Abstract and Applied Analysis
Volume 2010 (2010), Article ID 734126, 25 pages
doi:10.1155/2010/734126
Research Article

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

Siwaporn Saewan, Poom Kumam, and Kriengsak Wattanawitoon

Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand

Received 26 October 2009; Accepted 3 January 2010

Academic Editor: Simeon Reich

Copyright © 2010 Siwaporn Saewan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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

l i m 𝑡 → 0 ‖ 𝑥 + 𝑡 𝑦 ‖ − ‖ 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 ) = i n f 1 − 𝑥 + 𝑦 2 ‖ ‖ ‖ ∶ 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 ) ≥ 𝑐 𝜀 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 = 𝐽 2 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 , 𝐽 𝑦 ⟩ + ‖ 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 𝐶 𝑥 = 𝑥 , where 𝑥 is the solution to the minimization problem 𝜙 𝑥 , 𝑥 = i n f 𝑦 ∈ 𝐶 𝜙 ( 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 , 𝐽 𝑦 ⟩ = ‖ 𝑥 ‖ 2 = ‖ 𝐽 𝑦 ‖ 2 . From the definition of 𝐽 , one has 𝐽 𝑥 = 𝐽 𝑦 . 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 ⟨ 𝐴 𝑥 ∗ , 𝑦 − 𝑥 ∗ ⟩ ≥ 0 , ∀ 𝑦 ∈ 𝐶 . ( 1 . 7 ) The set of solutions of (1.4) is denoted by V I ( 𝐴 , 𝐶 ) . Recall that A is called

(i)monotone if ⟨ 𝐴 𝑥 − 𝐴 𝑦 , 𝑥 − 𝑦 ⟩ ≥ 0 , ∀ 𝑥 , 𝑦 ∈ 𝐶 , ( 1 . 8 ) (ii)an 𝛼 -inverse-strongly monotone if there exists a constant 𝛼 > 0 such that ⟨ 𝐴 𝑥 − 𝐴 𝑦 , 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 𝐴 𝑥 ∗ = 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 . 1 0 )

The set of solutions of (1.10) is denoted by E P ( 𝑓 ) . Given a mapping 𝑇 ∶ 𝐶 → 𝐸 ∗ , let 𝑓 ( 𝑥 , 𝑦 ) = ⟨ 𝑇 𝑥 , 𝑦 − 𝑥 ⟩ for all 𝑥 , 𝑦 ∈ 𝐶 . Then 𝑥 ∗ ∈ E P ( 𝑓 ) if and only if ⟨ 𝑇 𝑥 ∗ , 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

‖ 𝑇 𝑥 − 𝑇 𝑦 ‖ ≤ ‖ 𝑥 − 𝑦 ‖ , ∀ 𝑥 , 𝑦 ∈ 𝐶 . ( 1 . 1 1 )

A point 𝑥 ∈ 𝐶 is a fixed point  of 𝑇 provided that 𝑇 𝑥 = 𝑥 . Denote by 𝐹 ( 𝑇 ) the set of fixed points of 𝑇 ; that is, 𝐹 ( 𝑇 ) = { 𝑥 ∈ 𝐶 ∶ 𝑇 𝑥 = 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 l i m 𝑛 → ∞ ‖ 𝑥 𝑛 − 𝑇 𝑥 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 , 𝑇 𝑥 ) ≤ 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 ( 𝑇 𝑥 , 𝑇 𝑦 ) ≤ 𝜙 ( 𝑥 , 𝑦 ) for 𝑥 , 𝑦 ∈ 𝐶 . 𝑇 is said to be relatively quasi-nonexpansive if 𝐹 ( 𝑇 ) ≠ ∅ and 𝜙 ( 𝑝 , 𝑇 𝑥 ) ≤ 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 𝑇 𝑥 𝑛 → 𝑦 , then 𝑇 𝑥 = 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: 𝑥 1 = 𝑥 ∈ 𝐶 and

𝑥 𝑛 + 1 = 𝑃 𝐶 𝑥 𝑛 − 𝜆 𝑛 𝐴 𝑥 𝑛 , ( 1 . 1 2 ) 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 V I ( 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 : 𝑥 1 = 𝑥 ∈ 𝐶 and

𝑥 𝑛 + 1 = Π 𝐶 𝐽 − 1 𝐽 𝑥 𝑛 − 𝜆 𝑛 𝐴 𝑥 𝑛 ( 1 . 1 3 ) 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 V I ( 𝐴 , 𝐶 ) .

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

𝑥 𝑛 + 1 = Π 𝐶 𝐽 − 1 𝛼 𝑛 𝐽 𝑥 𝑛 + 1 − 𝛼 𝑛 𝐽 𝑇 𝑥 𝑛 , ( 1 . 1 4 ) where the initial guess element 𝑥 0 ∈ 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 :

𝑥 0 𝑦 ∈ 𝐶 c h o s e n a r b i t r a r i l y , 𝑛 = 𝐽 − 1 𝛼 𝑛 𝐽 𝑥 𝑛 + 1 − 𝛼 𝑛 𝐽 𝑇 𝑥 𝑛 , 𝐶 𝑛 = 𝑧 ∈ 𝐶 ∶ 𝜙 𝑧 , 𝑦 𝑛 ≤ 𝜙 𝑧 , 𝑥 𝑛 , 𝑄 𝑛 = 𝑧 ∈ 𝐶 ∶ ⟨ 𝑥 𝑛 − 𝑧 , 𝐽 𝑥 0 − 𝐽 𝑥 𝑛 , 𝑥 ⟩ ≥ 0 𝑛 + 1 = 𝐶 𝑛 ∩ 𝑄 𝑛 𝑥 0 . ( 1 . 1 5 )

They proved that { 𝑥 𝑛 } converges strongly to Π 𝐹 ( 𝑇 ) 𝑥 0 , 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:

𝑥 0 𝑦 = 𝑥 ∈ 𝐶 , 𝑛 = 𝐽 − 1 𝛼 𝑛 𝐽 𝑥 𝑛 + 1 − 𝛼 𝑛 𝐽 𝑆 𝑥 𝑛 , 𝑢 𝑛 𝑢 ∈ 𝐶 s u c h t h a t 𝑓 𝑛 + 1 , 𝑦 𝑟 𝑛 ⟨ 𝑦 − 𝑢 𝑛 , 𝐽 𝑢 𝑛 − 𝐽 𝑦 𝑛 𝐻 ⟩ ≥ 0 , ∀ 𝑦 ∈ 𝐶 , 𝑛 = 𝑧 ∈ 𝐶 ∶ 𝜙 𝑧 , 𝑢 𝑛 ≤ 𝜙 𝑧 , 𝑥 𝑛 , 𝑊 𝑛 = 𝑧 ∈ 𝐶 ∶ ⟨ 𝑥 𝑛 − 𝑧 , 𝐽 𝑥 − 𝐽 𝑥 𝑛 , 𝑥 ⟩ ≥ 0 𝑛 + 1 = 𝐻 𝑛 ∩ 𝑊 𝑛 𝑥 , ( 1 . 1 6 ) where 𝐽 is the duality mapping on 𝐸 . Then, { 𝑥 𝑛 } converges strongly to Π 𝐹 ( 𝑆 ) ∩ E P ( 𝑓 ) 𝑥 , where Π 𝐹 ( 𝑆 ) ∩ E P ( 𝑓 ) is the generalized projection of 𝐸 onto 𝐹 ( 𝑆 ) ∩ E P ( 𝑓 ) . Also, Takahashi and Zembayashi [24] proved the following iteration for a relatively nonexpansive mapping:

𝑦 𝑛 = 𝐽 − 1 𝛼 𝑛 𝐽 𝑥 𝑛 + 1 − 𝛼 𝑛 𝐽 𝑆 𝑥 𝑛 , 𝑢 𝑛 𝑢 ∈ 𝐶 s u c h t h a t 𝑓 𝑛 + 1 , 𝑦 𝑟 𝑛 ⟨ 𝑦 − 𝑢 𝑛 , 𝐽 𝑢 𝑛 − 𝐽 𝑦 𝑛 𝐶 ⟩ ≥ 0 , ∀ 𝑦 ∈ 𝐶 , 𝑛 + 1 = 𝑧 ∈ 𝐶 𝑛 ∶ 𝜙 𝑧 , 𝑢 𝑛 ≤ 𝜙 𝑧 , 𝑥 𝑛 , 𝑥 𝑛 + 1 = 𝐶 𝑛 + 1 𝑥 , ( 1 . 1 7 ) where 𝐽 is the duality mapping on 𝐸 . Then, { 𝑥 𝑛 } converges strongly to Π 𝐹 ( 𝑆 ) ∩ E P ( 𝑓 ) 𝑥 , where Π 𝐹 ( 𝑆 ) ∩ E P ( 𝑓 ) is the generalized projection of 𝐸 onto 𝐹 ( 𝑆 ) ∩ E P ( 𝑓 ) . Qin and Su [25] proved the following iteration for relatively nonexpansive mappings 𝑇 in a Banach space 𝐸 :

𝑥 0 𝑦 ∈ 𝐶 , c h o s e n a r b i t r a r i l y , 𝑛 = 𝐽 − 1 𝛼 𝑛 𝐽 𝑥 𝑛 + 1 − 𝛼 𝑛 𝐽 𝑇 𝑧 𝑛 , 𝑧 𝑛 = 𝐽 − 1 𝛽 𝑛 𝐽 𝑥 𝑛 + 1 − 𝛽 𝑛 𝐽 𝑇 𝑥 𝑛 , 𝐶 𝑛 = 𝑣 ∈ 𝐶 ∶ 𝜙 𝑣 , 𝑦 𝑛 ≤ 𝛼 𝑛 𝜙 𝑣 , 𝑥 𝑛 + 1 − 𝛼 𝑛 𝜙 𝑣 , 𝑧 𝑛 , 𝑄 𝑛 = 𝑣 ∈ 𝐶 ∶ ⟨ 𝐽 𝑥 0 − 𝐽 𝑥 𝑛 , 𝑥 𝑛 , 𝑥 − 𝑣 ⟩ ≥ 0 𝑛 + 1 = 𝐶 𝑛 ∩ 𝑄 𝑛 𝑥 0 , ( 1 . 1 8 ) the sequence { 𝑥 𝑛 } generated by (1.18) converges strongly to Π 𝐹 ( 𝑇 ) 𝑥 0 .

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

𝑥 0 ∈ 𝐶 , 𝐽 𝑧 𝑛 = 𝛼 𝑛 𝐽 𝑥 𝑛 + 1 − 𝛼 𝑛 𝐽 𝑇 𝑥 𝑛 , 𝐽 𝑢 𝑛 = 𝛽 𝑛 𝐽 𝑥 𝑛 + 1 − 𝛽 𝑛 𝐽 𝑆 𝑧 𝑛 , 𝐻 𝑛 = 𝑣 ∈ 𝐶 ∶ 𝜙 𝑣 , 𝑢 𝑛 ≤ 𝛽 𝑛 𝜙 𝑣 , 𝑥 𝑛 + 1 − 𝛽 𝑛 𝜙 𝑣 , 𝑧 𝑛 ≤ 𝜙 𝑣 , 𝑥 𝑛 , 𝑊 𝑛 = 𝑧 ∈ 𝐶 ∶ ⟨ 𝑧 − 𝑥 𝑛 , 𝐽 𝑥 0 − 𝐽 𝑥 𝑛 , 𝑥 ⟩ ≤ 0 𝑛 + 1 = 𝑄 𝐻 𝑛 ∩ 𝑊 𝑛 𝑥 0 ; ( 1 . 1 9 ) if { 𝛼 𝑛 } and { 𝛽 𝑛 } are sequences in [ 0 , 1 ) such that 𝛼 𝑛 ≤ 1 − 𝛿 1 and 𝛽 𝑛 ≤ 1 − 𝛿 2 for some 𝛿 1 , 𝛿 2 ∈ ( 0 , 1 ) , then { 𝑥 𝑛 } generated by (1.19) converges strongly to a point 𝑄 𝐹 ( 𝑇 ) ∩ 𝐹 ( 𝑆 ) 𝑥 0 , where the mapping 𝑄 𝐶 of 𝐸 onto 𝐶 is the generalized projection. Very recently, Cholamjiak [27] proved the following iteration:

𝑧 𝑛 = Π 𝐶 𝐽 − 1 𝐽 𝑥 𝑛 − 𝜆 𝑛 𝐴 𝑥 𝑛 , 𝑦 𝑛 = 𝐽 − 1 𝛼 𝑛 𝐽 𝑥 𝑛 + 𝛽 𝑛 𝐽 𝑇 𝑥 𝑛 + 𝛾 𝑛 𝐽 𝑆 𝑧 𝑛 , 𝑢 𝑛 𝑢 ∈ 𝐶 s u c h t h a t 𝑓 𝑛 + 1 , 𝑦 𝑟 𝑛 ⟨ 𝑦 − 𝑢 𝑛 , 𝐽 𝑢 𝑛 − 𝐽 𝑦 𝑛 𝐶 ⟩ ≥ 0 , ∀ 𝑦 ∈ 𝐶 , 𝑛 + 1 = 𝑧 ∈ 𝐶 𝑛 ∶ 𝜙 𝑧 , 𝑢 𝑛 ≤ 𝜙 𝑧 , 𝑥 𝑛 , 𝑥 𝑛 + 1 = 𝐶 𝑛 + 1 𝑥 0 , ( 1 . 2 0 ) 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 𝐹 𝑥 0 , where 𝐹 ∶ = 𝐹 ( 𝑇 ) ∩ 𝐹 ( 𝑆 ) ∩ E P ( 𝑓 ) ∩ V I ( 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 𝑥 0 ∈ 𝐸 with 𝑥 1 = Π 𝐶 1 𝑥 0 and 𝐶 1 = 𝐶 , define a sequence { 𝑥 𝑛 } as follows:

𝑤 𝑛 = Π 𝐶 𝐽 − 1 𝐽 𝑥 𝑛 − 𝜆 𝑛 𝐴 𝑥 𝑛 , 𝑧 𝑛 = 𝐽 − 1 𝛽 𝑛 𝐽 𝑥 𝑛 + 1 − 𝛽 𝑛 𝐽 𝑇 𝑤 𝑛 , 𝑦 𝑛 = 𝐽 − 1 𝛼 𝑛 𝐽 𝑥 𝑛 + 1 − 𝛼 𝑛 𝐽 𝑆 𝑧 𝑛 , 𝑢 𝑛 𝑢 ∈ 𝐶 s u c h t h a t 𝑓 𝑛 + 1 , 𝑦 𝑟 𝑛 ⟨ 𝑦 − 𝑢 𝑛 , 𝐽 𝑢 𝑛 − 𝐽 𝑦 𝑛 𝐶 ⟩ ≥ 0 , ∀ 𝑦 ∈ 𝐶 , 𝑛 + 1 = 𝑧 ∈ 𝐶 𝑛 ∶ 𝜙 𝑧 , 𝑢 𝑛 ≤ 𝛼 𝑛 𝜙 𝑧 , 𝑥 𝑛 + 1 − 𝛼 𝑛 𝜙 𝑧 , 𝑧 𝑛 ≤ 𝜙 𝑧 , 𝑥 𝑛 , 𝑥 𝑛 + 1 = 𝐶 𝑛 + 1 𝑥 0 , ∀ 𝑛 ≥ 1 , ( 1 . 2 1 ) 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 ) ∩ E P ( 𝑓 ) ∩ V I ( 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 ‖ ≤ 𝑐 2 ‖ 𝐽 𝑥 − 𝐽 𝑦 ‖ , ( 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 , 𝐽 𝑥 − 𝐽 𝑦 ⟩ ≥ 𝑝 2 𝑝 − 2 𝑝 ‖ 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, 𝑥 0 = Π 𝐶 𝑥 if and only if ⟨ 𝑥 0 − 𝑦 , 𝐽 𝑥 − 𝐽 𝑥 0 ⟩ ≥ 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 𝐶 𝑥 Π + 𝜙 𝐶 𝑥 , 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 , l i m 𝑡 ↓ 0 𝑓 ( 𝑡 𝑧 + ( 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 1 𝑓 ( 𝑧 , 𝑦 ) + 𝑟 ⟨ 𝑦 − 𝑧 , 𝐽 𝑧 − 𝐽 𝑥 ⟩ ≥ 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 𝑟 1 𝑥 = 𝑧 ∈ 𝐶 ∶ 𝑓 ( 𝑧 , 𝑦 ) + 𝑟 , ⟨ 𝑦 − 𝑧 , 𝐽 𝑧 − 𝐽 𝑥 ⟩ ≥ 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 𝑟 𝑥 − 𝑇 𝑟 𝑦 , 𝐽 𝑇 𝑟 𝑥 − 𝐽 𝑇 𝑟 𝑦 ⟩ ≤ ⟨ 𝑇 𝑟 𝑥 − 𝑇 𝑟 𝑦 , 𝐽 𝑥 − 𝐽 𝑦 ⟩ ; ( 2 . 8 ) (3) 𝐹 ( 𝑇 𝑟 ) = E P ( 𝑓 ) ; (4) E P ( 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 𝑟 𝑥 𝑇 + 𝜙 𝑟 𝑥 , 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 . 1 0 ) 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 ∗ 𝐽 + 2 − 1 𝑥 ∗ − 𝑥 , 𝑦 ∗ ≤ 𝑉 𝑥 , 𝑥 ∗ + 𝑦 ∗ ( 2 . 1 1 ) 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 ( 𝑡 𝑥 + ( 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 𝑥 ( 𝑣 ) = ∗ ∈ 𝐸 ∗ ∶ ⟨ 𝑣 − 𝑦 , 𝑥 ∗ ⟩ ≥ 0 , ∀ 𝑦 ∈ 𝐶 . ( 2 . 1 2 )

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: 𝑇 𝑣 = 𝐴 𝑣 + 𝑁 𝐶 ( 𝑣 ) , 𝑣 ∈ 𝐶 ; ∅ , o t h e r w i s e . ( 2 . 1 3 ) Then 𝑇 is maximal monotone and 𝑇 − 1 0 = V I ( 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 ‖ 𝐴 𝑦 ‖ ≤ ‖ 𝐴 𝑦 − 𝐴 𝑢 ‖ , f o r a l l 𝑦 ∈ 𝐶 and 𝑢 ∈ V I ( 𝐴 , 𝐶 ) ≠ ∅ . Let 𝑇 , 𝑆 ∶ 𝐶 → 𝐶 be closed relatively quasi-nonexpansive mappings such that Ω ∶ = 𝐹 ( 𝑇 ) ∩ 𝐹 ( 𝑆 ) ∩ E P ( 𝑓 ) ∩ V I ( 𝐴 , 𝐶 ) ≠ ∅ . For an initial point 𝑥 0 ∈ 𝐸 with 𝑥 1 = Π 𝐶 1 𝑥 0 and 𝐶 1 = 𝐶 , we define the sequence { 𝑥 𝑛 } as follows: 𝑤 𝑛 = Π 𝐶 𝐽 − 1 𝐽 𝑥 𝑛 − 𝜆 𝑛 𝐴 𝑥 𝑛 , 𝑧 𝑛 = 𝐽 − 1 𝛽 𝑛 𝐽 𝑥 𝑛 + 1 − 𝛽 𝑛 𝐽 𝑇 𝑤 𝑛 , 𝑦 𝑛 = 𝐽 − 1 𝛼 𝑛 𝐽 𝑥 𝑛 + 1 − 𝛼 𝑛 𝐽 𝑆 𝑧 𝑛 , 𝑢 𝑛 𝑢 ∈ 𝐶 s u c h t h a t 𝑓 𝑛 + 1 , 𝑦 𝑟 𝑛 ⟨ 𝑦 − 𝑢 𝑛 , 𝐽 𝑢 𝑛 − 𝐽 𝑦 𝑛 𝐶 ⟩ ≥ 0 , ∀ 𝑦 ∈ 𝐶 , 𝑛 + 1 = 𝑧 ∈ 𝐶 𝑛 ∶ 𝜙 𝑧 , 𝑢 𝑛 ≤ 𝛼 𝑛 𝜙 𝑧 , 𝑥 𝑛 + 1 − 𝛼 𝑛 𝜙 𝑧 , 𝑧 𝑛 ≤ 𝜙 𝑧 , 𝑥 𝑛 , 𝑥 𝑛 + 1 = 𝐶 𝑛 + 1 𝑥 0 , ∀ 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 − 𝛿 1 and 𝛽 𝑛 ≤ 1 − 𝛿 2 , for some 𝛿 1 , 𝛿 2 ∈ ( 0 , 1 ) , { 𝑟 𝑛 } ⊆ ( 0 , 2 𝛼 ) and { 𝜆 𝑛 } ⊂ [ 𝑎 , 𝑏 ] for some 𝑎 , 𝑏 with 0 < 𝑎 < 𝑏 < 𝑐 2 𝛼 / 2 , where 1 / 𝑐 is the 2-uniformly convexity constant of 𝐸 . Then { 𝑥 𝑛 } converges strongly to 𝑝 ∈ Ω , where 𝑝 = Π Ω 𝑥 0 .

Proof. We have several steps to prove this theorem as follows:Step 1. We show that 𝐶 𝑛 + 1 is closed and convex.
Clearly 𝐶 1 = 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 , 𝐽 𝑥 𝑛 − 𝐽 𝑢 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 𝑟 𝑛 𝑦 𝑛 for all 𝑛 ≥ 0 . On the other hand, from Lemma 2.8 one has 𝑇 𝑟 𝑛 is relatively quasi-nonexpansive mappings and Ω ⊂ 𝐶 1 = 𝐶 . Supposing Ω ⊂ 𝐶 𝑘 for