JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 524795 10.1155/2013/524795 524795 Research Article Hybrid Projection Algorithm for Two Countable Families of Hemirelatively Nonexpansive Mappings and Applications Wang Zi-Ming 1 Kumam Poom 2 La Torre Davide 1 Department of Foundation Shandong Yingcai University Jinan 250104 China ycxy.com 2 Department of Mathematics Faculty of Science King Mongkut’s University of Technology Thonburi (KMUTT) Bang Mod Thrung Khru Bangkok 10140 Thailand kmutt.ac.th 2013 7 10 2013 2013 29 05 2013 22 08 2013 2013 Copyright © 2013 Zi-Ming Wang and Poom Kumam. 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.

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 and the sets of positive integers and real numbers, respectively.

Let f:C×C be a bifunction and A:CE* a nonlinear mapping. The “so-called” generalized mixed equilibrium problem is to find xC such that (1)f(x,y)+Ax,y-x+φ(y)-φ(x)0,yC.

The set of solutions to (1) is denoted by GMEP(f,A,φ), that is, (2)GMEP(f,A,φ)={xC:f(x,y)+Ax,y-xGMEP(fkki,A,φ)+φ(y)-φ(x)0,yC}.

1.1. Analysis of Special Cases

(1) If φ(·)0, the problem (1) reduces to the generalized equilibrium problem, which is to find xC such that (3)f(x,y)+Ax,y-x0,yC. The set of solutions to (3) is denoted by GEP(f,A).

(2) If A0, the problem (1) reduces to the mixed equilibrium problem, which is to find xC such that (4)f(x,y)+φ(y)-φ(x)0,yC. 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 xC such that (5)Ax,y-x+φ(y)-φ(x)0,yC. 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 xC such that (6)Ax,y-x0,yC, which is called the Hartmann-Stampacchia variational inequality. The set of solutions to (6) is denoted by VI(A,C).

(5) If A0 in (3), the problem (3) reduces to the classic equilibrium problem, which is to find xC such that (7)f(x,y)0,yC. The set of solutions to (7) is denoted by EP(f). Given a mapping T:CE*, let f(x,y)=Tx,y-x for all x,yC. Then pEP(f) if and only if Tp,y-p0 for all yC; 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 xC such that (8)φ(y)-φ(x)0,yC. 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,  and references therein.

A Banach space E is said to be strictly convex if (x+y)/2<1 for all x,yE with x=y=1 and xy. Let SE={xE:x=1} be the unit sphere of E, and define f:SE×SE×{0} by (9)f(x,y,t)=x+ty-xt for x,ySE and t{0}. A Banach space E is said to be smooth if the limit limt0f(x,y,t) exists for each x,ySE. It is also said to be uniformly smooth if the limit limt0f(x,y,t) is attained uniformly for (x,y)SE×SE.

The modulus of convexity of E is the function δ:[0,2][0,1] defined by (10)δ(ε)=inf{1-x+y2:x,yE,δ(ε)=infx=y=1,x-yεx+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 p2. 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 p2 and 2-uniformly convex if 1<p2; see  for more details.

For each p>1, the generalized duality mapping Jp:E2E* is defined by (11)Jp(x)={xp-1x*E*:x,x*=xp,Jp(x)=x*=xp-1},xE. 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:

if E is smooth, then J is single valued;

if E is strictly convex, then J is one to one;

if E is reflexive, then J is surjective;

if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E;

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 ).

Let E be a smooth Banach space. Consider the function defined by (12)ϕ(x,y)=x2-2x,Jy+y2,x,yE. It is obvious from the definition of the function ϕ that (13)(x-y)2ϕ(x,y)(x+y)2,x,yE. We also know that ϕ(x,y)=0 if and only if x=y (see ). Moreover, if E is a Hilbert space, (12) reduces to ϕ(x,y)=x-y2, for any x,yE.

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  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  if C contains a sequence {xn} which converges strong to p such that limnxn-Txn=0. The set of strong asymptotic fixed points of T will be denoted by F~(T).

Let T:CC be a mapping, and recall the following definition:

T is called nonexpansive if (14)Tx-Tyx-y,x,yC;

T is called relatively nonexpansive if F^(T)=F(T) and (15)ϕ(p,Tx)ϕ(p,x),xC,pF(T);

a mapping T is said to be weak relatively nonexpansive if F~(T)=F(T) and (16)ϕ(p,Tx)ϕ(p,x),xC,pF(T);

a mapping T is called hemirelatively nonexpansive if F(T) and (17)ϕ(p,Tx)ϕ(p,x),xC,pF(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.

Let ΠC be the generalized projection from a smooth, strictly convex, and reflexive Banach space E onto a nonempty closed convex subset CE. Then ΠC is a relatively nonexpansive mapping, and then it is also a closed hemi-relatively nonexpansive mapping.

In 2005, Matsushita and Takahashi  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 [<xref ref-type="bibr" rid="B13">13</xref>, 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 {αn} be a sequence of real numbers such that 0αn<1 and limsupnαn<1. Suppose that {xn} is given by (18)x0=xC,yn=ΠC(αnJxn+(1-αn)JTxn),Cn={zC:ϕ(z,yn)ϕ(z,xn)},Qn={zC:xn-z,Jx-Jxn0},xn+1=ΠCnQnx,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, 1518], and for the hybrid algorithm projection methods for these problems, please see  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 {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=0F(Tn), where F(Tn) denote the set of fixed points of Tn, for all n{0}.

Recall that {Tn}n=0 is said to be uniformly closed, if pn=1F(Tn), whenever {xn}C converges strongly to p and xn-Tnxn0 as n (see  for more details).

A point pC 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 limnTnxn-xn=0. The set of asymptotic fixed points of {Tn}n=0 will be denoted by F^({Tn}n=0).

A point pC 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 limnTnxn-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.  introduced the following definitions.

Definition 3 (see Su et al. [<xref ref-type="bibr" rid="B19">46</xref>]).

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),xC,pF(Tn),n{0}.

Definition 4 (see Su et al. [<xref ref-type="bibr" rid="B19">46</xref>]).

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),xC,pF(Tn),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),xC,pF(Tn),n{0}.

Remark 6.

From Definitions 35, one has the following facts.

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 TnT for all n{0}.

Countable family of hemi-relatively nonexpansive mappings, which do not need the restriction F^({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.

Let E be any smooth Banach space and x0=(1+1/n)nx00 any element of E. Define a countable family of mappings Tn:EE as follows: for all n1, (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 that Tn has a unique fixed point 0; that is, F(Tn)={0} for all n1. In addition, one easily sees that (23)Tnxx,xE,n1. This implies that (24)Tnx2-x220,JTnx-Jx=2p,JTnx-Jx, for all pn=1F(Tn). It follows from the above inequality that (25)p2-2p,JTnx+Tnx2p2-2p,Jx+x2, for all pn=1F(Tn) and xE. That is, (26)ϕ(p,Tnx)ϕ(p,x), for all pn=1F(Tn) and xE. Hence, {Tn}n=1 is a countable family of hemi-relatively nonexpansive mappings. On the other hand, letting (27)xn=(12+12n)x0,n1, from the definition of Tn, one has (28)Tnxn=(12+12n+1)x0,n1, which implies that xn-Tnxn0 and xnex˘0(xnex˘0) as n. That is, ex˘0F^({Tn}n=0) but ex˘0F({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 [<xref ref-type="bibr" rid="B20">47</xref>]).

Let C be a convex subset of a smooth real Banach space E. Let xE and x0C. Then x0=ΠCx if and only if (29)z-x0,Jx0-Jx0,zC.

Lemma 9 (see Alber [<xref ref-type="bibr" rid="B20">47</xref>]).

Let C be a nonempty, closed, and convex subset of a reflexive, strictly convex, and smooth real Banach space E, and let xE. Then for each yC, (30)ϕ(y,ΠCx)+ϕ(ΠCx,x)ϕ(y,x).

Lemma 10 (see Kamimura and Takahashi [<xref ref-type="bibr" rid="B21">48</xref>]).

Let E 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-yn0.

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 {Sn}, {Tn} be two uniformly closed countable families of hemi-relatively nonexpansive mappings from C into itself such that (31)={n=1F(Sn)}{n=1F(Tn)}. For a point x0C chosen arbitrarily, let {xn} be a sequence generated by the following iterative algorithm: (32)C0=C,Cn+1={zCn: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=Πx0, where Π is the generalized projection from C onto .

Proof.

We first show that Cn+1 is closed and convex. It is obvious that Cn+1 is closed. Since (33)ϕ(z,Snyn)ϕ(z,Tnxn)Snyn2-Tnxn-2z,JSnyn-JTnxn0,(34)ϕ(z,Tnxn)ϕ(z,xn)Tnxn2-xn-2z,JTnxn-Jxn0,Cn+1 is convex. Therefore, Cn+1 is closed and convex for all n{0}.

Let u; 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 uCn+1. This implies that Cn+1 for arbitrary n{0}.

Noticing xn=ΠCnx0, from Lemma 8, we have (36)xn-z,Jx0-Jxn0,zCn. Since Cn for all n{0}, we arrive at (37)xn-p,Jx0-Jxn0,p. From Lemma 9, we have (38)ϕ(xn,x0)=ϕ(ΠCnx0,x0)ϕ(p,x0)-ϕ(p,xn)ϕ(p,x0) for each pCn and for all n{0}. So the sequence {ϕ(xn,x0)} is bounded. On the other hand, noticing that xn=ΠCnx0 and xn+1=ΠCn+1x0Cn+1Cn, we have (39)ϕ(xn,x0)ϕ(xn+1,x0),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=ΠCmx0CmCn for any positive integer mn. 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-xm0 as m,n. Hence {xn} is a Cauchy sequence. Therefore, there exists a point qC such that xnq as n.

Since xn+1=ΠCn+1x0Cn+1Cn, we have from the definition of Cn+1 that (41)ϕ(xn+1,Snyn)ϕ(xn+1,Tnxn)ϕ(xn+1,xn),n{0}. From the inequality above, we have (42)ϕ(xn+1,Tnxn)ϕ(xn+1,xn),      n{0},ϕ(xn+1,Snyn)ϕ(xn+1,xn),      n{0}. On the other hand, taking m=n+1 in (40), we have (43)limnϕ(xn+1,xn)=0. From (42) and (43), we have that (44)limnϕ(xn+1,Tnxn)=0,limnϕ(xn+1,Snyn)=0.

By using Lemma 10, the inequalities (43) and (44) follow that (45)limnxn+1-xn=0,(46)limnxn+1-Tnxn=0,(47)limnxn+1-Snyn=0. Respectively, noticing that (48)xn-Tnxn=xn-xn+1+xn+1-Tnxnxn-xn+1+xn+1-Tnxn. It follows from (45) and (46) that (49)limnxn-Tnxn=0. From uniform closedness of {Tn}, we get qn=1F(Tn). On the other hand, noticing that yn=Tnxn, we have (50)limnyn=limnxn=q,yn-Snyn=yn-xn+1+xn+1-Snynyn-xn+1+xn+1-SnynTnxn-xn+1+xn+1-Snyn. It follows from (46) and (47) that (51)limnyn-Snyn=0. From uniform closedness of {Sn}, we also have qn=1F(Sn). Therefore, q.

Finally, we show that q=Πx0. From xn=ΠCnx0, we have (52)xn-p,Jx0-Jxn0,pCn. Taking the limit as n in (52), we obtain (53)q-p,Jx0-Jq0,p, and hence p=Πx0 from Lemma 8. This completes the proof.

Remark 12.

Theorem 11 improves Theorem 3.15 of Zhang et al.  in the following senses:

from the class of a countable family of weak relatively nonexpansive mappings to the one of a countable family of hemi-relatively nonexpansive mappings;

from a single countable family of mappings to two countable families of mappings.

When Tn=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 {Sn} be a uniformly closed countable family of hemi-relatively nonexpansive mappings from C into itself such that (54)={n=1F(Sn)}. For a point x0C chosen arbitrarily, let {xn} be a sequence generated by the following iterative algorithm: (55)C0=C,Cn+1={zCn:ϕ(z,Snxn)ϕ(z,xn)},xn+1=ΠCn+1x0. Then the sequence {xn} converges strongly to a point q=Πx0, where Π is the generalized projection from C onto .

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 .

4. Applications to Convex Feasibility Problems

In this section, we consider the following convex feasibility problem (CFP): (56)finding  anxn=1Cn, where 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 , computer tomography , and radiation therapy treatment planning . 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 ).

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 {Ωn}n=0, {Ωn*}n=0 be two countable families of nonempty closed convex subset of C such that (57)Ω={n=0Ωn}{n=0Ωn*}. For a point x0C chosen arbitrarily, let {xn} be a sequence generated by the following iterative algorithm: (58)C0=C,Cn+1={zCn: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 Lemma 9, 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 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 {Ωn}n=0 be a countable family of nonempty closed convex subset of C such that (59)Ω={n=0Ωn}. For a point x0C chosen arbitrarily, let {xn} be a sequence generated by the following iterative algorithm: (60)C0=C,Cn+1={zCn: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.

Putting ΠΩn*I for all 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:CE* be a mapping. First, we recall the following definition:

(I) A is called monotone if (61)Ax-Ay,x-y0,x,yC;

(II) A is called α-inverse strongly monotone if there exists a constant α>0 such that (62)Ax-Ay,x-yαAx-Ay2,x,yC. 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:CE* is monotone and continuous, the function φ:C is convex and lower semicontinuous, and the bifunction f:C×C satisfies the following conditions:

f(x,x)=0, for all xC;

f is monotone, that is, f(x,y)+f(y,x)0, for all x,yC;

limsupt0f(x+t(z-x),y)f(x,y), for all x,y,zC;

the function yf(x,y) is convex and lower semicontinuous for all xC.

The following result can be found in Blum and Oettli .

Lemma 17 (see Blum and Oettli [<xref ref-type="bibr" rid="B1">1</xref>]).

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 satisfying (A1)–(A4), and let r>0 and xE. Then, there exists zC such that (63)f(z,y)+1ry-z,Jz-Jx0,yC.

Lemma 18.

Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let A:CE* be a monotone and continuous mapping, let the function φ:C be convex and lower semicontinuous, and let f be a bifunction from C×C to 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 xC. Since f(x,y) satisfies (A1), we have (64)F(x,x)=f(x,x)+Ax,x-xF(x,x)+φ(x)-φ(x)=f(x,x)=0,xC.

(II) We show that F is monotone; that is, F(x,y)+F(y,x)0, for all x,yC; 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-y0+Ax-Ay,y-x=-Ay-Az,y-x0.

(III) We show that limsupt0F(x+t(z-x),y)F(x,y), for all x,y,zC; Since A is continuous and φ is lower semicontinuous, we have (66)limsupt0F(x+t(z-x),y)=limsupt0f(x+t(z-x),y)+limsupt0A(x+t(z-x)),y-(x+t(z-x))+limsupt0[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 function yF(x,y) is convex and lower semicontinuous for each xC.

For each xC, for all t(0,1) and for all y,zC, 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 [<xref ref-type="bibr" rid="B17">17</xref>]).

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 satisfying (A1)–(A4). For r>0 and xE, define a mapping Tr:EC as follows: (68)Tr(x)={zC:f(z,y)+1ry-z,Jz-Jx0,yC+1r} for all xE. Then, the following properties hold:

Tr is single valued;

Tr is a firmly nonexpansive-type mapping; that is, for all x,yE, (69)Trx-Try,JTrx-JTryTrx-Try,Jx-Jy;

EP(f)=F(Tr)=F^(Tr);

EP(f) is closed and convex;

ϕ(q,Trx)+ϕ(Trx,x)ϕ(q,x), for all xE,qF(Tr).

Lemma 20 (see Zhang et al. [<xref ref-type="bibr" rid="B30">57</xref>]).

Let E be a p-uniformly convex with p0 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 satisfying (A1)–(A4). Let {rn} be a positive real sequence such that limnrn=r>0. Then the sequence of mappings Trn is uniformly closed.

Next, we shall apply Theorem 11 to solve two generalized mixed equilibrium problems. To accomplish this purpose, let A,B:CE* be two monotone and continuous mappings, let the function φ,ψ:C be convex and lower semicontinuous, and let f and g be a bifunction from C×C to satisfying (A1)–(A4). For r>0 and xE, define two mappings Jr,Kr:EC as follows: (70)Jr(x)={+1ry-z,Jz-Jx0,yC}zC:f(z,y)+Az,y-z+φ(y)-φ(z)Jr(x)+1ry-z,Jz-Jx0,yC},(71)Kr(x)={1rzC:g(z,y)+Bz,y-z+ψ(y)-ψ(z)Kr(x)+1ry-z,Jz-Jx0,yC}.

Theorem 21.

Let E be a p-uniformly convex with p2 and uniformly smooth Banach space, and let C be a nonempty closed convex subset of E. Let A,B:CE* be two monotone and continuous mappings, let the function φ,ψ:C be convex and lower semicontinuous, and let f and g be a bifunction from C×C to satisfying (A1)–(A4) such that =GMEP(f,A,φ)GMEP(g,B,ψ). For a point x0C chosen arbitrarily, let {xn} be a sequence generated by the following iterative algorithm: (72)C0=C,Cn+1={zCn:ϕ(z,vn)ϕ(z,un)ϕ(z,xn)},xn+1=ΠCn+1x0, where un=Jrnxn, vn=Krnun, and limnrn=r. Then the sequence {xn} converges strongly to a point q=Πx0, where Π is the generalized projection from C onto .

Proof.

From Lemmas 18 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.

Let E be a p-uniformly convex with p2 and uniformly smooth Banach space, and let C be a nonempty closed convex subset of E. Let A:CE* be a monotone and continuous mappings, let the function φ:C be convex and lower semicontinuous and let f be a bifunction from C×C to satisfying (A1)–(A4) such that =GMEP(f,A,φ). For a point x0C chosen arbitrarily, let {xn} be a sequence generated by the following iterative algorithm: (73)C0=C,Cn+1={zCn:ϕ(z,un)ϕ(z,xn)},xn+1=ΠCn+1x0, where un=Jrnxn and limnrn=r. Then the sequence {xn} converges strongly to a point q=Πx0, where Π is the generalized projection from C onto .

Proof.

From Lemmas 18 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 let f0, φ0 in (70) and B0, ψ0 in (71), the following corollary can be obtained by using Theorem 21.

Corollary 23.

Let E be a p-uniformly convex with p2 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 satisfying (A1)–(A4) and A:CE* a monotone and continuous mapping. Suppose that =VI(A,C)EP(g). For a point x0C chosen arbitrarily, let {xn} be a sequence generated by the following iterative algorithm: (74)C0=C,Cn+1={zCn:ϕ(z,vn)ϕ(z,un)ϕ(z,xn)},xn+1=ΠCn+1x0, where un=Jrnxn, vn=Krnun, and limnrn=r. Then the sequence {xn} converges strongly to a point q=Πx0, where Π is the generalized projection from C onto .

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)={zE:𝒜z} and range R(𝒜)={zE:zD(𝒜)}. An operator 𝒜 is said to be monotone if (75)x1-x2,y1-y20,x1,x2D(𝒜),  jjjjjjjjjjjjjjjjsssjjjjjjjjjjy1𝒜x1,y2𝒜x2.

A monotone operator 𝒜 is said to be maximal if its graph G(𝒜)={(x,y):y𝒜x} 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 [<xref ref-type="bibr" rid="B31">58</xref>]).

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 xE, there exists a unique xrD(𝒜) such that (76)JxJxr+r𝒜xr.

If Jrx=xr, then we can define a single-valued mapping Jr:ED(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.

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 {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=0F(Jrn)=𝒜-10. Firstly, we show Jrn is uniformly closed. Let {zn} be a sequence such that znp and limnzn-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 wD(𝒜) and w*𝒜w. Letting n, one has w-p,w*0 for all wD(𝒜) and w*𝒜w. Therefore, from the maximality of 𝒜, one obtains p𝒜-10=F(Jrn). Hence, Jrn is uniformly closed.

In addition, for any wE and pn=0F(Jrn), from the monotonicity of 𝒜, one has (80)ϕ(p,Jrnw)=p2-2p,JJrnw+Jrnw2=p2+2p,Jw-JJrnw-Jw+Jrnw2=p2+2p,Jw-JJrnw-2p,Jw+Jrnw2=p2-2Jrnw-p-Jrnw,Jw-JJrnw-Jw-2p,Jw+Jrnw2=p2-2Jrnw-p,Jw-JJrnw-Jw+2Jrnw,Jw-JJrnw-2p,Jw+Jrnw2p2+2Jrnw,Jw-JJrnw-2p,Jw+Jrnw2=p2-2p,Jw+w2-Jrnw2+2Jrnw,Jw-w2=ϕ(p,w)-ϕ(Jrnw,w)ϕ(p,w), for all 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.

Let C be a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space E. Let 𝒜, be two maximal monotone operators from E to E* such that =𝒜-10-100, and let {rn} be a sequence of positive real numbers which is bounded away from 0 such that Jrn𝒜=(I+rn𝒜)-1 and Jrn=(I+rn)-1. For a point x0C chosen arbitrarily, let {xn} be a sequence generated by the following iterative algorithm: (3.1)C0=C,Cn+1={zCn:ϕ(z,Jrnyn)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=Πx0, where Π is the generalized projection from C onto .

Proof.

From Lemma 26, we know that {Jrn𝒜} and {Jrn} 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 Πx0.

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).

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