₯β€–2=πœ™(𝑝,π‘₯).(3.26) That is 𝑝=π‘₯. This implies that 𝑝0=𝐽𝑝, and so 𝐽𝑒𝑛⇀𝐽𝑝. It follows from (3.24) and the Kadec-Klee property of πΈβˆ— that 𝐽𝑒𝑛𝑖→𝐽𝑝 (as π‘›β†’βˆž). Note that π½βˆ’1βˆΆπΈβˆ—β†’πΈ is hemi-continuous, it yields that 𝑒𝑛𝑖⇀𝑝. It follows from (3.23) and the Kadec-Klee property of 𝐸 that limπ‘›π‘–β†’βˆžπ‘’π‘›π‘–=𝑝.
By the similar way as given in the proof of (3.20), we can also prove that limπ‘›β†’βˆžπ‘’π‘›=𝑝.(3.27)
From (3.20) and (3.27) we have that β€–β€–π‘₯π‘›βˆ’π‘’π‘›β€–β€–βŸΆ0(asπ‘›βŸΆβˆž).(3.28) Since 𝐽 is uniformly continuous on any bounded subset of 𝐸, we have ‖‖𝐽π‘₯π‘›βˆ’π½π‘’π‘›β€–β€–βŸΆ0(asπ‘›βŸΆβˆž).(3.29) For any 𝑗β‰₯1 and any π‘’βˆˆβ„±, it follows from (3.13), (3.20), and (3.27) that 𝛼𝑛𝛼𝑛,0𝛼𝑛,𝑗𝑔‖‖𝐽π‘₯π‘›βˆ’π½π‘†π‘›π‘—π‘₯π‘›β€–β€–ξ€Έξ€·β‰€πœ™π‘’,π‘₯π‘›ξ€Έξ€·βˆ’πœ™π‘’,𝑒𝑛+π›Όπ‘›πœ‚π‘›.(3.30) Since πœ™ξ€·π‘’,π‘₯π‘›ξ€Έξ€·βˆ’πœ™π‘’,𝑒𝑛=β€–β€–π‘₯𝑛‖‖2βˆ’β€–β€–π‘’π‘›β€–β€–2βˆ’2βŸ¨π‘’,𝐽π‘₯π‘›βˆ’π½π‘’π‘›βŸ©β‰€||β€–β€–π‘₯𝑛‖‖2βˆ’β€–β€–π‘’π‘›β€–β€–2||β€–β€–+2‖𝑒‖⋅𝐽π‘₯π‘›βˆ’π½π‘’π‘›β€–β€–β‰€β€–β€–π‘₯π‘›βˆ’π‘’π‘›β€–β€–ξ€·β€–β€–π‘₯𝑛‖‖+‖‖𝑒𝑛‖‖‖‖+2‖𝑒‖⋅𝐽π‘₯π‘›βˆ’π½π‘’π‘›β€–β€–,(3.31) from (3.28) and (3.29), it follows that πœ™ξ€·π‘’,π‘₯π‘›ξ€Έξ€·βˆ’πœ™π‘’,π‘’π‘›ξ€ΈβŸΆ0(π‘›βŸΆβˆž).(3.32) In view of condition (b) and condition (c), we have that 𝑔‖‖𝐽π‘₯π‘›βˆ’π½π‘†π‘›π‘—π‘₯𝑛‖‖(⟢0asπ‘›βŸΆβˆž).(3.33) It follows from the property of 𝑔 that ‖‖𝐽π‘₯π‘›βˆ’π½π‘†π‘›π‘—π‘₯π‘›β€–β€–βŸΆ0,(asπ‘›βŸΆβˆž).(3.34) Since π‘₯𝑛→𝑝 and 𝐽 is uniformly continuous, it yields 𝐽π‘₯𝑛→𝐽𝑝. Hence from (3.34) we have 𝐽𝑆𝑛𝑗π‘₯π‘›βŸΆπ½π‘(asπ‘›βŸΆβˆž).(3.35) Since π½βˆ’1βˆΆπΈβˆ—β†’πΈ is hemicontinuous, it follows that 𝑆𝑛𝑗π‘₯𝑛⇀𝑝(βˆ€π‘—β‰₯1).(3.36) On the other hand, for each 𝑗β‰₯1 we have ||‖‖𝑆𝑛𝑗π‘₯𝑛‖‖||=||β€–β€–π½ξ€·π‘†βˆ’β€–π‘β€–π‘›π‘—π‘₯𝑛‖‖||β‰€β€–β€–βˆ’β€–π½π‘β€–π½π‘†π‘›π‘—π‘₯𝑛‖‖(βˆ’π½π‘βŸΆ0asπ‘›βŸΆβˆž).(3.37) This together with (3.36) shows that 𝑆𝑛𝑗π‘₯π‘›βŸΆπ‘(foreach𝑗β‰₯1).(3.38)
Furthermore, by the assumption that for each 𝑗β‰₯1,𝑆𝑗 is uniformly 𝐿𝑖-Lipschitz continuous, hence we have ‖‖𝑆𝑗𝑛+1π‘₯π‘›βˆ’π‘†π‘›π‘—π‘₯𝑛‖‖≀‖‖𝑆𝑗𝑛+1π‘₯π‘›βˆ’π‘†π‘—π‘›+1π‘₯𝑛+1β€–β€–+‖‖𝑆𝑗𝑛+1π‘₯𝑛+1βˆ’π‘₯𝑛+1β€–β€–+β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+β€–β€–π‘₯π‘›βˆ’π‘†π‘›π‘—π‘₯𝑛‖‖≀𝐿𝑗‖‖π‘₯+1𝑛+1βˆ’π‘₯𝑛‖‖+‖‖𝑆𝑗𝑛+1π‘₯𝑛+1βˆ’π‘₯𝑛+1β€–β€–+β€–β€–π‘₯π‘›βˆ’π‘†π‘›π‘—π‘₯𝑛‖‖.(3.39) This together with (3.20) and (3.38), yields ‖𝑆𝑗𝑛+1π‘₯π‘›βˆ’π‘†π‘›π‘—π‘₯𝑛‖→0 (as π‘›β†’βˆž). Hence from (3.36) we have 𝑆𝑗𝑛+1π‘₯𝑛→𝑝, that is, 𝑆𝑗𝑆𝑛𝑗π‘₯𝑛→𝑝. In view of (3.38) and the closeness of 𝑆𝑗, it yields that 𝑆𝑗𝑝=𝑝,forall𝑗β‰₯1. This implies that β‹‚π‘βˆˆβˆžπ‘–=1𝐹(𝑆𝑖).
Next, we prove that β‹‚π‘βˆˆπ‘€π‘–=1GMEP(𝐹𝑖,𝐴𝑖,πœ“π‘–). Denote that 𝑒𝑛(π‘š)=πΎΞ“π‘šπ‘Ÿπ‘š,π‘›πΎΞ“π‘šβˆ’1π‘Ÿπ‘šβˆ’1,𝑛⋯𝐾Γ2π‘Ÿ2,𝑛𝐾Γ1π‘Ÿ1,𝑛𝑦𝑛,π‘š=1,2,…,π‘€βˆ’1,𝑒𝑛(𝑀)=𝑒𝑛.(3.40) By the similar method as in the proof of (3.13), we can prove that πœ™ξ‚€π‘’,𝑒𝑛(π‘š)ξ‚ξ€·β‰€πœ™π‘’,π‘₯𝑛+πœ‚π‘›,π‘š=1,2,…,𝑀,π‘’βˆˆβ„±,βˆ€π‘›β‰₯1.(3.41) It follows from Lemma 2.6, (2.15), (3.32) that for any π‘’βˆˆβ„±, πœ™ξ‚€π‘’π‘›(𝑀),𝑒𝑛(π‘€βˆ’1)𝐾=πœ™Ξ“π‘€π‘Ÿπ‘›π‘’π‘›(π‘€βˆ’1),𝑒𝑛(π‘€βˆ’1)ξ‚ξ‚€β‰€πœ™π‘’,𝑒𝑛(π‘€βˆ’1)ξ‚ξ‚€βˆ’πœ™π‘’,πΎΞ“π‘€π‘Ÿπ‘›π‘’π‘›(π‘€βˆ’1)ξ‚ξ€·β‰€πœ™π‘’,π‘₯𝑛+πœ‚π‘›ξ‚€βˆ’πœ™π‘’,Ξ“π‘€π‘Ÿπ‘›π‘’π‘›(π‘€βˆ’1)=πœ™π‘’,π‘₯𝑛+πœ‚π‘›ξ‚€βˆ’πœ™π‘’,𝑒𝑛(𝑀)=πœ™π‘’,π‘₯𝑛+πœ‚π‘›ξ€·βˆ’πœ™π‘’,π‘’π‘›ξ€ΈβŸΆ0(asπ‘›βŸΆβˆž).(3.42) From (1.10) it yields (‖𝑒𝑛(𝑀)β€–βˆ’β€–π‘’π‘›(π‘€βˆ’1)β€–)2β†’0. Since ‖𝑒𝑛(𝑀)β€–=‖𝑒𝑛‖→‖𝑝‖, we have ‖‖𝑒𝑛(π‘€βˆ’1)β€–β€–βŸΆβ€–π‘β€–(asπ‘›βŸΆβˆž).(3.43)
Hence we have ‖‖𝐽𝑒𝑛(π‘€βˆ’1)‖‖→‖𝐽𝑝‖(asπ‘›βŸΆβˆž).(3.44)
This implies that {𝐽𝑒𝑛(π‘€βˆ’1)} is bounded in πΈβˆ—. Since 𝐸 is reflexive, and so πΈβˆ— is reflexive, there exists a subsequence {𝐽𝑒𝑛(π‘€βˆ’1)𝑖}βŠ‚{𝐽𝑒𝑛(π‘€βˆ’1)} such that 𝐽𝑒𝑛(π‘€βˆ’1)𝑖⇀𝑝0βˆˆπΈβˆ—. In view of the reflexive of 𝐸, we see that 𝐽(𝐸)=πΈβˆ—. Hence there exists π‘₯∈𝐸 such that 𝐽π‘₯=𝑝0. Since πœ™ξ‚€π‘’π‘›(𝑀)𝑖,𝑒𝑛(π‘βˆ’1)𝑖=‖‖𝑒𝑛(𝑀)𝑖‖‖2ξ‚¬π‘’βˆ’2𝑛(𝑀)𝑖,𝐽𝑒𝑛(π‘€βˆ’1)𝑖+‖‖𝑒𝑛(π‘€βˆ’1)𝑖‖‖2=‖‖𝑒𝑛(𝑀)𝑖‖‖2ξ‚¬π‘’βˆ’2𝑛(𝑀)𝑖,𝐽𝑒𝑛(π‘€βˆ’1)𝑖+‖‖𝐽𝑒𝑛(π‘€βˆ’1)𝑖‖‖2(3.45) taking liminfπ‘›π‘–β†’βˆž on the both sides of above equality and in view of the weak lower semicontinuity of norm β€–β‹…β€–, it yields that 0β‰₯‖𝑝‖2βˆ’2βŸ¨π‘,𝑝0β€–β€–π‘βŸ©+0β€–β€–2=‖𝑝‖2βˆ’2βŸ¨π‘,𝐽π‘₯⟩+‖𝐽π‘₯β€–2=‖𝑝‖2βˆ’2βŸ¨π‘,𝐽π‘₯⟩+β€–π‘₯β€–2=πœ™(𝑝,π‘₯).(3.46) This is, 𝑝=π‘₯. This implies that 𝑝0=𝐽𝑝, and so 𝐽𝑒𝑛(π‘€βˆ’1)𝑖⇀𝐽𝑝. It follows from (3.44) and the Kadec-Klee property of πΈβˆ— that 𝐽𝑒𝑛(π‘€βˆ’1)𝑖→𝐽𝑝 (as π‘›π‘–β†’βˆž). Note that π½βˆ’1βˆΆπΈβˆ—β†’πΈ is hemicontinuous it yields that 𝑒𝑛(π‘€βˆ’1)𝑖⇀𝑝. It follows from (3.43) and the Kadec-Klee property of 𝐸 that limπ‘›π‘–β†’βˆžπ‘’π‘›(π‘€βˆ’1)𝑖=𝑝.
By the similar way as given in the proof of (3.20), we can also prove that limπ‘›β†’βˆžπ‘’π‘›(π‘€βˆ’1)=𝑝.(3.47) From (3.27) and (3.47) we have that ‖‖𝑒𝑛(𝑀)βˆ’π‘’π‘›(π‘€βˆ’1)β€–β€–βŸΆ0(asπ‘›βŸΆβˆž).(3.48) Since 𝐽 is uniformly continuous on any bounded subset of 𝐸, we have ‖‖𝐽𝑒𝑛(𝑀)βˆ’π½π‘’π‘›(π‘€βˆ’1)β€–β€–βŸΆ0(asπ‘›βŸΆβˆž).(3.49)
Since 𝑒𝑛(𝑖)=πΎΞ“π‘–π‘Ÿπ‘›π‘’π‘›(π‘–βˆ’1),𝑖=2,3,…,𝑀,𝑒𝑛(0)=𝑦𝑛,𝑒𝑛(𝑀)=𝑒𝑛.(3.50)
By the similar way as above, we can also prove that 𝑒𝑛(𝑖)‖‖𝑒→𝑝,𝑛(𝑖)βˆ’π‘’π‘›(π‘–βˆ’1)β€–β€–β€–β€–βŸΆ0,𝐽𝑒𝑛(𝑖)βˆ’π½π‘’π‘›(π‘–βˆ’1)β€–β€–β€–β€–π‘¦βŸΆ0,𝑖=2,3,…,π‘€π‘›βˆ’π‘’π‘›(1)β€–β€–β€–β€–βŸΆ0,π½π‘¦π‘›βˆ’π½π‘’π‘›(1)β€–β€–βŸΆ0(asπ‘›βŸΆβˆž).(3.51) From (3.51) and the assumption that π‘Ÿπ‘›β‰₯𝑑,βˆ€π‘›β‰₯0, we have limπ‘›β†’βˆžβ€–β€–π½π‘’π‘›(𝑖)βˆ’π½π‘’π‘›(π‘–βˆ’1)β€–β€–π‘Ÿπ‘–,𝑛=0,𝑖=2,3,…,𝑀;limπ‘›β†’βˆžβ€–β€–π½π‘¦π‘›βˆ’π½π‘’π‘›(1)β€–β€–π‘Ÿ1,𝑛=0.(3.52) In the proof of Lemma 2.6 we have proved that the function Γ𝑖,𝑖=1,2,…,𝑀 defined by (3.6) satisfies the condition (A1)–(A4) and Γ𝑖𝑒𝑛(𝑖)+1,π‘¦π‘Ÿπ‘–,π‘›ξ‚¬π‘¦βˆ’π‘’π‘›(𝑖),𝐽𝑒𝑛(𝑖)βˆ’π½π‘’π‘›(π‘–βˆ’1)ξ‚­β‰₯0,βˆ€π‘¦βˆˆπΆ.(3.53) Therefore for any π‘¦βˆˆπΆ we have 1π‘Ÿπ‘–,π‘›ξ‚¬π‘¦βˆ’π‘’π‘›(𝑖),𝐽𝑒𝑛(𝑖)βˆ’π½π‘’π‘›(π‘–βˆ’1)ξ‚­β‰₯βˆ’Ξ“π‘–ξ‚€π‘’π‘›(𝑖),𝑦β‰₯Γ𝑖𝑦,𝑒𝑛(𝑖).(3.54) This implies that Γ𝑖𝑦,𝑒𝑛(𝑖)≀1π‘Ÿπ‘–,π‘›ξ‚¬π‘¦βˆ’π‘’π‘›(𝑖),𝐽𝑒𝑛(𝑖)βˆ’π½π‘’π‘›(π‘–βˆ’1)≀𝑀1ξ€Έβ€–β€–+β€–π‘¦β€–π½π‘’π‘–π‘›βˆ’π½π‘’π‘›(π‘–βˆ’1)β€–β€–π‘Ÿπ‘–,𝑛(3.55) for some constant 𝑀1>0. Since the function 𝑦↦Γ𝑖(π‘₯,𝑦) is convex and lower semi-continuous, letting π‘›β†’βˆž in (3.55), from (3.52) and (3.55), for each 𝑖, we have Γ𝑖(𝑦,𝑝)≀0,forallπ‘¦βˆˆπΆ.
For π‘‘βˆˆ(0,1] and π‘¦βˆˆπΆ, letting 𝑦𝑑=𝑑𝑦+(1βˆ’π‘‘)𝑝, there are π‘¦π‘‘βˆˆπΆ and Γ𝑖(𝑦𝑑,𝑝)≀0. By condition (A1) and (A4), we have 0=Γ𝑖𝑦𝑑,𝑦𝑑≀𝑑Γ𝑖𝑦𝑑+,𝑦(1βˆ’π‘‘)Γ𝑖𝑦𝑑,𝑝≀𝑑Γ𝑖𝑦𝑑.,𝑦(3.56) Dividing both sides of the above equation by 𝑑, we have Γ𝑖(𝑦𝑑,𝑦)β‰₯0,forallπ‘¦βˆˆπΆ. Letting 𝑑↓0, from condition (A3), we have Γ𝑖(𝑝,𝑦)β‰₯0,forallπ‘¦βˆˆπΆ,forall𝑖=1,2,…,𝑀, that is, for each 𝑖=1,2,…,𝑀, we have 𝐹𝑖(𝑝,𝑦)+βŸ¨π΄π‘–π‘,π‘¦βˆ’π‘βŸ©+πœ“π‘–(𝑦)βˆ’πœ“π‘–(𝑝)β‰₯0,βˆ€π‘¦βˆˆπΆ.(3.57) This implies that β‹‚π‘βˆˆπ‘€π‘—=1GMEP(𝐹𝑗,𝐴𝑗,πœ“π‘—). Therefore, we have that π‘βˆˆβ„±.(3.58)
(V) Now, we prove π‘₯𝑛→Πℱπ‘₯0.
Let 𝑀=Ξ β„±π‘₯0. From π‘€βˆˆβ„±βŠ‚πΆπ‘›+1 and π‘₯𝑛+1=Π𝐢𝑛+1π‘₯0, we have πœ™(π‘₯𝑛+1,π‘₯0)β‰€πœ™(𝑀,π‘₯0),forall𝑛β‰₯0. This implies that πœ™ξ€·π‘,π‘₯0ξ€Έ=limπ‘›β†’βˆžπœ™ξ€·π‘₯𝑛,π‘₯0ξ€Έξ€·β‰€πœ™π‘€,π‘₯0ξ€Έ.(3.59)
By the definition of Ξ β„±π‘₯0 and (3.59), we have 𝑝=𝑀. Therefore, π‘₯𝑛→Πℱπ‘₯0. This completes the proof of Theorem 3.1.

Theorem 3.2. Let 𝐸,𝐢,{πΎΞ“π‘–π‘Ÿπ‘–,𝑛}𝑀𝑖=1,{𝐴𝑖}𝑀𝑖=1,{𝐹𝑖}𝑀𝑖=1,{πœ“π‘–}𝑀𝑖=1,{GMEP(𝐹𝑗,𝐴𝑗,πœ“π‘—)}𝑀𝑗=1 be the same as above. Let {𝑆𝑖}βˆžπ‘–=1βˆΆπΆβ†’πΆ be an infinite family of closed and uniformly quasi-πœ™-asymptotically nonexpansive mappings with a sequence {π‘˜π‘›}βŠ‚[1,∞) and π‘˜π‘›β†’1. Suppose that for each 𝑖β‰₯1,𝑆𝑖 is uniformly L𝑖-Lipschitz continuous and that π’’βˆΆ=βˆžξ™π‘–=1𝐹𝑆𝑖𝑀𝑗=1𝐹GMEP𝑗,𝐴𝑗,πœ“π‘—ξ€Έ(3.60) is a nonempty and bounded subset of 𝐢. For any given π‘₯0∈𝐢, let {π‘₯𝑛} be the sequence generated by π‘₯0∈𝐢0𝑧=𝐢,𝑛=π½βˆ’1𝛼𝑛,0𝐽π‘₯𝑛+βˆžξ“π‘–=1𝛼𝑛,𝑖𝐽𝑆𝑛𝑖π‘₯𝑛ξƒͺ,𝑦𝑛=π½βˆ’1𝛼𝑛𝐽𝑧𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ½π‘₯𝑛,𝑒𝑛=πΎΞ“π‘€π‘Ÿπ‘€,π‘›πΎΞ“π‘€βˆ’1π‘Ÿπ‘€βˆ’1,𝑛⋯𝐾Γ2π‘Ÿ2,𝑛𝐾Γ1π‘Ÿ1,𝑛𝑦𝑛,𝐢𝑛+1=ξ€½π‘£βˆˆπΆπ‘›ξ€·βˆΆπœ™π‘£,π‘’π‘›ξ€Έξ€·β‰€πœ™π‘£,π‘₯𝑛+πœ‰π‘›ξ€Ύ,π‘₯𝑛+1=Π𝐢𝑛