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

Abstract

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

1. Introduction

In 1953, Mann [1] introduced the following iteration process which is now known as Mann's iteration: 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,(1.1) where 𝑇 is nonexpansive, the initial guess element 𝑥₁∈𝐶 is arbitrary, and {𝛼_𝑛} is a sequence in [0,1]. Mann iteration has been extensively investigated for nonexpansive mappings. In an infinite-dimensional Hilbert space, Mann iteration can conclude only weak conviergence (see [2, 3]).

Later, in 1967, Halpern [4] considered the following algorithm: 𝑥₁∈𝐶,𝑥_𝑛+1=𝛼_𝑛𝑥₁+1−𝛼_𝑛𝑇𝑥_𝑛,∀𝑛≥0,(1.2) where 𝑇 is nonexpansive. He proved the strong convergence theorem of {𝑥_𝑛} to a fixed point of 𝑇 under some control condition {𝛼_𝑛}. Many authors improved and studied the result of Halpern [4] such as Qin et al. [5], Wang et al. [6], and reference therein.

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

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

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

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

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

2. Preliminaries and Definitions

Let 𝐸 be a real Banach space with dual 𝐸^∗, and let 𝐶 be a nonempty closed and convex subset of 𝐸. Let {𝜃_𝑖}_𝑖∈Γ∶𝐶×𝐶→ℝ be a bifunction, where Γ is an arbitrary index set. The system of equilibrium problems is to find 𝑥∈𝐶 such that 𝜃_𝑖(𝑥,𝑦)≥0,𝑖∈Γ,∀𝑦∈𝐶.(2.1) If Γ is a singleton, then problem (2.1) reduces to the equilibrium problem, which is to find 𝑥∈𝐶 such that 𝜃(𝑥,𝑦)≥0,∀𝑦∈𝐶.(2.2) A mapping 𝑇 from 𝐶 into itself is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(2.3)𝑇 is said to be asymptotically nonexpansive if there exists a sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 as 𝑛→∞ such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝑘_𝑛‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(2.4)𝑇 is said to be total asymptotically nonexpansive if there exist nonnegative real sequences 𝜈_𝑛, 𝜇_𝑛 with 𝜈_𝑛→0, 𝜇_𝑛→0 as 𝑛→∞ and a strictly increasing continuous function 𝜑∶ℝ⁺→ℝ⁺ with 𝜑(0)=0 such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤‖𝑥−𝑦‖+𝜈_𝑛𝜓(‖𝑥−𝑦‖)+𝜇_𝑛,∀𝑥,𝑦∈𝐶,∀𝑛≥1.(2.5) A point 𝑥∈𝐶 is a fixed point of 𝑇 provided 𝑇𝑥=𝑥. Denote by 𝐹(𝑇) the fixed point set of 𝑇; that is, 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}. A point 𝑝 in 𝐶 is called an asymptotic fixed point of 𝑇 if 𝐶 contains a sequence {𝑥_𝑛} which converges weakly to 𝑝 such that lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. The asymptotic fixed point set of 𝑇 is denoted by 𝐹(𝑇).

The normalized duality mapping 𝐽∶𝐸→2^𝐸∗ is defined by 𝐽(𝑥)={𝑥^∗∈𝐸^∗∶⟨𝑥,𝑥^∗⟩=‖𝑥‖²,‖𝑥^∗‖=‖𝑥‖}. If 𝐸 is a Hilbert space, then 𝐽=𝐼, where 𝐼 is the identity mapping. Consider the functional defined by 𝜙(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖²,(2.6) where 𝐽 is the normalized duality mapping and ⟨⋅,⋅⟩ denote the duality pairing of 𝐸 and 𝐸^∗.

If 𝐸 is a Hilbert space, then 𝜙(𝑦,𝑥)=‖𝑦−𝑥‖². It is obvious from the definition of 𝜙 that (‖𝑦‖−‖𝑥‖)²≤𝜙(𝑦,𝑥)≤(‖𝑦‖+‖𝑥‖)²,∀𝑥,𝑦∈𝐸.(2.7) A mapping 𝑇 from 𝐶 into itself is said to be 𝜙-nonexpansive [14, 15] if 𝜙(𝑇𝑥,𝑇𝑦)≤𝜙(𝑥,𝑦),∀𝑥,𝑦∈𝐶.(2.8)𝑇 is said to be quasi-𝜙-nonexpansive [14, 15] if 𝐹(𝑇)≠∅ and 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥),∀𝑥∈𝐶,𝑝∈𝐹(𝑇).(2.9)𝑇 is said to be asymptotically 𝜙-nonexpansive [15] if there exists a sequence {𝑘_𝑛}⊂[0,∞) with 𝑘_𝑛→1 as 𝑛→∞ such that 𝜙(𝑇^𝑛𝑥,𝑇^𝑛𝑦)≤𝑘_𝑛𝜙(𝑥,𝑦),∀𝑥,𝑦∈𝐶.(2.10)𝑇 is said to be quasi-𝜙-asymptotically nonexpansive [15] if 𝐹(𝑇)≠∅ and there exists a sequence {𝑘_𝑛}⊂[0,∞) with 𝑘_𝑛→1 as 𝑛→∞ such that 𝜙(𝑝,𝑇^𝑛𝑥)≤𝑘_𝑛𝜙(𝑝,𝑥),∀𝑥∈𝐶,𝑝∈𝐹(𝑇),∀𝑛≥1.(2.11)𝑇 is said to be totally quasi-𝜙-asymptotically nonexpansive, if 𝐹(𝑇)≠∅ and there exist nonnegative real sequences 𝜈_𝑛, 𝜇_𝑛 with 𝜈_𝑛→0, 𝜇_𝑛→0 as 𝑛→∞ and a strictly increasing continuous function 𝜑∶ℝ⁺→ℝ⁺ with 𝜑(0)=0 such that 𝜙(𝑝,𝑇^𝑛𝑥)≤𝜙(𝑝,𝑥)+𝜈_𝑛𝜑(𝜙(𝑝,𝑥))+𝜇_𝑛,∀𝑛≥1,∀𝑥∈𝐶,𝑝∈𝐹(𝑇).(2.12) A mapping 𝑇 from 𝐶 into itself is said to be closed if for any sequence {𝑥_𝑛}⊂𝐶 such that lim_𝑛→∞𝑥_𝑛=𝑥₀ and lim_𝑛→∞𝑇𝑥_𝑛=𝑦₀, then 𝑇𝑥₀=𝑦₀.

Alber [16] introduced the generalized projection Π_𝐶∶𝐸→𝐶 is a map that assigns to an arbitrary point 𝑥∈𝐸 the minimum point of the functional 𝜙(𝑥,𝑦); that is, Π_𝐶𝑥=𝑥, where 𝑥 is the solution of the minimization problem: 𝜙𝑥,𝑥=inf_𝑦∈𝐶𝜙(𝑦,𝑥).(2.13) The existence and uniqueness of the operator Π_𝐶 follows from the properties of the functional 𝜙(𝑦,𝑥) and the strict monotonicity of the mapping 𝐽 (see, e.g., [16–20]). If 𝐸 is a Hilbert space, then 𝜙(𝑥,𝑦)=‖𝑥−𝑦‖² and Π_𝐶 becomes the metric projection 𝑃_𝐶∶𝐻→𝐶. If 𝐶 is a nonempty, closed, and convex subset of a Hilbert space 𝐻, then 𝑃_𝐶 is nonexpansive. This fact actually characterizes Hilbert spaces, and consequently, it is not available in more general Banach spaces. Later, Wu and Huang [21] introduced a new generalized 𝑓-projection operator in the Banach space. They extended the definition of the generalized projection operators and proved some properties of the generalized 𝑓-projection operator. Next, we recall the concept of the generalized 𝑓-projection operator. Let 𝐺∶𝐶×𝐸^∗→ℝ∪{+∞} be a functional defined by 𝐺(𝑦,𝜛)=‖𝑦‖²−2⟨𝑦,𝜛⟩+‖𝜛‖²+2𝜌𝑓(𝑦),(2.14) where 𝑦∈𝐶,  𝜛∈𝐸^∗,𝜌 is positive number, and 𝑓∶𝐶→ℝ∪{+∞} is proper, convex, and lower semicontinuous. From the definition of 𝐺, Wu and Huang [21] proved the following properties:(1)𝐺(𝑦,𝜛) is convex and continuous with respect to 𝜛 when 𝑦 is fixed; (2)𝐺(y,𝜛) is convex and lower semicontinuous with respect to 𝑦 when 𝜛 is fixed.

Definition 2.1. Let 𝐸 be a real Banach space with its dual 𝐸^∗. Let 𝐶 be a nonempty, closed, and convex subset of 𝐸. We say that 𝜋^𝑓𝐶∶𝐸^∗→2^𝐶 is a generalized 𝑓-projection operator if 𝜋^𝑓𝐶𝜛=𝑢∈𝐶∶𝐺(𝑢,𝜛)=inf_𝑦∈𝐶𝐺(𝑦,𝜛),∀𝜛∈𝐸^∗.(2.15)

A Banach space 𝐸 with norm ‖⋅‖ is called strictly convex if ‖(𝑥+𝑦)/2‖<1 for all 𝑥,𝑦∈𝐸 with ‖𝑥‖=‖𝑦‖=1 and 𝑥≠𝑦. Let 𝑈={𝑥∈𝐸∶‖𝑥‖=1} be the unit sphere of 𝐸. A Banach space 𝐸 is called smooth if the limit lim_𝑡→0((‖𝑥+𝑡𝑦‖−‖𝑥‖)/𝑡) exists for each 𝑥,𝑦∈𝑈. It is also called uniformly smooth if the limit exists uniformly for all 𝑥,𝑦∈𝑈. The modulus of smoothness of 𝐸 is the function 𝜌_𝐸∶[0,∞)→[0,∞) defined by 𝜌_𝐸(𝑡)=sup{(‖𝑥+𝑦‖+‖𝑥−𝑦‖)/2−1∶‖𝑥‖=1,‖𝑦‖≤𝑡}. The modulus of convexity of 𝐸 (see [22]) is the function 𝛿_𝐸∶[0,2]→[0,1] defined by 𝛿_𝐸(𝜀)=inf{1−‖(𝑥+𝑦)/2‖∶𝑥,𝑦∈𝐸,‖𝑥‖=‖𝑦‖=1,‖𝑥−𝑦‖≥𝜀}. In this paper we denote the strong convergence and weak convergence of a sequence {𝑥_𝑛} by 𝑥_𝑛→𝑥 and 𝑥_𝑛⇀𝑥, respectively.

Remark 2.2. The basic properties of 𝐸, 𝐸^∗, 𝐽, and 𝐽⁻¹ (see [18]) are as follows.(i)If 𝐸 is an arbitrary Banach space, then 𝐽 is monotone and bounded. (ii)If 𝐸 is a strictly convex, then 𝐽 is strictly monotone.(iii)If 𝐸 is a smooth, then 𝐽 is single valued and semicontinuous. (iv)If 𝐸 is uniformly smooth, then 𝐽 is uniformly norm-to-norm continuous on each bounded subset of 𝐸. (v)If 𝐸 is reflexive smooth and strictly convex, then the normalized duality mapping 𝐽 is single valued, one-to-one, and onto. (vi)If 𝐸 is a reflexive strictly convex and smooth Banach space and 𝐽 is the duality mapping from 𝐸 into 𝐸^∗, then 𝐽⁻¹ is also single valued, bijective, and is also the duality mapping from 𝐸^∗ into 𝐸, and thus 𝐽𝐽⁻¹=𝐼_𝐸^∗ and 𝐽⁻¹𝐽=𝐼_𝐸. (vii)If 𝐸 is uniformly smooth, then 𝐸 is smooth and reflexive.(viii)𝐸 is uniformly smooth if and only if 𝐸^∗ is uniformly convex.(ix)If 𝐸 is a reflexive and strictly convex Banach space, then 𝐽⁻¹ is norm-weak*-continuous.

Remark 2.3. If 𝐸 is a reflexive, strictly convex, and smooth Banach space, then 𝜙(𝑥,𝑦)=0, if and only if 𝑥=𝑦. It is sufficient to show that if 𝜙(𝑥,𝑦)=0 then 𝑥=𝑦. From (2.6), we have ‖𝑥‖=‖𝑦‖. This implies that ⟨𝑥,𝐽𝑦⟩=‖𝑥‖²=‖𝐽𝑦‖². From the definition of 𝐽, one has 𝐽𝑥=𝐽𝑦. Therefore, we have 𝑥=𝑦 (see [18, 20, 23] for more details).

Recall that a Banach space 𝐸 has the Kadec-Klee property [18, 20, 24], if for any sequence {𝑥_𝑛}⊂𝐸 and 𝑥∈𝐸 with 𝑥_𝑛⇀𝑥 and ‖𝑥_𝑛‖→‖𝑥‖, then ‖𝑥_𝑛−𝑥‖→0 as 𝑛→∞. It is well known that if 𝐸 is a uniformly convex Banach space, then 𝐸 has the Kadec-Klee property.

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

Lemma 2.4 (see Change et al. [25]). Let 𝐶 be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space 𝐸 with the Kadec-Klee property. Let 𝑇∶𝐶→𝐶 be a closed and total quasi-𝜙-asymptotically nonexpansive mapping with nonnegative real sequence 𝜈_𝑛 and 𝜇_𝑛 with 𝜈_𝑛→0, 𝜇_𝑛→0 as 𝑛→∞ and a strictly increasing continuous function 𝜁∶ℝ⁺→ℝ⁺ with 𝜁(0)=0. If 𝜇₁=0, then the fixed point set 𝐹(𝑇) is a closed convex subset of 𝐶.

Lemma 2.5 (see Wu and Hung [21]). Let 𝐸 be a real reflexive Banach space with its dual 𝐸^∗ and 𝐶 a nonempty, closed, and convex subset of 𝐸. The following statement hold:(1)𝜋^𝑓𝐶𝜛 is a nonempty, closed and convex subset of 𝐶 for all 𝜛∈𝐸^∗; (2)if 𝐸 is smooth, then for all 𝜛∈𝐸^∗, 𝑥∈𝜋^𝑓𝐶𝜛 if and only if ⟨𝑥−𝑦,𝜛−𝐽𝑥⟩+𝜌𝑓(𝑦)−𝜌𝑓(𝑥)≥0,∀𝑦∈𝐶;(2.16)(3)if 𝐸 is strictly convex and 𝑓∶𝐶→ℝ∪{+∞} is positive homogeneous (i.e., 𝑓(𝑡𝑥)=𝑡𝑓(𝑥) for all 𝑡>0 such that 𝑡𝑥∈𝐶 where 𝑥∈𝐶), then 𝜋^𝑓𝐶𝜛 is single-valued mapping.

Lemma 2.6 (see Fan et al. [26]). Let 𝐸 be a real reflexive Banach space with its dual 𝐸^∗ and 𝐶 be a nonempty, closed and convex subset of 𝐸. If 𝐸 is strictly convex, then 𝜋^𝑓𝐶𝜛 is single valued.

Recall that 𝐽 is single-valued mapping when 𝐸 is a smooth Banach space. There exists a unique element 𝜛∈𝐸^∗ such that 𝜛=𝐽𝑥 where 𝑥∈𝐸. This substitution in (2.14) gives 𝐺(𝑦,𝐽𝑥)=‖𝑦‖²−2⟨𝑦,𝐽𝑥⟩+‖𝑥‖²+2𝜌𝑓(𝑦).(2.17)

Now we consider the second generalized 𝑓 projection operator in Banach space (see [9]).

Definition 2.7. Let 𝐸 be a real smooth Banach space, and let 𝐶 be a nonempty, closed, and convex subset of 𝐸. We say that Π^𝑓𝐶∶𝐸→2^𝐶 is generalized 𝑓-projection operator if Π^𝑓𝐶𝑥=𝑢∈𝐶∶𝐺(𝑢,𝐽𝑥)=inf_𝑦∈𝐶𝐺(𝑦,𝐽𝑥),∀𝑥∈𝐸.(2.18)

Lemma 2.8 (see Deimling [27]). Let 𝐸 be a Banach space, and let 𝑓∶𝐸→ℝ∪{+∞} be a lower semicontinuous convex function. Then there exist 𝑥^∗∈𝐸^∗ and 𝛼∈ℝ such that 𝑓(𝑥)≥⟨𝑥,𝑥^∗⟩+𝛼,∀𝑥∈𝐸.(2.19)

Lemma 2.9 (see Li et al. [9]). Let 𝐸 be a reflexive smooth Banach space, and let 𝐶 be a nonempty, closed, and convex subset of 𝐸. The following statements hold: (1)Π^𝑓𝐶𝑥 is nonempty, closed and convex subset of 𝐶 for all 𝑥∈𝐸; (2)for all 𝑥∈𝐸, 𝑥̂∈Π^𝑓𝐶𝑥 if and only if ⟨𝑥̂−𝑦,𝐽𝑥−𝐽𝑥̂⟩+𝜌𝑓(𝑦)−𝜌𝑓(𝑥̂)≥0,∀𝑦∈𝐶;(2.20)(3)if 𝐸 is strictly convex, then Π^𝑓𝐶 is single-valued mapping.

Lemma 2.10 (see Li et al. [9]). Let 𝐸 be a real reflexive smooth Banach space, let 𝐶 be a nonempty, closed, and convex subset of 𝐸, 𝑥∈𝐸, and let 𝑥̂∈Π^𝑓𝐶𝑥. Then 𝜙(𝑦,𝑥̂)+𝐺(𝑥̂,𝐽𝑥)≤𝐺(𝑦,𝐽𝑥),∀𝑦∈𝐶.(2.21)

Remark 2.11. Let 𝐸 be a uniformly convex and uniformly smooth Banach space and 𝑓(𝑥)=0 for all 𝑥∈𝐸, then Lemma 2.10 reduces to the property of the generalized projection operator considered by Alber [16].

If 𝑓(𝑦)≥0 for all 𝑦∈𝐶 and 𝑓(0)=0, then the definition of totally quasi-𝜙-asymptotically nonexpansive 𝑇 is equivalent to if 𝐹(𝑇)≠∅, and there exist nonnegative real sequences 𝜈_𝑛, 𝜇_𝑛 with 𝜈_𝑛→0, 𝜇_𝑛→0 as 𝑛→∞ and a strictly increasing continuous function 𝜁∶ℝ⁺→ℝ⁺ with 𝜁(0)=0 such that 𝐺(𝑝,𝑇^𝑛𝑥)≤𝐺(𝑝,𝑥)+𝜈_𝑛𝜁𝐺(𝑝,𝑥)+𝜇_𝑛,∀𝑛≥1,∀𝑥∈𝐶,𝑝∈𝐹(𝑇).(2.22)

For solving the equilibrium problem for a bifunction 𝜃∶𝐶×𝐶→ℝ, let us assume that 𝜃 satisfies the following conditions: (A1)𝜃(𝑥,𝑥)=0 for all 𝑥∈𝐶; (A2)𝜃 is monotone; that is, 𝜃(𝑥,𝑦)+𝜃(𝑦,𝑥)≤0 for all 𝑥,𝑦∈𝐶; (A3)for each 𝑥,𝑦,𝑧∈𝐶, lim_𝑡↓0𝜃(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝜃(𝑥,𝑦);(2.23)(A4)for each 𝑥∈𝐶, 𝑦↦𝜃(𝑥,𝑦) is convex and lower semicontinuous.

For example, let 𝐴 be a continuous and monotone operator of 𝐶 into 𝐸^∗ and define 𝜃(𝑥,𝑦)=⟨𝐴𝑥,𝑦−𝑥⟩,∀𝑥,𝑦∈𝐶.(2.24) Then, 𝜃 satisfies (A1)–(A4). The following result is in Blum and Oettli [28].

Lemma 2.12 (see Blum and Oettli [28]). Let 𝐶 be a closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸, let 𝜃 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 𝜃(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶.(2.25)

Lemma 2.13 (see Takahashi and Zembayashi [8]). Let 𝐶 be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space 𝐸, and let 𝜃 be a bifunction from 𝐶×𝐶 to ℝ satisfying conditions (A1)–(A4). For all 𝑟>0 and 𝑥∈𝐸, define a mapping 𝑇^𝜃𝑟∶𝐸→𝐶 as follows: 𝑇^𝜃𝑟𝑥=𝑧∈𝐶∶𝜃(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶.(2.26) Then the following hold: (1)𝑇^𝜃𝑟 is single-valued; (2)𝑇^𝜃𝑟 is a firmly nonexpansive-type mapping [29]; that is, for all 𝑥,𝑦∈𝐸, ⟨𝑇^𝜃𝑟𝑥−𝑇^𝜃𝑟𝑦,𝐽𝑇^𝜃𝑟𝑥−𝐽𝑇^𝜃𝑟𝑦⟩≤⟨𝑇^𝜃𝑟𝑥−𝑇^𝜃𝑟𝑦,𝐽𝑥−𝐽𝑦⟩;(2.27)(3)𝐹(𝑇^𝜃𝑟)=EP(𝜃); (4)EP(𝜃) is closed and convex.

Lemma 2.14 (see Takahashi and Zembayashi [8]). Let 𝐶 be a closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸, let 𝜃 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4), and let 𝑟>0. Then, for 𝑥∈𝐸 and 𝑞∈𝐹(𝑇^𝜃𝑟), 𝜙𝑞,𝑇^𝜃𝑟𝑥+𝜙𝑇^𝜃𝑟𝑥,𝑥≤𝜙(𝑞,𝑥).(2.28)

3. Main Result

Theorem 3.1. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space 𝐸 with the Kadec-Klee property. For each 𝑗=1,2,…,𝑚, let 𝜃_j be a bifunction from 𝐶×𝐶 to ℝ which satisfies conditions (A1)–(A4). Let 𝑆∶𝐶→𝐶 be a closed totally quasi-𝜙-asymptotically nonexpansive mappings with nonnegative real sequences 𝜈_𝑛, 𝜇_𝑛 with 𝜈_𝑛→0, 𝜇_𝑛→0 as 𝑛→∞, and a strictly increasing continuous function 𝜓∶ℝ⁺→ℝ⁺ with 𝜓(0)=0. Let 𝑓∶𝐸→ℝ be a convex and lower semicontinuous function with 𝐶⊂int(𝐷(𝑓)) such that 𝑓(𝑥)≥0 for all 𝑥∈𝐶 and 𝑓(0)=0. Assume that ℱ∶=𝐹(𝑆)∩(∩^𝑚𝑗=1EP(𝜃_𝑗))≠∅. For an initial point 𝑥₁∈𝐸 and 𝐶₁=𝐶, one define the sequence {𝑥_𝑛} by 𝑧_𝑛𝑢_𝑛=𝑇^𝜃𝑚_{𝑟𝑚,𝑛}𝑇^𝜃𝑚−1_{𝑟𝑚−1,𝑛}𝑇^𝜃𝑚−2_{𝑟𝑚−2,𝑛}⋯𝑇^𝜃1_𝑟1,𝑛𝑥_𝑛,𝐶_𝑛+1=𝐽⁻¹𝛼_𝑛𝐽𝑥₁+1−𝛼_𝑛𝐽𝑆^𝑛𝑢_𝑛,=𝑣∈𝐶_𝑛∶𝐺𝑣,𝐽𝑧_𝑛≤𝑥𝐺_𝑛+1𝑣,𝐽𝑢_𝑛≤𝐺𝑣,𝐽𝑥₁+1−𝛼_𝑛𝐺𝑣,𝐽𝑥_𝑛+𝜁_𝑛,=Π^𝑓𝐶_𝑛+1𝑥₁,𝑛∈ℕ,(3.1) where {𝛼_𝑛} is a sequence in [0,1], 𝜁_𝑛=𝜈_𝑛sup_𝑞∈ℱ𝜓(𝐺(𝑞,𝑥_𝑛))+𝜇_𝑛 and {𝑟_𝑗,𝑛}⊂[𝑑,∞) for some 𝑑>0. If lim_𝑛→∞𝛼_𝑛=0, then {𝑥_𝑛} converges strongly to Π^𝑓ℱ𝑥₀.

Proof. We split the proof into four steps.

Step 1. First, we show that 𝐶_𝑛 is closed and convex for all 𝑛∈ℕ.
Clearly 𝐶₁=𝐶 is closed and convex. Suppose that 𝐶_𝑛 is closed and convex for all 𝑛∈ℕ. For any 𝜐∈𝐶_𝑛, we know that 𝐺(𝜐,𝐽𝑧_𝑛)≤𝐺(𝜐,𝐽𝑥_𝑛)+𝜁_𝑛 is equivalent to 2⟨𝜐,𝐽𝑥_𝑛−𝐽𝑧_𝑛⟩≤‖‖𝑥_𝑛‖‖²−‖‖𝑧_𝑛‖‖²+𝜁_𝑛.(3.2) So, 𝐶_𝑛+1 is closed and convex. Hence by induction 𝐶_𝑛 is closed and convex for all 𝑛≥1.

Step 2. We will show that the sequence {𝑥_𝑛} is well defined.
We will show by induction that ℱ⊂𝐶_𝑛 for all 𝑛∈ℕ. It is obvious that ℱ⊂𝐶₁=𝐶. Suppose that ℱ⊂𝐶_𝑛 for some 𝑛∈ℕ. Let 𝑞∈ℱ, put 𝑢_𝑛=𝐾^𝑚𝑛𝑥_𝑛,𝐾^𝑗𝑛=𝑇^𝜃𝑗_{𝑟𝑗,𝑛}𝑇^𝜃𝑗−1_{𝑟𝑗−1,𝑛}…𝑇^𝜃1_𝑟1,𝑛 for all 𝑗=1,2,3,…,𝑚, 𝐾^0𝑛=𝐼, we have that 𝐺𝑞,𝐽𝑢_𝑛=&𝐺𝑞,𝐽𝐾^𝑚𝑛𝑥_𝑛≤&𝐺𝑞,𝐽𝑥_𝑛.(3.3) From (3.3) and 𝑆 which is a totally quasi-𝜙 asymptotically nonexpansive mappings, it follows that 𝐺𝑞,𝐽𝑧_𝑛==‖𝐺𝑞‖𝑞₂,𝛼_𝑛𝐽𝑥₁+1−𝛼_𝑛𝐽𝑆^𝑛𝑢_𝑛+‖‖𝛼_𝑛−2𝛼_𝑛⟨𝑞,𝐽𝑥₁⟩−21−𝛼_𝑛⟨𝑞,𝐽𝑆^𝑛𝑢_𝑛⟩≤‖𝑞‖₂𝐽𝑥₁+(1−𝛼_𝑛)𝐽𝑆^𝑛𝑢_𝑛‖‖²+2𝜌𝑓(𝑞)+𝛼_𝑛−2𝛼_𝑛⟨𝑞,𝐽𝑥₁⟩−21−𝛼_𝑛⟨𝑞,𝐽𝑆^𝑛𝑢_𝑛⟩=𝛼_𝑛‖‖𝐽𝑥₁‖‖²+1−𝛼_𝑛‖‖𝐽𝑆^𝑛𝑢_𝑛‖‖²+2𝜌𝑓(𝑞)≤𝛼_𝑛𝐺𝑞,𝐽𝑥₁+1−𝛼_𝑛𝐺𝑞,𝐽𝑆^𝑛𝑢_𝑛≤𝛼_𝑛𝐺𝑞,𝐽𝑥₁+1−𝛼_𝑛𝐺𝑞,𝐽𝑢_𝑛+𝜈_𝑛𝜓𝐺𝑞,𝐽𝑢_𝑛+𝜇_𝑛𝐺𝑞,𝐽𝑥₁+1−𝛼_𝑛𝐺𝑞,𝐽𝑥_𝑛+𝜈_𝑛sup_𝑞∈ℱ𝜓𝐺𝑞,𝐽𝑥_𝑛+𝜇_𝑛=𝛼_𝑛𝐺𝑞,𝐽𝑥₁+1−𝛼_𝑛𝐺𝑞,𝐽𝑥_𝑛+𝜁_𝑛.(3.4)
This shows that 𝑞∈𝐶_𝑛+1 which implies that ℱ⊂𝐶_𝑛+1, and hence, ℱ⊂𝐶_𝑛 for all 𝑛∈ℕ. and the sequence {𝑥_𝑛} is well defined. From 𝑥_𝑛=Π^𝑓𝐶_𝑛𝑥₁, we see that ⟨𝑥_𝑛−𝑞,𝐽𝑥₁−𝐽𝑥_𝑛⟩+𝜌𝑓(𝑞)−𝜌𝑓𝑥_𝑛≥0,∀𝑞∈𝐶_𝑛.(3.5) Since ℱ⊂𝐶_𝑛 for each 𝑛∈ℕ, we arrive at ⟨𝑥_𝑛−𝑞,𝐽𝑥₁−𝐽𝑥_𝑛⟩+𝜌𝑓(𝑞)−𝜌𝑓𝑥_𝑛≥0,∀𝑞∈ℱ.(3.6) Hence, the sequence {𝑥_𝑛} is well defined.

Step 3. We will show that 𝑥_𝑛→𝑝∈ℱ∶=𝐹(𝑆)∩(∩^𝑚𝑗=1EP(𝜃_𝑗)).
Let 𝑓∶𝐸→ℝ is convex and lower semicontinuous function, follows from Lemma 2.8, there exist 𝑥^∗∈𝐸^∗ and 𝛼∈ℝ such that 𝑓(𝑦)≥⟨𝑦,𝑥^∗⟩+𝛼,∀𝑦∈𝐸.(3.7) Since 𝑥_𝑛∈𝐶_𝑛⊂𝐸, it follows that 𝐺𝑥_𝑛,𝐽𝑥₁≥=‖‖‖‖𝑥𝑥_𝑛𝑛‖‖²−2⟨𝑥_𝑛,𝐽𝑥₁⟩+‖‖𝑥₁‖‖²+2𝜌𝑓𝑥_𝑛=‖‖𝑥‖‖²−2⟨𝑥_𝑛,𝐽𝑥₁⟩+‖‖𝑥₁‖‖²+2𝜌⟨𝑥_𝑛,𝑥^∗⟩+2𝜌𝛼_𝑛≥‖‖𝑥‖‖²−2⟨𝑥_𝑛,𝐽𝑥₁−𝜌𝑥^∗⟩+‖‖𝑥₁‖‖²+2𝜌𝛼_𝑛=‖‖𝑥‖‖²−2‖‖𝑥_𝑛‖‖‖‖𝐽𝑥₁−𝜌𝑥^∗‖‖+‖‖𝑥₁‖‖²+2𝜌𝛼_𝑛‖‖−‖‖𝐽𝑥₁−𝜌𝑥^∗‖‖²+‖‖𝑥₁‖‖²−‖‖𝐽𝑥₁−𝜌𝑥^∗‖‖²+2𝜌𝛼.(3.8) For 𝑞∈ℱ and 𝑥_𝑛=Π^𝑓𝐶_𝑛𝑥₁, we have 𝐺𝑞,𝐽𝑥₁≥𝐺𝑥_𝑛,𝐽𝑥₁≥‖‖𝑥_𝑛‖‖−‖‖𝐽𝑥₁−𝜌𝑥^∗‖‖²+‖‖𝑥₁‖‖²−‖‖𝐽𝑥₁−𝜌𝑥^∗‖‖²+2𝜌𝛼.(3.9) This shows that {𝑥_𝑛} is bounded and so is {𝐺(𝑥_𝑛,𝐽𝑥₁)}. From the fact that 𝑥_𝑛+1=Π^𝑓𝐶_𝑛+1𝑥₁∈𝐶_𝑛+1⊂𝐶_𝑛 and 𝑥_𝑛=Π^𝑓𝐶_𝑛𝑥₁, it follows from Lemma 2.10 that 0≤(‖‖𝑥_𝑛+1−‖‖𝑥_𝑛‖)²≤𝜙𝑥_𝑛+1,𝑥_𝑛≤𝐺𝑥_𝑛+1,𝐽𝑥₁−𝐺𝑥_𝑛,𝐽𝑥₁.(3.10) That is, {𝐺(𝑥_𝑛,𝐽𝑥₁)} is nondecreasing. Hence, we obtain that lim_𝑛→∞𝐺(𝑥_𝑛,𝐽𝑥₁) exists. Taking 𝑛→∞, we obtain lim_𝑛→∞𝜙𝑥_𝑛+1,𝑥_𝑛=0.(3.11) Since 𝐸 is reflexive, {𝑥_𝑛} is bounded, and 𝐶_𝑛 is closed and convex for all 𝑛∈ℕ. Without loss of generality, we can assume that 𝑥_𝑛⇀𝑝∈𝐶_𝑛. From the fact that 𝑥_𝑛=Π^𝑓𝐶_𝑛𝑥₁, we get that 𝐺𝑥_𝑛,𝐽𝑥₁≤𝐺𝑝,𝐽𝑥₁,∀𝑛∈ℕ.(3.12) Since 𝑓 is convex and lower semicontinuous, we have liminf_𝑛→∞𝐺𝑥_𝑛,𝐽𝑥₁≥=l‖i𝑝m‖i₂nf_𝑛→∞‖‖𝑥^𝑛‖‖²−2⟨𝑥_𝑛,𝐽𝑥₁⟩+‖‖𝑥₁‖‖²+2𝜌𝑓𝑥_𝑛=𝐺𝑥_𝑛−2⟨𝑝,𝐽𝑥₁⟩+‖‖𝑥₁‖‖²+2𝜌𝑓(𝑝),𝐽𝑥₁.(3.13) By (3.12) and (3.13), we get 𝐺𝑝,𝐽𝑥₁≤liminf_𝑛→∞𝐺𝑥_𝑛,𝐽𝑥₁≤limsup_𝑛→∞𝐺𝑥_𝑛,𝐽𝑥₁≤𝐺𝑝,𝐽𝑥₁.(3.14) That is, lim_𝑛→∞𝐺(𝑥_𝑛,𝐽𝑥₁)=𝐺(𝑝,𝐽𝑥₁); this implies that ‖𝑥_𝑛‖→‖𝑝‖; by virtue of the Kadec-Klee property of 𝐸, we obtain that lim_𝑛→∞𝑥_𝑛=𝑝.(3.15) We also have lim_𝑛→∞𝑥_𝑛+1=𝑝.(3.16) From (3.15), we get that lim_𝑛→∞𝜁_𝑛=lim_𝑛→∞𝜈^𝑛sup_𝑞∈ℱ𝜓𝐺𝑞,𝑥_𝑛+𝜇_𝑛=0.(3.17)
(a) We show that 𝑝∈∩^𝑚𝑗=1EP(𝜃_𝑗).
Since 𝑥_𝑛+1=Π^𝑓𝐶_𝑛+1𝑥₁∈𝐶_𝑛+1⊂𝐶_𝑛 and the definition of 𝐶_𝑛+1, we have 𝐺𝑥_𝑛+1,𝐽𝑢_𝑛≤𝛼_𝑛𝐺𝑥_𝑛+1,𝐽𝑥₁+1−𝛼_𝑛𝐺𝑥_𝑛+1,𝐽𝑥_𝑛+𝜁_𝑛(3.18) is equivalent to 𝜙𝑥_𝑛+1,𝑢_𝑛≤𝛼_𝑛𝜙𝑥_𝑛+1,𝑥₁+1−𝛼_𝑛𝜙𝑥_𝑛+1,𝑥_𝑛+𝜁_𝑛.(3.19) From (3.11), (3.15), and (3.17), it follows that lim_𝑛→∞𝜙𝑥_𝑛+1,𝑢_𝑛=0.(3.20) From (2.7), we have ‖‖𝑥_𝑛+1‖‖−‖‖𝑢_𝑛‖‖²⟶0.(3.21) Since ‖𝑥_𝑛+1‖→‖𝑝‖, we have ‖‖𝑢_𝑛‖‖⟶‖𝑝‖as𝑛⟶∞.(3.22) It follow that ‖‖𝐽𝑢_𝑛‖‖⟶‖𝐽𝑝‖as𝑛⟶∞.(3.23) That is, {‖𝐽𝑢_𝑛‖} is bounded in 𝐸^∗ and 𝐸^∗ is reflexive; we assume that 𝐽𝑢_𝑛⇀𝑢^∗∈𝐸^∗. In view of 𝐽(𝐸)=𝐸^∗, there exists 𝑢∈𝐸 such that 𝐽𝑢=𝑢^∗. It follows that 𝜙𝑥_𝑛+1,𝑢_𝑛=‖‖𝑥_𝑛+1‖‖²−2⟨𝑥_𝑛+1,𝐽𝑦_𝑛⟩+‖‖𝑢_𝑛‖‖²=‖‖𝑥_𝑛+1‖‖²−2⟨𝑥_𝑛+1,𝐽𝑢_𝑛⟩+‖‖𝐽𝑢_𝑛‖‖².(3.24) Taking liminf_𝑛→∞ on both sides of the equality above and ‖⋅‖ is the weak lower semicontinuous, it yields that 0≥‖𝑝‖²−2⟨𝑝,𝑢^∗⟩+‖𝑢^∗‖²=‖𝑝‖²−2⟨𝑝,𝐽𝑢⟩+‖𝐽𝑢‖²=‖𝑝‖²−2⟨𝑝,𝐽𝑢⟩+‖𝑢‖²=𝜙(𝑝,𝑢).(3.25) That is, 𝑝=𝑢, which implies that 𝑢^∗=𝐽𝑝. It follows that 𝐽𝑢_𝑛⇀𝐽𝑝∈𝐸^∗. From (3.23) and the Kadec-Klee property of 𝐸^∗ we have 𝐽𝑢_𝑛→𝐽𝑝 as 𝑛→∞. Note that 𝐽⁻¹∶𝐸^∗→𝐸 is norm-weak∗-continuous; that is, 𝑢_𝑛⇀𝑝. From (3.22) and the Kadec-Klee property of 𝐸, we have lim_𝑛→∞𝑢_𝑛=𝑝.(3.26) For 𝑞∈𝐹⊂𝐶_𝑛, by nonexpansiveness, we observe that 𝜙𝑞,𝑢_𝑛≤=𝜙𝜙𝑞𝑞,,𝐾𝐾^𝑚𝑛𝑥_{𝑛𝑛𝑚−1}≤𝜙𝑞,𝐾_𝑛𝑚−2𝑥_𝑛≤⋮𝜙𝑞,𝐾_𝑗𝑛𝑥_𝑛𝑥_𝑛.(3.27) By Lemma 2.14, we have for 𝑗=1,2,3,…,𝑚𝜙𝐾^𝑗𝑛𝑥_𝑛,𝑥_𝑛&≤𝜙𝑞,𝑥_𝑛−𝜙𝑞,𝐾^𝑗𝑛𝑥_𝑛≤𝜙𝑞,𝑥_𝑛−𝜙𝑞,𝑢_𝑛.(3.28) Since 𝑥_𝑛,𝑢_𝑛→𝑝 as 𝑛→∞, we get 𝜙(𝐾^𝑗𝑛𝑥_𝑛,𝑥_𝑛)→0 as 𝑛→∞, for 𝑗=1,2,3,…,𝑚. From (2.7), it follow that ‖‖𝐾^𝑗𝑛𝑥_𝑛‖‖−‖‖𝑥_𝑛‖‖²⟶0.(3.29) Since ‖𝑥_𝑛‖→‖𝑝‖, we also have ‖‖𝐾^𝑗𝑛𝑥_𝑛‖‖⟶‖𝑝‖as𝑛⟶∞.(3.30) Since {𝐾^𝑗𝑛𝑥_𝑛} is bounded and 𝐸 is reflexive, without loss of generality we assume that 𝐾^𝑗𝑛𝑦_𝑛⇀ℎ. We know that 𝐶_𝑛 is closed and convex for each 𝑛≥1 it is obvious that ℎ∈𝐶_𝑛. Again since 𝜙𝐾^𝑗𝑛𝑥_𝑛,𝑥_𝑛=‖‖𝐾^𝑗𝑛𝑥_𝑛‖‖²−2𝐾^𝑗𝑛𝑥_𝑛,𝐽𝑥_𝑛+‖‖𝑥_𝑛‖‖²,(3.31) taking liminf_𝑛→∞ on the both sides of equality above, we have 0&≥‖ℎ‖²−2⟨ℎ,𝐽𝑝⟩+‖𝑝‖²=𝜙(ℎ,𝑝).(3.32) That is, ℎ=𝑝,for all 𝑗=1,2,3,…,𝑚; it follow that 𝐾^𝑗𝑛𝑥_𝑛⇀𝑝;(3.33) from (3.30), (3.33), and the Kadec-Klee property, it follows that lim_𝑛→∞𝐾^𝑗𝑛𝑥_𝑛=𝑝,∀𝑗=1,2,3,…,𝑚.(3.34) By using triangle inequality, we have ‖‖𝑥_𝑛−𝐾^𝑗𝑛𝑥_𝑛‖‖≤‖‖𝑥_𝑛−𝑝‖‖+‖‖𝑝−𝐾^𝑗𝑛𝑢_𝑛‖‖.(3.35) Since 𝑥_𝑛,𝐾^𝑗𝑛𝑥_𝑛→𝑝 as 𝑛→∞, we have lim_𝑛→∞‖‖𝑥_𝑛−𝐾^𝑗𝑛𝑥_𝑛‖‖=0,∀𝑗=1,2,3,…,𝑚.(3.36) Again by using triangle inequality, we have ‖‖𝐾^𝑗𝑛𝑥_𝑛−𝐾^𝑛𝑗−1𝑥_𝑛‖‖≤‖‖𝐾^𝑗𝑛𝑥_𝑛−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝐾^𝑛𝑗−1𝑥_𝑛‖‖.(3.37) From (3.36), we also have lim_𝑛→∞‖‖𝐾^𝑗𝑛𝑥_𝑛−𝐾^𝑛𝑗−1𝑥_𝑛‖‖=0,∀𝑗=1,2,3,…,𝑚.(3.38) Since 𝐽 is uniformly norm-to-norm continuous, we obtain lim_𝑛→∞‖‖𝐽𝐾^𝑗𝑛𝑥_𝑛−𝐽𝐾^𝑛𝑗−1𝑥_𝑛‖‖=0,∀𝑗=1,2,3,…,𝑚.(3.39) From 𝑟_𝑗,𝑛>0, we have ‖𝐽𝐾^𝑗𝑛𝑥_𝑛−𝐽𝐾^𝑛𝑗−1𝑥_𝑛‖/𝑟_𝑗,𝑛→0 as 𝑛→∞for all 𝑗=1,2,3,…,𝑚, and 𝜃_𝑗𝐾^𝑗𝑛𝑦_𝑛,𝑦+1𝑟_𝑗,𝑛⟨𝑦−𝐾^𝑗𝑛𝑥_𝑛,𝐽𝐾^𝑗𝑛𝑥_𝑛−𝐽𝐾^𝑛𝑗−1𝑥_𝑛⟩≥0,∀𝑦∈𝐶.(3.40) By (A2), that ‖‖𝑦−𝐾^𝑗𝑛𝑦_𝑛‖‖‖‖𝐽𝐾^𝑗𝑛𝑦_𝑛−𝐽𝐾^𝑛𝑗−1𝑥_𝑛‖‖𝑟_𝑛≥1𝑟_𝑗,𝑛≥−𝜃⟨_𝑗𝑦−𝐾^𝑗𝑛𝑥_𝑛,𝐽𝐾^𝑗𝑛𝑦_𝑛−𝐽𝐾^𝑛𝑗−1𝑥_𝑛⟩≥𝜃_𝑗𝐾^𝑗𝑛𝑥_𝑛,𝑦𝑦,𝐾^𝑗𝑛𝑥_𝑛,∀𝑦∈𝐶,(3.41) and 𝐾^𝑗𝑛𝑥_𝑛→𝑝 as 𝑛→∞, we get 𝜃_𝑗(𝑦,𝑝)≤0, for all 𝑦∈𝐶. For 0<𝑡<1, define 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑝, then 𝑦_𝑡∈𝐶 which imply that 𝜃_𝑗(𝑦_𝑡,𝑝)≤0. From (A1), we obtain that 0=𝜃_𝑗𝑦_𝑡,𝑦_𝑡≤𝑡𝜃_𝑗𝑦_𝑡,𝑦+(1−𝑡)𝜃_𝑗𝑦_𝑡,𝑝≤𝑡𝜃_𝑗𝑦_𝑡,𝑦.(3.42) We have that 𝜃_𝑗(𝑦_𝑡,𝑦)≥0. From (A3), we have 𝜃_𝑗(𝑝,𝑦)≥0, for all 𝑦∈𝐶 and 𝑗=1,2,3,…,𝑚. That is, 𝑝∈EP(𝜃_𝑗), for all𝑗=1,2,3,…,𝑚. This imply that 𝑝∈∩^𝑚𝑗=1EP(𝜃_𝑗).
(b) We show that 𝑝∈𝐹(𝑆).
Since 𝑥_𝑛+1=Π^𝑓𝐶_𝑛+1𝑥₁∈𝐶_𝑛+1⊂𝐶_𝑛 and the definition of 𝐶_𝑛+1, we have 𝐺𝑥_𝑛+1,𝐽𝑧_𝑛≤𝛼_𝑛𝐺𝑥_𝑛+1,𝐽𝑥₁+1−𝛼_𝑛𝐺𝑥_𝑛+1,𝐽𝑥_𝑛+𝜁_𝑛(3.43) is equivalent to 𝜙𝑥_𝑛+1,𝑧_𝑛≤𝛼_𝑛𝜙𝑥_𝑛+1,𝑥₁+1−𝛼_𝑛𝜙𝑥_𝑛+1,𝑥_𝑛+𝜁_𝑛.(3.44) Following (3.11), (3.15), and (3.17), we get that lim_𝑛→∞𝜙𝑥_𝑛+1,𝑧_𝑛=0.(3.45) From (2.7), we also have ‖‖𝑧_𝑛‖‖⟶‖𝑝‖as𝑛⟶∞.(3.46) It follows that ‖‖𝐽𝑧_𝑛‖‖⟶‖𝐽𝑝‖as𝑛⟶∞.(3.47) This implies that {‖𝐽𝑧_𝑛‖} is bounded in 𝐸^∗. Since 𝐸 is reflexive and 𝐸^∗ is also reflexive, we can assume that 𝐽𝑧_𝑛⇀𝑧^∗∈𝐸^∗. In view of the reflexive of 𝐸, we see that 𝐽(𝐸)=𝐸^∗. There exists 𝑧∈𝐸 such that 𝐽𝑧=𝑧^∗. It follows that 𝜙𝑥_𝑛+1,𝑧_𝑛=‖‖𝑥_𝑛+1‖‖²−2⟨𝑥_𝑛+1,𝐽𝑧_𝑛⟩+‖‖𝑧_𝑛‖‖²=‖‖𝑥_𝑛+1‖‖²−2⟨𝑥_𝑛+1,𝐽𝑧_𝑛⟩+‖‖𝐽𝑧_𝑛‖‖².(3.48) Taking liminf_𝑛→∞ on both sides of the equality above and in view of the weak lower semicontinuity of norm ‖⋅‖, it yields that 0≥‖𝑝‖²−2⟨𝑝,𝑧^∗⟩+‖𝑧^∗‖²=‖𝑝‖²−2⟨𝑝,𝐽𝑧⟩+‖𝐽𝑧‖²=‖𝑝‖²−2⟨𝑝,𝐽𝑧⟩+‖𝑧‖²=𝜙(𝑝,𝑧);(3.49) That is 𝑝=𝑧, which implies that 𝑧^∗=𝐽𝑝. It follows that 𝐽𝑧_𝑛⇀𝐽𝑝∈𝐸^∗.From (3.47) and the Kadec-Klee property of 𝐸^∗ we have 𝐽𝑧_𝑛→𝐽𝑝 as 𝑛→∞. Since 𝐽⁻¹∶𝐸^∗→𝐸 is norm-weak∗-continuous,𝑧_𝑛⇀𝑝 as 𝑛→∞. From (3.46) and the Kadec-Klee property of 𝐸, we have lim_𝑛→∞𝑧_𝑛=𝑝.(3.50) Since {𝑥_𝑛} is bounded, then a mapping 𝑆 is also bounded. From the condition lim_𝑛→∞𝛼_𝑛=0, we have that ‖‖𝐽𝑧_𝑛−𝐽𝑆^𝑛𝑢_𝑛‖‖=lim_𝑛→∞𝛼_𝑛‖‖𝐽𝑥₁−𝐽𝑆^𝑛𝑢_𝑛‖‖=0.(3.51) From (3.47), we get ‖‖𝐽𝑆^𝑛𝑢_𝑛‖‖⟶‖𝐽𝑝‖as𝑛⟶∞.(3.52) Since 𝐽⁻¹∶𝐸^∗→𝐸 is norm-weak*-continuous, 𝑆^𝑛𝑢_𝑛⇀𝑝as𝑛⟶∞.(3.53) On the other hand, we observe that ||‖‖𝑆^𝑛𝑢_𝑛‖‖−‖𝑝‖||=‖‖𝐽𝑆^𝑛𝑢_𝑛‖‖−‖𝐽𝑝‖≤‖‖𝐽𝑆^𝑛𝑢_𝑛−𝐽𝑝‖‖.(3.54) In view of (3.52), we obtain ‖𝑆^𝑛𝑢_𝑛‖→‖𝑝‖. Since 𝐸 has the Kadee-Klee property, we get 𝑆^𝑛𝑢_𝑛⟶𝑝foreach𝑛∈ℕ.(3.55) From 𝑆^𝑛𝑢_𝑛→𝑝, we get 𝑆^𝑛+1𝑢_𝑛→𝑝; that is, 𝑆𝑆^𝑛𝑢_𝑛→𝑝. In view of closeness of 𝑆, we have 𝑆𝑝=𝑝. This implies that 𝑝∈𝐹(𝑆). From (a) and (b), it follows that 𝑝∈∩^𝑚𝑗=1EP(𝜃_𝑗)∩𝐹(𝑆).

Step 4. We will show that 𝑝=Π^𝑓ℱ𝑥₁.
Since ℱ is closed and convex set from Lemma 2.9, we have Π^𝑓ℱ𝑥₁ which is single valued, denoted by 𝜐. By definition 𝑥_𝑛=Π^𝑓𝐶_𝑛𝑥₁ and 𝑣∈ℱ⊂𝐶_𝑛, we also have 𝐺𝑥_𝑛,𝐽𝑥₁≤G𝜐,𝐽𝑥₁,∀𝑛≥1.(3.56) By the definition of 𝐺 and 𝑓, we know that, for each given 𝑥,𝐺(𝜉,𝐽𝑥) is convex and lower semicontinuous with respect to 𝜉. So 𝐺𝑝,𝐽𝑥₁≤liminf_𝑛→∞𝐺𝑥_𝑛,𝐽𝑥₁≤limsup_𝑛→∞𝐺𝑥_𝑛,𝐽𝑥₁≤𝐺𝜐,𝐽𝑥₁.(3.57) From the definition of Π^𝑓ℱ𝑥₁ and since 𝑝∈ℱ, we conclude that 𝜐=𝑝=Π^𝑓ℱ𝑥₁ and 𝑥_𝑛→𝑝 as 𝑛→∞. The proof is completed.

Setting 𝜈_𝑛≡0 and 𝜇_𝑛≡0 in Theorem 3.1, then we have the following corollary.

Corollary 3.2. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space 𝐸 with the Kadec-Klee property. For each 𝑗=1,2,…,𝑚, let 𝜃_𝑗 be a bifunction from 𝐶×𝐶 to ℝ which satisfies conditions (A1)–(A4). Let 𝑆∶𝐶→𝐶 be a closed and quasi-𝜙-asymptotically nonexpansive mappings, and let 𝑓∶𝐸→ℝ be a convex and lower semicontinuous function with 𝐶⊂int(𝐷(𝑓)) such that 𝑓(𝑥)≥0 for all 𝑥∈𝐶 and 𝑓(0)=0. Assume that ℱ=𝐹(𝑆)∩(∩^𝑚𝑗=1EP(𝜃_𝑗))≠∅. For an initial point 𝑥₁∈𝐸 and 𝐶₁=𝐶, we define the sequence {𝑥_𝑛} by 𝑧_𝑛𝑢_𝑛=𝑇^𝜃𝑚_{𝑟𝑚,𝑛}𝑇^𝜃𝑚−1_{𝑟𝑚−1,𝑛}𝑇^𝜃𝑚−2_{𝑟𝑚−2,𝑛}⋯𝑇^𝜃1_𝑟1,𝑛𝑥_𝑛,𝐶_𝑛+1=𝐽⁻¹𝛼_𝑛𝐽𝑥₁+1−𝛼_𝑛𝐽𝑆^𝑛𝑢_𝑛,=𝑣∈𝐶_𝑛∶𝐺𝑣,𝐽𝑧_𝑛≤𝑥𝐺_𝑛+1𝑣,𝐽𝑢_𝑛≤𝐺𝑣,𝐽𝑥₁+1−𝛼_𝑛𝐺𝑣,𝐽𝑥_𝑛+𝜁_𝑛,=Π^𝑓𝐶_𝑛+1𝑥₁,𝑛∈ℕ,(3.58) where {𝛼_𝑛} is a sequence in [0,1], 𝜁_𝑛=𝜈_𝑛sup_𝑞∈ℱ𝜓(𝐺(𝑞,𝑥_𝑛))+𝜇_𝑛, and {𝑟_𝑗,𝑛}⊂[𝑑,∞) for some 𝑑>0. If lim_𝑛→∞𝛼_𝑛=0, then {𝑥_𝑛} converges strongly to Π^𝑓ℱ𝑥₁.

Let 𝐸 be a real Banach space, and let 𝐶 be a nonempty closed convex subset of 𝐸. Given a mapping 𝐴∶𝐶→𝐸^∗, let 𝜃(𝑥,𝑦)=⟨𝐴𝑥,𝑦−𝑥⟩ for all 𝑥,𝑦∈𝐶. Then 𝑥^∗∈EP(𝜃) if and only if ⟨𝐴𝑥^∗,𝑦−𝑥^∗⟩≥0 for all 𝑦∈𝐶; that is, 𝑥^∗ is a solution of the classical variational inequality problem. The set of this solution is denoted by VI(𝐴,𝐶). For each 𝑟>0 and 𝑥∈𝐸, we define the mapping 𝑇^𝜃𝑟𝑥 by 𝑇^𝜃𝑟𝑥=𝑧∈𝐶∶⟨𝐴𝑧,𝑦−𝑧⟩+1𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶.(3.59) Hence, we obtain the following corollary.

Corollary 3.3. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space 𝐸 with the Kadec-Klee property. For each 𝑗=1,2,…,𝑚, let{𝐴_𝑗} be a continuous monotone mapping of 𝐶 into 𝐸^∗. Let 𝑆∶𝐶→𝐶 be a closed totally quasi-𝜙-asymptotically nonexpansive mappings with nonnegative real sequences 𝜈_𝑛, 𝜇_𝑛 with 𝜈_𝑛→0, 𝜇_𝑛→0 as 𝑛→∞ and a strictly increasing continuous function 𝜓∶ℝ⁺→ℝ⁺ with 𝜓(0)=0, and let 𝑓∶𝐸→ℝ be a convex and lower semicontinuous function with 𝐶⊂int(𝐷(𝑓)) such that 𝑓(𝑥)≥0 for all 𝑥∈𝐶 and 𝑓(0)=0. Assume that ℱ=𝐹(𝑆)∩(∩^𝑚𝑗=1VI(𝐴_𝑗,𝐶))≠∅. For an initial point 𝑥₁∈𝐸 and 𝐶₁=𝐶, one defines the sequence {𝑥_𝑛} by 𝑧_𝑛𝑢_𝑛=𝑇^𝜃𝑚_{𝑟𝑚,𝑛}𝑇^𝜃𝑚−1_{𝑟𝑚−1,𝑛}𝑇^𝜃𝑚−2_{𝑟𝑚−2,𝑛}⋯𝑇^𝜃1_𝑟1,𝑛𝑥_𝑛,𝐶_𝑛+1=𝐽⁻¹𝛼_𝑛𝐽𝑥₁+1−𝛼_𝑛𝐽𝑆^𝑛𝑢_𝑛,=𝑣∈𝐶_𝑛∶𝐺𝑣,𝐽𝑧_𝑛≤𝑥𝐺_𝑛+1𝑣,𝐽𝑢_𝑛≤𝐺𝑣,𝐽𝑥₁+1−𝛼_𝑛𝐺𝑣,𝐽𝑥_𝑛+𝜁_𝑛,=Π^𝑓𝐶_𝑛+1𝑥₁,𝑛∈ℕ,(3.60) where 𝜁_𝑛=𝜈_𝑛sup_𝑞∈ℱ𝜓(𝐺(𝑞,𝑥_𝑛))+𝜇_𝑛, {𝛼_𝑛} is a sequence in [0,1], and {𝑟_𝑗,𝑛}⊂[𝑑,∞) for some 𝑑>0. If lim_𝑛→∞𝛼_𝑛=0, then {𝑥_𝑛} converges strongly to Π^𝑓ℱ𝑥₁.

If 𝑓(𝑥)=0 for all 𝑥∈𝐸, we have 𝐺(𝜉,𝐽𝑥)=𝜙(𝜉,𝑥) and Π^𝑓𝐶𝑥=Π_𝐶𝑥. From Theorem 3.1, we obtain the following corollary.

Corollary 3.4. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space 𝐸 with the Kadec-Klee property. For each 𝑗=1,2,…,𝑚, let 𝜃_𝑗 be a bifunction from 𝐶×𝐶 to ℝ which satisfies conditions (A1)–(A4). Let 𝑆∶𝐶→𝐶 be a closed totally quasi-𝜙-asymptotically nonexpansive mappings with nonnegative real sequences 𝜈_𝑛, 𝜇_𝑛 with 𝜈_𝑛→0, 𝜇_𝑛→0 as 𝑛→∞ and a strictly increasing continuous function 𝜓∶ℝ⁺→ℝ⁺ with 𝜓(0)=0. Assume that ℱ=𝐹(𝑆)∩(∩^𝑚𝑗=1EP(𝜃_𝑗))≠∅. For an initial point 𝑥₁∈𝐸 and 𝐶₁=𝐶, we define the sequence {𝑥_𝑛} by 𝑧_𝑛𝑢_𝑛=𝑇^𝜃𝑚_{𝑟𝑚,𝑛}𝑇^𝜃𝑚−1_{𝑟𝑚−1,𝑛}𝑇^𝜃𝑚−2_{𝑟𝑚−2,𝑛}⋯𝑇^𝜃1_𝑟1,𝑛𝑥_𝑛,𝐶_𝑛+1=𝐽⁻¹𝛼_𝑛𝐽𝑥₁+1−𝛼_𝑛𝐽𝑆^𝑛𝑢_𝑛,=𝑣∈𝐶_𝑛∶𝐺𝑣,𝐽𝑧_𝑛≤𝑥𝐺_𝑛+1𝑣,𝐽𝑢_𝑛≤𝐺𝑣,𝐽𝑥₁+1−𝛼_𝑛𝐺𝑣,𝐽𝑥_𝑛+𝜁_𝑛,=Π_𝐶𝑛+1𝑥₁,𝑛∈ℕ,(3.61) where {𝛼_𝑛} is a sequence in [0,1], 𝜁_𝑛=𝜈_𝑛sup_𝑞∈ℱ𝜓(𝐺(𝑞,𝑥_𝑛))+𝜇_𝑛, and {𝑟_𝑗,𝑛}⊂[𝑑,∞) for some 𝑑>0. If lim_𝑛→∞𝛼_𝑛=0, then {𝑥_𝑛} converges strongly to Π_ℱ𝑥₁.

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

Acknowledgments

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