Convergence Theorem Based on a New Hybrid Projection Method for Finding a Common Solution of Generalized Equilibrium and Variational Inequality Problems in Banach Spaces

Abstract

The purpose of this paper is to introduce a new hybrid projection method for finding a common element of the set of common fixed points of two relatively quasi-nonexpansive mappings, the set of the variational inequality for an 𝛼-inverse-strongly monotone, and the set of solutions of the generalized equilibrium problem in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2-uniformly convex and uniformly smooth Banach space. Base on this result, we also get some new and interesting results. The results in this paper generalize, extend, and unify some well-known strong convergence results in the literature.

1. Introduction

Let 𝐸 be a real Banach space, 𝐸^∗ the dual space of 𝐸. A Banach space 𝐸 is said to be 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. Then a Banach space 𝐸 is said to be smooth if the limit

lim_𝑡→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡 (1.1) exists for each 𝑥,𝑦∈𝑈. It is also said to be uniformly smooth if the limit is attained uniformly for 𝑥,𝑦∈𝑈. Let 𝐸 be a Banach space. The modulus of convexity of 𝐸 is the function 𝛿∶[0,2]→[0,1] defined by

𝛿(𝜀)=inf{1−‖𝑥+𝑦2‖∶𝑥,𝑦∈𝐸,‖𝑥‖=‖𝑦‖=1,‖𝑥−𝑦‖≥𝜀}. (1.2)

A Banach space 𝐸 is uniformly convex if and only if 𝛿(𝜀)>0 for all 𝜀∈(0,2]. Let 𝑝 be a fixed real number with 𝑝≥2. A Banach space 𝐸 is said to be 𝑝-uniformly convex if there exists a constant 𝑐>0 such that 𝛿(𝜀)≥𝑐𝜀^𝑝 for all 𝜀∈[0,2]; see [1, 2] for more details. Observe that every 𝑝-uniform convex is uniformly convex. One should note that no Banach space is 𝑝-uniform convex for 1<𝑝<2. It is well known that a Hilbert space is 2-uniformly convex, uniformly smooth. For each 𝑝>1, the generalized duality mapping 𝐽_𝑝∶𝐸→2^𝐸∗ is defined by

𝐽_𝑝(𝑥)={𝑥^∗∈𝐸^∗∶⟨𝑥,𝑥^∗⟩=‖𝑥‖^𝑝,‖𝑥^∗‖=‖𝑥‖^𝑝−1} (1.3) for all 𝑥∈𝐸. In particular, 𝐽=𝐽₂ is called the normalized duality mapping. If 𝐸 is a Hilbert space, then 𝐽=𝐼, where 𝐼 is the identity mapping. It is also known that if 𝐸 is uniformly smooth, then 𝐽 is uniformly norm-to-norm continuous on each bounded subset of 𝐸.

Let 𝐸 be a real Banach space with norm ‖⋅‖ and 𝐸^∗ denotes the dual space of 𝐸. Consider the functional defined by

𝜙(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖²∀𝑥,𝑦∈𝐸. (1.4)

Observe that, in a Hilbert space 𝐻, (1.4) reduces to 𝜙(𝑥,𝑦)=‖𝑥−𝑦‖², 𝑥,𝑦∈𝐻. 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 to the minimization problem 𝜙(𝑥,𝑥)=inf_𝑦∈𝐶𝜙(𝑦,𝑥); (1.5) existence and uniqueness of the mapping Π_𝐶 follow from the properties of the functional 𝜙(𝑥,𝑦) and strict monotonicity of the mapping 𝐽 (see, e.g., [3–7]). In Hilbert spaces, Π_𝐶=𝑃_𝐶. It is obvious from the definition of function 𝜙 that

(‖𝑦‖−‖𝑥‖)²≤𝜙(𝑦,𝑥)≤(‖𝑦‖+‖𝑥‖)²,∀𝑥,𝑦∈𝐸. (1.6)

Remark 1.1. If 𝐸 is a reflexive, strictly convex, and smooth Banach space, then for 𝑥,𝑦∈𝐸, 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 (2.13), 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 [5, 7] for more details.

Next, we give some examples which are closed relatively quasi-nonexpansive (see [8]).

Example 1.2. Let Π_𝐶 be the generalized projection from a smooth, strictly convex and reflexive Banach space 𝐸 onto a nonempty closed and convex subset 𝐶 of 𝐸. Then, Π_𝐶 is a closed relatively quasi-nonexpansive mapping from 𝐸 onto 𝐶 with 𝐹(Π_𝐶)=𝐶.

Let 𝐸 be a real Banach space and let 𝐶 be a nonempty closed and convex subset of 𝐸, and 𝐴∶𝐶→𝐸^∗ be a mapping. The classical variational inequality problem for a mapping A is to find 𝑥^∗∈𝐶 such that ⟨𝐴𝑥^∗,𝑦−𝑥^∗⟩≥0,∀𝑦∈𝐶. (1.7) The set of solutions of (1.4) is denoted by VI(𝐴,𝐶). Recall that A is called

(i)monotone if ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥0,∀𝑥,𝑦∈𝐶, (1.8)(ii)an 𝛼-inverse-strongly monotone if there exists a constant 𝛼>0 such that ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥𝛼‖𝑥−𝑦‖²,∀𝑥,𝑦∈𝐶. (1.9)

Such a problem is connected with the convex minimization problem, the complementary problem, and the problem of finding a point 𝑥^∗∈𝐸 satisfying 𝐴𝑥^∗=0.

Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ, where ℝ denotes the set of real numbers. The equilibrium problem (for short, EP) is to find 𝑥^∗∈𝐶 such that

𝑓(𝑥^∗,𝑦)≥0,∀𝑦∈𝐶. (1.10)

The set of solutions of (1.10) is denoted by EP(𝑓). Given a mapping 𝑇∶𝐶→𝐸^∗, let 𝑓(𝑥,𝑦)=⟨𝑇𝑥,𝑦−𝑥⟩ for all 𝑥,𝑦∈𝐶. Then 𝑥^∗∈EP(𝑓) if and only if ⟨𝑇𝑥^∗,𝑦−𝑥^∗⟩≥0 for all 𝑦∈𝐶; that is, 𝑥^∗ is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.10). Some methods have been proposed to solve the equilibrium problem; see, for instance, [9–11].

Let 𝐶 be a closed convex subset of 𝐸; a mapping 𝑇∶𝐶→𝐶 is said to be nonexpansive if

‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶. (1.11)

A point 𝑥∈𝐶 is a fixed point  of 𝑇 provided that 𝑇𝑥=𝑥. Denote by 𝐹(𝑇) the set of fixed points of 𝑇; that is, 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}. Recall that a point 𝑝 in 𝐶 is said to be an asymptotic fixed point of 𝑇 [12] if 𝐶 contains a sequence {𝑥_𝑛} which converges weakly to 𝑝 such that lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. The set of asymptotic fixed points of 𝑇 will be denoted by 𝐹(𝑇). A mapping 𝑇 from 𝐶 into itself is said to be relatively nonexpansive [13–15] if 𝐹(𝑇)=𝐹(𝑇) and 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥) for all 𝑥∈𝐶 and 𝑝∈𝐹(𝑇). The asymptotic behavior of a relatively nonexpansive mapping was studied in [16–18]. 𝑇 is said to be 𝜙-nonexpansive, if 𝜙(𝑇𝑥,𝑇𝑦)≤𝜙(𝑥,𝑦) for 𝑥,𝑦∈𝐶. 𝑇 is said to be relatively quasi-nonexpansive if 𝐹(𝑇)≠∅ and 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥) for 𝑥∈𝐶 and 𝑝∈𝐹(𝑇). A mapping 𝑇 in a Banach space 𝐸 is closed if 𝑥_𝑛→𝑥 and 𝑇𝑥_𝑛→𝑦, then 𝑇𝑥=𝑦.

Remark 1.3. The class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings [16–19] which requires the strong restriction 𝐹(𝑇)=𝐹(𝑇).

In Hilbert spaces 𝐻, Iiduka et al. [20] proved that the sequence {𝑥_𝑛} defined by: 𝑥₁=𝑥∈𝐶 and

𝑥_𝑛+1=𝑃_𝐶(𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛), (1.12) where 𝑃_𝐶 is the metric projection of 𝐻 onto 𝐶, and {𝜆_𝑛} is a sequence of positive real numbers, and converges weakly to some element of VI(𝐴,𝐶).

It is well know that if 𝐶 is a nonempty closed and convex subset of a Hilbert space 𝐻 and 𝑃_𝐶∶𝐻→𝐶 is the metric projection of 𝐻 onto 𝐶, then 𝑃_𝐶 is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. In this connection, Alber [4] recently introduced a generalized projection mapping Π_𝐶 in a Banach space 𝐸 which is an analogue of the metric projection in Hilbert spaces.

In 2008, Iiduka and Takahashi [21] introduced the following iterative scheme for finding a solution of the variational inequality problem for inverse-strongly monotone 𝐴 in a 2-uniformly convex and uniformly smooth Banach space 𝐸: 𝑥₁=𝑥∈𝐶 and

𝑥_𝑛+1=Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛) (1.13) for every 𝑛=1,2,3,…, where Π_𝐶 is the generalized metric projection from 𝐸 onto 𝐶, 𝐽 is the duality mapping from 𝐸 into 𝐸^∗, and {𝜆_𝑛} is a sequence of positive real numbers. They proved that the sequence {𝑥_𝑛} generated by (1.13) converges weakly to some element of VI(𝐴,𝐶).

Matsushita and Takahashi [22] introduced the following iteration: a sequence {𝑥_𝑛} defined by

𝑥_𝑛+1=Π_𝐶𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑇𝑥_𝑛), (1.14) where the initial guess element 𝑥₀∈𝐶 is arbitrary, {𝛼_𝑛} is a real sequence in [0,1], 𝑇 is a relatively nonexpansive mapping, and Π_𝐶 denotes the generalized projection from 𝐸 onto a closed convex subset 𝐶 of 𝐸. They proved that the sequence {𝑥_𝑛} converges weakly to a fixed point of 𝑇.

In 2005, Matsushita and Takahashi [19] proposed the following hybrid iteration method (it is also called the CQ method) with generalized projection for relatively nonexpansive mapping 𝑇 in a Banach space 𝐸:

𝑥₀𝑦∈𝐶chosenarbitrarily,_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑇𝑥_𝑛𝐶),_𝑛={𝑧∈𝐶∶𝜙(𝑧,𝑦_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑄)},_𝑛={𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛𝑥⟩≥0},_𝑛+1=_{𝐶𝑛∩𝑄𝑛}𝑥₀. (1.15)

They proved that {𝑥_𝑛} converges strongly to Π_𝐹(𝑇)𝑥₀, where Π_𝐹(𝑇) is the generalized projection from 𝐶 onto 𝐹(𝑇).

Recently, Takahashi and Zembayashi [23] proposed the following modification of iteration (1.15) for a relatively nonexpansive mapping:

𝑥₀𝑦=𝑥∈𝐶,_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑆𝑥_𝑛𝑢),_𝑛∈𝐶suchthat𝑓(𝑢_𝑛1,𝑦)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐻⟩≥0,∀𝑦∈𝐶,_𝑛={𝑧∈𝐶∶𝜙(𝑧,𝑢_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑊)},_𝑛={𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝐽𝑥−𝐽𝑥_𝑛𝑥⟩≥0},_𝑛+1=_{𝐻𝑛∩𝑊𝑛}𝑥, (1.16) where 𝐽 is the duality mapping on 𝐸. Then, {𝑥_𝑛} converges strongly to Π_{𝐹(𝑆)∩EP(𝑓)}𝑥, where Π_{𝐹(𝑆)∩EP(𝑓)} is the generalized projection of 𝐸 onto 𝐹(𝑆)∩EP(𝑓). Also, Takahashi and Zembayashi [24] proved the following iteration for a relatively nonexpansive mapping:

𝑦_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑆𝑥_𝑛𝑢),_𝑛∈𝐶suchthat𝑓(𝑢_𝑛1,𝑦)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶⟩≥0,∀𝑦∈𝐶,_𝑛+1={𝑧∈𝐶_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑥)},_𝑛+1=_𝐶𝑛+1𝑥, (1.17) where 𝐽 is the duality mapping on 𝐸. Then, {𝑥_𝑛} converges strongly to Π_{𝐹(𝑆)∩EP(𝑓)}𝑥, where Π_{𝐹(𝑆)∩EP(𝑓)} is the generalized projection of 𝐸 onto 𝐹(𝑆)∩EP(𝑓). Qin and Su [25] proved the following iteration for relatively nonexpansive mappings 𝑇 in a Banach space 𝐸:

𝑥₀𝑦∈𝐶,chosenarbitrarily,_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑇𝑧_𝑛𝑧),_𝑛=𝐽⁻¹(𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑇𝑥_𝑛𝐶),_𝑛={𝑣∈𝐶∶𝜙(𝑣,𝑦_𝑛)≤𝛼_𝑛𝜙(𝑣,𝑥_𝑛)+(1−𝛼_𝑛)𝜙(𝑣,𝑧_𝑛𝑄)},_𝑛={𝑣∈𝐶∶⟨𝐽𝑥₀−𝐽𝑥_𝑛,𝑥_𝑛𝑥−𝑣⟩≥0},_𝑛+1=_{𝐶𝑛∩𝑄𝑛}𝑥₀, (1.18) the sequence {𝑥_𝑛} generated by (1.18) converges strongly to Π_𝐹(𝑇)𝑥₀.

In 2009, Wei et al. [26] proved the following iteration for two relatively nonexpansive mappings in a Banach space 𝐸:

𝑥₀∈𝐶,𝐽𝑧_𝑛=𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑇𝑥_𝑛,𝐽𝑢_𝑛=(𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑆𝑧_𝑛𝐻),_𝑛={𝑣∈𝐶∶𝜙(𝑣,𝑢_𝑛)≤𝛽_𝑛𝜙(𝑣,𝑥_𝑛)+(1−𝛽_𝑛)𝜙(𝑣,𝑧_𝑛)≤𝜙(𝑣,𝑥_𝑛𝑊)},_𝑛={𝑧∈𝐶∶⟨𝑧−𝑥_𝑛,𝐽𝑥₀−𝐽𝑥_𝑛𝑥⟩≤0},_𝑛+1=𝑄_{𝐻𝑛∩𝑊𝑛}𝑥₀; (1.19) if {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1) such that 𝛼_𝑛≤1−𝛿₁ and 𝛽_𝑛≤1−𝛿₂ for some 𝛿₁,𝛿₂∈(0,1), then {𝑥_𝑛} generated by (1.19) converges strongly to a point 𝑄_{𝐹(𝑇)∩𝐹(𝑆)}𝑥₀, where the mapping 𝑄_𝐶 of 𝐸 onto 𝐶 is the generalized projection. Very recently, Cholamjiak [27] proved the following iteration:

𝑧_𝑛=Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛𝑦),_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+𝛽_𝑛𝐽𝑇𝑥_𝑛+𝛾_𝑛𝐽𝑆𝑧_𝑛𝑢),_𝑛∈𝐶suchthat𝑓(𝑢_𝑛1,𝑦)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶⟩≥0,∀𝑦∈𝐶,_𝑛+1={𝑧∈𝐶_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑥)},_𝑛+1=_𝐶𝑛+1𝑥₀, (1.20) where 𝐽 is the duality mapping on 𝐸. Assume that 𝛼_𝑛, 𝛽_𝑛, and 𝛾_𝑛 are sequences in [0,1]. Then {𝑥_𝑛} converges strongly to 𝑞=Π_𝐹𝑥₀, where 𝐹∶=𝐹(𝑇)∩𝐹(𝑆)∩EP(𝑓)∩VI(𝐴,𝐶).

Motivated and inspired by Iiduka and Takahashi [21], Takahashi and Zembayashi [23, 24], Wei et al. [26], Cholamjiak [27], and Kumam and Wattanawitoon [28], we introduce a new hybrid projection iterative scheme which is difference from the algorithm (1.20) of Cholamjiak in [27, Theorem  3.1] for two relatively quasi-nonexpansive mappings in a Banach space. For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π_𝐶1𝑥₀ and 𝐶₁=𝐶, define a sequence {𝑥_𝑛} as follows:

𝑤_𝑛=Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛𝑧),_𝑛=𝐽⁻¹(𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑇𝑤_𝑛𝑦),_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑆𝑧_𝑛𝑢),_𝑛∈𝐶suchthat𝑓(𝑢_𝑛1,𝑦)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶⟩≥0,∀𝑦∈𝐶,_𝑛+1={𝑧∈𝐶_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝛼_𝑛𝜙(𝑧,𝑥_𝑛)+(1−𝛼_𝑛)𝜙(𝑧,𝑧_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑥)},_𝑛+1=_𝐶𝑛+1𝑥₀,∀𝑛≥1, (1.21) where 𝐽 is the duality mapping on 𝐸. Then, we prove that under certain appropriate conditions on the parameters, the sequences {𝑥_𝑛} and {𝑢_𝑛} generated by (1.21) converge strongly to Π_{𝐹(𝑆)∩𝐹(𝑇)∩EP(𝑓)∩VI(𝐴,𝐶)}.

The results presented in this paper improve and extend the corresponding results announced by Iiduka and Takahashi [21], Wei et al. [26], Kumam and Wattanawitoon [28], and many other authors in the literature.

2. Preliminaries

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

Lemma 2.1 (Beauzamy [29] and Xu [30]). If 𝐸 is a 2-uniformly convex Banach space, then, for all 𝑥,𝑦∈𝐸 we have 2‖𝑥−𝑦‖≤𝑐²‖𝐽𝑥−𝐽𝑦‖, (2.1) where 𝐽 is the normalized duality mapping of 𝐸 and 0<𝑐≤1.

The best constant 1/𝑐 in the Lemma is called the 𝑝-uniformly convex constant of 𝐸.

Lemma 2.2 (Beauzamy [29] and Zǎlinescu [31]). If 𝐸 is a p-uniformly convex Banach space and 𝑝 is a given real number with 𝑝≥2, then for all 𝑥,𝑦∈𝐸,𝐽_𝑥∈𝐽_𝑝(𝑥), and 𝐽_𝑦∈𝐽_𝑝(𝑦),𝑐⟨𝑥−𝑦,𝐽𝑥−𝐽𝑦⟩≥^𝑝2^𝑝−2𝑝‖𝑥−𝑦‖^𝑝, (2.2) where 𝐽_𝑝 is the generalized duality mapping of 𝐸 and 1/𝑐 is the p-uniformly convexity constant of 𝐸.

Lemma 2.3 (Kamimura and Takahashi [6]). Let 𝐸 be a uniformly convex and smooth Banach space and let {𝑥_𝑛} and {𝑦_𝑛} be two sequences of 𝐸. If 𝜙(𝑥_𝑛,𝑦_𝑛)→0 and either {𝑥_𝑛} or {𝑦_𝑛} is bounded, then ‖𝑥_𝑛−𝑦_𝑛‖→0.

Lemma 2.4 (Alber [4]). Let 𝐶 be a nonempty closed and convex subset of a smooth Banach space 𝐸 and 𝑥∈𝐸. Then, 𝑥₀=Π_𝐶𝑥 if and only if ⟨𝑥₀−𝑦,𝐽𝑥−𝐽𝑥₀⟩≥0,∀𝑦∈𝐶. (2.3)

Lemma 2.5 (Alber [4]). Let 𝐸 be a reflexive, strictly convex, and smooth Banach space, let 𝐶 be a nonempty closed and convex subset of 𝐸, and let 𝑥∈𝐸. Then 𝜙(𝑦,Π_𝐶𝑥)+𝜙(Π_𝐶𝑥,𝑥)≤𝜙(𝑦,𝑥),∀𝑦∈𝐶. (2.4)

Lemma 2.6 (Qin et al. [8]). Let 𝐸 be a uniformly convex and smooth Banach space, let 𝐶 be a closed and convex subset of 𝐸, and let 𝑇 be a closed relatively quasi-nonexpansive mapping from 𝐶 into itself. Then 𝐹(𝑇) is a closed and convex subset of 𝐶.

For solving the equilibrium problem for a bifunction 𝑓∶𝐶×𝐶→ℝ, let us assume that 𝑓 satisfies the following conditions:

(A1)𝑓(𝑥,𝑥)=0 for all 𝑥∈𝐶; (A2)𝑓 is monotone, that is, 𝑓(𝑥,𝑦)+𝑓(𝑦,𝑥)≤0 for all 𝑥,𝑦∈𝐶; (A3)for each 𝑥,𝑦,𝑧∈𝐶, lim_𝑡↓0𝑓(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝑓(𝑥,𝑦); (2.5)(A4)for each 𝑥∈𝐶, 𝑦↦𝑓(𝑥,𝑦) is convex and lower semi-continuous.

Lemma 2.7 (Blum and Oettli [9]). Let 𝐶 be a closed and 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 1𝑓(𝑧,𝑦)+𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶. (2.6)

Lemma 2.8 (Combettes and Hirstoaga [10]). Let 𝐶 be a closed and convex subset of a uniformly smooth, strictly convex and reflexive Banach space 𝐸 and let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4). For 𝑟>0 and 𝑥∈𝐸, define a mapping 𝑇_𝑟∶𝐸→𝐶 as follows: 𝑇_𝑟1𝑥={𝑧∈𝐶∶𝑓(𝑧,𝑦)+𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶}, (2.7) for all 𝑥∈𝐶. Then the following holds: (1)𝑇_𝑟 is single-valued; (2)𝑇_𝑟 is a firmly nonexpansive-type mapping, for all 𝑥,𝑦∈𝐸, ⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽𝑇_𝑟𝑥−𝐽𝑇_𝑟𝑦⟩≤⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽𝑥−𝐽𝑦⟩; (2.8)(3)𝐹(𝑇_𝑟)=EP(𝑓);(4)EP(𝑓) is closed and convex.

Lemma 2.9 (Takahashi and Zembayashi [24]). Let 𝐶 be a closed and 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. Then, for 𝑥∈𝐸 and 𝑞∈𝐹(𝑇_𝑟),𝜙(𝑞,𝑇_𝑟𝑥)+𝜙(𝑇_𝑟𝑥,𝑥)≤𝜙(𝑞,𝑥). (2.9)

Let 𝐸 be a reflexive, strictly convex, and smooth Banach space and 𝐽 the duality mapping from 𝐸 into 𝐸^∗. Then 𝐽⁻¹ is also single value, one-to-one, surjective, and it is the duality mapping from 𝐸^∗ into 𝐸. We make use of the following mapping 𝑉 studied in Alber [4]:

𝑉(𝑥,𝑥^∗)=‖𝑥‖²−2⟨𝑥,𝑥^∗⟩+‖𝑥^∗‖² (2.10) for all 𝑥∈𝐸 and 𝑥^∗∈𝐸^∗, that is, 𝑉(𝑥,𝑥^∗)=𝜙(𝑥,𝐽⁻¹(𝑥^∗)).

Lemma 2.10 (Alber [4]). Let 𝐸 be a reflexive, strictly convex, and smooth Banach space and let 𝑉 be as in (2.10). Then 𝑉(𝑥,𝑥^∗)+2⟨𝐽⁻¹(𝑥^∗)−𝑥,𝑦^∗⟩≤𝑉(𝑥,𝑥^∗+𝑦^∗) (2.11) for all 𝑥∈𝐸 and 𝑥^∗,𝑦^∗∈𝐸^∗.

Let 𝐴 be an inverse-strongly monotone of 𝐶 into 𝐸^∗ which is said to be hemicontinuous if for all 𝑥,𝑦∈𝐶, the mapping 𝐹 of [0,1] into 𝐸^∗, defined by 𝐹(𝑡)=𝐴(𝑡𝑥+(1−𝑡)𝑦), is continuous with respect to the weak∗ topology of 𝐸^∗. We define by 𝑁_𝐶(𝑣)the normal cone for 𝐶 at a point 𝑣∈𝐶; that is,

𝑁_𝐶(𝑣)={𝑥^∗∈𝐸^∗∶⟨𝑣−𝑦,𝑥^∗⟩≥0,∀𝑦∈𝐶}. (2.12)

Theorem 2.11 (Rockafellar [32]). Let 𝐶 be a nonempty, closed and convex subset of a Banach space 𝐸, and 𝐴 a monotone, hemicontinuous mapping of 𝐶 into 𝐸^∗. Let 𝑇⊂𝐸×𝐸^∗ be a mapping defined as follows: 𝑇𝑣=𝐴𝑣+𝑁_𝐶(𝑣),𝑣∈𝐶;∅,otherwise. (2.13) Then 𝑇 is maximal monotone and 𝑇⁻¹0=VI(𝐴,𝐶).

3. Main Results

In this section, we establish a new hybrid iterative scheme for finding a common element of the set of solutions of an equilibrium problems, the common fixed point set of two relatively quasi-nonexpansive mappings, and the solution set of variational inequalities for 𝛼-inverse strongly monotone in a 2-uniformly convex and uniformly smooth Banach space.

Theorem 3.1. Let 𝐶 be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4) and let 𝐴 be an 𝛼-inverse-strongly monotone mapping of 𝐶 into 𝐸^∗ satisfying ‖𝐴𝑦‖≤‖𝐴𝑦−𝐴𝑢‖,forall𝑦∈𝐶 and 𝑢∈VI(𝐴,𝐶)≠∅. Let 𝑇,𝑆∶𝐶→𝐶 be closed relatively quasi-nonexpansive mappings such that Ω∶=𝐹(𝑇)∩𝐹(𝑆)∩EP(𝑓)∩VI(𝐴,𝐶)≠∅. For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π_𝐶1𝑥₀ and 𝐶₁=𝐶, we define the sequence {𝑥_𝑛} as follows: 𝑤_𝑛=Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛𝑧),_𝑛=𝐽⁻¹(𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑇𝑤_𝑛𝑦),_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑆𝑧_𝑛𝑢),_𝑛∈𝐶suchthat𝑓(𝑢_𝑛1,𝑦)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶⟩≥0,∀𝑦∈𝐶,_𝑛+1={𝑧∈𝐶_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝛼_𝑛𝜙(𝑧,𝑥_𝑛)+(1−𝛼_𝑛)𝜙(𝑧,𝑧_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑥)},_𝑛+1=_𝐶𝑛+1𝑥₀,∀𝑛≥1, (3.1) where 𝐽 is the duality mapping on 𝐸, {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1] such that 𝛼_𝑛≤1−𝛿₁ and 𝛽_𝑛≤1−𝛿₂, for some 𝛿₁,𝛿₂∈(0,1), {𝑟_𝑛}⊆(0,2𝛼) and {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<𝑐²𝛼/2, where 1/𝑐 is the 2-uniformly convexity constant of 𝐸. Then {𝑥_𝑛} converges strongly to 𝑝∈Ω, where 𝑝=Π_Ω𝑥₀.

Proof. We have several steps to prove this theorem as follows:Step 1. We show that 𝐶_𝑛+1 is closed and convex.
Clearly 𝐶₁=𝐶 is closed and convex. Suppose that 𝐶_𝑛 is closed and convex for each 𝑛∈ℕ. Since for any 𝑧∈𝐶_𝑛, we know that
𝜙(𝑧,𝑢_𝑛)≤𝜙(𝑧,𝑥_𝑛) (3.2) is equivalent to 2⟨𝑧,𝐽𝑥_𝑛−𝐽𝑢_𝑛⟩≤‖𝑥_𝑛‖²−‖𝑢_𝑛‖². (3.3) So, 𝐶_𝑛+1 is closed and convex. Then, by induction, 𝐶_𝑛 is closed and convex for all 𝑛≥1.Step 2. We show that {𝑥_𝑛} is well defined.
Put 𝑢_𝑛=𝑇_𝑟𝑛𝑦_𝑛 for all 𝑛≥0. On the other hand, from Lemma 2.8 one has 𝑇_𝑟𝑛 is relatively quasi-nonexpansive mappings and Ω⊂𝐶₁=𝐶. Supposing Ω⊂𝐶_𝑘 for 𝑘∈ℕ, by the convexity of ‖⋅‖², for each 𝑞∈Ω⊂𝐶_𝑘, we have
𝜙(𝑞,𝑢_𝑘)=𝜙(𝑞,𝑇_𝑟𝑘𝑦_𝑘)≤𝜙(𝑞,𝑦_𝑘)=𝜙(𝑞,𝐽⁻¹(𝛼_𝑘𝐽𝑥_𝑘+(1−𝛼_𝑘)𝐽𝑆𝑧_𝑘))=‖𝑞‖²−2⟨𝑞,𝛼_𝑘𝐽𝑥_𝑘+(1−𝛼_𝑘)𝐽𝑆𝑧_𝑘⟩+‖𝛼_𝑘𝐽𝑥_𝑘+(1−𝛼_𝑘)𝐽𝑆𝑧_𝑘‖²≤‖𝑞‖²−2𝛼_𝑘⟨𝑞,𝐽𝑥_𝑘⟩−2(1−𝛼_𝑘)⟨𝑞,𝐽𝑆𝑧_𝑘⟩+𝛼_𝑘‖𝑥_𝑘‖²+(1−𝛼_𝑘)‖𝑆𝑧_𝑘‖²=𝛼_𝑘𝜙(𝑞,𝑥_𝑘)+(1−𝛼_𝑘)𝜙(𝑞,𝑆𝑧_𝑘)≤𝛼_𝑘𝜙(𝑞,𝑥_𝑘)+(1−𝛼_𝑘)𝜙(𝑞,𝑧_𝑘), (3.4) and so 𝜙(𝑞,𝑧_𝑘)=𝜙(𝑞,𝐽⁻¹(𝛽_𝑘𝐽𝑥_𝑘+(1−𝛽_𝑘)𝐽𝑇𝑤_𝑘))=‖𝑞‖²−2⟨𝑞,𝛽_𝑘𝐽𝑥_𝑘+(1−𝛽_𝑘)𝐽𝑇𝑤_𝑘⟩+‖𝛽_𝑘𝐽𝑥_𝑘+(1−𝛽_𝑘)𝐽𝑇𝑤_𝑘‖²≤‖𝑞‖²−2𝛽_𝑘⟨𝑞,𝐽𝑥_𝑘⟩−2(1−𝛽_𝑘)⟨𝑞,𝐽𝑇𝑤_𝑘⟩+𝛽_𝑘‖𝐽𝑥_𝑘‖²+(1−𝛽_𝑘)‖𝐽𝑇𝑤_𝑘‖²=𝛽_𝑘𝜙(𝑞,𝑥_𝑘)+(1−𝛽_𝑘)𝜙(𝑞,𝑇𝑤_𝑘)≤𝛽_𝑘𝜙(𝑞,𝑥_𝑘)+(1−𝛽_𝑘)𝜙(𝑞,𝑤_𝑘). (3.5) For all 𝑞∈Ω, we know from Lemma 2.10, that 𝜙(𝑞,𝑤_𝑘)=𝜙(𝑞,Π_𝐶𝐽⁻¹(𝐽𝑥_𝑘−𝜆_𝑘𝐴𝑥_𝑘))≤𝜙(𝑞,𝐽⁻¹(𝐽𝑥_𝑘−𝜆_𝑘𝐴𝑥_𝑘))=𝑉(𝑞,𝐽𝑥_𝑘−𝜆_𝑘𝐴𝑥_𝑘)≤𝑉(𝑞,(𝐽𝑥_𝑘−𝜆_𝑘𝐴𝑥_𝑘)+𝜆_𝑘𝐴𝑥_𝑘)−2⟨𝐽⁻¹(𝐽𝑥_𝑘−𝜆_𝑘𝐴𝑥_𝑘)−𝑞,𝜆_𝑘𝐴𝑥_𝑘⟩=𝑉(𝑞,𝐽𝑥_𝑘)−2𝜆_𝑘⟨𝐽⁻¹(𝐽𝑥_𝑘−𝜆_𝑘𝐴𝑥_𝑘)−𝑞,𝐴𝑥_𝑘⟩=𝜙(𝑞,𝑥_𝑘)−2𝜆_𝑘⟨𝑥_𝑘−𝑞,𝐴𝑥_𝑘⟩+2⟨𝐽⁻¹(𝐽𝑥_𝑘−𝜆_𝑘𝐴𝑥_𝑘)−𝑥_𝑘,−𝜆_𝑘𝐴𝑥_𝑘⟩. (3.6) Since 𝑞∈VI(𝐴,𝐶) and from 𝐴 being an 𝛼-inverse-strongly monotone, we get −2𝜆_𝑘⟨𝑥_𝑘−𝑞,𝐴𝑥_𝑘⟩=−2𝜆_𝑘⟨𝑥_𝑘−𝑞,𝐴𝑥_𝑘−𝐴𝑞⟩−2𝜆_𝑘⟨𝑥_𝑘−𝑞,𝐴𝑞⟩≤−2𝜆_𝑘⟨𝑥_𝑘−𝑞,𝐴𝑥_𝑘−𝐴𝑞⟩=−2𝛼𝜆_𝑘‖𝐴𝑥_𝑘−𝐴𝑞‖². (3.7) From Lemma 2.1 and 𝐴 being an 𝛼-inverse-strongly monotone, we obtain 2⟨𝐽⁻¹(𝐽𝑥_𝑘−𝜆_𝑘𝐴𝑥_𝑘)−𝑥_𝑘,−𝜆_𝑘𝐴𝑥_𝑘⟩=2⟨𝐽⁻¹(𝐽𝑥_𝑘−𝜆_𝑘𝐴𝑥_𝑘)−𝐽⁻¹(𝐽𝑥_𝑘),−𝜆_𝑘𝐴𝑥_𝑘⟩≤2‖𝐽⁻¹(𝐽𝑥_𝑘−𝜆_𝑘𝐴𝑥_𝑘)−𝐽⁻¹(𝐽𝑥_𝑘)‖‖𝜆_𝑘𝐴𝑥_𝑘‖≤4𝑐²‖𝐽𝐽⁻¹(𝐽𝑥_𝑘−𝜆_𝑘𝐴𝑥_𝑘)−𝐽𝐽⁻¹(𝐽𝑥_𝑘)‖‖𝜆_𝑘𝐴𝑥_𝑘‖=4𝑐²‖𝐽𝑥_𝑘−𝜆_𝑘𝐴𝑥_𝑘−𝐽𝑥_𝑘‖‖𝜆_𝑘𝐴𝑥_𝑘‖=4𝑐²‖𝜆_𝑘𝐴𝑥_𝑘‖²=4𝑐²𝜆²_𝑘‖𝐴𝑥_𝑘‖²≤4𝑐²𝜆²_𝑘‖𝐴𝑥_𝑘−𝐴𝑞‖². (3.8) Substituting (3.7) and (3.8) into (3.6), we have 𝜙(𝑞,𝑤_𝑘)≤𝜙(𝑞,𝑥_𝑘)−2𝛼𝜆_𝑘‖𝐴𝑥_𝑘−𝐴𝑞‖²+4𝑐²𝜆²_𝑘‖𝐴𝑥_𝑘−𝐴𝑞‖²=𝜙(𝑞,𝑥_𝑘)+2𝜆_𝑘(2𝑐²𝜆_𝑘−𝛼)‖𝐴𝑥_𝑘−𝐴𝑞‖²≤𝜙(𝑞,𝑥_𝑘). (3.9) Replacing (3.9) into (3.5), we get 𝜙(𝑞,𝑧_𝑘)≤𝜙(𝑞,𝑥_𝑘). (3.10) Substituting (3.10) into (3.4), we also have 𝜙(𝑞,𝑢_𝑘)≤𝛼_𝑘𝜙(𝑞,𝑥_𝑘)+(1−𝛼_𝑘)𝜙(𝑞,𝑥_𝑘),=𝜙(𝑞,𝑥_𝑘). (3.11)
This shows that 𝑞∈𝐶_𝑘+1 and hence, Ω⊂𝐶_𝑘+1. Hence, Ω⊂𝐶_𝑛 for all 𝑛≥1. This implies that the sequence {𝑥_𝑛} is well defined.Step 3. We show that lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₀) exists and {𝑥_𝑛} is bounded.
From 𝑥_𝑛=Π_𝐶𝑛𝑥₀ and 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀, we have
𝜙(𝑥_𝑛,𝑥₀)≤𝜙(𝑥_𝑛+1,𝑥₀),∀𝑛≥1, (3.12) and from Lemma 2.5, we have 𝜙(𝑥_𝑛,𝑥₀)=𝜙(Π_𝐶𝑛(𝑥₀),𝑥₀)≤𝜙(𝑝,𝑥₀)−𝜙(𝑝,𝑥_𝑛)≤𝜙(𝑝,𝑥₀),∀𝑝∈Ω. (3.13) From (3.12) and (3.13), then {𝜙(𝑥_𝑛,𝑥₀)} are nondecreasing and bounded. So, we obtain that lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₀) exists. In particular, by (1.6), the sequence {(‖𝑥_𝑛‖−‖𝑥₀‖)²} is bounded. This implies that {𝑥_𝑛} is also bounded.Step 4. We show that {𝑥_𝑛} is a Cauchy sequence in 𝐶.
Since 𝑥_𝑚=Π_𝐶𝑚𝑥₀∈𝐶_𝑚⊂𝐶_𝑛, for 𝑚>𝑛, by Lemma 2.5, we have
𝜙(𝑥_𝑚,𝑥_𝑛)=𝜙(𝑥_𝑚,Π_𝐶𝑛𝑥₀)≤𝜙(𝑥_𝑚,𝑥₀)−𝜙(Π_𝐶𝑛𝑥₀,𝑥₀)=𝜙(𝑥_𝑚,𝑥₀)−𝜙(𝑥_𝑛,𝑥₀). (3.14)
 Taking 𝑚,𝑛→∞, we have 𝜙(𝑥_𝑚,𝑥_𝑛)→0. We have lim_𝑛→∞𝜙(𝑥_𝑛+1,𝑥₀)=0. From Lemma 2.3, we get lim_𝑛→∞‖𝑥_𝑛+1−𝑥₀‖=0. Thus {𝑥_𝑛} is a Cauchy sequence.Step 5. We cliam that ‖𝐽𝑢_𝑛−𝐽𝑥_𝑛‖→0, as 𝑛→∞.
By the completeness of 𝐸, the closedness of 𝐶 and {𝑥_𝑛} is a Cauchy sequence (from Step 4); we can assume that there exists 𝑝∈𝐶 such that {𝑥_𝑛}→𝑝 as 𝑛→∞.
By definition of Π_𝐶𝑛𝑥₀, we have
𝜙(𝑥_𝑛+1,𝑥_𝑛)=𝜙(𝑥_𝑛+1,Π_𝐶𝑛𝑥₀)≤𝜙(𝑥_𝑛+1,𝑥₀)−𝜙(Π_𝐶𝑛𝑥₀,𝑥₀)=𝜙(𝑥_𝑛+1,𝑥₀)−𝜙(𝑥_𝑛,𝑥₀). (3.15) Since lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₀) exists, we get lim_𝑛→∞𝜙(𝑥_𝑛+1,𝑥_𝑛)=0. (3.16) It follow form Lemma 2.3, that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. (3.17) Since 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1⊂𝐶_𝑛 and from the definition of 𝐶_𝑛+1, we have 𝜙(𝑥_𝑛+1,𝑢_𝑛)≤𝜙(𝑥_𝑛+1,𝑥_𝑛),∀𝑛≥1 (3.18) and so lim_𝑛→∞𝜙(𝑥_𝑛+1,𝑢_𝑛)=0. (3.19) Hence lim_𝑛→∞‖𝑥_𝑛+1−𝑢_𝑛‖=0. (3.20) By using the triangle inequality, we obtain ‖𝑢_𝑛−𝑥_𝑛‖=‖𝑢_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛‖≤‖𝑢_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑥_𝑛‖. (3.21) By (3.17), (3.20), we get lim_𝑛→∞‖𝑢_𝑛−𝑥_𝑛‖=0. (3.22) Since 𝐽 is uniformly norm-to-norm continuous on bounded subsets of 𝐸, we have lim_𝑛→∞‖𝐽𝑢_𝑛−𝐽𝑥_𝑛‖=0. (3.23)Step 6. Show that 𝑥_𝑛→𝑝∈EP(𝑓).
Applying (3.4) and (3.11), we get 𝜙(𝑝,𝑦_𝑛)≤𝜙(𝑝,𝑥_𝑛). From Lemma 2.9 and 𝑢_𝑛=𝑇_𝑟𝑛𝑦_𝑛, we observe that
𝜙(𝑢_𝑛,𝑦_𝑛)=𝜙(𝑇_𝑟𝑛𝑦_𝑛,𝑦_𝑛)≤𝜙(𝑝,𝑦_𝑛)−𝜙(𝑝,𝑇_𝑟𝑛𝑦_𝑛)≤𝜙(𝑝,𝑥_𝑛)−𝜙(𝑝,𝑇_𝑟𝑛𝑦_𝑛)=𝜙(𝑝,𝑥_𝑛)−𝜙(𝑝,𝑢_𝑛)=‖𝑝‖²−2⟨𝑝,𝐽𝑥_𝑛⟩+‖𝑥_𝑛‖²−(‖𝑝‖²−2⟨𝑝,𝐽𝑢_𝑛⟩+‖𝑢_𝑛‖²)=‖𝑥_𝑛‖²−‖𝑢_𝑛‖²−2⟨𝑝,𝐽𝑥_𝑛−𝐽𝑢_𝑛⟩≤‖𝑥_𝑛−𝑢_𝑛‖(‖𝑥_𝑛+𝑢_𝑛‖)+2‖𝑝‖‖𝐽𝑥_𝑛−𝐽𝑢_𝑛‖. (3.24) From (3.22), (3.23) and Lemma 2.3, we get lim_𝑛→∞‖𝑢_𝑛−𝑦_𝑛‖=0. (3.25) Since 𝐽 is uniformly norm-to-norm continuous, we obtain lim_𝑛→∞‖𝐽𝑢_𝑛−𝐽𝑦_𝑛‖=0. (3.26) From 𝑟_𝑛>0, we have ‖𝐽𝑢_𝑛−𝐽𝑦_𝑛‖/𝑟_𝑛→0 as 𝑛→∞ and 𝑓(𝑢_𝑛1,𝑦)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥0,∀𝑦∈𝐶. (3.27) By (A2), that ‖𝑦−𝑢_𝑛‖‖𝐽𝑢_𝑛−𝐽𝑦_𝑛‖𝑟_𝑛≥1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥−𝑓(𝑢_𝑛,𝑦)≥𝑓(𝑦,𝑢_𝑛),∀𝑦∈𝐶 (3.28) and 𝑢_𝑛→𝑝, we get 𝑓(𝑦,𝑝)≤0 for all 𝑦∈𝐶. For 0<𝑡<1, define 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑝. Then 𝑦_𝑡∈𝐶 which implies that 𝑓(𝑦_𝑡,𝑝)≤0. From (A1), we obtain that 0=𝑓(𝑦_𝑡,𝑦_𝑡)≤𝑡𝑓(𝑦_𝑡,𝑦)+(1−𝑡)𝑓(𝑦_𝑡,𝑝)≤𝑡𝑓(𝑦_𝑡,𝑦). (3.29) Thus 𝑓(𝑦_𝑡,𝑦)≥0. From (A3), we have 𝑓(𝑝,𝑦)≥0 for all 𝑦∈𝐶. Hence 𝑝∈EP(𝑓).Step 7. We show that 𝑥_𝑛→𝑝∈𝐹(𝑇)∩𝐹(𝑆).
From definition of 𝐶_𝑛, we have
𝛼_𝑛𝜙(𝑧,𝑥_𝑛)+(1−𝛼_𝑛)𝜙(𝑧,𝑧_𝑛)≤𝜙(𝑧,𝑥_𝑛)⇔𝜙(𝑧,𝑧_𝑛)≤𝜙(𝑧,𝑥_𝑛). (3.30) Since 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1, we have 𝜙(𝑥_𝑛+1,𝑧_𝑛)≤𝜙(𝑥_𝑛+1,𝑥_𝑛). (3.31) It follows from (3.16) that lim_𝑛→∞𝜙(𝑥_𝑛+1,𝑧_𝑛)=0, (3.32) again from Lemma 2.3, we get lim_𝑛→∞‖𝑥_𝑛+1−𝑧_𝑛‖=0. (3.33) By using the triangle inequality, we get ‖𝑧_𝑛−𝑥_𝑛‖≤‖𝑧_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑥_𝑛‖. (3.34) Again by (3.17) and (3.33), we also have lim_𝑛→∞‖𝑧_𝑛−𝑥_𝑛‖=0. (3.35) Since 𝐽 is uniformly norm-to-norm continuous, we obtain lim_𝑛→∞‖𝐽𝑧_𝑛−𝐽𝑥_𝑛‖=0. (3.36) Since ‖𝑦_𝑛−𝑧_𝑛‖≤‖𝑦_𝑛−𝑢_𝑛‖+‖𝑢_𝑛−𝑥_𝑛‖+‖𝑥_𝑛−𝑧_𝑛‖, (3.37) from (3.22), (3.25), and (3.35), we have lim_𝑛→∞‖𝑦_𝑛−𝑧_𝑛‖=0. (3.38) Since 𝐽 is uniformly norm-to-norm continuous, we also have lim_𝑛→∞‖𝐽𝑦_𝑛−𝐽𝑧_𝑛‖=0. (3.39) From (3.1), we get ‖𝐽𝑦_𝑛−𝐽𝑧_𝑛‖=‖𝛼_𝑛(𝐽𝑥_𝑛−𝐽𝑧_𝑛)+(1−𝛼_𝑛)(𝐽𝑆𝑧_𝑛−𝐽𝑧_𝑛)‖=‖(1−𝛼_𝑛)(𝐽𝑆𝑧_𝑛−𝐽𝑧_𝑛)−𝛼_𝑛(𝐽𝑧_𝑛−𝐽𝑥_𝑛)‖≥(1−𝛼_𝑛)‖𝐽𝑆𝑧_𝑛−𝐽𝑧_𝑛‖−𝛼_𝑛‖𝐽𝑧_𝑛−𝐽𝑥_𝑛‖; (3.40) it follows that (1−𝛼_𝑛)‖𝐽𝑆𝑧_𝑛−𝐽𝑧_𝑛‖≤‖𝐽𝑦_𝑛−𝐽𝑧_𝑛‖+𝛼_𝑛‖𝐽𝑧_𝑛−𝐽𝑥_𝑛‖, (3.41) and hence ‖𝐽𝑆𝑧_𝑛−𝐽𝑧_𝑛1‖≤1−𝛼_𝑛(‖𝐽𝑦_𝑛−𝐽𝑧_𝑛‖+𝛼_𝑛‖𝐽𝑧_𝑛−𝐽𝑥_𝑛‖). (3.42) Since 𝛼_𝑛≤1−𝛿₁ for some 𝛿₁∈(0,1), (3.36), and (3.39), one has lim_𝑛→∞‖𝐽𝑆𝑧_𝑛−𝐽𝑧_𝑛‖=0. Since 𝐽⁻¹ is uniformly norm-to-norm continuous, we get lim_𝑛→∞‖𝑆𝑧_𝑛−𝑧_𝑛‖=0. (3.43) Since ‖𝑆𝑥_𝑛−𝑥_𝑛‖≤‖𝑆𝑥_𝑛−𝑆𝑧_𝑛‖+‖𝑆𝑧_𝑛−𝑧_𝑛‖+‖𝑧_𝑛−𝑥_𝑛‖≤‖𝑥_𝑛−𝑧_𝑛‖+‖𝑆𝑧_𝑛−𝑧_𝑛‖+‖𝑧_𝑛−𝑥_𝑛‖, (3.44) from (3.35) and (3.43), we obtain lim_𝑛→∞‖𝑆𝑥_𝑛−𝑥_𝑛‖=0. (3.45) Since 𝑆 is closed and 𝑥_𝑛→𝑝, we have 𝑝∈𝐹(𝑆).
On the other hand, we note that
𝜙(𝑞,𝑥_𝑛)−𝜙(𝑞,𝑢_𝑛)=‖𝑥_𝑛‖²−‖𝑢_𝑛‖²−2⟨𝑞,𝐽𝑥_𝑛−𝐽𝑢_𝑛⟩≤‖𝑥_𝑛−𝑢_𝑛‖(‖𝑥_𝑛+𝑢_𝑛‖)+2‖𝑞‖‖𝐽𝑥_𝑛−𝐽𝑢_𝑛‖. (3.46) It follows from ‖𝑥_𝑛−𝑢_𝑛‖→0 and ‖𝐽𝑥_𝑛−𝐽𝑢_𝑛‖→0, that 𝜙(𝑞,𝑥_𝑛)−𝜙(𝑞,𝑢_𝑛)→0. (3.47) Furthermore, from (3.4) and (3.5), 𝜙(𝑞,𝑢_𝑛)≤𝜙(𝑞,𝑦_𝑛)≤𝛼_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛼_𝑛)𝜙(𝑞,𝑧_𝑛)≤𝛼_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛼_𝑛)[𝛽_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛽_𝑛)𝜙(𝑞,𝑤_𝑛)]=𝛼_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛼_𝑛)𝛽_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛼_𝑛)(1−𝛽_𝑛)𝜙(𝑞,𝑤_𝑛)≤𝛼_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛼_𝑛)𝛽_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛼_𝑛)(1−𝛽_𝑛)×[𝜙(𝑞,𝑥_𝑛)−2𝜆_𝑛2(𝛼−𝑐²𝜆_𝑛)‖𝐴𝑥_𝑛−𝐴𝑞‖²]=𝛼_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛼_𝑛)𝛽_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛼_𝑛)(1−𝛽_𝑛)𝜙(𝑞,𝑥_𝑛)−(1−𝛼_𝑛)(1−𝛽_𝑛)2𝜆_𝑛2(𝛼−𝑐²𝜆_𝑛)‖𝐴𝑥_𝑛−𝐴𝑞‖²=𝜙(𝑞,𝑥_𝑛)−(1−𝛼_𝑛)(1−𝛽_𝑛)2𝜆_𝑛2(𝛼−𝑐²𝜆_𝑛)‖𝐴𝑥_𝑛−𝐴𝑞‖², (3.48) and hence 𝛿₁𝛿₂2𝑎(𝛼−2𝑎𝑐²)‖𝐴𝑥_𝑛−𝐴𝑞‖²≤(1−𝛼_𝑛)(1−𝛽_𝑛)2𝜆_𝑛2(𝛼−𝑐²𝜆_𝑛)‖𝐴𝑥_𝑛−𝐴𝑞‖²≤𝜙(𝑞,𝑥_𝑛)−𝜙(𝑞,𝑢_𝑛). (3.49) From (3.47) and (3.49), we have ‖𝐴𝑥_𝑛−𝐴𝑞‖→0. (3.50) From Lemma 2.5, Lemma 2.10, and (3.8), we compute 𝜙(𝑥_𝑛,𝑤_𝑛)=𝜙(𝑥_𝑛,Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛))≤𝜙(𝑥_𝑛,𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛))=𝑉(𝑥_𝑛,𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛)≤𝑉(𝑥_𝑛,(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛)+𝜆_𝑛𝐴𝑥_𝑛)−2⟨𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛)−𝑥_𝑛,𝜆_𝑛𝐴𝑥_𝑛⟩=𝜙(𝑥_𝑛,𝑥_𝑛)+2⟨𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛)−𝑥_𝑛,−𝜆_𝑛𝐴𝑥_𝑛⟩=2⟨𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛)−𝑥_𝑛,−𝜆_𝑛𝐴𝑥_𝑛⟩≤4𝜆²_𝑛𝑐²‖𝐴𝑥_𝑛−𝐴𝑞‖²≤4𝑏²𝑐²‖𝐴𝑥_𝑛−𝐴𝑞‖². (3.51)
Applying Lemmas 2.3 and (3.50), we obtain that
‖𝑥_𝑛−𝑤_𝑛‖→0. (3.52)
Again since 𝐽 is uniformly norm-to-norm continuous on bounded set, we have
‖𝐽𝑥_𝑛−𝐽𝑤_𝑛‖→0. (3.53)
Since
‖𝑧_𝑛−𝑤_𝑛‖≤‖𝑧_𝑛−𝑥_𝑛‖+‖𝑥_𝑛−𝑤_𝑛‖, (3.54)
by (3.35) and (3.52), we have
lim_𝑛→∞‖𝑧_𝑛−𝑤_𝑛‖=0, (3.55) and hence lim_𝑛→∞‖𝐽𝑧_𝑛−𝐽𝑤_𝑛‖=0. (3.56)From (3.1) we obtain that‖𝐽𝑧_𝑛−𝐽𝑤_𝑛‖=‖𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑇𝑤_𝑛−𝐽𝑤_𝑛‖≥(1−𝛽_𝑛)‖𝐽𝑇𝑤_𝑛−𝐽𝑤_𝑛‖−𝛽_𝑛‖𝐽𝑤_𝑛−𝐽𝑥_𝑛‖, (3.57) and hence (1−𝛽_𝑛)‖𝐽𝑇𝑤_𝑛−𝐽𝑤_𝑛‖≤‖𝐽𝑧_𝑛−𝐽𝑤_𝑛‖+𝛽_𝑛‖𝐽𝑤_𝑛−𝐽𝑥_𝑛‖, (3.58) so ‖𝐽𝑇𝑤_𝑛−𝐽𝑤_𝑛1‖≤1−𝛽_𝑛‖𝐽𝑧_𝑛−𝐽𝑤_𝑛‖+𝛽_𝑛‖𝐽𝑤_𝑛−𝐽𝑥_𝑛‖. (3.59)By (3.53), (3.56) andcondition 𝛽_𝑛≤1−𝛿₂ for some 𝛿₂∈(0,1), we obtain ‖𝐽𝑇𝑤_𝑛−𝐽𝑤_𝑛‖→0. (3.60) Since 𝐽⁻¹ is uniformly norm-to-norm continuous on bounded set, we obtain ‖𝑇𝑤_𝑛−𝑤_𝑛‖→0. (3.61) Since 𝑥_𝑛→𝑤_𝑛, then ‖𝑇𝑥_𝑛−𝑥_𝑛‖→0. Thus by the closedness of 𝑇 and 𝑥_𝑛→𝑝, we get 𝑝∈𝐹(𝑇). Hence 𝑝∈𝐹(𝑇)∩𝐹(𝑆).Step 8. We show that 𝑥_𝑛→𝑝∈VI(𝐴,𝐶).
Define 𝑇⊂𝐸×𝐸^∗ by Theorem 2.11; 𝑇 is maximal monotone and 𝑇⁻¹0=VI(𝐴,𝐶). Let (𝑣,𝑤)∈𝐺(𝑇). Since 𝑤∈𝑇𝑣=𝐴𝑣+𝑁_𝐶(𝑣), we get 𝑤−𝐴𝑣∈𝑁_𝐶(𝑣).
From 𝑤_𝑛∈𝐶, we have
⟨𝑣−𝑤_𝑛,𝑤−𝐴𝑣⟩≥0. (3.62) On the other hand, since 𝑤_𝑛=Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛), then by Lemma 2.4, we have ⟨𝑣−𝑤_𝑛,𝐽𝑤_𝑛−(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛)⟩≥0, (3.63) and hence ⟨𝑣−𝑤_𝑛,𝐽𝑥_𝑛−𝐽𝑤_𝑛𝜆_𝑛−𝐴𝑥_𝑛⟩≤0. (3.64) It follows from (3.62) and (3.64), that ⟨𝑣−𝑤_𝑛,𝑤⟩≥⟨𝑣−𝑤_𝑛,𝐴𝑣⟩≥⟨𝑣−𝑤_𝑛,𝐴𝑣⟩+⟨𝑣−𝑤_𝑛,𝐽𝑥_𝑛−𝐽𝑤_𝑛𝜆_𝑛−𝐴𝑥_𝑛⟩=⟨𝑣−𝑤_𝑛,𝐴𝑣−𝐴𝑥_𝑛⟩+⟨𝑣−𝑤_𝑛,𝐽𝑥_𝑛−𝐽𝑤_𝑛𝜆_𝑛⟩=⟨𝑣−𝑤_𝑛,𝐴𝑣−𝐴𝑤_𝑛⟩+⟨𝑣−𝑤_𝑛,𝐴𝑤_𝑛−𝐴𝑥_𝑛⟩+⟨𝑣−𝑤_𝑛,𝐽𝑥_𝑛−𝐽𝑤_𝑛𝜆_𝑛⟩≥−‖𝑣−𝑤_𝑛‖‖𝑤_𝑛−𝑥_𝑛‖𝛼−‖𝑣−𝑤_𝑛‖‖𝐽𝑥_𝑛−𝐽𝑤_𝑛‖𝑎≥−𝑀(‖𝑤_𝑛−𝑥_𝑛‖𝛼+‖𝐽𝑥_𝑛−𝐽𝑤_𝑛‖𝑎). (3.65) Where 𝑀=sup_𝑛≥1‖𝑣−𝑤_𝑛‖. Taking the limit as 𝑛→∞ and (3.53), we obtain ⟨𝑣−𝑝,𝑤⟩≥0. By the maximality of 𝑇, we have 𝑝∈𝑇⁻¹0; that is, 𝑝∈VI(𝐴,𝐶).Step 9. We show that 𝑝=Π_Ω𝑥₀.
From 𝑥_𝑛=Π_𝐶𝑛𝑥₀, we have ⟨𝐽𝑥₀−𝐽𝑥_𝑛,𝑥_𝑛−𝑧⟩≥0, ∀𝑧∈𝐶_𝑛. Since Ω⊂𝐶_𝑛, we also have
⟨𝐽𝑥₀−𝐽𝑥_𝑛,𝑥_𝑛−𝑦⟩≥0,∀𝑦∈Ω. (3.66) By taking limit 𝑛→∞, we obtain that ⟨𝐽𝑥₀−𝐽𝑝,𝑝−𝑦⟩≥0,∀𝑦∈Ω. (3.67) By Lemma 2.4, we can conclude that 𝑝=Π_Ω𝑥₀ and 𝑥_𝑛→𝑝 as 𝑛→∞. This completes the proof.

Setting 𝑆≡𝑇 in Theorem 3.1., so, we obtain the following corollary.

Corollary 3.2. Let 𝐶 be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4) and let 𝐴 be an 𝛼-inverse-strongly monotone mapping of 𝐶 into 𝐸^∗ satisfying ‖𝐴𝑦‖≤‖𝐴𝑦−𝐴𝑢‖,forall𝑦∈𝐶 and 𝑢∈VI(𝐴,𝐶)≠∅. Let 𝑇∶𝐶→𝐶 be closed relatively quasi-nonexpansive mappings such that Ω∶=𝐹(𝑇)∩EP(𝑓)∩VI(𝐴,𝐶)≠∅. For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π_𝐶1𝑥₀ and 𝐶₁=𝐶, define a sequence {𝑥_𝑛} as follows: 𝑤_𝑛=Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛𝑧),_𝑛=𝐽⁻¹(𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑇𝑤_𝑛𝑦),_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑇𝑧_𝑛𝑢),_𝑛∈𝐶suchthat𝑓(𝑢_𝑛1,𝑦)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶⟩≥0,∀𝑦∈𝐶,_𝑛+1={𝑧∈𝐶_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝛼_𝑛𝜙(𝑧,𝑥_𝑛)+(1−𝛼_𝑛)𝜙(𝑧,𝑧_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑥)},_𝑛+1=_𝐶𝑛+1𝑥₀,∀𝑛≥1, (3.68) where 𝐽 is the duality mapping on 𝐸. Assume that {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1] such that 𝛼_𝑛≤1−𝛿₁ and 𝛽_𝑛≤1−𝛿₂, for some 𝛿₁,𝛿₂∈(0,1), {𝑟_𝑛}⊆(0,2𝛼), and {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<𝑐²𝛼/2, where 1/𝑐 is the 2-uniformly convexity constant of 𝐸. Then {𝑥_𝑛} converges strongly to 𝑝∈Ω, where 𝑝=Π_Ω𝑥₀.

If 𝐴≡0 in Theorem 3.1, then we obtain the following corollary.

Corollary 3.3. Let 𝐶 be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4). Let 𝑇,𝑆∶𝐶→𝐶 is closed relatively quasi-nonexpansive mappings such that Ω∶=𝐹(𝑇)∩𝐹(𝑆)∩EP(𝑓)≠∅. For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π_𝐶1𝑥₀ and 𝐶₁=𝐶, define a sequence {𝑥_𝑛} as follows: 𝑧_𝑛=𝐽⁻¹(𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑇𝑤_𝑛𝑦),_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑆𝑧_𝑛𝑢),_𝑛∈𝐶suchthat𝑓(𝑢_𝑛1,𝑦)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶⟩≥0,∀𝑦∈𝐶,_𝑛+1={𝑧∈𝐶_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝛼_𝑛𝜙(𝑧,𝑥_𝑛)+(1−𝛼_𝑛)𝜙(𝑧,𝑧_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑥)},_𝑛+1=_𝐶𝑛+1𝑥₀,∀𝑛≥1, (3.69) where 𝐽 is the duality mapping on 𝐸. Assume that {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1] such that 𝛼_𝑛≤1−𝛿₁ and 𝛽_𝑛≤1−𝛿₂, for some 𝛿₁,𝛿₂∈(0,1) and {𝑟_𝑛}⊆(0,2𝛼). Then {𝑥_𝑛} converges strongly to 𝑝∈Ω, where 𝑝=Π_Ω𝑥₀.

4. Application

4.1. Complementarity Problem

Let 𝐾 be a nonempty, closed and convex cone 𝐸, A a mapping of 𝐾 into 𝐸^∗. We define its polar in 𝐸^∗ to be the set

𝐾^∗={𝑦^∗∈𝐸^∗∶⟨𝑥,𝑦^∗⟩≥0,∀𝑥∈𝐾}. (4.1) Then the element 𝑢∈𝐾 is called a solution of the complementarity problem if

𝐴𝑢∈𝐾^∗,⟨𝑢,𝐴𝑢⟩=0. (4.2) The set of solutions of the complementarity problem is denoted by 𝐶(𝐾,𝐴).

Theorem 4.1. Let 𝐾 be a nonempty and closed convex subset of a 2-uniformly convex and uniformly smooth Banach space 𝐸. Let 𝑓 be a bifunction from 𝐾×𝐾 to ℝ satisfying (A1)–(A4) and let 𝐴 be an 𝛼-inverse-strongly monotone of 𝐸 into 𝐸^∗ satisfying ‖𝐴𝑦‖≤‖𝐴𝑦−𝐴𝑢‖,forall𝑦∈𝐾 and 𝑢∈𝐶(𝐾,𝐴)≠∅. Let 𝑇,𝑆∶𝐾→𝐾 be closed relatively quasi-nonexpansive mappings and Ω∶=𝐹(𝑇)∩𝐹(𝑆)∩EP(𝑓)∩𝐶(𝐾,𝐴)≠∅. For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π_𝐾1 and 𝐾₁=𝐾, we define the sequence {𝑥_𝑛} as follows: 𝑤_𝑛=Π_𝐾𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛𝑧),_𝑛=𝐽⁻¹(𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑇𝑤_𝑛𝑦),_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑆𝑧_𝑛𝑢),_𝑛∈𝐶suchthat𝑓(𝑢_𝑛1,𝑦)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶⟩≥0,∀𝑦∈𝐾,_𝑛+1={𝑧∈𝐶_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝛼_𝑛𝜙(𝑧,𝑥_𝑛)+(1−𝛼_𝑛)𝜙(𝑧,𝑧_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑥)},_𝑛+1=_𝐶𝑛+1𝑥₀,∀𝑛≥1, (4.3) where 𝐽 is the duality mapping on 𝐸, {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1] such that 𝛼_𝑛≤1−𝛿₁ and 𝛽_𝑛≤1−𝛿₂, for some 𝛿₁,𝛿₂∈(0,1), {𝑟_𝑛}⊆(0,2𝛼), and {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<𝑐²𝛼/2, where 1/𝑐 is the 2-uniformly convexity constant of 𝐸. Then {𝑥_𝑛} converges strongly to 𝑝∈Ω, where 𝑝=Π_Ω𝑥₀.

Proof. As in the proof of Takahashi in [7, Lemma 7.11], we get that VI(𝐾,𝐴)=𝐶(𝐾,𝐴). So, we obtain the result.

4.2. Approximation of a Zero of a Maximal Monotone Operator

Let 𝐵 be a multivalued mapping from 𝐸 to 𝐸^∗ with domain 𝐷(𝐵)={𝑧∈𝐸∶𝐴𝑧≠∅} and range 𝑅(𝐵)=∪{𝐵𝑧∶𝑧∈𝐷(𝐵)}. A mapping 𝐵 is said to be a monotone operator if ⟨𝑥₁−𝑥₂,𝑦₁−𝑦₂⟩≥0 for each 𝑥_𝑖∈𝐷(𝐵) and 𝑦_𝑖∈𝐴𝑥_𝑖,𝑖=1,2. A monotone operator 𝐵 is said to be maximal if its graph 𝐺(𝐵)={(𝑥,𝑦)∶𝑦∈𝐴𝑥} is not property contained in the graph of any other monotone operator. We know that if 𝐵 is a maximal monotone operator, then 𝐵⁻¹(0) is closed and convex. Let 𝐸 be a reflexive, strictly convex, and smooth Banach space, and let 𝐵 be a monotone operator from 𝐸 to 𝐸^∗, we know that 𝐵 is maximal if and only if 𝑅(𝐽+𝑟𝐵)=𝐸^∗ for all 𝑟>0. Let 𝐽_𝑟∶𝐸→𝐷(𝐵)be defined by 𝐽_𝑟=(𝐽+𝑟𝐵)⁻¹𝐽and such a 𝐽_𝑟 is called the resolvent of 𝐵. We know that 𝐽_𝑟 is a relatively nonexpansive (closed relatively quasi-nonexpansive for example; see [8]), and 𝐵⁻¹(0)=𝐹(𝐽_𝑟) for all 𝑟>0 (see [7, 33–35] for more details).

Theorem 4.2. Let 𝐶 be a nonempty and closed convex subset of a 2-uniformly convex and uniformly smooth Banach space 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4) and let 𝐴 be 𝛼-inverse-strongly monotone of 𝐸 into 𝐸^∗ satisfying ‖𝐴𝑦‖≤‖𝐴𝑦−𝐴𝑢‖,forall𝑦∈𝐶 and 𝑢∈VI(𝐴,𝐶)≠∅. Let 𝐵 be a maximal monotone operator of 𝐸 into 𝐸^∗ and let 𝐽_𝑟 be a resolvent of 𝐵 and a closed mapping such that Ω∶=𝐵⁻¹(0)∩𝐹(𝑆)∩EP(𝑓)∩VI(𝐴,𝐶)≠∅. For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π_𝐶1 and 𝐶₁=𝐶, we define the sequence {𝑥_𝑛} as follows: 𝑤_𝑛=Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛𝑧),_𝑛=𝐽⁻¹(𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝐽_𝑟𝑤_𝑛𝑦),_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑆𝑧_𝑛𝑢),_𝑛∈𝐶suchthat𝑓(𝑢_𝑛1,𝑦)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶⟩≥0,∀𝑦∈𝐶,_𝑛+1={𝑧∈𝐶_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝛼_𝑛𝜙(𝑧,𝑥_𝑛)+(1−𝛼_𝑛)𝜙(𝑧,𝑧_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑥)},_𝑛+1=_𝐶𝑛+1𝑥₀,∀𝑛≥1, (4.4) where 𝐽 is the duality mapping on 𝐸, {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1] such that 𝛼_𝑛≤1−𝛿₁ and 𝛽_𝑛≤1−𝛿₂, for some 𝛿₁,𝛿₂∈(0,1), {𝑟_𝑛}⊆(0,2𝛼) and {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<𝑐²𝛼/2, where 1/𝑐 is the 2-uniformly convexity constant of 𝐸. Then {𝑥_𝑛} converges strongly to 𝑝∈Ω, where 𝑝=Π_Ω𝑥₀.

Proof. Since 𝐽_𝑟 is a closed relatively nonexpansive mapping and 𝐵⁻¹0=𝐹(𝐽_𝑟). So, we obtain the result.

Corollary 4.3. Let 𝐶 be a nonempty and closed convex subset of a 2-uniformly convex and uniformly smooth Banach space 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4) and let 𝐴 be 𝛼-inverse-strongly monotone of 𝐸 into 𝐸^∗ satisfying ‖𝐴𝑦‖≤‖𝐴𝑦−𝐴𝑢‖,forall𝑦∈𝐶 and 𝑢∈VI(𝐴,𝐶)≠∅. Let 𝐵 be a maximal monotone operator of 𝐸 into 𝐸^∗ and let 𝐽_𝑟 be a resolvent of 𝐵 and closed such that Ω∶=𝐵⁻¹(0)∩EP(𝑓)∩VI(𝐴,𝐶)≠∅. For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π_𝐶1 and 𝐶₁=𝐶, we define the sequence {𝑥_𝑛} as follows: 𝑤_𝑛=Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛𝑧),_𝑛=𝐽⁻¹(𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝐽_𝑟𝑤_𝑛𝑦),_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝐽_𝑟𝑧_𝑛𝑢),_𝑛∈𝐶suchthat𝑓(𝑢_𝑛1,𝑦)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶⟩≥0,∀𝑦∈𝐶,_𝑛+1={𝑧∈𝐶_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝛼_𝑛𝜙(𝑧,𝑥_𝑛)+(1−𝛼_𝑛)𝜙(𝑧,𝑧_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑥)},_𝑛+1=_𝐶𝑛+1𝑥₀,∀𝑛≥1, (4.5) where 𝐽 is the duality mapping on 𝐸, {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1] such that 𝛼_𝑛≤1−𝛿₁ and 𝛽_𝑛≤1−𝛿₂, for some 𝛿₁,𝛿₂∈(0,1), {𝑟_𝑛}⊆(0,2𝛼) and {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<𝑐²𝛼/2, where 1/𝑐 is the 2-uniformly convexity constant of 𝐸. Then {𝑥_𝑛} converges strongly to 𝑝∈Ω, where 𝑝=Π_Ω𝑥₀.

Acknowledgments

The authors would like to thank the referee for the valuable suggestions on the manuscript. Siwaporn Saewan would like to thank the Office of the Higher Education Commission, Thailand for supporting by grant fund under the program Strategic Scholarships for Frontier Research Network for the Join Ph.D. Program Thai Doctoral degree for this research. Moreover, Poom Kumam was supported by the Thailand Research Fund and the Commission on Higher Education (MRG5180034).