# Viscosity Approximations by the Shrinking Projection Method of Quasi-Nonexpansive Mappings for Generalized Equilibrium Problems

Authors: Rabian Wangkeeree; Nimit Nimana
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

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.


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## Body

## 1. Introduction

LetH 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:C→H is said to be nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈C. A mapping T:C→H is said to be quasi-nonexpansive if ∥Tx-y∥≤∥x-y∥ for all x∈C and y∈F(T):={x∈C:Tx=x}. Recall that a mapping Ψ:C→H is said to be δ-inverse strongly monotone if there exists a positive real number δ such that
(1.1)〈Ψx-Ψy,x-y〉≥δ∥Ψx-Ψy∥2,∀x,y∈C.
If Ψ is an δ-inverse strongly monotone mapping of C into H, then it is obvious that Ψ is 1/δ-Lipschitz continuous.LetF:C×C→ℝ be a bifunction and Ψ:C→H be δ-inverse strongly monotone mapping. The generalized equilibrium problem (for short, GEP) for F and Ψ is to find z∈C such that
(1.2)F(z,y)+〈Ψz,y-z〉≥0,∀y∈C.
The problem (1.2) was studied by Moudafi [1]. The set of solutions for the problem (1.2) is denoted by GEP(F,Ψ), that is,
(1.3)GEP(F,Ψ)={z∈C:F(z,y)+〈Ψz,y-z〉≥0,∀y∈C}.
If Ψ≡0 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)={z∈C:F(z,y)≥0,∀y∈C}.
If F≡0 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)={z∈C:〈Ψz,y-z〉≥0,∀y∈C}.
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 [2] and Moudafi [3].In 2005, Combettes and Hirstoaga [4] 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 [5] 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 [6] 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 [7] introduced an iterative algorithm for equilibrium problems and fixed point problems. Wangkeeree [8] 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 [9] 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 [10] 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∈ℕ,
where the initial point x1 is taken in C arbitrarily and {αn} is a sequence in [0,1]. Wittmann [11] obtained the strong convergence results of the sequence {xn} defined by (1.6) to PFx1 under the following assumptions:  (C1)
limn→∞αn=0;(C2)
∑n=1∞‍αn=∞;(C3)
∑n=1∞‍|αn+1-αn|<∞, where PF(T) is the metric projection of H onto F(T). In 2000, Moudafi [12] introduced the viscosity approximation method for nonexpansive mappings (see [13] for further developments in both Hilbert and Banach spaces). Let f be a contraction on H. Starting with an arbitrary initial x1∈H, define a sequence {xn} recursively by
(1.7)xn+1=αnf(xn)+(1-αn)Txn,n≥1,
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,x∈C.
Suzuki [14] 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 [5] 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:C→H be a nonexpansive mapping. Starting with arbitrary initial x1∈H, define sequences {xn} and {un} recursively by
(1.9)F(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnf(xn)+(1-αn)Tun,∀n∈ℕ.
They proved that under certain appropriate conditions imposed on {αn} and {rn}, the sequences {xn} and {un} converge strongly to z∈F(T)∩EP(F), where z=PF(T)∩EP(F)f(z).On the other hand, in 2008, Takahashi et al. [15] has adapted Nakajo and Takahashi’s [16] 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: x0∈H, C1=C, u1=PC1x0 and
(1.10)yn=αnun+(1-αn)Tnun,Cn+1={z∈Cn:∥yn-z∥≤∥un-z∥},un+1=PCn+1x0,n∈ℕ,
where 0≤αn≤a<1 for all n∈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 [17] considered viscosity approximations by using the shrinking projection method established by Takahashi et al. [15] and the modified shrinking projection method proposed by Qin et al. [18], 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 byℕ the set of positive integers and by ℝ 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 x∈H by xn→x and xn⇀x, respectively. From [19], we know the following basic properties. For x,y∈H and λ∈ℝ we have
(2.1)∥λx+(1-λ)y∥2=λ∥x∥2+(1-λ)∥y∥2-λ(1-λ)∥x-y∥2.
We also know that for u,v,x,y∈H, we have
(2.2)2〈u-v,x-y〉=∥u-y∥2+∥v-x∥2-∥u-x∥2-∥v-y∥2.For every pointx∈H, there exists a unique nearest point of C, denoted by PCx, such that ∥x-PCx∥≤∥x-y∥ for all y∈C. PC is called the metric projection from H onto C. It is well known that z=PCx⇔〈x-z,z-y〉≥0, for all x∈H and z,y∈C. We also know that PC is firmly nonexpansive mapping from H onto C, that is,
(2.3)∥PCx-PCy∥2≤〈PCx-PCy,x-y〉,∀x,y∈H,
and so is nonexpansive mapping.For solving the generalized equilibrium problem, let us assume thatF satisfies the following conditions: (A1)
F(x,x)=0 for all x∈C;(A2)
F is monotone, that is, F(x,y)+F(y,x)≤0 for all x,y∈C;(A3)
for eachx,y,z∈C,limt↓0F(tz+(1-t)x,y)≤F(x,y);(A4)
for eachx∈C,y↦F(x,y) is convex and lower semicontinuous.In order to prove our main results, we also need the following lemmas.Lemma 2.1 (see [2]).
LetC be a nonempty closed convex subset of H and let F be a bifunction from C×C into ℝ satisfying (A1), (A2), (A3), and (A4). Then, for any r>0 and x∈H, there exists a unique z∈C such that
(2.4)F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C.Lemma 2.2 (see [4]).
LetC be a nonempty closed convex subset of H and let F be a bifunction from C×C into ℝ satisfying (A1), (A2), (A3), and (A4). Then, for any r>0 and x∈H, define a mapping Trx:H→C as follows:
(2.5)Trx={z∈C:F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C}∀x∈H,r∈ℝ.
Then the following hold:(i)
Tr is single-valued;(ii)
Tr is firmly nonexpansive, that is,
(2.6)∥Trx-Try∥2≤〈Trx-Try,x-y〉,∀x,y∈H;(iii)
F(Tr)=EP(F);(iv)
EP(F) is closed and convex.Remark 2.3 (see [20]).
Using (ii) in Lemma2.2 and (2.2), we have
(2.7)2∥Trx-Try∥2≤2〈Trx-Try,x-y〉=∥Trx-y∥2+∥Try-x∥2-∥Trx-x∥2-∥Try-y∥2.
So, for y∈F(Tr) and x∈H, we have
(2.8)∥Trx-u∥2+∥Trx-x∥2≤∥x-u∥2.Remark 2.4.
For anyx∈H and r>0, by Lemma 2.1, there exists z∈C such that
(2.9)F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C.
Replacing x with x-rΨx∈H in (2.9), we have
(2.10)F(z,y)+〈Ψx,y-z〉+1r〈y-z,z-x〉≥0,∀y∈C,
where Ψ:C→H 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 xn→x and that xn∈Cn for all n∈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 yi⇀y and that yi∈Cni for all i∈ℕ.IfC0 satisfies
(2.11)C0=s-LinCn=w-LsnCn,
it is said that {Cn} converges to C0 in the sense of Mosco [21] and we write C0=M-limn→∞Cn. It is easy to show that if {Cn} is nonincreasing with respect to inclusion, then {Cn} converges to ⋂n=1∞‍Cn in the sense of Mosco. For more details, see [21]. Tsukada [22] proved the following theorem for the metric projection.Theorem 2.5 (see Tsukada [22]).
LetH be a Hilbert space. Let {Cn} be a sequence of nonempty closed convex subsets of H. If C0=M-limn→∞Cn exists and is nonempty, then for each x∈H, {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 mappingf 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,y∈C. It is well known that f has a unique fixed point [23]. Meir-Keeler [24] defined the following mapping called Meir-Keeler contraction. Let (X,d) be a complete metric space. A mapping f:X→X is called a Meir-Keeler contraction if for all ɛ>0, there exists δ>0 such that ɛ≤d(x,y)<ɛ+δ implies d(f(x),f(y))<ɛ for all x,y∈X. It is well known that Meir-Keeler contraction is a generalization of contraction and the following result is proved in [24].Theorem 2.6 (see Meir-Keeler [24]).
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 [14].Theorem 2.7 (see Suzuki [14]).
Letf 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-y∥≥ɛ implies ∥f(x)-f(y)∥≤r∥x-y∥ for all x,y∈C.Lemma 2.8 (see Suzuki [14]).
LetC 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. (i)
T∘f is a Meir-Keeler contraction on C.(ii)
For eachα∈(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. [15], 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.
LetC be a nonempty closed convex subset of a Hilbert space H and δ>0 and let Ψ:C→H be δ-inverse strongly monotone. If 0<λ≤2δ, then I-λΨ is a nonexpansive mapping.Proof.
Forx,y∈C, we can calculate
(3.1)∥(I-λΨ)x-(I-λΨ)y∥2=∥x-y-λ(Ψx-Ψy)∥2=∥x-y∥2-2λ〈x-y,Ψx-Ψy〉+λ2∥Ψx-Ψy∥2≤∥x-y∥2-2λδ∥Ψx-Ψy∥2+λ2∥Ψx-Ψy∥2=∥x-y∥2+λ(λ-2δ)∥Ψx-Ψy∥2≤∥x-y∥2.
Therefore I-λΨ is nonexpansive. This completes the proof.Lemma 3.2.
LetC 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 thatF(T) is closed. Let {zn} be any sequence in F(T) with zn→z. We claim that z∈F(T). Since C is closed, we have z∈C. We observe that
(3.2)∥z-Tz∥≤∥z-zn∥+∥zn-Tz∥≤∥z-zn∥+∥z-zn∥=2∥z-zn∥.
Since zn→z, we obtain that ∥z-Tz∥≤0 and hence z=Tz. This show that z∈F(T).
Next, we show thatF(T) is convex. Let x,y∈F(T) and α∈[0,1]. We claim that αx+(1-α)y∈F(T). Putting z=αx+(1-α)y, we have
(3.3)∥z-Tz∥2=∥αx+(1-α)y-Tz∥2=∥α(x-Tz)+(1-α)(y-Tz)∥2=α∥x-Tz∥2+(1-α)∥y-Tz∥2-α(1-α)∥x-y∥2≤α∥x-z∥2+(1-α)∥y-z∥2-α(1-α)∥x-y∥2=α∥(1-α)(x-y)∥2+(1-α)∥α(y-x)∥2-α(1-α)∥x-y∥2={α(1-α)2+α2(1-α)-α(1-α)}∥x-y∥2=α(1-α)(1-α+α-1)∥x-y∥2=0.
Hence F(T) is convex. This completes the proof.Theorem 3.3.
LetH be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C→ℝ 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:C→C be a quasi-nonexpansive mapping which is demiclosed on C, that is, if {wk}⊂C,wk⇀w and (I-T)wk→0, then w∈F(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=x∈C,F(zn,y)+〈Ψxn,y-zn〉+1λn〈zn-xn,y-zn〉≥0,∀y∈C,yn=αnxn+(1-αn)Tzn,Cn+1={z∈Cn:∥yn-z∥≤∥xn-z∥},xn+1=PCn+1f(xn),∀n∈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)liminfn→∞αn<1,0<a≤λn≤b<2δ,
for some a,b∈ℝ. Then, {xn} converges strongly to z0∈Ω, which satisfies z0=PΩf(z0).Proof.
SinceΩ 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 z0∈C such that z0=PΩf(z0). Next, we observe that zn=Tλn(xn-λnΨxn) for each n∈ℕ and take z∈F(T)∩GEP(F,Ψ). From z=Tλn(z-λnΨz) and Lemma 2.2, we have that for any n∈ℕ,
(3.6)∥zn-z∥2=∥Tλn(xn-λnΨxn)-Tλn(z-λnΨz)∥2≤∥(xn-λnΨxn)-(z-λnΨz)∥2≤∥xn-z∥2.
Next, we divide the proof into several steps.
Step
1. Cn is closed convex and {xn} is well defined for every n∈ℕ.
It is obvious from the assumption thatC1:=C is closed convex and Ω⊂C1. For any k∈ℕ, suppose that Ck is closed and convex, and Ω⊂Ck. Note that for all z∈Ck,
(3.7)∥yk-z∥2≤∥xk-z∥2⟺∥yk-z∥2-∥xk-z∥2≤0⟺∥yk∥2+∥z∥2-2〈yk,z〉-∥xk∥2-∥z∥2+2〈xk,z〉≤0⟺∥yk∥2-2〈yk-xk,z〉-∥xk∥2≤0.
It is easy to see that Ck+1 is closed. Next, we prove that Ck+1 is convex. For any u,v∈Ck+1 and α∈[0,1], we claim that z:=αu+(1-α)v∈Ck+1. Since u∈Ck+1, we have ∥yk-u∥≤∥xk-u∥ and so ∥yk-u∥2≤∥xk-u∥2, that is, ∥yk∥2-2〈yk-xk,u〉-∥xk∥2≤0. Similarly, v∈Ck+1, we get ∥yk∥2-2〈yk-xk,v〉-∥xk∥2≤0.
Thus,(3.8)α∥yk∥2-2〈yk-xk,αu〉-α∥xk∥2≤0,(1-α)∥yk∥2-2〈yk-xk,(1-α)v〉-(1-α)∥xk∥2≤0.
Combining the above inequalities, we obtain
(3.9)∥yk∥2-2〈yk-xk,αu+(1-α)v〉-∥xk∥2≤0.
Therefore ∥yk-z∥≤∥xk-z∥. This shows that z∈Ck+1 and hence Ck+1 is convex. Therefore Cn is closed and convex for all n∈ℕ.
Next, we show thatΩ⊂Cn, for all n∈ℕ. For any k∈ℕ, suppose that v∈Ω⊂Ck. Since T is quasi-nonexpansive and from (3.6), we have
(3.10)∥yk-v∥2=∥αkxk+(1-αk)Tzk-v∥2=∥αk(xk-v)+(1-αk)(Tzk-v)∥2≤αk∥xk-v∥2+(1-αk)∥Tzk-v∥2≤αk∥xk-v∥2+(1-αk)∥zk-v∥2≤αk∥xk-v∥2+(1-αk)∥xk-v∥2=∥xk-v∥2.
So, we have v∈Ck+1. By principle of mathematical induction, we can conclude that Cn is closed and convex, and Ω⊂Cn, for all n∈ℕ. Hence, we have
(3.11)∅≠Ω⊂Cn+1⊂Cn,
for all n∈ℕ. Therefore {xn} is well defined.
Step
2. limn→∞∥xn-u∥=0 for some u∈⋂n=1∞‍Cn and 〈f(u)-u,u-y〉≥0 for all y∈Ω.
Since⋂n=1∞‍Cn is closed convex, we also have that P⋂n=1∞‍Cn is well defined and so P⋂n=1∞‍Cnf is a Meir-Keeler contraction on C. By Theorem 2.6, there exists a unique fixed point u∈⋂n=1∞‍Cn of P⋂n=1∞‍Cnf. Since Cn is a nonincreasing sequence of nonempty closed convex subsets of H with respect to inclusion, it follows that
(3.12)∅≠Ω⊂⋂n=1∞‍Cn=M-limn→∞Cn.
Setting un:=PCnf(u) and applying Theorem 2.5, we can conclude that
(3.13)limn→∞un=P⋂n=1∞Cnf(u)=u.
Next, we will prove that limn→∞∥xn-u∥=0. Assume to contrary that limsupn→∞∥xn-u∥≠0, there exists ɛ>0 and a subsequence {∥xnj-u∥} of {∥xn-u∥} such that
(3.14)∥xnj-u∥≥ɛ,∀j∈ℕ,
which gives that
(3.15)limsupj→∞∥xnj-u∥≥ɛ>0.
We choose a positive number ɛ′>0 such that
(3.16)limsupj→∞∥xnj-u∥>ɛ′>0.
For such ɛ′, by the definition of Meir-Keeler contraction, there exists δɛ'>0 with
(3.17)ɛ′+δɛ′<limsupj→∞∥xnj-u∥,
such that
(3.18)∥x-y∥<ɛ′+δɛ′implies∥f(x)-f(y)∥<ɛ′,
for all x,y∈C. Again for such ɛ', by Theorem 2.7, there exists rɛ'∈(0,1) such that
(3.19)∥x-y∥≥ɛ′+δɛ′implies∥f(x)-f(y)∥<rɛ′∥x-y∥.
Since un→u, there exists n0∈ℕ such that
(3.20)∥un-u∥<δɛ′,∀n≥n0.
By the idea of Suzuki [14] and Kimura and Nakajo [17], we consider the following two cases.
Case
I. Assume that there exists n1≥n0 such that
(3.21)∥xn1-u∥<ɛ′+δɛ′.
Thus, we get
(3.22)∥xn1+1-u∥≤∥xn1+1-un1+1∥+∥un1+1-u∥=∥PCn1+1f(xn1)-PCn1+1f(u)∥+∥un1+1-u∥≤∥f(xn1)-f(u)∥+∥un1+1-u∥<ɛ′+δɛ′.
By induction on {n}, we can obtain that
(3.23)∥xn-u∥<ɛ′+δɛ′,
for all n≥n0. In particular, for all j≥n0, we have nj≥j≥n0 and
(3.24)∥xnj-u∥<ɛ′+δɛ′.
This implies that
(3.25)limsupj→∞∥xnj-u∥≤ɛ′+δɛ′<limsupj→∞∥xnj-u∥,
which is a contradiction. Therefore, we conclude that ∥xn-u∥→0 as n→∞.
Case
I
I. Assume that
(3.26)∥xn-u∥≥ɛ′+δɛ′,∀n≥n0.
By (3.19), we have
(3.27)∥f(xn)-f(u)∥<rɛ′∥xn-u∥,∀n≥n0.
Thus, we have
(3.28)∥xn+1-un+1∥=∥PCn+1f(xn)-PCn+1f(u)∥≤∥f(xn)-f(u)∥≤rɛ′∥xn-u∥≤rɛ′(∥xn-un∥+∥un-u∥),
for every n≥n0. In particular, we have
(3.29)∥xnj+1-unj+1∥≤rɛ′(∥xnj-unj∥+∥unj-u∥),
for every j≥n0(nj≥j≥n0). Let us consider
(3.30)limsupn→∞∥xnj-unj∥=limsupj→∞∥xnj+1-unj+1∥≤rɛ′limsupj→∞(∥xnj-unj∥+∥unj-u∥)≤rɛ′limsupj→∞∥xnj-unj∥+rɛ′limsupj→∞∥unj-u∥=rɛ′limsupj→∞∥xnj-unj∥<limsupj→∞∥xnj-unj∥,
which gives a contradiction. Hence, we obtain that
(3.31)limn→∞∥xn-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-y〉≥0,∀y∈Cn+1.
Since Ω⊂Cn+1, we get
(3.33)〈f(xn)-xn+1,xn+1-y〉≥0,∀n∈ℕ,y∈Ω.
We have from xn→u that
(3.34)〈f(u)-u,u-y〉≥0,∀y∈Ω.
Step
3. There exists a subsequence {∥xni-zni∥} of {∥xn-zn∥} such that ∥xni-zni∥→0 as i→∞.
We have from (3.13) and (3.31) that
(3.35)∥xn-xn+1∥≤∥xn-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+1∈Cn+1, we have that
(3.36)∥yn-xn+1∥≤∥xn-xn+1∥,
and so ∥yn-xn+1∥→0. We also have
(3.37)∥yn-xn∥≤∥yn-xn+1∥+∥xn+1-xn∥→0.
From liminfn→∞αn<1, there exists a subsequence {αni} of {αn} and α0 with 0≤α0<1 such that αni→α0. Since ∥xn-yn∥=∥xn-αnxn-(1-α)Tzn∥=(1-αn)∥xn-Tzn∥, we have
(3.38)∥Tzni-xni∥→0asi→∞.
Using Lemma 2.2 (ii) and (3.6), we have
(3.39)∥zn-z∥2=∥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-z∥2-∥(xn-λnΨxn)-(z-λnΨz)+(z-zn)∥2)≤12(∥xn-z∥2+∥zn-z∥2-∥(xn-zn)-λn(Ψxn-Ψz)∥2)=12(∥xn-z∥2+∥zn-z∥2-∥xn-zn∥2+2λn〈xn-zn,Ψxn-Ψz〉-λn2∥Ψxn-Ψz∥2).
So, we have
(3.40)∥zn-z∥2≤∥xn-z∥2-∥xn-zn∥2+2λn〈xn-zn,Ψxn-Ψz〉-λn2∥Ψxn-Ψz∥2.
Let us consider
(3.41)∥yn-z∥2=∥αn(xn-z)+(1-αn)(Tzn-z)∥2≤αn∥xn-z∥2+(1-αn)∥Tzn-z∥2≤αn∥xn-z∥2+(1-αn)∥zn-z∥2=αn∥xn-z∥2+(1-αn)∥Tλn(I-λnΨ)xn-Tλn(I-λnΨ)z∥2≤αn∥xn-z∥2+(1-αn)∥(I-λnΨ)xn-(I-λnΨ)z∥2=αn∥xn-z∥2+(1-αn)∥(xn-z)-λn(Ψxn-Ψz)∥2=αn∥xn-z∥2+(1-αn)∥xn-z∥2+(1-αn)λn2∥Ψxn-Ψz∥2-2(1-αn)λn〈xn-z,Ψxn-Ψz〉≤∥xn-z∥2+(1-αn)λn2∥Ψxn-Ψz∥2-2(1-αn)λnδ∥Ψxn-Ψz∥2=∥xn-z∥2+(1-αn)(λn-2δ)λn∥Ψxn-Ψz∥2≤∥xn-z∥2+(1-αn)(b-2δ)b∥Ψxn-Ψz∥2.
In particular, we have
(3.42)(1-αni)(2δ-b)b∥Ψxni-Ψz∥2≤∥xni-z∥2-∥yni-z∥2≤∥xni-yni∥2+2∥xni-yni∥∥yni-z∥.
Since αni→α0 with α0<1 and ∥xni-yni∥→0, we obtain that
(3.43)∥Ψxni-Ψz∥→0.
Using (3.40), we have
(3.44)∥yn-z∥2≤αn∥xn-z∥2+(1-αn)∥zn-z∥2≤αn∥xn-z∥2+(1-αn)×(∥xn-z∥2-∥xn-zn∥2+2λn〈xn-zn,Ψxn-Ψz〉-λn2∥Ψxn-Ψz∥2)≤αn∥xn-z∥2+(1-αn)(∥xn-z∥2-∥xn-zn∥2+2λn∥xn-zn∥∥Ψxn-Ψz∥)≤∥xn-z∥2-(1-αn)∥xn-zn∥2+2(1-αn)λn(∥xn∥+∥zn∥)∥Ψxn-Ψz∥≤∥xn-z∥2-(1-αn)∥xn-zn∥2+2(1-αn)bM∥Ψxn-Ψz∥,
where M:=sup{∥xn∥+∥yn∥:n∈ℕ}.
So, we have(3.45)(1-αni)∥xni-zni∥2≤∥xni-z∥2-∥yni-z∥2+2(1-αni)bM∥Ψxni-Ψz∥≤∥xni-yni∥2+2∥xni-yni∥∥yni-z∥+2(1-αni)bM∥Ψxni-Ψz∥.
We have from αni→α0, (3.37), and (3.43) that
(3.46)∥xni-zni∥→0.
Step
4. Finally, we prove that u∈Ω:=F(T)∩GEP(F,Ψ).
Sinceyn=αnxn+(1-αn)Tzn, we have yn-Tzn=αn(xn-Tzn). So, from (3.38) we have
(3.47)∥yni-Tzni∥=αni∥xni-Tzni∥→0.
Since ∥zni-Tzni∥≤∥zni-xni∥+∥xni-yni∥+∥yni-Tzni∥, from (3.37), (3.46), and (3.47) we have
(3.48)∥zni-Tzni∥→0.
Since xni→u, we have zni→u. So, from (3.48) and the demiclosed property of T, we have
(3.49)u∈F(T).
We next show that u∈GEP(F,Ψ). Since zn=Tλn(xn-λnΨxn), for any y∈C we have
(3.50)F(zn,y)+〈Ψxn,y-zn〉+1λn〈y-zn,zn-xn〉≥0.
From (A2), we have
(3.51)-F(y,zn)+〈Ψxn,y-zn〉+1λn〈y-zn,zn-xn〉≥0
and so
(3.52)〈Ψxn,y-zn〉+1λn〈y-zn,zn-xn〉≥F(y,zn).
Replacing n by ni, we have
(3.53)〈Ψxni,y-zni〉+〈y-zni,zni-xniλni〉≥F(y,zni).
Note that Ψ is 1/δ-Lipschitz continuous, and from (3.46), we have
(3.54)∥Ψzni-Ψxni∥→0.
For t∈(0,1] and y∈C, 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*-Ψzni〉≥0, 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*-zni∥∥Ψzni-Ψxni∥≥-∥zt*-zni∥∥zni-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)t〈y-u,Ψzt*〉,
and hence
(3.60)0≤F(zt*,y)+(1-t)〈y-u,Ψzt*〉.
Letting t↓0 and from (A3), we have that for each y∈C,
(3.61)0≤limt↓0(F(zt*,y)+(1-t)〈y-u,Ψzt*〉)=limt↓0(F(ty+(1-t)u,y)+(1-t)〈y-u,tΨy+(1-t)Ψu〉)≤F(u,y)+〈y-u,Ψu〉.
This implies that u∈GEP(F,Ψ). So, we have u∈F(T)∩GEP(F,Ψ). We obtain from (3.34) that u=z0 and hence, {xn} converges strongly to z0. This completes the proof.By Theorem3.3, we can obtain some new and interesting strong convergence theorems. Now we give some examples as follows.Settingf(xn)=x,∀n∈ℕ in Theorem 3.3, we obtain the following result.Corollary 3.4.
LetH be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C→ℝ 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:C→C 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=x∈C and
(3.62)F(zn,y)+〈Ψxn,y-zn〉+1λn〈zn-xn,y-zn〉≥0,∀y∈C,yn=αnxn+(1-αn)Tzn,Cn+1={z∈Cn:∥yn-z∥≤∥xn-z∥},xn+1=PCn+1x,∀n∈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)liminfn→∞αn<1,0<a≤λn≤b<2δ,
for some a∈ℝ. Then {xn} converges strongly to z0=PΩz0.SettingΨ≡0 in Theorem 3.3, we obtain the following result.Corollary 3.5.
LetH be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4). Let T:C→C 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=x∈C and
(3.64)F(zn,y)+1λn〈zn-xn,y-zn〉≥0,∀y∈C,yn=αnxn+(1-αn)Tzn,Cn+1={z∈Cn:∥yn-z∥≤∥xn-z∥},xn+1=PCn+1f(xn),∀n∈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)liminfn→∞αn<1,0<a≤λn,
for some a∈ℝ. Then {xn} converges strongly to z0∈F(T)∩EP(F).SettingΨ≡0 and f(xn)=x for all n∈ℕ in Theorem 3.3, we obtain the following result.Corollary 3.6.
LetH be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4). Let T:C→C 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=x∈C and
(3.66)F(zn,y)+1λn〈zn-xn,y-zn〉≥0,∀y∈C,yn=αnxn+(1-αn)Tzn,Cn+1={z∈Cn:∥yn-z∥≤∥xn-z∥},xn+1=PCn+1x,∀n∈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)liminfn→∞αn<1,0<a≤λn,
for some a∈ℝ. Then {xn} converges strongly to z0∈F(T)∩EP(F).Next, using the CQ hybrid method introduced by Nakajo and Takahashi [16], we prove a strong convergence theorem of a quasi-nonexpansive mapping for solving the generalized equilibrium problem.Theorem 3.7.
LetH be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C→ℝ 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:C→C 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=x∈C and
(3.68)F(zn,y)+〈Ψxn,y-zn〉+1λn〈zn-xn,y-zn〉≥0,∀y∈C,yn=αnxn+(1-αn)Tzn,Cn={z∈C:∥yn-z∥≤∥xn-z∥},Qn={z∈Qn-1:〈f(xn-1)-xn,xn-z〉≥0},xn+1=PCn∩Qnf(xn),∀n∈ℕ,
where PCn∩Qn is the metric projection of H onto Cn∩Qn and {αn}⊂(0,1) and {λn}⊂(0,2δ) satisfy
(3.69)0≤αn≤b<1,0<c≤λn≤d<2δ,
for some b,c,d∈ℝ. Then, {xn} converges strongly to z0∈Ω, which satisfies z0=PΩf(z0).Proof.
As in the proof of Theorem3.3, 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 z0∈C 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∈ℕ. For any n∈ℕ, let {zk} be a sequence in Qn such that zk→z. For each k∈ℕ, we observe that
(3.70)0≤〈xn-zk,f(xn-1)-xn〉=12(∥f(xn-1)-zk∥2-∥xn-zk∥2-∥f(xn-1)-xn∥2).
Taking k→∞, we get 〈xn-z,f(xn-1)-xn〉≥0 and then z∈Qn. Therefore Qn is closed.
Next, we will show thatQn is convex. For any u,v∈Qn, and α∈[0,1], put z=αu+(1-α)v. We claim that z∈Qn. Since u∈Qn, we have 〈αxn-αu,f(xn-1)-xn〉≥0. Similarly, since v∈Qn, we have 〈(1-α)xn-(1-α)v,f(xn-1)-xn〉≥0. Thus,
(3.71)0≤〈αxn-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 z∈Qn, and therefore we have that Qn is convex. We obtain from both Cn and Qn which are closed convex sets for every n∈ℕ that Cn∩Qn is closed and convex for every n∈ℕ.
Next, we will show thatCn∩Qn is nonempty. Let z∈F(T)∩GEP(F,Ψ). We will show that z∈Cn for any n∈ℕ. We notice that zn=Tλn(xn-λnΨxn) for each n∈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-z∥≤∥xn-z∥,foranyn∈ℕ.
Since T is quasi-nonexpansive with the fixed point z and from (3.72), we have
(3.73)∥yn-z∥2≤∥xn-z∥2.
So, we have z∈Cn. Therefore F(T)∩GEP(F,Ψ)⊂Cn, for all n∈ℕ.
Next, we will show that(3.74)F(T)∩GEP(F,Ψ)⊂Cn∩Qn,∀n∈ℕ.
It is obvious that F(T)∩GEP(F,Ψ)⊂C=Q1. Hence
(3.75)F(T)∩GEP(F,Ψ)⊂C1∩Q1.
For any k∈ℕ, suppose that
(3.76)F(T)∩GEP(F,Ψ)⊂Ck∩Qk.
Since xk+1=PCk∩Qkf(xk), we have
(3.77)〈f(xk)-xk+1,xk+1-z〉≥0,∀z∈Ck∩Qk.
In particular, for any z∈F(T)∩GEP(F,Ψ), we obtain that
(3.78)〈f(xk)-xk+1,xk+1-z〉≥0.
This shows that z∈Qk+1. Hence F(T)∩GEP(F,Ψ)⊂Qk+1. Therefore, we conclude that
(3.79)F(T)∩GEP(F,Ψ)⊂Ck+1∩Qk+1.
By principle of mathematical induction, we can conclude that
(3.80)F(T)∩GEP(F,Ψ)⊂Cn∩Qn,∀n∈ℕ.
Hence {xn} is well defined. Since P⋂n=1∞‍Qnf is a Meir-Keeler contraction on C, there exists a unique element u∈C such that u=P⋂n=1∞‍Qnf(u). For each n, let un=PQnf(u). Since F(T)∩GEP(F,Ψ)⊂Qn+1⊂Qn, we have from Theorem 2.5 that un→u. Notice that xn=PQnf(xn-1). Thus, as in the proof of Theorem 3.3, we get xn→u and hence {xn} is bounded. Moreover,
(3.81)limn→∞∥xn-xn+1∥=0,limn→∞∥yn-xn∥=0.
As the proof of Theorem 3.3, we have that
(3.82)∥yn-z∥2≤∥xn-z∥2+(1-αn)(λn-2δ)λn∥Ψxn-Ψz∥2.
Thus,
(3.83)∥yn-z∥2≤∥xn-z∥2+(1-a)(d-2δ)d∥Ψxn-Ψz∥2
and so
(3.84)(1-a)(2δ-d)d∥Ψxn-Ψz∥2≤∥xn-z∥2-∥yn-z∥2≤∥xn-yn∥2+2∥xn-yn∥∥yn-z∥.
Since (1-a)(2δ-d)d>0 and ∥xn-yn∥→0, we obtain that
(3.85)∥Ψxn-Ψz∥→0.
Using (3.40) in Theorem 3.3, we have
(3.86)∥yn-z∥2≤∥xn-z∥2-(1-b)∥xn-zn∥2+2(1-a)dM∥Ψxn-Ψz∥,
where M:=sup{∥xn∥+∥yn∥}. So, we have
(3.87)(1-b)∥xn-zn∥2≤∥xn-z∥2-∥yn-z∥2+2(1-a)dM∥Ψxn-Ψz∥≤∥xn-yn∥2+2∥xn-yn∥∥yn-z∥+2(1-a)dM∥Ψxn-Ψz∥.
We have from 1-b>0, ∥xn-yn∥→0 and (3.85) that
(3.88)∥xn-zn∥→0,
which implies that
(3.89)zn→u.
Notice that
(3.90)∥xn-yn∥=∥xn-αnxn-(1-α)Tzn∥=(1-αn)∥xn-Tzn∥,
and from limsupn→∞αn≤b<1, we have
(3.91)∥Tzn-xn∥→0.
Since yn=αnxn+(1-αn)Tzn, we have yn-Tzn=αn(xn-Tzn). So, from 0<a≤αn≤b<1 and (3.91), we obtain
(3.92)∥yn-Tzn∥→0
and hence
(3.93)∥zn-Tzn∥→0.
From (3.89), (3.93), and the demiclosed property of T, we have u∈F(T). As in the proof of Theorem 3.3 we have that u∈F(T)∩GEP(F,Ψ). Since F(T)∩GEP(F,Ψ)⊂Qn+1, we get
(3.94)〈f(xn)-xn+1,xn+1-y〉≥0,
for all n∈ℕ and y∈F(T)∩GEP(F,Ψ). We have from xn→u that
(3.95)〈f(u)-u,u-y〉≥0,
for all y∈F(T)∩GEP(F,Ψ), which implies that u=PΩf(u). It follows that u=z0, since z0∈F(T)∩GEP(F,Ψ) of PΩf is unique. Hence, {xn} converges strongly to z0. This completes the proof.Settingf(xn)=x, for all n∈ℕ in Theorem 3.7, we obtain the following result.Corollary 3.8.
LetH be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C→ℝ 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:C→C 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=x∈C and
(3.96)F(zn,y)+〈Ψxn,y-zn〉+1λn〈zn-xn,y-zn〉≥0,∀y∈C,yn=αnxn+(1-αn)Tzn,Cn={z∈C:∥yn-z∥≤∥xn-z∥},Qn={z∈Qn-1:〈x-xn,xn-z〉≥0},xn+1=PCn∩Qnx,∀n∈ℕ,
where PCn∩Qn is the metric projection of H onto Cn∩Qn and {αn}⊂(0,1) and {λn}⊂(0,2δ) satisfy
(3.97)0≤αn≤b<1,0<c≤λn≤d<2δ,
for some a,b,c,d∈ℝ. Then {xn} converges strongly to z0=PΩf(z0).SettingΨ≡0 in Theorem 3.7, we obtain the following result.Corollary 3.9.
LetH be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4) and let T:C→C 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=x∈C and
(3.98)F(zn,y)+1λn〈zn-xn,y-zn〉≥0,∀y∈C,yn=αnxn+(1-αn)Tzn,Cn={z∈C:∥yn-z∥≤∥xn-z∥},Qn={z∈Qn-1:〈f(xn-1)-xn,xn-z〉≥0},xn+1=PCn∩Qnf(xn),∀n∈ℕ,
where PCn∩Qn is the metric projection of H onto Cn∩Qn and {αn}⊂(0,1) and {λn}⊂(0,∞) satisfy
(3.99)0≤αn≤b<1,0<c≤λn,
for some a,b,c∈ℝ. Then {xn} converges strongly to z0∈F(T)∩EP(F).SettingΨ≡0 and f(xn)=x,∀n∈ℕ in Theorem 3.7, we obtain the following result.Corollary 3.10.
LetH be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4) and let T:C→C 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=x∈C and
(3.100)F(zn,y)+1λn〈zn-xn,y-zn〉≥0,∀y∈C,yn=αnxn+(1-αn)Tzn,Cn={z∈C:∥yn-z∥≤∥xn-z∥},Qn={z∈Qn-1:〈x-xn,xn-z〉≥0},xn+1=PCn∩Qnx,∀n∈ℕ,
where PCn∩Qn is the metric projection of H onto Cn∩Qn and {αn}⊂(0,1) and {λn}⊂(0,∞) satisfy
(3.101)0<a≤αn≤b<1,0<c≤λn,
for some a,b,c∈ℝ. Then {xn} converges strongly to z0∈F(T)∩EP(F).


## 4. Applications

In this section, we present some convergence theorems deduced from the results in the previous section. Recall that a mappingT:C→H is said to be nonspreading if
(4.1)2∥Tx-Ty∥2≤∥Tx-y∥2+∥Ty-x∥2
for all x,y∈C. Further, a mapping T:C→H is said to be hybrid if
(4.2)3∥Tx-Ty∥2≤∥x-y∥2+∥Tx-y∥2+∥Ty-x∥2,
for all x,y∈C. These mappings are deduced from a firmly nonexpansive mapping in a Hilbert space.A mappingF:C→H is said to be firmly nonexpansive if
(4.3)∥Fx-Fy∥2≤〈x-y,Fx-Fy〉,
for all x,y∈C; see, for instance, Browder [25] and Goebel and Kirk [26]. We also know that a firmly nonexpansive mapping F can be deduced from an equilibrium problem in a Hilbert space.Recently, Kocourek et al. [27] introduced a more broad class of nonlinear mappings called generalized hybrid if there are α,β∈ℝ such that
(4.4)α∥Tx-Ty∥2+(1-α)∥x-Ty∥2≤β∥Tx-y∥2+(1-β)∥x-y∥2,
for all x,y∈C. Very recently, they defined a more broad class of mappings than the class of generalized hybrid mappings in a Hilbert space. A mapping S:C→H is called super hybrid if there are α,β,γ∈ℝ such that
(4.5)α∥Sx-Sy∥2+(1-α+γ)∥x-Sy∥2≤(β+(β-α)γ)∥Sx-y∥2+(1-β-(β-α-1)γ)∥x-y∥2+(α-β)γ∥x-Sx∥2+γ∥y-Sy∥2,
for all x,y∈C. 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 [20]. Before proving, we need the following lemmas.Lemma 4.1 (see [20]).
LetC 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 [20]).
LetH be a Hilbert space and let C be a nonempty closed convex subset of H. Let T:C→H be a generalized hybrid mapping. Then T is demiclosed on C.SettingT:=(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.
LetH be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C→ℝ 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:C→H 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=x∈C and
(4.6)F(zn,y)+〈Ψxn,y-zn〉+1λn〈zn-xn,y-zn〉≥0,∀y∈C,yn=αnxn+(1-αn)(11+γSzn+γ1+γzn),Cn+1={z∈Cn:∥yn-z∥≤∥xn-z∥},xn+1=PCn+1f(xn),∀n∈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)liminfn→∞αn<1,0<a≤λn≤b<2δ,
for some a,b∈ℝ. Then {xn} converges strongly to z0=PF(S)∩GEP(F,Ψ)f(z0).Proof.
PutT=(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.Settingf(xn)=x,∀n∈ℕ in Theorem 4.3, we obtain the following result.Corollary 4.4.
LetH be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C→ℝ 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:C→H 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=x∈C and
(4.8)F(zn,y)+〈Ψxn,y-zn〉+1λn〈zn-xn,y-zn〉≥0,∀y∈C,yn=αnxn+(1-αn)(11+γSzn+γ1+γzn),Cn+1={z∈Cn:∥yn-z∥≤∥xn-z∥},xn+1=PCn+1x,∀n∈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)liminfn→∞αn<1,0<a≤λn≤b<2δ,
for some a,b∈ℝ. Then {xn} converges strongly to z0∈F(S)∩GEP(F,Ψ).SettingΨ≡0 in Theorem 4.3, we obtain the following result.Corollary 4.5.
LetH be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4). Let α,β, and γ be real numbers with γ≠-1 and let S:C→H 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=x∈C and
(4.10)F(zn,y)+1λn〈zn-xn,y-zn〉≥0,∀y∈C,yn=αnxn+(1-αn)(11+γSzn+γ1+γzn),Cn+1={z∈Cn:∥yn-z∥≤∥xn-z∥},xn+1=PCn+1f(xn),∀n∈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)liminfn→∞αn<1,0<a≤λn,
for some a∈ℝ. Then {xn} converges strongly to z0∈F(S)∩EP(F).SettingΨ≡0 and f(xn)=x, for all n∈ℕ in Theorem 4.3, we obtain the following result.Corollary 4.6 (see [20], Theorem  5.2).
LetH be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4). Let α,β, and γ be real numbers with γ≠-1 and let S:C→H 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=x∈C and
(4.12)F(zn,y)+1λn〈zn-xn,y-zn〉≥0,∀y∈C,yn=αnxn+(1-αn)(11+γSzn+γ1+γzn),Cn+1={z∈Cn:∥yn-z∥≤∥xn-z∥},xn+1=PCn+1x,∀n∈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)liminfn→∞αn<1,0≤λn,
for some a∈ℝ. Then {xn} converges strongly to z0∈F(S)∩EP(F).SettingT:=(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.
LetH be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C→ℝ 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:C→H 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=x∈C and
(4.14)F(zn,y)+〈Ψxn,y-zn〉+1λn〈zn-xn,y-zn〉≥0,∀y∈C,yn=αnxn+(1-αn)(11+γSzn+γ1+γzn),Cn={z∈C:∥yn-z∥≤∥xn-z∥},Qn={z∈Qn-1:〈f(xn-1)-xn,xn-z〉≥0},xn+1=PCn∩Qnf(xn),∀n∈ℕ,
where PCn∩Qn is the metric projection of H onto Cn∩Qn and {αn}⊂(0,1) and {λn}⊂(0,2δ) satisfy
(4.15)0≤αn≤b<1,0<c≤λn≤d<2δ,
for some b,c,d∈ℝ. Then {xn} converges strongly to z0=PF(S)∩GEP(F,Ψ)f(z0).Proof.
PutT=(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.Settingf(xn)=x,∀n∈ℕ in Theorem 4.7, we obtains the following result.Corollary 4.8.
LetH be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C→ℝ 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:C→H 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=x∈C and
(4.16)F(zn,y)+〈Ψxn,y-zn〉+1λn〈zn-xn,y-zn〉≥0,∀y∈C,yn=αnxn+(1-αn)(11+γSzn+γ1+γzn),Cn={z∈C:∥yn-z∥≤∥xn-z∥},Qn={z∈C:〈x-xn,xn-z〉≥0},xn+1=PCn∩Qnx,∀n∈ℕ,
where PCn∩Qn is the metric projection of H onto Cn∩Qn and {αn}⊂(0,1) and {λn}⊂(0,2δ) satisfy
(4.17)0≤αn≤b<1,0<c≤λn≤d<2δ,
for some b,c,d∈ℝ. Then {xn} converges strongly to z0∈F(S)∩GEP(F,Ψ).SettingΨ≡0 in Theorem 4.7, we obtain the following result.Corollary 4.9.
LetH be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4). Let α,β, and γ be real numbers with γ≠-1 and let S:C→H 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=x∈C and
(4.18)F(zn,y)+1λn〈zn-xn,y-zn〉≥0,∀y∈C,yn=αnxn+(1-αn)(11+γSzn+γ1+γzn),Cn={z∈C:∥yn-z∥≤∥xn-z∥},Qn={z∈Qn-1:〈f(xn-1)-xn,xn-z〉≥0},xn+1=PCn∩Qnf(xn),∀n∈ℕ,
where PCn∩Qn is the metric projection of H onto Cn∩Qn and {αn}⊂(0,1) and {λn}⊂(0,∞) satisfy
(4.19)0≤αn≤b<1,0<c≤λn,
for some b,c∈ℝ. Then {xn} converges strongly to z0∈F(S)∩EP(F).SettingΨ≡0 and f(xn)=x,∀n∈ℕ in Theorem 4.7, we obtain the following result.Corollary 4.10 (see [20], Theorem  5.1).
LetH be a Hilbert space and let C be a nonempty closed convex subset of H. Let F:C×C→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4). Let α,β, and γ be real numbers with γ≠-1 and let S:C→H be an (α,β,γ)-super hybrid mapping such that EP(F)∩F(S)≠∅. Let {xn}⊂C be a sequence generated by x1=x∈C and
(4.20)F(zn,y)+1λn〈zn-xn,y-zn〉≥0,∀y∈C,yn=αnxn+(1-αn)(11+γSzn+γ1+γzn),Cn={z∈C:∥yn-z∥≤∥xn-z∥},Qn={z∈C:〈xn-z,x-zn〉≥0},xn+1=PCn∩Qnx,∀n∈ℕ,
where PCn∩Qn is the metric projection of H onto Cn∩Qn and {αn}⊂(0,1) and {λn}⊂(0,∞) satisfy
(4.21)0≤αn≤b<1,0<c≤λn,
for some b,c∈ℝ. Then {xn} converges strongly to z0∈F(S)∩EP(F).



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*Source: 235474-2012-09-24.xml*