JAMJournal of Applied Mathematics1687-00421110-757XHindawi Publishing Corporation30879110.1155/2012/308791308791Research ArticleStrong Convergence to Solutions of Generalized Mixed Equilibrium Problems with ApplicationsCholamjiakPrasit1, 2SuantaiSuthep2, 3ChoYeol Je4YaoYonghong1School of ScienceUniversity of PhayaoPhayao 56000Thailandup.ac.th2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400Thailandcmu.ac.th3Department of MathematicsFaculty of Science, Chiang Mai UniversityChiang Mai 50200Thailandcmu.ac.th4Department of Mathematics Education and the RINSGyeongsang National UniversityJinju 660-701Republic of Koreagnu.ac.kr20121612012201221102011231120112012Copyright © 2012 Prasit Cholamjiak et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduce a Halpern-type iteration for a generalized mixed equilibrium problem in uniformly smooth and uniformly convex Banach spaces. Strong convergence theorems are also established in this paper. As applications, we apply our main result to mixed equilibrium, generalized equilibrium, and mixed variational inequality problems in Banach spaces. Finally, examples and numerical results are also given.

1. Introduction

Let E be a real Banach space, C a nonempty, closed, and convex subset of E, and E* the dual space of E. Let T:CC be a nonlinear mapping. The fixed points set of T is denoted by F(T), that is, F(T)={xC:  x=Tx}.

One classical way often used to approximate a fixed point of a nonlinear self-mapping T on C was firstly introduced by Halpern  which is defined by x1=xC and xn+1=αnx+(1-αn)Txn,n1, where {αn} is a real sequence in [0,1]. He proved, in a real Hilbert space, a strong convergence theorem for a nonexpansive mapping T when αn=n-a for any a(0,1).

Subsequently, motivated by Halpern , many mathematicians devoted time to study algorithm (1.1) in different styles. Several strong convergence results for nonlinear mappings were also continuously established in some certain Banach spaces (see also ).

Let f:C×C be a bifunction, A:CE* a mapping, and φ:C a real-valued function. The generalized mixed equilibrium problem is to find x̂C such that f(x̂,y)+Ax̂,y-x̂+φ(y)φ(x̂),yC. The solutions set of (1.2) is denoted by GMEP(f,A,φ) (see Peng and Yao ).

If A0, then the generalized mixed equilibrium problem (1.2) reduces to the following mixed equilibrium problem: finding x̂C such that f(x̂,y)+φ(y)φ(x̂),yC. The solutions set of (1.3) is denoted by MEP(f,φ) (see Ceng and Yao ).

If f0, then the generalized mixed equilibrium problem (1.2) reduces to the following mixed variational inequality problem: finding x̂C such that Ax̂,y-x̂+φ(y)φ(x̂),yC. The solutions set of (1.4) is denoted by VI(C,A,φ) (see Noor ).

If φ0, then the generalized mixed equilibrium problem (1.2) reduces to the following generalized equilibrium problem: finding x̂C such that f(x̂,y)+Ax̂,y-x̂0,yC. The solutions set of (1.5) is denoted by GEP(f,A) (see Moudafi ).

If φ0, then the mixed equilibrium problem (1.3) reduces to the following equilibrium problem: finding x̂C such that f(x̂,y)0,yC. The solutions set of (1.6) is denoted by EP(f) (see Combettes and Hirstoaga ).

If f0, then the mixed equilibrium problem (1.3) reduces to the following convex minimization problem: finding x̂C such that φ(y)φ(x̂),yC. The solutions set of (1.7) is denoted by CMP(φ).

If φ0, then the mixed variational inequality problem (1.4) reduces to the following variational inequality problem: finding x̂C such that Ax̂,y-x̂0,yC. The solutions set of (1.8) is denoted by VI(C,A) (see Stampacchia ).

The problem (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others. For more details on these topics, see, for instance, .

For solving the generalized mixed equilibrium problem, let us assume the following :

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

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

for all x,y,zC, limsupt0f(tz+(1-t)x,y)f(x,y);

for all xC,  f(x,·) is convex and lower semicontinuous.

The purpose of this paper is to investigate strong convergence of Halpern-type iteration for a generalized mixed equilibrium problem in uniformly smooth and uniformly convex Banach spaces. As applications, our main result can be deduced to mixed equilibrium, generalized equilibrium, mixed variational inequality problems, and so on. Examples and numerical results are also given in the last section.

2. Preliminaries and Lemmas

In this section, we need the following preliminaries and lemmas which will be used in our main theorem.

Let E be a real Banach space and let U={xE:x=1} be the unit sphere of E. A Banach space E is said to be strictly convex if, for any x,yU, xy  implies  x+y2<1. It is also said to be uniformly convex if, for any ε(0,2], there exists δ>0 such that, for any x,yU, x-yε  implies  x+y2<1-δ. It is known that a uniformly convex Banach space is reflexive and strictly convex. Define a function δ:[0,2][0,1] called the modulus of convexity of E as follows: δ(ε)=inf{1-x+y2:  x,yE,  x=y=1,  x-yε}. Then E is uniformly convex if and only if δ(ε)>0 for all ε(0,2]. A Banach space E is said to be smooth if the limit limt0x+ty-xt exists for all x,yU. It is also said to be uniformly smooth if the limit (2.4) is attained uniformly for x,yU. The normalized duality mapping J:E2E* is defined by J(x)={x*E*:x,x*=x2=x*2} for all xE. It is also known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E (see ).

Let E be a smooth Banach space. The function ϕ:E×E is defined by ϕ(x,y)=x2-2x,Jy+y2,x,yE.

Remark 2.1.

We know the following: for any x,y,zE,

(x-y)2ϕ(x,y)(x+y)2;

ϕ(x,y)=ϕ(x,z)+ϕ(z,y)+2x-z,Jz-Jy;

ϕ(x,y)=x-y2 in a real Hilbert space.

Lemma 2.2 (see [<xref ref-type="bibr" rid="B36">36</xref>]).

Let E be a uniformly convex and smooth Banach space and let {xn} and {yn} be sequences of E such that {xn} or {yn} is bounded and limnϕ(xn,yn)=0. Then limnxn-yn=0.

Let E be a reflexive, strictly convex, and smooth Banach space and let C be a nonempty closed and convex subset of E. The generalized projection mapping, introduced by Alber , is a mapping ΠC:EC, that assigns to an arbitrary point xE the minimum point of the functional ϕ(y,x), that is, ΠCx=x¯, where x¯ is the solution to the minimization problem: ϕ(x¯,x)=min{ϕ(y,x):yC}.

In fact, we have the following result.

Lemma 2.3 (see [<xref ref-type="bibr" rid="B37">37</xref>]).

Let C be a nonempty, closed, and convex subset of a reflexive, strictly convex, and smooth Banach space E and let xE. Then there exists a unique element x0C such that ϕ(x0,x)=min{ϕ(z,x):zC}.

Lemma 2.4 (see [<xref ref-type="bibr" rid="B36">36</xref>, <xref ref-type="bibr" rid="B37">37</xref>]).

Let C be a nonempty closed and convex subset of a reflexive, strictly convex, and smooth Banach space E, xE, and zC. Then z=ΠCx if and only if Jx-Jz,y-z0,yC.

Lemma 2.5 (see [<xref ref-type="bibr" rid="B36">36</xref>, <xref ref-type="bibr" rid="B37">37</xref>]).

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

Lemma 2.6 (see [<xref ref-type="bibr" rid="B38">38</xref>]).

Let E be a uniformly convex and uniformly smooth Banach space and C a nonempty, closed, and convex subset of E. Then ΠC is uniformly norm-to-norm continuous on every bounded set.

We make use of the following mapping V studied in Alber : V(x,x*)=x2-2x,x*+x*2 for all xE and x*E*, that is, V(x,x*)=ϕ(x,J-1(x*)).

Lemma 2.7 (see [<xref ref-type="bibr" rid="B39">39</xref>]).

Let E be a reflexive, strictly convex, smooth Banach space. Then V(x,x*)+2J-1x*-x,y*V(x,x*+y*) for all xE and x*,y*E*.

Lemma 2.8 (see [<xref ref-type="bibr" rid="B25">25</xref>]).

Let C be a closed and convex subset of a smooth, strictly convex, and reflexive Banach space E, let f be a bifunction from C×C to which satisfies conditions (A1)–(A4), and let r>0 and xE. Then there exists zC such that f(z,y)+1rJz-Jx,y-z0,yC.

Following [25, 40], we know the following lemma.

Lemma 2.9 (see [<xref ref-type="bibr" rid="B41">41</xref>]).

Let C be a nonempty closed and convex subset of a smooth, strictly convex, and reflexive Banach space E. Let A:CE* be a continuous and monotone mapping, let f be a bifunction from C×C to satisfying (A1)–(A4), and let φ be a lower semicontinuous and convex function from C to . For all r>0 and xE, there exists zC such that f(z,y)+Az,y-z+φ(y)+1rJz-Jx,y-zφ(z),yC. Define the mapping Tr:E2C as follows: Tr(x)={zC:f(z,y)+Az,y-z+φ(y)+1rJz-Jx,y-zφ(z),  yC}. Then, the followings hold:

Tr is single-valued;

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

F(Tr)=GMEP(f,A,φ);

GMEP(f,A,φ) is closed and convex.

Remark 2.10.

It is known that T is of firmly nonexpansive type if and only if ϕ(Tx,Ty)+ϕ(Ty,Tx)+ϕ(Tx,x)+ϕ(Ty,y)ϕ(Tx,y)+ϕ(Ty,x) for all x,y dom T (see ).

The following lemmas give us some nice properties of real sequences.

Lemma 2.11 (see [<xref ref-type="bibr" rid="B43">43</xref>]).

Assume that {an} is a sequence of nonnegative real numbers such that an+1(1-αn)an+bn,n1, where {αn} is a sequence in (0,1) and {bn} is a sequence such that

n=1αn=+;

limsupnbn/αn0 or n=1|bn|<+.

Then limnan=0.

Lemma 2.12 (see [<xref ref-type="bibr" rid="B44">44</xref>]).

Let {γn} be a sequence of real numbers such that there exists a subsequence {γnj} of {γn} such that γnj<γnj+1 for all j1. Then there exists a nondecreasing sequence {mk} of such that limkmk= and the following properties are satisfied by all (sufficiently large) numbers k1: γmkγmk+1,γkγmk+1. In fact, mk is the largest number n in the set {1,2,,k} such that the condition γn<γn+1 holds.

3. Main Results

In this section, we prove our main theorem in this paper. To this end, we need the following proposition.

Proposition 3.1.

Let C be a nonempty closed and convex subset of a reflexive, strictly convex, and uniformly smooth Banach space E. Let f be a bifunction from C×C to satisfying (A1)–(A4), A:CE* a continuous and monotone mapping, and φ a lower semicontinuous and convex function from C to such that GMEP(f,A,φ). Let {rn}(0,) be such that liminfnrn>0. For each n1, let Trn be defined as in Lemma 2.9. Suppose that xC and {xn} is a bounded sequence in C such that limnxn-Trnxn=0. Then limsupnJx-Jp,xn-p0, where p=ΠGMEP(f,A,φ)x and ΠGMEP(f,A,φ) is the generalized projection of C onto GMEP(f,A,φ).

Proof.

Let xC and put p=ΠGMEP(f,A,φ)x. Since E is reflexive and {xn} is bounded, there exists a subsequence {xnk} of {xn} such that xnkvC and limsupnJx-Jp,xn-p=Jx-Jp,v-p. Put yn=Trnxn. Since limkxnk-ynk=0, we have ynkv. On the other hand, since E is uniformly smooth, J is uniformly norm-to-norm continuous on bounded subsets of E. So we have limkJxnk-Jynk=0. Since liminfkrnk>0, limkJxnk-Jynkrnk=0. By the definition of Trnk, for any yC, we see that f(ynk,y)+Aynk,y-ynk+φ(y)+1rnkJynk-Jxnk,y-ynkφ(ynk). By (A2), for each yC, we obtain f(y,ynk)+φ(ynk)-f(ynk,y)+φ(ynk)Aynk,y-ynk+φ(y)+1rnkJynk-Jxnk,y-ynk. For any t(0,1) and yC, we define yt=ty+(1-t)v. Then ytC. It follows by the monotonicity of A that f(yt,ynk)+φ(ynk)Aynk-Ayt,yt-ynk+Ayt,yt-ynk+  φ(yt)+1rnkJynk-Jxnk,yt-ynkAyt,yt-ynk+φ(yt)+1rnkJynk-Jxnk,yt-ynk. By (A4), (3.4), and the weakly lower semicontinuity of φ, letting k, we obtain f(yt,v)+φ(v)Ayt,yt-v+φ(yt). By (A1), (A4), and the convexity of φ, we have 0=f(yt,yt)+φ(yt)-φ(yt)tf(yt,y)+(1-t)f(yt,v)+tφ(y)+(1-t)φ(v)-φ(yt)=t(f(yt,y)+φ(y)-φ(yt))+(1-t)(f(yt,v)+φ(v)-φ(yt))t(f(yt,y)+φ(y)-φ(yt))+(1-t)Ayt,yt-v=t(f(yt,y)+φ(y)-φ(yt))+(1-t)tAyt,y-v. It follows that f(yt,y)+φ(y)-φ(yt)+(1-t)Ayt,y-v0. By (A3), the weakly lower semicontinuity of φ, and the continuity of A, letting t0, we obtain f(v,y)+φ(y)-φ(v)+Av,y-v0,yC. This shows that vGMEP(f,A,φ). By Lemma 2.4, we have limsupnJx-Jp,xn-p=Jx-Jp,v-p0. This completes the proof.

Theorem 3.2.

Let C be nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space E. Let f be a bifunction from C×C to satisfying (A1)–(A4), A:CE* a continuous and monotone mapping, and φ a lower semicontinuous and convex function from C to such that GMEP(f,A,φ). Define the sequence {xn} as follows: x1=xC and f(yn,y)+Ayn,y-yn+φ(y)+1rnJyn-Jxn,y-ynφ(yn),yC,xn+1=ΠCJ-1(αnJx+(1-αn)Jyn),n1, where {αn}(0,1) and {rn}(0,) satisfy the following conditions:

limnαn=0;

n=1αn=;

liminfnrn>0.

Then {xn} converges strongly to ΠGMEP(f,A,φ)x, where ΠGMEP(f,A,φ) is the generalized projection of C onto GMEP(f,A,φ).

Proof.

From Lemma 2.9(4), we know that GMEP(f,A,φ) is closed and convex. Let p=ΠGMEP(f,A,φ)x. Put yn=Trnxn and zn=J-1(αnJx+(1-αn)Jyn) for all n. So, by Lemma 2.5, we have ϕ(p,xn+1)ϕ(p,zn)αnϕ(p,x)+(1-αn)ϕ(p,yn)αnϕ(p,x)+(1-αn)ϕ(p,xn). By induction, we can show that ϕ(p,xn)ϕ(p,x) for each n. Hence {ϕ(p,xn)} is bounded and thus {xn} is also bounded.

We next show that if there exists a subsequence {xnk} of {xn} such that limk(ϕ(p,xnk+1)-ϕ(p,xnk))=0, then limk(ϕ(p,ynk)-ϕ(p,xnk))=0. Since αnk0, limkJznk-Jynk=limkαnkJx-Jynk=0. Since J is uniformly norm-to-norm continuous on bounded subsets of E, so is J-1. It follows that limkznk-ynk=0. Since E is uniformly smooth and uniformly convex, by Lemma 2.6, ΠC is uniformly norm-to-norm continuous on bounded sets. So we obtain limkxnk+1-ynk=limkΠCznk-ΠCynk=0, and hence limkJxnk+1-Jynk=0. Furthermore, limkϕ(xnk+1,ynk)=0. Indeed, by the definition of ϕ, we observe that ϕ(xnk+1,ynk)=xnk+12-2xnk+1,Jynk+ynk2=xnk+1,Jxnk+1-Jynk+ynk-xnk+1,Jynk. It follows from (3.19) and (3.20) that limkϕ(xnk+1,ynk)=0. On the other hand, from Remark 2.1(2), we have ϕ(p,ynk)-ϕ(p,xnk)=(ϕ(p,xnk+1)-ϕ(p,xnk))+(ϕ(p,ynk)-ϕ(p,xnk+1))=(ϕ(p,xnk+1)-ϕ(p,xnk))+  ϕ(xnk+1,ynk)+2p-xnk+1,Jxnk+1-Jynk. It follows from (3.20) and (3.21) thatlimk(ϕ(p,ynk)-ϕ(p,xnk))=0.

We next consider the following two cases.

Case 1.

ϕ(p,xn+1)ϕ(p,xn) for all sufficiently large n. Hence the sequence {ϕ(p,xn)} is bounded and nonincreasing. So limnϕ(p,xn) exists. This shows that limn(ϕ(p,xn+1)-ϕ(p,xn))=0 and hence limn(ϕ(p,yn)-ϕ(p,xn))=0. Since Trn is of firmly nonexpansive type, by Remark 2.10, we have ϕ(yn,p)+ϕ(p,yn)+ϕ(yn,xn)+ϕ(Trnp,p)ϕ(yn,p)+ϕ(p,xn), which implies ϕ(p,yn)+ϕ(yn,xn)ϕ(p,xn). Hence ϕ(yn,xn)ϕ(p,xn)-ϕ(p,yn)0 as n. By Lemma 2.2, we obtain limnxn-yn=0. Proposition 3.1 yields that limsupnJx-Jp,xn-p0. It also follows that limsupnJx-Jp,yn-p0.

Finally, we show that xnp. Using Lemma 2.7, we see that ϕ(p,xn+1)ϕ(p,zn)=V(p,αnJx+(1-αn)Jyn)V(p,αnJx+(1-αn)Jyn-αn(Jx-Jp))+αn(Jx-Jp),zn-p=V(p,αnJp+(1-αn)Jyn)+αnJx-Jp,zn-pαnV(p,Jp)+(1-αn)V(p,Jyn)+αnJx-Jp,zn-p=(1-αn)ϕ(p,yn)+αnJx-Jp,zn-p(1-αn)ϕ(p,xn)+αnJx-Jp,zn-p=(1-αn)ϕ(p,xn)+αn(Jx-Jp,zn-yn+Jx-Jp,yn-p). Set an=ϕ(p,xn) and bn=αn(Jx-Jp,zn-yn+Jx-Jp,yn-p). We see that limsupnbn/αn0. By Lemma 2.11, since n=1αn=+, we conclude that limnϕ(p,xn)=0. Hence xnp as n.

Case 2.

There exists a subsequence {ϕ(p,xnj)} of {ϕ(p,xn)} such that ϕ(p,xnj)<ϕ(p,xnj+1) for all j. By Lemma 2.12, there exists a strictly increasing sequence {mk} of positive integers such that the following properties are satisfied by all numbers k: ϕ(p,xmk)ϕ(p,xmk+1),ϕ(p,xk)ϕ(p,xmk+1). So we have 0limk(ϕ(p,xmk+1)-ϕ(p,xmk))limsupn(ϕ(p,xn+1)-ϕ(p,xn))limsupn(ϕ(p,zn)-ϕ(p,xn))limsupn(αnϕ(p,x)+(1-αn)ϕ(p,yn)-ϕ(p,xn))=limsupn(αn(ϕ(p,x)-ϕ(p,yn))+(ϕ(p,yn)-ϕ(p,xn)))limsupnαn(ϕ(p,x)-ϕ(p,yn))=0. This shows that limk(ϕ(p,xmk+1)-ϕ(p,xmk))=0. Following the proof line in Case 1, we can show that limsupkJx-Jp,ymk-p0,ϕ(p,xmk+1)(1-αmk)ϕ(p,xmk)+αmk(Jx-Jp,zmk-ymk+Jx-Jp,ymk-p). This implies αmkϕ(p,xmk)ϕ(p,xmk)-ϕ(p,xmk+1)+  αmk(Jx-Jp,zmk-ymk+Jx-Jp,ymk-p)αmk(Jx-Jp,zmk-ymk+Jx-Jp,ymk-p). Hence limkϕ(p,xmk)=0. Using this and (3.33) together, we conclude that limsupkϕ(p,xk)limkϕ(p,xmk+1)=0. This completes the proof.

As a direct consequence of Theorem 3.2, we obtain the following results.

Corollary 3.3.

Let C be nonempty closed and convex subset of a uniformly smooth and uniformly convex Banach space E. Let f be a bifunction from C×C to satisfying (A1)–(A4) and φ a lower semicontinuous and convex function from C to such that MEP(f,φ). Define the sequence {xn} as follows: x1=xC and f(yn,y)+φ(y)+1rnJyn-Jxn,y-ynφ(yn),yC,xn+1=ΠCJ-1(αnJx+(1-αn)Jyn),n1, where {αn}(0,1) and {rn}(0,) satisfy the following conditions:

limnαn=0;

n=1αn=;

liminfnrn>0.

Then {xn} converges strongly to ΠMEP(f,φ)x, where ΠMEP(f,φ) is the generalized projection of C onto MEP(f,φ).

Corollary 3.4.

Let C be nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space E. Let f be a bifunction from C×C to satisfying (A1)–(A4), and A:CE* a continuous and monotone mapping such that GEP(f,A). Define the sequence {xn} as follows: x1=xC and f(yn,y)+Ayn,y-yn+1rnJyn-Jxn,y-yn0,yC,xn+1=ΠCJ-1(αnJx+(1-αn)Jyn),n1, where {αn}(0,1) and {rn}(0,) satisfy the following conditions:

limnαn=0;

n=1αn=;

liminfnrn>0.

Then {xn} converges strongly to ΠGEP(f,A)x, where ΠGEP(f,A) is the generalized projection of C onto GEP(f,A).

Corollary 3.5.

Let C be nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space E. Let A:CE* be a continuous and monotone mapping, and φ a lower semicontinuous and convex function from C to such that VI(C,A,φ). Define the sequence {xn} as follows: x1=xC and Ayn,y-yn+φ(y)+1rnJyn-Jxn,y-ynφ(yn),yC,xn+1=ΠCJ-1(αnJx+(1-αn)Jyn),n1, where {αn}(0,1) and {rn}(0,) satisfy the following conditions:

limnαn=0;

n=1αn=;

liminfnrn>0.

Then {xn} converges strongly to ΠVI(C,A,φ)x, where ΠVI(C,A,φ) is the generalized projection of C onto VI(C,A,φ).

4. Examples and Numerical Results

In this section, we give examples and numerical results for our main theorem.

Example 4.1.

Let E= and C=[-1,1]. Let f(x,y)=-9x2+xy+8y2, φ(x)=3x2, and Ax=2x. Find x̂[-1,1] such that f(x̂,y)+Ax̂,y-x̂+φ(y)φ(x̂),y[-1,1].

Solution 4.

It is easy to check that f, φ, and A satisfy all conditions in Theorem 3.2. For each r>0 and x[-1,1], Lemma 2.9 ensures that there exists z[-1,1] such that, for any y[-1,1],   f(z,y)+Az,y-z+φ(y)+1rz-x,y-zφ(z)-9z2+yz+8y2+2z(y-z)+3y2+1r(z-x)(y-z)3z211ry2+(3rz+z-x)y-(14rz2+z2-xz)0. Put G(y)=11ry2+(3rz+z-x)y-(14rz2+z2-xz). Then G is a quadratic function of y with coefficient a=11r, b=(3rz+z-x), and c=-(14rz2+z2-xz). We next compute the discriminant Δ of G as follows:   Δ=b2-4ac=[(3r+1)z-x]2+44r(14rz2+z2-xz)=x2-2(3r+1)xz+(3r+1)2z2+616r2z2+44rz2-44rxz=x2-50rxz-2xz+625r2z2+50rz2+z2=x2-2(25rz+z)x+(625r2z2+50rz2+z2)=[x-(25rz+z)]2. We know that G(y)0 for all y[-1,1] if it has at most one solution in [-1,1]. So Δ0 and hence x=25rz+z. Now we have z=Trx=x/(25r+1).

Let {xn}n=1 be the sequence generated by x1=x[-1,1] and f(yn,y)+Ayn,y-yn+φ(y)+1rnyn-xn,y-ynφ(yn),y[-1,1],xn+1=αnx+(1-αn)yn,n1, and, equivalently, xn+1=αnx+(1-αn)Trnxn,n1.

We next give two numerical results for algorithm (4.5).

Algorithm 4.2.

Let αn=1/80n and rn=n/(n+1). Choose x1=x=1. Then algorithm (4.5) becomes xn+1=180n+(1-180n)(n+126n+1)xn,n1.

Numerical Result I

See Table 1.

nxn
1 1.0000
2 0.0856
3 0.0111
4 0.0047
5 0.0033
261 0.0001
262 0.0000
Algorithm 4.3.

Let αn=1/100n and rn=(n+1)/2n. Choose x1=x=-1. Then algorithm (4.5) becomes xn+1=-1100n+(1-1100n)(2n27n+25)xn,n1.

Numerical Result II

See Table 2.

nxn
1 −1.0000
2 −0.0481
3 −0.0074
4 −0.0038
5 −0.0027
217 −0.0001
218 0.0000
5. Conclusion

Tables 1 and 2 show that the sequence {xn} converges to 0 which solves the generalized mixed equilibrium problem. On the other hand, using Lemma 2.9(3), we can check that GMEP(f,A,φ)=F(Tr)={0}.

Remark 5.1.

In the view of computation, our algorithm is simple in order to get strong convergence for generalized mixed equilibrium problems.

Acknowledgments

The first and the second authors wish to thank the Thailand Research Fund and the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The third author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2011-0021821).

HalpernB.Fixed points of nonexpanding mapsBulletin of the American Mathematical Society196773957961021893810.1090/S0002-9904-1967-11864-0ZBL0177.19101ChoY. J.KangS. M.ZhouH.Some control conditions on iterative methodsCommunications on Applied Nonlinear Analysis200512227342129053ZBL1088.47053LionsP.-L.Approximation de points fixes de contractions197728421A1357A13590470770ZBL0349.47046NilsrakooW.SaejungS.Strong convergence theorems by Halpern-Mann iterations for relatively nonexpansive mappings in Banach spacesApplied Mathematics and Computation20112171465776586277324410.1016/j.amc.2011.01.040ZBL1215.65104ReichS.Approximating fixed points of nonexpansive mappingsPanamerican Mathematical Journal19944223281274185ZBL0856.47032SaejungS.Halpern's iteration in Banach spacesNonlinear Analysis: Theory, Methods & Applications201073103431343910.1016/j.na.2010.07.0312680036StampacchiaG.Formes bilinéaires coercitives sur les ensembles convexesComptes Rendus de lAcademie des Sciences1964258441344160166591ZBL0124.06401WittmannR.Approximation of fixed points of nonexpansive mappingsArchiv der Mathematik1992585486491115658110.1007/BF01190119ZBL0797.47036XuH.-K.Another control condition in an iterative method for nonexpansive mappingsBulletin of the Australian Mathematical Society2002651109113188938410.1017/S0004972700020116ZBL1030.47036PengJ.-W.YaoJ.-C.A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problemsTaiwanese Journal of Mathematics2008126140114322444865ZBL1185.47079CengL.-C.YaoJ.-C.A hybrid iterative scheme for mixed equilibrium problems and fixed point problemsJournal of Computational and Applied Mathematics20082141186201239168210.1016/j.cam.2007.02.022ZBL1143.65049NoorM. A.An implicit method for mixed variational inequalitiesApplied Mathematics Letters199811410911310.1016/S0893-9659(98)00066-41631142ZBL0941.49005MoudafiA.Weak convergence theorems for nonexpansive mappings and equilibrium problemsJournal of Nonlinear and Convex Analysis20089137432408333ZBL1167.47049CombettesP. L.HirstoagaS. A.Equilibrium programming in Hilbert spacesJournal of Nonlinear and Convex Analysis2005611171362138105ZBL1109.90079AgarwalR. P.ChoY. J.PetrotN.Systems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spacesFixed Point Theory Application20112011, article 3110.1186/1687-1812-2011-31ChoY. J.QinX.KangJ. I.Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problemsNonlinear Analysis: Theory, Methods & Applications20097194203421410.1016/j.na.2009.02.1062536325ChoY. J.ArgyrosI. K.PetrotN.Approximation methods for common solutions of generalized equilibrium, systems of nonlinear variational inequalities and fixed point problemsComputers & Mathematics with Applications201060822922301272532010.1016/j.camwa.2010.08.021ZBL1205.65185ChoY. J.PetrotN.On the system of nonlinear mixed implicit equilibrium problems in Hilbert spacesJournal of Inequalities and Applications201020101243797610.1155/2010/4379762586607ZBL1184.49003ChoY. J.PetrotN.An optimization problem related to generalized equilibrium and fixed point problems with applicationsFixed Point Theory20101122372502743778ChoY. J.PetrotN.Regularization and iterative method for general variational inequality problem in Hilbert spacesJournal of Inequalities and Applications20112011, article 212823621HeH.LiuS.ChoY. J.An explicit method for systems of equilibrium problems and fixed points of infinite family of nonexpansive mappingsJournal of Computational and Applied Mathematics20112351441284139280143410.1016/j.cam.2011.03.003QinX.ChangS.-S.ChoY. J.Iterative methods for generalized equilibrium problems and fixed point problems with applicationsNonlinear Analysis: Real World Applications20101142963297210.1016/j.nonrwa.2009.10.0172661959ZBL1192.58010YaoY.ChoY. J.LiouY.-C.Iterative algorithms for variational inclusions, mixed equilibrium and fixed point problems with application to optimization problemsCentral European Journal of Mathematics201193640656278403510.2478/s11533-011-0021-3YaoY.ChoY. J.LiouY.-C.Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problemsEuropean Journal of Operational Research20112122242250278420210.1016/j.ejor.2011.01.042BlumE.OettliW.From optimization and variational inequalities to equilibrium problemsThe Mathematics Student1994631–41231451292380ZBL0888.49007CengL.-C.YaoJ.-C.A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problemNonlinear Analysis: Theory, Methods & Applications2010723-41922193710.1016/j.na.2009.09.0332577590ZBL1179.49003SaewanS.KumamP.A hybrid iterative scheme for a maximal monotone operator and two countable families of relatively quasi-nonexpansive mappings for generalized mixed equilibrium and variational inequality problemsAbstract and Applied Analysis201020103112302710.1155/2010/1230272735003ZBL1204.65062TadaA.TakahashiW.Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problemJournal of Optimization Theory and Applications20071333359370233382010.1007/s10957-007-9187-zZBL1147.47052TakahashiS.TakahashiW.Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert spaceNonlinear Analysis: Theory, Methods & Applications20086931025103310.1016/j.na.2008.02.0422428774ZBL1142.47350YaoY.LiouY.-C.YaoJ.-C.New relaxed hybrid-extragradient method for fixed point problems, a general system of variational inequality problems and generalized mixed equilibrium problemsOptimization201160339541210.1080/023319309031969412780926YaoY.yaoyonghong@yahoo.cnLiouY.-C.simplex_liou@hotmail.comKangS. M.smkang@gnu.ac.krTwo-step projection methods for a system of variational inequality problems in Banach spacesJournal of Global Optimization. In press10.1007/s10898-011-9804-0YaoY.yaoyonghong@yahoo.cnShahzadN.nshahzad@kau.edu.saStrong convergence of a proximal point algorithm with general errorsOptimization Letters. In press10.1007/s11590-011-0286-2YaoY.NoorM. A.LiouY.-C.Strong convergence of a modified extra-gradient method to the minimum-norm solution of variational inequalitiesAbstract and Applied Analysis. In pressYaoY.yaoyonghong@yahoo.cnChenR.chenrd@tjpu.edu.cnLiouY.-C.simplex_liou@hotmail.comA unified implicit algorithm for solving the triple-hierarchical constrained optimization problemMathematical and Computer Modelling2012553-41506151510.1016/j.mcm.2011.10.041TakahashiW.Nonlinear Functional Analysis: Fixed Point Theory and Its Applications2000Yokohama, JapanYokohama Publishersiv+2761864294KamimuraS.TakahashiW.Strong convergence of a proximal-type algorithm in a Banach spaceSIAM Journal on Optimization200213393894510.1137/S105262340139611X1972223AlberY. I.KartsatosA. G.Metric and generalized projection operators in Banach spaces: properties and applicationsTheory and Applications of Nonlinear Operators of Accretive and Monotone Type1996178New York, NY, USADekker1550Lecture Notes in Pure and Appl. Math.1386667ZBL0883.47083AoyamaK.KohsakaF.TakahashiW.Strongly relatively nonexpansive sequences in Banach spaces and applicationsJournal of Fixed Point Theory and Applications200952201224252949710.1007/s11784-009-0108-7KohsakaF.TakahashiW.Strong convergence of an iterative sequence for maximal monotone operators in a Banach spaceAbstract and Applied Analysis2004323924910.1155/S10853375043090362058504ZBL1064.47068TakahashiW.ZembayashiK.Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spacesNonlinear Analysis: Theory, Methods & Applications2009701455710.1016/j.na.2007.11.0312468217ZBL1170.47049ZhangS.-s.Generalized mixed equilibrium problem in Banach spacesApplied Mathematics and Mechanics. English Edition200930911051112256915510.1007/s10483-009-0904-6ZBL1178.47051KohsakaF.TakahashiW.Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spacesSIAM Journal on Optimization2008192824835244891510.1137/070688717ZBL1168.47047XuH. K.An iterative approach to quadratic optimizationJournal of Optimization Theory and Applications20031163659678197775610.1023/A:1023073621589ZBL1043.90063MaingéP.-E.The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spacesComputers & Mathematics with Applications20105917479257549310.1016/j.camwa.2009.09.003ZBL1189.49011