# Iterative Algorithms with Perturbations for Solving the Systems of Generalized Equilibrium Problems and the Fixed Point Problems of Two Quasi-Nonexpansive Mappings

Authors: Rabian Wangkeeree; Uraiwan Boonkong
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

We introduce new iterative algorithms with perturbations for finding a common element of the set of solutions of the system of generalized equilibrium problems and the set of common fixed points of two quasi-nonexpansive mappings in a Hilbert space. Under suitable conditions, strong convergence theorems are obtained. Furthermore, we also consider the iterative algorithms with perturbations for finding a common element of the solution set of the systems of generalized equilibrium problems and the common fixed point set of the super hybrid mappings in Hilbert spaces.


---

## 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}. It is well known that the set F(T) of fixed points of a quasi-nonexpansive mapping T is closed and convex; see Itoh and Takahashi [1]. A mapping T:C→H is called nonspreading [2] if
(1.1)2∥Tx-Ty∥2≤∥Tx-y∥2+∥Ty-x∥2,
for all x,y∈C. We remark that nonlinear every nonspreading mappings are quasi-nonexpansive mappings if the set of fixed points is nonempty.Recall that a mappingΨ:C→H is said to be μ-inverse strongly monotone if there exists a positive real number μ such that
(1.2)〈Ψx-Ψy,x-y〉≥μ∥Ψx-Ψy∥2,∀x,y∈C.
If Ψ is a μ-inverse strongly monotone mapping of C into H, then it is obvious that Ψ is 1/μ-Lipschitz continuous.LetG:C×C→ℝ be a bifuction and Ψ:C→H be μ-inverse strongly monotone mapping. The generalized equilibrium problem (for short, GEP) for F and Ψ is to find z∈C such that
(1.3)G(z,y)+〈Ψz,y-z〉≥0,∀y∈C.
The problem (1.3) was studied by Moudafi [3]. The set of solutions for problem (1.3) is denoted by GEP(F,Ψ), that is,
(1.4)GEP(F,Ψ)={z∈C:G(z,y)+〈Ψz,y-z〉≥0,∀y∈C}.
If Ψ≡0 in (1.3), then GEP reduces to the classical equilibrium problem and GEP(G,0) is denoted by EP(G), that is,
(1.5)EP(G)={z∈C:G(z,y)≥0,∀y∈C}.
If G≡0 in (1.3), then GEP reduces to the classical variational inequality and GEP(0,Ψ) is denoted by VI(Ψ,C), that is,
(1.6)VI(Ψ,C)={z∈C:〈Ψz,y-z〉≥0,∀y∈C}.
The problem (1.3) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, Min–Max problems, the Nash equilibrium problems in noncooperative games, and others; see, for example, Blum and Oettli [4] and Moudafi [3].In 2005, Combettes and Hirstoaga [5] 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 [6] 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 [7] 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. Mainge and Moudafi [8] introduced an iterative algorithm for equilibrium problems and fixed point problems. Wangkeeree [9] 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 [10] 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 1967, Wittmann [11] (see also [12]) proved the strong convergence theorem of Halpern’s type [13] {xn} defined by, for any x1=x∈C,
(1.7)xn+1=αnx+(1-αn)Txn,∀n∈ℕ,
where {αn}⊂(0,1) satisfies limn→∞αn=0,∑n=1∞‍αn=∞, and ∑n=1∞‍|αn-αn+1|<∞. In [14], Kurokawa and Takahashi also studied the following Halpern’s type for nonspreading mappings in a Hilbert space; see also Hojo and Takahashi [15]. Let T be a nonspreading mapping of C into itself. Let u∈C and define two sequences {xn} and {zn} in C as follows: x1=x∈C and
(1.8)xn+1=αnu+(1-αn)zn,wherezn=1n∑k=0n-1‍Tkxn
for all n=1,2,…, where {αn}⊂(0,1), limn→∞αn=0 and ∑n=1∞αn=∞. If F(T) is nonempty, they proved that {xn} and {zn} converge strongly to PF(T)u, where PF(T) is the metric projection of H onto F(T). Recently, Yao and Shahzad [16] gave the following iteration process for nonexpansive mappings with perturbation: x1∈C and
(1.9)xn+1=(1-βn)xn+βnPC(αnun+(1-αn)Txn),∀n∈ℕ,
where {αn} and {βn} are sequences in [0,1], and the sequence {un}⊆H is a small perturbation for the n-step iteration satisfying ∥un∥→0 as n→∞. In fact, there are perturbations always occurring in the iterative processes because the manipulations are inaccurate.On the other hand, very recently, Chuang et al. [17] considered the following iteration process for finding a common element of the set of solutions of the equilibrium problem and the set of common fixed points for a quasi-nonexpansive mapping T:C→H with perturbation
(1.10)q1∈H,xn∈C,suchthatG(xn,y)+1rn〈y-xn,xn-qn〉≥0,∀y∈C,yn=βnxn+(1-βn)Txn,qn+1=αnun+(1-αn)yn,∀n∈ℕ,
where C is a nonempty closed convex subset of H, G:C×C→ℝ is a function, {αn} and {βn} are real sequences in (0,1), and {un}⊂H is a convergent sequence and {rn}⊂[a,∞) for some a>0. They obtained a strong convergence theorem for such iterations.In this paper, inspired and motivated by Yao and Shahzad [16], S. Takahashi and W. Takahashi [18] and Chuang et al. [17], we introduce a new iterative algorithms with perturbations for finding a common element of the set of solutions of the system of generalized equilibrium problems and the set of common fixed points of two quasi-nonexpansive mappings in a Hilbert space. Under suitable conditions, strong convergence theorems are obtained. Furthermore, we also consider the iterative algorithms with perturbations for finding a common element of the solution set of the system of generalized equilibrium problems and the common fixed point set of the super hybrid mappings in a Hilbert space.


## 2. Preliminaries

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥. We denote the strongly convergence and the weak convergence of {xn} to x∈H by xn→x and xn⇀x, respectively. In a Hilbert space, it is known that
(2.1)∥λx+(1-λ)y∥2=λ∥x∥2+(1-λ)∥y∥2-λ(1-λ)∥x-y∥2,
for all x,y∈H and λ∈ℝ; see [19]. Furthermore, we have that for any x,y,u,v∈H(2.2)2〈x-y,u-v〉=∥x-v∥2+∥y-u∥2-∥x-u∥2-∥y-v∥2.
Let C be a nonempty closed convex subset of H and x∈H. We know that there exists a unique nearest point z∈C such that ∥x-z∥=infy∈C∥x-y∥. We denote such a correspondence by z=PCx. The mapping PC is called the metric projection of H onto C. It is known that PC is nonexpansive and
(2.3)〈x-PCx,PCx-u〉≥0;
for all x∈H and u∈C; see [19, 20] for more details.LetC be a nonempty, closed and convex subset of H and let G:C×C→ℝ be a bifunction. For solving the generalized equilibrium problem, let us assume that the bifunction G:C×C→ℝ satisfies the following conditions:(A1)
G(x,x)=0 for all x∈C;(A2)
G is monotone, that is, G(x,y)+G(y,x)≤0 for any x,y∈C;(A3)
for eachx,y,z∈C(2.4)limt↓0G(tz+(1-t)x,y)≤G(x,y);(A4)
for eachx∈C,G(x,·) is convex and lower semicontinuous.We know the following lemma which appears implicitly in Blum and Oettli [4].Lemma 2.1 (see [4]).
LetC be a nonempty closed convex subset of H and let G be a bifunction of C×C into ℝ satisfying (A1)–(A4). Let r>0 and x∈H. Then, there exists a unique z∈C such that
(2.5)G(z,y)+1r〈y-z,z-x〉≥0,∀y∈C.The following lemma was also given in Combettes and Hirstoaga [5].Lemma 2.2 (see [5]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let G:C×C→ℝ be a bifunction which satisfies conditions (A1)–(A4). For r>0 and x∈H, define a mapping Tr:H→C as follows:
(2.6)Tr(x)={z∈C:G(z,y)+1r〈y-z,z-x〉≥0,∀y∈C}
for all x∈H. Then the following hold:(i)
Tr is single-valued;(ii)
Tr is firmly nonexpansive, that is, for any x,y∈C,
(2.7)∥Trx-Try∥2≤〈Trx-Try,x-y〉;(iii)
(EP) is a closed convex subset of C;(iv)
F(Tr)=EP(G).Remark 2.3.
For anyx∈H and r>0, by Lemma 2.2 (i), there exists u∈H such that
(2.8)G(u,y)+1r〈y-u,u-x〉≥0,∀y∈H.
Replacing x with x-rΨx∈H in (2.8), we have
(2.9)G(u,y)+〈Ψx,y-u〉+1r〈y-u,u-x〉≥0,∀y∈H,
where Ψ:H→H is an inverse strongly monotone mapping.Lemma 2.4 (see [21]).
Let{Γn} be a sequence of real numbers that does not decrease at infinity in the sense that there exists a subsequence {Γni} of {Γn} which satisfies Γni<Γni+1 for all i∈ℕ. Define the sequence {τ(n)}n≥n0 of integers as follows:
(2.10)τ(n)=max{k≤n:Γk<Γk+1},
where n0∈ℕ such that {k≤n0:Γk<Γk+1}≠∅. Then, the following hold:(i)
τ(1)≤τ(2)≤⋯ and τ(n)→∞;(ii)
Γτ(n)≤Γτ(n)+1 and Γn≤Γτ(n)+1,∀n∈ℕ.Lemma 2.5 (see [22]).
Let{an}n∈ℕ be a sequence of nonnegative real numbers, {αn} a sequence of real numbers in [0,1] with ∑n=1∞αn=∞,{un} a sequence of nonnegative real numbers with ∑n=1∞un<∞,{tn} a sequence of real numbers with limsuptn≤0. Suppose that
(2.11)an+1≤(1-αn)an+αntn+un,∀n∈ℕ.
Then limn→∞an=0.


## 3. Main Results

LetC be a nonempty closed convex subset of a Hilbert space H. For each i=1,2,…,k, let Gi:C×C→ℝ be a bifunction satisfying (A1)–(A4) and Ψi a μi-inverse strongly monotone mapping. For each j=1,2, let Tj:C→H be two mappings. Let {xn} be a sequence generated in the following manner:
(3.1)x1∈H,G1(un,1,y)+〈Ψ1xn,y-un,1〉+1rn〈y-un,1,un,1-xn〉≥0,∀y∈C,G2(un,2,y)+〈Ψ2xn,y-un,2〉+1rn〈y-un,2,un,2-xn〉≥0,∀y∈C,⋮Gk(un,k,y)+〈Ψkxn,y-un,k〉+1rn〈y-un,k,un,k-xn〉≥0,∀y∈C,ωn=1k∑i=1k‍un,i,yn=γnωn+(1-γn)T1ωn,zn=βnyn+(1-βn)T2ωn,xn+1=αnun+(1-αn)zn,∀n∈ℕ,
where {αn},{βn},{γn} are sequences in (0, 1) and {un}⊂H is a sequence and {rn}⊂[a,2μi) for some a>0 and for all i∈{1,2,…,k}. Under certain appropriate assumptions imposed on the sequences {αn},{βn},{γn}, the strong convergence theorem of {xn} defined by (3.1) is studied in the following theorem.Theorem 3.1.
LetC be a nonempty closed convex subset of a Hilbert space H. For each i=1,2,…,k, let Gi:C×C→ℝ be a bifunction satisfying (A1)–(A4) and Ψi a μi-inverse strongly monotone mapping. For each j=1,2, let Tj:C→H be two quasi-nonexpansive mappings such that I-Tj are demiclosed at zero with Ω:=F(T1)∩F(T2)∩(∩i=1kGEP(Gi,Ψi))≠∅. Let the sequences {xn},  {yn}, and {zn} be defined by (3.1), where {αn},{βn},{γn}, and {un} satisfy the following conditions:(C1)
limn→∞αn=0 and ∑n=1∞αn=∞;(C2)
liminfn→∞βn(1-βn)>0;(C3)
liminfn→∞γn(1-γn)>0;(C4)
limn→∞un=u for some u∈H.Then {xn} converges strongly to x*, where x*=PΩu.Proof.
We first have that for alli=1,2,…,k, I-rnΨi is a nonexpansive mapping. Indeed, for all x,y∈C, we obtain
(3.2)∥(I-rnΨi)x-(I-rnΨi)y∥2=∥(x-y)-rn(Ψix-Ψiy)∥2=∥x-y∥2-2rn〈Ψix-Ψiy,x-y〉+rn2∥Ψix-Ψiy∥2≤∥x-y∥2-rn(2μi-rn)∥Ψix-Ψiy∥2≤∥x-y∥2.
Thus I-rnΨi is nonexpansive for each i∈{1,2,…,k}. Now, let w∈Ω be arbitrary. By (C4), {un} is a bounded sequence, there exists M≤0 such that
(3.3)supn∈ℕ∥un-w∥≤M.
For each i=1,2,…,k and n∈ℕ, we have from un,i=Trn,i(xn-rnΨixn) that
(3.4)∥un,i-w∥=∥Trn,i(xn-rnΨixn)-Trn,i(w-rnΨiw)∥≤∥(xn-rnΨixn)-(w-rnΨiw)∥≤∥xn-w∥,
which gives also that
(3.5)∥ωn-w∥≤1k∑i=1k‍∥un,i-w∥≤∥xn-w∥∀w∈Ω.
Since T1 is quasi-nonexpansive, we have
(3.6)∥yn-w∥=∥γnωn+(1-γn)T1ωn-w∥=∥γn(ωn-w)+(1-γn)(T1ωn-w)∥≤γn∥ωn-w∥+(1-γn)∥T1ωn-w∥≤∥ωn-w∥.
So, we have from (3.5) and (3.6) and the quasi-nonexpansiveness of T2 that
(3.7)∥xn+1-w∥=∥αn(un-w)+(1-αn)(zn-w)∥≤αn∥un-w∥+(1-αn)∥zn-w∥≤αn∥un-w∥+(1-αn){βn∥yn-w∥+(1-βn)∥T2ωn-w∥}≤αn∥un-w∥+(1-αn){βn∥ωn-w∥+(1-βn)∥ωn-w∥}≤αn∥un-w∥+(1-αn)∥ωn-w∥≤αn∥un-w∥+(1-αn)∥xn-w∥≤max{M,∥xn-w∥}.
By Induction, we have that
(3.8)∥xn-w∥≤max{∥x1-w∥,M},∀n∈ℕ.
Thus we obtain that {∥xn-w∥} is bounded, so also {xn},{yn},{zn},{ωn},{T1ωn}, and {T2ωn} are bounded. Since Ω is closed and convex, we can take x*=PΩu. It follows that
(3.9)∥yn-x*∥2=∥γn(ωn-x*)+(1-γn)(T1ωn-x*)∥2=γn∥ωn-x*∥2+(1-γn)∥T1ωn-x*∥2-γn(1-γn)∥ωn-T1ωn∥2≤γn∥ωn-x*∥2+(1-γn)∥ωn-x*∥2-γn(1-γn)∥ωn-T1ωn∥2=∥ωn-x*∥2-γn(1-γn)∥ωn-T1ωn∥2≤∥ωn-x*∥2.
From (3.9), we have
(3.10)∥zn-x*∥2=∥βn(yn-x*)+(1-βn)(T2ωn-x*)∥2=βn∥yn-x*∥2+(1-βn)∥T2ωn-x*∥2-βn(1-βn)∥yn-T2ωn∥2≤βn∥ωn-x*∥2+(1-βn)∥ωn-x*∥2-βn(1-βn)∥yn-T2ωn∥2=∥ωn-x*∥2-βn(1-βn)∥yn-T2ωn∥2≤∥ωn-x*∥2.
Hence we have from (3.5), (3.9), and (3.10) that
(3.11)∥ωn+1-x*∥2≤∥xn+1-x*∥2=∥αn(un-x*)+(1-αn)(zn-x*)∥2=αn∥un-x*∥2+(1-αn)∥zn-x*∥2-αn(1-αn)∥un-zn∥2≤αn∥un-x*∥2+(1-αn)∥zn-x*∥2=αn∥un-x*∥2+(1-αn){βn∥yn-x*∥2+(1-βn)∥T2ωn-x*∥2-βn(1-βn)∥yn-T2ωn∥2}≤αn∥un-x*∥2+βn{γn∥ωn-x*∥2+(1-γn)∥T1ωn-x*∥2-γn(1-γn)∥ωn-T1ωn∥2}+(1-βn)∥T2ωn-x*∥2-βn(1-βn)∥yn-T2ωn∥2≤αn∥un-x*∥2+βn(γn∥ωn-x*∥2+(1-γn)∥ωn-x*∥2-γn(1-γn)∥ωn-T1ωn∥2)+(1-βn)∥ωn-x*∥2-βn(1-βn)∥yn-T2ωn∥2=αn∥un-x*∥2+∥ωn-x*∥2-γn(1-γn)∥ωn-T1ωn∥2-βn(1-βn)∥yn-T2ωn∥2.
We also have that
(3.12)γn(1-γn)∥ωn-T1ωn∥2≤αn∥un-x*∥2+∥ωn-x*∥2-∥ωn+1-x*∥2,(3.13)βn(1-βn)∥yn-T2ωn∥2≤αn∥un-x*∥2+∥ωn-x*∥2-∥ωn+1-x*∥2.
Furthemore, we have from yn=γnωn+(1-γn)T1ωn that
(3.14)∥ωn-T2ωn∥≤∥ωn-yn∥+∥yn-T2ωn∥=∥ωn-γnωn-(1-γn)T1ωn∥+∥yn-T2xn∥=(1-γn)∥ωn-T1ωn∥+∥yn-T2ωn∥.
On the other hand, since xn+1-x*=αn(un-x*)+(1-αn)(zn-x*), we have
(3.15)∥ωn+1-x*∥2≤∥xn+1-x*∥2≤(1-αn)∥zn-x*∥2+2αn〈un-x*,xn+1-x*〉≤(1-αn)∥ωn-x*∥2+2αn〈un-x*,xn+1-x*〉=(1-αn)∥ωn-x*∥2+2αn〈un-u,xn+1-x*〉+2αn〈u-x*,xn+1-x*〉=(1-αn)∥ωn-x*∥2+2αn〈un-u,xn+1-x*〉+2αn〈u-x*,xn+1-ωn〉+2αn〈u-x*,ωn-x*〉.
We also have that
(3.16)∥xn+1-ωn∥≤∥xn+1-yn∥+∥yn-ωn∥=∥αn(un-yn)+(1-αn)(zn-yn)∥+∥(1-γn)(ωn-T1ωn)∥≤αn∥un-yn∥+(1-αn)∥βnyn+(1-βn)T2ωn-yn∥+(1-γn)∥ωn-T1ωn∥=αn∥un-yn∥+(1-αn)(1-βn)∥yn-T2ωn∥+(1-γn)∥ωn-T1ωn∥.
Moreover, for any i∈{1,2,…,k}, we have from un,i=Trn,i(xn-rnΨixn) that
(3.17)∥un,i-x*∥2≤∥(xn-x*)-rn(Ψixn-Ψix*)∥2=∥xn-x*∥2-2rn〈xn-x*,Ψixn-Ψix*〉+rn2∥Ψixn-Ψix*∥2≤∥xn-x*∥2-rn(2μi-rn)∥Ψixn-Ψix*∥2.
It follows that
(3.18)∥ωn-x*∥2=∥∑i=1k‍1k(un,i-x*)∥2≤1k∑i=1k‍∥un,i-x*∥2≤∥xn-x*∥2-1k∑i=1k‍rn(2μi-rn)∥Ψixn-Ψix*∥2.
This implies that
(3.19)∥xn+1-x*∥2=∥αn(un-x*)+(1-αn)(zn-x*)∥2≤αn∥un-x*∥2+(1-αn)∥ωn-x*∥2≤αn∥un-x*∥2+(1-αn)∥xn-x*∥2-(1-αn)1k∑i=1k‍rn(2μi-rn)∥Ψixn-Ψix*∥2,
and hence
(3.20)(1-αn)1k∑i=1k‍rn(2μi-rn)∥Ψixn-Ψix*∥2≤αn∥un-x*∥2+∥xn-x*∥2-∥xn+1-x*∥2.
Furthermore, we have from Lemma 2.2 that for any i∈1,2,…,k, we have
(3.21)∥un,i-x*∥2≤〈(xn-rnΨixn)-(x*-rnΨix*),un,i-x*〉=12{∥(xn-rnΨixn)-(x*-rnΨix*)∥2+∥un,i-x*∥2-∥(xn-rnΨixn)-(x*-rnΨix*)-(un,i-x*)∥2}≤12{∥xn-x*∥2+∥un,i-x*∥2-∥(xn-un,i)-rn(Ψixn-Ψix*)∥2}=12{∥xn-x*∥2+∥un,i-x*∥2-∥xn-un,i∥2-rn2∥Ψixn-Ψix*∥2gg+2rn〈xn-un,i,Ψixn-Ψix*〉∥un,i-x*∥2}.
This implies that
(3.22)∥un,i-x*∥2≤∥xn-x*∥2-∥xn-un,i∥2+2rn∥xn-un,i∥∥Ψixn-Ψix*∥.
Then we have from (3.22) that
(3.23)∥ωn-x*∥2≤1k∑i=1k‍∥un,i-x*∥2≤∥xn-x*∥2-1k∑i=1k‍∥un,i-xn∥2+1k∑i=1k‍2rn∥xn-un,i∥∥Ψixn-Ψix*∥.
Hence we have from (3.23) that
(3.24)∥xn+1-x*∥2≤αn∥un-x*∥2+(1-αn)∥ωn-x*∥2≤αn∥un-x*∥2+(1-αn)(∥xn-x*∥2-1k∑i=1k‍∥un,i-xn∥2)+(1-αn)(1k∑i=1k‍2rn∥xn-un,i∥∥Ψixn-Ψix*∥).
It follows that
(3.25)(1-αn)1k∑i=1k‍∥un,i-xn∥2≤αn∥un-x*∥2+∥xn-x*∥2-∥xn+1-x*∥2+(1-αn)(1k∑i=1k‍2rn∥xn-un,i∥∥Ψixn-Ψix*∥).
Next, we will consider the following two cases.
Case A. Put Γn=∥ωn-x*∥2 for all n∈ℕ. Suppose that Γn+1≤Γn for all n∈ℕ. In this case limn→∞Γn exists and then limn→∞(Γn+1-Γn)=0. By (C1), (C3), and (3.12), we have
(3.26)limn→∞∥ωn-T1ωn∥=0.
Similarly by (C1), (C2), and (3.13), we also have
(3.27)limn→∞∥yn-T2ωn∥=0.
So, we have from (3.14), (3.26), and (3.27) that
(3.28)limn→∞∥ωn-T2ωn∥=0.
Since limn→∞∥ωn-x*∥ exists, we have from (3.11) and (3.26)
(3.29)limn→∞∥ωn-x*∥=limn→∞∥xn-x*∥.
We also have from (C1), (3.16), (3.26), and (3.27) that
(3.30)limn→∞∥xn+1-ωn∥=0.
Since limn→∞∥xn-x*∥ exists we have from (C1) and (3.20) that
(3.31)limn→∞∥Ψixn-Ψix*∥=0,∀i=1,2,…,k.
This together with (3.25) and the existence of limn→∞∥xn-x*∥ implies that
(3.32)limn→∞∥un,i-xn∥=0,∀i=1,2,…,k,
which gives that
(3.33)∥ωn-xn∥≤1k∑i=1k‍∥un,i-xn∥→0asn→∞.
So, from (3.30), limn→∞∥xn+1-xn∥=0. Furthermore, we have from (3.33) that
(3.34)∥ωn+1-ωn∥≤∥ωn+1-xn+1∥+∥xn+1-xn∥+∥xn-ωn∥→0asn→∞;
that is
(3.35)limn→∞∥ωn+1-ωn∥=0.
Now, since {ωn} is a bounded sequence, there exists a subsequence {ωnj} of {ωn} such that
(3.36)limsupn→∞〈u-x*,ωn-x*〉=limj→∞〈u-x*,ωnj-x*〉.
Without loss of generality, we may assume that ωnj⇀v. Since T1 is demiclosed at zero and by (3.26), we conclude that v∈F(T1). Similarly, since T2 is demiclosed at zero and by (3.28), we have v∈F(T2). Therefore, we get that
(3.37)v∈F(T1)∩F(T2).
Next, we show that v∈∩i=1kGEP(Gi,Ψi). For each i∈{1,2,…,k}, since un,i=Trn,i(xn-rnΨixn), we have
(3.38)Gi(un,i,y)+〈Ψixn,y-un,i〉+1rn〈y-un,i,un,i-xn〉≥0,∀y∈C.
From (A2), we also have
(3.39)〈Ψixn,y-un,i〉+1rn〈y-un,i,un,i-xn〉≥Gi(y,un,i).
Replacing n by nj, we have
(3.40)〈Ψixnj,y-unj,i〉+〈y-unj,i,unj,i-xnjrnj〉≥Gi(y,unj,i).
Put yt=ty+(1-t)v for all t∈(0,1] and y∈C. Since v∈C, then yt∈C and
(3.41)〈yt-unj,i,Ψiyt〉≥〈yt-unj,i,Ψiyt〉-〈yt-unj,i,Ψixnj〉-〈yt-unj,i,unj,i-xnjrnj〉+Gi(yt,unj,i)=〈yt-unj,i,Ψiyt-Ψiunj,i〉+〈yt-unj,i,Ψiunj,i-Ψixnj〉-〈yt-unj,i,unj,i-xnjrnj〉+Gi(yt,unj,i).
Since ∥unj,i-xnj∥→0 as j→∞, we obtain that ∥Ψiunj,i-Ψixnj∥→0 as j→∞. Furthermore, by the monotonicity of Ψi, we obtain that
(3.42)〈yt-unj,i,Ψiyt-Ψiunj,i〉≥0.
Taking j→∞ in (3.41), we have from (A4) that
(3.43)〈yt-v,Ψiyt〉≥Gi(yt,v).
Now, from (A1), (A4), and (3.43), we also have
(3.44)0=Gi(yt,yt)≤tGi(yt,y)+(1-t)Gi(yt,v)≤tGi(yt,y)+(1-t)〈yt-v,Ψiyt〉=tGi(yt,y)+(1-t)t〈y-v,Ψiyt〉,
which yields that
(3.45)Gi(yt,y)+(1-t)〈y-v,Ψiyt〉≥0.
Taking t→0, we have, for each y∈C(3.46)Gi(v,y)+〈y-v,Ψiv〉≥0,∀i∈{1,2,…,k}.
This shows v∈GEP(Gi,Ψi), for all i=1,2,…,k. Then, v∈∩i=1kGEP(Gi,Ψi). Hence we have v∈F(T1)∩F(T2)∩(∩i=1kGEP(Gi,Ψi)):=Ω. So, we have from (3.36) that
(3.47)limsupn→∞〈u-x*,ωn-x*〉=〈u-x*,v-x*〉≤0.
By (C1), (C4), (3.15), (3.30), (3.47), and Lemma 2.5, we obtain that limn→∞∥ωn-x*∥=0. Hence we have from (3.29) that {xn} converges to x*, where x*=PΩu.
Case B. Assume that there exists a subsequence {Γni}i≥0 of {Γn}n≥0 such that Γni<Γni+1 for all i∈ℕ. In this case, it follows from Lemma 2.4 that there exists a subsequence {Γτ(n)} of {Γn} such that Γτ(n)+1>Γτ(n), where τ:ℕ→ℕ is defined by
(3.48)τ(n)=max{k≤n:Γk<Γk+1},∀n∈ℕ.
So, from (3.12), that
(3.49)∥ωτ(n)+1-x*∥2-∥ωτ(n)-x*∥2+γτ(n)(1-γτ(n))∥ωτ(n)-T1ωτ(n)∥2≤ατ(n)∥uτ(n)-x*∥2.
Since ∥ωτ(n)-x*∥2:=Γτ(n)<Γτ(n)+1:=∥ωτ(n)+1-x*∥2, we have
(3.50)γτ(n)(1-γτ(n))∥ωτ(n)-T1ωτ(n)∥2≤ατ(n)∥uτ(n)-x*∥2.
By (C1) and (C3), we have
(3.51)limn→∞∥ωτ(n)-T1ωτ(n)∥=0.
By (3.15), we have
(3.52)∥ωτ(n)+1-x*∥2≤(1-ατ(n))∥ωτ(n)-x*∥2+2ατ(n)〈uτ(n)-x*,xτ(n)+1-x*〉.
Now, in view of Γτ(n)<Γτ(n)+1, we see that
(3.53)∥ωτ(n)-x*∥2≤2〈uτ(n)-x*,xτ(n)+1-x*〉=2〈uτ(n)-u,xτ(n)+1-x*〉+2〈u-x*,xτ(n)+1-ωτ(n)〉+2〈u-x*,ωτ(n)-x*〉.
Furthermore, we also have from (3.13) that
(3.54)βτ(n)(1-βτ(n))∥yτ(n)-T2ωτ(n)∥2≤ατ(n)∥uτ(n)-x*∥2+∥ωτ(n)-x*∥2-∥ωτ(n)+1-x*∥2≤ατ(n)∥uτ(n)-x*∥2.
Applying (C1) and (C2) to the last inequality, we get that
(3.55)limn→∞∥yτ(n)-T2ωτ(n)∥=0.
By (C1), (3.16), (3.51), and (3.55), we have
(3.56)limn→∞∥xτ(n)+1-ωτ(n)∥=0.
By (3.33), we have
(3.57)limn→∞∥ωτ(n)+1-xτ(n)+1∥=0.
It follows from (3.56) and (3.57) that
(3.58)limn→∞∥ωτ(n)+1-ωτ(n)∥=0.
Since {ωτ(n)} is a bounded sequence, there exists a subsequence {ωτ(nj)} such that
(3.59)limsupn→∞〈u-x*,ωτ(n)-x*〉=limj→∞〈u-x*,ωτ(nj)-x*〉.
Following the same argument as the proof of Case A for {ωτ(nj)}, we have that
(3.60)limsupn→∞〈u-x*,ωτ(n)-x*〉≤0.
Using (C4), (3.53), (3.56), and (3.60), we have that
(3.61)limn→∞∥ωτ(n)-x*∥=0.
By (3.58) and (3.61), we have that
(3.62)limn→∞∥ωτ(n)+1-x*∥=0.
By Lemma 2.4 (ii), we get limn→∞Γn=0; that is limn→∞∥ωn-x*∥=0. We observe that
(3.63)∥xn+1-x*∥2≤αn∥un-x*∥2+(1-αn)∥ωn-x*∥2.
Applying (C1), (C4), and limn→∞∥ωn-x*∥2=0, we have immediately
(3.64)limn→∞∥xn-x*∥=0;
that is, {xn} converges strongly to x*, where x*=PΩu. This completes the proof.SettingΨi≡0 for all i=1,2,…,k in Theorem 3.1, we obtain the following result.Corollary 3.2.
LetC be a nonempty closed convex subset of a Hilbert space H. For each i=1,2,…,k, let Gi:C×C→ℝ be a bifunction satisfying(A1)–(A4). For each j=1,2, let Tj:C→H be two quasi-nonexpansive mappings such that I-Tj are demiclosed at zero with Ω:=F(T1)∩F(T2)∩(∩i=1kEP(Gi))≠∅. Let the sequences {xn},   {yn}, and {zn} be defined by
(3.65)x1∈H,G1(un,1,y)+1rn〈y-un,1,un,1-xn〉≥0,∀y∈C,G2(un,2,y)+1rn〈y-un,2,un,2-xn〉≥0,∀y∈C,⋮Gk(un,k,y)+1rn〈y-un,k,un,k-xn〉≥0,∀y∈C,ωn=1k∑i=1k‍un,i,yn=γnωn+(1-γn)T1ωn,zn=βnyn+(1-βn)T2ωn,xn+1=αnun+(1-αn)zn,∀n∈ℕ,
where {αn},{βn},{γn} satisfy the following conditions.(C1)
limn→∞αn=0 and ∑n=1∞αn=∞;(C2)
liminfn→∞βn(1-βn)>0;(C3)
liminfn→∞γn(1-γn)>0;(C4)
limn→∞un=u for some u∈H.Then {xn} converges strongly to x*, where x*=PΩu.In the next results, using Theorem3.1, we have new strong convergence theorems for two nonexpansive mappings in a Hilbert space.Corollary 3.3.
LetC be a nonempty closed convex subset of a Hilbert space H. For each i=1,2,…,k, let Gi:C×C→ℝ be a bifunction satisfying (A1)–(A4) and Ψi a μi-inverse strongly monotone mapping. For each j=1,2, let Tj:C→H be two nonexpansive mappings such that Ω:=F(T1)∩F(T2)∩(∩i=1kGEP(Gi,Ψi))≠∅. Let the sequences {xn},  {yn}, and {zn} be defined by (3.1), where {αn},{βn},{γn} satisfy the following conditions.(C1)
limn→∞αn=0 and ∑n=1∞αn=∞;(C2)
liminfn→∞βn(1-βn)>0;(C3)
liminfn→∞γn(1-γn)>0;(C4)
limn→∞un=u for some u∈H.Then {xn} converges strongly to x*, where x*=PΩu.


## 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 [2] 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 [23] 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 [24] and Goebel and Kirk [25]. We also know that a firmly nonexpansive mapping F can be deduced from an equilibrium problem in a Hilbert space.Recently, Kocourek et al. [26] introduced a more broad class of nonlinear mappings call 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 [27]. Before proving, we need the following lemmas.Lemma 4.1 (see [27]).
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 S=(1/(1+γ))T+(γ/(1+γ))I. Then, T is (α,β,γ)-super hybrid if and only if S is (α,β)-generalized hybrid. In this case, F(S)=F(T).Lemma 4.2 (see [27]).
LetH be a Hilbert space and let C be a nonempty closed convex subset of H. Let S:C→H be a generalized hybrid mapping. Then S is demiclosed on C.
SettingSj:=(1/(1+γj))Tj+(γj/(1+γj))I in Theorem 3.1, where Tj is a super hybrid mapping and γj is a real number, we obtain the following result.Theorem 4.3.
LetC be a nonempty closed convex subset of a Hilbert space H. For each i=1,2,…,k, let Gi:C×C→ℝ be a bifunction satisfying (A1)–(A4) and Ψi a μi-inverse strongly monotone mapping. For each j=1,2, let Tj:C→H be (αj,βj,γj)-super hybrid mappings such that Ω:=F(T1)∩F(T2)∩(∩i=1kGEP(Gi,Ψi))≠∅. Let the sequences {xn},  {yn}, and {zn} be defined by
(4.6)x1∈H,G1(un,1,y)+〈Ψ1xn,y-un,1〉+1rn〈y-un,1,un,1-xn〉≥0,∀y∈C,G2(un,2,y)+〈Ψ2xn,y-un,2〉+1rn〈y-un,2,un,2-xn〉≥0,∀y∈C,⋮Gk(un,k,y)+〈Ψkxn,y-un,k〉+1rn〈y-un,k,un,k-xn〉≥0,∀y∈C,ωn=1k∑i=1k‍un,i,yn=γnωn+(1-γn)(11+γ1T1ωn+γ11+γ1ωn),zn=βnyn+(1-βn)(11+γ2T2ωn+γ21+γ2ωn),xn+1=αnun+(1-αn)zn,∀n∈ℕ,
where {αn},{βn},{γn} are sequences in (0,1) and {un}⊂H is a sequence and {rn}⊂[a,2μi) for some a>0 and for all i∈{1,2,…,k}. Suppose the following conditions are satisfied.(C1)
limn→∞αn=0 and ∑n=1∞αn=∞;(C2)
liminfn→∞βn(1-βn)>0;(C3)
liminfn→∞γn(1-γn)>0;(C4)
limn→∞un=u for some u∈H.Then {xn} converges strongly to x*, where x*=PΩu.Proof.
For eachj=1,2, setting
(4.7)Sj=11+γjTj+γj1+γjI,
we have from Lemma 4.1 that each Sj is a generalized hybrid mapping and F(Sj)=F(Tj). Since F(Sj)≠∅, we have that each Sj is quasi-nonexpansive. Following the proof of Theorem 3.1 and applying Lemma 4.2, we have the desired result. This completes the proof.SettingΨ≡0 in Theorem 4.3, we obtains the following result.Corollary 4.4.
LetC be a nonempty closed convex subset of a Hilbert space H. For each i=1,2,…,k, let Gi:C×C→ℝ be a bifunction satisfying (A1)-(A4). For each j=1,2, let Tj:C→H be (αj,βj,γj)-super hybrid mappings such that Ω:=F(T1)∩F(T2)∩(∩i=1kEP(Gi))≠∅. Let the sequences {xn},{yn}, and {zn} be defined by
(4.8)x1∈H,G1(un,1,y)+1rn〈y-un,1,un,1-xn〉≥0,∀y∈C,G2(un,2,y)+1rn〈y-un,2,un,2-xn〉≥0,∀y∈C,⋮Gk(un,k,y)+1rn〈y-un,k,un,k-xn〉≥0,∀y∈C,ωn=1k∑i=1k‍un,i,yn=γnωn+(1-γn)(11+γ1T1ωn+γ11+γ1ωn),zn=βnyn+(1-βn)(11+γ2T2ωn+γ21+γ2ωn),xn+1=αnun+(1-αn)zn,∀n∈ℕ,
where {αn},{βn},{γn} are sequences in (0, 1) and {un}⊂H is a sequence and {rn}⊂[a,∞) for some a>0. Suppose the following conditions are satisfied.(C1)
limn→∞αn=0 and ∑n=1∞αn=∞;(C2)
liminfn→∞βn(1-βn)>0;(C3)
liminfn→∞γn(1-γn)>0;(C4)
limn→∞un=u for some u∈H.Then {xn} converges strongly to x*, where x*=PΩu.In Corollary4.4, put Gi(x,y)=0 for all x,y∈C and rn=1 for all n∈ℕ. Then we have that un,i=xn for all i=1,2,…,k, which gives that ωn=(1/k)∑i=1k‍un,i=xn. Thus we obtain the following results from Corollary 4.4.Corollary 4.5.
LetC be a nonempty closed convex subset of a Hilbert space H. For each j=1,2, let Tj:C→H be (αj,βj,γj)-super hybrid mappings such that F(T1)∩F(T2)≠∅. Let the sequences {xn},   {yn}, and {zn} be defined by
(4.9)x1∈H,yn=γnxn+(1-γn)(11+γ1T1xn+γ11+γ1xn),zn=βnyn+(1-βn)(11+γ2T2xn+γ21+γ2xn),xn+1=αnun+(1-αn)zn,∀n∈ℕ,
where {αn},{βn},{γn} are sequences in (0,1) and {un}⊂H is a sequence. Suppose the following conditions are satisfied.(C1)
limn→∞αn=0 and ∑n=1∞αn=∞;(C2)
liminfn→∞βn(1-βn)>0;(C3)
liminfn→∞γn(1-γn)>0;(C4)
limn→∞un=u for some u∈H.Then {xn} converges strongly to x*, where x*=PF(T1)∩F(T2)u.In Corollary4.5, put T1=I, the identity mapping, and T2:=T, an (α,β,γ)-super hybrid mapping. Thus we obtain the following results.Corollary 4.6.
LetC be a nonempty closed convex subset of a Hilbert space H. Let T be an (α,β,γ)-super hybrid mapping such that F(T)≠∅. Let the sequences {xn},{yn}, and {zn} be defined by
(4.10)x1∈H,(4.11)zn=βnxn+(1-βn)(11+γTxn+γ1+γxn),(4.12)xn+1=αnun+(1-αn)zn,∀n∈ℕ,
where {αn} and {βn} are sequences in (0,1) and {un}⊂H is a sequence. Suppose the following conditions are satisfied.(C1)
limn→∞αn=0 and ∑n=1∞αn=∞;(C2)
liminfn→∞βn(1-βn)>0;(C3)
limn→∞un=u for some u∈H.Then {xn} converges strongly to x*, where x*=PF(T)u.



---

*Source: 161897-2012-12-05.xml*