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Journal of Applied Mathematics
Volume 2012, Article ID 516897, 19 pages
http://dx.doi.org/10.1155/2012/516897
Research Article

Strong Convergence of a Hybrid Iteration Scheme for Equilibrium Problems, Variational Inequality Problems and Common Fixed Point Problems, of Quasi-ϕ-Asymptotically Nonexpansive Mappings in Banach Spaces

1College of Science, Civil Aviation University of China, Tianjin 300300, China
2Tianjin Key Laboratory for Advanced Signal Processing, Civil Aviation University of China, Tianjin 300300, China

Received 3 April 2012; Accepted 5 June 2012

Academic Editor: Hong-Kun Xu

Copyright © 2012 Jing Zhao. 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

We introduce an iterative algorithm for finding a common element of the set of common fixed points of a finite family of closed quasi-ϕ-asymptotically nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality problem for a γ-inverse strongly monotone mapping in Banach spaces. Then we study the strong convergence of the algorithm. Our results improve and extend the corresponding results announced by many others.

1. Introduction and Preliminary

Let 𝐸 be a Banach space with the dual 𝐸. A mapping 𝐴𝐷(𝐴)𝐸𝐸 is said to be monotone if, for each 𝑥,𝑦𝐷(𝐴), the following inequality holds: 𝐴𝑥𝐴𝑦,𝑥𝑦0.(1.1)𝐴is said to be 𝛾-inverse strongly monotone if there exists a positive real number 𝛾 such that 𝑥𝑦,𝐴𝑥𝐴𝑦𝛾𝐴𝑥𝐴𝑦2,𝑥,𝑦𝐷(𝐴).(1.2) If 𝐴 is 𝛾-inverse strongly monotone, then it is Lipschitz continuous with constant 1/𝛾, that is, 𝐴𝑥𝐴𝑦(1/𝛾)𝑥𝑦, for all 𝑥,𝑦𝐷(𝐴), and hence uniformly continuous.

Let 𝐶 be a nonempty closed convex subset of 𝐸 and 𝑓𝐶×𝐶 a bifunction, where is the set of real numbers. The equilibrium problem for 𝑓 is to find ̂𝑥𝐶 such that 𝑓(̂𝑥,𝑦)0(1.3) for all 𝑦𝐶. The set of solutions of (1.3) is denoted by EP(𝑓). Given a mapping 𝑇𝐶𝐸, let 𝑓(𝑥,𝑦)=𝑇𝑥,𝑦𝑥 for all 𝑥,𝑦𝐶. Then ̂𝑥EP(𝑓) if and only if 𝑇̂𝑥,𝑦̂𝑥0 for all 𝑦𝐶; that is, ̂𝑥 is a solution of the variational inequality. Numerous problems in physics, optimization, engineering, and economics reduce to find a solution of (1.3). Some methods have been proposed to solve the equilibrium problem; see, for example, Blum and Oettli [1] and Moudafi [2]. For solving the equilibrium problem, let us assume that 𝑓 satisfies the following conditions: (A1)𝑓(𝑥,𝑥)=0 for all 𝑥𝐶;(A2)𝑓 is monotone, that is, 𝑓(𝑥,𝑦)+𝑓(𝑦,𝑥)0 for all 𝑥,𝑦𝐶;(A3) for each 𝑥,𝑦,𝑧𝐶, lim𝑡0𝑓(𝑡𝑧+(1𝑡)𝑥,𝑦)𝑓(𝑥,𝑦);(A4) for each 𝑥𝐶, the function 𝑦𝑓(𝑥,𝑦) is convex and lower semicontinuous.

Let 𝐸 be a Banach space with the dual 𝐸. We denote by 𝐽 the normalized duality mapping from 𝐸 to 2𝐸 defined by 𝑥𝐽(𝑥)=𝐸𝑥,𝑥=𝑥2=𝑥2,(1.4) where , denotes the generalized duality pairing. Let dim 𝐸2, and the modulus of smoothness of 𝐸 is the function 𝜌𝐸[0,)[0,) defined by 𝜌𝐸(𝜏)=sup𝑥+𝑦+𝑥𝑦21𝑥=1;𝑦=𝜏.(1.5) The space 𝐸 is said to be smooth if 𝜌𝐸(𝜏)>0, for all 𝜏>0 and 𝐸 is called uniformly smooth if and only if lim𝑡0+(𝜌𝐸(𝑡)/𝑡)=0. A Banach space 𝐸 is said to be strictly convex if 𝑥+𝑦/2<1 for 𝑥,𝑦𝐸 with 𝑥=𝑦=1 and 𝑥𝑦. The modulus of convexity of 𝐸 is the function 𝛿𝐸(0,2][0,1] defined by 𝛿𝐸(𝜖)=inf1𝑥+𝑦2.𝑥=𝑦=1;𝜖=𝑥𝑦(1.6)𝐸 is called uniformly convex if and only if 𝛿𝐸(𝜖)>0 for every 𝜖(0,2]. Let 𝑝>1, then 𝐸 is said to be 𝑝-uniformly convex if there exists a constant 𝑐>0 such that 𝛿𝐸(𝜖)𝑐𝜖𝑝 for all 𝜖(0,2]. Observe that every 𝑝-uniformly convex space is uniformly convex. We know that if 𝐸 is uniformly smooth, strictly convex, and reflexive, then the normalized duality mapping 𝐽 is single-valued, one-to-one, onto and uniformly norm-to-norm continuous on each bounded subset of 𝐸. Moreover, if 𝐸 is a reflexive and strictly convex Banach space with a strictly convex dual, then 𝐽1 is single-valued, one-to-one, surjective, and it is the duality mapping from 𝐸 into 𝐸 and thus 𝐽𝐽1=𝐼𝐸 and 𝐽1𝐽=𝐼𝐸 (see, [3]). It is also well known that 𝐸 is uniformly smooth if and only if 𝐸 is uniformly convex.

Let 𝐶 be a nonempty closed convex subset of a Banach space 𝐸 and 𝑇𝐶𝐶 a mapping. A point 𝑥𝐶 is said to be a fixed point of 𝑇 provided 𝑇𝑥=𝑥. A point 𝑥𝐶 is said to be an asymptotic fixed point of 𝑇 provided 𝐶 contains a sequence {𝑥𝑛} which converges weakly to 𝑥 such that lim𝑛𝑥𝑛𝑇𝑥𝑛=0. In this paper, we use 𝐹(𝑇) and 𝐹(𝑇) to denote the fixed point set and the asymptotic fixed point set of 𝑇 and use to denote the strong convergence and weak convergence, respectively. Recall that a mapping 𝑇𝐶𝐶 is called nonexpansive if 𝑇𝑥𝑇𝑦𝑥𝑦,𝑥,𝑦𝐶.(1.7) A mapping 𝑇𝐶𝐶 is called asymptotically nonexpansive if there exists a sequence {𝑘𝑛} of real numbers with 𝑘𝑛1 as 𝑛 such that 𝑇𝑛𝑥𝑇𝑛𝑦𝑘𝑛𝑥𝑦,𝑥,𝑦𝐶,𝑛1.(1.8)

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [4] in 1972. They proved that if 𝐶 is a nonempty bounded closed convex subset of a uniformly convex Banach space 𝐸, then every asymptotically nonexpansive self-mapping 𝑇 of 𝐶 has a fixed point. Further, the set 𝐹(𝑇) is closed and convex. Since 1972, a host of authors have studied the weak and strong convergence problems of the iterative algorithms for such a class of mappings (see, e.g., [46] and the references therein).

It is well known that if 𝐶 is a nonempty closed convex subset of a Hilbert space 𝐻 and 𝑃𝐶𝐻𝐶 is the metric projection of 𝐻 onto 𝐶, then 𝑃𝐶 is nonexpansive. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces. In this connection, Alber [7] recently introduced a generalized projection operator Π𝐶 in a Banach space 𝐸 which is an analogue of the metric projection in Hilbert spaces.

Next, we assume that 𝐸 is a smooth Banach space. Consider the functional defined by 𝜙(𝑥,𝑦)=𝑥22𝑥,𝐽𝑦+𝑦2,𝑥,𝑦𝐸.(1.9) Following Alber [7], the generalized projection Π𝐶𝐸𝐶 is a mapping that assigns to an arbitrary point 𝑥𝐸 the minimum point of the functional 𝜙(𝑦,𝑥), that is, Π𝐶𝑥=𝑥, where 𝑥 is the solution to the following minimization problem: 𝜙𝑥,𝑥=inf𝑦𝐶𝜙(𝑦,𝑥).(1.10) It follows from the definition of the function 𝜙 that ()𝑦𝑥2)𝜙(𝑦,𝑥)(𝑦+𝑥2,𝑥,𝑦𝐸.(1.11) If 𝐸 is a Hilbert space, then 𝜙(𝑦,𝑥)=𝑦𝑥2 and Π𝐶=𝑃𝐶 is the metric projection of 𝐻 onto 𝐶.

Remark 1.1 (see [8, 9]). If 𝐸 is a reflexive, strictly convex, and smooth Banach space, then for 𝑥,𝑦𝐸, 𝜙(𝑥,𝑦)=0 if and only if 𝑥=𝑦.

Let 𝐶 be a nonempty, closed, and convex subset of a smooth Banach 𝐸 and 𝑇 a mapping from 𝐶 into itself. The mapping 𝑇 is said to be relatively nonexpansive if 𝐹(𝑇)=𝐹(𝑇),𝜙(𝑝,𝑇𝑥)𝜙(𝑝,𝑥),forall𝑥𝐶,𝑝𝐹(𝑇). The mapping 𝑇 is said to be 𝜙-nonexpansive if 𝜙(𝑇𝑥,𝑇𝑦)𝜙(𝑥,𝑦),forall𝑥,𝑦𝐶. The mapping 𝑇 is said to be quasi-𝜙-nonexpansive if 𝐹(𝑇),𝜙(𝑝,𝑇𝑥)𝜙(𝑝,𝑥),forall𝑥𝐶,𝑝𝐹(𝑇). The mapping 𝑇 is said to be relatively asymptotically nonexpansive if there exists some real sequence {𝑘𝑛} with 𝑘𝑛1 and 𝑘𝑛1 as 𝑛 such that 𝐹(𝑇)=𝐹(𝑇),𝜙(𝑝,𝑇𝑛𝑥)𝑘𝑛𝜙(𝑝,𝑥),forall𝑥𝐶,𝑝𝐹(𝑇). The mapping 𝑇 is said to be 𝜙-asymptotically nonexpansive if there exists some real sequence {𝑘𝑛} with 𝑘𝑛1 and 𝑘𝑛1 as 𝑛 such that 𝜙(𝑇𝑛𝑥,𝑇𝑛𝑦)𝑘𝑛𝜙(𝑥,𝑦),forall𝑥,𝑦𝐶. The mapping 𝑇 is said to be quasi-𝜙-asymptotically nonexpansive if there exists some real sequence {𝑘𝑛} with 𝑘𝑛1 and 𝑘𝑛1 as 𝑛 such that 𝐹(𝑇),𝜙(𝑝,𝑇𝑛𝑥)𝑘𝑛𝜙(𝑝,𝑥),forall𝑥𝐶,𝑝𝐹(𝑇). The mapping 𝑇 is said to be asymptotically regular on 𝐶 if, for any bounded subset 𝐾 of 𝐶, limsup𝑛{𝑇𝑛+1𝑥𝑇𝑛𝑥𝑥𝐾}=0. The mapping 𝑇 is said to be closed on 𝐶 if, for any sequence {𝑥𝑛} such that lim𝑛𝑥𝑛=𝑥0 and lim𝑛𝑇𝑥𝑛=𝑦0, 𝑇𝑥0=𝑦0.

We remark that a 𝜙-asymptotically nonexpansive mapping with a nonempty fixed point set 𝐹(𝑇) is a quasi-𝜙-asymptotically nonexpansive mapping, but the converse may be not true. The class of quasi-𝜙-nonexpansive mappings and quasi-𝜙-asymptotically nonexpansive mappings is more general than the class of relatively nonexpansive mappings and relatively asymptotically nonexpansive mappings, respectively.

Recently, many authors studied the problem of finding a common element of the set of fixed points of nonexpansive or relatively nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of variational inequalities in the frame work of Hilbert spaces and Banach spaces, respectively; see, for instance, [1021] and the references therein.

In 2009, Takahashi and Zembayashi [22] introduced the following iterative process: 𝑥0𝑦=𝑥𝐶,𝑛=𝐽1𝛼𝑛𝐽𝑥𝑛+1𝛼𝑛𝐽𝑆𝑥𝑛,𝑢𝑛𝑢𝐶,𝑓𝑛+1,𝑦𝑟𝑛𝑦𝑢𝑛,𝐽𝑢𝑛𝐽𝑦𝑛𝐻0,𝑦𝐶,𝑛=𝑧𝐶𝜙𝑧,𝑢𝑛𝜙𝑧,𝑥𝑛,𝑊𝑛=𝑧𝐶𝑥𝑛𝑧,𝐽𝑥𝐽𝑥𝑛,𝑥0𝑛+1=Π𝐻𝑛𝑊𝑛𝑥,𝑛1,(1.12) where 𝑓𝐶×𝐶 is a bifunction satisfying (A1)–(A4), 𝐽 is the normalized duality mapping on 𝐸 and 𝑆𝐶𝐶 is a relatively nonexpansive mapping. They proved the sequence {𝑥𝑛} defined by (1.12) converges strongly to a common point of the set of solutions of the equilibrium problem (1.3) and the set of fixed points of 𝑆 provided the control sequences {𝛼𝑛} and {𝑟𝑛} satisfy appropriate conditions in Banach spaces.

Qin et al. [8] introduced the following iterative scheme on the equilibrium problem (1.3) and a family of quasi-𝜙-nonexpansive mapping: 𝑥0𝐸,𝐶1=𝐶,𝑥1=Π𝐶1𝑥0,𝑦𝑛=𝐽1𝛼𝑛,0𝐽𝑥𝑛+Σ𝑁𝑖=1𝛼𝑛,𝑖𝐽𝑇𝑖𝑥𝑛,𝑢𝑛𝑢𝐶,𝑓𝑛+1,𝑦𝑟𝑛𝑦𝑢𝑛,𝐽𝑢𝑛𝐽𝑦𝑛𝐶0,𝑦𝐶,𝑛+1=𝑧𝐶𝑛𝜙𝑧,𝑢𝑛𝜙𝑧,𝑥𝑛,𝑥𝑛+1=Π𝐶𝑛+1𝑥0,𝑛1.(1.13) Strong convergence theorems of common elements are established in a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property.

Very recently, for finding a common element of 𝑟𝑖=1𝐹(𝑇𝑖)EP(𝑓,𝐵)VI(𝐴,𝐶) Zegeye [23] proposed the following iterative algorithm: 𝑥0𝐶0𝑧=𝐶,𝑛=Π𝐶𝐽1𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛,𝑦𝑛=𝐽1𝛼0𝐽𝑥𝑛+Σ𝑟𝑖=1𝛼𝑖𝐽𝑇𝑖𝑧𝑛,𝑢𝑛𝑢𝐶,𝑓𝑛+1,𝑦𝑟𝑛𝑦𝑢𝑛,𝐽𝑢𝑛𝐽𝑦𝑛𝐶0,𝑦𝐶,𝑛+1=𝑧𝐶𝑛𝜙𝑧,𝑢𝑛𝜙𝑧,𝑥𝑛,𝑥𝑛+1=Π𝐶𝑛+1𝑥0,𝑛1,(1.14) where 𝑇𝑖𝐶𝐶 is closed quasi-𝜙-nonexpansive mapping (𝑖=1,,𝑟), 𝑓𝐶×𝐶 is a bifunction satisfying (A1)–(A4) and 𝐴 is a 𝛾-inverse strongly monotone mapping of 𝐶 into 𝐸. Strong convergence theorems for iterative scheme (1.14) are obtained under some conditions on parameters in 2-uniformly convex and uniformly smooth real Banach space 𝐸.

In this paper, inspired and motivated by the works mentioned above, we introduce an iterative process for finding a common element of the set of common fixed points of a finite family of closed quasi-𝜙-asymptotically nonexpansive mappings, the solution set of equilibrium problem, and the solution set of the variational inequality problem for a 𝛾-inverse strongly monotone mapping in Banach spaces. The results presented in this paper improve and generalize the corresponding results announced by many others.

In order to the main results of this paper, we need the following lemmas.

Lemma 1.2 (see [24]). Let 𝐸 be a 2-uniformly convex and smooth Banach space. Then, for all 𝑥,𝑦𝐸, one has 2𝑥𝑦𝑐2𝐽𝑥𝐽𝑦,(1.15) where 𝐽 is the normalized duality mapping of 𝐸 and 1/𝑐(0<𝑐1) is the 2-uniformly convex constant of 𝐸.

Lemma 1.3 (see [7, 25]). Let 𝐶 be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space 𝐸. Then 𝜙𝑥,Π𝐶𝑦Π+𝜙𝐶𝑦,𝑦𝜙(𝑥,𝑦),𝑥𝐶,𝑦𝐸.(1.16)

Lemma 1.4 (see [25]). Let 𝐸 be a smooth and uniformly convex Banach space and let {𝑥𝑛} and {𝑦𝑛} be sequences in 𝐸 such that either {𝑥𝑛} or {𝑦𝑛} is bounded. If lim𝑛𝜙(𝑥𝑛,𝑦𝑛)=0, then lim𝑛𝑥𝑛𝑦𝑛=0.

Lemma 1.5 (see [7]). Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝐸, let 𝑥𝐸 and let 𝑧𝐶. Then 𝑧=Π𝐶𝑥𝑦𝑧,𝐽𝑥𝐽𝑧0,𝑦𝐶.(1.17)

We denote by 𝑁𝐶(𝑣) the normal cone for 𝐶𝐸 at a point 𝑣𝐶, that is, 𝑁𝐶(𝑣)={𝑥𝐸𝑣𝑦,𝑥0,forall𝑦𝐶}. We shall use the following lemma.

Lemma 1.6 (see [26]). Let 𝐶 be a nonempty closed convex subset of a Banach space 𝐸 and let 𝐴 be a monotone and hemicontinuous operator of 𝐶 into 𝐸 with 𝐶=𝐷(𝐴). Let 𝑆𝐸×𝐸 be an operator defined as follows: 𝑆𝑣=𝐴𝑣+𝑁𝐶(𝑣),𝑣𝐶,,𝑣𝐶.(1.18) Then 𝑆 is maximal monotone and 𝑆1(0)=VI(𝐶,𝐴).

We make use of the function 𝑉𝐸×𝐸 defined by 𝑉𝑥,𝑥=𝑥22𝑥,𝑥+𝑥2,(1.19) for all 𝑥𝐸 and 𝑥𝐸 (see [7]). That is, 𝑉(𝑥,𝑥)=𝜙(𝑥,𝐽1𝑥) for all 𝑥𝐸 and 𝑥𝐸.

Lemma 1.7 (see [7]). Let 𝐸 be a reflexive, strictly convex, and smooth Banach space with 𝐸 as its dual. Then, 𝑉𝑥,𝑥𝐽+21𝑥𝑥,𝑦𝑉𝑥,𝑥+𝑦(1.20) for all 𝑥𝐸 and 𝑥,𝑦𝐸.

Lemma 1.8 (see [1]). Let 𝐶 be a closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸, let 𝑓 be a bifunction from 𝐶×𝐶 to satisfying (A1)–(A4), and let 𝑟>0 and 𝑥𝐸. Then, there exists 𝑧𝐶 such that 1𝑓(𝑧,𝑦)+𝑟𝑦𝑧,𝐽𝑧𝐽𝑥0,𝑦𝐶.(1.21)

Lemma 1.9 (see [22]). Let 𝐶 be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to satisfying (A1)–(A4). For 𝑟>0 and 𝑥𝐸, define a mapping 𝑇𝑟𝐸𝐶 as follows: 𝑇𝑟1(𝑥)=𝑧𝐶𝑓(𝑧,𝑦)+𝑟𝑦𝑧,𝐽𝑧𝐽𝑥0,𝑦𝐶(1.22) for all 𝑥𝐸. Then, the following hold:(1)𝑇𝑟 is single-valued;(2)𝑇𝑟 is firmly nonexpansive, that is, for any 𝑥,𝑦𝐸, 𝑇𝑟𝑥𝑇𝑟𝑦,𝐽𝑇𝑟𝑥𝐽𝑇𝑟𝑦𝑇𝑟𝑥𝑇𝑟𝑦,𝐽𝑥𝐽𝑦;(1.23)(3)𝐹(𝑇𝑟)=EP(𝑓);(4)EP(𝑓) is closed and convex;(5)𝜙(𝑞,𝑇𝑟𝑥)+𝜙(𝑇𝑟𝑥,𝑥)𝜙(𝑞,𝑥),forall𝑞𝐹(𝑇𝑟).

Lemma 1.10 (see [8, 23]). Let 𝐸 be a uniformly convex Banach space, 𝑠>0 a positive number and 𝐵𝑠(0) a closed ball of 𝐸. Then there exists a strictly increasing, continuous, and convex function 𝑔[0,)[0,) with 𝑔(0)=0 such that 𝑁𝑖=0𝛼𝑖𝑥𝑖2𝑁𝑖=0𝛼𝑖𝑥𝑖2𝛼𝑘𝛼𝑙𝑔𝑥𝑘𝑥𝑙(1.24) for any 𝑘,𝑙{0,1,,𝑁}, for all 𝑥0,𝑥1,,𝑥𝑁𝐵𝑠(0)={𝑥𝐸𝑥𝑠} and 𝛼0,𝛼1,,𝛼𝑛[0,1] such that 𝑁𝑖=0𝛼𝑖=1.

Lemma 1.11 (see [27]). Let 𝐸 be a uniformly convex and uniformly smooth Banach space, 𝐶 a nonempty, closed, and convex subset of 𝐸, and 𝑇 a closed quasi-𝜙-asymptotically nonexpansive mapping from 𝐶 into itself. Then 𝐹(𝑇) is a closed convex subset of 𝐶.

2. Main Results

Theorem 2.1. Let 𝐶 be a nonempty, closed, and convex subset of a 2-uniformly convex and uniformly smooth real Banach space 𝐸 and 𝑇𝑖𝐶𝐶 a closed quasi-𝜙-asymptotically nonexpansive mapping with sequence {𝑘𝑛,𝑖}[1,) such that lim𝑛𝑘𝑛,𝑖=1 for each 1𝑖𝑁. Let 𝑓 be a bifunction from 𝐶×𝐶 to satisfying (A1)–(A4). Let 𝐴 be a 𝛾-inverse strongly monotone mapping of 𝐶 into 𝐸 with constant 𝛾>0 such that 𝐹=(𝑁𝑖=1𝐹(𝑇𝑖))EP(𝑓)VI(𝐶,𝐴) and 𝐹 is bounded. Assume that 𝑇𝑖 is asymptotically regular on C for each 1𝑖𝑁 and 𝐴𝑥𝐴𝑥𝐴𝑝 for all 𝑥𝐶 and 𝑝𝐹. Define a sequence {𝑥𝑛} in 𝐶 in the following manner: 𝑥0𝐶0𝑧=𝐶chosenarbitrarily,𝑛=Π𝐶𝐽1𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛,𝑦𝑛=𝐽1𝛼𝑛,0𝐽𝑥𝑛+𝛼𝑛,1𝐽𝑇𝑛1𝑧𝑛++𝛼𝑛,𝑁𝐽𝑇𝑛𝑁𝑧𝑛,𝑢𝑛𝑢𝐶,𝑓𝑛+1,𝑦𝑟𝑛𝑦𝑢𝑛,𝐽𝑢𝑛𝐽𝑦𝑛𝐶0,𝑦𝐶,𝑛+1=𝑧𝐶𝑛𝜙𝑧,𝑢𝑛𝜙𝑧,𝑥𝑛+𝑁𝑖=1𝛼𝑛,𝑖𝑘𝑛,𝑖𝐿1𝑛,𝑥𝑛+1=Π𝐶𝑛+1𝑥0(2.1) for every 𝑛0, where {𝑟𝑛} is a real sequence in [𝑎,) for some 𝑎>0, 𝐽 is the normalized duality mapping on 𝐸 and 𝐿𝑛=sup{𝜙(𝑝,𝑥𝑛)𝑝𝐹}<. Assume that {𝛼𝑛,0}, {𝛼𝑛,1}, , {𝛼𝑛,𝑁} are real sequences in (0,1) such that 𝑁𝑖=0𝛼𝑛,𝑖=1 and liminf𝑛𝛼𝑛,0𝛼𝑛,𝑖>0, for all 𝑖{1,2,,𝑁}. Let {𝜆𝑛} be a sequence in [𝑠,𝑡] for some 0<𝑠<𝑡<𝑐2𝛾/2, where 1/𝑐 is the 2-uniformly convex constant of 𝐸. Then the sequence {𝑥𝑛} converges strongly to Π𝐹𝑥0.

Proof. We break the proof into nine steps.

Step 1. Π𝐹𝑥0 is well defined for 𝑥0𝐶.
By Lemma 1.11 we know that 𝐹(𝑇𝑖) is a closed convex subset of 𝐶 for every 1𝑖𝑁. Hence 𝐹=(𝑁𝑖=1𝐹(𝑇𝑖))EP(𝑓)VI(𝐶,𝐴) is a nonempty closed convex subset of 𝐶. Consequently, Π𝐹𝑥0 is well defined for 𝑥0𝐶.

Step 2. 𝐶𝑛 is closed and convex for each 𝑛0.
It is obvious that 𝐶0=𝐶 is closed and convex. Suppose that 𝐶𝑛 is closed and convex for some integer 𝑛. Since the defining inequality in 𝐶𝑛+1 is equivalent to the inequality: 2𝑧,𝐽𝑥𝑛𝐽𝑢𝑛𝑥𝑛2𝑢𝑛2+𝑁𝑖=1𝛼𝑛,𝑖𝑘𝑛,𝑖𝐿1𝑛,(2.2) we have that 𝐶𝑛+1 is closed and convex. So 𝐶𝑛 is closed and convex for each 𝑛0. This in turn shows that Π𝐶𝑛+1𝑥0 is well defined.

Step 3. 𝐹𝐶𝑛 for all 𝑛0.
We do this by induction. For 𝑛=0, we have 𝐹𝐶=𝐶0. Suppose that 𝐹𝐶𝑛 for some 𝑛0. Let 𝑝𝐹𝐶. Putting 𝑢𝑛=𝑇𝑟𝑛𝑦𝑛 for all 𝑛0, we have that 𝑇𝑟𝑛 is quasi-𝜙-nonexpansive from Lemma 1.9. Since 𝑇𝑖 is quasi-𝜙-asymptotically nonexpansive, we have 𝜙𝑝,𝑢𝑛=𝜙𝑝,𝑇𝑟𝑛𝑦𝑛𝜙𝑝,𝑦𝑛=𝜙𝑝,𝐽1𝛼𝑛,0𝐽𝑥𝑛+𝑁𝑖=1𝛼𝑛,𝑖𝐽𝑇𝑛𝑖𝑧𝑛=𝑝22𝑝,𝛼𝑛,0𝐽𝑥𝑛+𝑁𝑖=1𝛼𝑛,𝑖𝐽𝑇𝑛𝑖𝑧𝑛+𝛼𝑛,0𝐽𝑥𝑛+𝑁𝑖=1𝛼𝑛,𝑖𝐽𝑇𝑛𝑖𝑧𝑛2𝑝22𝛼𝑛,0𝑝,𝐽𝑥𝑛2𝑁𝑖=1𝛼𝑛,𝑖𝑝,𝐽𝑇𝑛𝑖𝑧𝑛+𝛼𝑛,0𝑥𝑛2+𝑁𝑖=1𝛼𝑛,𝑖𝑇𝑛𝑖𝑧𝑛2=𝛼𝑛,0𝜙𝑝,𝑥𝑛+𝑁𝑖=1𝛼𝑛,𝑖𝜙𝑝,𝑇𝑛𝑖𝑧𝑛𝛼𝑛,0𝜙𝑝,𝑥𝑛+𝑁𝑖=1𝛼𝑛,𝑖𝑘𝑛,𝑖𝜙𝑝,𝑧𝑛.(2.3) Moreover, by Lemmas 1.3 and 1.7, we get that 𝜙𝑝,𝑧𝑛=𝜙𝑝,Π𝐶𝐽1𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝜙𝑝,𝐽1𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛=𝑉𝑝,𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝑉𝑝,𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛+𝜆𝑛𝐴𝑥𝑛𝐽21𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝑝,𝜆𝑛𝐴𝑥𝑛=𝑉𝑝,𝐽𝑥𝑛2𝜆𝑛𝐽1𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝑝,𝐴𝑥𝑛=𝜙𝑝,𝑥𝑛2𝜆𝑛𝑥𝑛𝑝,𝐴𝑥𝑛2𝜆𝑛𝐽1𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝑥𝑛,𝐴𝑥𝑛𝜙𝑝,𝑥𝑛2𝜆𝑛𝑥𝑛𝑝,𝐴𝑥𝑛𝐴𝑝2𝜆𝑛𝑥𝑛𝐽𝑝,𝐴𝑝+21𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝑥𝑛,𝜆𝑛𝐴𝑥𝑛.(2.4) Thus, since 𝑝VI(𝐶,𝐴) and 𝐴 is 𝛾-inverse strongly monotone, we have from (2.4) that 𝜙𝑝,𝑧𝑛𝜙𝑝,𝑥𝑛2𝜆𝑛𝛾𝐴𝑥𝑛𝐴𝑝2𝐽+21𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝑥𝑛,𝜆𝑛𝐴𝑥𝑛.(2.5) Therefore, from (2.5), Lemma 1.2 and the fact that 𝜆𝑛<𝑐2𝛾/2 and 𝐴𝑥𝐴𝑥𝐴𝑝 for all 𝑥𝐶 and 𝑝𝐹, we have 𝜙𝑝,𝑧𝑛𝜙𝑝,𝑥𝑛2𝜆𝑛𝛾𝐴𝑥𝑛𝐴𝑝2+4𝑐2𝜆2𝑛𝐴𝑥𝑛𝐴𝑝2=𝜙𝑝,𝑥𝑛+2𝜆𝑛2𝑐2𝜆𝑛𝛾𝐴𝑥𝑛𝐴𝑝2𝜙𝑝,𝑥𝑛.(2.6) Substituting (2.6) into (2.3), we get 𝜙𝑝,𝑢𝑛𝜙𝑝,𝑥𝑛+𝑁𝑖=1𝛼𝑛,𝑖𝑘𝑛,𝑖𝐿1𝑛,(2.7) that is, 𝑝𝐶𝑛+1. By induction, 𝐹𝐶𝑛 and the iteration algorithm generated by (2.1) is well defined.

Step 4. lim𝑛𝜙(𝑥𝑛,𝑥0) exists and {𝑥𝑛} is bounded.
Noticing that 𝑥𝑛=Π𝐶𝑛𝑥0 and Lemma 1.3, we have 𝜙𝑥𝑛,𝑥0Π=𝜙𝐶𝑛𝑥0,𝑥0𝜙𝑝,𝑥0𝜙𝑝,𝑥𝑛𝜙𝑝,𝑥0(2.8) for all 𝑝𝐹 and 𝑛0. This shows that the sequence {𝜙(𝑥𝑛,𝑥0)} is bounded. From 𝑥𝑛=Π𝐶𝑛𝑥0 and 𝑥𝑛+1=Π𝐶𝑛+1𝑥0𝐶𝑛+1𝐶𝑛, we obtain that 𝜙𝑥𝑛,𝑥0𝑥𝜙𝑛+1,𝑥0,𝑛0,(2.9) which implies that {𝜙(𝑥𝑛,𝑥0)} is nondecreasing. Therefore, the limit of {𝜙(𝑥𝑛,𝑥0)} exists and {𝑥𝑛} is bounded.

Step 5. We have 𝑥𝑛𝑥𝐶.
By Lemma 1.3, we have, for any positive integer 𝑚𝑛, that 𝜙𝑥𝑚,𝑥𝑛𝑥=𝜙𝑚,Π𝐶𝑛𝑥0𝑥𝜙𝑚,𝑥0Π𝜙𝐶𝑛𝑥0,𝑥0𝑥=𝜙𝑚,𝑥0𝑥𝜙𝑛,𝑥0.(2.10) In view of Step  4 we deduce that 𝜙(𝑥𝑚,𝑥𝑛)0 as 𝑚,𝑛. It follows from Lemma 1.4 that 𝑥𝑚𝑥𝑛0 as 𝑚,𝑛. Hence {𝑥𝑛} is a Cauchy sequence of 𝐶. Since 𝐸 is a Banach space and 𝐶 is closed subset of 𝐸, there exists a point 𝑥𝐶 such that 𝑥𝑛𝑥(𝑛).

Step 6. We have𝑥𝑁𝑖=1𝐹(𝑇𝑖).
By taking 𝑚=𝑛+1 in (2.10), we have lim𝑛𝜙𝑥𝑛+1,𝑥𝑛=0.(2.11) From Lemma 1.4, it follows that lim𝑛𝑥𝑛+1𝑥𝑛=0.(2.12) Noticing that 𝑥𝑛+1𝐶𝑛+1, we obtain 𝜙𝑥𝑛+1,𝑢𝑛𝑥𝜙𝑛+1,𝑥𝑛+𝑁𝑖=1𝛼𝑛,𝑖𝑘𝑛,𝑖𝐿1𝑛.(2.13) From (2.11), lim𝑛𝑘𝑛,𝑖=1 for any 1𝑖𝑁, and Lemma 1.4, we know lim𝑛𝑥𝑛+1𝑢𝑛=0.(2.14) Notice that 𝑥𝑛𝑢𝑛𝑥𝑛𝑥𝑛+1+𝑥𝑛+1𝑢𝑛(2.15) for all 𝑛0. It follows from (2.12) and (2.14) that lim𝑛𝑥𝑛𝑢𝑛=0,(2.16) which implies that 𝑢𝑛𝑥 as 𝑛. Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, from (2.16), we have lim𝑛𝐽𝑥𝑛𝐽𝑢𝑛=0.(2.17) Let 𝑠=sup{𝑥𝑛,𝑇𝑛1𝑥𝑛,𝑇𝑛2𝑥𝑛,,𝑇𝑛𝑁𝑥𝑛𝑛}. Since 𝐸 is uniformly smooth Banach space, we know that 𝐸 is a uniformly convex Banach space. Therefore, from Lemma 1.10 we have, for any 𝑝𝐹, that 𝜙𝑝,𝑢𝑛=𝜙𝑝,𝑇𝑟𝑛𝑦𝑛𝜙𝑝,𝑦𝑛=𝜙𝑝,𝐽1𝛼𝑛,0𝐽𝑥𝑛+𝑁𝑖=1𝛼𝑛,𝑖𝐽𝑇𝑛𝑖𝑧𝑛=𝑝22𝛼𝑛,0𝑝,𝐽𝑥𝑛2𝑁𝑖=1𝛼𝑛,𝑖𝑝,𝐽𝑇𝑛𝑖𝑧𝑛+𝛼𝑛,0𝐽𝑥𝑛+𝑁𝑖=1𝛼𝑛,𝑖𝐽𝑇𝑛𝑖𝑧𝑛2𝑝22𝛼𝑛,0𝑝,𝐽𝑥𝑛2𝑁𝑖=1𝛼𝑛,𝑖𝑝,𝐽𝑇𝑛𝑖𝑧𝑛+𝛼𝑛,0𝑥𝑛2+𝑁𝑖=1𝛼𝑛,𝑖𝑇𝑛𝑖𝑧𝑛2𝛼𝑛,0𝛼𝑛,1𝑔𝐽𝑥𝑛𝐽𝑇𝑛1𝑧𝑛=𝛼𝑛,0𝜙𝑝,𝑥𝑛+𝑁𝑖=1𝛼𝑛,𝑖𝜙𝑝,𝑇𝑛𝑖𝑧𝑛𝛼𝑛,0𝛼𝑛,1𝑔𝐽𝑥𝑛𝐽𝑇𝑛1𝑧𝑛𝛼𝑛,0𝜙𝑝,𝑥𝑛+𝑁𝑖=1𝛼𝑛,𝑖𝑘𝑛,𝑖𝜙𝑝,𝑧𝑛𝛼𝑛,0𝛼𝑛,1𝑔𝐽𝑥𝑛𝐽𝑇𝑛1𝑧𝑛. Therefore, from (2.6) and (2.18), we have 𝜙𝑝,𝑢𝑛𝜙𝑝,𝑥𝑛+𝑁𝑖=1𝛼𝑛,𝑖𝑘𝑛,𝑖𝜙1𝑝,𝑥𝑛𝛼𝑛,0𝛼𝑛,1𝑔𝐽𝑥𝑛𝐽𝑇𝑛1𝑧𝑛+2𝜆𝑛2𝑐2𝜆𝑛𝛾𝐴𝑥𝑛𝐴𝑝2𝑁𝑖=1𝛼𝑛,𝑖𝑘𝑛,𝑖.(2.19) It follows from 𝜆𝑛<𝑐2𝛾/2 that 𝛼𝑛,0𝛼𝑛,1𝑔𝐽𝑥𝑛𝐽𝑇𝑛1𝑧𝑛𝜙𝑝,𝑥𝑛𝜙𝑝,𝑢𝑛+𝑁𝑖=1𝛼𝑛,𝑖𝑘𝑛,𝑖𝜙1𝑝,𝑥𝑛.(2.20) On the other hand, we have ||𝜙𝑝,𝑥𝑛𝜙𝑝,𝑢𝑛||=||𝑥𝑛2𝑢𝑛22𝑝,𝐽𝑥𝑛𝐽𝑢𝑛||||𝑥𝑛𝑢𝑛||𝑥𝑛+𝑢𝑛+2𝐽𝑥𝑛𝐽𝑢𝑛𝑥𝑝𝑛𝑢𝑛𝑥𝑛+𝑢𝑛+2𝐽𝑥𝑛𝐽𝑢𝑛𝑝.(2.21) It follows from (2.16) and (2.17) that lim𝑛𝜙𝑝,𝑥𝑛𝜙𝑝,𝑢𝑛=0.(2.22) Since lim𝑛𝑘𝑛,𝑖=1 and liminf𝑛𝛼𝑛,0𝛼𝑛,1>0, from (2.20) and (2.22) we have lim𝑛𝑔𝐽𝑥𝑛𝐽𝑇𝑛1𝑧𝑛=0.(2.23) Therefore, from the property of 𝑔, we obtain lim𝑛𝐽𝑥𝑛𝐽𝑇𝑛1𝑧𝑛=0.(2.24) Since 𝐽1 is uniformly norm-to-norm continuous on bounded sets, we have lim𝑛𝑥𝑛𝑇𝑛1𝑧𝑛=0,(2.25) and hence 𝑇𝑛1𝑧𝑛𝑥 as 𝑛. Since 𝑇1𝑛+1𝑧𝑛𝑥𝑇1𝑛+1𝑧𝑛𝑇𝑛1𝑧𝑛+𝑇𝑛1𝑧𝑛𝑥, it follows from the asymptotic regularity of 𝑇1 that lim𝑛𝑇1𝑛+1𝑧𝑛𝑥=0.(2.26) That is, 𝑇1(𝑇𝑛1𝑧𝑛)𝑥 as 𝑛. From the closedness of 𝑇1, we get 𝑇1𝑥=𝑥. Similarly, one can obtain that 𝑇𝑖𝑥=𝑥 for 𝑖=2,,𝑁. So, 𝑥𝑁𝑖=1𝐹(𝑇𝑖).
Moreover, from (2.19) we have that 2𝜆𝑛2𝛾𝑐2𝜆𝑛𝐴𝑥𝑛𝐴𝑝21𝛼𝑛,02𝜆𝑛2𝛾𝑐2𝜆𝑛𝐴𝑥𝑛𝐴𝑝2𝑁𝑖=1𝛼𝑛,𝑖𝑘𝑛,𝑖𝜙𝑝,𝑥𝑛𝜙𝑝,𝑢𝑛+𝑁𝑖=1𝛼𝑛,𝑖𝑘𝑛,𝑖𝜙1𝑝,𝑥𝑛,(2.27) which implies that lim𝑛𝐴𝑥𝑛𝐴𝑝=0.(2.28) Now, Lemmas 1.3 and 1.7 imply that 𝜙𝑥𝑛,𝑧𝑛𝑥=𝜙𝑛,Π𝐶𝐽1𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝑥𝜙𝑛,𝐽1𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝑥=𝑉𝑛,𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝑥𝑉𝑛,𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛+𝜆𝑛𝐴𝑥𝑛𝐽21𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝑥𝑛,𝜆𝑛𝐴𝑥𝑛𝑥=𝜙𝑛,𝑥𝑛𝐽+21𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝑥𝑛,𝜆𝑛𝐴𝑥𝑛𝐽=21𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝑥𝑛,𝜆𝑛𝐴𝑥𝑛𝐽21𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝐽1𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛.(2.29) In view of Lemma 1.2 and the fact that 𝐴𝑥𝐴𝑥𝐴𝑝 for all 𝑥𝐶, 𝑝𝐹, we have 𝜙𝑥𝑛,𝑧𝑛4𝑐2𝜆2𝑛𝐴𝑥𝑛𝐴𝑝24𝑐2𝑡2𝐴𝑥𝑛𝐴𝑝2.(2.30) From (2.28) and Lemma 1.4 we get lim𝑛𝑥𝑛𝑧𝑛=0,(2.31) and hence 𝑧𝑛𝑥 as 𝑛.

Step 7. We have 𝑥VI(𝐶,𝐴).
Let 𝑆𝐸×𝐸 be an operator as follows: 𝑆𝑣=𝐴𝑣+𝑁𝐶(𝑣),𝑣𝐶,,𝑣𝐶.(2.32) By Lemma 1.6, 𝑆 is maximal monotone and 𝑆1(0)=VI(𝐶,𝐴). Let (𝑣,𝑤)𝐺(𝑆). Since 𝑤𝑆𝑣=𝐴𝑣+𝑁𝐶(𝑣), we have 𝑤𝐴𝑣𝑁𝐶(𝑣). It follows from 𝑧𝑛𝐶 that 𝑣𝑧𝑛,𝑤𝐴𝑣0.(2.33) On the other hand, from 𝑧𝑛=Π𝐶𝐽1(𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛) and Lemma 1.5 we obtain that 𝑣𝑧𝑛,𝐽𝑧𝑛𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛0,(2.34) and hence 𝑣𝑧𝑛,𝐽𝑥𝑛𝐽𝑧𝑛𝜆𝑛𝐴𝑥𝑛0.(2.35) Then, from (2.33) and (2.35), we have 𝑣𝑧𝑛,𝑤𝑣𝑧𝑛,𝐴𝑣𝑣𝑧𝑛,𝐴𝑣+𝑣𝑧𝑛,𝐽𝑥𝑛𝐽𝑧𝑛𝜆𝑛𝐴𝑥𝑛=𝑣𝑧𝑛,𝐴𝑣𝐴𝑥𝑛+𝐽𝑥𝑛𝐽𝑧𝑛𝜆𝑛=𝑣𝑧𝑛,𝐴𝑣𝐴𝑧𝑛+𝑣𝑧𝑛,𝐴𝑧𝑛𝐴𝑥𝑛+𝑣𝑧𝑛,𝐽𝑥𝑛𝐽𝑧𝑛𝜆𝑛𝑣𝑧𝑛𝐴𝑧𝑛𝐴𝑥𝑛𝑣𝑧𝑛𝐽𝑥𝑛𝐽𝑧𝑛𝑠.(2.36) Hence we have 𝑣𝑥,𝑤0 as 𝑛, since the uniform continuity of 𝐽 and 𝐴 imply that the right side of (2.36) goes to 0 as 𝑛. Thus, since 𝑆 is maximal monotone, we have 𝑥𝑆1(0) and hence 𝑥VI(𝐶,𝐴).

Step 8. We have 𝑥EP(𝑓)=𝐹(𝑇𝑟).
Let 𝑝𝐹. From 𝑢𝑛=𝑇𝑟𝑛𝑦𝑛, (2.3), (2.6) and Lemma 1.9 we obtain that 𝜙𝑢𝑛,𝑦𝑛𝑇=𝜙𝑟𝑛𝑦𝑛,𝑦𝑛𝜙𝑝,𝑦𝑛𝜙𝑝,𝑇𝑟𝑛𝑦𝑛𝜙𝑝,𝑥𝑛+𝑁𝑖=1𝛼𝑛,𝑖𝑘𝑛,𝑖𝜙1𝑝,𝑥𝑛𝜙𝑝,𝑢𝑛.(2.37) It follows from (2.22) and 𝑘𝑛,𝑖1 that 𝜙(𝑢𝑛,𝑦𝑛)0 as 𝑛. Now, by Lemma 1.4 we have that 𝑢𝑛𝑦𝑛0 as 𝑛. Consequently, we obtain that 𝐽𝑢𝑛𝐽𝑦𝑛0 and 𝑦𝑛𝑥 from 𝑢𝑛𝑥 as 𝑛. From the assumption 𝑟𝑛>𝑎, we get lim𝑛𝐽𝑢𝑛𝐽𝑦𝑛𝑟𝑛=0.(2.38) Noting that 𝑢𝑛=𝑇𝑟𝑛𝑦𝑛, we obtain 𝑓𝑢𝑛+1,𝑦𝑟𝑛𝑦𝑢𝑛,𝐽𝑢𝑛𝐽𝑦𝑛0,𝑦𝐶.(2.39) From (A2), we have 𝑦𝑢𝑛,𝐽𝑢𝑛𝐽𝑦𝑛𝑟𝑛𝑢𝑓𝑛,𝑦𝑓𝑦,𝑢𝑛,𝑦𝐶.(2.40) Letting 𝑛, we have from 𝑢𝑛𝑥, (2.38) and (A4) that 𝑓(𝑦,𝑥)0(forall𝑦𝐶). For 𝑡 with 0<𝑡1 and 𝑦𝐶, let 𝑦𝑡=𝑡𝑦+(1𝑡)𝑥. Since 𝑦𝐶 and 𝑥𝐶, we have 𝑦𝑡𝐶 and hence 𝑓(𝑦𝑡,𝑥)0. Now, from (A1) and (A4) we have 𝑦0=𝑓𝑡,𝑦𝑡𝑦𝑡𝑓𝑡+𝑦,𝑦(1𝑡)𝑓𝑡,𝑥𝑦𝑡𝑓𝑡,𝑦(2.41) and hence 𝑓(𝑦𝑡,𝑦)0. Letting 𝑡0, from (A3), we have 𝑓(𝑥,𝑦)0. This implies that 𝑥EP(𝑓). Therefore, in view of Steps  6, 7, and 8 we have 𝑥𝐹.

Step 9. We have 𝑥=Π𝐹𝑥0.
From 𝑥𝑛=Π𝐶𝑛𝑥0, we get 𝑥𝑛𝑧,𝐽𝑥0𝐽𝑥𝑛0,𝑧𝐶𝑛.(2.42) Since 𝐹𝐶𝑛 for all 𝑛1, we arrive at 𝑥𝑛𝑝,𝐽𝑥0𝐽𝑥𝑛0,𝑝𝐹.(2.43) Letting 𝑛, we have 𝑥𝑝,𝐽𝑥0𝐽𝑥0,𝑝𝐹,(2.44) and hence 𝑥=Π𝐹𝑥0 by Lemma 1.5. This completes the proof.

Strong convergence theorem for approximating a common element of the set of solutions of the equilibrium problem and the set of fixed points of a finite family of closed quasi-𝜙-asymptotically nonexpansive mappings in Banach spaces may not require that 𝐸 be 2-uniformly convex. In fact, we have the following theorem.

Theorem 2.2. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space 𝐸 and 𝑇𝑖𝐶𝐶 a closed quasi-𝜙-asymptotically nonexpansive mapping with sequence {𝑘𝑛,𝑖}[1,) such that lim𝑛𝑘𝑛,𝑖=1 for each 1𝑖𝑁. Let 𝑓 be a bifunction from 𝐶×𝐶 to satisfying (A1)–(A4) such that 𝐹=(𝑁𝑖=1𝐹(𝑇𝑖))EP(𝑓) and 𝐹 is bounded. Assume that 𝑇𝑖 is asymptotically regular on 𝐶 for each 1𝑖𝑁. Define a sequence {𝑥𝑛} in 𝐶 in the following manner: 𝑥0𝐶0𝑦=𝐶𝑐𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,𝑛=𝐽1𝛼𝑛,0𝐽𝑥𝑛+𝛼𝑛,1𝐽𝑇𝑛1𝑥𝑛++𝛼𝑛,𝑁𝐽𝑇𝑛𝑁𝑥𝑛,𝑢𝑛𝑢𝐶,𝑓𝑛+1,𝑦𝑟𝑛𝑦𝑢𝑛,𝐽𝑢𝑛𝐽𝑦𝑛𝐶0,𝑦𝐶,𝑛+1=𝑧𝐶𝑛𝜙𝑧,𝑢𝑛𝜙𝑧,𝑥𝑛+𝑁𝑖=1𝛼𝑛,𝑖𝑘𝑛,𝑖𝐿1𝑛,𝑥𝑛+1=Π𝐶𝑛+1𝑥0(2.45) for every 𝑛0, where {𝑟𝑛} is a real sequence in [𝑎,) for some 𝑎>0, 𝐽 is the normalized duality mapping on 𝐸 and 𝐿𝑛=sup{𝜙(𝑝,𝑥𝑛)𝑝𝐹}<. Assume that {𝛼𝑛,0}, {𝛼𝑛,1}, , {𝛼𝑛,𝑁} are real sequences in (0,1) such that 𝑁𝑖=0𝛼𝑛,𝑖=1 and liminf𝑛𝛼𝑛,0𝛼𝑛,𝑖>0, for all 𝑖{1,2,,𝑁}. Then the sequence {𝑥𝑛} converges strongly to Π𝐹𝑥0.

Proof. Put 𝐴0 in Theorem 2.1. We have 𝑧𝑛=𝑥𝑛. Thus, the method of proof of Theorem 2.1 gives the required assertion without the requirement that 𝐸 is 2-uniformly convex.

As some corollaries of Theorems 2.1 and 2.2, we have the following results immediately.

Corollary 2.3. Let 𝐶 be a nonempty, closed, and convex subset of a Hilbert space 𝐻 and 𝑇𝑖𝐶𝐶 a closed quasi-𝜙-asymptotically nonexpansive mapping with sequence {𝑘𝑛,𝑖}[1,) such that lim𝑛𝑘𝑛,𝑖=1 for each 1𝑖𝑁. Let 𝑓 be a bifunction from 𝐶×𝐶 to satisfying (A1)–(A4). Let 𝐴 be a 𝛾-inverse strongly monotone mapping of 𝐶 into 𝐻 with constant 𝛾>0 such that 𝐹=(𝑁𝑖=1𝐹(𝑇𝑖))EP(𝑓)VI(𝐶,𝐴) and 𝐹 is bounded. Assume that 𝑇𝑖 is asymptotically regular on C for each 1𝑖𝑁 and 𝐴𝑥𝐴𝑥𝐴𝑝 for all 𝑥𝐶 and 𝑝𝐹. Define a sequence {𝑥𝑛} in 𝐶 in the following manner: 𝑥0𝐶0𝑧=𝐶𝑐𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,𝑛=𝑃𝐶𝑥𝑛𝜆𝑛𝐴𝑥𝑛,𝑦𝑛=𝛼𝑛,0𝑥𝑛+𝛼𝑛,1𝑇𝑛1𝑧𝑛++𝛼𝑛,𝑁𝑇𝑛𝑁𝑧𝑛,𝑢𝑛𝑢𝐶,𝑓𝑛+1,𝑦𝑟𝑛𝑦𝑢𝑛,𝑢𝑛𝑦𝑛𝐶0,𝑦𝐶,𝑛+1=𝑧𝐶𝑛𝑧𝑢𝑛2𝑧𝑥𝑛2+𝑁𝑖=1𝛼𝑛,𝑖𝑘𝑛,𝑖𝐿1𝑛,𝑥𝑛+1=𝑃𝐶𝑛+1𝑥0(2.46) for every 𝑛0, where {𝑟𝑛} is a real sequence in [𝑎,) for some 𝑎>0 and 𝐿𝑛=sup{𝑥𝑛𝑝2𝑝𝐹}<. Assume that {𝛼𝑛,0}, {𝛼𝑛,1}, , {𝛼𝑛,𝑁} are real sequences in (0,1) such that 𝑁𝑖=0𝛼𝑛,𝑖=1 and liminf𝑛𝛼𝑛,0𝛼𝑛,𝑖>0, for all 𝑖{1,2,,𝑁}. Let {𝜆𝑛} be a sequence in [𝑠,𝑡] for some 0<𝑠<𝑡<𝛾/2. Then the sequence {𝑥𝑛} converges strongly to 𝑃𝐹𝑥0.

Corollary 2.4. Let 𝐶 be a nonempty, closed, and convex subset of a Hilbert space 𝐻 and 𝑇𝑖𝐶𝐶 a closed quasi-𝜙-asymptotically nonexpansive mapping with sequence {𝑘𝑛,𝑖}[1,) such that lim𝑛𝑘𝑛,𝑖=1 for each 1𝑖𝑁. Let 𝑓 be a bifunction from 𝐶×𝐶 to satisfying (A1)–(A4) such that 𝐹=(𝑁𝑖=1𝐹(𝑇𝑖))EP(𝑓) and 𝐹 is bounded. Assume that 𝑇𝑖 is asymptotically regular on C for each 1𝑖𝑁. Define a sequence {𝑥𝑛} in 𝐶 in the following manner: 𝑥0𝐶0𝑦=𝐶𝑐𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,𝑛=𝛼𝑛,0𝑥𝑛+𝛼𝑛,1𝑇𝑛1𝑥𝑛++𝛼𝑛,𝑁𝑇𝑛𝑁𝑥𝑛,𝑢𝑛𝑢𝐶,𝑓𝑛+1,𝑦𝑟𝑛𝑦𝑢𝑛,𝑢𝑛𝑦𝑛𝐶0,𝑦𝐶,𝑛+1=𝑧𝐶𝑛𝑧𝑢𝑛2𝑧𝑥𝑛2+𝑁𝑖=1𝛼𝑛,𝑖𝑘𝑛,𝑖𝐿1𝑛,𝑥𝑛+1=𝑃𝐶𝑛+1𝑥0(2.47) for every 𝑛0, where {𝑟𝑛} is a real sequence in [𝑎,) for some 𝑎>0 and 𝐿𝑛=sup{𝑥𝑛𝑝2𝑝𝐹}<. Assume that {𝛼𝑛,0}, {𝛼𝑛,1}, , {𝛼𝑛,𝑁} are real sequences in (0,1) such that 𝑁𝑖=0𝛼𝑛,𝑖=1 and liminf𝑛𝛼𝑛,0𝛼𝑛,𝑖>0, for all 𝑖{1,2,,𝑁}. Let {𝜆𝑛} be a sequence in [𝑠,𝑡] for some 0<𝑠<𝑡<𝛾/2. Then the sequence {𝑥𝑛} converges strongly to 𝑃𝐹𝑥0.

Remark 2.5. Theorems 2.1 and 2.2 extend the main results of [23] from quasi-𝜙-nonexpansive mappings to more general quasi-𝜙-asymptotically nonexpansive mappings.

Acknowledgments

The research was supported by Fundamental Research Funds for the Central Universities (Program no. ZXH2012K001), supported in part by the Foundation of Tianjin Key Lab for Advanced Signal Processing and it was also supported by the Science Research Foundation Program in Civil Aviation University of China (2012KYM04).

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