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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 859492, 21 pages
doi:10.1155/2012/859492
Research Article

Existence and Strong Convergence Theorems for Generalized Mixed Equilibrium Problems of a Finite Family of Asymptotically Nonexpansive Mappings in Banach Spaces

Rabian Wangkeeree,1,2 Hossein Dehghan,3 and Pakkapon Preechasilp1

1Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
3Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Gava Zang, Zanjan 45137-66731, Iran

Received 5 April 2012; Accepted 10 May 2012

Academic Editor: Giuseppe Marino

Copyright © 2012 Rabian Wangkeeree 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

We first prove the existence of solutions for a generalized mixed equilibrium problem under the new conditions imposed on the given bifunction and introduce the algorithm for solving a common element in the solution set of a generalized mixed equilibrium problem and the common fixed point set of finite family of asymptotically nonexpansive mappings. Next, the strong convergence theorems are obtained, under some appropriate conditions, in uniformly convex and smooth Banach spaces. The main results extend various results existing in the current literature.

1. Introduction

Let 𝐸 be a real Banach space with the dual 𝐸 ∗ and 𝐶 be a nonempty closed convex subset of 𝐸 . We denote by ℕ and ℝ the sets of positive integers and real numbers, respectively. Also, we denote by 𝐽 the normalized duality mapping from 𝐸 to 2 𝐸 ∗ defined by 𝑥 𝐽 𝑥 = ∗ ∈ 𝐸 ∗ ∶ ⟨ 𝑥 , 𝑥 ∗ ⟩ = ‖ 𝑥 ‖ 2 = ‖ 𝑥 ∗ ‖ 2 , ∀ 𝑥 ∈ 𝐸 , ( 1 . 1 ) where ⟨ ⋅ , ⋅ ⟩ denotes the generalized duality pairing. Recall that if 𝐸 is smooth, then 𝐽 is single valued and if 𝐸 is uniformly smooth, then 𝐽 is uniformly norm-to-norm continuous on bounded subsets of 𝐸 . We will still denote by 𝐽 the single-valued duality mapping.

A mapping 𝑆 ∶ 𝐶 → 𝐸 is called nonexpansive if ‖ 𝑆 𝑥 − 𝑆 𝑦 ‖ ≤ ‖ 𝑥 − 𝑦 ‖ for all 𝑥 , 𝑦 ∈ 𝐶 . Also a mapping 𝑆 ∶ 𝐶 → 𝐶 is called asymptotically nonexpansive if there exists a sequence { 𝑘 𝑛 } ⊂ [ 1 , ∞ ) with 𝑘 𝑛 → 1 as 𝑛 → ∞ such that ‖ 𝑆 𝑛 𝑥 − 𝑆 𝑛 𝑦 ‖ ≤ 𝑘 𝑛 ‖ 𝑥 − 𝑦 ‖ for all 𝑥 , 𝑦 ∈ 𝐶 and for each 𝑛 ≥ 1 . Denote by 𝐹 ( 𝑆 ) the set of fixed points of 𝑆 , that is, 𝐹 ( 𝑆 ) = { 𝑥 ∈ 𝐶 ∶ 𝑆 𝑥 = 𝑥 } . The following example shows that the class of asymptotically nonexpansive mappings which was first introduced by Goebel and Kirk [1] is wider than the class of nonexpansive mappings.

Example 1.1 (see [2]). Let 𝐵 𝐻 be the closed unit ball in the Hilbert space 𝐻 = 𝑙 2 and 𝑆 ∶ 𝐵 𝐻 → 𝐵 𝐻 a mapping defined by 𝑆 𝑥 1 , 𝑥 2 , 𝑥 3 = , … 0 , 𝑥 2 1 , 𝑎 2 𝑥 2 , 𝑎 3 𝑥 3 , , … ( 1 . 2 ) where { 𝑎 𝑛 } is a sequence of real numbers such that 0 < 𝑎 𝑖 < 1 and ∏ ∞ 𝑖 = 2 𝑎 𝑖 = 1 / 2 . Then ‖ 𝑆 𝑥 − 𝑆 𝑦 ‖ ≤ 2 ‖ 𝑥 − 𝑦 ‖ , ∀ 𝑥 , 𝑦 ∈ 𝐵 𝐻 . ( 1 . 3 ) That is, 𝑆 is Lipschitzian but not nonexpansive. Observe that ‖ 𝑆 𝑛 𝑥 − 𝑆 𝑛 𝑦 ‖ ≤ 2 𝑛 𝑖 = 2 𝑎 𝑖 ‖ 𝑥 − 𝑦 ‖ , ∀ 𝑥 , 𝑦 ∈ 𝐵 𝐻 , 𝑛 ≥ 2 . ( 1 . 4 ) Here 𝑘 𝑛 ∏ = 2 𝑛 𝑖 = 2 𝑎 𝑖 → 1 as 𝑛 → ∞ . Therefore, 𝑆 is asymptotically nonexpansive but not nonexpansive.

A mapping 𝑇 ∶ 𝐶 → 𝐸 ∗ is said to be relaxed 𝜂 - 𝜉 monotone if there exist a mapping 𝜂 ∶ 𝐶 × 𝐶 → 𝐸 and a function 𝜉 ∶ 𝐸 → ℝ positively homogeneous of degree 𝑝 , that is, 𝜉 ( 𝑡 𝑧 ) = 𝑡 𝑝 𝜉 ( 𝑧 ) for all 𝑡 > 0 and 𝑧 ∈ 𝐸 such that ⟨ 𝑇 𝑥 − 𝑇 𝑦 , 𝜂 ( 𝑥 , 𝑦 ) ⟩ ≥ 𝜉 ( 𝑥 − 𝑦 ) , ∀ 𝑥 , 𝑦 ∈ 𝐶 , ( 1 . 5 ) where 𝑝 > 1 is a constant; see [3]. In the case of 𝜂 ( 𝑥 , 𝑦 ) = 𝑥 − 𝑦 for all 𝑥 ,  𝑦 ∈ 𝐶 , 𝑇 is said to be relaxed 𝜉 -monotone. In the case of 𝜂 ( 𝑥 , 𝑦 ) = 𝑥 − 𝑦 for all 𝑥 , 𝑦 ∈ 𝐶 and 𝜉 ( 𝑧 ) = 𝑘 ‖ 𝑧 ‖ 𝑝 , where 𝑝 > 1 and 𝑘 > 0 , 𝑇 is said to be 𝑝 -monotone; see [4–6]. In fact, in this case, if 𝑝 = 2 , then 𝑇 is a 𝑘 -strongly monotone mapping. Moreover, every monotone mapping is relaxed 𝜂 - 𝜉 monotone with 𝜂 ( 𝑥 , 𝑦 ) = 𝑥 − 𝑦 for all 𝑥 , 𝑦 ∈ 𝐶 and 𝜉 = 0 . The following is an example of 𝜂 - 𝜉 monotone mapping which can be found in [3]. Let 𝐶 = ( − ∞ , ∞ ) ,  𝑇 𝑥 = − 𝑥 , and 𝜂 ( 𝑥 , 𝑦 ) = − 𝑐 ( 𝑥 − 𝑦 ) , 𝑥 ≥ 𝑦 , 𝑐 ( 𝑥 − 𝑦 ) , 𝑥 < 𝑦 , ( 1 . 6 ) where 𝑐 > 0 is a constant. Then, 𝑇 is relaxed 𝜂 - 𝜉 monotone with 𝜉 ( 𝑧 ) = 𝑐 𝑧 2 , 𝑧 ≥ 0 , − 𝑐 𝑧 2 , 𝑧 < 0 . ( 1 . 7 ) A mapping 𝑇 ∶ 𝐶 → 𝐸 ∗ is said to be 𝜂 -hemicontinuous if, for each fixed 𝑥 , 𝑦 ∈ 𝐶 , the mapping 𝑓 ∶ [ 0 , 1 ] → ( − ∞ , + ∞ ) defined by 𝑓 ( 𝑡 ) = ⟨ 𝑇 ( 𝑥 + 𝑡 ( 𝑦 − 𝑥 ) ) , 𝜂 ( 𝑦 , 𝑥 ) ⟩ is continuous at 0 + . For a real Banach space 𝐸 with the dual 𝐸 ∗ and for 𝐶 a nonempty closed convex subset of 𝐸 , let 𝑓 ∶ 𝐶 × 𝐶 → ℝ be a bifunction, 𝜑 ∶ 𝐶 → ℝ a real-valued function and 𝑇 ∶ 𝐶 → 𝐸 ∗ be a relaxed 𝜂 - 𝜉 monotone mapping. Recently, Kamraksa and Wangkeeree [7] introduced the following generalized mixed equilibrium problem (GMEP). F i n d 𝑥 ∈ 𝐶 s u c h t h a t 𝑓 ( 𝑥 , 𝑦 ) + ⟨ 𝑇 𝑥 , 𝜂 ( 𝑦 , 𝑥 ) ⟩ + 𝜑 ( 𝑦 ) ≥ 𝜑 ( 𝑥 ) , ∀ 𝑦 ∈ 𝐶 . ( 1 . 8 ) The set of such 𝑥 ∈ 𝐶 is denoted by G M E P ( 𝑓 , 𝑇 ) , that is, G M E P ( 𝑓 , 𝑇 ) = { 𝑥 ∈ 𝐶 ∶ 𝑓 ( 𝑥 , 𝑦 ) + ⟨ 𝑇 𝑥 , 𝜂 ( 𝑦 , 𝑥 ) ⟩ + 𝜑 ( 𝑦 ) ≥ 𝜑 ( 𝑥 ) , ∀ 𝑦 ∈ 𝐶 } . ( 1 . 9 )

Special Cases
(1) If 𝑇 is monotone that is 𝑇 is relaxed 𝜂 - 𝜉 monotone with 𝜂 ( 𝑥 , 𝑦 ) = 𝑥 − 𝑦 for all 𝑥 , 𝑦 ∈ 𝐶 and 𝜉 = 0 , (1.8) is reduced to the following generalized equilibrium problem (GEP). F i n d 𝑥 ∈ 𝐶 s u c h t h a t 𝑓 ( 𝑥 , 𝑦 ) + ⟨ 𝑇 𝑥 , 𝑦 − 𝑥 ⟩ + 𝜑 ( 𝑦 ) ≥ 𝜑 ( 𝑥 ) , ∀ 𝑦 ∈ 𝐶 . ( 1 . 1 0 ) The solution set of (1.10) is denoted by G E P ( 𝑓 ) , that is, G E P ( 𝑓 ) = { 𝑥 ∈ 𝐶 ∶ 𝑓 ( 𝑥 , 𝑦 ) + ⟨ 𝑇 𝑥 , 𝑦 − 𝑥 ⟩ + 𝜑 ( 𝑦 ) ≥ 𝜑 ( 𝑥 ) , ∀ 𝑦 ∈ 𝐶 } . ( 1 . 1 1 )
(2) In the case of 𝑇 ≡ 0 and 𝜑 ≡ 0 , (1.8) is reduced to the following classical equilibrium problem F i n d 𝑥 ∈ 𝐶 s u c h t h a t 𝑓 ( 𝑥 , 𝑦 ) ≥ 0 , ∀ 𝑦 ∈ 𝐶 . ( 1 . 1 2 ) The set of all solution of (1.12) is denoted by E P ( 𝑓 ) , that is, E P ( 𝑓 ) = { 𝑥 ∈ 𝐶 ∶ 𝑓 ( 𝑥 , 𝑦 ) ≥ 0 , ∀ 𝑦 ∈ 𝐶 } . ( 1 . 1 3 )
(3) In the case of 𝑓 ≡ 0 , (1.8) is reduced to the following variational-like inequality problem [3]. F i n d 𝑥 ∈ 𝐶 s u c h t h a t ⟨ 𝑇 𝑥 , 𝜂 ( 𝑦 , 𝑥 ) ⟩ + 𝜑 ( 𝑦 ) − 𝜑 ( 𝑥 ) ≥ 0 , ∀ 𝑦 ∈ 𝐶 . ( 1 . 1 4 )

The generalized mixed equilibrium problem (GMEP) (1.8) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, and Nash equilibrium problems. Using the KKM technique introduced by Kanster et al. [8] and 𝜂 - 𝜉 monotonicity of the mapping 𝜑 , Kamraksa and Wangkeeree [7] obtained the existence of solutions of generalized mixed equilibrium problem (1.8) in a real reflexive Banach space.

Some methods have been proposed to solve the equilibrium problem in a Hilbert space; see, for instance, Blum and Oettli [9], Combettes and Hirstoaga [10], and Moudafi [11]. On the other hand, there are several methods for approximation fixed points of a nonexpansive mapping; see, for instance, [12–17]. Recently, Tada and Takahashi [13, 16] and S. Takahashi and W. Takahashi [17] obtained weak and strong convergence theorems for finding a common elements in the solution set of an equilibrium problem and the set of fixed point of a nonexpansive mapping in a Hilbert space. In particular, Tada and Takahashi [16] established a strong convergence theorem for finding a common element of two sets by using the hybrid method introduced in Nakajo and Takahashi [18]. They also proved such a strong convergence theorem in a uniformly convex and uniformly smooth Banach space.

On the other hand, in 1953, Mann [12] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping 𝑆 in a Hilbert space 𝐻 : 𝑥 𝑛 + 1 = 𝛼 𝑛 𝑥 𝑛 + 1 − 𝛼 𝑛 𝑆 𝑥 𝑛 , ∀ 𝑛 ∈ ℕ , ( 1 . 1 5 ) where the initial point 𝑥 0 is taken in 𝐶 arbitrarily and { 𝛼 𝑛 } is a sequence in [ 0 , 1 ] . However, we note that Manns iteration process (1.15) has only weak convergence, in general; for instance, see [19–21]. In 2003, Nakajo and Takahashi [18] proposed the following sequence for a nonexpansive mapping 𝑆 in a Hilbert space: 𝑥 0 𝑦 = 𝑥 ∈ 𝐶 , 𝑛 = 𝛼 𝑛 𝑥 𝑛 + 1 − 𝛼 𝑛 𝑆 𝑥 𝑛 , 𝐶 𝑛 = ‖ ‖ 𝑦 𝑧 ∈ 𝐶 ∶ 𝑛 ‖ ‖ ≤ ‖ ‖ 𝑥 − 𝑧 𝑛 ‖ ‖ , 𝑄 − 𝑧 𝑛 = 𝑧 ∈ 𝐶 ∶ ⟨ 𝑥 𝑛 − 𝑧 , 𝑥 0 − 𝑥 𝑛 , 𝑥 ⟩ ≥ 0 𝑛 + 1 = 𝑃 𝐶 𝑛 ∩ 𝑄 𝑛 𝑥 0 , ( 1 . 1 6 ) where 0 ≤ 𝛼 𝑛 ≤ 𝑎 < 1 for all 𝑛 ∈ ℕ , and 𝑃 𝐶 𝑛 ∩ 𝐷 𝑛 is the metric projection from 𝐸 onto 𝐶 𝑛 ∩ 𝐷 𝑛 . Then, they proved that { 𝑥 𝑛 } converges strongly to 𝑃 𝐹 ( 𝑇 ) 𝑥 0 . Recently, motivated by Nakajo and Takahashi [18] and Xu [22], Matsushita and Takahashi [14] introduced the iterative algorithm for finding fixed points of nonexpansive mappings in a uniformly convex and smooth Banach space: 𝑥 0 = 𝑥 ∈ 𝐶 and 𝐶 𝑛 = c o 𝑧 ∈ 𝐶 ∶ ‖ 𝑧 − 𝑆 𝑧 ‖ ≤ 𝑡 𝑛 ‖ ‖ 𝑥 𝑛 − 𝑆 𝑥 𝑛 ‖ ‖ , 𝐷 𝑛 = 𝑥 𝑧 ∈ 𝐶 ∶ 𝑛 − 𝑧 , 𝐽 𝑥 − 𝑥 𝑛 , 𝑥 ≥ 0 𝑛 + 1 = 𝑃 𝐶 𝑛 ∩ 𝐷 𝑛 𝑥 , 𝑛 ≥ 0 , ( 1 . 1 7 ) where c o 𝐷 denotes the convex closure of the set 𝐷 , { 𝑡 𝑛 } is a sequence in (0, 1) with 𝑡 𝑛 → 0 . They proved that { 𝑥 𝑛 } generated by (1.17) converges strongly to a fixed point of 𝑆 . Very recently, Dehghan [23] investigated iterative schemes for finding fixed point of an asymptotically nonexpansive mapping and proved strong convergence theorems in a uniformly convex and smooth Banach space. More precisely, he proposed the following algorithm: 𝑥 1 = 𝑥 ∈ 𝐶 , 𝐶 0 = 𝐷 0 = 𝐶 and 𝐶 𝑛 = c o 𝑧 ∈ 𝐶 𝑛 − 1 ∶ ‖ 𝑧 − 𝑆 𝑛 𝑧 ‖ ≤ 𝑡 𝑛 ‖ ‖ 𝑥 𝑛 − 𝑆 𝑛 𝑥 𝑛 ‖ ‖ , 𝐷 𝑛 = 𝑧 ∈ 𝐷 𝑛 − 1 ∶ 𝑥 𝑛 − 𝑧 , 𝐽 𝑥 − 𝑥 𝑛 , 𝑥 ≥ 0 𝑛 + 1 = 𝑃 𝐶 𝑛 ∩ 𝐷 𝑛 𝑥 , 𝑛 ≥ 0 , ( 1 . 1 8 ) where { 𝑡 𝑛 } is a sequence in ( 0 , 1 ) with 𝑡 𝑛 → 0 as 𝑛 → ∞ and 𝑆 is an asymptotically nonexpansive mapping. It is proved in [23] that { 𝑥 𝑛 } converges strongly to a fixed point of 𝑆 .

On the other hand, recently, Kamraksa and Wangkeeree [7] studied the hybrid projection algorithm for finding a common element in the solution set of the GMEP and the common fixed point set of a countable family of nonexpansive mappings in a uniformly convex and smooth Banach space.

Motivated by the above mentioned results and the on-going research, we first prove the existence results of solutions for GMEP under the new conditions imposed on the bifunction 𝑓 . Next, we introduce the following iterative algorithm for finding a common element in the solution set of the GMEP and the common fixed point set of a finite family of asymptotically nonexpansive mappings { 𝑆 1 , 𝑆 2 , … , 𝑆 𝑁 } in a uniformly convex and smooth Banach space: 𝑥 0 ∈ 𝐶 , 𝐷 0 = 𝐶 0 = 𝐶 , and 𝑥 1 = 𝑃 𝐶 0 ∩ 𝐷 0 𝑥 0 = 𝑃 𝐶 𝑥 0 , 𝐶 1 = ‖ ‖ c o 𝑧 ∈ 𝐶 ∶ 𝑧 − 𝑆 1 𝑧 ‖ ‖ ≤ 𝑡 1 ‖ ‖ 𝑥 1 − 𝑆 1 𝑥 1 ‖ ‖ , 𝑢 1 𝑢 ∈ 𝐶 s u c h t h a t 𝑓 1 , 𝑦 + 𝜑 ( 𝑦 ) + 𝑇 𝑢 1 , 𝜂 𝑦 , 𝑢 1 + 1 𝑟 1 𝑦 − 𝑢 1 𝑢 , 𝐽 1 − 𝑥 1 𝐷 , ∀ 𝑦 ∈ 𝐶 , 1 = 𝑢 𝑧 ∈ 𝐶 ∶ 1 𝑥 − 𝑧 , 𝐽 1 − 𝑢 1 , 𝑥 ≥ 0 2 = 𝑃 𝐶 1 ∩ 𝐷 1 𝑥 0 , ⋮ 𝐶 𝑁 = c o 𝑧 ∈ 𝐶 𝑁 − 1 ∶ ‖ ‖ 𝑧 − 𝑆 𝑁 𝑧 ‖ ‖ ≤ 𝑡 1 ‖ ‖ 𝑥 𝑁 − 𝑆 𝑁 𝑥 𝑁 ‖ ‖ , 𝑢 𝑁 𝑢 ∈ 𝐶 s u c h t h a t 𝑓 𝑁 , 𝑦 + 𝜑 ( 𝑦 ) + 𝑇 𝑢 𝑁 , 𝜂 𝑦 , 𝑢 𝑁 + 1 𝑟 𝑁 𝑦 − 𝑢 𝑁 𝑢 , 𝐽 𝑁 − 𝑥 𝑁 𝐷 , ∀ 𝑦 ∈ 𝐶 , 𝑁 = 𝑧 ∈ 𝐷 𝑁 − 1 ∶ 𝑢 𝑁 𝑥 − 𝑧 , 𝐽 𝑁 − 𝑢 𝑁 , 𝑥 ≥ 0 𝑁 + 1 = 𝑃 𝐶 𝑁 ∩ 𝐷 𝑁 𝑥 0 , 𝐶 𝑁 + 1 = c o 𝑧 ∈ 𝐶 𝑁 ∶ ‖ ‖ 𝑧 − 𝑆 2 1 𝑧 ‖ ‖ ≤ 𝑡 1 ‖ ‖ 𝑥 𝑁 + 1 − 𝑆 2 1 𝑥 𝑁 + 1 ‖ ‖ , 𝑢 𝑁 + 1 𝑢 ∈ 𝐶 s u c h t h a t 𝑓 𝑁 + 1 , 𝑦 + 𝜑 ( 𝑦 ) + 𝑇 𝑢 𝑁 + 1 , 𝜂 𝑦 , 𝑢 𝑁 + 1 + 1 𝑟 𝑁 + 1 𝑦 − 𝑢 𝑁 + 1 𝑢 , 𝐽 𝑁 + 1 − 𝑥 𝑁 + 1 𝐷 , ∀ 𝑦 ∈ 𝐶 , 𝑁 + 1 = 𝑧 ∈ 𝐷 𝑁 ∶ 𝑢 𝑁 + 1 𝑥 − 𝑧 , 𝐽 𝑁 + 1 − 𝑢 𝑁 + 1 , 𝑥 ≥ 0 𝑁 + 2 = 𝑃 𝐶 𝑁 + 1 ∩ 𝐷 𝑁 + 1 𝑥 0 , ⋮ 𝐶 2 𝑁 = c o 𝑧 ∈ 𝐶 2 𝑁 − 1 ∶ ‖ ‖ 𝑧 − 𝑆 2 𝑁 𝑧 ‖ ‖ ≤ 𝑡 1 ‖ ‖ 𝑥 2 𝑁 − 𝑆 2 𝑁 𝑥 2 𝑁 ‖ ‖ , 𝑢 2 𝑁 𝑢 ∈ 𝐶 s u c h t h a t 𝑓 2 𝑁 , 𝑦 + 𝜑 ( 𝑦 ) + 𝑇 𝑢 2 𝑁 , 𝜂 𝑦 , 𝑢 2 𝑁 + 1 𝑟 2 𝑁 𝑦 − 𝑢 2 𝑁 𝑢 , 𝐽 2 𝑁 − 𝑥 2 𝑁 𝐷 , ∀ 𝑦 ∈ 𝐶 , 2 𝑁 = 𝑧 ∈ 𝐷 2 𝑁 − 1 ∶ 𝑢 2 𝑁 𝑥 − 𝑧 , 𝐽 2 𝑁 − 𝑢 2 𝑁 , 𝑥 ≥ 0 2 𝑁 + 1 = 𝑃 𝐶 2 𝑁 ∩ 𝐷 2 𝑁 𝑥 0 , 𝐶 2 𝑁 + 1 = c o 𝑧 ∈ 𝐶 2 𝑁 ∶ ‖ ‖ 𝑧 − S 3 1 𝑧 ‖ ‖ ≤ 𝑡 1 ‖ ‖ 𝑥 2 𝑁 + 1 − 𝑆 3 1 𝑥 2 𝑁 + 1 ‖ ‖ , 𝑢 2 𝑁 + 1 𝑢 ∈ 𝐶 s u c h t h a t 𝑓 2 𝑁 + 1 , 𝑦 + 𝜑 ( 𝑦 ) + 𝑇 𝑢 2 𝑁 + 1 , 𝜂 𝑦 , 𝑢 2 𝑁 + 1 + 1 𝑟 2 𝑁 + 1 𝑦 − 𝑢 2 𝑁 + 1 𝑢 , 𝐽 2 𝑁 + 1 − 𝑥 2 𝑁 + 1 𝐷 , ∀ 𝑦 ∈ 𝐶 , 2 𝑁 + 1 = 𝑧 ∈ 𝐷 2 𝑁 ∶ 𝑢 2 𝑁 + 1 𝑥 − 𝑧 , 𝐽 2 𝑁 + 1 − 𝑢 2 𝑁 + 1 , 𝑥 ≥ 0 2 𝑁 + 2 = 𝑃 𝐶 2 𝑁 + 1 ∩ 𝐷 2 𝑁 + 1 𝑥 0 , ⋮ ( 1 . 1 9 ) The above algorithm is called the hybrid iterative algorithm for a finite family of asymptotically nonexpansive mappings from 𝐶 into itself. Since, for each 𝑛 ≥ 1 , it can be written as 𝑛 = ( ℎ − 1 ) 𝑁 + 𝑖 , where 𝑖 = 𝑖 ( 𝑛 ) ∈ { 1 , 2 , … , 𝑁 } , ℎ = ℎ ( 𝑛 ) ≥ 1 is a positive integer and ℎ ( 𝑛 ) → ∞ as 𝑛 → ∞ . Hence the above table can be written in the following form: 𝑥 0 ∈ 𝐶 , 𝐷 0 = 𝐶 0 𝐶 = 𝐶 , 𝑛 = c o 𝑧 ∈ 𝐶 𝑛 − 1 ∶ ‖ ‖ 𝑧 − 𝑆 ℎ ( 𝑛 ) 𝑖 ( 𝑛 ) 𝑧 ‖ ‖ ≤ 𝑡 𝑛 ‖ ‖ 𝑥 𝑛 − 𝑆 ℎ ( 𝑛 ) 𝑖 ( 𝑛 ) 𝑥 𝑛 ‖ ‖ 𝑢 , 𝑛 ≥ 1 , 𝑛 𝑢 ∈ 𝐶 s u c h t h a t 𝑓 𝑛 , 𝑦 + 𝜑 ( 𝑦 ) + 𝑇 𝑢 𝑛 , 𝜂 𝑦 , 𝑢 𝑛 + 1 𝑟 𝑛 𝑦 − 𝑢 𝑛 𝑢 , 𝐽 𝑛 − 𝑥 𝑛 𝐷 , ∀ 𝑦 ∈ 𝐶 , 𝑛 ≥ 1 , 𝑛 = 𝑧 ∈ 𝐷 𝑛 − 1 ∶ 𝑢 𝑛 𝑥 − 𝑧 , 𝐽 𝑛 − 𝑢 𝑛 𝑥 ≥ 0 , 𝑛 ≥ 1 , 𝑛 + 1 = 𝑃 𝐶 𝑛 ∩ 𝐷 𝑛 𝑥 0 , 𝑛 ≥ 0 . ( 1 . 2 0 ) Strong convergence theorems are obtained in a uniformly convex and smooth Banach space. The results presented in this paper extend and improve the corresponding Kimura and Nakajo [24], Kamraksa and Wangkeeree [7], Dehghan [23], and many others.

2. Preliminaries

Let 𝐸 be a real Banach space and let 𝑈 = { 𝑥 ∈ 𝐸 ∶ ‖ 𝑥 ‖ = 1 } be the unit sphere of 𝐸 . A Banach space 𝐸 is said to be strictly convex if for any 𝑥 , 𝑦 ∈ 𝑈 , 𝑥 ≠ 𝑦 i m p l i e s ‖ 𝑥 + 𝑦 ‖ < 2 . ( 2 . 1 ) It is also said to be uniformly convex if for each 𝜀 ∈ ( 0 , 2 ] , there exists 𝛿 > 0 such that for any 𝑥 , 𝑦 ∈ 𝑈 , ‖ 𝑥 − 𝑦 ‖ ≥ 𝜀 i m p l i e s ‖ 𝑥 + 𝑦 ‖ < 2 ( 1 − 𝛿 ) . ( 2 . 2 ) It is known that a uniformly convex Banach space is reflexive and strictly convex. Define a function 𝛿 ∶ [ 0 , 2 ] → [ 0 , 1 ] called the modulus of convexity of 𝐸 as follows: ‖ ‖ ‖ 𝛿 ( 𝜀 ) = i n f 1 − 𝑥 + 𝑦 2 ‖ ‖ ‖ . ∶ 𝑥 , 𝑦 ∈ 𝐸 , ‖ 𝑥 ‖ = ‖ 𝑦 ‖ = 1 , ‖ 𝑥 − 𝑦 ‖ ≥ 𝜀 ( 2 . 3 ) Then 𝐸 is uniformly convex if and only if 𝛿 ( 𝜀 ) > 0 for all 𝜀 ∈ ( 0 , 2 ] . A Banach space 𝐸 is said to be smooth if the limit l i m 𝑡 → 0 ‖ 𝑥 + 𝑡 𝑦 ‖ − ‖ 𝑥 ‖ 𝑡 ( 2 . 4 ) exists for all 𝑥 , 𝑦 ∈ 𝑈 . Let 𝐶 be a nonempty, closed, and convex subset of a reflexive, strictly convex, and smooth Banach space 𝐸 . Then for any 𝑥 ∈ 𝐸 , there exists a unique point 𝑥 0 ∈ 𝐶 such that ‖ ‖ 𝑥 0 ‖ ‖ − 𝑥 ≤ m i n 𝑦 ∈ 𝐶 ‖ 𝑦 − 𝑥 ‖ . ( 2 . 5 ) The mapping 𝑃 𝐶 ∶ 𝐸 → 𝐶 defined by 𝑃 𝐶 𝑥 = 𝑥 0 is called the metric projection from 𝐸 onto 𝐶 . Let 𝑥 ∈ 𝐸 and 𝑢 ∈ 𝐶 . The following theorem is well known.

Theorem 2.1. Let 𝐶 be a nonempty convex subset of a smooth Banach space 𝐸 and let 𝑥 ∈ 𝐸 and 𝑦 ∈ 𝐶 . Then the following are equivalent: (a) 𝑦 is a best approximation to 𝑥 ∶ 𝑦 = 𝑃 𝐶 𝑥 ,(b) 𝑦 is a solution of the variational inequality: ⟨ 𝑦 − 𝑧 , 𝐽 ( 𝑥 − 𝑦 ) ⟩ ≥ 0 ∀ 𝑧 ∈ 𝐶 , ( 2 . 6 ) where 𝐽 is a duality mapping and 𝑃 𝐶 is the metric projection from 𝐸 onto 𝐶 .

It is well known that if 𝑃 𝐶 is a metric projection from a real Hilbert space 𝐻 onto a nonempty, closed, and convex subset 𝐶 , then 𝑃 𝐶 is nonexpansive. But, in a general Banach space, this fact is not true.

In the sequel one will need the following lemmas.

Lemma 2.2 (see [25]). Let 𝐸 be a uniformly convex Banach space, let { 𝛼 𝑛 } be a sequence of real numbers such that 0 < 𝑏 ≤ 𝛼 𝑛 ≤ 𝑐 < 1 for all 𝑛 ≥ 1 , and let { 𝑥 𝑛 } and { 𝑦 𝑛 } be sequences in 𝐸 such that l i m s u p 𝑛 → ∞ ‖ 𝑥 𝑛 ‖ ≤ 𝑑 , l i m s u p 𝑛 → ∞ ‖ 𝑦 𝑛 ‖ ≤ 𝑑 and l i m 𝑛 → ∞ ‖ 𝛼 𝑛 𝑥 𝑛 + ( 1 − 𝛼 𝑛 ) 𝑦 𝑛 ‖ = 𝑑 . Then l i m 𝑛 → ∞ ‖ 𝑥 𝑛 − 𝑦 𝑛 ‖ = 0 .

Dehghan [23] obtained the following useful result.

Theorem 2.3 (see [23]). Let 𝐶 be a bounded, closed, and convex subset of a uniformly convex Banach space 𝐸 . Then there exists a strictly increasing, convex, and continuous function 𝛾 ∶ [ 0 , ∞ ) → [ 0 , ∞ ) such that 𝛾 ( 0 ) = 0 and 𝛾 1 𝑘 𝑚 ‖ ‖ ‖ ‖ 𝑆 𝑚 𝑛 𝑖 = 1 𝜆 𝑖 𝑥 𝑖 − 𝑛 𝑖 = 1 𝜆 𝑖 𝑆 𝑚 𝑥 𝑖 ‖ ‖ ‖ ‖ ≤ m a x 1 ≤ 𝑗 ≤ 𝑘 ≤ 𝑛 ‖ ‖ 𝑥 𝑗 − 𝑥 𝑘 ‖ ‖ − 1 𝑘 𝑚 ‖ ‖ 𝑆 𝑚 𝑥 𝑗 − 𝑆 𝑚 𝑥 𝑘 ‖ ‖ ( 2 . 7 ) for any asymptotically nonexpansive mapping 𝑆 of 𝐶 into 𝐶 with { 𝑘 𝑛 } , any elements 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ∈ 𝐶 , any numbers 𝜆 1 , 𝜆 2 , … , 𝜆 𝑛 ≥ 0 with ∑ 𝑛 𝑖 = 1 𝜆 𝑖 = 1 and each 𝑚 ≥ 1 .

Lemma 2.4 (see [26, Lemma 1.6]). Let 𝐸 be a uniformly convex Banach space, 𝐶 be a nonempty closed convex subset of 𝐸 and 𝑆 ∶ 𝐶 → 𝐶 be an asymptotically nonexpansive mapping. Then ( 𝐼 − 𝑆 ) is demiclosed at 0 , that is, if 𝑥 𝑛 ⇀ 𝑥 and ( 𝐼 − 𝑆 ) 𝑥 𝑛 → 0 , then 𝑥 ∈ 𝐹 ( 𝑆 ) .

The following lemma can be found in [7].

Lemma 2.5 (see [7, Lemma 3.2]). Let 𝐶 be a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸 , let 𝑇 ∶ 𝐶 → 𝐸 ∗ be an 𝜂 -hemicontinuous and relaxed 𝜂 − 𝜉 monotone mapping. Let 𝑓 be a bifunction from 𝐶 × 𝐶 to ℛ satisfying (A1), (A3), and (A4) and let 𝜑 be a lower semicontinuous and convex function from 𝐶 to ℛ . Let 𝑟 > 0 and 𝑧 ∈ 𝐶 . Assume that (i) 𝜂 ( 𝑥 , 𝑦 ) + 𝜂 ( 𝑦 , 𝑥 ) = 0 for all 𝑥 , 𝑦 ∈ 𝐶 ; (ii)for any fixed 𝑢 , 𝑣 ∈ 𝐶 , the mapping 𝑥 ↦ ⟨ 𝑇 𝑣 , 𝜂 ( 𝑥 , 𝑢 ) ⟩ is convex and lower semicontinuous; (iii) 𝜉 ∶ 𝐸 → ℛ is weakly lower semicontinuous, that is, for any net { 𝑥 𝛽 } , 𝑥 𝛽 converges to 𝑥 in 𝜎 ( 𝐸 , 𝐸 ∗ ) which implies that 𝜉 ( 𝑥 ) ≤ l i m i n f 𝜉 ( 𝑥 𝛽 ) . Then there exists 𝑥 0 ∈ 𝐶 such that 𝑓 𝑥 0 + , 𝑦 𝑇 𝑥 0 , 𝜂 𝑦 , 𝑥 0 1 + 𝜑 ( 𝑦 ) + 𝑟 𝑦 − 𝑥 0 𝑥 , 𝐽 0 𝑥 − 𝑧 ≥ 𝜑 0 , ∀ 𝑦 ∈ 𝐶 . ( 2 . 8 )

Lemma 2.6 (see [7, Lemma 3.3]). Let 𝐶 be a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸 , let 𝑇 ∶ 𝐶 → 𝐸 ∗ be an 𝜂 -hemicontinuous and relaxed 𝜂 - 𝜉 monotone mapping. Let 𝑓 be a bifunction from 𝐶 × 𝐶 to ℛ satisfying (A1)–(A4) and let 𝜑 be a lower semicontinuous and convex function from 𝐶 to ℛ . Let 𝑟 > 0 and define a mapping Φ 𝑟 ∶ 𝐸 → 𝐶 as follows: Φ 𝑟 1 ( 𝑥 ) = 𝑧 ∈ 𝐶 ∶ 𝑓 ( 𝑧 , 𝑦 ) + ⟨ 𝑇 𝑧 , 𝜂 ( 𝑦 , 𝑧 ) ⟩ + 𝜑 ( 𝑦 ) + 𝑟 ⟨ 𝑦 − 𝑧 , 𝐽 ( 𝑧 − 𝑥 ) ⟩ ≥ 𝜑 ( 𝑧 ) , ∀ 𝑦 ∈ C ( 2 . 9 ) for all 𝑥 ∈ 𝐸 . Assume that (i) 𝜂 ( 𝑥 , 𝑦 ) + 𝜂 ( 𝑦 , 𝑥 ) = 0 , for all 𝑥 , 𝑦 ∈ 𝐶 ; (ii)for any fixed 𝑢 , 𝑣 ∈ 𝐶 , the mapping 𝑥 ↦ ⟨ 𝑇 𝑣 , 𝜂 ( 𝑥 , 𝑢 ) ⟩ is convex and lower semicontinuous and the mapping 𝑥 ↦ ⟨ 𝑇 𝑢 , 𝜂 ( 𝑣 , 𝑥 ) ⟩ is lower semicontinuous; (iii) 𝜉 ∶ 𝐸 → ℛ is weakly lower semicontinuous; (iv)for any 𝑥 , 𝑦 ∈ 𝐶 , 𝜉 ( 𝑥 − 𝑦 ) + 𝜉 ( 𝑦 − 𝑥 ) ≥ 0 . Then, the following holds: (1) Φ 𝑟 is single valued; (2) ⟨ Φ 𝑟 𝑥 − Φ 𝑟 𝑦 , 𝐽 ( Φ 𝑟 𝑥 − 𝑥 ) ⟩ ≤ ⟨ Φ 𝑟 𝑥 − Φ 𝑟 𝑦 , 𝐽 ( Φ 𝑟 𝑦 − 𝑦 ) ⟩ for all 𝑥 , 𝑦 ∈ 𝐸 ; (3) 𝐹 ( Φ 𝑟 ) = E P ( 𝑓 , 𝑇 ) ; (4) E P ( 𝑓 , 𝑇 ) is nonempty closed and convex.

3. Existence of Solutions for GMEP

In this section, we prove the existence results of solutions for GMEP under the new conditions imposed on the bifunction 𝑓 .

Theorem 3.1. Let 𝐶 be a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸 , let 𝑇 ∶ 𝐶 → 𝐸 ∗ be an 𝜂 -hemicontinuous and relaxed 𝜂 - 𝜉 monotone mapping. Let 𝑓 be a bifunction from 𝐶 × 𝐶 to ℝ satisfying the following conditions (A1)–(A4): (A1) 𝑓 ( 𝑥 , 𝑥 ) = 0 for all 𝑥 ∈ 𝐶 ; (A2) 𝑓 ( 𝑥 , 𝑦 ) + 𝑓 ( 𝑦 , 𝑥 ) ≤ m i n { 𝜉 ( 𝑥 − 𝑦 ) , 𝜉 ( 𝑦 − 𝑥 ) } for all 𝑥 , 𝑦 ∈ 𝐶 ; (A3)for all 𝑦 ∈ 𝐶 , 𝑓 ( ⋅ , 𝑦 ) is weakly upper semicontinuous; (A4)for all 𝑥 ∈ 𝐶 , 𝑓 ( 𝑥 , ⋅ ) is convex. For any 𝑟 > 0 and 𝑥 ∈ 𝐸 , define a mapping Φ 𝑟 ∶ 𝐸 → 𝐶 as follows: Φ 𝑟 1 ( 𝑥 ) = 𝑧 ∈ 𝐶 ∶ 𝑓 ( 𝑧 , 𝑦 ) + ⟨ 𝑇 𝑧 , 𝜂 ( 𝑦 , 𝑧 ) ⟩ + 𝜑 ( 𝑦 ) + 𝑟 , ⟨ 𝑦 − 𝑧 , 𝐽 ( 𝑧 − 𝑥 ) ⟩ ≥ 𝜑 ( 𝑧 ) , ∀ 𝑦 ∈ 𝐶 ( 3 . 1 ) where 𝜑 is a lower semicontinuous and convex function from 𝐶 to ℝ . Assume that (i) 𝜂 ( 𝑥 , 𝑦 ) + 𝜂 ( 𝑦 , 𝑥 ) = 0 , for all 𝑥 , 𝑦 ∈ 𝐶 ; (ii)for any fixed 𝑢 , 𝑣 ∈ 𝐶 , the mapping 𝑥 ↦ ⟨ 𝑇 𝑣 , 𝜂 ( 𝑥 , 𝑢 ) ⟩ is convex and lower semicontinuous and the mapping 𝑥 ↦ ⟨ 𝑇 𝑢 , 𝜂 ( 𝑣 , 𝑥 ) ⟩ is lower semicontinuous; (iii) 𝜉 ∶ 𝐸 → ℝ is weakly lower semicontinuous. Then, the following holds: (1) Φ 𝑟 is single valued; (2) ⟨ Φ 𝑟 𝑥 − Φ 𝑟 𝑦 , 𝐽 ( Φ 𝑟 𝑥 − 𝑥 ) ⟩ ≤ ⟨ Φ 𝑟 𝑥 − Φ 𝑟 𝑦 , 𝐽 ( Φ 𝑟 𝑦 − 𝑦 ) ⟩ for all 𝑥 , 𝑦 ∈ 𝐸 ; (3) 𝐹 ( Φ 𝑟 ) = G M E P ( 𝑓 , 𝑇 ) ; (4) G M E P ( 𝑓 , 𝑇 ) is nonempty closed and convex.

Proof. For each 𝑥 ∈ 𝐸 . It follows from Lemma 2.5 that Φ 𝑟 ( 𝑥 ) is nonempty.
(1) We prove that Φ 𝑟 is single valued. Indeed, for 𝑥 ∈ 𝐸 and 𝑟 > 0 , let 𝑧 1 , 𝑧 2 ∈ Φ 𝑟 𝑥 . Then 𝑓 𝑧 1 , 𝑧 2 + 𝑇 𝑧 2 𝑧 , 𝜂 2 , 𝑧 1 𝑧 + 𝜑 2 + 1 𝑟 𝑧 1 − 𝑧 2 𝑧 , 𝐽 1 𝑧 − 𝑥 ≥ 𝜑 1 , 𝑓 𝑧 2 , 𝑧 1 + 𝑇 𝑧 1 𝑧 , 𝜂 1 , 𝑧 2 𝑧 + 𝜑 1 + 1 𝑟 𝑧 2 − 𝑧 1 𝑧 , 𝐽 2 𝑧 − 𝑥 ≥ 𝜑 2 . ( 3 . 2 ) Adding the two inequalities, from (i) we have 𝑓 𝑧 2 , 𝑧 1 𝑧 + 𝑓 1 , 𝑧 2 + ⟨ 𝑇 𝑧 1 − 𝑇 𝑧 2 𝑧 , 𝜂 2 , 𝑧 1 1 ⟩ + 𝑟 𝑧 2 − 𝑧 1 𝑧 , 𝐽 1 𝑧 − 𝑥 − 𝐽 2 − 𝑥 ≥ 0 . ( 3 . 3 ) Setting Δ ∶ = m i n { 𝜉 ( 𝑧 1 − 𝑧 2 ) , 𝜉 ( 𝑧 2 − 𝑧 1 ) } and using (A2), we have Δ + 𝑇 𝑧 1 − 𝑇 𝑧 2 𝑧 , 𝜂 2 , 𝑧 1 + 1 𝑟 𝑧 2 − 𝑧 1 𝑧 , 𝐽 1 𝑧 − 𝑥 − 𝐽 2 − 𝑥 ≥ 0 , ( 3 . 4 ) that is, 1 𝑟 𝑧 2 − 𝑧 1 𝑧 , 𝐽 1 𝑧 − 𝑥 − 𝐽 2 ≥ − 𝑥 𝑇 𝑧 2 − 𝑇 𝑧 1 𝑧 , 𝜂 2 , 𝑧 1 − Δ . ( 3 . 5 ) Since 𝑇 is relaxed 𝜂 - 𝜉 monotone and 𝑟 > 0 , one has 𝑧 2 − 𝑧 1 𝑧 , 𝐽 1 𝑧 − 𝑥 − 𝐽 2 𝜉 𝑧 − 𝑥 ≥ 𝑟 2 − 𝑧 1 − Δ ≥ 0 . ( 3 . 6 ) In (3.5) exchanging the position of 𝑧 1 and 𝑧 2 , we get 1 𝑟 𝑧 1 − 𝑧 2 𝑧 , 𝐽 2 𝑧 − 𝑥 − 𝐽 1 ≥ − 𝑥 𝑇 𝑧 1 − 𝑇 𝑧 2 𝑧 , 𝜂 1 , 𝑧 2 − Δ , ( 3 . 7 ) that is, 𝑧 1 − 𝑧 2 𝑧 , 𝐽 2 𝑧 − 𝑥 − 𝐽 1 𝜉 𝑧 − 𝑥 ≥ 𝑟 1 − 𝑧 2 − Δ ≥ 0 . ( 3 . 8 ) Now, adding the inequalities (3.6) and (3.8), we have 2 𝑧 2 − 𝑧 1 𝑧 , 𝐽 1 𝑧 − 𝑥 − 𝐽 2 − 𝑥 ≥ 0 . ( 3 . 9 ) Hence, 𝑧 0 ≤ 2 − 𝑧 1 𝑧 , 𝐽 1 𝑧 − 𝑥 − 𝐽 2 = 𝑧 − 𝑥 2 − 𝑧 − 𝑥 1 𝑧 − 𝑥 , 𝐽 1 𝑧 − 𝑥 − 𝐽 2 − 𝑥 . ( 3 . 1 0 ) Since 𝐽 is monotone and 𝐸 is strictly convex, we obtain that 𝑧 1 − 𝑥 = 𝑧 2 − 𝑥 and hence 𝑧 1 = 𝑧 2 . Therefore 𝑆 𝑟 is single valued.
(2) For 𝑥 , 𝑦 ∈ 𝐶 , we have 𝑓 Φ 𝑟 𝑥 , Φ 𝑟 𝑦 + 𝑇 Φ 𝑟 Φ 𝑥 , 𝜂 𝑟 𝑦 , Φ 𝑟 𝑥 Φ + 𝜑 𝑟 𝑦 Φ − 𝜑 𝑟 𝑥 + 1 𝑟 Φ 𝑟 𝑦 − Φ 𝑟 Φ 𝑥 , 𝐽 𝑟 𝑓 Φ 𝑥 − 𝑥 ≥ 0 , 𝑟 𝑦 , Φ 𝑟 𝑥 + 𝑇 Φ 𝑟 Φ 𝑦 , 𝜂 𝑟 𝑥 , Φ 𝑟 𝑦 Φ + 𝜑 𝑟 𝑥 Φ − 𝜑 𝑟 𝑦 + 1 𝑟 Φ 𝑟 𝑥 − Φ 𝑟 Φ 𝑦 , 𝐽 𝑟 𝑦 − 𝑦 ≥ 0 . ( 3 . 1 1 ) Setting Λ 𝑥 , 𝑦 ∶ = m i n { 𝜉 ( Φ 𝑟 𝑥 − Φ 𝑟 𝑦 ) , 𝜉 ( Φ 𝑟 𝑦 − Φ 𝑟 𝑥 ) } and applying (A2), we get 𝑇 Φ 𝑟 𝑥 − 𝑇 Φ 𝑟 Φ 𝑦 , 𝜂 𝑟 𝑦 , Φ 𝑟 𝑥 + 1 𝑟 Φ 𝑟 𝑦 − Φ 𝑟 Φ 𝑥 , 𝐽 𝑟 Φ 𝑥 − 𝑥 − 𝐽 𝑟 𝑦 − 𝑦 ≥ − Λ 𝑥 , 𝑦 , ( 3 . 1 2 ) that is, 1 𝑟 Φ 𝑟 𝑦 − Φ 𝑟 Φ 𝑥 , 𝐽 𝑟 Φ 𝑥 − 𝑥 − 𝐽 𝑟 ≥ 𝑦 − 𝑦 𝑇 Φ 𝑟 𝑦 − 𝑇 Φ 𝑟 Φ 𝑥 , 𝜂 𝑟 𝑦 , Φ 𝑟 𝑥 − Λ 𝑥 , 𝑦 Φ ≥ 𝜉 𝑟 𝑦 − Φ 𝑟 𝑥 − Λ 𝑥 , 𝑦 ≥ 0 . ( 3 . 1 3 ) In (3.13) exchanging the position of Φ 𝑟 𝑥 and Φ 𝑟 𝑦 , we get 1 𝑟 Φ 𝑟 𝑥 − Φ 𝑟 Φ 𝑦 , 𝐽 𝑟 Φ 𝑦 − 𝑦 − 𝐽 𝑟 𝑥 − 𝑥 ≥ 0 . ( 3 . 1 4 ) Adding the inequalities (3.13) and (3.14), we have 2 𝑟 Φ 𝑟 𝑦 − Φ 𝑟 Φ 𝑥 , 𝐽 𝑟 Φ 𝑥 − 𝑥 − 𝐽 𝑟 𝑦 − 𝑦 ≥ 0 . ( 3 . 1 5 ) It follows that Φ 𝑟 𝑦 − Φ 𝑟 Φ 𝑥 , 𝐽 𝑟 Φ 𝑥 − 𝑥 − 𝐽 𝑟 𝑦 − 𝑦 ≥ 0 . ( 3 . 1 6 ) Hence Φ 𝑟 𝑥 − Φ 𝑟 Φ 𝑦 , 𝐽 𝑟 ≤ Φ 𝑥 − 𝑥 𝑟 𝑥 − Φ 𝑟 Φ 𝑦 , 𝐽 𝑟 𝑦 − 𝑦 . ( 3 . 1 7 ) The conclusions (3), (4) follow from Lemma 2.6.

Example 3.2. Define 𝜉 ∶ ℝ → ℝ and 𝑓 ∶ ℝ × ℝ → ℝ by 𝑓 ( 𝑥 , 𝑦 ) = ( 𝑥 − 𝑦 ) 2 2 , 𝜉 ( 𝑥 ) = 𝑥 2 ∀ 𝑥 , 𝑦 ∈ ℝ . ( 3 . 1 8 ) It is easy to see that 𝑓 satisfies (A1), (A3), (A4), and (A2): 𝑓 ( 𝑥 , 𝑦 ) + 𝑓 ( 𝑦 , 𝑥 ) ≤ m i n { 𝜉 ( 𝑥 − 𝑦 ) , 𝜉 ( 𝑥 − 𝑦 ) } , for  all ( 𝑥 , 𝑦 ) ∈ ℝ × ℝ .

Remark 3.3. Theorem 3.1 generalizes and improves [7, Lemma 3.3] in the following manners.(1)The condition 𝑓 ( 𝑥 , 𝑦 ) + 𝑓 ( 𝑦 , 𝑥 ) ≤ 0 has been weakened by (A2) that is 𝑓 ( 𝑥 , 𝑦 ) + 𝑓 ( 𝑦 , 𝑥 ) ≤ m i n { 𝜉 ( 𝑥 − 𝑦 ) , 𝜉 ( 𝑦 − 𝑥 ) } for all 𝑥 , 𝑦 ∈ 𝐶 .(2)The control condition 𝜉 ( 𝑥 − 𝑦 ) + 𝜉 ( 𝑦 − 𝑥 ) ≥ 0 imposed on the mapping 𝜉 in [7, Lemma 3.3] can be removed.
If 𝑇 is monotone that is 𝑇 is relaxed 𝜂 - 𝜉 monotone with 𝜂 ( 𝑥 , 𝑦 ) = 𝑥 − 𝑦 for all 𝑥 , 𝑦 ∈ 𝐶 and 𝜉 = 0 , we have the following results.

Corollary 3.4. Let 𝐶 be a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸 . Let 𝑇 ∶ 𝐶 → 𝐸 ∗ be a monotone mapping and 𝑓 be a bifunction from 𝐶 × 𝐶 to ℝ satisfying the following conditions (i)–(iv): (i) 𝑓 ( 𝑥 , 𝑥 ) = 0 for all 𝑥 ∈ 𝐶 ; (ii) 𝑓 ( 𝑥 , 𝑦 ) + 𝑓 ( 𝑦 , 𝑥 ) ≤ 0 for all 𝑥 , 𝑦 ∈ 𝐶 ; (iii)for all 𝑦 ∈ 𝐶 , 𝑓 ( ⋅ , 𝑦 ) is weakly upper semicontinuous; (iv)for all 𝑥 ∈ 𝐶 , 𝑓 ( 𝑥 , ⋅ ) is convex. For any 𝑟 > 0 and 𝑥 ∈ 𝐸 , define a mapping Φ 𝑟 ∶ 𝐸 → 𝐶 as follows: Φ 𝑟 1 ( 𝑥 ) = 𝑧 ∈ 𝐶 ∶ 𝑓 ( 𝑧 , 𝑦 ) + ⟨ 𝑇 𝑧 , 𝑦 − 𝑧 ⟩ + 𝜑 ( 𝑦 ) + 𝑟 , ⟨ 𝑦 − 𝑧 , 𝐽 ( 𝑧 − 𝑥 ) ⟩ ≥ 𝜑 ( 𝑧 ) , ∀ 𝑦 ∈ 𝐶 ( 3 . 1 9 ) where 𝜑 is a lower semicontinuous and convex function from 𝐶 to ℝ . Then, the following holds: (1) Φ 𝑟 is single valued; (2) ⟨ Φ 𝑟 𝑥 − Φ 𝑟 𝑦 , 𝐽 ( Φ 𝑟 𝑥 − 𝑥 ) ⟩ ≤ ⟨ Φ 𝑟 𝑥 − Φ 𝑟 𝑦 , 𝐽 ( Φ 𝑟 𝑦 − 𝑦 ) ⟩ for all 𝑥 , 𝑦 ∈ 𝐸 ; (3) 𝐹 ( Φ 𝑟 ) = G E P ( 𝑓 ) ;(4) G E P ( 𝑓 ) is nonempty closed and convex.

4. Strong Convergence Theorems

In this section, we prove the strong convergence theorem of the sequence { 𝑥 𝑛 } defined by (1.20) for solving a common element in the solution set of a generalized mixed equilibrium problem and the common fixed point set of a finite family of asymptotically nonexpansive mappings.

Theorem 4.1. Let 𝐸 be a uniformly convex and smooth Banach space and let 𝐶 be a nonempty, bounded, closed, and convex subset of 𝐸 . Let 𝑓 be a bifunction from 𝐶 × 𝐶 to ℝ satisfying (A1)–(A4). Let 𝑇 ∶ 𝐶 → 𝐸 ∗ be an 𝜂 -hemicontinuous and relaxed 𝜂 - 𝜉 monotone mapping and 𝜑 a lower semicontinuous and convex function from 𝐶 to ℝ . Let, for each 1 ≤ 𝑖 ≤ 𝑁 , 𝑆 𝑖 ∶ 𝐶 → 𝐶 be an asymptotically nonexpansive mapping with a sequence { 𝑘 𝑛 , 𝑖 } ∞ 𝑛 = 1 , respectively, such that 𝑘 𝑛 , 𝑖 → 1 as 𝑛 → ∞ . Assume that ⋂ Ω ∶ = 𝑁 𝑖 = 1 𝐹 ( 𝑆 𝑖 ) ∩ G M E P ( 𝑓 , 𝑇 ) is nonempty. Let { 𝑥 𝑛 } be a sequence generated by (1.20), where { 𝑡 𝑛 } and { 𝑟 𝑛 } are real sequences in ( 0 , 1 ) satisfying l i m 𝑛 → ∞ 𝑡 𝑛 = 0 and l i m i n f 𝑛 → ∞ 𝑟 𝑛 > 0 . Then { 𝑥 𝑛 } converges strongly, as 𝑛 → ∞ , to 𝑃 Ω 𝑥 0 , where 𝑃 Ω is the metric projection of 𝐸 onto Ω .

Proof. First, define the sequence { 𝑘 𝑛 } by 𝑘 𝑛 ∶ = m a x { 𝑘 𝑛 , 𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑁 } and so 𝑘 𝑛 → 1 as 𝑛 → ∞ and ‖ ‖ 𝑆 ℎ ( 𝑛 ) 𝑖 ( 𝑛 ) 𝑥 − 𝑆 ℎ ( 𝑛 ) 𝑖 ( 𝑛 ) 𝑦 ‖ ‖ ≤ 𝑘 𝑛 ‖ 𝑥 − 𝑦 ‖ ∀ 𝑥 , 𝑦 ∈ 𝐶 , ( 4 . 1 ) where ℎ ( 𝑛 ) = 𝑗 + 1 if 𝑗 𝑁 < 𝑛 ≤ ( 𝑗 + 1 ) 𝑁 , 𝑗 = 1 , 2 … , 𝑁 and 𝑛 = 𝑗 𝑁 + 𝑖 ( 𝑛 ) ; 𝑖 ( 𝑛 ) ∈ { 1 , 2 , … , 𝑁 } . Next, we rewrite the algorithm (1.20) as the following relation: 𝑥 0 ∈ 𝐶 , 𝐷 0 = 𝐶 0 𝐶 = 𝐶 , 𝑛 = c o 𝑧 ∈ 𝐶 𝑛 − 1 ∶ ‖ ‖ 𝑧 − 𝑆 ℎ ( 𝑛 ) 𝑖 ( 𝑛 ) 𝑧 ‖ ‖ ≤ 𝑡 𝑛 ‖ ‖ 𝑥 𝑛 − 𝑆 ℎ ( 𝑛 ) 𝑖 ( 𝑛 ) 𝑥 𝑛 ‖ ‖ 𝐷 , 𝑛 ≥ 0 , 𝑛 = 𝑧 ∈ 𝐷 𝑛 − 1 ∶ Φ 𝑟 𝑛 𝑥 𝑛 𝑥 − 𝑧 , 𝐽 𝑛 − Φ 𝑟 𝑛 𝑥 𝑛 𝑥 ≥ 0 , 𝑛 ≥ 1 , 𝑛 + 1 = 𝑃 𝐶 𝑛 ∩ 𝐷 𝑛 𝑥 0 , 𝑛 ≥ 0 , ( 4 . 2 ) where Φ 𝑟 is the mapping defined by (3.19). We show that the sequence { 𝑥 𝑛 } is well defined. It is easy to verify that 𝐶 𝑛 ∩ 𝐷 𝑛 is closed and convex and Ω ⊂ 𝐶 𝑛 for all 𝑛 ≥ 0 . Next, we prove that Ω ⊂ 𝐶 𝑛 ∩ 𝐷 𝑛 . Indeed, since 𝐷 0 = 𝐶 , we also have Ω ⊂ 𝐶 0 ∩ 𝐷 0 . Assume that Ω ⊂ 𝐶 𝑘 − 1 ∩ 𝐷 𝑘 − 1 for 𝑘 ≥ 2 . Utilizing Theorem 3.1 (2), we obtain Φ 𝑟 𝑘 𝑥 𝑘 − Φ 𝑟 𝑘 Φ 𝑢 , 𝐽 𝑟 𝑘 Φ 𝑢 − 𝑢 − 𝐽 𝑟 𝑘 𝑥 𝑘 − 𝑥 𝑘 ≥ 0 , ∀ 𝑢 ∈ Ω , ( 4 . 3 ) which gives that Φ 𝑟 𝑘 𝑥 𝑘 𝑥 − 𝑢 , 𝐽 𝑘 − Φ 𝑟 𝑘 𝑥 𝑘 ≥ 0 , ∀ 𝑢 ∈ Ω , ( 4 . 4 ) hence Ω ⊂ 𝐷 𝑘 . By the mathematical induction, we get that Ω ⊂ 𝐶 𝑛 ∩ 𝐷 𝑛 for each 𝑛 ≥ 0 and hence { 𝑥 𝑛 } is well defined. Now, we show that l i m 𝑛 → ∞ ‖ ‖ 𝑥 𝑛 − 𝑥 𝑛 + 𝑗 ‖ ‖ = 0 , ∀ 𝑗 = 1 , 2 , … , 𝑁 . ( 4 . 5 ) Put 𝑤 = 𝑃 Ω 𝑥 0 , since Ω ⊂ 𝐶 𝑛 ∩ 𝐷 𝑛 and 𝑥 𝑛 + 1 = 𝑃 𝐶 𝑛 ∩ 𝐷 𝑛 , we have ‖ ‖ 𝑥 𝑛 + 1 − 𝑥 0 ‖ ‖ ≤ ‖ ‖ 𝑤 − 𝑥 0 ‖ ‖ , ∀ 𝑛 ≥ 0 . ( 4 . 6 ) Since 𝑥 𝑛 + 2 ∈ 𝐷 𝑛 + 1 ⊂ 𝐷 𝑛 and 𝑥 𝑛 + 1 = 𝑃 𝐶 𝑛 ∩ 𝐷 𝑛 𝑥 0 , we have ‖ ‖ 𝑥 𝑛 + 1 − 𝑥 0 ‖ ‖ ≤ ‖ ‖ 𝑥 𝑛 + 2 − 𝑥 0 ‖ ‖ . ( 4 . 7 ) Hence the sequence { ‖ 𝑥 𝑛 − 𝑥 0 ‖ } is bounded and monotone increasing and hence there exists a constant 𝑑 such that l i m 𝑛 → ∞ ‖ ‖ 𝑥 𝑛 − 𝑥 0 ‖ ‖ = 𝑑 . ( 4 . 8 ) Moreover, by the convexity of 𝐷 𝑛 , we also have 1 / 2 ( 𝑥 𝑛 + 1 + 𝑥 𝑛 + 2 ) ∈ 𝐷 𝑛 and hence ‖ ‖ 𝑥 0 − 𝑥 𝑛 + 1 ‖ ‖ ≤ ‖ ‖ ‖ 𝑥 0 − 𝑥 𝑛 + 1 + 𝑥 𝑛 + 2 2 ‖ ‖ ‖ ≤ 1 2 ‖ ‖ 𝑥 0 − 𝑥 𝑛 + 1 ‖ ‖ + ‖ ‖ 𝑥 0 − 𝑥 𝑛 + 2 ‖ ‖ . ( 4 . 9 ) This implies that l i m 𝑛 → ∞ ‖ ‖ ‖ 1 2 𝑥 0 − 𝑥 𝑛 + 1 + 1 2 𝑥 0 − 𝑥 𝑛 + 2 ‖ ‖ ‖ = l i m 𝑛 → ∞ ‖ ‖ ‖ 𝑥 0 − 𝑥 𝑛 + 1 + 𝑥 𝑛 + 2 2 ‖ ‖ ‖ = 𝑑 . ( 4 . 1 0 ) By Lemma 2.2, we have l i m 𝑛 → ∞ ‖ ‖ 𝑥 𝑛 − 𝑥 𝑛 + 1 ‖ ‖ = 0 . ( 4 . 1 1 ) Furthermore, we can easily see that l i m 𝑛 → ∞ ‖ ‖ 𝑥 𝑛 − 𝑥 𝑛 + 𝑗 ‖ ‖ = 0 , ∀ 𝑗 = 1 , 2 , … , 𝑁 . ( 4 . 1 2 ) Next, we show that l i m 𝑛 → ∞ ‖ ‖ 𝑥 𝑛 − 𝑆 ℎ ( 𝑛 − 𝜅 ) 𝑖 ( 𝑛 − 𝜅 ) 𝑥 𝑛 ‖ ‖ = 0 , f o r a n y 𝜅 ∈ { 1 , 2 , … , 𝑁 } . ( 4 . 1 3 ) Fix 𝜅 ∈ { 1 , 2 , … , 𝑁 } and put 𝑚 = 𝑛 − 𝜅 . Since 𝑥 𝑛 = 𝑃 𝐶 𝑛 − 1 ∩ 𝐷 𝑛 − 1 𝑥 , we have 𝑥 𝑛 ∈ 𝐶 𝑛 − 1 ⊆ ⋯ ⊆ 𝐶 𝑚 . Since 𝑡 𝑚 > 0 , there exists 𝑦 1 , … , 𝑦 𝑃 ∈ 𝐶 and a nonnegative number 𝜆 1 , … , 𝜆 𝑃 with 𝜆 1 + ⋯ + 𝜆 𝑃 = 1 such that ‖ ‖ ‖ ‖ 𝑥 𝑛 − 𝑃 𝑖 = 1 𝜆 𝑖 𝑦 𝑖 ‖ ‖ ‖ ‖ < 𝑡 𝑚 , ‖ ‖ 𝑦 ( 4 . 1 4 ) 𝑖 − 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑦 𝑖 ‖ ‖ ≤ 𝑡 𝑚 ‖ ‖ 𝑥 𝑚 − 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑥 𝑚 ‖ ‖ , ∀ 𝑖 ∈ { 1 , … , 𝑃 } . ( 4 . 1 5 ) By the boundedness of 𝐶 and { 𝑘 𝑛 } , we can put the following: 𝑀 = s u p 𝑥 ∈ 𝐶 ‖ 𝑥 ‖ , 𝑢 = 𝑃 ⋂ 𝑁 𝑖 = 1 𝐹 ( 𝑆 𝑖 ) 𝑥 0 , 𝑟 0 = s u p 𝑛 ≥ 1 1 + 𝑘 𝑛 ‖ ‖ 𝑥 𝑛 ‖ ‖ . − 𝑢 ( 4 . 1 6 ) This together with (4.14) implies that ‖ ‖ ‖ ‖ 𝑥 𝑛 − 1 𝑘 𝑚 𝑃 𝑖 = 1 𝜆 𝑖 𝑦 𝑖 ‖ ‖ ‖ ‖ ≤ 1 1 − 𝑘 𝑚 1 ‖ 𝑥 ‖ + 𝑘 𝑚 ‖ ‖ ‖ ‖ 𝑥 𝑛 − 𝑃 𝑖 = 1 𝜆 𝑖 𝑦 𝑖 ‖ ‖ ‖ ‖ ≤ 1 1 − 𝑘 𝑚 𝑀 + 𝑡 𝑚 , ‖ ‖ 𝑦 𝑖 − 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑦 𝑖 ‖ ‖ ≤ 𝑡 𝑚 ‖ ‖ 𝑥 𝑚 − 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑥 𝑚 ‖ ‖ ≤ 𝑡 𝑚 ‖ ‖ 𝑥 𝑚 − 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑢 ‖ ‖ + 𝑡 𝑚 ‖ ‖ 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑢 − 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑥 𝑚 ‖ ‖ ≤ 𝑡 𝑚 ‖ ‖ 𝑥 𝑚 ‖ ‖ − 𝑢 + 𝑡 𝑚 𝑘 𝑚 ‖ ‖ 𝑢 − 𝑥 𝑚 ‖ ‖ ≤ 𝑡 𝑚 1 + 𝑘 𝑚 ‖ ‖ 𝑥 𝑚 ‖ ‖ − 𝑢 ≤ 𝑡 𝑚 𝑟 0 , ( 4 . 1 7 ) for all 𝑖 ∈ { 1 , … , 𝑁 } . Therefore, for each 𝑖 ∈ { 1 , … , 𝑃 } , we get ‖ ‖ ‖ 𝑦 𝑖 − 1 𝑘 𝑚 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑦 𝑖 ‖ ‖ ‖ ≤ ‖ ‖ 𝑦 𝑖 − 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑦 𝑖 ‖ ‖ + ‖ ‖ ‖ 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑦 𝑖 − 1 𝑘 𝑚 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑦 𝑖 ‖ ‖ ‖ ≤ 𝑟 0 𝑡 𝑚 + 1 1 − 𝑘 𝑚 𝑀 . ( 4 . 1 8 ) Moreover, since each 𝑆 𝑖 , 𝑖 ∈ { 1 , 2 , … , 𝑁 } , is asymptotically nonexpansive, we can obtain that ‖ ‖ ‖ ‖ 1 𝑘 𝑚 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑃 𝑖 = 1 𝜆 𝑖 𝑦 𝑖 − 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑥 𝑛 ‖ ‖ ‖ ‖ ≤ ‖ ‖ ‖ ‖ 1 𝑘 𝑚 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑃 𝑖 = 1 𝜆 𝑖 𝑦 𝑖 − 1 𝑘 𝑚 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑥 𝑛 ‖ ‖ ‖ ‖ + ‖ ‖ ‖ 1 𝑘 𝑚 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑥 𝑛 − 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑥 𝑛 ‖ ‖ ‖ ≤ ‖ ‖ ‖ ‖ 𝑃 𝑖 = 1 𝜆 𝑖 𝑦 𝑖 − 𝑥 𝑛 ‖ ‖ ‖ ‖ + 1 1 − 𝑘 𝑚 𝑀 = 𝑡 𝑚 + 1 1 − 𝑘 𝑚 𝑀 . ( 4 . 1 9 ) It follows from Theorem 2.3 and the inequalities (4.17)–(4.19) that ‖ ‖ 𝑥 𝑛 − 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑥 𝑛 ‖ ‖ ≤ ‖ ‖ ‖ ‖ 𝑥 𝑛 − 1 𝑘 𝑚 𝑃 𝑖 = 1 𝜆 𝑖 𝑦 𝑖 ‖ ‖ ‖ ‖ + 1 𝑘 𝑚 ‖ ‖ ‖ ‖ 𝑃 𝑖 = 1 𝜆 𝑖 𝑦 𝑖 − 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑦 𝑖 ‖ ‖ ‖ ‖ + 1 𝑘 𝑚 ‖ ‖ ‖ ‖ 𝑃 𝑖 = 1 𝜆 𝑖 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑦 𝑖 − 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑃 𝑖 = 1 𝜆 𝑖 𝑦 𝑖 ‖ ‖ ‖ ‖ + ‖ ‖ ‖ ‖ 1 𝑘 𝑚 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑃 𝑖 = 1 𝜆 𝑖 𝑦 𝑖 − 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑥 𝑛 ‖ ‖ ‖ ‖ 1 ≤ 2 1 − 𝑘 𝑚 𝑀 + 𝑡 𝑚 + 𝑟 0 𝑡 𝑚 𝑘 𝑚 + 𝛾 − 1 m a x 1 ≤ 𝑖 ≤ 𝑗 ≤ 𝑁 ‖ ‖ 𝑦 𝑖 − 𝑦 𝑗 ‖ ‖ − 1 𝑘 𝑚 ‖ ‖ 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑦 𝑖 − 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑦 𝑗 ‖ ‖ 1 = 2 1 − 𝑘 𝑚 𝑀 + 2 𝑡 𝑚 + 𝑟 0 𝑡 𝑚 𝑘 𝑚 + 𝛾 − 1 m a x 1 ≤ 𝑖 ≤ 𝑗 ≤ 𝑁 ‖ ‖ 𝑦 𝑖 − 𝑦 𝑗 ‖ ‖ − 1 𝑘 𝑚 ‖ ‖ 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑦 𝑖 − 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑦 𝑗 ‖ ‖ 1 ≤ 2 1 − 𝑘 𝑚 𝑀 + 2 𝑡 𝑚 + 𝑟 0 𝑡 𝑚 𝑘 𝑚 + 𝛾 − 1 m a x 1 ≤ 𝑖 ≤ 𝑗 ≤ 𝑁 ‖ ‖ ‖ 𝑦 𝑖 − 1 𝑘 𝑚 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑦 𝑖 ‖ ‖ ‖ + ‖ ‖ ‖ 𝑦 𝑗 − 1 𝑘 𝑚 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑦 𝑗 ‖ ‖ ‖ 1 ≤ 2 1 − 𝑘 𝑚 𝑀 + 2 𝑡 𝑚 + 𝑟 0 𝑡 𝑚 𝑘 𝑚 + 𝛾 − 1 2 1 1 − 𝑘 𝑚 𝑀 + 2 𝑟 0 𝑡 𝑚 . ( 4 . 2 0 ) Since l i m 𝑛 → ∞ 𝑘 𝑛 = 1 and l i m 𝑛 → ∞ 𝑡 𝑛 = 0 , it follows from the above inequality that l i m 𝑛 → ∞ ‖ ‖ 𝑥 𝑛 − 𝑆 ℎ ( 𝑚 ) 𝑖 ( 𝑚 ) 𝑥 𝑛 ‖ ‖ = 0 . ( 4 . 2 1 ) Hence (4.13) is proved. Next, we show that l i m 𝑛 → ∞ ‖ ‖ 𝑥 𝑛 − 𝑆 𝑙 𝑥 𝑛 ‖ ‖ = 0 ; ∀ 𝑙 = 1 , 2 , … , 𝑁 . ( 4 . 2 2 ) From the construction of 𝐶 𝑛 , one can easily see that ‖ ‖ 𝑥 𝑛 + 1 − 𝑆 ℎ ( 𝑛 ) 𝑖 ( 𝑛 ) 𝑥 𝑛 + 1 ‖ ‖ ≤ 𝑡 𝑛 ‖ ‖ 𝑥 𝑛 − 𝑆 ℎ ( 𝑛 ) 𝑖 ( 𝑛 ) 𝑥 𝑛 ‖ ‖ . ( 4 . 2 3 ) The boundedness of 𝐶 and l i m 𝑛 → ∞ 𝑡 𝑛 = 0 implies that l i m 𝑛 → ∞ ‖ ‖ 𝑥 𝑛 + 1 − 𝑆 ℎ ( 𝑛 ) 𝑖 ( 𝑛 ) 𝑥 𝑛 + 1 ‖ ‖ = 0 . ( 4 . 2 4 ) On the other hand, since for any positive integer 𝑛 > 𝑁 , 𝑛 = ( 𝑛 − 𝑁 ) ( m o d 𝑁 ) and 𝑛 = ( ℎ ( 𝑛 ) − 1 ) 𝑁 + 𝑖 ( 𝑛 ) , we have 𝑛 − 𝑁 = ( ℎ ( 𝑛 ) − 1 ) 𝑁 + 𝑖 ( 𝑛 ) = ( ℎ ( 𝑛 − 𝑁 ) − 1 ) 𝑁 + 𝑖 ( 𝑛 − 𝑁 ) ( 4 . 2 5 ) that is ℎ ( 𝑛 − 𝑁 ) = ℎ ( 𝑛 ) − 1 , 𝑖 ( 𝑛 − 𝑁 ) = 𝑖 ( 𝑛 ) . ( 4 . 2 6 ) Thus, ‖ ‖ 𝑥 𝑛 − 𝑆 𝑛 𝑥 𝑛 ‖ ‖ ≤ ‖ ‖ 𝑥 𝑛 − 𝑥 𝑛 + 1 ‖ ‖ + ‖ ‖ 𝑥 𝑛 + 1 − 𝑆 ℎ ( 𝑛 ) 𝑖 ( 𝑛 ) 𝑥 𝑛 + 1 ‖ ‖ + ‖ ‖ 𝑆 ℎ ( 𝑛 ) 𝑖 ( 𝑛 ) 𝑥 𝑛 + 1 − S 𝑛 𝑥 𝑛 ‖ ‖ ≤ ‖ ‖ 𝑥 𝑛 − 𝑥 𝑛 + 1 ‖ ‖ + ‖ ‖ 𝑥 𝑛 + 1 − 𝑆 ℎ ( 𝑛 ) 𝑖 ( 𝑛 ) 𝑥 𝑛 + 1 ‖ ‖ + ‖ ‖ 𝑆 ℎ ( 𝑛 ) 𝑖 ( 𝑛 ) 𝑥 𝑛 + 1 − 𝑆 𝑛 𝑥 𝑛 + 1 ‖ ‖ + ‖ ‖ 𝑆 𝑛 𝑥 𝑛 + 1 − 𝑆 𝑛 𝑥 𝑛 ‖ ‖ ≤ 1 + 𝑘 1 ‖ ‖ 𝑥 𝑛 − 𝑥 𝑛 + 1 ‖ ‖ + ‖ ‖ 𝑥 𝑛 + 1 − 𝑆 ℎ ( 𝑛 ) 𝑖 ( 𝑛 ) 𝑥 𝑛 + 1 ‖ ‖ + 𝑘 1 ‖ ‖ 𝑆 ℎ ( 𝑛 ) − 1 𝑖 ( 𝑛 ) 𝑥 𝑛 + 1 − 𝑥 𝑛 + 1 ‖ ‖ ≤ 1 + 𝑘 1 ‖ ‖ 𝑥 𝑛 − 𝑥 𝑛 + 1 ‖ ‖ + ‖ ‖ 𝑥 𝑛 + 1 − 𝑆 ℎ ( 𝑛 ) 𝑖 ( 𝑛 ) 𝑥 𝑛 + 1 ‖ ‖ + 𝑘 1 ‖ ‖ 𝑆 ℎ ( 𝑛 ) − 1 𝑖 ( 𝑛 ) 𝑥 𝑛 + 1 − 𝑆 ℎ ( 𝑛 ) − 1 𝑖 ( 𝑛 ) 𝑥 𝑛 ‖ ‖ + ‖ ‖ 𝑆 ℎ ( 𝑛 ) − 1 𝑖 ( 𝑛 ) 𝑥 𝑛 − 𝑥 𝑛 ‖ ‖ + ‖ ‖ 𝑥 𝑛 − 𝑥 𝑛 + 1 ‖ ‖ ≤ 1 + 2 𝑘 1 ‖ ‖ 𝑥 𝑛 − 𝑥 𝑛 + 1 ‖ ‖ + ‖ ‖ 𝑥 𝑛 + 1 − 𝑆 ℎ ( 𝑛 ) 𝑖 ( 𝑛 ) 𝑥 𝑛 + 1 ‖ ‖ + 𝑘 1 ‖ ‖ 𝑆 ℎ ( 𝑛 − 𝑁 ) 𝑖 ( 𝑛 − 𝑁 ) 𝑥 𝑛 + 1 − 𝑆 ℎ ( 𝑛 − 𝑁 ) 𝑖 ( 𝑛 − 𝑁 ) 𝑥 𝑛 ‖ ‖ + 𝑘 1 ‖ ‖ 𝑆 ℎ ( 𝑛 − 𝑁 ) 𝑖 ( 𝑛 − 𝑁 ) 𝑥 𝑛 − 𝑥 𝑛 ‖ ‖ ≤ 1 + 2 𝑘 1 ‖ ‖ 𝑥 𝑛 − 𝑥 𝑛 + 1 ‖ ‖ + ‖ ‖ 𝑥 𝑛 + 1 − 𝑆 ℎ ( 𝑛 ) 𝑖 ( 𝑛 ) 𝑥 𝑛 + 1 ‖ ‖ + 𝑘 1 𝑘 𝑛 − 𝑁 ‖ ‖ 𝑥 𝑛 + 1 − 𝑥 𝑛 ‖ ‖ + 𝑘 1 ‖ ‖ 𝑆 ℎ ( 𝑛 − 𝑁 ) 𝑖 ( 𝑛 − 𝑁 ) 𝑥 𝑛 − 𝑥 𝑛 ‖ ‖ ≤ 1 + 2 𝑘 1 + 𝑘 1 𝑘 𝑛 − 𝑁 ‖ ‖ 𝑥 𝑛 − 𝑥 𝑛 + 1 ‖ ‖ + ‖ ‖ 𝑥 𝑛 + 1 − 𝑆 ℎ ( 𝑛 ) 𝑖 ( 𝑛 ) 𝑥 𝑛 + 1 ‖ ‖ + 𝑘 1 ‖ ‖ 𝑆 ℎ ( 𝑛 − 𝑁 ) 𝑖 ( 𝑛 − 𝑁 ) 𝑥 𝑛 − 𝑥 𝑛 ‖ ‖ . ( 4 . 2 7 ) Applying the facts (4.11), (4.13), and (4.24) to the above inequality, we obtain l i m 𝑛 → ∞ ‖ ‖ 𝑥 𝑛 − 𝑆 𝑛 𝑥 𝑛 ‖ ‖ = 0 . ( 4 . 2 8 ) Therefore, for any 𝑗 = 1 , 2 , … , 𝑁 , we have ‖ ‖ 𝑥 𝑛 − 𝑆 𝑛 + 𝑗 𝑥 𝑛 ‖ ‖ ≤ ‖ ‖ 𝑥 𝑛 − 𝑥 𝑛 + 𝑗 ‖ ‖ + ‖ ‖ 𝑥 𝑛 + 𝑗 − 𝑆 𝑛 + 𝑗 𝑥 𝑛 + 𝑗 ‖ ‖ + ‖ ‖ 𝑆 𝑛 + 𝑗 𝑥 𝑛 + 𝑗 − 𝑆 𝑛 + 𝑗 𝑥 𝑛 ‖ ‖ ≤ ‖ ‖ 𝑥 𝑛 − 𝑥 𝑛 + 𝑗 ‖ ‖ + ‖ ‖ 𝑥 𝑛 + 𝑗 − 𝑆 𝑛 + 𝑗 𝑥 𝑛 + 𝑗 ‖ ‖ + 𝑘 1 ‖ ‖ 𝑥 𝑛 + 𝑗 − 𝑥 𝑛 ‖ ‖ = 1 + 𝑘 1 ‖ ‖ 𝑥 𝑛 − 𝑥 𝑛 + 𝑗 ‖ ‖ + ‖ ‖ 𝑥 𝑛 + 𝑗 − 𝑆 𝑛 + 𝑗 𝑥 𝑛 + 𝑗 ‖ ‖ ⟶ 0 a s 𝑛 ⟶ ∞ , ( 4 . 2 9 ) which gives that l i m 𝑛 → ∞ ‖ ‖ 𝑥 𝑛 − 𝑆 𝑙 𝑥 𝑛 ‖ ‖ = 0 ; ∀ 𝑙 = 1 , 2 , … , 𝑁 , ( 4 . 3 0 ) as required. Since { 𝑥 𝑛 } is bounded, there exists a subsequence { 𝑥 𝑛 𝑖 } of { 𝑥 𝑛 } such that 𝑥 𝑛 𝑖 ⇀ ̃ 𝑥 ∈ 𝐶 . It follows from Lemma 2.4 that ̃ 𝑥 ∈ 𝐹 ( 𝑆 𝑙 ) for all 𝑙 = 1 , 2 , … , 𝑁 . That is ⋂ 𝑥 ∈ 𝑁 𝑖 = 1 𝐹 ( 𝑆 𝑖 ) .
Next, we show that ̃ 𝑥 ∈ G M E P ( 𝑓 , 𝑇 ) . By the construction of 𝐷 𝑛 , we see from Theorem 2.1 that Φ 𝑟 𝑛 𝑥 𝑛 = 𝑃 𝐷 𝑛 𝑥 𝑛 . Since 𝑥 𝑛 + 1 ∈ 𝐷 𝑛 , we get ‖ ‖ 𝑥 𝑛 − Φ 𝑟 𝑛 𝑥 𝑛 ‖ ‖ ≤ ‖ ‖ 𝑥 𝑛 − 𝑥 𝑛 + 1 ‖ ‖ ⟶ 0 . ( 4 . 3 1 ) Furthermore, since l i m i n f 𝑛 → ∞ 𝑟 𝑛 > 0 , we have 1 𝑟 𝑛 ‖ ‖ 𝐽 𝑥 𝑛 − Φ 𝑟 𝑛 𝑥 𝑛 ‖ ‖ = 1 𝑟 𝑛 ‖ ‖ 𝑥 𝑛 − Φ 𝑟 𝑛 𝑥 𝑛 ‖ ‖ ⟶ 0 , ( 4 . 3 2 ) as 𝑛 → ∞ . By (4.32), we also have Φ 𝑟 𝑛 𝑖 𝑥 𝑛 𝑖 ⇀ ̃ 𝑥 . By the definition of Φ 𝑟 𝑛 𝑖 , for each 𝑦 ∈ 𝐶 , we obtain 𝑓 Φ 𝑟 𝑛 𝑖 𝑥 𝑛 𝑖 + , 𝑦 𝑇 Φ 𝑟 𝑛 𝑖 𝑥 𝑛 𝑖 , 𝜂 𝑦 , Φ 𝑟 𝑛 𝑖 𝑥 𝑛 𝑖 1 + 𝜑 ( 𝑦 ) + 𝑟 𝑛 𝑖 𝑦 − Φ 𝑟 𝑛 𝑖 𝑥 𝑛 𝑖 Φ , 𝐽 𝑟 𝑛 𝑖 𝑥 𝑛 𝑖 − 𝑥 𝑛 𝑖 Φ ≥ 𝜑 𝑟 𝑛 𝑖 𝑥 𝑛 𝑖 . ( 4 . 3 3 ) By (A3), (4.32), (ii), the weakly lower semicontinuity of 𝜑 and 𝜂 -hemicontinuity of 𝑇 , we have 𝜑 ( ̃ 𝑥 ) ≤ l i m i n f 𝑖 → ∞ 𝜑 Φ 𝑟 𝑛 𝑖 𝑥 𝑛 𝑖 ≤ l i m i n f 𝑖 → ∞ 𝑓 Φ 𝑟 𝑛 𝑖 𝑥 𝑛 𝑖 , 𝑦 + l i m i n f 𝑖 → ∞</