Modified Hybrid Block Iterative Algorithm for Uniformly Quasi-𝜙-Asymptotically Nonexpansive Mappings

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-𝜙-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 𝐶 be a nonempty closed convex subspace of a real Banach space 𝐸. A mapping 𝐴∶𝐷(𝐴)⊂𝐸→𝐸^∗ is said to be monotone if for each 𝑥,𝑦∈𝐷(𝐴), the following inequality holds: ⟨𝑥−𝑦,𝐴𝑥−𝐴𝑦⟩≥0.(1.1)

A mapping 𝐴∶𝐶→𝐸^∗ is called 𝛼-inverse-strongly monotone if there exists 𝛼>0 such that ⟨𝑥−𝑦,𝐴𝑥−𝐴𝑦⟩≥𝛼‖𝐴𝑥−𝐴𝑦‖².(1.2)

A monotone mapping 𝑇 is said to be maximal monotone if 𝑅(𝐽+𝑟𝑇)=𝐸^∗, for all 𝑟>0, where 𝐽 is the normalized duality mapping. We denote by 𝑇⁻¹(0)={𝑥∈𝐸∶0∈𝑇𝑥} the set of zero points of 𝑇.

Remark 1.1. It is well know that if 𝐴∶𝐶→𝐸^∗ 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 𝐶 be a nonempty closed convex subspace of a real Banach space 𝐸 with dual 𝐸^∗ and ⟨⋅,⋅⟩ is the pairing between 𝐸 and 𝐸^∗. Let 𝑓∶𝐶×𝐶→𝑅 be a bifunction and 𝐴∶𝐶→𝐸^∗ be a monotone mapping. The generalized equilibrium problem means that finding a 𝑢∈𝐶 such that 𝑓(𝑢,𝑦)+⟨𝐴𝑢,𝑦−𝑢⟩≥0,∀𝑦∈𝐶.(1.3)

The set of solutions of (1.3) is denoted by GEP(𝑓,𝐴), that is, GEP(𝑓,𝐴)={𝑢∈𝐶∶𝑓(𝑢,𝑦)+⟨𝐴𝑢,𝑦−𝑢⟩≥0,∀𝑦∈𝐶}.(1.4)

If 𝐴=0, then the problem (1.3) is equivalent to that of finding a 𝑢∈𝐶 such that 𝑓(𝑢,𝑦)≥0,∀𝑦∈𝐶,(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 𝑢∈𝐶 such that ⟨𝐴𝑢,𝑦−𝑢⟩≥0,∀𝑦∈𝐶,(1.6) which is called the variational inequality of Browder type. The solution of (1.6) is denoted by VI(𝐶,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 𝐶_𝑗,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: 𝑥𝑦_0𝑛=𝑥∈𝐶,arbitrarily,𝑢_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑥_𝑛;∈𝐶s.t.𝑓𝑢_𝑛,𝑦+1𝑟_𝑛𝐶⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩,∀𝑦∈𝐶_𝑛+1𝑥=𝑧∈𝐶_𝑛∶𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑥_𝑛_𝑛+1;=Π_𝐶𝑛+1𝑥,𝑛=1,2,…,(1.7) where 𝑇 is a relatively nonexpansive mapping and 𝑓 is a bifunction from 𝐶×𝐶 into 𝑅. They prove that the sequence {𝑥_𝑛} converges strongly to 𝑞=Π_{𝐹(𝑇)∩EP(𝑓)} under appropriate conditions.

In the same year, Qin et al. [7] introduced a hybrid projection algorithm to two quasi-𝜙-nonexpansive mappings in Banach spaces as follows: 𝑥𝐶₀₁=𝑥∈𝐶,arbitrarily,𝑥=𝐶;₁𝑦_𝑛=Π_𝐶1𝑥₀;𝑢_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+𝛽_𝑛𝐽𝑇𝑥_𝑛+𝛾_𝑛𝐽𝑆𝑥_𝑛;∈𝐶s.t.𝑓𝑢_𝑛,𝑦+1𝑟_𝑛𝐶⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥0,∀𝑦∈𝐶;_𝑛+1𝑥=𝑧∈𝐶_𝑛∶𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑥_𝑛_𝑛+1;=Π_𝐶𝑛+1𝑥,𝑛=1,2,…,(1.8) where Π_𝐶𝑛+1 is the generalized projection from 𝐸 onto 𝐶_𝑛+1. They proved that the sequence {𝑥_𝑛} converges strongly to Π_{𝐹(𝑆)∩𝐹(𝑇)∩EP(𝑓)}𝑥₀. Then Petrot et al. [17] improved the notion from a relatively nonexpansive mapping or a quasi-𝜙-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-𝜙-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. 𝐶𝑥₀₁=𝑥∈𝐸,arbitrarily,𝑥=𝐶;₁𝑦_𝑛=Π_𝐶1𝑥₀;𝑧_𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑧_𝑛;=𝐽⁻¹𝛼^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛𝑖𝑥_𝑛𝑢_𝑛;∈𝐶s.t.𝑓𝑢_𝑛,𝑦+⟨𝐴𝑦_𝑛,𝑦−𝑢_𝑛⟩+1𝑟_𝑛𝐶⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥0;_𝑛+1𝑥_𝑛+1=𝑧∈𝐶_𝑛∶𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑥_𝑛+𝜃_𝑛;=Π_𝐶𝑛+1𝑥,𝑛=1,2,….(1.9)

They proved that {𝑥_𝑛} converges strongly to Π_∩^∞𝑖=1_{𝐹(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-𝜙-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-𝜙-asymptotically nonexpansive mappings which is more general than closed quasi-𝜙-nonexpansive mappings in a uniformly smooth and strictly convex Banach space 𝐸. In (1.9) we find iterative step 𝑦_𝑛 is not essential, so we combine 𝑦_𝑛 with 𝑧_𝑛 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 lim_𝑡→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡(2.1) exists for all 𝑥,𝑦∈𝑈={𝑧∈𝐸∶‖𝑧‖=1}, and 𝐸 is said to be uniformly smooth if the limit (2.1) exists uniformly for all 𝑥,𝑦∈𝑈. Then a Banach space 𝐸 is said to be strictly convex if ‖𝑥+𝑦‖/2≤1 for all 𝑥,𝑦∈𝑈 and 𝑥≠𝑦. It is said to be uniformly convex if for each 𝜀∈(0,2], there exists 𝛿>0 such that ‖𝑥+𝑦‖/2≤1−𝛿 for all 𝑥,𝑦∈𝑈 with ‖𝑥−𝑦‖≥𝜀.

Let 𝐸 be a Banach space and let 𝐸^∗ be the topological dual of 𝐸. For all 𝑥∈𝐸 and 𝑥^∗∈𝐸^∗, we denote the value of 𝑥^∗ at 𝑥 by ⟨𝑥,𝑥^∗⟩. Then, the duality mapping 𝐽∶𝐸→2^𝐸∗ is defined by 𝐽(𝑥)=𝑓∈𝐸^∗∶⟨𝑥,𝑓⟩=‖𝑥‖²=‖𝑓‖²,(2.2) for every 𝑥∈𝐸. By the Hahn-Banach theorem, 𝐽(𝑥) 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 𝐽⁻¹ is norm-weak∗- continuous.(iii)If 𝐸 is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping 𝐽∶𝐸→2^𝐸∗ is single-valued, one-to-one, and onto.(iv)A Banach space 𝐸 is uniformly smooth if and only if 𝐸^∗ is uniformly convex.

Let 𝐸 be a smooth, strictly convex, and reflexive Banach space and let 𝐶 be a nonempty closed convex subset of 𝐸. Throughout this paper, we denote by 𝜙 the function defined by 𝜙(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖²,∀𝑥,𝑦∈𝐸.(2.3) Following Alber [21], the generalized projection Π_𝐶 from 𝐸 onto 𝐶 is defined by Π_𝐶(𝑥)=argmin_𝑢∈𝐶𝜙(𝑢,𝑥),∀𝑥∈𝐸.(2.4)

If 𝐸 is a Hilbert space, then 𝜙(𝑥,𝑦)=‖𝑥−𝑦‖² and Π_𝐶 is the metric projection of 𝐻 onto 𝐶. We know the following lemmas for generalized projections.

Lemma 2.1 (see Alber [21] and Kamimura and Takahashi [22]). Let 𝐶 be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space 𝐸. Then 𝜙𝑥,Π_𝐶𝑦+𝜙Π_𝐶𝑦,𝑦≤𝜙(𝑥,𝑦),∀𝑥∈𝐶,𝑦∈𝐸.(2.5)

Lemma 2.2 (see Alber [21], Kamimura and Takahashi [22]). Let 𝐶 be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space and let 𝑥∈𝐸 and 𝑧∈𝐶. Then 𝑧=Π_𝐶𝑥⟺⟨𝑦−𝑧,𝐽𝑥−𝐽𝑧⟩≤0,∀𝑦∈𝐶.(2.6)

Lemma 2.3 (see Kamimura and Takahashi [22]). Let 𝐸 be a smooth and uniformly convex Banach space and let {𝑥_𝑛} and {𝑦_𝑛} be sequences in 𝐸 such that either {𝑥_𝑛} or {𝑦_𝑛} is bounded. If lim_𝑛→∞𝜙(𝑥_𝑛,𝑦_𝑛)=0, then lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.

Let 𝐶 be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸 and let 𝑇 be a mapping from 𝐶 into itself. We denoted by 𝐹(𝑇) the set of fixed points of 𝑇, that is 𝐹(𝑇)={𝑥∶𝑇𝑥=𝑥}. A point 𝑝∈𝐶 is said to be an asymptotic fixed point of 𝑇 if there exists {𝑥_𝑛} in 𝐶 which converges weakly to 𝑝 and lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. We denote the set of all asymptotic fixed points of 𝑇 by 𝐹(𝑇).

A mapping 𝑇 from 𝐶 into itself is said to be relatively nonexpansive [23] if the following conditions are satisfied:(1)𝐹(𝑇) is nonempty,(2)𝜙(𝑢,𝑇𝑥)≤𝜙(𝑢,𝑥),forall𝑢∈𝐹(𝑇),𝑥∈𝐸,(3)𝐹(𝑇)=𝐹(𝑇).

A mapping 𝑇 from 𝐶 into itself is said to be relatively quasi-nonexpansive if the following conditions are satisfied:(1)𝐹(𝑇) is nonempty,(2)𝜙(𝑢,𝑇𝑥)≤𝜙(𝑢,𝑥),forall𝑢∈𝐹(𝑇),𝑥∈𝐸,

The asymptotic behavior of a relatively nonexpansive mapping was studied in [24]. 𝑇 is said to be 𝜙-nonexpansive, if 𝜙(𝑇𝑥,𝑇𝑦)≤𝜙(𝑥,𝑦) for 𝑥,𝑦∈𝐶. 𝑇 is said to be quasi-𝜙-asymptotically nonexpansive if the following conditions are satisfied:(1)𝐹(𝑇) is nonempty,(2)𝜙(𝑢,𝑇^𝑛𝑥)≤𝑘_𝑛𝜙(𝑢,𝑥),forall𝑢∈𝐹(𝑇),𝑥∈𝐸 and 𝑛≥1,where {𝑘_𝑛} is a real sequence within [1,∞) and 𝑘_𝑛→1 as 𝑛→∞.

A mapping 𝑇 is said to be closed if for any sequence {𝑥_𝑛}⊂𝐶 with 𝑥_𝑛→𝑥 and 𝑇𝑥_𝑛→𝑦, then 𝑇𝑥=𝑦. It is easy to know that each relatively nonexpansive mapping is closed. The class of quasi-𝜙-asymptotically nonexpansive mappings contains properly the class of quasi-𝜙-nonexpansive mappings as a subclass and the class of quasi-𝜙-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 𝐶 be a closed convex subset of 𝐸, and let 𝑇 be a closed and quasi-𝜙-asymptotically nonexpansive mapping from 𝐶 into itself with a sequence {𝑘_𝑛}⊂[1,∞) and 𝑘_𝑛→1 as 𝑛→∞. Then 𝐹(𝑇) is a closed and convex subset of 𝐶.

For solving the equilibrium problem, let us assume that a bifunction 𝑓 satisfies the following conditions:(A1)𝑓(𝑥,𝑥)=0,forall𝑥∈𝐸,(A2)𝑓 is monotone, that is, 𝑓(𝑥,𝑦)+𝑓(𝑦,𝑥)≤0,forall𝑥,𝑦∈𝐸,(A3)for all 𝑥,𝑦,𝑧∈𝐸, limsup_𝑡↓0𝑓(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝑓(𝑥,𝑦),(A4)for all 𝑥∈𝐶, 𝑦↦𝑓(𝑥,𝑦) is convex and lower semicontinuous.

Lemma 2.5 (see Blum and Oettli [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 𝑅=(−∞,+∞) satisfying (A1)–(A4), and let 𝑟>0 and 𝑥∈𝐸. Then, there exists 𝑧∈𝐶 such that 𝑓(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶.(2.7)

Lemma 2.6 (see Kumam [5]). 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 𝑅=(−∞,+∞) satisfying (A1)–(A4) and let 𝐴 be a monotone mapping from 𝐶 into 𝐸^∗. For 𝑟>0, define a mapping 𝑇_𝑟∶𝐶→𝐶 as follows: 𝑇_𝑟(𝑥)=𝑧∈𝐶∶𝑓(𝑧,𝑦)+⟨𝐴𝑥,𝑦−𝑧⟩+1𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶,(2.8) for all 𝑥∈𝐸. Then, the following hold:(1)𝑇_𝑟 is single-valued,(2)𝑇_𝑟 is a firmly nonexpansive-type mapping [6], that is, for all 𝑥,𝑦∈𝐸, ⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽𝑇_𝑟𝑥−𝐽𝑇_𝑟𝑦⟩≤⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽𝑥−𝐽𝑦⟩,(2.9)(3)𝐹(𝑇_𝑟)=GEP(𝑓,𝐴),(4)GEP(𝑓,𝐴) is closed and convex.

Lemma 2.7 (see Kumam [5]). 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 𝑅=(−∞,+∞) satisfying (A1)–(A4) and let 𝐴 be a monotone mapping from 𝐶 into 𝐸^∗. For 𝑥∈𝐸, 𝑞∈𝐹(𝑇_𝑟), then the following holds: 𝜙𝑞,𝑇_𝑟𝑥+𝜙𝑇_𝑟𝑥,𝑥≤𝜙(𝑞,𝑥).(2.10)

Lemma 2.8 (see Chang et al. [19]). Let E be a uniformly convex Banach space, 𝑟>0 a positive number, and 𝐵_𝑟(0) a closed ball of 𝐸. Then, for any given sequence {𝑥_𝑖}^∞𝑖=1⊂𝐵_𝑟(0) and for any given sequence {𝜆_𝑖}^∞𝑖=1 of positive number with ∑^∞𝑛=1𝜆_𝑛=1, there exists a continuous, strictly increasing, and convex function 𝑔∶[0,2𝑟)→[0,∞) with 𝑔(0)=0 such that, for any positive integer 𝑖,𝑗 with 𝑖≠𝑗, ‖‖‖‖^∞_𝑛=1𝜆_𝑛𝑥_𝑛‖‖‖‖²≤^∞_𝑛=1𝜆_𝑛‖‖𝑥_𝑛‖‖²−𝜆_𝑖𝜆_𝑖𝑔‖‖𝑥_𝑖−𝑥_𝑗‖‖.(2.11)

Definition 2.9. A mapping 𝑆 from 𝐶 into itself is said to be equally continuous if it is follows that lim_𝑛→∞‖‖𝑥_𝑛−𝑦_𝑛‖‖=0⇒lim_𝑛→∞‖‖𝑆^𝑛𝑥_𝑛−𝑆^𝑛𝑦_𝑛‖‖=0,∀𝑥_𝑛,𝑦_𝑛∈𝐶.(2.12)
A mapping 𝑆 from 𝐶 into itself is said to be uniformly 𝐿-Lipschitz continuous if there exists a constant 𝐿>0 such that ‖𝑆^𝑛𝑥−𝑆^𝑛𝑦‖≤𝐿‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(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 {𝑆_𝑖}^∞𝑖=1∶𝐶→𝐶 be a sequence of mapping. {𝑆_𝑖}^∞𝑖=1 is said to be a family of uniformly quasi-𝜙-asymptotically nonexpansive mappings, if ∩^∞𝑖=1𝐹(𝑆_𝑖)≠∅, and there exists a sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 such that for each 𝑖≥1, 𝜙𝑝,𝑆^𝑛𝑖𝑥≤𝑘_𝑛𝜙(𝑝,𝑥),∀𝑝∈∩^∞𝑖=1𝐹𝑆_𝑖,𝑥∈𝐶,∀𝑛≥1.(2.14)

3. Main Results

Theorem 3.1. Let 𝐸 be a uniformly smooth and uniformly convex Banach space, and let 𝐶 be a nonempty closed convex subset of 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to 𝑅=(−∞,+∞) satisfying (𝐴1)−(𝐴4) and let 𝐴 be a continuous monotone mapping of 𝐶 into 𝐸^∗. Let {𝑆_𝑖}^∞𝑖=1∶𝐶→𝐶 be an infinite family of closed equally continuous and uniformly quasi-𝜙-asymptotically nonexpansive mappings with a sequence {𝑘_𝑛}⊂[1,∞),𝑘_𝑛→1 such that 𝐹∶=∩^∞𝑖=1𝐹(𝑆_𝑖)∩GEP(𝑓,𝐴) is a nonempty and bounded subset in 𝐶. Let {𝑥_𝑛} be a sequence generated by 𝑥𝐶₀∈𝐸𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,₁𝑦_𝑛=𝐶,𝑥₁=Π_𝐶1𝑥₀,=𝐽⁻¹𝛼^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛𝑖𝑥_𝑛𝑢_𝑛,𝐶=_𝑛+1𝑇_𝑟𝑛𝑦_𝑛,𝑥_𝑛+1=𝑧∈𝐶_𝑛∶𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑥_𝑛+𝜃_𝑛,=Π_𝐶𝑛+1𝑥₀,∀𝑛=1,2,3,…,(3.1) where 𝐽 is the duality mapping on 𝐸, {𝛼_𝑛,𝑖}^∞𝑖=0 are sequences in [0,1] which satisfies ∑^∞𝑖=0𝛼_𝑛,𝑖=1, 𝜃_𝑛=sup_𝑝∈𝐹(𝑘_𝑛−1)𝜙(𝑝,𝑥_𝑛), and 𝑟_𝑛∈[𝑎,+∞) for some 𝑎>0. If liminf_𝑛→∞𝛼_𝑛,0𝛼_𝑛,𝑖>0 for all 𝑛≥0, then {𝑥_𝑛} converges strongly to Π_𝐹𝑥₀, where Π_𝐹 is the generalized projection from 𝐶 onto 𝐹.

Proof. We first show that 𝐶_𝑛 is closed and convex. It is obvious that 𝐶_𝑛 is closed. In addition, since 𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑥_𝑛+𝜃_𝑛⟺‖‖𝑢_𝑛‖‖²−‖‖𝑥_𝑛‖‖²−2⟨𝑧,𝐽𝑢_𝑛−𝐽𝑥_𝑛⟩−𝜃_𝑛≤0,(3.2)
so 𝐶_𝑛 is convex, therefore, 𝐶_𝑛 is a closed convex subset of 𝐸 for all 𝑛≥0.
Next, we show that 𝐹⊂𝐶_𝑛 for all 𝑛≥1. It is clear that 𝐹⊂𝐶₁=𝐶. Suppose 𝐹∈𝐶_𝑛 for 𝑛>1, by the property of 𝜙, ∑^∞𝑖=0𝛼_𝑛,𝑖=1, Lemmas 2.6 and 2.8, and uniformly quasi-𝜙-asymptotically nonexpansive of 𝑆_𝑛 for each 𝑢∈𝐹⊂𝐶_𝑛, then we have𝜙𝑢,𝑢_𝑛=≤𝜙𝜙𝑢𝑢,,𝑦𝑇_𝑟𝑛𝑛𝑦_𝑛=𝜙𝑢,𝐽₋₁𝛼^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛𝑖𝑥_𝑛=‖𝑢‖₂−2𝑢,𝛼_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛𝑖𝑥_𝑛+‖‖‖‖𝛼_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛𝑖𝑥_𝑛‖‖‖‖²≤‖𝑝‖²−2𝛼_𝑛,0⟨𝑝,𝐽𝑥_𝑛⟩−2^∞_𝑖=1𝛼_𝑛,𝑖𝑢,𝐽𝑆^𝑛𝑖𝑥_𝑛+𝛼_𝑛,0‖‖𝑥_𝑛‖‖²+^∞_𝑖=1≤𝛼_𝑛,0𝛼_𝑛,𝑖‖‖𝑆^𝑛𝑖𝑥_𝑛‖‖²−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑗𝑥_𝑛‖‖𝜙𝑢,𝑥_𝑛+^∞_𝑖=1≤𝛼_𝑛,0𝛼_𝑛,𝑖𝜙𝑢,𝑆^𝑛𝑖𝑥_𝑛𝜙𝑢,𝑥_𝑛+^∞_𝑖=1≤𝑘_𝑛𝛼_𝑛,𝑖𝑘_𝑛𝜙𝑢,𝑥_𝑛≤𝜙𝜙𝑢,𝑥𝑢,_𝑛𝑥_𝑛+𝜃_𝑛.(3.3)
This shows that 𝑢∈𝐶_𝑛+1 implies that 𝐹⊂𝐶_𝑛 for all 𝑛≥1 by induction. On the one hand, since 𝑥_𝑛+1=Π_𝐶𝑛𝑥₀ and 𝐶_𝑛+1⊂𝐶_𝑛 for all 𝑛≥1, we have 𝜙𝑥_𝑛,𝑥₀≤𝜙𝑥_𝑛+1,𝑥₀.(3.4)
Therefore {𝜙(𝑥_𝑛,𝑥₀)} is nondecreasing. In the other hand, by Lemma 2.1, we have 𝜙𝑥_𝑛,𝑥₀=≤𝜙𝜙𝑢Π,𝑥_𝐶𝑛0𝑥₀,𝑥₀≤𝜙𝑢,𝑥₀−𝜙𝑢,𝑥_𝑛,(3.5) for each 𝑢∈𝐹(𝑇)⊂𝐶_𝑛 for all 𝑛≥0. Therefore, {𝜙(𝑥_𝑛,𝑥₀)} is bounded; this together with (3.4) implies that the limit of {𝜙(𝑥_𝑛,𝑥₀)} exists.
Since {𝜙(𝑥_𝑛,𝑥₀)} is bounded, so {𝑥_𝑛} is bounded by (1.7), together with lim_𝑛→∞𝑘_𝑛=1, we have that lim_𝑛→∞𝜃_𝑛=0.(3.6)
From Lemma 2.1, we have, for any positive integers 𝑛,𝑚, that 𝜙𝑥_𝑛+𝑚,𝑥_𝑛=≤𝜙𝜙𝑥𝑥_{𝑛+𝑚𝑛+𝑚},Π_𝐶𝑛𝑥₀=𝜙𝑥_𝑛+𝑚,𝑥₀−𝜙Π_𝐶𝑛𝑥₀,𝑥₀,𝑥₀−𝜙𝑥_𝑛,𝑥₀.(3.7)
Because the limit of {𝜙(𝑥_𝑛,𝑥₀)} exists, then we have lim_𝑛→∞𝜙𝑥_𝑛+𝑚,𝑥_𝑛=0(3.8) uniformly for positive integers 𝑚>1. Since {𝑥_𝑛} is a bounded sequence, by using Lemma 2.4, we have lim_𝑛→∞‖‖𝑥_𝑛+𝑚−𝑥_𝑛‖‖=0(3.9) uniformly for positive integers 𝑚>1. Hence {𝑥_𝑛} is a Cauchy sequence, therefore, there exists a point 𝑝∈𝐶 such that {𝑥_𝑛} converges strongly to 𝑝.
In addition, from (3.7) we have lim_𝑛→∞𝜙(𝑥_𝑛+1,𝑥_𝑛)=0, this together with the fact 𝑥_𝑛+1∈𝐶_𝑛 implies that 𝜙𝑥_𝑛+1,𝑢_𝑛≤𝜙𝑥_𝑛+1,𝑥_𝑛+𝜃_𝑛.(3.10)
Taking limit on both side of (3.10) and from (3.6),we get that lim_𝑛→∞𝜙𝑥_𝑛+1,𝑢_𝑛=0.(3.11)
By using Lemma 2.4, we have lim_𝑛→∞‖‖𝑥_𝑛+1−𝑢_𝑛‖‖=0,(3.12) which implies that {𝑢_𝑛} converges strongly to 𝑝.
From (3.3), we have 𝜙(𝑢,𝑦_𝑛)≤(𝑢,𝑥_𝑛)+𝜃_𝑛, together with 𝑢_𝑛=𝑇_𝑟𝑛𝑦_𝑛 and Lemma 2.7, we have 𝜙𝑢_𝑛,𝑦_𝑛≤=𝜙𝜙𝑢𝑇,_𝑟𝑛𝑦𝑦_𝑛𝑛,𝑦_𝑛≤𝜙𝑢,𝑥_𝑛−𝜙𝑢,𝑇_𝑟𝑛𝑦_𝑛−𝜙𝑢,𝑇_𝑟𝑛𝑦_𝑛+𝜃_𝑛=𝜙𝑢,𝑥_𝑛−𝜙𝑢,𝑢_𝑛+𝜃_𝑛(3.13)
for any 𝑢∈𝐹. This implies that lim_𝑛→∞𝜙𝑢_𝑛,𝑦_𝑛=0.(3.14)
Therefore, we have lim_𝑛→∞‖‖𝑢_𝑛−𝑦_𝑛‖‖=0,(3.15) which implies that {𝑦_𝑛} converges strongly to 𝑝. Thus we have proved that 𝑥_𝑛→𝑝,𝑢_𝑛→𝑝,𝑦_𝑛→𝑝,(3.16) as 𝑛→∞, where 𝑝∈𝐶. From (3.1) ‖‖𝐽𝑥_𝑛−𝐽𝑦_𝑛‖‖=‖‖‖‖𝐽𝑥_𝑛−𝛼_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛𝑖𝑥_𝑛=‖‖‖‖_∞‖‖‖‖_𝑖=1≤_∞𝛼_𝑛,𝑖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑖𝑥_𝑛‖‖‖‖_𝑖=1𝛼_𝑛,𝑖‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑖𝑥_𝑛‖‖,(3.17) and hence ‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑖𝑥_𝑛‖‖≤1∑^∞𝑖=1𝛼_𝑛,𝑖‖‖𝐽𝑥_𝑛−𝐽𝑦_𝑛‖‖.(3.18)
Taking limit on both side of above inequality, by liminf_𝑛→∞∑^∞𝑖=1𝛼_𝑛,𝑖>0 and from (3.16), we have lim_𝑛→∞‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑖𝑥_𝑛‖‖=0.(3.19)
Since 𝐽⁻¹ is uniformly norm-to-norm continuous on bounded sets, we have lim_𝑛→∞‖‖𝑥_𝑛−𝑆^𝑛𝑖𝑥_𝑛‖‖=0,(3.20) for each 𝑖≥1, together with (3.16), we get that lim_𝑛→∞𝑆^𝑛𝑖𝑥_𝑛=𝑝,(3.21) for each 𝑖≥1. Since 𝑆_𝑖 is equally continuous, we have ‖‖𝑆^𝑖𝑛+1𝑥_𝑛−𝑆^𝑛𝑖𝑥_𝑛‖‖=+‖‖𝑆‖‖𝑥^𝑖𝑛+1𝑥_𝑛𝑛+1−𝑆^𝑖𝑛+1𝑥_𝑛+1‖‖+‖‖𝑆^𝑖𝑛+1𝑥_𝑛+1−𝑥_𝑛+1‖‖≤𝐿_𝑖−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑆^𝑖𝑛+1𝑥_𝑛‖‖+1‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+‖‖𝑆^𝑖𝑛+1𝑥_𝑛+1−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛−𝑆^𝑖𝑛+1𝑥_𝑛‖‖.(3.22)
Together with (3.16) and (3.20), we have lim_𝑛→∞‖𝑆^𝑖𝑛+1𝑥_𝑛−𝑆^𝑛𝑖𝑥_𝑛‖=0. From (3.21), we have 𝑆^𝑖𝑛+1𝑥_𝑛→𝑝, that is, 𝑆_𝑖𝑆^𝑛𝑖𝑥_𝑛→𝑝. In view of closeness of 𝑆_𝑖, we have 𝑆_𝑖𝑝=𝑝, for all 𝑖≥1. This implies that 𝑝∈∩^∞𝑖=1𝐹(𝑆_𝑖).
Next we show 𝑝∈GEP(𝑓,𝐴). By 𝑢_𝑛=𝑇_𝑟𝑛𝑦_𝑛, we have 𝑓𝑢_𝑛,𝑦+⟨𝐴𝑦_𝑛,𝑦−𝑢_𝑛⟩+1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥0,∀𝑦∈𝐶.(3.23)
From (A2), we get that ⟨𝐴𝑦_𝑛,𝑦−𝑢_𝑛⟩+1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥−𝑓𝑢_𝑛,𝑦≥𝑓𝑦,𝑢_𝑛,∀𝑦∈𝐶,(3.24) and hence 0≥−⟨𝐴𝑦_𝑛,𝑦−𝑢_𝑛⟩−𝑦−𝑢_𝑛,𝐽𝑢^𝑛−𝐽𝑦_𝑛𝑟_𝑛+𝑓𝑦,𝑢^𝑛,∀𝑦∈𝐶.(3.25)
For 𝑡 with 0<𝑡<1 and 𝑦∈𝐶, let 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑝, then 𝑦_𝑡∈𝐶, from (3.25) we have ⟨𝐴𝑦_𝑡,𝑦_𝑡−𝑢_𝑛⟩≥⟨𝐴𝑦_𝑡,𝑦_𝑡−𝑢_𝑛⟩−⟨𝐴𝑦_𝑛,𝑦_𝑡−𝑢_𝑛⟩−𝑦_𝑡−𝑢_𝑛,𝐽𝑢^𝑛−𝐽𝑦_𝑛𝑟_𝑛+𝑓𝑦^𝑡=⟨𝐴𝑦_𝑡,𝑢_𝑛−𝑦_𝑡−𝐴𝑢_𝑛,𝑦_𝑡−𝑢_𝑛⟩+⟨𝐴𝑢_𝑛−𝐴𝑦_𝑛,𝑦_𝑡−𝑢_𝑛⟩−𝑢_𝑛,𝐽𝑢^𝑛−𝐽𝑦_𝑛𝑟_𝑛+𝑓𝑦^𝑡,𝑢_𝑛.(3.26) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, 𝐴 is monotone and (3.16), we have ⟨𝐴𝑦_𝑡,𝑦_𝑡−𝑢_𝑛⟩≥0.(3.27)
It follows from (A4) that 𝑓𝑦_𝑡,𝑝≤≤lliimm_𝑛→∞inf_𝑛→∞𝑓𝑦_𝑡,𝑢_𝑛=⟨𝐴𝑦_𝑡⟨𝐴𝑦_𝑡,𝑦_𝑡−𝑢_𝑛⟩=𝑡⟨𝐴𝑦,_𝑡𝑦_𝑡−𝑝⟩,𝑦−𝑝⟩.(3.28)
From the conditions (A1) and (A4), we have 0=≤𝑓𝑡𝑓𝑦𝑦_𝑡,_𝑡𝑦_𝑡≤𝑡𝑓𝑦_𝑡,𝑦+(1−𝑡)𝑓𝑦_𝑡,𝑝≤𝑓𝑦_𝑡,𝑦+(1−𝑡)𝑡⟨𝐴𝑦_𝑡,𝑦−𝑝⟩,𝑦+(1−𝑡)⟨𝐴𝑦_𝑡,𝑦−𝑝⟩.(3.29)
Letting 𝑡→0, we get 𝑓(𝑝,𝑦)+⟨𝐴𝑝,𝑦−𝑝⟩≥0,∀𝑦∈𝐶.(3.30)
This implies that 𝑝∈GEP(𝑓,𝐴).
Finally, we show that 𝑝=Π_𝐹𝑥₀. Let 𝑤=Π_𝐹𝑥₀, from 𝑝∈𝐹, we have 𝜙𝑝,𝑥₀≥𝜙𝑤,𝑥₀.(3.31)
Since 𝑥_𝑛=Π_𝐶𝑛𝑥₀ and 𝑤∈𝐹⊂𝐶_𝑛, 𝜙𝑥_𝑛,𝑥₀≤𝜙𝑤,𝑥₀,(3.32)
together with above two hands and lim_𝑛→∞𝑥_𝑛=𝑝, we obtain 𝜙𝑝,𝑥₀=𝜙𝑤,𝑥₀.(3.33)
that is 𝑝=𝑤=Π_𝐹𝑥₀. 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 𝐶 be a nonempty closed convex subset of 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to 𝑅=(−∞,+∞) satisfying (𝐴1)−(𝐴4). Let {𝑆_𝑖}^∞𝑖=1∶𝐶→𝐶 be an infinite family of closed equally continuous and uniformly quasi-𝜙-asymptotically nonexpansive mappings with a sequence {𝑘_𝑛}⊂[1,∞),𝑘_𝑛→1 such that 𝐹∶=∩^∞𝑖=1𝐹(𝑆_𝑖)∩EP(𝑓) is a nonempty and bounded subset in 𝐶. Let {𝑥_𝑛} be a sequence generated by 𝑥₀∈𝐸𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,𝑦𝐶_1𝑛=𝐶,𝑥₁=Π_𝐶1𝑥₀,=𝐽⁻¹𝛼^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛𝑖𝑥_𝑛𝑢_𝑛,∈𝐶s.t.𝑓𝑢_𝑛,𝑦+1𝑟_𝑛𝐶⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛+1𝑥_𝑛+1=𝑧∈𝐶_𝑛∶𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑥_𝑛+𝜃_𝑛,=Π_𝐶𝑛+1𝑥₀,∀𝑛=1,2,3,…,(3.34) where 𝐽 is the duality mapping on 𝐸, {𝛼_𝑛,𝑖}^∞𝑖=0 are sequences in [0,1] which satisfies ∑^∞𝑖=0𝛼_𝑛,𝑖=1, 𝜃_𝑛=sup_𝑝∈𝐹(𝑘_𝑛−1)𝜙(𝑝,𝑥_𝑛), and 𝑟_𝑛∈[𝑎,+∞) for some 𝑎>0. If liminf_𝑛→∞𝛼_𝑛,0𝛼_𝑛,𝑖>0 for all 𝑛≥0, then {𝑥_𝑛} converges strongly to Π_𝐹𝑥₀, where Π_𝐹 is the generalized projection from 𝐶 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 𝐶 be a nonempty closed convex subset of 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to 𝑅=(−∞,+∞) satisfying (𝐴1)−(𝐴4) and let 𝐴 be a continuous monotone mapping of 𝐶 into 𝐸^∗. Let 𝑆∶𝐶→𝐶 be an infinite family of closed equally continuous and quasi-𝜙-asymptotically nonexpansive mappings with a sequence {𝑘_𝑛}⊂[1,∞),𝑘_𝑛→1 such that 𝐹∶=𝐹(𝑆)∩GEP(𝑓,𝐴) is a nonempty and bounded subset in 𝐶. Let {𝑥_𝑛} be a sequence generated by 𝑥𝐶₀∈𝐸𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,₁𝑦_𝑛=𝐶,𝑥₁=Π_𝐶1𝑥₀,𝑢_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑆_𝑛𝑥_𝑛,𝐶=_𝑛+1𝑇_𝑟𝑛𝑦_𝑛,𝑥_𝑛+1=𝑧∈𝐶_𝑛∶𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑥_𝑛+𝜃_𝑛,=Π_𝐶𝑛+1𝑥₀,∀𝑛=1,2,3,…,(3.35) where 𝐽 is the duality mapping on 𝐸, {𝛼_𝑛} are sequences in [0,1] which satisfies liminf_𝑛→∞𝛼_𝑛(1−𝛼_𝑛)>0 for all 𝑛≥0, 𝜃_𝑛=sup_𝑝∈𝐹(𝑘_𝑛−1)𝜙(𝑝,𝑥_𝑛) and 𝑟_𝑛∈[𝑎,+∞) for some 𝑎>0. Then {𝑥_𝑛} converges strongly to Π_𝐹𝑥₀, where Π_𝐹 is the generalized projection from 𝐶 onto 𝐹.

Proof. In Theorem 3.1, put 𝑆_𝑖=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: maxℎ(𝑥),𝑥∈𝐶,(4.1) where ℎ(𝑥) is a convex and lower semicontinuous functional defined on a closed convex subset 𝐶 of a Banach space 𝐻. We denoted by 𝑆 the set of solutions of (4.1). Let 𝐹 be a bifunction from 𝐶×𝐶 to 𝑅 defined by 𝑓(𝑥,𝑦)=ℎ(𝑥)−ℎ(𝑦). We consider the following equilibrium problem, that is, to find 𝑥∈𝐶 such that 𝑓(𝑥,𝑦)≥0,∀𝑦∈𝐶.(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(𝑥,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 𝐶 be a nonempty closed convex subset of 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to 𝑅=(−∞,+∞) satisfying (𝐴1)−(𝐴4) and let 𝐴 be a continuous monotone mapping of 𝐶 into 𝐸^∗. Let {𝑆_𝑖}^∞𝑖=1∶𝐶→𝐶 be an infinite family of closed equally continuous and uniformly quasi-𝜙-asymptotically nonexpansive mappings with a sequence {𝑘_𝑛}⊂[1,∞),𝑘_𝑛→1 such that 𝐹∶=∩^∞𝑖=1𝐹(𝑆_𝑖)∩𝑆 is a nonempty and bounded subset in 𝐶. Let {𝑥_𝑛} be a sequence generated by 𝑥𝐶₀∈𝐸𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,₁𝑦_𝑛=𝐶,𝑥₁=Π_𝐶1𝑥₀,=𝐽⁻¹𝛼^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛𝑖𝑥_𝑛𝑢_𝑛,∈𝐶,s.t.ℎ𝑢_𝑛−ℎ(𝑦)+1𝑟_𝑛𝐶⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑦_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛+1𝑥_𝑛+1=𝑧∈𝐶_𝑛∶𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑥_𝑛+𝜃_𝑛,=Π_𝐶𝑛+1𝑥₀∀𝑛=1,2,3,…,(4.3) where 𝐽 is the duality mapping on 𝐸, {𝛼_𝑛,𝑖}^∞𝑖=0 are sequences in [0,1] which satisfies ∑^∞𝑖=0𝛼_𝑛,𝑖=1, 𝜃_𝑛=sup_𝑝∈𝐹(𝑘_𝑛−1)𝜙(𝑝,𝑥_𝑛), and 𝑟_𝑛∈[𝑎,+∞) for some 𝑎>0. If liminf_𝑛→∞𝛼_𝑛,0𝛼_𝑛,𝑖>0 for all 𝑛≥0, then {𝑥_𝑛} converges strongly to Π_𝐹𝑥₀, where Π_𝐹 is the generalized projection from 𝐶 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-𝜙-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).