Strong Convergence of a Hybrid Projection Algorithm for Equilibrium Problems, Variational Inequality Problems and Fixed Point Problems in a Banach Space

Abstract

We introduce and study a new hybrid projection algorithm for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed points of relatively quasi-nonexpansive mappings, and the set of solutions of the variational inequality for an inverse-strongly-monotone operator in a Banach space. Under suitable assumptions, we show a strong convergence theorem. Using this result, we obtain some applications in a Banach space. The results obtained in this paper extend and improve the several recent results in this area.

1. Introduction

Let 𝐸 be a real Banach space with norm ‖⋅‖ and 𝐸^∗ the dual space of 𝐸. Let 𝐶 be a nonempty closed and convex subset of 𝐸, and 𝐴 a monotone operator of 𝐶 into 𝐸^∗. A mapping 𝑇∶𝐶→𝐶 is called nonexpansive if ‖𝑇𝑥−𝑇𝑦‖<‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶. We denote by 𝐹(𝑇)={𝑥∈𝐸∶𝑇𝑥=𝑥} the set of fixed points of 𝑇. The classical variational inequality problem [1, 2], denoted by 𝑉𝐼(𝐶,𝐴), is to find 𝑢∈𝐶 such that ⟨𝐴𝑢,𝑣−𝑢⟩≥0 (1.1) for all 𝑣∈𝐶. One can see that the variational inequality problem (1.1) is connected with the convex minimization problem, the complementarity problem, the problem of finding a point 𝑢∈𝐸 satisfying 0=𝐴𝑢, and so on.

Recall that an operator 𝐴 is called monotone if ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥0 for all 𝑥,𝑦∈𝐶. An operator 𝐴 of 𝐶 into 𝐸^∗ is said to be 𝛼-inverse-strongly-monotone [3–5] if each 𝑥,𝑦∈𝐶. We have ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥𝛼‖𝐴𝑥−𝐴𝑦‖², for a constant 𝛼>0.

Assume that the following hold:

(C1)𝐴 is 𝛼-inverse-strongly-monotone,(C2)𝑉𝐼(𝐶,𝐴)≠∅,(C3)‖𝐴𝑦‖≤‖𝐴𝑦−𝐴𝑢‖ for all 𝑦∈𝐶 and 𝑢∈𝑉𝐼(𝐶,𝐴).

For finding a solution of the variational inequality problem for an operator 𝐴 that satisfies conditions (C1)–(C3) in a 2-uniformly convex and uniformly smooth Banach space 𝐸, Iiduka and Takahashi [6] introduced and studied the following algorithm: 𝑥₁=𝑥∈𝐶, define a sequence {𝑥_𝑛} by 𝑥_𝑛+1=Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛) (1.2) for every 𝑛=1,2,…, where 𝐽 is the duality mapping from 𝐸 into 𝐸^∗, Π_𝐶 is the generalized projection from 𝐸 onto 𝐶, and {𝜆_𝑛} is a sequence of positive real numbers. They proved that under certain appropriate conditions imposed on {𝜆_𝑛}, and 𝐽 is weakly sequentially continuous, the sequence {𝑥_𝑛} generated by (1.2) converges weakly to some element 𝑧 in 𝑉𝐼(𝐶,𝐴), where 𝑧=lim_𝑛→∞Π_{𝑉𝐼(𝐶,𝐴)}(𝑥_𝑛).

In 2004, Matsushita and Takahashi [7] introduced the following iteration: 𝑥₀∈𝐶 chosen arbitrarily, 𝑥_𝑛+1=Π_𝐶𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑇𝑥_𝑛), (1.3) where {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 prove that the sequence {𝑥_𝑛} generated by (1.3) converges weakly to a fixed point of 𝑇.

The problem of finding a common element of the set of the variational inequalities for monotone mappings in the framework of Hilbert spaces and Banach spaces has been intensively studied by many authors; see, for instance, [3–5] and the references cited therein.

Let 𝑓 be a bifunction of 𝐶×𝐶 into ℝ, where ℝ is the set of real numbers. The equilibrium problem for 𝑓∶𝐶×𝐶→ℝ is to find 𝑥∈𝐶 such that 𝑓(𝑥,𝑦)≥0 (1.4) for all 𝑦∈𝐶. The set of solutions of (1.4) is denoted by 𝐸𝑃(𝑓), that is, 𝐸𝑃(𝑓)={𝑥∈𝐶∶𝑓(𝑥,𝑦)≥0forall𝑦∈𝐶}. 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. Many problems in physics, optimization, and economics reduce to finding a solution of (1.4). Equilibrium problems have been studied extensively; see, for instance, [8, 9]. Combettes and Hirstoaga [8] introduced an iterative scheme for finding the best approximation to the initial data when 𝐸𝑃(𝑓) is nonempty and proved a strong convergence theorem.

For solving the equilibrium problem, let us assume that a bifunction 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 all 𝑥,𝑦,𝑧∈𝐶, limsup_𝑡↓0𝑓(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝑓(𝑥,𝑦); (1.5)(A4)for all 𝑥∈𝐶,𝑓(𝑥,⋅) is convex and lower semicontinuous.

The problem of finding a common element of the set of fixed points and the set of solutions of an equilibrium problem in the framework of Hilbert spaces and Banach spaces has been studied by many authors; see [7, 8, 10–18].

In 2008, Takahashi and Zembayashi [15] introduced the shrinking projection method which is the modification of (1.3) for a relatively nonexpansive mapping. It is given as follows: 𝑥₀=𝑥∈𝐶,𝐶₀𝑦=𝐶,_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑇𝑥_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=Π_𝐶𝑛+1𝑥₀ (1.6) for every 𝑛∈ℕ∪{0}, where 𝐽 is the duality mapping on 𝐸,{𝛼_𝑛}⊂[0,1] satisfying liminf_𝑛→∞𝛼_𝑛(1−𝛼_𝑛)>0 and {𝑟_𝑛}⊂[𝑠,∞) for some 𝑠>0. They proved that the sequence {𝑥_𝑛} generated by (1.6) converges strongly to 𝑞=Π_{𝐹(𝑇)∩𝐸𝑃(𝑓)}𝑥₀, where Π_{𝐹(𝑇)∩𝐸𝑃(𝑓)} is the generalized projection of 𝐸 onto 𝐹(𝑇)∩𝐸𝑃(𝑓).

In the same year, Qin et al. [19] extend the iteration process (1.6) from a single relatively nonexpansive mapping to two relatively quasi-nonexpansive mappings: 𝑥₀∈𝐸,𝐶₁=𝐶 and 𝑥₁=Π_𝐶1𝑥₀,𝑦_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+𝛽_𝑛𝐽𝑇𝑥_𝑛+𝛾_𝑛𝐽𝑆𝑥_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=Π_𝐶𝑛+1𝑥₀ (1.7) for every 𝑛∈ℕ∪{0}. Under appropriate conditions imposed on {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛}, and {𝑟_𝑛}, they obtain that the sequence {𝑥_𝑛} generated by (1.7) converges strongly to 𝑞=Π_{𝐹(𝑇)∩𝐹(𝑆)∩𝐸𝑃(𝑓)}𝑥₀.

In 2009, Wattanawitoon and Kumam [17] introduced the following iterative scheme which is the modification of (1.6) and (1.7) in a Banach space: 𝑥₀∈𝐸,𝐶₁=𝐶, and 𝑥₁=Π_𝐶1𝑥₀,𝑦_𝑛=𝐽⁻¹(𝛿_𝑛𝐽𝑥_𝑛+(1−𝛿_𝑛)𝐽𝑧_𝑛𝑧),_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+𝛽_𝑛𝐽𝑇𝑥_𝑛+𝛾_𝑛𝐽𝑆𝑥_𝑛𝑢),_𝑛∈𝐶suchthat𝑓(𝑢_𝑛1,𝑦)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑧_𝑛𝐶⟩≥0,∀𝑦∈𝐶,_𝑛+1={𝑧∈𝐶_𝑛∶𝜙(𝑧,𝑦_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑥)},_𝑛+1=Π_𝐶𝑛+1𝑥₀ (1.8) for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of two relatively quasi-nonexpansive mappings in a Banach space. They proved that under certain appropriate conditions imposed on {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝛿_𝑛}, and {𝑟_𝑛}, the sequences {𝑥_𝑛} and {𝑢_𝑛} generated by (1.8) converge strongly to 𝑞∈𝐹(𝑇)∩𝐹(𝑆)∩𝐸𝑃(𝑓), where 𝑞=Π_{𝐹(𝑇)∩𝐹(𝑆)∩𝐸𝑃(𝑓)}𝑥₀.

For finding common elements of the set of the equilibrium problem, the set of the variational inequality problem for an inverse-strongly-monotone operator and the set of common fixed points for relatively quasi-nonexpansive mappings. Cholamjiak [20] introduced an iterative scheme by using the new hybrid method in a Banach space. The scheme is defined as follows: 𝑥₀∈𝐸,𝐶₁=𝐶 and 𝑥₁∈Π_𝐶1𝑥₀,, 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=Π_𝐶𝑛+1𝑥₀ (1.9) for every 𝑛∈ℕ, where 𝐽 is the duality mapping on 𝐸 and {𝛼_𝑛},{𝛽_𝑛}, and {𝛾_𝑛} are sequences in [0,1]. He proved that under certain appropriate conditions imposed on {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝑟_𝑛}, and {𝜆_𝑛}, the sequences {𝑥_𝑛} and {𝑢_𝑛} generated by (1.9) converge strongly to 𝑞=Π_{𝐹(𝑇)∩𝐹(𝑆)∩𝐸𝑃(𝑓)∩𝑉𝐼(𝐶,𝐴)}𝑥₀.

Motivated by the recent works, we introduce an iterative scheme by a new hybrid method as follows: 𝑥₀∈𝐸,𝐶₁=𝐶 and 𝑥₁=Π_𝐶1𝑥₀, 𝑤_𝑛=Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛𝑦),_𝑛=𝐽⁻¹(𝛿_𝑛𝐽𝑥_𝑛+(1−𝛿_𝑛)𝐽𝑤_𝑛𝑧),_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+𝛽_𝑛𝐽𝑇𝑥_𝑛+𝛾_𝑛𝐽𝑆𝑦_𝑛𝑢),_𝑛∈𝐶suchthat𝑓(𝑢_𝑛1,𝑦)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑧_𝑛𝐶⟩≥0,∀𝑦∈𝐶,_𝑛+1={𝑧∈𝐶_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑥)},_𝑛+1=Π_𝐶𝑛+1𝑥₀ (1.10) for every 𝑛∈ℕ, where 𝐽 is the duality mapping on 𝐸, Π_𝐶 is the generalized projection from 𝐸 onto a closed convex subset 𝐶 of 𝐸, 𝑓 is an equilibrium bifunction satisfying (A1)–(A4), 𝐴 is an operator of 𝐶 into 𝐸^∗ satisfying (C1)–(C3), 𝑇,𝑆∶𝐶→𝐶 are two closed relatively quasi-nonexpansive mappings, {𝛼_𝑛},{𝛽_𝑛,{𝛾_𝑛}, and {𝛿_𝑛} are sequences in [0,1] such that 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1,limsup_𝑛→∞𝛿_𝑛<1, liminf_𝑛→∞𝛼_𝑛𝛽_𝑛>0, and liminf_𝑛→∞𝛼_𝑛𝛾_𝑛>0,{𝑟_𝑛}⊂[𝑠,∞) for some 𝑠>0 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. We prove that the sequences {𝑥_𝑛} and {𝑢_𝑛} generated by the above iterative scheme converge strongly to 𝑞=Π_{𝐹(𝑇)∩𝐹(𝑆)∩𝐸𝑃(𝑓)∩𝑉𝐼(𝐶,𝐴)}𝑥₀.

2. Preliminaries

In this section, we recall some well know concepts and results.

Let 𝐸 be a real Banach space with dimension 𝐸≥2. The modulus of 𝐸 is the function 𝛿_𝐸∶(0,2]→[0,1] defined by 𝛿_𝐸(𝜀)=inf{1−‖𝑥+𝑦2‖∶𝑥,𝑦∈𝐸,‖𝑥‖=‖𝑦‖=1,‖𝑥−𝑦‖≥𝜀}. (2.1) Banach space 𝐸 is uniformly convex if and only if 𝛿_𝐸(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 [21–23] for more details. A Banach space 𝐸 is said to be smooth if the limit lim_𝑡→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡 (2.2) exists for all 𝑥,𝑦∈𝑈, where 𝑈 denotes the unit sphere of 𝐸 (i.e., 𝑈={𝑥∈𝐸∶‖𝑥‖=1}). It is also said to be uniformly smooth if the limit (2.2) is attained uniformly for 𝑥,𝑦∈𝑈. One should note that no Banach space is 𝑝-uniformly convex for 1<𝑝<2; see [23]. 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} (2.3) 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. See [24, 25] for more details.

Let 𝐸 be a smooth, strictly convex, and reflexive Banach space and let 𝐶 be a nonempty closed convex subset of 𝐸. We denote by 𝜙 the function defined by 𝜙(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖² (2.4) for all 𝑥,𝑦∈𝐸. Following Alber [26], 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). (2.5) The existence and uniqueness of the operator Π_𝐶 follows from the properties of the functional 𝜙(𝑥,𝑦) and strict monotonicity of the mapping 𝐽 (see, e.g., [24, 26–29]. In Hilbert space, Π_𝐶=𝑃_𝐶. It is obvious from the definition of function 𝜙 that (‖𝑦‖−‖𝑥‖)²≤𝜙(𝑦,𝑥)=(‖𝑦‖+‖𝑥‖)² (2.6) for all 𝑥,𝑦∈𝐸. If 𝐸 is a Hilbert space, then 𝜙(𝑥,𝑦)=‖𝑥−𝑦‖².

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 (2.6), 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 [24, 28] for more details.

Recall that a point 𝑝 in a closed convex subset 𝐶 of 𝐸 is said to be an asymptotic fixed point of 𝑇 [30] 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 point of 𝑇 will be denoted by 𝐹(𝑇). A mapping 𝑇∶𝐶→𝐶 is called relatively nonexpansive [11, 31–33] if 𝑇 satisfies the following conditions:

(1)𝐹(𝑇)≠∅;(2)𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥) for all 𝑝∈𝐹(𝑇) and 𝑥∈𝐶;(3)𝐹(𝑇)=𝐹(𝑇).

The asymptotic behavior of a relatively nonexpansive mapping was studied in [31–33].

A mapping 𝑇 is said to be relatively quasi-nonexpansive if 𝑇 satisfies conditions (1) and (2). It is easy to see that the class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings [11, 31–33] which requires the strong restriction: 𝐹(𝑇)=𝐹(𝑇).

We give some examples which are closed relatively quasi-nonexpansive; see [19].

Example 2.1. Let 𝐸 be a uniformly smooth and strictly convex Banach space and 𝐴⊂𝐸×𝐸^∗ be a maximal monotone mapping such that its zero set 𝐴⁻¹0≠∅. Then, 𝐽_𝑟=(𝐽+𝑟𝐴)⁻¹𝐽 is a closed relatively quasi-nonexpansive mapping from 𝐸 onto 𝐷(𝐴) and 𝐹(𝐽_𝑟)=𝐴⁻¹0.

Example 2.2. Let Π_𝐶 be the generalized projection from a smooth, strictly convex, and reflexive Banach space 𝐸 onto a nonempty closed convex subset 𝐶 of 𝐸. Then, Π_𝐶 is a closed relatively quasi-nonexpansive mapping with 𝐹(Π_𝐶)=𝐶.

An operator 𝐴 of 𝐶 into 𝐸^∗ 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}. (2.7)

Let 𝐶 be a nonempty, closed convex subset of a Banach space 𝐸 and 𝐴 a monotone, hemicontinuous operator of 𝐶 into 𝐸^∗. Let 𝑇_𝑒⊂𝐸×𝐸^∗ be an operator defined as follows: 𝑇_𝑒𝑣=𝐴𝑣+𝑁_𝐶(𝑣),𝑣∈𝐶,∅,otherwise. (2.8) Then, 𝑇_𝑒 is maximal monotone and 𝑇_𝑒⁻¹0=𝑉𝐼(𝐶,𝐴); see [34].

In the sequel, the following lemmas are needed to prove our main results.

Lemma 2.3 ([22, 35]). Let 𝑝 be a given real number with 𝑝≥2 and 𝐸 a 𝑝-uniformly convex Banach space. Then, for all 𝑥,𝑦∈𝐸, 𝑗_𝑥∈𝐽_𝑝(𝑥) and 𝑗_𝑦∈𝐽_𝑝(𝑦), ⟨𝑥−𝑦,𝑗_𝑥−𝑗_𝑦𝑐⟩≥^𝑝2^𝑝−2𝑝‖𝑥−𝑦‖^𝑝, (2.9) where 𝐽_𝑝 is the generalized duality mapping of 𝐸 and 1/𝑐 is the 𝑝-uniformly convexity constant of 𝐸.

Lemma 2.4 ([29]). 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 2.5 ([26]). Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝐸 and 𝑥∈𝐸. Then 𝑥₀=Π_𝐶𝑥 if and only if ⟨𝑥₀−𝑦,𝐽𝑥−𝐽𝑥₀⟩≥0 for all 𝑦∈𝐶.

Lemma 2.6 ([26]). Let 𝐶 be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space 𝐸 and let 𝑥∈𝐸. Then 𝜙(𝑦,Π_𝐶𝑥)+𝜙(Π_𝐶𝑥,𝑥)≤𝜙(𝑦,𝑥),∀𝑦∈𝐶. (2.10)

Lemma 2.7 ([19]). Let 𝐸 be a uniformly convex, smooth Banach space, let 𝐶 be a closed convex subset of 𝐸, let 𝑇 be a closed and relatively quasi-nonexpansive mapping from 𝐶 into itself. Then 𝐹(𝑇) is a closed convex subset of 𝐶.

Lemma 2.8 ([36]). Let 𝐸 be a uniformly convex Banach space and 𝐵_𝑟(0) be a closed ball of 𝐸. Then there exists a continuous strictly increasing convex function 𝑔∶[0,∞)→[0,∞) with 𝑔(0)=0 such that ‖𝛼𝑥+𝛽𝑦+𝛾𝑧‖²≤𝛼‖𝑥‖²+𝛽‖𝑦‖²+𝛾‖𝑧‖²−𝛼𝛽𝑔(‖𝑥−𝑦‖) (2.11) for all 𝑥,𝑦,𝑧∈𝐵_𝑟(0) and 𝛼,𝛽,𝛾∈[0,1] with 𝛼+𝛽+𝛾=1.

Lemma 2.9 ([10]). 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 1𝑓(𝑧,𝑦)+𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶. (2.12)

Lemma 2.10 ([19]). Let 𝐶 be a closed 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 all 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}. (2.13)

Then, the following hold:

(1)𝑇_𝑟 is single-valued;(2)𝑇_𝑟 is a firmly nonexpansive-type mapping [37], that is, for all 𝑥,𝑦∈𝐸, ⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽𝑇_𝑟𝑥−𝐽𝑇_𝑟𝑦⟩≤⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽𝑥−𝐽𝑦⟩; (2.14)(3)𝐹(𝑇_𝑟)=𝐸𝑃(𝑓);(4)𝐸𝑃(𝑓) is closed and convex.

Lemma 2.11 ([16]). Let 𝐶 be a closed convex subset of a smooth, strictly, and reflexive Banach space 𝐸, let 𝑓 be a bifucntion from 𝐶×𝐶 to 𝑅 satisfying (A1)–(A4), let 𝑟>0. Then, for all 𝑥∈𝐸 and 𝑞∈𝐹(𝑇_𝑟), 𝜙(𝑞,𝑇_𝑟𝑥)+𝜙(𝑇_𝑟𝑥,𝑥)≤𝜙(𝑞,𝑥). (2.15)

We make use of the following mapping 𝑉 studied in Alber [26]: 𝑉(𝑥,𝑥^∗)=‖𝑥‖²−2⟨𝑥,𝑥^∗⟩+‖𝑥^∗‖² (2.16) for all 𝑥∈𝐸 and 𝑥^∗∈𝐸^∗, that is, 𝑉(𝑥,𝑥^∗)=𝜙(𝑥,𝐽⁻¹(𝑥^∗)).

Lemma 2.12 ([26]). Let 𝐸 be a reflexive, strictly convex, smooth Banach space and let 𝑉 be as in (2.16). Then 𝑉(𝑥,𝑥^∗)+2⟨𝐽⁻¹(𝑥^∗)−𝑥,𝑦^∗⟩≤𝑉(𝑥,𝑥^∗+𝑦^∗) (2.17) for all 𝑥∈𝐸 and 𝑥^∗,𝑦^∗∈𝐸^∗.

3. Main Results

In this section, we prove strong convergence theorems by hybrid methods which solves the problem of finding a common element of the set of solutions of an equilibrium problem, the set of common fixed points of relatively quasi-nonexpansive mappings and the set of solutions of the variational inequality of an 𝛼-inverse-strongly-monotone mapping in a 2-uniformly convex, uniformly smooth Banach space.

Theorem 3.1. Let 𝐸 be a 2-uniformly convex, uniformly smooth Banach space, 𝐶 a nonempty closed convex subset of 𝐸, 𝑓 a bifunction from 𝐶×𝐶 to ℝ which satisfies (A1)–(A4), 𝐴 an operator of 𝐶 into 𝐸^∗ satisfying (C1)–(C3), and 𝑇,𝑆 two closed relatively quasi-nonexpansive mappings from 𝐶 into itself such that the set 𝐹∶=𝐹(𝑇)∩𝐹(𝑆)∩𝐸𝑃(𝑓)∩𝑉𝐼(𝐶,𝐴)≠∅. For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π_𝐶1𝑥₀ and 𝐶₁=𝐶, define sequences {𝑥_𝑛} and {𝑢_𝑛} of 𝐶 as follows: 𝑤_𝑛=Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛𝑦),_𝑛=𝐽⁻¹(𝛿_𝑛𝐽𝑥_𝑛+(1−𝛿_𝑛)𝐽𝑤_𝑛𝑧),_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+𝛽_𝑛𝐽𝑇𝑥_𝑛+𝛾_𝑛𝐽𝑆𝑦_𝑛𝑢),_𝑛∈𝐶suchthat𝑓(𝑢_𝑛1,𝑦)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑧_𝑛𝐶⟩≥0,∀𝑦∈𝐶,_𝑛+1={𝑧∈𝐶_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑥)},_𝑛+1=Π_𝐶𝑛+1𝑥₀ (3.1) for every 𝑛∈ℕ, where 𝐽 is the duality mapping on 𝐸. Suppose that {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛}, and {𝛿_𝑛} are sequences in [0,1] satisfying the restrictions:(B1)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1;(B2)limsup_𝑛→∞𝛿_𝑛<1, liminf_𝑛→∞𝛼_𝑛𝛽_𝑛>0 and liminf_𝑛→∞𝛼_𝑛𝛾_𝑛>0;(B3){𝑟_𝑛}⊂[𝑠,∞) for some 𝑠>0;(B4){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} and {𝑢_𝑛} converge strongly to 𝑞=Π_𝐹𝑥₀.

Proof. We divide the proof into eight steps.Step 1. Show that Π_𝐹𝑥₀ and Π_𝐶𝑛+1𝑥₀ are well defined.
It is obvious that 𝑉𝐼(𝐶,𝐴) is a closed convex subset of 𝐶. By Lemma 2.7, we know that 𝐹(𝑇)∩𝐹(𝑆) is closed and convex. From Lemma 2.10 (4), we also have 𝐸𝑃(𝑓) is closed and convex. Hence, 𝐹∶=𝐹(𝑇)∩𝐹(𝑆)∩𝐸𝑃(𝑓)∩𝑉𝐼(𝐶,𝐴) is a nonempty, closed, and convex subset of 𝐶; consequently, Π_𝐹𝑥₀ is well defined.
Next, we show that 𝐶_𝑛 is closed and convex for all 𝑛∈ℕ.
It is obvious that, 𝐶₁=𝐶 is closed and convex. Suppose that 𝐶_𝑘 is closed and convex for some 𝑘∈ℕ. For all 𝑧∈𝐶_𝑘, we know that 𝜙(𝑧,𝑢_𝑘)≤𝜙(𝑧,𝑥_𝑘) is equivalent to 2⟨𝑧,𝐽𝑥_𝑘−𝐽𝑢_𝑘⟩≤‖𝑥_𝑘‖²−‖𝑢_𝑘‖². (3.2) So, 𝐶_𝑘+1 is closed and convex. Then, for any 𝑛∈ℕ, 𝐶_𝑛 is closed and convex. This implies that Π_𝐶𝑛+1𝑥₀ is well defined.Step 2. We prove by induction that 𝐹⊂𝐶_𝑛 for all 𝑛∈ℕ.
Putting 𝑣_𝑛=𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛). First, we observe that 𝑢_𝑛=𝑇_𝑟𝑛𝑧_𝑛 for all 𝑛∈ℕ and 𝐹⊂𝐶₁=𝐶. Suppose that 𝐹⊂𝐶_𝑘 for some 𝑘∈ℕ. Let 𝑢∈𝐹⊂𝐶_𝑘. From Lemmas 2.6 and 2.12, we have 𝜙(𝑢,𝑤_𝑘)=𝜙(𝑢,Π_𝐶𝑣_𝑘)≤𝜙(𝑢,𝑣_𝑘)=𝜙(𝑢,𝐽⁻¹(𝐽𝑥_𝑘−𝜆_𝑘𝐴𝑥_𝑘))=𝑉(𝑢,𝐽𝑥_𝑘−𝜆_𝑘𝐴𝑥_𝑘)≤𝑉(𝑢,(𝐽𝑥_𝑘−𝜆_𝑘𝐴𝑥_𝑘)+𝜆_𝑘𝐴𝑥_𝑘)−2⟨𝐽⁻¹(𝐽𝑥_𝑘−𝜆_𝑘𝐴𝑥_𝑘)−𝑢,𝜆_𝑘𝐴𝑥_𝑘⟩=𝑉(𝑢,𝐽𝑥_𝑘)−2𝜆_𝑘⟨𝑣_𝑘−𝑢,𝐴𝑥_𝑘⟩=𝜙(𝑢,𝑥_𝑘)−2𝜆_𝑘⟨𝑥_𝑘−𝑢,𝐴𝑥_𝑘⟩+2⟨𝑣_𝑘−𝑥_𝑘,−𝜆_𝑘𝐴𝑥_𝑘⟩. (3.3) Using (C1) and 𝑢∈𝑉𝐼(𝐶,𝐴), we have −2𝜆_𝑘⟨𝑥_𝑘−𝑢,𝐴𝑥_𝑘⟩=−2𝜆_𝑘⟨𝑥_𝑘−𝑢,𝐴𝑥_𝑘−𝐴𝑢+𝐴𝑢⟩=−2𝜆_𝑘⟨𝑥_𝑘−𝑢,𝐴𝑥_𝑘−𝐴𝑢⟩−2𝜆_𝑘⟨𝑥_𝑘−𝑢,𝐴𝑢⟩≤−2𝛼𝜆_𝑘‖𝐴𝑥_𝑘−𝐴𝑢‖². (3.4) By using Lemma 2.3 and (C3), we have 2⟨𝑣_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≤4‖𝑐²‖𝐽𝐽⁻¹(𝐽𝑥_𝑘−𝜆_𝑘𝐴𝑥_𝑘)−𝐽𝐽⁻¹(𝐽𝑥_𝑘)‖‖𝜆_𝑘𝐴𝑥_𝑘=4‖𝑐²‖𝐽𝑥_𝑘−𝜆_𝑘𝐴𝑥_𝑘−𝐽𝑥_𝑘‖‖𝜆_𝑘𝐴𝑥_𝑘=4‖𝑐²𝜆²_𝑘‖𝐴𝑥_𝑘‖²≤4𝑐²𝜆²_𝑘‖𝐴𝑥_𝑘−𝐴𝑢‖². (3.5)(1)(2)(3)(4)(5) Replacing (3.4) and (3.5) into (3.3) and using (B4), we get 𝜙(𝑢,𝑤_𝑘)≤𝜙(𝑢,𝑥_𝑘2)+2𝑎(𝑐²𝑏−𝛼)‖𝐴𝑥_𝑘−𝐴𝑢‖²≤𝜙(𝑢,𝑥_𝑘). (3.6)
By the convexity of ‖⋅‖² and (3.6), for each 𝑢∈𝐹⊂𝐶_𝑘, we have 𝜙(𝑢,𝑦_𝑘)=𝜙(𝑢,𝐽⁻¹(𝛿_𝑘𝐽𝑥_𝑘+(1−𝛿_𝑘)𝐽𝑤_𝑘))=𝑉(𝑢,𝛿_𝑘𝐽𝑥_𝑘+(1−𝛿_𝑘)𝐽𝑤_𝑘)=‖𝑢‖²−2⟨𝑢,𝛿_𝑘𝐽𝑥_𝑘+(1−𝛿_𝑘)𝐽𝑤_𝑘⟩+‖𝛿_𝑘𝐽𝑥_𝑘+(1−𝛿_𝑘)𝐽𝑤_𝑘‖²≤‖𝑢‖²−2𝛿_𝑘⟨𝑢,𝐽𝑥_𝑘⟩−2(1−𝛿_𝑘)⟨𝑢,𝐽𝑤_𝑘⟩+𝛿_𝑘‖𝐽𝑥_𝑘‖²+(1−𝛿_𝑘)‖𝐽𝑤_𝑘‖²=(𝛿_𝑘+(1−𝛿_𝑘))‖𝑢‖²−2𝛿_𝑘⟨𝑢,𝐽𝑥_𝑘⟩−2(1−𝛿_𝑘)⟨𝑢,𝐽𝑤_𝑘⟩+𝛿_𝑘‖𝐽𝑥_𝑘‖²+(1−𝛿_𝑘)‖𝐽𝑤_𝑘‖²=𝛿_𝑘(‖𝑢‖²−2⟨𝑢,𝐽𝑥_𝑘⟩+‖𝐽𝑥_𝑘‖²)+(1−𝛿_𝑘)(‖𝑢‖²−2⟨𝑢,𝐽𝑤_𝑘⟩+‖𝐽𝑤_𝑘‖²)=𝛿_𝑘𝜙(𝑢,𝑥_𝑘)+(1−𝛿_𝑘)𝜙(𝑢,𝑤_𝑘)≤𝛿_𝑘𝜙(𝑢,𝑥_𝑘)+(1−𝛿_𝑘)𝜙(𝑢,𝑥_𝑘)=𝜙(𝑢,𝑥_𝑘), (3.7) and so 𝜙(𝑢,𝑢_𝑘)=𝜙(𝑢,𝑇_𝑟𝑘𝑧_𝑘)≤𝜙(𝑢,𝑧_𝑘)=𝜙(𝑢,𝐽⁻¹(𝛼_𝑘𝐽𝑥_𝑘+𝛽_𝑘𝐽𝑇𝑥_𝑘+𝛾_𝑘𝐽𝑆𝑦_𝑘))=𝑉(𝑢,𝛼_𝑘𝐽𝑥_𝑘+𝛽_𝑘𝐽𝑇𝑥_𝑘+𝛾_𝑘𝐽𝑆𝑦_𝑘)=‖𝑢‖²−2𝛼_𝑘⟨𝑢,𝐽𝑥_𝑘⟩−2𝛽_𝑘⟨𝑢,𝐽𝑇𝑥_𝑘⟩−2𝛾_𝑘⟨𝑢,𝐽𝑆𝑦_𝑘⟩+‖𝛼_𝑘𝐽𝑥_𝑘+𝛽_𝑘𝐽𝑇𝑥_𝑘+𝛾_𝑘𝐽𝑆𝑦_𝑘‖²≤‖𝑢‖²−2𝛼_𝑘⟨𝑢,𝐽𝑥_𝑘⟩−2𝛽_𝑘⟨𝑢,𝐽𝑇𝑥_𝑘⟩−2𝛾_𝑘⟨𝑢,𝐽𝑆𝑦_𝑘⟩+𝛼_𝑘‖𝐽𝑥_𝑘‖²+𝛽_𝑘‖𝐽𝑇𝑥_𝑘‖²+𝛾_𝑘‖𝐽𝑆𝑦_𝑘‖²=(𝛼_𝑘+𝛽_𝑘+𝛾_𝑘)‖𝑢‖²−2𝛼_𝑘⟨𝑢,𝐽𝑥_𝑘⟩−2𝛽_𝑘⟨𝑢,𝐽𝑇𝑥_𝑘⟩−2𝛾_𝑘⟨𝑢,𝐽𝑆𝑦_𝑘⟩+𝛼_𝑘‖𝐽𝑥_𝑘‖²+𝛽_𝑘‖𝐽𝑇𝑥_𝑘‖²+𝛾_𝑘‖𝐽𝑆𝑦_𝑘‖²=𝛼_𝑘(‖𝑢‖²−2⟨𝑢,𝐽𝑥_𝑘⟩+‖𝐽𝑥_𝑘‖²)+𝛽_𝑘(‖𝑢‖²−2⟨𝑢,𝐽𝑇𝑥_𝑘⟩+‖𝐽𝑇𝑥_𝑘‖²)+𝛾_𝑘(‖𝑢‖²−2⟨𝑢,𝐽𝑆𝑦_𝑘⟩+‖𝐽𝑆𝑦_𝑘‖²)=𝛼_𝑘𝜙(𝑢,𝑥_𝑘)+𝛽_𝑘𝜙(𝑢,𝑇𝑥_𝑘)+𝛾_𝑘𝜙(𝑢,𝑆𝑦_𝑘)≤𝛼_𝑘𝜙(𝑢,𝑥_𝑘)+𝛽_𝑘𝜙(𝑢,𝑥_𝑘)+𝛾_𝑘𝜙(𝑢,𝑦_𝑘)≤𝛼_𝑘𝜙(𝑢,𝑥_𝑘)+𝛽_𝑘𝜙(𝑢,𝑥_𝑘)+𝛾_𝑘𝜙(𝑢,𝑥_𝑘)=(𝛼_𝑘+𝛽_𝑘+𝛾_𝑘)𝜙(𝑢,𝑥_𝑘)=𝜙(𝑢,𝑥_𝑘). (3.8)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) This shows that 𝑢∈𝐶_𝑘+1 and hence 𝐹⊂𝐶_𝑘+1. This implies that 𝐹⊂𝐶_𝑛 for all 𝑛∈ℕ.Step 3. Show that lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₀) exists.
From 𝑥_𝑛=Π_𝐶𝑛𝑥₀ and 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1⊂𝐶_𝑛, we have 𝜙(𝑥_𝑛,𝑥₀)≤𝜙(𝑥_𝑛+1,𝑥₀),∀𝑛∈ℕ. (3.9) Therefore, {𝜙(𝑥_𝑛,𝑥₀)} is nondecreasing. From Lemma 2.6, we have 𝜙(𝑥_𝑛,𝑥₀)=𝜙(Π_𝐶𝑛𝑥₀,𝑥₀)≤𝜙(𝑢,𝑥₀)−𝜙(𝑢,𝑥_𝑛)≤𝜙(𝑢,𝑥₀). (3.10) Then the sequence {𝜙(𝑥_𝑛,𝑥₀)} is bounded. It follows that lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₀) exists.Step 4. Show that {𝑥_𝑛} is a Cauchy sequence in 𝐶.
Since 𝑥_𝑚=Π_𝐶𝑚𝑥₀∈𝐶_𝑚⊂𝐶_𝑛 for any positive integer 𝑚≥𝑛, by Lemma 2.6, we also 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). (3.11) Letting 𝑚,𝑛→∞ in (3.11), we have 𝜙(𝑥_𝑚,𝑥_𝑛)→0. It follows from Lemma 2.4 that ‖𝑥_𝑚−𝑥_𝑛‖→0 as 𝑚,𝑛→∞. Hence, {𝑥_𝑛} is a Cauchy sequence. By the completeness of 𝐸 and the closedness of 𝐶, one can assume that 𝑥_𝑛→𝑞∈𝐶 as 𝑛→∞. Further, we obtain lim_𝑛→∞𝜙(𝑥_𝑛+1,𝑥_𝑛)=0. (3.12) Since 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1, we have 𝜙(𝑥_𝑛+1,𝑢_𝑛)≤𝜙(𝑥_𝑛+1,𝑥_𝑛)→0(as𝑛→∞). (3.13) Applying Lemma 2.4 to (3.12) and (3.13), we have ‖𝑢_𝑛−𝑥_𝑛‖=‖𝑢_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛‖≤‖𝑢_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑥_𝑛‖→0(as𝑛→∞). (3.14) This implies that 𝑢_𝑛→𝑞 as 𝑛→∞. Since 𝐽 is uniformly norm-to-norm continuous on bounded subsets of 𝐸, we also obtain lim_𝑛→∞‖𝐽𝑢_𝑛−𝐽𝑥_𝑛‖=0. (3.15)Step 5. Show that 𝑥_𝑛→𝑞∈𝐹(𝑇)∩𝐹(𝑆).
Let 𝑟=sup_𝑛≥1{‖𝑥_𝑛‖,‖𝑇𝑥_𝑛‖,‖𝑆𝑤_𝑛‖}. From (3.7) and Lemma 2.8, we obtain 𝜙(𝑢,𝑢_𝑛)=𝜙(𝑢,𝑇_𝑟𝑛𝑧_𝑛)≤𝜙(𝑢,𝑧_𝑛)=𝜙(𝑢,𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+𝛽_𝑛𝐽𝑇𝑥_𝑛+𝛾_𝑛𝐽𝑆𝑦_𝑛))=𝑉(𝑢,𝛼_𝑛𝐽𝑥_𝑛+𝛽_𝑛𝐽𝑇𝑥_𝑛+𝛾_𝑛𝐽𝑆𝑦_𝑛)=‖𝑢‖²−2𝛼_𝑛⟨𝑢,𝐽𝑥_𝑛⟩−2𝛽_𝑛⟨𝑢,𝐽𝑇𝑥_𝑛⟩−2𝛾_𝑛⟨𝑢,𝐽𝑆𝑦_𝑛⟩+‖𝛼_𝑛𝐽𝑥_𝑛+𝛽_𝑛𝐽𝑇𝑥_𝑛+𝛾_𝑛𝐽𝑆𝑦_𝑛‖²≤‖𝑢‖²−2𝛼_𝑛⟨𝑢,𝐽𝑥_𝑛⟩−2𝛽_𝑛⟨𝑢,𝐽𝑇𝑥_𝑛⟩−2𝛾_𝑛⟨𝑢,𝐽𝑆𝑦_𝑛⟩+𝛼_𝑛‖𝐽𝑥_𝑛‖²+𝛽_𝑛‖𝐽𝑇𝑥_𝑛‖²+𝛾_𝑛‖𝐽𝑆𝑦_𝑛‖²−𝛼_𝑛𝛽_𝑛𝑔(‖𝐽𝑥_𝑛−𝐽𝑇𝑥_𝑛‖)=(𝛼_𝑛+𝛽_𝑛+𝛾_𝑛)‖𝑢‖²−2𝛼_𝑛⟨𝑢,𝐽𝑥_𝑛⟩−2𝛽_𝑛⟨𝑢,𝐽𝑇𝑥_𝑛⟩−2𝛾_𝑛⟨𝑢,𝐽𝑆𝑦_𝑛⟩+𝛼_𝑛‖𝐽𝑥_𝑛‖²+𝛽_𝑛‖𝐽𝑇𝑥_𝑛‖²+𝛾_𝑛‖𝐽𝑆𝑦_𝑛‖²−𝛼_𝑛𝛽_𝑛𝑔(‖𝐽𝑥_𝑛−𝐽𝑇𝑥_𝑛‖)=𝛼_𝑛𝜙(𝑢,𝑥_𝑛)+𝛽_𝑛𝜙(𝑢,𝑇𝑥_𝑛)+𝛾_𝑛𝜙(𝑢,𝑆𝑦_𝑛)−𝛼_𝑛𝛽_𝑛𝑔(‖𝐽𝑥_𝑛−𝐽𝑇𝑥_𝑛‖)≤𝛼_𝑛𝜙(𝑢,𝑥_𝑛)+𝛽_𝑛𝜙(𝑢,𝑥_𝑛)+𝛾_𝑛𝜙(𝑢,𝑦_𝑛)−𝛼_𝑛𝛽_𝑛𝑔(‖𝐽𝑥_𝑛−𝐽𝑇𝑥_𝑛‖)≤𝛼_𝑛𝜙(𝑢,𝑥_𝑛)+𝛽_𝑛𝜙(𝑢,𝑥_𝑛)+𝛾_𝑛𝜙(𝑢,𝑥_𝑛)−𝛼_𝑛𝛽_𝑛𝑔(‖𝐽𝑥_𝑛−𝐽𝑇𝑥_𝑛‖)=𝜙(𝑢,𝑥_𝑛)−𝛼_𝑛𝛽_𝑛𝑔(‖𝐽𝑥_𝑛−𝐽𝑇𝑥_𝑛‖). (3.16) This implies that 𝛼_𝑛𝛽_𝑛𝑔(‖𝐽𝑥_𝑛−𝐽𝑇𝑥_𝑛‖)≤𝜙(𝑢,𝑥_𝑛)−𝜙(𝑢,𝑢_𝑛)=‖𝑥_𝑛‖²−‖𝑢_𝑛‖²−2⟨𝑢,𝐽𝑥_𝑛−𝐽𝑢_𝑛⟩≤‖𝑥_𝑛−𝑢_𝑛‖(‖𝑥_𝑛‖+‖𝑢_𝑛‖)+2‖𝑢‖‖𝐽𝑥_𝑛−𝐽𝑢_𝑛‖. (3.17) It follows from (3.14), (3.15), and (B2) that lim_𝑛→∞𝑔(‖𝐽𝑥_𝑛−𝐽𝑇𝑥_𝑛‖)=0. (3.18) Since 𝑔 is strictly increasing and continuous at 0 with 𝑔(0)=0, it follows that lim_𝑛→∞‖𝐽𝑥_𝑛−𝐽𝑇𝑥_𝑛‖=0. (3.19) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, so is 𝐽⁻¹. Then lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=lim_𝑛→∞‖𝐽⁻¹(𝐽𝑥_𝑛)−𝐽⁻¹(𝐽𝑇𝑥_𝑛)‖=0. (3.20) In the same manner, we can show that lim_𝑛→∞‖𝑥_𝑛−𝑆𝑦_𝑛‖=0. (3.21) In addition, 𝜙(𝑢,𝑦_𝑛)≤𝛿_𝑛𝜙(𝑢,𝑥_𝑛)+(1−𝛿_𝑛)𝜙(𝑢,𝑤_𝑛), using (3.6), we have 𝜙(𝑢,𝑢_𝑛)≤𝛼_𝑛𝜙(𝑢,𝑥_𝑛)+𝛽_𝑛𝜙(𝑢,𝑥_𝑛)+𝛾_𝑛𝜙(𝑢,𝑦_𝑛)≤𝛼_𝑛𝜙(𝑢,𝑥_𝑛)+𝛽_𝑛𝜙(𝑢,𝑥_𝑛)+𝛾_𝑛(𝛿_𝑛𝜙(𝑢,𝑥_𝑛)+(1−𝛿_𝑛)𝜙(𝑢,𝑤_𝑛))=𝛼_𝑛𝜙(𝑢,𝑥_𝑛)+𝛽_𝑛𝜙(𝑢,𝑥_𝑛)+𝛾_𝑛𝛿_𝑛𝜙(𝑢,𝑥_𝑛)+𝛾_𝑛(1−𝛿_𝑛)𝜙(𝑢,𝑤_𝑛)≤𝛼_𝑛𝜙(𝑢,𝑥_𝑛)+𝛽_𝑛𝜙(𝑢,𝑥_𝑛)+𝛾_𝑛𝛿_𝑛𝜙(𝑢,𝑥_𝑛)+𝛾_𝑛(1−𝛿_𝑛)(𝜙(𝑢,𝑥_𝑛2)+2𝑎(𝑐²𝑏−𝛼)‖𝐴𝑥_𝑛−𝐴𝑢‖²)=𝛼_𝑛𝜙(𝑢,𝑥_𝑛)+𝛽_𝑛𝜙(𝑢,𝑥_𝑛)+𝛾_𝑛𝛿_𝑛𝜙(𝑢,𝑥_𝑛)+𝛾_𝑛(1−𝛿_𝑛)𝜙(𝑢,𝑥_𝑛)+𝛾_𝑛(1−𝛿_𝑛2)2𝑎(𝑐²𝑏−𝛼)‖𝐴𝑥_𝑛−𝐴𝑢‖²=(𝛼_𝑛+𝛽_𝑛+𝛾_𝑛)𝜙(𝑢,𝑥_𝑛)−𝛾_𝑛(1−𝛿_𝑛2)2𝑎(𝛼−𝑐²𝑏)‖𝐴𝑥_𝑛−𝐴𝑢‖²=𝜙(𝑢,𝑥_𝑛)−𝛾_𝑛(1−𝛿_𝑛2)2𝑎(𝛼−𝑐²𝑏)‖𝐴𝑥_𝑛−𝐴𝑢‖², (3.22) which leads to the following: 𝛾_𝑛(1−𝛿_𝑛2)2𝑎(𝛼−𝑐²𝑏)‖𝐴𝑥_𝑛−𝐴𝑢‖²≤𝜙(𝑢,𝑥_𝑛)−𝜙(𝑢,𝑢_𝑛). (3.23) Since limsup_𝑛→∞𝛿_𝑛<1,0<liminf_𝑛→∞𝛼_𝑛𝛾_𝑛≤liminf_𝑛→∞𝛾_𝑛 and from (3.17), we observe lim_𝑛→∞(𝜙(𝑢,𝑥_𝑛)−𝜙(𝑢,𝑢_𝑛))=0, (3.24) which yields that lim_𝑛→∞‖𝐴𝑥_𝑛−𝐴𝑢‖=0. (3.25) From Lemmas 2.6 and 2.12, and (3.5), we have 𝜙(𝑥_𝑛,𝑤_𝑛)=𝜙(𝑥_𝑛,Π_𝐶𝑣_𝑛)≤𝜙(𝑥_𝑛,𝑣_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⟨𝑣_𝑛−𝑥_𝑛,−𝜆_𝑛𝐴𝑥_𝑛⟩=2⟨𝑣_𝑛−𝑥_𝑛,−𝜆_𝑛𝐴𝑥_𝑛⟩≤4𝑐²𝑏²‖𝐴𝑥_𝑛−𝐴𝑢‖²→0(as𝑛→∞). (3.26) It follows from Lemma 2.4 and (3.25) that lim_𝑛→∞‖𝑥_𝑛−𝑤_𝑛‖=0. (3.27) Hence 𝑤_𝑛→𝑞 as 𝑛→∞ and lim_𝑛→∞‖𝐽𝑥_𝑛−𝐽𝑤_𝑛‖=0. (3.28) By using (3.26), we have 𝜙(𝑥_𝑛,𝑦_𝑛)=𝜙(𝑥_𝑛,𝐽⁻¹(𝛿_𝑛𝐽𝑥_𝑛+(1−𝛿_𝑛)𝐽𝑤_𝑛))=𝑉(𝑢,𝛿_𝑛𝐽𝑥_𝑛+(1−𝛿_𝑛)𝐽𝑤_𝑛)=‖𝑥_𝑛‖²−2⟨𝑥_𝑛,𝛿_𝑛𝐽𝑥_𝑛+(1−𝛿_𝑛)𝐽𝑤_𝑛⟩+‖𝛿_𝑛𝐽𝑥_𝑛+(1−𝛿_𝑛)𝐽𝑤_𝑛‖²≤‖𝑥_𝑛‖²−2𝛿_𝑛⟨𝑥_𝑛,𝐽𝑥_𝑛⟩−2(1−𝛿_𝑛)⟨𝑥_𝑛,𝐽𝑤_𝑛⟩+𝛿_𝑛‖𝐽𝑥_𝑛‖²+(1−𝛿_𝑛)‖𝐽𝑤_𝑛‖²=(𝛿_𝑛+(1−𝛿_𝑛))‖𝑥_𝑛‖²−2𝛿_𝑛⟨𝑥_𝑛,𝐽𝑥_𝑛⟩−2(1−𝛿_𝑛)⟨𝑥_𝑛,𝐽𝑤_𝑛⟩+𝛿_𝑛‖𝐽𝑥_𝑛‖²+(1−𝛿_𝑛)‖𝐽𝑤_𝑛‖²=𝛿_𝑛‖𝑥_𝑛‖²+(1−𝛿_𝑛)‖𝑥_𝑛‖²−2𝛿_𝑛⟨𝑥_𝑛,𝐽𝑥_𝑛⟩−2(1−𝛿_𝑛)⟨𝑥_𝑛,𝐽𝑤_𝑛⟩+𝛿_𝑛‖𝐽𝑥_𝑛‖²+(1−𝛿_𝑛)‖𝐽𝑤_𝑛‖²=𝛿_𝑛𝜙(𝑥_𝑛,𝑥_𝑛)+(1−𝛿_𝑛)𝜙(𝑥_𝑛,𝑤_𝑛)≤𝜙(𝑥_𝑛,𝑤_𝑛)→0(as𝑛→∞). (3.29) Applying Lemma 2.4, we get lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0. (3.30) Hence, 𝑦_𝑛→𝑞 as 𝑛→∞.
In addition, ‖𝑆𝑦_𝑛−𝑦_𝑛‖=‖𝑆𝑦_𝑛−𝑥_𝑛+𝑥_𝑛−𝑦_𝑛‖≤‖𝑆𝑦_𝑛−𝑥_𝑛‖+‖𝑥_𝑛−𝑦_𝑛‖. (3.31)
It follows from (3.21), (3.30), and (3.31) that lim_𝑛→∞‖𝑆𝑦_𝑛−𝑦_𝑛‖=0. (3.32) From (3.20), (3.32) and by the closedness of 𝑇 and 𝑆, we get 𝑞∈𝐹(𝑇)∩𝐹(𝑆).Step 6. Show that 𝑥_𝑛→𝑞∈𝐸𝑃(𝑓).
From (3.16), we have 𝜙(𝑢,𝑧_𝑛)≤𝜙(𝑢,𝑥_𝑛). (3.33) Note that 𝑢_𝑛=𝑇_𝑟𝑛𝑧_𝑛. From (3.33) and Lemma 2.11, we have 𝜙(𝑢_𝑛,𝑧_𝑛)=𝜙(𝑇_𝑟𝑛𝑧_𝑛,𝑧_𝑛)≤𝜙(𝑢,𝑧_𝑛)−𝜙(𝑢,𝑇_𝑟𝑛𝑧_𝑛)≤𝜙(𝑢,𝑥_𝑛)−𝜙(𝑢,𝑇_𝑟𝑛𝑧_𝑛)=𝜙(𝑢,𝑥_𝑛)−𝜙(𝑢,𝑢_𝑛). (3.34) Using (3.24), we have lim_𝑛→∞𝜙(𝑢_𝑛,𝑧_𝑛)=0. By Lemma 2.4, we obtain lim_𝑛→∞‖𝑢_𝑛−𝑧_𝑛‖=0. (3.35) Since 𝑟_𝑛≥𝑠, we have ‖𝐽𝑢_𝑛−𝐽𝑧_𝑛‖𝑟_𝑛→0as𝑛→∞. (3.36) From 𝑢_𝑛=𝑇_𝑟𝑛𝑧_𝑛, we have 𝑓(𝑢_𝑛1,𝑦)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑧_𝑛⟩≥0,∀𝑦∈𝐶. (3.37) By using (A2), we have ‖𝑦−𝑢_𝑛‖‖𝐽𝑢_𝑛−𝐽𝑧_𝑛‖𝑟_𝑛≥1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑧_𝑛⟩≥−𝑓(𝑢_𝑛,𝑦)≥𝑓(𝑦,𝑢_𝑛),∀𝑦∈𝐶. (3.38) From (A4) and 𝑢_𝑛→𝑞, we get 𝑓(𝑦,𝑞)≤0 for all 𝑦∈𝐶. For 𝑡 with 0<𝑡≤1 and 𝑦∈𝐶, let 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑞. Since 𝑦∈𝐶 and 𝑞∈𝐶, we have 𝑦_𝑡∈𝐶 and hence 𝑓(𝑦_𝑡,𝑞)≤0. So from (A1), we have 0=𝑓(𝑦_𝑡,𝑦_𝑡)≤𝑡𝑓(𝑦_𝑡,𝑦)+(1−𝑡)𝑓(𝑦_𝑡,𝑞)≤𝑡𝑓(𝑦_𝑡,𝑦). That is, 𝑓(𝑦_𝑡,𝑦)≥0. It follows from (A3) that 𝑓(𝑞,𝑦)≥0 for all 𝑦∈𝐶 and hence 𝑞∈𝐸𝑃(𝑓).Step 7. Show that 𝑥_𝑛→𝑞∈𝑉𝐼(𝐶,𝐴).
Define 𝑇_𝑒⊂𝐸×𝐸^∗ be as in (2.8), which yields that 𝑇_𝑒 is maximal monotone and 𝑇_𝑒⁻¹0=𝑉𝐼(𝐶,𝐴). Let (𝑣,𝑤)∈𝐺(𝑇_𝑒). Since 𝑤∈𝑇_𝑒𝑣=𝐴𝑣+𝑁_𝐶(𝑣), we get 𝑤−𝐴𝑣∈𝑁_𝐶(𝑣). From 𝑤_𝑛∈𝐶, we have ⟨𝑣−𝑤_𝑛,𝑤−𝐴𝑣⟩≥0. (3.39) On the other hand, from 𝑤_𝑛=Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛) and Lemma 2.5, we have ⟨𝑣−𝑤_𝑛,𝐽𝑤_𝑛−(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛)⟩≥0, and hence ⟨𝑣−𝑤_𝑛,𝐽𝑥_𝑛−𝐽𝑤_𝑛𝜆_𝑛−𝐴𝑥_𝑛⟩≤0. (3.40) It follows from (3.39) and (3.40) 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), (3.41) where 𝑀=sup_𝑛≥1{‖𝑣−𝑤_𝑛‖}. By taking the limit as 𝑛→∞ and from (3.27) and (3.28), we obtain ⟨𝑣−𝑞,𝑤⟩≥0. By the maximality of 𝑇_𝑒, we have 𝑞∈𝑇_𝑒⁻¹0 and hence 𝑞∈𝑉𝐼(𝐶,𝐴). That is, 𝑞∈𝐹.Step 8. Show that 𝑞=Π_𝐹𝑥₀.
From 𝑥_𝑛=Π_𝐶𝑛𝑥₀, we have ⟨𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛⟩≥0,∀𝑧∈𝐶_𝑛. (3.42) Since 𝐹⊂𝐶_𝑛, we also have ⟨𝑥_𝑛−𝑢,𝐽𝑥₀−𝐽𝑥_𝑛⟩≥0,∀𝑢∈𝐹. (3.43) By taking the limit in (3.43), we obtain that ⟨𝑞−𝑢,𝐽𝑥₀−𝐽𝑞⟩≥0,∀𝑢∈𝐹. (3.44) By Lemma 2.5, we can conclude that 𝑞=Π_𝐹𝑥₀. From (3.14), we have lim_𝑛→∞‖𝑢_𝑛−𝑥_𝑛‖=0, and it follows that lim_𝑛→∞‖𝑢_𝑛−𝑞‖=0. This completes the proof.

Finally, we prove two strong convergence theorems in a 2-uniformly convex, uniformly smooth Banach space by using Theorem 3.1.

First, we consider the problem of finding a zero point of an inverse-strongly-monotone operator of 𝐸 into 𝐸^∗. Assume that 𝐴 satisfies the conditions:

(D1)𝐴 is 𝛼-inverse-strongly monotone,(D2)𝐴⁻¹(0)={𝑢∈𝐸∶𝐴𝑢=0}≠∅.

Theorem 3.2. Let 𝐸 be a 2-uniformly convex, uniformly smooth Banach space, 𝑓 a bifunction from 𝐸×𝐸 to ℝ which satisfies (A1)–(A4), 𝐴 an operator of 𝐸 into 𝐸^∗ satisfying (D1)–(D2), and 𝑇,𝑆 two closed relatively quasi-nonexpansive mappings from 𝐸 into itself such that 𝐹∶=𝐹(𝑇)∩𝐹(𝑆)∩𝐸𝑃(𝑓)∩𝐴⁻¹(0)≠∅. For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π_𝐶1𝑥₀ and 𝐶₁=𝐸, define sequences {𝑥_𝑛} and {𝑢_𝑛} as follows: 𝑤_𝑛=𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛𝑦),_𝑛=𝐽⁻¹(𝛿_𝑛𝐽𝑥_𝑛+(1−𝛿_𝑛)𝐽𝑤_𝑛𝑧),_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+𝛽_𝑛𝐽𝑇𝑥_𝑛+𝛾_𝑛𝐽𝑆𝑦_𝑛𝑢),_𝑛∈𝐶suchthat𝑓(𝑢_𝑛1,𝑦)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑧_𝑛𝐶⟩≥0,∀𝑦∈𝐸,_𝑛+1={𝑧∈𝐶_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑥)},_𝑛+1=Π_𝐶𝑛+1𝑥₀ (3.45) for every 𝑛∈ℕ, where 𝐽 is the duality mapping on 𝐸. Suppose that {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛}, and {𝛿_𝑛} are sequences in [0,1] satisfying conditions (B1)–(B4) of Theorem 3.1.
Then, {𝑥_𝑛} and {𝑢_𝑛} converge strongly to 𝑞=Π_𝐹𝑥₀.

Proof. Putting 𝐶=𝐸 in Theorem 3.1, we have Π_𝐸=𝐼. We also have 𝑉𝐼(𝐸,𝐴)=𝐴⁻¹(0) and then condition (C3) of Theorem 3.1 holds for all 𝑦∈𝐸 and 𝑢∈𝐴⁻¹(0). So, we obtain the desired result.

Next, let 𝐾 be a nonempty closed convex cone in 𝐸 and 𝐴 an operator of 𝐾 into 𝐸^∗. We define its polar in 𝐸^∗ to be the set 𝐾^∗={𝑦^∗∈𝐸^∗∶⟨𝑥,𝑦^∗⟩≥0∀𝑥∈𝐾}. (3.46) Then the element 𝑢∈𝐾 is called a solution of the complementarity problem if 𝐴𝑢∈𝐾^∗,⟨𝑢,𝐴𝑢⟩=0. (3.47) The set of solutions of the complementarity problem is denoted by 𝐶(𝐾,𝐴).

Assume that 𝐴 is an operator satisfying the conditions:

(E1)𝐴 is 𝛼-inverse-strongly-monotone,(E2)𝐶(𝐾,𝐴)≠∅,(E3)‖𝐴𝑦‖≤‖𝐴𝑦−𝐴𝑢‖ for all 𝑦∈𝐾 and 𝑢∈𝐶(𝐾,𝐴).

Theorem 3.3. Let 𝐸 be a 2-uniformly convex, uniformly smooth Banach space, 𝐾 a nonempty closed convex cone in 𝐸, 𝑓 a bifunction from 𝐾×𝐾 to ℝ which satisfies (A1)–(A4), 𝐴 an operator of 𝐾 into 𝐸^∗ satisfying (E1)–(E3), and 𝑇,𝑆 two closed relatively quasi-nonexpansive mappings from 𝐾 into itself such that 𝐹∶=𝐹(𝑇)∩𝐹(𝑆)∩𝐸𝑃(𝑓)∩𝐶(𝐾,𝐴)≠∅. For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π_𝐶1𝑥₀ and 𝐶₁=𝐾, define sequences {𝑥_𝑛} and {𝑢_𝑛} as follows: 𝑤_𝑛=Π_𝐾𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛𝑦),_𝑛=𝐽⁻¹(𝛿_𝑛𝐽𝑥_𝑛+(1−𝛿_𝑛)𝐽𝑤_𝑛𝑧),_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+𝛽_𝑛𝐽𝑇𝑥_𝑛+𝛾_𝑛𝐽𝑆𝑦_𝑛𝑢),_𝑛∈𝐶suchthat𝑓(𝑢_𝑛1,𝑦)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑧_𝑛𝐶⟩≥0,∀𝑦∈𝐾,_𝑛+1={𝑧∈𝐶_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑥)},_𝑛+1=Π_𝐶𝑛+1𝑥₀ (3.48) for every 𝑛∈ℕ, where 𝐽 is the duality mapping on 𝐸. Suppose that {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛}, and {𝛿_𝑛} are sequences in [0,1] satisfying conditions (B1)–(B4) of Theorem 3.1.
Then, {𝑥_𝑛} and {𝑢_𝑛} converge strongly to 𝑞=Π_𝐹𝑥₀.

Proof. From [24, Lemma 7.1.1], we have 𝑉𝐼(𝐾,𝐴)=𝐶(𝐾,𝐴). So by Theorem 3.1, we obtain the desired result.

Acknowledgments

The authors would like to thank the Thailand Research Fund, The Commission on Higher Education, and Naresuan Phayao University, Phayao, Thailand, for financial support during the preparation of this paper.