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

Authors: Zi-Ming Wang; Poom Kumam
Journal: Journal of Applied Mathematics
 (2013)

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## 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.


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## Body

## 1. Introduction

Throughout this paper, we always assume thatE 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.Letf: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
(1)f(x,y)+〈Ax,y-x〉+φ(y)-φ(x)≥0,∀y∈C.The set of solutions to (1) is denoted by GMEP(f,A,φ), that is,
(2)GMEP(f,A,φ)={x∈C:f(x,y)+〈Ax,y-x〉GMEP(fkki,A,φ)+φ(y)-φ(x)≥0,∀y∈C}.


### 1.1. Analysis of Special Cases

(1) Ifφ(·)≡0, the problem (1) reduces to the generalized equilibrium problem, which is to find x∈C such that
(3)f(x,y)+〈Ax,y-x〉≥0,∀y∈C.
The set of solutions to (3) is denoted by GEP(f,A).(2) IfA≡0, the problem (1) reduces to the mixed equilibrium problem, which is to find x∈C such that
(4)f(x,y)+φ(y)-φ(x)≥0,∀y∈C.
The set of solutions to (4) is denoted by MEP(f,φ).(3) Iff(·,·)≡0, the problem (1) reduces to the mixed variational inequality of Browder type, which is to find x∈C such that
(5)〈Ax,y-x〉+φ(y)-φ(x)≥0,∀y∈C.
The set of solutions to (5) is denoted by MVI(A,φ,C).(4) Iff(·,·)≡0 in (3), the problem (3) reduces to the classic variational inequality, which is to find x∈C such that
(6)〈Ax,y-x〉≥0,∀y∈C,
which is called the Hartmann-Stampacchia variational inequality. The set of solutions to (6) is denoted by VI(A,C).(5) IfA≡0 in (3), the problem (3) reduces to the classic equilibrium problem, which is to find x∈C such that
(7)f(x,y)≥0,∀y∈C.
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) Iff(·,·)≡0 in (4), the problem (4) reduces to the minimize problem, which is to find x∈C such that
(8)φ(y)-φ(x)≥0,∀y∈C.
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 spaceE is said to be strictly convex if ∥(x+y)/2∥<1 for all x,y∈E with ∥x∥=∥y∥=1 and x≠y. Let SE={x∈E:∥x∥=1} be the unit sphere of E, and define f:SE×SE×ℝ∖{0}→ℝ by
(9)f(x,y,t)=∥x+ty∥-∥x∥t
for x,y∈SE and t∈ℝ∖{0}. A Banach space E is said to be smooth if the limit limt→0f(x,y,t) exists for each x,y∈SE. It is also said to be uniformly smooth if the limit limt→0f(x,y,t) is attained uniformly for (x,y)∈SE×SE.The modulus of convexity ofE is the function δ:[0,2]→[0,1] defined by
(10)δ(ε)=inf{1-∥x+y2∥:x,y∈E,δ(ε)=inf∥x∥=∥y∥=1,∥x-y∥≥ε∥x+y2∥}.
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)≥cεp 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 Lp(lp) or Wmp is p-uniformly convex if p≥2 and 2-uniformly convex if 1<p≤2; see [8] for more details.For eachp>1, the generalized duality mapping Jp:E→2E* is defined by
(11)Jp(x)={∥x∥p-1x*∈E*:〈x,x*〉=∥x∥p,Jp(x)=∥x*∥=∥x∥p-1},∀x∈E.
In particular, if p=2, Jp is called the normalized duality mapping. If E is a Hilbert space, then Jp=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)
ifE is smooth, then J is single valued;(ii)
ifE is strictly convex, then J is one to one;(iii)
ifE is reflexive, then J is surjective;(iv)
ifE is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E;(v)
ifE* 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]).LetE be a smooth Banach space. Consider the function defined by
(12)ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2,∀x,y∈E.
It is obvious from the definition of the function ϕ that
(13)(∥x∥-∥y∥)2≤ϕ(x,y)≤(∥x∥+∥y∥)2,∀x,y∈E.
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.LetC 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 {xn} which converges weakly to p such that the strong limn→∞(xn-Txn)=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 {xn} which converges strong to p such that limn→∞∥xn-Txn∥=0. The set of strong asymptotic fixed points of T will be denoted by F~(T).LetT:C→C be a mapping, and recall the following definition:(a)
T is called nonexpansive if
(14)∥Tx-Ty∥≤∥x-y∥,∀x,y∈C;(b)
T is called relatively nonexpansive if F^(T)=F(T)≠∅ and
(15)ϕ(p,Tx)≤ϕ(p,x),∀x∈C,p∈F(T);(c)
a mappingT is said to be weak relatively nonexpansive if F~(T)=F(T)≠∅ and
(16)ϕ(p,Tx)≤ϕ(p,x),∀x∈C,p∈F(T);(d)
a mappingT is called hemirelatively nonexpansive if F(T)≠∅ and
(17)ϕ(p,Tx)≤ϕ(p,x),∀x∈C,p∈F(T).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.
LetΠC be the generalized projection from a smooth, strictly convex, and reflexive Banach space E onto a nonempty closed convex subset C⊂E. Then ΠC 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]).
LetE 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 {αn} be a sequence of real numbers such that 0≤αn<1 and limsupn→∞αn<1. Suppose that {xn} is given by
(18)x0=x∈C,yn=ΠC(αnJxn+(1-αn)JTxn),Cn={z∈C:ϕ(z,yn)≤ϕ(z,xn)},Qn={z∈C:〈xn-z,Jx-Jxn〉≥0},xn+1=ΠCn∩Qnx,∀n∈ℕ∪{0},
where J is the duality mapping on E. If F(T) is nonempty, then {xn} converges strongly to ΠF(T)x, where ΠF(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.


## 1.1. Analysis of Special Cases

(1) Ifφ(·)≡0, the problem (1) reduces to the generalized equilibrium problem, which is to find x∈C such that
(3)f(x,y)+〈Ax,y-x〉≥0,∀y∈C.
The set of solutions to (3) is denoted by GEP(f,A).(2) IfA≡0, the problem (1) reduces to the mixed equilibrium problem, which is to find x∈C such that
(4)f(x,y)+φ(y)-φ(x)≥0,∀y∈C.
The set of solutions to (4) is denoted by MEP(f,φ).(3) Iff(·,·)≡0, the problem (1) reduces to the mixed variational inequality of Browder type, which is to find x∈C such that
(5)〈Ax,y-x〉+φ(y)-φ(x)≥0,∀y∈C.
The set of solutions to (5) is denoted by MVI(A,φ,C).(4) Iff(·,·)≡0 in (3), the problem (3) reduces to the classic variational inequality, which is to find x∈C such that
(6)〈Ax,y-x〉≥0,∀y∈C,
which is called the Hartmann-Stampacchia variational inequality. The set of solutions to (6) is denoted by VI(A,C).(5) IfA≡0 in (3), the problem (3) reduces to the classic equilibrium problem, which is to find x∈C such that
(7)f(x,y)≥0,∀y∈C.
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) Iff(·,·)≡0 in (4), the problem (4) reduces to the minimize problem, which is to find x∈C such that
(8)φ(y)-φ(x)≥0,∀y∈C.
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 spaceE is said to be strictly convex if ∥(x+y)/2∥<1 for all x,y∈E with ∥x∥=∥y∥=1 and x≠y. Let SE={x∈E:∥x∥=1} be the unit sphere of E, and define f:SE×SE×ℝ∖{0}→ℝ by
(9)f(x,y,t)=∥x+ty∥-∥x∥t
for x,y∈SE and t∈ℝ∖{0}. A Banach space E is said to be smooth if the limit limt→0f(x,y,t) exists for each x,y∈SE. It is also said to be uniformly smooth if the limit limt→0f(x,y,t) is attained uniformly for (x,y)∈SE×SE.The modulus of convexity ofE is the function δ:[0,2]→[0,1] defined by
(10)δ(ε)=inf{1-∥x+y2∥:x,y∈E,δ(ε)=inf∥x∥=∥y∥=1,∥x-y∥≥ε∥x+y2∥}.
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)≥cεp 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 Lp(lp) or Wmp is p-uniformly convex if p≥2 and 2-uniformly convex if 1<p≤2; see [8] for more details.For eachp>1, the generalized duality mapping Jp:E→2E* is defined by
(11)Jp(x)={∥x∥p-1x*∈E*:〈x,x*〉=∥x∥p,Jp(x)=∥x*∥=∥x∥p-1},∀x∈E.
In particular, if p=2, Jp is called the normalized duality mapping. If E is a Hilbert space, then Jp=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)
ifE is smooth, then J is single valued;(ii)
ifE is strictly convex, then J is one to one;(iii)
ifE is reflexive, then J is surjective;(iv)
ifE is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E;(v)
ifE* 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]).LetE be a smooth Banach space. Consider the function defined by
(12)ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2,∀x,y∈E.
It is obvious from the definition of the function ϕ that
(13)(∥x∥-∥y∥)2≤ϕ(x,y)≤(∥x∥+∥y∥)2,∀x,y∈E.
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.LetC 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 {xn} which converges weakly to p such that the strong limn→∞(xn-Txn)=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 {xn} which converges strong to p such that limn→∞∥xn-Txn∥=0. The set of strong asymptotic fixed points of T will be denoted by F~(T).LetT:C→C be a mapping, and recall the following definition:(a)
T is called nonexpansive if
(14)∥Tx-Ty∥≤∥x-y∥,∀x,y∈C;(b)
T is called relatively nonexpansive if F^(T)=F(T)≠∅ and
(15)ϕ(p,Tx)≤ϕ(p,x),∀x∈C,p∈F(T);(c)
a mappingT is said to be weak relatively nonexpansive if F~(T)=F(T)≠∅ and
(16)ϕ(p,Tx)≤ϕ(p,x),∀x∈C,p∈F(T);(d)
a mappingT is called hemirelatively nonexpansive if F(T)≠∅ and
(17)ϕ(p,Tx)≤ϕ(p,x),∀x∈C,p∈F(T).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.
LetΠC be the generalized projection from a smooth, strictly convex, and reflexive Banach space E onto a nonempty closed convex subset C⊂E. Then ΠC 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]).
LetE 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 {αn} be a sequence of real numbers such that 0≤αn<1 and limsupn→∞αn<1. Suppose that {xn} is given by
(18)x0=x∈C,yn=ΠC(αnJxn+(1-αn)JTxn),Cn={z∈C:ϕ(z,yn)≤ϕ(z,xn)},Qn={z∈C:〈xn-z,Jx-Jxn〉≥0},xn+1=ΠCn∩Qnx,∀n∈ℕ∪{0},
where J is the duality mapping on E. If F(T) is nonempty, then {xn} converges strongly to ΠF(T)x, where ΠF(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

LetC be a closed convex subset of E, and let {Tn}n=0∞ be a countable family of mappings from C into itself. We denote by F the set of common fixed points of {Tn}n=0∞. That is, F=⋂n=0∞‍F(Tn), where F(Tn) denote the set of fixed points of Tn, for all n∈ℕ∪{0}.Recall that{Tn}n=0∞ is said to be uniformly closed, if p∈⋂n=1∞‍F(Tn), whenever {xn}⊂C converges strongly to p and ∥xn-Tnxn∥→0 as n→∞ (see [45] for more details).A pointp∈C is said to be an asymptotic fixed point of {Tn}n=0∞ if C contains a sequence {xn} which converges weakly to p such that limn→∞∥Tnxn-xn∥=0. The set of asymptotic fixed points of {Tn}n=0∞ will be denoted by F^({Tn}n=0∞).A pointp∈C is said to be a strong asymptotic fixed point of {Tn}n=0∞ if C contains a sequence {xn} which converges strongly to p such that limn→∞∥Tnxn-xn∥=0. The set of strong asymptotic fixed points of {Tn}n=0∞ will be denoted by F~({Tn}n=0∞).Using the definition of (strong) asymptotic fixed point of{Tn}n=0∞, Su et al. [46] introduced the following definitions.Definition 3 (see Su et al. [46]).
Countable family of mappings{Tn} is said to be countable family of relatively nonexpansive mappings if F^({Tn}n=0∞)=F({Tn}n=0∞)≠∅ and
(19)ϕ(p,Tnx)≤ϕ(p,x),∀x∈C,p∈F(Tn),n∈ℕ∪{0}.Definition 4 (see Su et al. [46]).
Countable family of mappings{Tn} is said to be countable family of weak relatively nonexpansive mappings if F~({Tn}n=0∞)=F({Tn}n=0∞)≠∅ and
(20)ϕ(p,Tnx)≤ϕ(p,x),∀x∈C,p∈F(Tn),n∈ℕ∪{0}.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{Tn} is said to be countable family of hemi-relatively nonexpansive mappings if F({Tn}n=0∞)≠∅ and
(21)ϕ(p,Tnx)≤ϕ(p,x),∀x∈C,p∈F(Tn),n∈ℕ∪{0}.Remark 6.
From Definitions3–5, 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 Definitions3, 4, and 5 as Tn≡T for all n∈ℕ∪{0}.(2)
Countable family of hemi-relatively nonexpansive mappings, which do not need the restrictionF^({Tn}n=0∞)=F({Tn}n=0∞) (or F~({Tn}n=0∞)=F({Tn}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.
LetE be any smooth Banach space and x0=(1+1/n)nx0≠0 any element of E. Define a countable family of mappings Tn:E→E as follows: for all n≥1,
(22)Tn(x)={(12+12n+1)x0,ifx=(12+12n)x0,-x,ifx≠(12+12n)x0.
Then {Tn}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 thatTn has a unique fixed point 0; that is, F(Tn)={0} for all n≥1. In addition, one easily sees that
(23)∥Tnx∥≤∥x∥,∀x∈E,n≥1.
This implies that
(24)∥Tnx∥2-∥x∥2≤2〈0,JTnx-Jx〉=2〈p,JTnx-Jx〉,
for all p∈⋂n=1∞‍F(Tn). It follows from the above inequality that
(25)∥p∥2-2〈p,JTnx〉+∥Tnx∥2≤∥p∥2-2〈p,Jx〉+∥x∥2,
for all p∈⋂n=1∞‍F(Tn) and x∈E. That is,
(26)ϕ(p,Tnx)≤ϕ(p,x),
for all p∈⋂n=1∞‍F(Tn) and x∈E. Hence, {Tn}n=1∞ is a countable family of hemi-relatively nonexpansive mappings. On the other hand, letting
(27)xn=(12+12n)x0,∀n≥1,
from the definition of Tn, one has
(28)Tnxn=(12+12n+1)x0,∀n≥1,
which implies that ∥xn-Tnxn∥→0 and xn→ex˘0(xn⇀ex˘0) as n→∞. That is, ex˘0∈F^({Tn}n=0∞) but ex˘0∉F({Tn}n=0∞), which shows that {Tn}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]).
LetC be a convex subset of a smooth real Banach space E. Let x∈E and x0∈C. Then x0=ΠCx if and only if
(29)〈z-x0,Jx0-Jx〉≥0,∀z∈C.Lemma 9 (see Alber [47]).
LetC 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,
(30)ϕ(y,ΠCx)+ϕ(ΠCx,x)≤ϕ(y,x).Lemma 10 (see Kamimura and Takahashi [48]).
LetE be a uniformly convex and smooth real Banach space, and let {xn}, {yn} be two sequences of E. If ϕ(xn,yn)→0 and either {xn} or {yn} is bounded, then ∥xn-yn∥→0.


## 3. Main Results

Now, we give our main results in this paper.Theorem 11.
LetC be a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space E. Let {Sn}, {Tn} be two uniformly closed countable families of hemi-relatively nonexpansive mappings from C into itself such that
(31)ℱ={⋂n=1∞‍F(Sn)}⋂‍{⋂n=1∞‍F(Tn)}≠∅.
For a point x0∈C chosen arbitrarily, let {xn} be a sequence generated by the following iterative algorithm:
(32)C0=C,Cn+1={z∈Cn:ϕ(z,Snyn)≤ϕ(z,Tnxn)≤ϕ(z,xn)},xn+1=ΠCn+1x0,
where the sequences yn=Tnxn. Then the sequence {xn} converges strongly to a point q=Πℱx0, where Πℱ is the generalized projection from C onto ℱ.Proof.
We first show thatCn+1 is closed and convex. It is obvious that Cn+1 is closed. Since
(33)ϕ(z,Snyn)≤ϕ(z,Tnxn)⟺∥Snyn∥2-∥Tnxn∥-2〈z,JSnyn-JTnxn〉≥0,(34)ϕ(z,Tnxn)≤ϕ(z,xn)⟺∥Tnxn∥2-∥xn∥-2〈z,JTnxn-Jxn〉≥0,Cn+1 is convex. Therefore, Cn+1 is closed and convex for all n∈ℕ∪{0}.
Letu∈ℱ; from the definition of Sn and Tn, we have
(35)ϕ(u,Snyn)≤ϕ(u,yn)=ϕ(u,Tnxn)≤ϕ(u,xn).
Hence, we have u∈Cn+1. This implies that ℱ⊂Cn+1 for arbitrary n∈ℕ∪{0}.
Noticingxn=ΠCnx0, from Lemma 8, we have
(36)〈xn-z,Jx0-Jxn〉≥0,∀z∈Cn.
Since ℱ⊂Cn for all n∈ℕ∪{0}, we arrive at
(37)〈xn-p,Jx0-Jxn〉≥0,∀p∈ℱ.
From Lemma 9, we have
(38)ϕ(xn,x0)=ϕ(ΠCnx0,x0)≤ϕ(p,x0)-ϕ(p,xn)≤ϕ(p,x0)
for each p∈ℱ⊂Cn and for all n∈ℕ∪{0}. So the sequence {ϕ(xn,x0)} is bounded. On the other hand, noticing that xn=ΠCnx0 and xn+1=ΠCn+1x0∈Cn+1⊂Cn, we have
(39)ϕ(xn,x0)≤ϕ(xn+1,x0),∀n∈ℕ∪{0}.
This implies that the sequence {ϕ(xn,x0)} is nondecreasing. It follows that the limit of {ϕ(xn,x0)} exists. By the construction of Cn, we have that xm=ΠCmx0∈Cm⊂Cn for any positive integer m≥n. It follows that
(40)ϕ(xm,xn)=ϕ(xm,ΠCnx0)≤ϕ(xm,x0)-ϕ(xn,x0).
Letting m,n→∞ in (40), by the existence of the limit of {ϕ(xn,x0)}, we have ϕ(xm,xn)→0. It follows from Lemma 10 that xn-xm→0 as m,n→∞. Hence {xn} is a Cauchy sequence. Therefore, there exists a point q∈C such that xn→q as n→∞.
Sincexn+1=ΠCn+1x0∈Cn+1⊂Cn, we have from the definition of Cn+1 that
(41)ϕ(xn+1,Snyn)≤ϕ(xn+1,Tnxn)≤ϕ(xn+1,xn),∀n∈ℕ∪{0}.
From the inequality above, we have
(42)ϕ(xn+1,Tnxn)≤ϕ(xn+1,xn),∀n∈ℕ∪{0},ϕ(xn+1,Snyn)≤ϕ(xn+1,xn),∀n∈ℕ∪{0}.
On the other hand, taking m=n+1 in (40), we have
(43)limn→∞ϕ(xn+1,xn)=0.
From (42) and (43), we have that
(44)limn→∞ϕ(xn+1,Tnxn)=0,limn→∞ϕ(xn+1,Snyn)=0.
By using Lemma10, the inequalities (43) and (44) follow that
(45)limn→∞∥xn+1-xn∥=0,(46)limn→∞∥xn+1-Tnxn∥=0,(47)limn→∞∥xn+1-Snyn∥=0.
Respectively, noticing that
(48)∥xn-Tnxn∥=∥xn-xn+1+xn+1-Tnxn∥≤∥xn-xn+1∥+∥xn+1-Tnxn∥.
It follows from (45) and (46) that
(49)limn→∞∥xn-Tnxn∥=0.
From uniform closedness of {Tn}, we get q∈⋂n=1∞‍F(Tn). On the other hand, noticing that yn=Tnxn, we have
(50)limn→∞yn=limn→∞xn=q,∥yn-Snyn∥=∥yn-xn+1+xn+1-Snyn∥≤∥yn-xn+1∥+∥xn+1-Snyn∥≤∥Tnxn-xn+1∥+∥xn+1-Snyn∥.
It follows from (46) and (47) that
(51)limn→∞∥yn-Snyn∥=0.
From uniform closedness of {Sn}, we also have q∈⋂n=1∞‍F(Sn). Therefore, q∈ℱ.
Finally, we show thatq=Πℱx0. From xn=ΠCnx0, we have
(52)〈xn-p,Jx0-Jxn〉≥0,∀p∈ℱ⊂Cn.
Taking the limit as n→∞ in (52), we obtain
(53)〈q-p,Jx0-Jq〉≥0,∀p∈ℱ,
and hence p=Πℱx0 from Lemma 8. This completes the proof.Remark 12.
Theorem11 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.
WhenTn=I in (32), we can obtain the following corollary immediately.Corollary 13.
LetC be a nonempty, closed and convex subset of a uniformly smooth and uniformly convex Banach space E. Let {Sn} be a uniformly closed countable family of hemi-relatively nonexpansive mappings from C into itself such that
(54)ℱ={⋂n=1∞‍F(Sn)}≠∅.
For a point x0∈C chosen arbitrarily, let {xn} be a sequence generated by the following iterative algorithm:
(55)C0=C,Cn+1={z∈Cn:ϕ(z,Snxn)≤ϕ(z,xn)},xn+1=ΠCn+1x0.
Then the sequence {xn} converges strongly to a point q=Πℱx0, where Πℱ is the generalized projection from C onto ℱ.Remark 14.
We notice that if{Sn} 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):(56)findinganx∈⋂n=1∞‍Cn,
where n∈ℕ∪{0}, and {Cn}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 Cn to which the unknown image should belong (see [56]).Using Theorem11, we discuss the convex feasibility problems as an application.Theorem 15.
LetC be a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space E. Let {Ωn}n=0∞, {Ωn*}n=0∞ be two countable families of nonempty closed convex subset of C such that
(57)Ω={⋂n=0∞‍Ωn}⋂‍{⋂n=0∞‍Ωn*}≠∅.
For a point x0∈C chosen arbitrarily, let {xn} be a sequence generated by the following iterative algorithm:
(58)C0=C,Cn+1={z∈Cn:ϕ(z,ΠΩnyn)≤ϕ(z,ΠΩn*xn)≤ϕ(z,xn)},xn+1=ΠCn+1x0,
where the sequences yn=ΠΩn*xn. Then the sequence {xn} converges strongly to a point q=ΠΩx0, where ΠΩ is the generalized projection from C onto Ω.Proof.
From Lemma9, we easily have that {ΠΩn} and {ΠΩn*} are two countable families of hemi-relatively nonexpansive mappings. In view of the continuity of ΠΩn and ΠΩn*, we have that {ΠΩn} and {ΠΩn*} are two uniformly closed countable families of hemi-relatively nonexpansive mappings. Thus, by using Theorem 11, we have that the sequence {xn} converges strongly to a point q=ΠΩx0. This completes the proof.If we only consider a countable family of nonempty closed convex subset ofC, the following corollary can be obtained by using Theorem 15.Corollary 16.
LetC be a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space E. Let {Ωn}n=0∞ be a countable family of nonempty closed convex subset of C such that
(59)Ω={⋂n=0∞‍Ωn}≠∅.
For a point x0∈C chosen arbitrarily, let {xn} be a sequence generated by the following iterative algorithm:
(60)C0=C,Cn+1={z∈Cn:ϕ(z,ΠΩnxn)≤ϕ(z,xn)},xn+1=ΠCn+1x0.
Then the sequence {xn} converges strongly to a point q=ΠΩx0, where ΠΩ is the generalized projection from C onto Ω.Proof.
PuttingΠΩn*≡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 spaceE.LetA:C→E* be a mapping. First, we recall the following definition:(I)A is called monotone if
(61)〈Ax-Ay,x-y〉≥0,∀x,y∈C;(II)A is called α-inverse strongly monotone if there exists a constant α>0 such that
(62)〈Ax-Ay,x-y〉≥α∥Ax-Ay∥2,∀x,y∈C.
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)
limsupt↓0f(x+t(z-x),y)≤f(x,y), for all x,y,z∈C;(A4)
the functiony↦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]).
LetC 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
(63)f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C.Lemma 18.
LetC 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 setF(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 thatF(x,x)=0, for all x∈C. Since f(x,y) satisfies (A1), we have
(64)F(x,x)=f(x,x)+〈Ax,x-x〉F(x,x)+φ(x)-φ(x)=f(x,x)=0,∀x∈C.
(II) We show thatF 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
(65)F(x,y)+F(y,x)=f(x,y)+〈Ax,y-x〉+φ(y)-φ(x)+f(y,x)+〈Ay,x-y〉+φ(x)-φ(y)=f(x,y)+〈Ax,y-x〉+f(y,x)+〈Ay,x-y〉≤0+〈Ax-Ay,y-x〉=-〈Ay-Az,y-x〉≤0.
(III) We show thatlimsupt↓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
(66)limsupt↓0F(x+t(z-x),y)=limsupt↓0f(x+t(z-x),y)+limsupt↓0〈A(x+t(z-x)),y-(x+t(z-x))〉+limsupt↓0[φ(y)-φ(x+t(z-x))]≤f(x,y)+〈Ax,y-x〉+φ(y)-φ(x)=F(x,y).
(IV) We show that the functiony↦F(x,y) is convex and lower semicontinuous for each x∈C.
For eachx∈C, for all t∈(0,1) and for all y,z∈C, since f satisfies (A4) and φ is convex, we have
(67)F(x,ty+(1-t)z)=f(x,ty+(1-t)z)+〈Ax,ty+(1-t)z-x〉+φ(ty+(1-t)z)-φ(x)=t[f(x,y)+〈Ax,y-x〉+φ(y)-φ(x)]+(1-t)[f(x,t)+〈Ax,z-x〉+φ(z)-φ(x)]=tF(x,y)+(1-t)F(x,z).
This completes the proof.Lemma 19 (see Takahashi and Zembayashi [17]).
LetC 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 Tr:E→C as follows:
(68)Tr(x)={z∈C:f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C+1r}
for all x∈E. Then, the following properties hold:(1)
Tr is single valued;(2)
Tr is a firmly nonexpansive-type mapping; that is, for all x,y∈E,
(69)〈Trx-Try,JTrx-JTry〉≤〈Trx-Try,Jx-Jy〉;(3)
EP(f)=F(Tr)=F^(Tr);(4)
EP(f) is closed and convex;(5)
ϕ(q,Trx)+ϕ(Trx,x)≤ϕ(q,x), for all x∈E,q∈F(Tr).Lemma 20 (see Zhang et al. [57]).
LetE 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 {rn} be a positive real sequence such that limn→∞rn=r>0. Then the sequence of mappings Trn is uniformly closed.Next, we shall apply Theorem11 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 Jr,Kr:E→C as follows:
(70)Jr(x)={+1r〈y-z,Jz-Jx〉≥0,∀y∈C}z∈C:f(z,y)+〈Az,y-z〉+φ(y)-φ(z)Jr(x)+1r〈y-z,Jz-Jx〉≥0,∀y∈C},(71)Kr(x)={1rz∈C:g(z,y)+〈Bz,y-z〉+ψ(y)-ψ(z)Kr(x)+1r〈y-z,Jz-Jx〉≥0,∀y∈C}.Theorem 21.
LetE 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 {xn} be a sequence generated by the following iterative algorithm:
(72)C0=C,Cn+1={z∈Cn:ϕ(z,vn)≤ϕ(z,un)≤ϕ(z,xn)},xn+1=ΠCn+1x0,
where un=Jrnxn, vn=Krnun, and limn→∞rn=r. Then the sequence {xn} converges strongly to a point q=Πℑx0, where Πℑ is the generalized projection from C onto ℑ.Proof.
From Lemmas18 and 20, we learn that {Jrn} and {Krn} are uniformly closed. And by Lemma 19 (5), one can easily get that {Jrn} and {Krn} 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.
LetE 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 {xn} be a sequence generated by the following iterative algorithm:
(73)C0=C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)},xn+1=ΠCn+1x0,
where un=Jrnxn and limn→∞rn=r. Then the sequence {xn} converges strongly to a point q=Πℑx0, where Πℑ is the generalized projection from C onto ℑ.Proof.
From Lemmas18 and 20, we learn that {Jrn} is uniformly closed. And by Lemma 19(5), one can easily get that {Jrn} 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 letf≡0, φ≡0 in (70) and B≡0, ψ≡0 in (71), the following corollary can be obtained by using Theorem 21.Corollary 23.
LetE 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 {xn} be a sequence generated by the following iterative algorithm:
(74)C0=C,Cn+1={z∈Cn:ϕ(z,vn)≤ϕ(z,un)≤ϕ(z,xn)},xn+1=ΠCn+1x0,
where un=Jrnxn, vn=Krnun, and limn→∞rn=r. Then the sequence {xn} converges strongly to a point q=Πℑx0, where Πℑ 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 Theorems21 and 22 in sequence. Here, we do not itemize these results.


## 6. Applications to Maximal Monotone Operators

Let𝒜 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
(75)〈x1-x2,y1-y2〉≥0,∀x1,x2∈D(𝒜),jjjjjjjjjjjjjjjjsssjjjjjjjjjjy1∈𝒜x1,y2∈𝒜x2.A monotone operator𝒜 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]).
LetE 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.LetE 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 xr∈D(𝒜) such that
(76)Jx∈Jxr+r𝒜xr.IfJrx=xr, then we can define a single-valued mapping Jr:E→D(𝒜) by Jr=(J+r𝒜)-1J and such a Jr is called the resolvent of 𝒜. We know that 𝒜-10=F(Jr) 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.
LetE 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 {rn} be a sequence of positive real numbers which is bounded away from 0 such that Jrn=(I+rn𝒜)-1. Then {Jrn} is a uniformly closed countable family of hemi-relatively nonexpansive mappings.Proof.
One has⋂n=0∞‍F(Jrn)=𝒜-10≠∅. Firstly, we show Jrn is uniformly closed. Let {zn} be a sequence such that zn→p and limn→∞∥zn-Jrnzn∥=0. Since J is uniformly norm-to-norm continuous on bounded sets, we obtain
(77)1rn(Jzn-JJrnzn)⟶0,asn⟶∞.
It follows from
(78)1rn(Jzn-JJrnzn)∈𝒜Jrnzn
and the monotonicity of B that
(79)〈w-Jrnzn,w*-1rn(Jzn-JJrnzn)〉≥0
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(Jrn). Hence, Jrn is uniformly closed.
In addition, for anyw∈E and p∈⋂n=0∞‍F(Jrn), from the monotonicity of 𝒜, one has
(80)ϕ(p,Jrnw)=∥p∥2-2〈p,JJrnw〉+∥Jrnw∥2=∥p∥2+2〈p,Jw-JJrnw-Jw〉+∥Jrnw∥2=∥p∥2+2〈p,Jw-JJrnw〉-2〈p,Jw〉+∥Jrnw∥2=∥p∥2-2〈Jrnw-p-Jrnw,Jw-JJrnw-Jw〉-2〈p,Jw〉+∥Jrnw∥2=∥p∥2-2〈Jrnw-p,Jw-JJrnw-Jw〉+2〈Jrnw,Jw-JJrnw〉-2〈p,Jw〉+∥Jrnw∥2≤∥p∥2+2〈Jrnw,Jw-JJrnw〉-2〈p,Jw〉+∥Jrnw∥2=∥p∥2-2〈p,Jw〉+∥w∥2-∥Jrnw∥2+2〈Jrnw,Jw〉-∥w∥2=ϕ(p,w)-ϕ(Jrnw,w)≤ϕ(p,w),
for all n∈ℕ∪{0}. This implies that {Jrn} is a countable family of hemi-relatively nonexpansive mappings. Hence, {Jrn} 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.
LetC 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 {rn} be a sequence of positive real numbers which is bounded away from 0 such that Jrn𝒜=(I+rn𝒜)-1 and Jrnℬ=(I+rnℬ)-1. For a point x0∈C chosen arbitrarily, let {xn} be a sequence generated by the following iterative algorithm:
(3.1)C0=C,Cn+1={z∈Cn:ϕ(z,Jrnℬyn)≤ϕ(z,Jrn𝒜xn)≤ϕ(z,xn)},xn+1=ΠCn+1x0,
where the sequences yn=Jrn𝒜xn. Then the sequence {xn} converges strongly to a point q=Πℱx0, where Πℱ is the generalized projection from C onto ℱ.Proof.
From Lemma26, we know that {Jrn𝒜} and {Jrnℬ} are two uniformly closed countable families of hemi-relatively nonexpansive mappings. Furthermore, applying Theorem 11, one sees that the sequence {xn} converges strongly to a point Πℱx0.



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