Hybrid Projection Algorithm for Two Countable Families of Hemirelatively Nonexpansive Mappings and Applications

Abstract

Two countable families of hemirelatively nonexpansive mappings are considered based on a hybrid projection algorithm. Strong convergence theorems of iterative sequences are obtained in an uniformly convex and uniformly smooth Banach space. As applications, convex feasibility problems, equilibrium problems, variational inequality problems, and zeros of maximal monotone operators are studied.

1. Introduction

Throughout this paper, we always assume that 𝐸 is a real Banach space, 𝐸^∗ is the dual space of 𝐸, 𝐶 is a nonempty closed convex subset of 𝐸 and ⟨⋅,⋅⟩ is the pairing between 𝐸, and 𝐸^∗. We denote by ℕ and ℝ the sets of positive integers and real numbers, respectively.

Let 𝑓∶𝐶×𝐶→ℝ be a bifunction and 𝐴∶𝐶→𝐸^∗ a nonlinear mapping. The “so-called” generalized mixed equilibrium problem is to find 𝑥∈𝐶 such that 𝑓𝑥,𝑦+⟨𝐴𝑥,𝑦−𝑥⟩+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (1)

The set of solutions to (1) is denoted by GMEP(𝑓,𝐴,𝜑), that is, =+𝑦(𝑥).GMEP𝑓,𝐴,𝜑𝑥∈𝐶∶𝑓𝑥,𝑦𝐴𝑥,𝑦−𝑥+𝜑−𝜑≥0,∀𝑦∈𝐶 (2)

1.1. Analysis of Special Cases

(1) If 𝜑(⋅)≡0, the problem (1) reduces to the generalized equilibrium problem, which is to find 𝑥∈𝐶 such that 𝑓𝑥,𝑦+𝐴𝑥,𝑦−𝑥≥0,∀𝑦∈𝐶. (3) The set of solutions to (3) is denoted by GEP(𝑓,𝐴).

(2) If 𝐴≡0, the problem (1) reduces to the mixed equilibrium problem, which is to find 𝑥∈𝐶 such that 𝑓𝑥,𝑦+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (4) The set of solutions to (4) is denoted by MEP(𝑓,𝜑).

(3) If 𝑓(⋅,⋅)≡0, the problem (1) reduces to the mixed variational inequality of Browder type, which is to find 𝑥∈𝐶 such that 𝐴𝑥,𝑦−𝑥+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (5) The set of solutions to (5) is denoted by MVI(𝐴,𝜑,𝐶).

(4) If 𝑓(⋅,⋅)≡0 in (3), the problem (3) reduces to the classic variational inequality, which is to find 𝑥∈𝐶 such that 𝐴𝑥,𝑦−𝑥≥0,∀𝑦∈𝐶, (6) which is called the Hartmann-Stampacchia variational inequality. The set of solutions to (6) is denoted by VI(𝐴,𝐶).

(5) If 𝐴≡0 in (3), the problem (3) reduces to the classic equilibrium problem, which is to find 𝑥∈𝐶 such that 𝑓𝑥,𝑦≥0,∀𝑦∈𝐶. (7) The set of solutions to (7) is denoted by EP(𝑓). Given a mapping 𝑇∶𝐶→𝐸^∗, let 𝑓(𝑥,𝑦)=⟨𝑇𝑥,𝑦−𝑥⟩ for all 𝑥,𝑦∈𝐶. Then 𝑝∈EP(𝑓) if and only if ⟨𝑇𝑝,𝑦−𝑝⟩≥0 for all 𝑦∈𝐶; that is, 𝑝 is a solution of the variational inequality.

(6) If 𝑓(⋅,⋅)≡0 in (4), the problem (4) reduces to the minimize problem, which is to find 𝑥∈𝐶 such that 𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (8) The set of solutions to (8) is denoted by Argmin(𝜑).

The problem (1) is very general in the sense that it includes, as special case, optimization problems, variational inequalities, minimax problems, monotone inclusion problems, saddle point problems, vector equilibrium problems, and the Nash equilibrium problem in noncooperative games. Numerous problems in physics, optimization, and economics reduce to finding a solution of some special case or the problem (1). Some solution methods have been proposed to solve the problems (1), (3)–(8) in Hilbert spaces and Banach spaces; see, for example, [1–7] and references therein.

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 𝑆_𝐸={𝑥∈𝐸∶‖𝑥‖=1} be the unit sphere of 𝐸, and define 𝑓∶𝑆_𝐸×𝑆_𝐸×ℝ⧵{0}→ℝ by 𝑓=−‖𝑥‖𝑥,𝑦,𝑡𝑥+𝑡𝑦𝑡 (9) for 𝑥,𝑦∈𝑆_𝐸 and 𝑡∈ℝ⧵{0}. A Banach space 𝐸 is said to be smooth if the limit lim_𝑡→0𝑓(𝑥,𝑦,𝑡) exists for each 𝑥,𝑦∈𝑆_𝐸. It is also said to be uniformly smooth if the limit lim_𝑡→0𝑓(𝑥,𝑦,𝑡) is attained uniformly for (𝑥,𝑦)∈𝑆_𝐸×𝑆_𝐸.

The modulus of convexity of 𝐸 is the function 𝛿∶[0,2]→[0,1] defined by 𝛿(𝜀)=inf1−𝑥+𝑦2‖𝑥‖=𝑦.∶𝑥,𝑦∈𝐸,=1,𝑥−𝑦≥𝜀 (10) 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]. Observe that every 𝑝-uniformly convex is uniformly convex. One should note that no Banach space is 𝑝-uniformly convex for 1<𝑝<2. It is well known that 𝐿_𝑝(𝑙_𝑝) or 𝑊^𝑝_𝑚 is 𝑝-uniformly convex if 𝑝≥2 and 2-uniformly convex if 1<𝑝≤2; see [8] for more details.

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,∀𝑥∈𝐸. (11) In particular, if 𝑝=2, 𝐽_𝑝 is called the normalized duality mapping. If 𝐸 is a Hilbert space, then 𝐽_𝑝=𝐼, where 𝐼 is the identity mapping. In this paper, We denote by 𝐽 the normalized duality mapping. It is known that the duality mapping 𝐽 has the following properties:(i)if 𝐸 is smooth, then 𝐽 is single valued; (ii)if 𝐸 is strictly convex, then 𝐽 is one to one;(iii)if 𝐸 is reflexive, then 𝐽 is surjective;(iv)if 𝐸 is uniformly smooth, then 𝐽 is uniformly norm-to-norm continuous on each bounded subset of 𝐸;(v)if 𝐸^∗ is uniformly convex, then 𝐽 is uniformly continuous on bounded subsets of 𝐸 and 𝐽 is single valued and also one to one (see [9–12]).

Let 𝐸 be a smooth Banach space. Consider the function defined by 𝜙𝑥,𝑦=‖𝑥‖²−2𝑥,𝐽𝑦+𝑦²,∀𝑥,𝑦∈𝐸. (12) It is obvious from the definition of the function 𝜙 that ‖𝑥‖−𝑦²≤𝜙𝑥,𝑦≤‖𝑥‖+𝑦²,∀𝑥,𝑦∈𝐸. (13) We also know that 𝜙(𝑥,𝑦)=0 if and only if 𝑥=𝑦 (see [13]). Moreover, if 𝐸 is a Hilbert space, (12) reduces to 𝜙(𝑥,𝑦)=‖𝑥−𝑦‖², for any 𝑥,𝑦∈𝐸.

Let 𝐶 be a closed convex subset of 𝐸, and let 𝑇 be a mapping from 𝐶 into itself. We denote by 𝐹(𝑇) the set of fixed points of 𝑇. A point 𝑝 in 𝐶 is said to be an asymptotic fixed point of 𝑇 [14] if 𝐶 contains a sequence {𝑥_𝑛} which converges weakly to 𝑝 such that the strong lim_𝑛→∞(𝑥_𝑛−𝑇𝑥_𝑛)=0. The set of asymptotic fixed points of 𝑇 will be denoted by 𝐹(𝑇). A point 𝑝 in 𝐶 is said to be a strong asymptotic fixed point of 𝑇 [14] if 𝐶 contains a sequence {𝑥_𝑛} which converges strong to 𝑝 such that lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. The set of strong asymptotic fixed points of 𝑇 will be denoted by 𝐹(𝑇).

Let 𝑇∶𝐶→𝐶 be a mapping, and recall the following definition:(a)𝑇 is called nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶; (14)(b)𝑇 is called relatively nonexpansive if 𝐹(𝑇)=𝐹(𝑇)≠∅ and 𝜙𝑝,𝑇𝑥≤𝜙𝑝,𝑥,∀𝑥∈𝐶,𝑝∈𝐹(𝑇); (15)(c)a mapping 𝑇 is said to be weak relatively nonexpansive if 𝐹(𝑇)=𝐹(𝑇)≠∅ and 𝜙𝑝,𝑇𝑥≤𝜙𝑝,𝑥,∀𝑥∈𝐶,𝑝∈𝐹(𝑇); (16)(d)a mapping 𝑇 is called hemirelatively nonexpansive if 𝐹(𝑇)≠∅ and 𝜙𝑝,𝑇𝑥≤𝜙𝑝,𝑥,∀𝑥∈𝐶,𝑝∈𝐹(𝑇). (17)

Remark 1. From the definitions, it is obvious that a relatively nonexpansive mapping is a weak relatively nonexpansive mapping, and a weak relatively nonexpansive mapping is a hemi-relatively nonexpansive mapping, but the converse is not true.

Next, we give an example which is a closed hemirelatively nonexpansive mapping.

Example 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⊂𝐸. Then Π_𝐶 is a relatively nonexpansive mapping, and then it is also a closed hemi-relatively nonexpansive mapping.

In 2005, Matsushita and Takahashi [13] obtained strong convergence theorems for a single relatively nonexpansive mapping in a uniformly convex and uniformly smooth Banach space 𝐸. To be more precise, they proved the following theorem.

Theorem MT (see Matsushita and Takahashi [13, Theorem 3.1]). Let 𝐸 be precisely a uniformly convex and uniformly smooth Banach space and 𝐶 a nonempty closed convex subset of 𝐸, and let 𝑇 be a relatively nonexpansive mapping from 𝐶 into itself, and let {𝛼_𝑛} be a sequence of real numbers such that 0≤𝛼_𝑛<1 and limsup_𝑛→∞𝛼_𝑛<1. Suppose that {𝑥_𝑛} is given by 𝑥₀𝑦=𝑥∈𝐶,_𝑛=Π_𝐶𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝐽𝑥−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥,∀𝑛∈ℕ∪{0}, (18) where 𝐽 is the duality mapping on 𝐸. If 𝐹(𝑇) is nonempty, then {𝑥_𝑛} converges strongly to Π_𝐹(𝑇)𝑥, where Π_𝐹(𝑇) is the generalized projection from 𝐶 onto 𝐹(𝑇).

Since then, algorithms constructed for solving the same equilibrium problem, variational inequality problems, and fixed point of relatively nonexpansive mappings (or weak relatively nonexpansive mappings or hemi-relatively nonexpanisve mappings) have been further developed by many authors. For a part of works related to these problems, please see [4, 15–18], and for the hybrid algorithm projection methods for these problems, please see [19–44] and the references therein.

Motivated and inspired by the results in the literature, in this paper we focus our attention on finding a common fixed point of two countable families of hemi-relatively nonexpansive mappings (we shall give the definition of a countable family of hemi-relatively nonexpansive mappings in the next section) by using a simple hybrid algorithm. Furthermore, we will give some applications of our main result in equilibrium problems, variational inequality problems, and convex feasibility problems.

2. Preliminaries

Let 𝐶 be a closed convex subset of 𝐸, and let {𝑇_𝑛}^∞_𝑛=0 be a countable family of mappings from 𝐶 into itself. We denote by 𝐹 the set of common fixed points of {𝑇_𝑛}^∞_𝑛=0. That is, 𝐹=⋂^∞_𝑛=0𝐹(𝑇_𝑛), where 𝐹(𝑇_𝑛) denote the set of fixed points of 𝑇_𝑛, for all 𝑛∈ℕ∪{0}.

Recall that {𝑇_𝑛}^∞_𝑛=0 is said to be uniformly closed, if 𝑝∈⋂^∞_𝑛=1𝐹(𝑇_𝑛), whenever {𝑥_𝑛}⊂𝐶 converges strongly to 𝑝 and ‖𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖→0 as 𝑛→∞ (see [45] for more details).

A point 𝑝∈𝐶 is said to be an asymptotic fixed point of {𝑇_𝑛}^∞_𝑛=0 if 𝐶 contains a sequence {𝑥_𝑛} which converges weakly to 𝑝 such that lim_𝑛→∞‖𝑇_𝑛𝑥_𝑛−𝑥_𝑛‖=0. The set of asymptotic fixed points of {𝑇_𝑛}^∞_𝑛=0 will be denoted by 𝐹({𝑇_𝑛}^∞_𝑛=0).

A point 𝑝∈𝐶 is said to be a strong asymptotic fixed point of {𝑇_𝑛}^∞_𝑛=0 if 𝐶 contains a sequence {𝑥_𝑛} which converges strongly to 𝑝 such that lim_𝑛→∞‖𝑇_𝑛𝑥_𝑛−𝑥_𝑛‖=0. The set of strong asymptotic fixed points of {𝑇_𝑛}^∞_𝑛=0 will be denoted by 𝐹({𝑇_𝑛}^∞_𝑛=0).

Using the definition of (strong) asymptotic fixed point of {𝑇_𝑛}^∞_𝑛=0, Su et al. [46] introduced the following definitions.

Definition 3 (see Su et al. [46]). Countable family of mappings {𝑇_𝑛} is said to be countable family of relatively nonexpansive mappings if 𝐹({𝑇_𝑛}^∞_𝑛=0)=𝐹({𝑇_𝑛}^∞_𝑛=0)≠∅ and 𝜙𝑝,𝑇_𝑛𝑥≤𝜙𝑝,𝑥,∀𝑥∈𝐶,𝑝∈𝐹𝑇_𝑛,𝑛∈ℕ∪{0}. (19)

Definition 4 (see Su et al. [46]). Countable family of mappings {𝑇_𝑛} is said to be countable family of weak relatively nonexpansive mappings if 𝐹({𝑇_𝑛}^∞_𝑛=0)=𝐹({𝑇_𝑛}^∞_𝑛=0)≠∅ and 𝜙𝑝,𝑇_𝑛𝑥≤𝜙𝑝,𝑥,∀𝑥∈𝐶,𝑝∈𝐹𝑇_𝑛,𝑛∈ℕ∪{0}. (20)

Now, we introduce the definition of countable family of hemi-relatively nonexpansive mappings which is more general than countable family of relatively nonexpansive mappings and countable family of weak relatively nonexpansive mappings.

Definition 5. Countable family of mappings {𝑇_𝑛} is said to be countable family of hemi-relatively nonexpansive mappings if 𝐹({𝑇_𝑛}^∞_𝑛=0)≠∅ and 𝜙𝑝,𝑇_𝑛𝑥≤𝜙𝑝,𝑥,∀𝑥∈𝐶,𝑝∈𝐹𝑇_𝑛,𝑛∈ℕ∪{0}. (21)

Remark 6. From Definitions 3–5, one has the following facts.(1)The definitions of relatively nonexpansive mapping, weak relatively nonexpansive mapping, and hemi-relatively nonexpansive mapping are special cases of Definitions 3, 4, and 5 as 𝑇_𝑛≡𝑇 for all 𝑛∈ℕ∪{0}.(2)Countable family of hemi-relatively nonexpansive mappings, which do not need the restriction 𝐹({𝑇_𝑛}^∞_𝑛=0)=𝐹({𝑇_𝑛}^∞_𝑛=0) (or 𝐹({𝑇_𝑛}^∞_𝑛=0)=𝐹({𝑇_𝑛}^∞_𝑛=0)), is more general than countable family of relatively nonexpansive mappings (or countable family of weak relatively nonexpansive mappings).

Next we give an example which is a countable family of hemi-relatively nonexpansive mappings but not a countable family of relatively nonexpansive mappings.

Example 7. Let 𝐸 be any smooth Banach space and 𝑥₀=(1+1/𝑛)^𝑛𝑥₀≠0 any element of 𝐸. Define a countable family of mappings 𝑇_𝑛∶𝐸→𝐸 as follows: for all 𝑛≥1, 𝑇_𝑛(𝑥)=12+12^𝑛+1𝑥₀1,if𝑥=2+12^𝑛𝑥₀,1−𝑥,if𝑥≠2+12^𝑛𝑥₀. (22) Then {𝑇_𝑛}^∞_𝑛=1 is a countable family of hemi-relatively nonexpansive mappings but not a countable family of relatively nonexpansive mappings.

Proof. First, it is obvious that 𝑇_𝑛 has a unique fixed point 0; that is, 𝐹(𝑇_𝑛)={0} for all 𝑛≥1. In addition, one easily sees that 𝑇_𝑛𝑥≤‖𝑥‖,∀𝑥∈𝐸,𝑛≥1. (23) This implies that 𝑇_𝑛𝑥²−‖𝑥‖²≤20,𝐽𝑇_𝑛𝑥−𝐽𝑥=2𝑝,𝐽𝑇_𝑛𝑥−𝐽𝑥, (24) for all 𝑝∈⋂^∞_𝑛=1𝐹(𝑇_𝑛). It follows from the above inequality that 𝑝²−2𝑝,𝐽𝑇_𝑛𝑥+𝑇_𝑛𝑥²≤𝑝²−2𝑝,𝐽𝑥+‖𝑥‖², (25) for all 𝑝∈⋂^∞_𝑛=1𝐹(𝑇_𝑛) and 𝑥∈𝐸. That is, 𝜙𝑝,𝑇_𝑛𝑥≤𝜙𝑝,𝑥, (26) for all 𝑝∈⋂^∞_𝑛=1𝐹(𝑇_𝑛) and 𝑥∈𝐸. Hence, {𝑇_𝑛}^∞_𝑛=1 is a countable family of hemi-relatively nonexpansive mappings. On the other hand, letting 𝑥_𝑛=12+12^𝑛𝑥₀,∀𝑛≥1, (27) from the definition of 𝑇_𝑛, one has 𝑇_𝑛𝑥_𝑛=12+12^𝑛+1𝑥₀,∀𝑛≥1, (28) which implies that ‖𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖→0 and 𝑥_𝑛→𝑒𝑥̆₀(𝑥_𝑛⇀𝑒𝑥̆₀) as 𝑛→∞. That is, 𝑒𝑥̆₀∈𝐹({𝑇_𝑛}^∞_𝑛=0) but 𝑒𝑥̆₀∉𝐹({𝑇_𝑛}^∞_𝑛=0), which shows that {𝑇_𝑛}^∞_𝑛=1 is not a countable family of relatively nonexpansive mappings.

In what follows, we will need the following lemmas.

Lemma 8 (see Alber [47]). Let 𝐶 be a convex subset of a smooth real Banach space 𝐸. Let 𝑥∈𝐸 and 𝑥₀∈𝐶. Then 𝑥₀=Π_𝐶𝑥 if and only if 𝑧−𝑥₀,𝐽𝑥₀−𝐽𝑥≥0,∀𝑧∈𝐶. (29)

Lemma 9 (see Alber [47]). Let 𝐶 be a nonempty, closed, and convex subset of a reflexive, strictly convex, and smooth real Banach space 𝐸, and let 𝑥∈𝐸. Then for each 𝑦∈𝐶, 𝜙𝑦,Π_𝐶𝑥+𝜙Π_𝐶𝑥,𝑥≤𝜙𝑦,𝑥. (30)

Lemma 10 (see Kamimura and Takahashi [48]). Let 𝐸 be a uniformly convex and smooth real Banach space, and let {𝑥_𝑛}, {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.

3. Main Results

Now, we give our main results in this paper.

Theorem 11. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space 𝐸. Let {𝑆_𝑛}, {𝑇_𝑛} be two uniformly closed countable families of hemi-relatively nonexpansive mappings from 𝐶 into itself such that ℱ=^∞_𝑛=1𝐹𝑆_𝑛^∞_𝑛=1𝐹𝑇_𝑛≠∅. (31) For a point 𝑥₀∈𝐶 chosen arbitrarily, let {𝑥_𝑛} be a sequence generated by the following iterative algorithm: 𝐶₀𝐶=𝐶,_𝑛+1=𝑧∈𝐶_𝑛∶𝜙𝑧,𝑆_𝑛𝑦_𝑛≤𝜙𝑧,𝑇_𝑛𝑥_𝑛≤𝜙𝑧,𝑥_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀, (32) where the sequences 𝑦_𝑛=𝑇_𝑛𝑥_𝑛. Then the sequence {𝑥_𝑛} converges strongly to a point 𝑞=Π_ℱ𝑥₀, where Π_ℱ is the generalized projection from 𝐶 onto ℱ.

Proof. We first show that 𝐶_𝑛+1 is closed and convex. It is obvious that 𝐶_𝑛+1 is closed. Since 𝜙𝑧,𝑆_𝑛𝑦_𝑛≤𝜙𝑧,𝑇_𝑛𝑥_𝑛⟺𝑆_𝑛𝑦_𝑛²−𝑇_𝑛𝑥_𝑛−2𝑧,𝐽𝑆_𝑛𝑦_𝑛−𝐽𝑇_𝑛𝑥_𝑛≥0,𝜙𝑧,𝑇_𝑛𝑥_𝑛≤𝜙𝑧,𝑥_𝑛⟺𝑇_𝑛𝑥_𝑛²−𝑥_𝑛−2𝑧,𝐽𝑇_𝑛𝑥_𝑛−𝐽𝑥_𝑛≥0, (33)(34)𝐶_𝑛+1 is convex. Therefore, 𝐶_𝑛+1 is closed and convex for all 𝑛∈ℕ∪{0}.
Let 𝑢∈ℱ; from the definition of 𝑆_𝑛 and 𝑇_𝑛, we have 𝜙𝑢,𝑆_𝑛𝑦_𝑛≤𝜙𝑢,𝑦_𝑛=𝜙𝑢,𝑇_𝑛𝑥_𝑛≤𝜙𝑢,𝑥_𝑛. (35) Hence, we have 𝑢∈𝐶_𝑛+1. This implies that ℱ⊂𝐶_𝑛+1 for arbitrary 𝑛∈ℕ∪{0}.
Noticing 𝑥_𝑛=Π_𝐶𝑛𝑥₀, from Lemma 8, we have 𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛≥0,∀𝑧∈𝐶_𝑛. (36) Since ℱ⊂𝐶_𝑛 for all 𝑛∈ℕ∪{0}, we arrive at 𝑥_𝑛−𝑝,𝐽𝑥₀−𝐽𝑥_𝑛≥0,∀𝑝∈ℱ. (37) From Lemma 9, we have 𝜙𝑥_𝑛,𝑥₀=𝜙Π_𝐶𝑛𝑥₀,𝑥₀≤𝜙𝑝,𝑥₀−𝜙𝑝,𝑥_𝑛≤𝜙𝑝,𝑥₀ (38) for each 𝑝∈ℱ⊂𝐶_𝑛 and for all 𝑛∈ℕ∪{0}. So the sequence {𝜙(𝑥_𝑛,𝑥₀)} is bounded. On the other hand, noticing that 𝑥_𝑛=Π_𝐶𝑛𝑥₀ and 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1⊂𝐶_𝑛, we have 𝜙𝑥_𝑛,𝑥₀≤𝜙𝑥_𝑛+1,𝑥₀,∀𝑛∈ℕ∪{0}. (39) This implies that the sequence {𝜙(𝑥_𝑛,𝑥₀)} is nondecreasing. It follows that the limit of {𝜙(𝑥_𝑛,𝑥₀)} exists. By the construction of 𝐶_𝑛, we have that 𝑥_𝑚=Π_𝐶𝑚𝑥₀∈𝐶_𝑚⊂𝐶_𝑛 for any positive integer 𝑚≥𝑛. It follows that 𝜙𝑥_𝑚,𝑥_𝑛=𝜙𝑥_𝑚,Π_𝐶𝑛𝑥₀≤𝜙𝑥_𝑚,𝑥₀−𝜙𝑥_𝑛,𝑥₀. (40) Letting 𝑚,𝑛→∞ in (40), by the existence of the limit of {𝜙(𝑥_𝑛,𝑥₀)}, we have 𝜙(𝑥_𝑚,𝑥_𝑛)→0. It follows from Lemma 10 that 𝑥_𝑛−𝑥_𝑚→0 as 𝑚,𝑛→∞. Hence {𝑥_𝑛} is a Cauchy sequence. Therefore, there exists a point 𝑞∈𝐶 such that 𝑥_𝑛→𝑞 as 𝑛→∞.
Since 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1⊂𝐶_𝑛, we have from the definition of 𝐶_𝑛+1 that 𝜙𝑥_𝑛+1,𝑆_𝑛𝑦_𝑛≤𝜙𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛≤𝜙𝑥_𝑛+1,𝑥_𝑛,∀𝑛∈ℕ∪{0}. (41) From the inequality above, we have 𝜙𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛≤𝜙𝑥_𝑛+1,𝑥_𝑛{0},𝜙𝑥,∀𝑛∈ℕ∪_𝑛+1,𝑆_𝑛𝑦_𝑛≤𝜙𝑥_𝑛+1,𝑥_𝑛,∀𝑛∈ℕ∪{0}. (42) On the other hand, taking 𝑚=𝑛+1 in (40), we have lim_𝑛→∞𝜙𝑥_𝑛+1,𝑥_𝑛=0. (43) From (42) and (43), we have that lim_𝑛→∞𝜙𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛=0,lim_𝑛→∞𝜙𝑥_𝑛+1,𝑆_𝑛𝑦_𝑛=0. (44)
By using Lemma 10, the inequalities (43) and (44) follow that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0,lim_𝑛→∞𝑥_𝑛+1−𝑇_𝑛𝑥_𝑛=0,lim_𝑛→∞𝑥_𝑛+1−𝑆_𝑛𝑦_𝑛=0. (45)(46)(47) Respectively, noticing that 𝑥_𝑛−𝑇_𝑛𝑥_𝑛=𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑇_𝑛𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑇_𝑛𝑥_𝑛. (48) It follows from (45) and (46) that lim_𝑛→∞𝑥_𝑛−𝑇_𝑛𝑥_𝑛=0. (49) From uniform closedness of {𝑇_𝑛}, we get 𝑞∈⋂^∞_𝑛=1𝐹(𝑇_𝑛). On the other hand, noticing that 𝑦_𝑛=𝑇_𝑛𝑥_𝑛, we have lim_𝑛→∞𝑦_𝑛=lim_𝑛→∞𝑥_𝑛𝑦=𝑞,_𝑛−𝑆_𝑛𝑦_𝑛=𝑦_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑆_𝑛𝑦_𝑛≤𝑦_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑆_𝑛𝑦_𝑛≤𝑇_𝑛𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑆_𝑛𝑦_𝑛. (50) It follows from (46) and (47) that lim_𝑛→∞𝑦_𝑛−𝑆_𝑛𝑦_𝑛=0. (51) From uniform closedness of {𝑆_𝑛}, we also have 𝑞∈⋂^∞_𝑛=1𝐹(𝑆_𝑛). Therefore, 𝑞∈ℱ.
Finally, we show that 𝑞=Π_ℱ𝑥₀. From 𝑥_𝑛=Π_𝐶𝑛𝑥₀, we have 𝑥_𝑛−𝑝,𝐽𝑥₀−𝐽𝑥_𝑛≥0,∀𝑝∈ℱ⊂𝐶_𝑛. (52) Taking the limit as 𝑛→∞ in (52), we obtain 𝑞−𝑝,𝐽𝑥₀−𝐽𝑞≥0,∀𝑝∈ℱ, (53) and hence 𝑝=Π_ℱ𝑥₀ from Lemma 8. This completes the proof.

Remark 12. Theorem 11 improves Theorem 3.15 of Zhang et al. [49] in the following senses:(1)from the class of a countable family of weak relatively nonexpansive mappings to the one of a countable family of hemi-relatively nonexpansive mappings;(2)from a single countable family of mappings to two countable families of mappings.
When 𝑇_𝑛=𝐼 in (32), we can obtain the following corollary immediately.

Corollary 13. Let 𝐶 be a nonempty, closed and convex subset of a uniformly smooth and uniformly convex Banach space 𝐸. Let {𝑆_𝑛} be a uniformly closed countable family of hemi-relatively nonexpansive mappings from 𝐶 into itself such that ℱ=^∞_𝑛=1𝐹𝑆_𝑛≠∅. (54) For a point 𝑥₀∈𝐶 chosen arbitrarily, let {𝑥_𝑛} be a sequence generated by the following iterative algorithm: 𝐶₀𝐶=𝐶,_𝑛+1=𝑧∈𝐶_𝑛∶𝜙𝑧,𝑆_𝑛𝑥_𝑛≤𝜙𝑧,𝑥_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀. (55) Then the sequence {𝑥_𝑛} converges strongly to a point 𝑞=Π_ℱ𝑥₀, where Π_ℱ is the generalized projection from 𝐶 onto ℱ.

Remark 14. We notice that if {𝑆_𝑛} is a countable family of weak relatively nonexpansive mappings, Corollary 13 is still held. Therefore, Corollary 13 extends and improves Theorem 3.15 in [49].

4. Applications to Convex Feasibility Problems

In this section, we consider the following convex feasibility problem (CFP): findingan𝑥∈^∞_𝑛=1𝐶_𝑛, (56) where 𝑛∈ℕ∪{0}, and {𝐶_𝑛}^∞_𝑛=0 is an intersecting closed convex subset sequence of a Banach space 𝐸. This problem is a frequently appearing problem in diverse areas of mathematical and physical sciences. There is a considerable investigation on (CFP) in the framework of Hilbert spaces which captures applications in various disciplines such as image restoration [50–53], computer tomography [54], and radiation therapy treatment planning [55]. In computer tomography with limited data, in which an unknown image has to be reconstructed from a priori knowledge and from measured results, each piece of information gives a constraint which in turn gives rise to a convex set 𝐶_𝑛 to which the unknown image should belong (see [56]).

Using Theorem 11, we discuss the convex feasibility problems as an application.

Theorem 15. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space 𝐸. Let {Ω_𝑛}^∞_𝑛=0, {Ω^∗_𝑛}^∞_𝑛=0 be two countable families of nonempty closed convex subset of 𝐶 such that Ω=^∞_𝑛=0Ω_𝑛^∞_𝑛=0Ω^∗_𝑛≠∅. (57) For a point 𝑥₀∈𝐶 chosen arbitrarily, let {𝑥_𝑛} be a sequence generated by the following iterative algorithm: 𝐶₀𝐶=𝐶,_𝑛+1=𝑧∈𝐶_𝑛∶𝜙𝑧,Π_Ω𝑛𝑦_𝑛≤𝜙𝑧,Π_Ω^∗_𝑛𝑥_𝑛≤𝜙𝑧,𝑥_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀, (58) where the sequences 𝑦_𝑛=Π_Ω^∗_𝑛𝑥_𝑛. Then the sequence {𝑥_𝑛} converges strongly to a point 𝑞=Π_Ω𝑥₀, where Π_Ω is the generalized projection from 𝐶 onto Ω.

Proof. From Lemma 9, we easily have that {Π_Ω𝑛} and {Π_Ω^∗_𝑛} are two countable families of hemi-relatively nonexpansive mappings. In view of the continuity of Π_Ω𝑛 and Π_Ω^∗_𝑛, we have that {Π_Ω𝑛} and {Π_Ω^∗_𝑛} are two uniformly closed countable families of hemi-relatively nonexpansive mappings. Thus, by using Theorem 11, we have that the sequence {𝑥_𝑛} converges strongly to a point 𝑞=Π_Ω𝑥₀. This completes the proof.

If we only consider a countable family of nonempty closed convex subset of 𝐶, the following corollary can be obtained by using Theorem 15.

Corollary 16. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space 𝐸. Let {Ω_𝑛}^∞_𝑛=0 be a countable family of nonempty closed convex subset of 𝐶 such that Ω=^∞_𝑛=0Ω_𝑛≠∅. (59) For a point 𝑥₀∈𝐶 chosen arbitrarily, let {𝑥_𝑛} be a sequence generated by the following iterative algorithm: 𝐶₀𝐶=𝐶,_𝑛+1=𝑧∈𝐶_𝑛∶𝜙𝑧,Π_Ω𝑛𝑥_𝑛≤𝜙𝑧,𝑥_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀. (60) Then the sequence {𝑥_𝑛} converges strongly to a point 𝑞=Π_Ω𝑥₀, where Π_Ω is the generalized projection from 𝐶 onto Ω.

Proof. Putting Π_Ω^∗_𝑛≡𝐼 for all 𝑛∈ℕ∪{0} in algorithm (58), the conclusion can be obtained from Theorem 15 immediately.

5. Applications to Generalized Mixed Equilibrium Problems

In this section, we apply our main results to prove some strong convergence theorems concerning generalized mixed equilibrium problems in a Banach space 𝐸.

Let 𝐴∶𝐶→𝐸^∗ be a mapping. First, we recall the following definition:

(I) 𝐴 is called monotone if 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥0,∀𝑥,𝑦∈𝐶; (61)

(II) 𝐴 is called 𝛼-inverse strongly monotone if there exists a constant 𝛼>0 such that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝛼𝐴𝑥−𝐴𝑦²,∀𝑥,𝑦∈𝐶. (62) We remark here that an 𝛼-inverse strongly monotone 𝐴 is (1/𝛼)-Lipschitz continuous.

For solving the generalized mixed equilibrium problem (1), let us assume that the nonlinear mapping 𝐴∶𝐶→𝐸^∗ is monotone and continuous, the function 𝜑∶𝐶→ℝ is convex and lower semicontinuous, and the bifunction 𝑓∶𝐶×𝐶→ℝ 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)limsup_𝑡↓0𝑓(𝑥+𝑡(𝑧−𝑥),𝑦)≤𝑓(𝑥,𝑦), for all 𝑥,𝑦,𝑧∈𝐶;(𝐴₄) the function 𝑦↦𝑓(𝑥,𝑦) is convex and lower semicontinuous for all 𝑥∈𝐶.

The following result can be found in Blum and Oettli [1].

Lemma 17 (see Blum and Oettli [1]). 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 𝑓+1𝑧,𝑦𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶. (63)

Lemma 18. Let 𝐶 be a closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸, let 𝐴∶𝐶→𝐸^∗ be a monotone and continuous mapping, let the function 𝜑∶𝐶→ℝ be convex and lower semicontinuous, and let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (𝐴₁)–(𝐴₄). Then, 𝑓(𝑥,𝑦)+⟨𝐴𝑥,𝑦−𝑥⟩+𝜑(𝑦)−𝜑(𝑥) satisfies (𝐴₁)–(𝐴₄).

Proof. For convenience, we set 𝐹(𝑥,𝑦)=𝑓(𝑥,𝑦)+⟨𝐴𝑥,𝑦−𝑥⟩+𝜑(𝑦)−𝜑(𝑥). So, we only need to prove that 𝐹(𝑥,𝑦) satisfies (𝐴₁)–(𝐴₄).
(I) We show that 𝐹(𝑥,𝑥)=0, for all 𝑥∈𝐶. Since 𝑓(𝑥,𝑦) satisfies (𝐴₁), we have 𝐹()()+⟨⟩(𝑥)(𝑥)()𝑥,𝑥=𝑓𝑥,𝑥𝐴𝑥,𝑥−𝑥+𝜑−𝜑=𝑓𝑥,𝑥=0,∀𝑥∈𝐶. (64)
(II) We show that 𝐹 is monotone; that is, 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)≤0, for all 𝑥,𝑦∈𝐶; since 𝐴 is continuous and monotone, from (𝐴₂), we have 𝐹+𝑦(𝑥)+(𝑥)𝑦++𝑥,𝑦+𝐹𝑦,𝑥=𝑓𝑥,𝑦𝐴𝑥,𝑦−𝑥+𝜑−𝜑+𝑓𝑦,𝑥𝐴𝑦,𝑥−𝑦+𝜑−𝜑=𝑓𝑥,𝑦𝐴𝑥,𝑦−𝑥+𝑓𝑦,𝑥𝐴𝑦,𝑥−𝑦≤0+𝐴𝑥−𝐴𝑦,𝑦−𝑥=−𝐴𝑦−𝐴𝑧,𝑦−𝑥≤0. (65)
(III) We show that limsup_𝑡↓0𝐹(𝑥+𝑡(𝑧−𝑥),𝑦)≤𝐹(𝑥,𝑦), for all 𝑥,𝑦,𝑧∈𝐶; Since 𝐴 is continuous and 𝜑 is lower semicontinuous, we have limsup_𝑡↓0𝐹𝑥+𝑡(𝑧−𝑥),𝑦=limsup_𝑡↓0𝑓𝑥+𝑡(𝑧−𝑥),𝑦+limsup_𝑡↓0𝐴(𝑥+𝑡(𝑧−𝑥)),𝑦−(𝑥+𝑡(𝑧−𝑥))+limsup_𝑡↓0𝜑𝑦((+𝑦(𝑥).−𝜑𝑥+𝑡𝑧−𝑥))≤𝑓𝑥,𝑦𝐴𝑥,𝑦−𝑥+𝜑−𝜑=𝐹𝑥,𝑦 (66)
(IV) We show that the function 𝑦↦𝐹(𝑥,𝑦) is convex and lower semicontinuous for each 𝑥∈𝐶.
For each 𝑥∈𝐶, for all 𝑡∈(0,1) and for all 𝑦,𝑧∈𝐶, since 𝑓 satisfies (𝐴₄) and 𝜑 is convex, we have 𝐹()𝑧()𝑧+()()𝑧(𝑥)𝑓+𝑦(𝑥+(𝑓()+⟨⟩(𝑧)(𝑥+()𝐹().𝑥,𝑡𝑦+1−𝑡=𝑓𝑥,𝑡𝑦+1−𝑡𝐴𝑥,𝑡𝑦+1−𝑡𝑧−𝑥+𝜑𝑡𝑦+1−𝑡−𝜑=𝑡𝑥,𝑦𝐴𝑥,𝑦−𝑥+𝜑−𝜑)1−𝑡)𝑥,𝑡𝐴𝑥,𝑧−𝑥+𝜑−𝜑)=𝑡𝐹𝑥,𝑦1−𝑡𝑥,𝑧 (67) This completes the proof.

Lemma 19 (see Takahashi and Zembayashi [17]). 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 𝑟>0 and 𝑥∈𝐸, define a mapping 𝑇_𝑟∶𝐸→𝐶 as follows: 𝑇_𝑟(𝑥)=+1𝑧∈𝐶∶𝑓𝑧,𝑦𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶 (68) for all 𝑥∈𝐸. Then, the following properties hold:(1)𝑇_𝑟 is single valued;(2)𝑇_𝑟 is a firmly nonexpansive-type mapping; that is, for all 𝑥,𝑦∈𝐸, 𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽𝑇_𝑟𝑥−𝐽𝑇_𝑟𝑦≤𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽𝑥−𝐽𝑦; (69)(3)EP(𝑓)=𝐹(𝑇_𝑟)=𝐹(𝑇_𝑟);(4)EP(𝑓) is closed and convex;(5)𝜙(𝑞,𝑇_𝑟𝑥)+𝜙(𝑇_𝑟𝑥,𝑥)≤𝜙(𝑞,𝑥), for all 𝑥∈𝐸,𝑞∈𝐹(𝑇_𝑟).

Lemma 20 (see Zhang et al. [57]). Let 𝐸 be a 𝑝-uniformly convex with 𝑝≥0 and uniformly smooth Banach space, and let 𝐶 be a nonempty closed convex subset of 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (𝐴₁)–(𝐴₄). Let {𝑟_𝑛} be a positive real sequence such that lim_𝑛→∞𝑟_𝑛=𝑟>0. Then the sequence of mappings 𝑇_𝑟𝑛 is uniformly closed.

Next, we shall apply Theorem 11 to solve two generalized mixed equilibrium problems. To accomplish this purpose, let 𝐴,𝐵∶𝐶→𝐸^∗ be two monotone and continuous mappings, let the function 𝜑,𝜓∶𝐶→ℝ be convex and lower semicontinuous, and let 𝑓 and 𝑔 be a bifunction from 𝐶×𝐶 to ℝ satisfying (𝐴₁)–(𝐴₄). For 𝑟>0 and 𝑥∈𝐸, define two mappings 𝐽_𝑟,𝐾_𝑟∶𝐸→𝐶 as follows: 𝐽_𝑟(𝑥)=+𝑦(𝑧)+1𝑧∈𝐶∶𝑓𝑧,𝑦𝐴𝑧,𝑦−𝑧+𝜑−𝜑𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶,𝐾_𝑟(𝑥)=+𝑦(𝑧)+1𝑧∈𝐶∶𝑔𝑧,𝑦𝐵𝑧,𝑦−𝑧+𝜓−𝜓𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶. (70)(71)

Theorem 21. Let 𝐸 be a 𝑝-uniformly convex with 𝑝≥2 and uniformly smooth Banach space, and let 𝐶 be a nonempty closed convex subset of 𝐸. Let 𝐴,𝐵∶𝐶→𝐸^∗ be two monotone and continuous mappings, let the function 𝜑,𝜓∶𝐶→ℝ be convex and lower semicontinuous, and let 𝑓 and 𝑔 be a bifunction from 𝐶×𝐶 to ℝ satisfying (𝐴₁)–(𝐴₄) such that ℑ=GMEP(𝑓,𝐴,𝜑)⋂GMEP(𝑔,𝐵,𝜓)≠∅. For a point 𝑥₀∈𝐶 chosen arbitrarily, let {𝑥_𝑛} be a sequence generated by the following iterative algorithm: 𝐶₀𝐶=𝐶,_𝑛+1=𝑧∈𝐶_𝑛∶𝜙𝑧,𝑣_𝑛≤𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑥_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀, (72) where 𝑢_𝑛=𝐽_𝑟𝑛𝑥_𝑛, 𝑣_𝑛=𝐾_𝑟𝑛𝑢_𝑛, and lim_𝑛→∞𝑟_𝑛=𝑟. Then the sequence {𝑥_𝑛} converges strongly to a point 𝑞=Π_ℑ𝑥₀, where Π_ℑ is the generalized projection from 𝐶 onto ℑ.

Proof. From Lemmas 18 and 20, we learn that {𝐽_𝑟𝑛} and {𝐾_𝑟𝑛} are uniformly closed. And by Lemma 19 (5), one can easily get that {𝐽_𝑟𝑛} and {𝐾_𝑟𝑛} are uniformly closed countable families of hemi-relatively nonexpansive mappings. Notice that if 𝐸 is 𝑝-uniformly convex, it must be uniformly convex. Therefore, by using Theorem 11, we can obtain the conclusion of Theorem 21. This completes the proof.

Theorem 22. Let 𝐸 be a 𝑝-uniformly convex with 𝑝≥2 and uniformly smooth Banach space, and let 𝐶 be a nonempty closed convex subset of 𝐸. Let 𝐴∶𝐶→𝐸^∗ be a monotone and continuous mappings, let the function 𝜑∶𝐶→ℝ be convex and lower semicontinuous and let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (𝐴₁)–(𝐴₄) such that ℑ=GMEP(𝑓,𝐴,𝜑)≠∅. For a point 𝑥₀∈𝐶 chosen arbitrarily, let {𝑥_𝑛} be a sequence generated by the following iterative algorithm: 𝐶₀𝐶=𝐶,_𝑛+1=𝑧∈𝐶_𝑛∶𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑥_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀, (73) where 𝑢_𝑛=𝐽_𝑟𝑛𝑥_𝑛 and lim_𝑛→∞𝑟_𝑛=𝑟. Then the sequence {𝑥_𝑛} converges strongly to a point 𝑞=Π_ℑ𝑥₀, where Π_ℑ is the generalized projection from 𝐶 onto ℑ.

Proof. From Lemmas 18 and 20, we learn that {𝐽_𝑟𝑛} is uniformly closed. And by Lemma 19(5), one can easily get that {𝐽_𝑟𝑛} is an uniformly closed countable family of hemi-relatively nonexpansive mappings. Notice that if 𝐸 is 𝑝-uniformly convex, it must be uniformly convex. Therefore, by using Corollary 13, we can obtain the conclusion of Theorem 21. This completes the proof.

If we let 𝑓≡0, 𝜑≡0 in (70) and 𝐵≡0, 𝜓≡0 in (71), the following corollary can be obtained by using Theorem 21.

Corollary 23. Let 𝐸 be a 𝑝-uniformly convex with 𝑝≥2 and uniformly smooth Banach space, and let 𝐶 be a nonempty closed convex subset of 𝐸. Let 𝑔 be a bifunction from 𝐶×𝐶 to ℝ satisfying (𝐴₁)–(𝐴₄) and 𝐴∶𝐶→𝐸^∗ a monotone and continuous mapping. Suppose that ℑ=VI(𝐴,𝐶)⋂EP(𝑔)≠∅. For a point 𝑥₀∈𝐶 chosen arbitrarily, let {𝑥_𝑛} be a sequence generated by the following iterative algorithm: 𝐶₀𝐶=𝐶,_𝑛+1=𝑧∈𝐶_𝑛∶𝜙𝑧,𝑣_𝑛≤𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑥_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀, (74) where 𝑢_𝑛=𝐽_𝑟𝑛𝑥_𝑛, 𝑣_𝑛=𝐾_𝑟𝑛𝑢_𝑛, and lim_𝑛→∞𝑟_𝑛=𝑟. Then the sequence {𝑥_𝑛} converges strongly to a point 𝑞=Π_ℑ𝑥₀, where Π_ℑ is the generalized projection from 𝐶 onto ℑ.

Remark 24. By analysis of special cases for generalized mixed equilibrium problem, we can obtain the corresponding results based on Theorems 21 and 22 in sequence. Here, we do not itemize these results.

6. Applications to Maximal Monotone Operators

Let 𝒜 be a multivalued operator from 𝐸 to 𝐸^∗ with domain 𝐷(𝒜)={𝑧∈𝐸∶𝒜𝑧≠∅} and range 𝑅(𝒜)={𝑧∈𝐸∶𝑧∈𝐷(𝒜)}. An operator 𝒜 is said to be monotone if 𝑥₁−𝑥₂,𝑦₁−𝑦₂≥0,∀𝑥₁,𝑥₂(𝒜),𝑦∈𝐷₁∈𝒜𝑥₁,𝑦₂∈𝒜𝑥₂. (75)

A monotone operator 𝒜 is said to be maximal if its graph 𝐺(𝒜)={(𝑥,𝑦)∶𝑦∈𝒜𝑥} is not properly contained in the graph of any other monotone operator. It is well known that if 𝒜 is a maximal monotone operator, then 𝒜⁻¹0 is closed and convex.

The following result is also well known.

Lemma 25 (see Rockafellar [58]). Let 𝐸 be a reflexive, strictly convex, and smooth Banach space and 𝒜 a monotone operator from 𝐸 to 𝐸^∗. Then 𝒜 is maximal if and only if 𝑅(𝐽+𝑟𝒜)=𝐸^∗ for all 𝑟>0.

Let 𝐸 be a reflexive, strictly convex, and smooth Banach space and 𝒜 a maximal monotone operator from 𝐸 to 𝐸^∗. Using Lemma 25 and the strict convexity of 𝐸, it follows that, for all 𝑟>0 and 𝑥∈𝐸, there exists a unique 𝑥_𝑟∈𝐷(𝒜) such that 𝐽𝑥∈𝐽𝑥_𝑟+𝑟𝒜𝑥_𝑟. (76)

If 𝐽_𝑟𝑥=𝑥_𝑟, then we can define a single-valued mapping 𝐽_𝑟∶𝐸→𝐷(𝒜) by 𝐽_𝑟=(𝐽+𝑟𝒜)⁻¹𝐽 and such a 𝐽_𝑟 is called the resolvent of 𝒜. We know that 𝒜⁻¹0=𝐹(𝐽_𝑟) for all 𝑟>0 (see [10, 59] for more details).

First, we give an important lemma for this section and remark that the following lemma can be as example of a countable family of hemi-relatively nonexpansive mappings.

Lemma 26. Let 𝐸 be a strictly convex and uniformly smooth Banach space, let 𝒜 be a maximal monotone operator from 𝐸 to 𝐸^∗ such that 𝒜⁻¹0 is nonempty, and let {𝑟_𝑛} be a sequence of positive real numbers which is bounded away from 0 such that 𝐽_𝑟𝑛=(𝐼+𝑟_𝑛𝒜)⁻¹. Then {𝐽_𝑟𝑛} is a uniformly closed countable family of hemi-relatively nonexpansive mappings.

Proof. One has ⋂^∞_𝑛=0𝐹(𝐽_𝑟𝑛)=𝒜⁻¹0≠∅. Firstly, we show 𝐽_𝑟𝑛 is uniformly closed. Let {𝑧_𝑛} be a sequence such that 𝑧_𝑛→𝑝 and lim_𝑛→∞‖𝑧_𝑛−𝐽_𝑟𝑛𝑧_𝑛‖=0. Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, we obtain 1𝑟_𝑛𝐽𝑧_𝑛−𝐽𝐽_𝑟𝑛𝑧_𝑛⟶0,as𝑛⟶∞. (77) It follows from 1𝑟_𝑛𝐽𝑧_𝑛−𝐽𝐽_𝑟𝑛𝑧_𝑛∈𝒜𝐽_𝑟𝑛𝑧_𝑛 (78) and the monotonicity of 𝐵 that 𝑤−𝐽_𝑟𝑛𝑧_𝑛,𝑤^∗−1𝑟_𝑛𝐽𝑧_𝑛−𝐽𝐽_𝑟𝑛𝑧_𝑛≥0 (79) for all 𝑤∈𝐷(𝒜) and 𝑤^∗∈𝒜𝑤. Letting 𝑛→∞, one has ⟨𝑤−𝑝,𝑤^∗⟩≥0 for all 𝑤∈𝐷(𝒜) and 𝑤^∗∈𝒜𝑤. Therefore, from the maximality of 𝒜, one obtains 𝑝∈𝒜⁻¹0=𝐹(𝐽_𝑟𝑛). Hence, 𝐽_𝑟𝑛 is uniformly closed.
In addition, for any 𝑤∈𝐸 and 𝑝∈⋂^∞_𝑛=0𝐹(𝐽_𝑟𝑛), from the monotonicity of 𝒜, one has 𝜙𝑝,𝐽_𝑟𝑛𝑤=𝑝²−2𝑝,𝐽𝐽_𝑟𝑛𝑤+𝐽_𝑟𝑛𝑤²=𝑝²+2𝑝,𝐽𝑤−𝐽𝐽_𝑟𝑛𝑤−𝐽𝑤+𝐽_𝑟𝑛𝑤²=𝑝²+2⟨𝑝,𝐽𝑤−𝐽𝐽_𝑟𝑛𝑤⟩−2𝑝,𝐽𝑤+𝐽_𝑟𝑛𝑤²=𝑝²−2𝐽_𝑟𝑛𝑤−𝑝−𝐽_𝑟𝑛𝑤,𝐽𝑤−𝐽𝐽_𝑟𝑛+𝐽𝑤−𝐽𝑤−2𝑝,𝐽𝑤_𝑟𝑛𝑤²=𝑝²−2𝐽_𝑟𝑛𝑤−𝑝,𝐽𝑤−𝐽𝐽_𝑟𝑛𝐽𝑤−𝐽𝑤+2_𝑟𝑛𝑤,𝐽𝑤−𝐽𝐽_𝑟𝑛𝑤−2𝑝,𝐽𝑤+𝐽_𝑟𝑛𝑤²≤𝑝²+2𝐽_𝑟𝑛𝑤,𝐽𝑤−𝐽𝐽_𝑟𝑛𝑤−2𝑝,𝐽𝑤+𝐽_𝑟𝑛𝑤²=𝑝²−2𝑝,𝐽𝑤+‖𝑤‖²−𝐽_𝑟𝑛𝑤²+2𝐽_𝑟𝑛𝑤,𝐽𝑤−‖𝑤‖²=𝜙𝑝,𝑤−𝜙𝐽_𝑟𝑛𝑤,𝑤≤𝜙𝑝,𝑤, (80) for all 𝑛∈ℕ∪{0}. This implies that {𝐽_𝑟𝑛} is a countable family of hemi-relatively nonexpansive mappings. Hence, {𝐽_𝑟𝑛} is a uniformly closed countable family of hemi-relatively nonexpansive mappings.

We consider the problem of strong convergence concerning maximal monotone operators in a Banach space. Such a problem has been also studied in [4, 13, 49]. Using Theorem 11, we obtain the following result.

Theorem 27. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space 𝐸. Let 𝒜, ℬ be two maximal monotone operators from 𝐸 to 𝐸^∗ such that ℱ=𝒜⁻¹0⋂ℬ⁻¹0≠0, and let {𝑟_𝑛} be a sequence of positive real numbers which is bounded away from 0 such that 𝐽^𝒜_𝑟𝑛=(𝐼+𝑟_𝑛𝒜)⁻¹ and 𝐽^ℬ_𝑟𝑛=(𝐼+𝑟_𝑛ℬ)⁻¹. For a point 𝑥₀∈𝐶 chosen arbitrarily, let {𝑥_𝑛} be a sequence generated by the following iterative algorithm: 𝐶₀𝐶=𝐶,_𝑛+1=𝑧∈𝐶_𝑛∶𝜙𝑧,𝐽^ℬ_𝑟𝑛𝑦_𝑛≤𝜙𝑧,𝐽^𝒜_𝑟𝑛𝑥_𝑛≤𝜙𝑧,𝑥_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀, (3.1) where the sequences 𝑦_𝑛=𝐽^𝒜_𝑟𝑛𝑥_𝑛. Then the sequence {𝑥_𝑛} converges strongly to a point 𝑞=Π_ℱ𝑥₀, where Π_ℱ is the generalized projection from 𝐶 onto ℱ.

Proof. From Lemma 26, we know that {𝐽^𝒜_𝑟𝑛} and {𝐽^ℬ_𝑟𝑛} are two uniformly closed countable families of hemi-relatively nonexpansive mappings. Furthermore, applying Theorem 11, one sees that the sequence {𝑥_𝑛} converges strongly to a point Π_ℱ𝑥₀.

Acknowledgments

The authors are thankful to an anonymous referee for his useful comments on this paper. This research was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission under the Computational Science and Engineering Research Cluster (CSEC-KMUTT) (Grant Project no. NRU56000508). The first author is supported by the Project of Shandong Province Higher Educational Science and Technology Program (Grant no. J13LI51) and the Foundation of Shandong Yingcai University (Grant no. 12YCZDZR03).