JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 235474 10.1155/2012/235474 235474 Research Article Viscosity Approximations by the Shrinking Projection Method of Quasi-Nonexpansive Mappings for Generalized Equilibrium Problems Wangkeeree Rabian Nimana Nimit Torregrosa Juan Department of Mathematics Faculty of Science Naresuan University Phitsanulok 65000 Thailand nu.ac.th 2012 24 9 2012 2012 16 07 2012 27 08 2012 2012 Copyright © 2012 Rabian Wangkeeree and Nimit Nimana. 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 viscosity approximations by using the shrinking projection method established by Takahashi, Takeuchi, and Kubota, for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of a quasi-nonexpansive mapping. Furthermore, we also consider the viscosity shrinking projection method for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of the super hybrid mappings in Hilbert spaces.

1. Introduction

Let H be a real Hilbert space with inner product ·,· and norm · and C a nonempty closed convex subset of H and let T be a mapping of C into H. Then, T:CH is said to be nonexpansive if Tx-Tyx-y for all x,yC. A mapping T:CH is said to be quasi-nonexpansive if Tx-yx-y for all xC and yF(T):={xC:Tx=x}. Recall that a mapping Ψ:CH is said to be δ-inverse strongly monotone if there exists a positive real number δ such that (1.1)Ψx-Ψy,x-yδΨx-Ψy2,x,yC. If Ψ is an δ-inverse strongly monotone mapping of C into H, then it is obvious that Ψ is 1/δ-Lipschitz continuous.

Let F:C×C be a bifunction and Ψ:CH be δ-inverse strongly monotone mapping. The generalized equilibrium problem (for short, GEP) for F and Ψ is to find zC such that (1.2)F(z,y)+Ψz,y-z0,yC. The problem (1.2) was studied by Moudafi . The set of solutions for the problem (1.2) is denoted by GEP(F,Ψ), that is, (1.3)GEP(F,Ψ)={zC:F(z,y)+Ψz,y-z0,yC}. If Ψ0 in (1.2), then GEP reduces to the classical equilibrium problem and GEP(F,0) is denoted by EP(F), that is, (1.4)EP(F)={zC:F(z,y)0,yC}. If F0 in (1.2), then GEP reduces to the classical variational inequality and GEP(0,Ψ) is denoted by VI(Ψ,C), that is, (1.5)VI(Ψ,C)={zC:Ψz,y-z0,yC}. The problem (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, min-max problems, and the Nash equilibrium problems in noncooperative games, see, for example, Blum and Oettli  and Moudafi .

In 2005, Combettes and Hirstoaga  introduced an iterative algorithm of finding the best approximation to the initial data and proved a strong convergence theorem. In 2007, by using the viscosity approximation method, S. Takahashi and W. Takahashi  introduced another iterative scheme for finding a common element of the set of solutions of the equilibrium problem and the set of fixed points of a nonexpansive mapping. Subsequently, algorithms constructed for solving the equilibrium problems and fixed point problems have further developed by some authors. In particular, Ceng and Yao  introduced an iterative scheme for finding a common element of the set of solutions of the mixed equilibrium problem and the set of common fixed points of finitely many nonexpansive mappings. Maingé and Moudafi  introduced an iterative algorithm for equilibrium problems and fixed point problems. Wangkeeree  introduced a new iterative scheme for finding the common element of the set of common fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality. Wangkeeree and Kamraksa  introduced an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of a general system of variational inequalities for a cocoercive mapping in a real Hilbert space. Their results extend and improve many results in the literature.

In 1953, Mann  introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping T in a Hilbert space H as follows: (1.6)xn+1=αnxn+(1-αn)Txn,n, where the initial point x1 is taken in C arbitrarily and {αn} is a sequence in [0,1]. Wittmann  obtained the strong convergence results of the sequence {xn} defined by (1.6) to PFx1 under the following assumptions:

limnαn=0;

n=1αn=;

n=1|αn+1-αn|<,

where PF(T) is the metric projection of H onto F(T). In 2000, Moudafi  introduced the viscosity approximation method for nonexpansive mappings (see  for further developments in both Hilbert and Banach spaces). Let f be a contraction on H. Starting with an arbitrary initial x1H, define a sequence {xn} recursively by (1.7)xn+1=αnf(xn)+(1-αn)Txn,n1, where {αn} is a sequence in (0,1). It is proved [12, 13] that under conditions (C1), (C2), and (C3) imposed on {αn}, the sequence {xn} generated by (1.7) strongly converges to the unique fixed point x* of PF(T)f which is a unique solution of the variational inequality (1.8)(I-f)x*,x-x*0,xC. Suzuki  considered the Meir-Keeler contractions, which is extended notion of contractions and studied equivalency of convergence of these approximation methods.

Using the viscosity approximation method, in 2007, S. Takahashi and W. Takahashi  introduced an iterative scheme for finding a common element of the solution set of the classical equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. Let T:CH be a nonexpansive mapping. Starting with arbitrary initial x1H, define sequences {xn} and {un} recursively by (1.9)F(un,y)+1rny-un,un-xn0,yC,xn+1=αnf(xn)+(1-αn)Tun,n. They proved that under certain appropriate conditions imposed on {αn} and {rn}, the sequences {xn} and {un} converge strongly to zF(T)EP(F), where z=PF(T)EP(F)f(z).

On the other hand, in 2008, Takahashi et al.  has adapted Nakajo and Takahashi’s  idea to modify the process (1.6) so that strong convergence has been guaranteed. They proposed the following modification for a family of nonexpansive mappings in a Hilbert space: x0H, C1=C, u1=PC1x0 and (1.10)yn=αnun+(1-αn)Tnun,Cn+1={zCn:yn-zun-z},un+1=PCn+1x0,n, where 0αna<1 for all n. They proved that if {Tn} satisfies the appropriate conditions, then {un} generated by (1.10) converges strongly to a common fixed point of Tn.

Very recently, Kimura and Nakajo  considered viscosity approximations by using the shrinking projection method established by Takahashi et al.  and the modified shrinking projection method proposed by Qin et al. , for finding a common fixed point of countably many nonlinear mappings, and they obtained some strong convergence theorems.

Motivated by these results, we introduce the viscosity shrinking projection method for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of a quasi-nonexpansive mapping. Furthermore, we also consider the viscosity shrinking projection method for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of the super hybrid mappings in Hilbert spaces.

2. Preliminaries

Throughout this paper, we denote by the set of positive integers and by the set of real numbers. Let H be a real Hilbert space with inner product ·,· and norm ·. We denote the strong convergence and the weak convergence of {xn} to xH by xnx and xnx, respectively. From , we know the following basic properties. For x,yH and λ we have (2.1)λx+(1-λ)y2=λx2+(1-λ)y2-λ(1-λ)x-y2. We also know that for u,v,x,yH, we have (2.2)2u-v,x-y=u-y2+v-x2-u-x2-v-y2.

For every point xH, there exists a unique nearest point of C, denoted by PCx, such that x-PCxx-y for all yC. PC is called the metric projection from H onto C. It is well known that z=PCxx-z,z-y0, for all xH and z,yC. We also know that PC is firmly nonexpansive mapping from H onto C, that is, (2.3)PCx-PCy2PCx-PCy,x-y,x,yH, and so is nonexpansive mapping.

For solving the generalized equilibrium problem, let us assume that F 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;

for each x,y,zC,limt0F(tz+(1-t)x,y)F(x,y);

for each xC,yF(x,y) is convex and lower semicontinuous.

In order to prove our main results, we also need the following lemmas.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B2">2</xref>]).

Let C be a nonempty closed convex subset of H and let F be a bifunction from C×C into satisfying (A1), (A2), (A3), and (A4). Then, for any r>0 and xH, there exists a unique zC such that (2.4)F(z,y)+1ry-z,z-x0,yC.

Lemma 2.2 (see [<xref ref-type="bibr" rid="B5">4</xref>]).

Let C be a nonempty closed convex subset of H and let F be a bifunction from C×C into satisfying (A1), (A2), (A3), and (A4). Then, for any r>0 and xH, define a mapping Trx:HC as follows: (2.5)Trx={zC:F(z,y)+1ry-z,z-x0,yC}xH,r. Then the following hold:

Tr is single-valued;

Tr is firmly nonexpansive, that is, (2.6)Trx-Try2Trx-Try,x-y,x,yH;

F(Tr)=EP(F);

EP(F) is closed and convex.

Remark 2.3 (see [<xref ref-type="bibr" rid="B13">20</xref>]).

Using (ii) in Lemma 2.2 and (2.2), we have (2.7)2Trx-Try22Trx-Try,x-y=Trx-y2+Try-x2-Trx-x2-Try-y2. So, for yF(Tr) and xH, we have (2.8)Trx-u2+Trx-x2x-u2.

Remark 2.4.

For any xH and r>0, by Lemma 2.1, there exists zC such that (2.9)F(z,y)+1ry-z,z-x0,yC. Replacing x with x-rΨxH in (2.9), we have (2.10)F(z,y)+Ψx,y-z+1ry-z,z-x0,yC, where Ψ:CH is an inverse-strongly monotone mapping.

For a sequence {Cn} of nonempty closed convex subsets of a Hilbert space H, define s-LinCn and w-LsnCn as follows.

x s - L i n C n if and only if there exists {xn}H such that xnx and that xnCn for all n.

x w - L s n C n if and only if there exists a subsequence {Cni} of {Cn} and a subsequence {yi}H such that yiy and that yiCni for all i.

If C0 satisfies (2.11)C0=s-LinCn=w-LsnCn, it is said that {Cn} converges to C0 in the sense of Mosco  and we write C0=M-limnCn. It is easy to show that if {Cn} is nonincreasing with respect to inclusion, then {Cn} converges to n=1Cn in the sense of Mosco. For more details, see . Tsukada  proved the following theorem for the metric projection.

Theorem 2.5 (see Tsukada [<xref ref-type="bibr" rid="B22">22</xref>]).

Let H be a Hilbert space. Let {Cn} be a sequence of nonempty closed convex subsets of H. If C0=M-limnCn exists and is nonempty, then for each xH, {PCnx} converges strongly to PC0x, where PCn and PC0 are the metric projections of H onto Cn and C0, respectively.

On the other hand, a mapping f of a complete metric space (X,d) into itself is said to be a contraction with coefficient r(0,1) if d(f(x),f(y))rd(x,y) for all x,yC. It is well known that f has a unique fixed point . Meir-Keeler  defined the following mapping called Meir-Keeler contraction. Let (X,d) be a complete metric space. A mapping f:XX is called a Meir-Keeler contraction if for all ɛ>0, there exists δ>0 such that ɛd(x,y)<ɛ+δ implies d(f(x),f(y))<ɛ for all x,yX. It is well known that Meir-Keeler contraction is a generalization of contraction and the following result is proved in .

Theorem 2.6 (see Meir-Keeler [<xref ref-type="bibr" rid="B15">24</xref>]).

A Meir-Keeler contraction defined on a complete metric space has a unique fixed point.

We have the following results for Meir-Keeler contractions defined on a Banach space by Suzuki .

Theorem 2.7 (see Suzuki [<xref ref-type="bibr" rid="B16">14</xref>]).

Let f be a Meir-Keeler contraction on a convex subset C of a Banach space E. Then, for every ɛ>0, there exists r(0,1) such that x-yɛ implies f(x)-f(y)rx-y for all x,yC.

Lemma 2.8 (see Suzuki [<xref ref-type="bibr" rid="B16">14</xref>]).

Let C be a convex subset of a Banach space E. Let T be a nonexpansive mapping on C, and let f be a Meir-Keeler contraction on C. Then the following hold.

Tf is a Meir-Keeler contraction on C.

For each α(0,1), a mapping x(1-α)Tx+αfx is a Meir-Keeler contraction on C.

3. Main Results

In this section, using the shrinking projection method by Takahashi et al. , we prove a strong convergence theorem for a quasi-nonexpansive mapping with a generalized equilibrium problem in a Hilbert space. Before proving it, we need the following lemmas.

Lemma 3.1.

Let C be a nonempty closed convex subset of a Hilbert space H and δ>0 and let Ψ:CH be δ-inverse strongly monotone. If 0<λ2δ, then I-λΨ is a nonexpansive mapping.

Proof.

For x,yC, we can calculate (3.1)(I-λΨ)x-(I-λΨ)y2=x-y-λ(Ψx-Ψy)2=x-y2-2λx-y,Ψx-Ψy+λ2Ψx-Ψy2x-y2-2λδΨx-Ψy2+λ2Ψx-Ψy2=x-y2+λ(λ-2δ)Ψx-Ψy2x-y2. Therefore I-λΨ is nonexpansive. This completes the proof.

Lemma 3.2.

Let C be a nonempty closed convex subset of H, and let T be a quasi-nonexpansive mapping of C into H. Then, F(T) is closed and convex.

Proof.

We first show that F(T) is closed. Let {zn} be any sequence in F(T) with znz. We claim that zF(T). Since C is closed, we have zC. We observe that (3.2)z-Tzz-zn+zn-Tzz-zn+z-zn=2z-zn. Since znz, we obtain that z-Tz0 and hence z=Tz. This show that zF(T).

Next, we show that F(T) is convex. Let x,yF(T) and α[0,1]. We claim that αx+(1-α)yF(T). Putting z=αx+(1-α)y, we have (3.3)z-Tz2=αx+(1-α)y-Tz2=α(x-Tz)+(1-α)(y-Tz)2=αx-Tz2+(1-α)y-Tz2-α(1-α)x-y2αx-z2+(1-α)y-z2-α(1-α)x-y2=α(1-α)(x-y)2+(1-α)α(y-x)2-α(1-α)x-y2={α(1-α)2+α2(1-α)-α(1-α)}x-y2=α(1-α)(1-α+α-1)x-y2=0. Hence F(T) is convex. This completes the proof.

Theorem 3.3.

Let H be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C be a bifunction satisfying (A1), (A2), (A3), and (A4) and let Ψ be a δ-inverse strongly monotone mapping from C into H. Let T:CC be a quasi-nonexpansive mapping which is demiclosed on C, that is, if {wk}C,wkw and (I-T)wk0, then wF(T). Assume that Ω:=GEP(F,Ψ)F(T) and f is a Meir-Keeler contraction of C into itself. Let the sequence {xn}C be defined by(3.4)C1=C,x1=xC,F(zn,y)+Ψxn,y-zn+1λnzn-xn,y-zn0,yC,yn=αnxn+(1-αn)Tzn,Cn+1={zCn:yn-zxn-z},xn+1=PCn+1f(xn),n, where PCn+1 is the metric projection of H onto Cn+1 and {αn}[0,1] and {λn}(0,2δ) are real sequences satisfying (3.5)liminfnαn<1,0<aλnb<2δ, for some a,b. Then, {xn} converges strongly to z0Ω, which satisfies z0=PΩf(z0).

Proof.

Since Ω is a closed convex subset of C, we have that PΩ is well defined and nonexpansive. Furthermore, we know that f is Meir-Keeler contraction and we know from Lemma 2.8 (i) that PΩf of C onto Ω is a Meir-Keeler contraction on C. By Theorem 2.6, there exists a unique fixed point z0C such that z0=PΩf(z0). Next, we observe that zn=Tλn(xn-λnΨxn) for each n and take zF(T)GEP(F,Ψ). From z=Tλn(z-λnΨz) and Lemma 2.2, we have that for any n, (3.6)zn-z2=Tλn(xn-λnΨxn)-Tλn(z-λnΨz)2(xn-λnΨxn)-(z-λnΨz)2xn-z2. Next, we divide the proof into several steps.

Step 1. Cn is closed convex and {xn} is well defined for every n.

It is obvious from the assumption that C1:=C is closed convex and ΩC1. For any k, suppose that Ck is closed and convex, and ΩCk. Note that for all zCk, (3.7)yk-z2xk-z2yk-z2-xk-z20yk2+z2-2yk,z-xk2-z2+2xk,z0yk2-2yk-xk,z-xk20. It is easy to see that Ck+1 is closed. Next, we prove that Ck+1 is convex. For any u,vCk+1 and α[0,1], we claim that z:=αu+(1-α)vCk+1. Since uCk+1, we have yk-uxk-u and so yk-u2xk-u2, that is, yk2-2yk-xk,u-xk20. Similarly, vCk+1, we get yk2-2yk-xk,v-xk20.

Thus, (3.8)αyk2-2yk-xk,αu-αxk20,(1-α)yk2-2yk-xk,(1-α)v-(1-α)xk20. Combining the above inequalities, we obtain (3.9)yk2-2yk-xk,αu+(1-α)v-xk20. Therefore yk-zxk-z. This shows that zCk+1 and hence Ck+1 is convex. Therefore Cn is closed and convex for all n.

Next, we show that ΩCn, for all n. For any k, suppose that vΩCk. Since T is quasi-nonexpansive and from (3.6), we have (3.10)yk-v2=αkxk+(1-αk)Tzk-v2=αk(xk-v)+(1-αk)(Tzk-v)2αkxk-v2+(1-αk)Tzk-v2αkxk-v2+(1-αk)zk-v2αkxk-v2+(1-αk)xk-v2=xk-v2. So, we have vCk+1. By principle of mathematical induction, we can conclude that Cn is closed and convex, and ΩCn, for all n. Hence, we have (3.11)ΩCn+1Cn, for all n. Therefore {xn} is well defined.

Step 2. limnxn-u=0 for some un=1Cn and f(u)-u,u-y0 for all yΩ.

Since n=1Cn is closed convex, we also have that Pn=1Cn is well defined and so Pn=1Cnf is a Meir-Keeler contraction on C. By Theorem 2.6, there exists a unique fixed point un=1Cn of Pn=1Cnf. Since Cn is a nonincreasing sequence of nonempty closed convex subsets of H with respect to inclusion, it follows that (3.12)Ωn=1Cn=M-limnCn. Setting un:=PCnf(u) and applying Theorem 2.5, we can conclude that (3.13)limnun=Pn=1Cnf(u)=u. Next, we will prove that limnxn-u=0. Assume to contrary that limsupnxn-u0, there exists ɛ>0 and a subsequence {xnj-u} of {xn-u} such that (3.14)xnj-uɛ,j, which gives that (3.15)limsupjxnj-uɛ>0. We choose a positive number ɛ>0 such that (3.16)limsupjxnj-u>ɛ>0. For such ɛ, by the definition of Meir-Keeler contraction, there exists δɛ'>0 with (3.17)ɛ+δɛ<limsupjxnj-u, such that (3.18)x-y<ɛ+δɛimpliesf(x)-f(y)<ɛ, for all x,yC. Again for such ɛ', by Theorem 2.7, there exists rɛ'(0,1) such that (3.19)x-yɛ+δɛimpliesf(x)-f(y)<rɛx-y. Since unu, there exists n0 such that (3.20)un-u<δɛ,nn0. By the idea of Suzuki  and Kimura and Nakajo , we consider the following two cases.

Case I . Assume that there exists n1n0 such that (3.21)xn1-u<ɛ+δɛ. Thus, we get (3.22)xn1+1-uxn1+1-un1+1+un1+1-u=PCn1+1f(xn1)-PCn1+1f(u)+un1+1-uf(xn1)-f(u)+un1+1-u<ɛ+δɛ. By induction on {n}, we can obtain that (3.23)xn-u<ɛ+δɛ, for all nn0. In particular, for all jn0, we have njjn0 and (3.24)xnj-u<ɛ+δɛ. This implies that (3.25)limsupjxnj-uɛ+δɛ<limsupjxnj-u, which is a contradiction. Therefore, we conclude that xn-u0 as n.

Case I I . Assume that (3.26)xn-uɛ+δɛ,nn0. By (3.19), we have (3.27)f(xn)-f(u)<rɛxn-u,nn0. Thus, we have (3.28)xn+1-un+1=PCn+1f(xn)-PCn+1f(u)f(xn)-f(u)rɛxn-urɛ(xn-un+un-u), for every nn0. In particular, we have (3.29)xnj+1-unj+1rɛ(xnj-unj+unj-u), for every jn0  (njjn0). Let us consider (3.30)limsupnxnj-unj=limsupjxnj+1-unj+1rɛlimsupj(xnj-unj+unj-u)rɛlimsupjxnj-unj+rɛlimsupjunj-u=rɛlimsupjxnj-unj<limsupjxnj-unj, which gives a contradiction. Hence, we obtain that (3.31)limnxn-u=0, and therefore {xn} is bounded. Moreover, {f(xn)},{zn}, and {yn} are also bounded. Since xn+1=PCn+1f(xn), we have (3.32)f(xn)-xn+1,xn+1-y0,yCn+1. Since ΩCn+1, we get (3.33)f(xn)-xn+1,xn+1-y0,n,yΩ. We have from xnu that (3.34)f(u)-u,u-y0,yΩ.

Step 3. There exists a subsequence {xni-zni} of {xn-zn} such that xni-zni0 as i.

We have from (3.13) and (3.31) that (3.35)xn-xn+1xn-u+u-un+1+un+1-xn+1=xn-u+u-un+1+PCn+1f(xn)-PCn+1f(u)xn-u+u-un+1+f(xn)-f(u)0. From xn+1Cn+1, we have that (3.36)yn-xn+1xn-xn+1, and so yn-xn+10. We also have (3.37)yn-xnyn-xn+1+xn+1-xn0. From liminfnαn<1, there exists a subsequence {αni} of {αn} and α0 with 0α0<1 such that αniα0. Since xn-yn=xn-αnxn-(1-α)Tzn=(1-αn)xn-Tzn, we have (3.38)Tzni-xni0asi. Using Lemma 2.2 (ii) and (3.6), we have (3.39)zn-z2=Tλn(xn-λnΨxn)-Tλn(z-λnΨz)2(xn-λnΨxn)-(z-λnΨz),zn-z=-(xn-λnΨxn)-(z-λnΨz),z-zn=12((xn-λnΨxn)-(z-λnΨz)2+zn-z2-(xn-λnΨxn)-(z-λnΨz)+(z-zn)2)12(xn-z2+zn-z2-(xn-zn)-λn(Ψxn-Ψz)2)=12(xn-z2+zn-z2-xn-zn2+2λnxn-zn,Ψxn-Ψz-λn2Ψxn-Ψz2). So, we have (3.40)zn-z2xn-z2-xn-zn2+2λnxn-zn,Ψxn-Ψz-λn2Ψxn-Ψz2. Let us consider (3.41)yn-z2=αn(xn-z)+(1-αn)(Tzn-z)2αnxn-z2+(1-αn)Tzn-z2αnxn-z2+(1-αn)zn-z2=αnxn-z2+(1-αn)Tλn(I-λnΨ)xn-Tλn(I-λnΨ)z2αnxn-z2+(1-αn)(I-λnΨ)xn-(I-λnΨ)z2=αnxn-z2+(1-αn)(xn-z)-λn(Ψxn-Ψz)2=αnxn-z2+(1-αn)xn-z2+(1-αn)λn2Ψxn-Ψz2-2(1-αn)λnxn-z,Ψxn-Ψzxn-z2+(1-αn)λn2Ψxn-Ψz2-2(1-αn)λnδΨxn-Ψz2=xn-z2+(1-αn)(λn-2δ)λnΨxn-Ψz2xn-z2+(1-αn)(b-2δ)bΨxn-Ψz2. In particular, we have (3.42)(1-αni)(2δ-b)bΨxni-Ψz2xni-z2-yni-z2xni-yni2+2xni-yniyni-z. Since αniα0 with α0<1 and xni-yni0, we obtain that (3.43)Ψxni-Ψz0. Using (3.40), we have (3.44)yn-z2αnxn-z2+(1-αn)zn-z2αnxn-z2+(1-αn)×(xn-z2-xn-zn2+2λnxn-zn,Ψxn-Ψz-λn2Ψxn-Ψz2)αnxn-z2+(1-αn)(xn-z2-xn-zn2+2λnxn-znΨxn-Ψz)xn-z2-(1-αn)xn-zn2+2(1-αn)λn(xn+zn)Ψxn-Ψzxn-z2-(1-αn)xn-zn2+2(1-αn)bMΨxn-Ψz, where M:=sup{xn+yn:n}.

So, we have (3.45)(1-αni)xni-zni2xni-z2-yni-z2+2(1-αni)bMΨxni-Ψzxni-yni2+2xni-yniyni-z+2(1-αni)bMΨxni-Ψz. We have from αniα0, (3.37), and (3.43) that (3.46)xni-zni0.

Step 4. Finally, we prove that uΩ:=F(T)GEP(F,Ψ).

Since yn=αnxn+(1-αn)Tzn, we have yn-Tzn=αn(xn-Tzn). So, from (3.38) we have (3.47)yni-Tzni=αnixni-Tzni0. Since zni-Tznizni-xni+xni-yni+yni-Tzni, from (3.37), (3.46), and (3.47) we have (3.48)zni-Tzni0. Since xniu, we have zniu. So, from (3.48) and the demiclosed property of T, we have (3.49)uF(T). We next show that uGEP(F,Ψ). Since zn=Tλn(xn-λnΨxn), for any yC we have (3.50)F(zn,y)+Ψxn,y-zn+1λny-zn,zn-xn0. From (A2), we have (3.51)-F(y,zn)+Ψxn,y-zn+1λny-zn,zn-xn0 and so (3.52)Ψxn,y-zn+1λny-zn,zn-xnF(y,zn). Replacing n by ni, we have (3.53)Ψxni,y-zni+y-zni,zni-xniλniF(y,zni). Note that Ψ is 1/δ-Lipschitz continuous, and from (3.46), we have (3.54)Ψzni-Ψxni0. For t(0,1] and yC, let zt*=ty+(1-t)u. Since C is convex, we have zt*C. So, from (3.53) we have (3.55)zt*-zni,Ψzt*zt*-zni,Ψzt*-zt*-zni,Ψxni-zt*-zni,zni-xniλni+F(zt*,zni)=zt*-zni,Ψzt*-Ψxni-zt*-zni,zni-xniλni+F(zt*,zni)=zt*-zni,Ψzt*-Ψzni+zt*-zni,Ψzni-Ψxni-zt*-zni,zni-xniλni+F(zt*,zni). From zt*-zni,Ψzt*-Ψzni0, we have (3.56)zt*-zni,Ψzt*zt*-zni,Ψzni-Ψxni-zt*-zni,zni-xniλni+F(zt*,zni). Thus, (3.57)zt*-zni,Ψzt*-zt*-zniΨzni-Ψxni-zt*-znizni-xniλni+F(zt*,zni). From Step  3 and (3.54), we obtain (3.58)zt*-u,Ψzt*F(zt*,u). From (A1), (A4), and (3.58), we have (3.59)0=F(zt*,zt*)=F(zt*,ty+(1-t)u)tF(zt*,y)+(1-t)F(zt*,u)tF(zt*,y)+(1-t)zt*-u,Ψzt*tF(zt*,y)+(1-t)ty-u,Ψzt*, and hence (3.60)0F(zt*,y)+(1-t)y-u,Ψzt*. Letting t0 and from (A3), we have that for each yC, (3.61)0limt0(F(zt*,y)+(1-t)y-u,Ψzt*)=limt0(F(ty+(1-t)u,y)+(1-t)y-u,tΨy+(1-t)Ψu)F(u,y)+y-u,Ψu. This implies that uGEP(F,Ψ). So, we have uF(T)GEP(F,Ψ). We obtain from (3.34) that u=z0 and hence, {xn} converges strongly to z0. This completes the proof.

By Theorem 3.3, we can obtain some new and interesting strong convergence theorems. Now we give some examples as follows.

Setting f(xn)=x,n in Theorem 3.3, we obtain the following result.

Corollary 3.4.

Let H be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C be a bifunction satisfying (A1), (A2), (A3), and (A4) and let Ψ be a δ-inverse strongly monotone mapping from C into H. Let T:CC be a quasi-nonexpansive mapping which is demiclosed on C. Assume that Ω and let C1=C and {xn}C be a sequence generated by x1=xC and (3.62)F(zn,y)+Ψxn,y-zn+1λnzn-xn,y-zn0,yC,yn=αnxn+(1-αn)Tzn,Cn+1={zCn:yn-zxn-z},xn+1=PCn+1x,n, where PCn+1 is the metric projection of H onto Cn+1 and {αn}[0,1] and {λn}[0,2δ) are sequences such that (3.63)liminfnαn<1,0<aλnb<2δ, for some a. Then {xn} converges strongly to z0=PΩz0.

Setting Ψ0 in Theorem 3.3, we obtain the following result.

Corollary 3.5.

Let H be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C be a bifunction satisfying (A1), (A2), (A3), and (A4). Let T:CC be a quasi-nonexpansive mapping which is demiclosed on C. Assume that EP(F)F(T) and f is a Meir-Keeler contraction of C into itself. Let C1=C and {xn}C be a sequence generated by x1=xC and (3.64)F(zn,y)+1λnzn-xn,y-zn0,yC,yn=αnxn+(1-αn)Tzn,Cn+1={zCn:yn-zxn-z},xn+1=PCn+1f(xn),n, where PCn+1 is the metric projection of H onto Cn+1 and {αn}[0,1] and {λn}[0,) are sequences such that (3.65)liminfnαn<1,0<aλn, for some a. Then {xn} converges strongly to z0F(T)EP(F).

Setting Ψ0 and f(xn)=x for all n in Theorem 3.3, we obtain the following result.

Corollary 3.6.

Let H be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C be a bifunction satisfying (A1), (A2), (A3), and (A4). Let T:CC be an quasi-nonexpansive mapping which is demiclosed on C and assume that EP(F)F(T). Let C1=C and {xn}C be a sequence generated by x1=xC and (3.66)F(zn,y)+1λnzn-xn,y-zn0,yC,yn=αnxn+(1-αn)Tzn,Cn+1={zCn:yn-zxn-z},xn+1=PCn+1x,n, where PCn+1 is the metric projection of H onto Cn+1 and {αn}[0,1] and {λn}[0,) are sequences such that (3.67)liminfnαn<1,0<aλn, for some a. Then {xn} converges strongly to z0F(T)EP(F).

Next, using the CQ hybrid method introduced by Nakajo and Takahashi , we prove a strong convergence theorem of a quasi-nonexpansive mapping for solving the generalized equilibrium problem.

Theorem 3.7.

Let H be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C be a bifunction satisfying (A1), (A2), (A3), and (A4) and let Ψ be a δ-inverse strongly monotone mapping from C into H. Let T:CC be an quasi-nonexpansive mapping which is demiclosed on C. Assume that Ω and f is a Meir-Keeler contraction of C into itself. Let Q1=C and {xn}C be a sequence generated by x1=xC and (3.68)F(zn,y)+Ψxn,y-zn+1λnzn-xn,y-zn0,yC,yn=αnxn+(1-αn)Tzn,Cn={zC:yn-zxn-z},Qn={zQn-1:f(xn-1)-xn,xn-z0},xn+1=PCnQnf(xn),n, where PCnQn is the metric projection of H onto CnQn and {αn}(0,1) and {λn}(0,2δ) satisfy (3.69)0αnb<1,0<cλnd<2δ, for some b,c,d. Then, {xn} converges strongly to z0Ω, which satisfies z0=PΩf(z0).

Proof.

As in the proof of Theorem 3.3, we have that the mapping PΩf of C onto Ω is a Meir-Keeler contraction on C. By Theorem 2.6, there exists a unique fixed point z0C such that z0=PΩf(z0). Next, it is clear that Cn is closed and convex. Next, we will show that Qn is closed and convex for all n. For any n, let {zk} be a sequence in Qn such that zkz. For each k, we observe that (3.70)0xn-zk,f(xn-1)-xn=12(f(xn-1)-zk2-xn-zk2-f(xn-1)-xn2). Taking k, we get xn-z,f(xn-1)-xn0 and then zQn. Therefore Qn is closed.

Next, we will show that Qn is convex. For any u,vQn, and α[0,1], put z=αu+(1-α)v. We claim that zQn. Since uQn, we have αxn-αu,f(xn-1)-xn0. Similarly, since vQn, we have (1-α)xn-(1-α)v,f(xn-1)-xn0. Thus, (3.71)0αxn-αu+(1-α)xn-(1-α)v,f(xn-1)-xn=xn-αu-(1-α)v,f(xn-1)-xn=xn-z,f(xn-1)-xn. It follows that zQn, and therefore we have that Qn is convex. We obtain from both Cn and Qn which are closed convex sets for every n that CnQn is closed and convex for every n.

Next, we will show that CnQn is nonempty. Let zF(T)GEP(F,Ψ). We will show that zCn for any n. We notice that zn=Tλn(xn-λnΨxn) for each n and z=Tλn(z-λnΨz). From Ψ which is an inverse strongly monotone mapping Lemma 2.2 (ii), Lemma 3.1, we obtain (3.72)zn-zxn-z,foranyn. Since T is quasi-nonexpansive with the fixed point z and from (3.72), we have (3.73)yn-z2xn-z2. So, we have zCn. Therefore F(T)GEP(F,Ψ)Cn, for all n.

Next, we will show that (3.74)F(T)GEP(F,Ψ)CnQn,n. It is obvious that F(T)GEP(F,Ψ)C=Q1. Hence (3.75)F(T)GEP(F,Ψ)C1Q1. For any k, suppose that (3.76)F(T)GEP(F,Ψ)CkQk. Since xk+1=PCkQkf(xk), we have (3.77)f(xk)-xk+1,xk+1-z0,zCkQk. In particular, for any zF(T)GEP(F,Ψ), we obtain that (3.78)f(xk)-xk+1,xk+1-z0. This shows that zQk+1. Hence F(T)GEP(F,Ψ)Qk+1. Therefore, we conclude that (3.79)F(T)GEP(F,Ψ)Ck+1Qk+1. By principle of mathematical induction, we can conclude that (3.80)F(T)GEP(F,Ψ)CnQn,n. Hence {xn} is well defined. Since Pn=1Qnf is a Meir-Keeler contraction on C, there exists a unique element uC such that u=Pn=1Qnf(u). For each n, let un=PQnf(u). Since F(T)GEP(F,Ψ)Qn+1Qn, we have from Theorem 2.5 that unu. Notice that xn=PQnf(xn-1). Thus, as in the proof of Theorem 3.3, we get xnu and hence {xn} is bounded. Moreover, (3.81)limnxn-xn+1=0,limnyn-xn=0. As the proof of Theorem 3.3, we have that (3.82)yn-z2xn-z2+(1-αn)(λn-2δ)λnΨxn-Ψz2. Thus, (3.83)yn-z2xn-z2+(1-a)(d-2δ)dΨxn-Ψz2 and so (3.84)(1-a)(2δ-d)dΨxn-Ψz2xn-z2-yn-z2xn-yn2+2xn-ynyn-z. Since (1-a)(2δ-d)d>0 and xn-yn0, we obtain that (3.85)Ψxn-Ψz0. Using (3.40) in Theorem 3.3, we have (3.86)yn-z2xn-z2-(1-b)xn-zn2+2(1-a)dMΨxn-Ψz, where M:=sup{xn+yn}. So, we have (3.87)(1-b)xn-zn2xn-z2-yn-z2+2(1-a)dMΨxn-Ψzxn-yn2+2xn-ynyn-z+2(1-a)dMΨxn-Ψz. We have from 1-b>0, xn-yn0 and (3.85) that (3.88)xn-zn0, which implies that (3.89)znu. Notice that (3.90)xn-yn=xn-αnxn-(1-α)Tzn=(1-αn)xn-Tzn, and from limsupnαnb<1, we have (3.91)Tzn-xn0. Since yn=αnxn+(1-αn)Tzn, we have yn-Tzn=αn(xn-Tzn). So, from 0<aαnb<1 and (3.91), we obtain (3.92)yn-Tzn0 and hence (3.93)zn-Tzn0. From (3.89), (3.93), and the demiclosed property of T, we have uF(T). As in the proof of Theorem 3.3 we have that uF(T)GEP(F,Ψ). Since F(T)GEP(F,Ψ)Qn+1, we get (3.94)f(xn)-xn+1,xn+1-y0, for all n and yF(T)GEP(F,Ψ). We have from xnu that (3.95)f(u)-u,u-y0, for all yF(T)GEP(F,Ψ), which implies that u=PΩf(u). It follows that u=z0, since z0F(T)GEP(F,Ψ) of PΩf is unique. Hence, {xn} converges strongly to z0. This completes the proof.

Setting f(xn)=x, for all n in Theorem 3.7, we obtain the following result.

Corollary 3.8.

Let H be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C be a bifunction satisfying (A1), (A2), (A3), and (A4) and let Ψ be a δ-inverse strongly monotone mapping from C into H and let T:CC be a quasi-nonexpansive mapping which is demiclosed on C. Assume that Ω:=GEP(F,Ψ)F(T). Let Q1=C and {xn}C be a sequence generated by x1=xC and (3.96)F(zn,y)+Ψxn,y-zn+1λnzn-xn,y-zn0,yC,yn=αnxn+(1-αn)Tzn,Cn={zC:yn-zxn-z},Qn={zQn-1:x-xn,xn-z0},xn+1=PCnQnx,n, where PCnQn is the metric projection of H onto CnQn and {αn}(0,1) and {λn}(0,2δ) satisfy (3.97)0αnb<1,0<cλnd<2δ, for some a,b,c,d. Then {xn} converges strongly to z0=PΩf(z0).

Setting Ψ0 in Theorem 3.7, we obtain the following result.

Corollary 3.9.

Let H be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C be a bifunction satisfying (A1), (A2), (A3), and (A4) and let T:CC be an quasi-nonexpansive mapping which is demiclosed on C. Assume that EP(F)F(T) and f is a Meir-Keeler contraction of C into itself. Let Q1=C and {xn}C be a sequence generated by x1=xC and (3.98)F(zn,y)+1λnzn-xn,y-zn0,yC,yn=αnxn+(1-αn)Tzn,Cn={zC:yn-zxn-z},Qn={zQn-1:f(xn-1)-xn,xn-z0},xn+1=PCnQnf(xn),n, where PCnQn is the metric projection of H onto CnQn and {αn}(0,1) and {λn}(0,) satisfy (3.99)0αnb<1,0<cλn, for some a,b,c. Then {xn} converges strongly to z0F(T)EP(F).

Setting Ψ0 and f(xn)=x,n in Theorem 3.7, we obtain the following result.

Corollary 3.10.

Let H be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C be a bifunction satisfying (A1), (A2), (A3), and (A4) and let T:CC be a quasi-nonexpansive mapping which is demiclosed on H. Assume that EP(F)F(T). Let Q1=C and {xn}C be a sequence generated by x1=xC and (3.100)F(zn,y)+1λnzn-xn,y-zn0,yC,yn=αnxn+(1-αn)Tzn,Cn={zC:yn-zxn-z},Qn={zQn-1:x-xn,xn-z0},xn+1=PCnQnx,n, where PCnQn is the metric projection of H onto CnQn and {αn}(0,1) and {λn}(0,) satisfy (3.101)0<aαnb<1,0<cλn, for some a,b,c. Then {xn} converges strongly to z0F(T)EP(F).

4. Applications

In this section, we present some convergence theorems deduced from the results in the previous section. Recall that a mapping T:CH is said to be nonspreading if (4.1)2Tx-Ty2Tx-y2+Ty-x2 for all x,yC. Further, a mapping T:CH is said to be hybrid if (4.2)3Tx-Ty2x-y2+Tx-y2+Ty-x2, for all x,yC. These mappings are deduced from a firmly nonexpansive mapping in a Hilbert space.

A mapping F:CH is said to be firmly nonexpansive if (4.3)Fx-Fy2x-y,Fx-Fy, for all x,yC; see, for instance, Browder  and Goebel and Kirk . We also know that a firmly nonexpansive mapping F can be deduced from an equilibrium problem in a Hilbert space.

Recently, Kocourek et al.  introduced a more broad class of nonlinear mappings called generalized hybrid if there are α,β such that (4.4)αTx-Ty2+(1-α)x-Ty2βTx-y2+(1-β)x-y2, for all x,yC. Very recently, they defined a more broad class of mappings than the class of generalized hybrid mappings in a Hilbert space. A mapping S:CH is called super hybrid if there are α,β,γ such that (4.5)αSx-Sy2+(1-α+γ)x-Sy2(β+(β-α)γ)Sx-y2+(1-β-(β-α-1)γ)x-y2+(α-β)γx-Sx2+γy-Sy2, for all x,yC. We call such a mapping an (α,β,γ)-super hybrid mapping. We notice that an (α,β,0)-super hybrid mapping is (α,β)-generalized hybrid. So, the class of super hybrid mappings contains the class of generalized hybrid mappings. A super hybrid mapping is not quasi-nonexpansive generally. For more details, see . Before proving, we need the following lemmas.

Lemma 4.1 (see [<xref ref-type="bibr" rid="B13">20</xref>]).

Let C be a nonempty subset of a Hilbert space H and let α,β, and γ be real numbers with γ-1. Let S and T be mappings of C into H such that T=(1/(1+γ))S+(γ/(1+γ))I. Then, S is (α,β,γ)-super hybrid if and only if T is (α,β)-generalized hybrid. In this case, F(S)=F(T).

Lemma 4.2 (see [<xref ref-type="bibr" rid="B13">20</xref>]).

Let H be a Hilbert space and let C be a nonempty closed convex subset of H. Let T:CH be a generalized hybrid mapping. Then T is demiclosed on C.

Setting T:=(1/(1+γ))S+(γ/(1+γ))I in Theorem 3.3, where S is a super hybrid mapping and γ is a real number, we obtain the following result.

Theorem 4.3.

Let H be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C be a bifunction satisfying (A1), (A2), (A3), and (A4) and let Ψ be a δ-inverse strongly monotone mapping from C into H. Let α,β and γ be real numbers with γ-1 and let S:CH be an (α,β,γ)-super hybrid mapping such that GEP(F,Ψ)F(S) and let f be a Meir-Keeler contraction of C into itself. Let C1=C and {xn}C be a sequence generated by x1=xC and (4.6)F(zn,y)+Ψxn,y-zn+1λnzn-xn,y-zn0,yC,yn=αnxn+(1-αn)(11+γSzn+γ1+γzn),Cn+1={zCn:yn-zxn-z},xn+1=PCn+1f(xn),n, where PCn+1 is the metric projection of H onto Cn+1 and {αn}(0,1) and {λn}(0,2δ) are sequences such that (4.7)liminfnαn<1,0<aλnb<2δ, for some a,b. Then {xn} converges strongly to z0=PF(S)GEP(F,Ψ)f(z0).

Proof.

Put T=(1/(1+γ))S+(γ/(1+γ))I; we have from Lemma 4.1 that T is a generalized hybrid mapping and F(T)=F(S). Since F(T), we have that T is quasi-nonexpansive. Following the proof of Theorem 3.3 and applying Lemma 4.2, we have the following result.

Setting f(xn)=x,n in Theorem 4.3, we obtain the following result.

Corollary 4.4.

Let H be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C be a bifunction satisfying (A1), (A2), (A3), and (A4) and let Ψ be a δ-inverse strongly monotone mapping from C into H. Let α,β, and γ be real numbers with γ-1 and let S:CH be an (α,β,γ)-super hybrid mapping such that GEP(F,Ψ)F(S). Let C1=C and {xn}C be a sequence generated by x1=xC and (4.8)F(zn,y)+Ψxn,y-zn+1λnzn-xn,y-zn0,yC,yn=αnxn+(1-αn)(11+γSzn+γ1+γzn),Cn+1={zCn:yn-zxn-z},xn+1=PCn+1x,n, where PCn+1 is the metric projection of H onto Cn+1 and {αn}(0,1) and {λn}(0,2δ) are sequences such that (4.9)liminfnαn<1,0<aλnb<2δ, for some a,b. Then {xn} converges strongly to z0F(S)GEP(F,Ψ).

Setting Ψ0 in Theorem 4.3, we obtain the following result.

Corollary 4.5.

Let H be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C be a bifunction satisfying (A1), (A2), (A3), and (A4). Let α,β, and γ be real numbers with γ-1 and let S:CH be an (α,β,γ)-super hybrid mapping such that EP(F)F(S) and let f be a Meir-Keeler contraction of C into itself. Let C1=C and {xn}C be a sequence generated by x1=xC and (4.10)F(zn,y)+1λnzn-xn,y-zn0,yC,yn=αnxn+(1-αn)(11+γSzn+γ1+γzn),Cn+1={zCn:yn-zxn-z},xn+1=PCn+1f(xn),n, where PCn+1 is the metric projection of H onto Cn+1 and {αn}(0,1) and {λn}(0,) are sequences such that (4.11)liminfnαn<1,0<aλn, for some a. Then {xn} converges strongly to z0F(S)EP(F).

Setting Ψ0 and f(xn)=x, for all n in Theorem 4.3, we obtain the following result.

Corollary 4.6 (see [<xref ref-type="bibr" rid="B13">20</xref>], Theorem  5.2).

Let H be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C be a bifunction satisfying (A1), (A2), (A3), and (A4). Let α,β, and γ be real numbers with γ-1 and let S:CH be an (α,β,γ)-super hybrid mapping such that EP(F)F(S). Let C1=C and let {xn}C be a sequence generated by x1=xC and (4.12)F(zn,y)+1λnzn-xn,y-zn0,yC,yn=αnxn+(1-αn)(11+γSzn+γ1+γzn),Cn+1={zCn:yn-zxn-z},xn+1=PCn+1x,n, where PCn+1 is the metric projection of H onto Cn+1 and {αn}(0,1) and {λn}(0,) are sequences such that (4.13)liminfnαn<1,0λn, for some a. Then {xn} converges strongly to z0F(S)EP(F).

Setting T:=(1/(1+γ))S+(γ/(1+γ))I in Theorem 3.7, where S is an super hybrid mapping and γ is a real number, we obtain the following result.

Theorem 4.7.

Let H be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C be a bifunction satisfying (A1), (A2), (A3), and (A4) and let Ψ be a δ-inverse strongly monotone mapping from C into H. Let α,β, and γ be real numbers with γ-1 and let S:CH be an (α,β,γ)-super hybrid mapping such that GEP(F,Ψ)F(S) and let f be a Meir-Keeler contraction of C into itself. Let Q1=C and {xn}C be a sequence generated by x1=xC and (4.14)F(zn,y)+Ψxn,y-zn+1λnzn-xn,y-zn0,yC,yn=αnxn+(1-αn)(11+γSzn+γ1+γzn),Cn={zC:yn-zxn-z},Qn={zQn-1:f(xn-1)-xn,xn-z0},xn+1=PCnQnf(xn),n, where PCnQn is the metric projection of H onto CnQn and {αn}(0,1) and {λn}(0,2δ) satisfy (4.15)0αnb<1,0<cλnd<2δ, for some b,c,d. Then {xn} converges strongly to z0=PF(S)GEP(F,Ψ)f(z0).

Proof.

Put T=(1/(1+γ))S+(γ/(1+γ))I; we have from Lemma 4.1 that T is a generalized hybrid mapping and F(T)=F(S). Since F(T), we have that T is quasi-nonexpansive. Following the proof of Theorem 3.7 and applying Lemma 4.2, we obtain the following result.

Setting f(xn)=x,n in Theorem 4.7, we obtains the following result.

Corollary 4.8.

Let H be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C be a bifunction satisfying (A1), (A2), (A3), and (A4) and let Ψ be a δ-inverse strongly monotone mapping from C into H. Let α,β, and γ be real numbers with γ-1 and let S:CH be an (α,β,γ)-super hybrid mapping such that GEP(F,Ψ)F(S) and let f be a Meir-Keeler contraction of C into itself. Let {xn}C be a sequence generated by x1=xC and (4.16)F(zn,y)+Ψxn,y-zn+1λnzn-xn,y-zn0,yC,yn=αnxn+(1-αn)(11+γSzn+γ1+γzn),Cn={zC:yn-zxn-z},Qn={zC:x-xn,xn-z0},xn+1=PCnQnx,n, where PCnQn is the metric projection of H onto CnQn and {αn}(0,1) and {λn}(0,2δ) satisfy (4.17)0αnb<1,0<cλnd<2δ, for some b,c,d. Then {xn} converges strongly to z0F(S)GEP(F,Ψ).

Setting Ψ0 in Theorem 4.7, we obtain the following result.

Corollary 4.9.

Let H be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C be a bifunction satisfying (A1), (A2), (A3), and (A4). Let α,β, and γ be real numbers with γ-1 and let S:CH be an (α,β,γ)-super hybrid mapping such that EP(F)F(S) and let f be a Meir-Keeler contraction of C into itself. Let Q1=C and {xn}C be a sequence generated by x1=xC and (4.18)F(zn,y)+1λnzn-xn,y-zn0,yC,yn=αnxn+(1-αn)(11+γSzn+γ1+γzn),Cn={zC:yn-zxn-z},Qn={zQn-1:f(xn-1)-xn,xn-z0},xn+1=PCnQnf(xn),n, where PCnQn is the metric projection of H onto CnQn and {αn}(0,1) and {λn}(0,) satisfy (4.19)0αnb<1,0<cλn, for some b,c. Then {xn} converges strongly to z0F(S)EP(F).

Setting Ψ0 and f(xn)=x,n in Theorem 4.7, we obtain the following result.

Corollary 4.10 (see [<xref ref-type="bibr" rid="B13">20</xref>], Theorem  5.1).

Let H be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C be a bifunction satisfying (A1), (A2), (A3), and (A4). Let α,β, and γ be real numbers with γ-1 and let S:CH be an (α,β,γ)-super hybrid mapping such that EP(F)F(S). Let {xn}C be a sequence generated by x1=xC and (4.20)F(zn,y)+1λnzn-xn,y-zn0,yC,yn=αnxn+(1-αn)(11+γSzn+γ1+γzn),Cn={zC:yn-zxn-z},Qn={zC:xn-z,x-zn0},xn+1=PCnQnx,n, where PCnQn is the metric projection of H onto CnQn and {αn}(0,1) and {λn}(0,) satisfy (4.21)0αnb<1,0<cλn, for some b,c. Then {xn} converges strongly to z0F(S)EP(F).

Acknowledgment

The authors would like to thank Naresuan University for financial support.

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