# Weak Convergence Theorem for Finding Fixed Points and Solution of Split Feasibility and Systems of Equilibrium Problems

Authors: Kamonrat Sombut; Somyot Plubtieng
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The purpose of this paper is to introduce an iterative algorithm for finding a common element of the set of fixed points of quasi-nonexpansive mappings and the solution of split feasibility problems (SFP) and systems of equilibrium problems (SEP) in Hilbert spaces. We prove that the sequences generated by the proposed algorithm converge weakly to a common element of the fixed points set of quasi-nonexpansive mappings and the solution of split feasibility problems and systems of equilibrium problems under mild conditions. Our main result improves and extends the recent ones announced by Ceng et al. (2012)  and many others.


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## Body

## 1. Introduction

LetC be a nonempty closed convex subset of a real Hilbert space H. A mapping T:C→C is said to be nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈C. Denote the set of fixed points of T by F(T). On the other hand, a mapping T:C→C is said to be quasi-nonexpansive if F(T)≠∅ and ∥Tx-q∥≤∥x-q∥ for all x∈C and q∈F(T). If T:C→C is nonexpansive and the set F(T) of fixed points of T is nonempty, then T is quasi-nonexpansive. Fixed point iterations process for nonexpansive mappings and quasi-nonexpansive mappings in Banach spaces including Mann and Ishikawa iterations process have been studied extensively by many authors to solve the nonlinear operator equations (see [1–4]).LetF be a bifunction of C×C into ℝ, where ℝ is the set of real numbers. The equilibrium problem for F:C×C→ℝ is to find x∈C such that
(1)F(x,y)≥0∀y∈C.
The set of solutions of (1) is denoted by EP(F). Numerous problems in physics, optimization, and economics reduce to find a solution of (1) in Hilbert spaces; see, for instance, Blum and Oettli [5],  Flam and Antipin [6], and Moudafi [7]. Moreover,  Flam and Antipin [6] introduced an iterative scheme of finding the best approximation to the solution of equilibrium problem, when EP(F) is nonempty, and proved a strong convergence theorem (see also in [8–11]). Let F1,F2:C×C→ℝ be two-monotone bifunction and λ>0 is a constant. Recently, Moudafi [12] considered the following of a system of equilibrium problem, denoting the set of solution of SEP by Ω, for finding (x*,y*)∈C×C such that
(2)λF1(x*,z)+〈y*-x*,x*-z〉≥0,∀z∈C,λF2(y*,z)+〈x*-y*,y*-z〉≥0,∀z∈C.
He also proved the weak convergence theorem of this problem (some related work can be found in [13, 14]).The split feasibility problem (SFP) in Hilbert spaces for modeling inverse problems which arise from phase retrievals and in medical image reconstruction was first introduced by Censor and Elfving [15] (see, e.g. [16, 17]). It has been found that the SFP can also be used to model the intensity-modulated radiation therapy (see [18, 19]). In this work, the SFP is formulated as finding a point x* with the property
(3)x*∈C,Ax*∈Q,
where C and Q are the nonempty closed convex subsets of the infinite-dimensional real Hilbert spaces H1 and H2, respectively, and A∈B(H1,H2) (i.e., A is a bounded linear operator from H1 to H2). Very recently, there are related works which we can find in [16, 18, 20–26] and the references therein.A special case of the SFP is called the convex constrained linear inverse problem (see [27]), that is, the problem to finding an element x such that
(4)x∈C,Ax=b∈Q.
In fact, it has been extensively investigated in the literature using the projected Landweber iterative method [27, 28]. Throughout this paper, we assume that the solution set Γ of the SFP is nonempty.Motivated and inspired by the regularization method and extragradient method due to Ceng et al. [29], we introduce and analyze an extragradient method with regularization for finding a common element of the fixed points set of quasi-nonexpansive mappings and the solution of split feasibility problems (SFP) and systems of equilibrium problems (SEP) in Hilbert spaces. Our results represent the improvement, extension, and development of the corresponding results in [14, 29].


## 2. Preliminaries

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, and let C be a closed convex subset of H. We write xn⇀x to indicate that the sequence {xn} converges weakly to x and xn→x to indicate that the sequence {xn} converges strongly to x. For every point x∈H, there exists a unique nearest point in C, denoted by PCx, such that
(5)∥x-PCx∥=infy∈C∥x-y∥≤∥x-y∥∀y∈C.PC is called the metric projection of H onto C.Some important properties of projections are gathered in the following proposition.Proposition 1 (see [29]).
For givenx∈H and z∈C:  (i)
z=PCxifandonlyif〈x-z,y-z〉≤0,forally∈C;(ii)
z=PCxifandonlyif∥x-z∥2≤∥x-y∥2-∥y-z∥2,forally∈C.Definition 2 (see [30, 31]).
LetT be a nonlinear operator with domain D(T)⊆H and range R(T)⊆H, and let β>0 and v>0 be given constants. The operator T is called(a)
monotone if
(6)〈x-y,Tx-Ty〉≥0,∀x,y∈D(T);(b)
β-strongly  monotone if
(7)〈x-y,Tx-Ty〉≥β∥x-y∥2,∀x,y∈D(T);(c)
v-inverse  strongly  monotone (v-ism) if
(8)〈x-y,Tx-Ty〉≥v∥Tx-Ty∥2,∀x,y∈D(T).We can easily see that ifS is nonexpansive, then I-S is monotone. It is also easy to see that a projection PC is a 1-ism.Definition 3 (see [29]).
A mappingT:H→H is said to be an averaged mapping if it can be written as the average of the identity I and a nonexpansive mapping, that is,
(9)T≡(1-α)I+αS,
where α∈(0,1) and S:H→H is nonexpansive. More precisely, when (9) holds, we say that T is α-averaged. It is easly to see that if T is an averaged mapping, then T is nonexpansive.Proposition 4 (see [20]).
LetT:H→H be a given mapping. Then consider the following.(i)
T is nonexpansive if and only if the complement I-T is (1/2)-ism.(ii)
T is averaged if and only if the complement I-T is v-ism for some v>1/2. Indeed, for α∈(0,1),T is α-averaged if and only if I-T is (1/2α)-ism.(iii)
The composite of finitely many averaged mappings is averaged. That is, if each of the mappings{Ti}Ni=1 is averaged, then so is the composite T1∘T2∘⋯∘TN. In particular, if T1 is α1-averaged and T2isα2-averaged, where α1,α2∈(0,1), then the composite T1∘T2 is α-averaged, where α=α1+α2-α1α2.In this paper, we use an equilibrium bifunctionF:C×C→ℝ for solving the equilibrium problems, let us assume that F satisfies the following conditions:  (A1)
F(x,x)=0forallx∈C;(A2)
F is monotone, that is, F(x,y)+F(y,x)≤0forallx,y∈C;(A3)
for eachy∈C,x↦F(x,y) is weakly upper semicontinuous;(A4)
for eachx∈C,y↦F(x,y) is convex; semicontinuous.Lemma 5 (see [6]).
Assume thatF:C×C→R satisfies (A1)–(A4). For r>0 and x∈H, define a mapping Tr:H→C as follows:
(10)Tr(x)={z∈C:F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C},
for all z∈H. Then, the following hold: (i)
Tr is single-valued;(ii)
Tr is firmly nonexpansive, that is, for any x,y∈H,∥Trx-Try∥2≤〈Trx-Try,x-y〉;(iii)
F(Tr)=EP(F);(iv)
EP(F) is closed and convex.Lemma 6 (see [32]).
LetC be a closed convex subset of a real Hilbert space H. Let F1 and F2 be two mappings from C×C→R satisfying (A1)–(A4) and let T1,λ and T2,μ are defined as in Lemma 5 associated to F1 and F2, respectively. For given x*,y*∈C,(x*,y*) is a solution of problem (2) if and only if x* is a fixed point of the mapping G:C→C defined by
(11)G(x)=T1,λ(T2,μx),∀x∈C,
where y*=T2,μx*.Lemma 7 (see [33]).
Let{an}n=1∞,{bn}n=1∞ and {δn}n=1∞ be sequences of nonnegative real numbers satisfying the inequality
(12)an+1≤(1+δn)an+bn,∀n≥1.
If ∑n=1∞‍δn<∞ and ∑n=1∞‍bn<∞, then limn→∞an exists. If, in addition, {an}n=1∞ has a subsequence which converges to zero, then limn→∞an=0.


## 3. Weak Convergence Theorem

In this section, we prove a weak convergence theorem by an extragradient methods for finding a common element of the fixed points set of quasi-nonexpansive mappings and the solution of split feasibility problems and systems of equilibrium problems in Hilbert spaces. The functionf:H→ℝ is a continuous differentiable function with the minimization problem given by
(13)minx∈Cf(x):=12∥Ax-PQAx∥2.
In 2010, Xu [17] considered the following Tikhonov regularized problem:
(14)minx∈Cfα(x):=12∥Ax-PQAx∥2+12α∥x∥2,
where α>0 is the regularization parameter. The gradient given by
(15)∇fα(x)=∇f(x)+αI=A*(I-PQ)A+αI
is (α+∥A∥2)-Lipschitz continuous and α-strongly monotone (see [29] for the details).Lemma 8 (see [17, 29]).
The following hold:(i)
Γ=F(PC(I-λ∇f))=VI(C,∇f) for any λ>0, where F(PC(I-λ∇f)) and VI(C,∇f) denote the set of fixed points of PC(I-λ∇f) and the solution set of VIP;(ii)
PC(I-λ∇fα) is ξ-averaged for each λ∈(0,2/(α+∥A∥2)), where ξ=(2+λ(α+∥A∥2))/4.Theorem 9.
LetC be a nonempty closed convex subset in a real Hilbert space H. Let μ>0, F1 and F2 be two bifunctions from C×C→ℝ satisfying (A1)–(A4). Let S be a quasi-nonexpansive mapping of C into itself such that I-S be demiclosed at zero, that is, if {wn}⊂C,wn⇀w and (I-S)wn→0, then w∈F(S), with F(S)∩Γ∩Ω≠∅. Let {xn}, {yn}, {zn}, and {wn} be the sequence in C generated by the following extragradient algorithm:
(16)x0=x∈C,wn∈C;F2(wn,z)+φ(z)-φ(wn)+1μ〈z-wn,wn-xn〉≥0,∀z∈C,zn∈C;F1(zn,z)+φ(z)-φ(zn)+1μ〈z-zn,zn-wn〉≥0,∀z∈C,yn=PC(I-λn∇fαn)zn,xn+1=βnxn+(1-βn)SPC(xn-λn∇fαn(yn)),
where ∑n=0∞‍αn<∞, {λn}⊂[a,b] for some a,b∈(0,1/∥A∥2) and {βn}⊂[c,d] for some c,d∈(0,1). Then, the sequences {xn} and {yn} converge weakly to an element x~∈F(S)∩Γ∩Ω.Proof.
By Lemma8, we have that PC(I-λ∇fα) is ξ-averaged for each λ∈(0,2/(α+∥A∥2)), where ξ=(2+λ(α+∥A∥2))/4. Hence, by Proposition 2.4, PC(I-λ∇fα) is nonexpansive. From {λn}⊂[a,b] and a,b∈(0,1/∥A∥2), we have a≤infn≥0λn≤supn≥0λn≤b<1/∥A∥2=limn→∞1/(αn+∥A∥2). Without loss of generality, we assume that a≤infn≥0λn≤supn≥0λn≤b<1/(αn+∥A∥2),foralln≥0. Hence, for each n≥0,PC(I-λn∇fαn) is ξn-averaged with
(17)ξn=12+λn(αn+∥A∥2)2-12·λn(αn+∥A∥2)2=2+λn(αn+∥A∥2)4∈(0,1).
This implies that PC(I-λn∇fαn) is nonexpansive for all n≥0.
Next, we show that the sequence{xn} is bounded. Indeed, take a fixed p∈F(S)∩Γ∩Ω arbitrarily. Let T1,μ and T2,μ be defined as in Lemma 5 associated to F1 and F2, respectively. Thus, we get p=Sp=SPC(p),foralln≥0,p=PC(I-λ∇f)p,forallλ∈(0,2/∥A∥2) and p=T1,μ(T2,μp),forallμ>0. Put y*=T2,μp,zn=T1,μwn and wn=T2,μxn. From (29), we have
∥yn-p∥=∥PC(I-λn∇fαn)zn-PC(I-λn∇f)p∥≤∥PC(I-λn∇fαn)zn-PC(I-λn∇fαn)p∥+∥PC(I-λn∇fαn)p-PC(I-λn∇f)p∥≤∥zn-p∥+∥(I-λn∇fαn)p-(I-λn∇f)p∥=∥zn-p∥+∥λnp(∇f-∇fαn)∥=∥zn-p∥+∥λnαnp∥=∥zn-p∥+λnαn∥p∥.
This implies that ∥zn-p∥=∥T1,μwn-T1,μy*∥≤∥wn-y*∥=∥T2,μxn-T2,μp∥≤∥xn-p∥. Thus, we obtain ∥yn-p∥≤∥xn-p∥+λnαn∥p∥. Put ln=PC(xn-λn∇fαn(yn)) for each n≥0. Then, by Proposition 1(ii), we have
(19)∥ln-p∥2≤∥xn-λn∇fαn(yn)-p∥2-∥xn-λn∇fαn(yn)-ln∥2=∥xn-p∥2-∥xn-ln∥2+2λn〈∇fαn(yn),p-ln〉≤∥xn-p∥2-∥xn-ln∥2+2λn[〈∇fαn(yn),yn-ln〉+〈∇fαnp,p-yn〉]≤∥xn-p∥2-∥xn-ln∥2+2λn[αn〈p,p-yn〉+〈∇fαn(yn),yn-ln〉]≤∥xn-p∥2-∥xn-yn∥2-2〈xn-yn,yn-ln〉-∥yn-ln∥2+2λn[αn〈p,p-yn〉+〈∇fαn(yn),yn-ln〉]=∥xn-p∥2-∥xn-yn∥2-∥yn-ln∥2+2〈xn-λn∇fαn(yn)-yn,ln-yn〉+2λnαn〈p,p-yn〉.
Hence, by Proposition 1(i), we have
(20)〈xn-λn∇fαn(yn)-yn,ln-yn〉=〈xn-λn∇fαn(xn)-yn,ln-yn〉+〈λn∇fαn(xn)-λn∇fαn(yn),ln-yn〉≤〈λn∇fαn(xn)-λn∇fαn(yn),ln-yn〉≤λn∥∇fαn(xn)-λn∇fαn(yn)∥∥ln-yn∥≤λn(αn+∥A∥2)∥xn-yn∥∥ln-yn∥.
So, we have
(21)∥ln-p∥2≤∥xn-p∥2-∥xn-yn∥2-∥yn-ln∥2+2λn(αn+∥A∥2)∥xn-yn∥∥ln-yn∥+2λnαn∥p∥∥p-yn∥≤∥xn-p∥2-∥xn-yn∥2-∥yn-ln∥2+λn2(αn+∥A∥2)2∥xn-yn∥2+∥ln-yn∥2+2λnαn∥p∥∥p-yn∥=∥xn-p∥2+2λnαn∥p∥∥p-yn∥+[λn2(αn+∥A∥2)2-1]∥xn-yn∥2≤∥xn-p∥2+2λnαn∥p∥∥p-yn∥≤∥xn-p∥2+2λnαn∥p∥[∥zn-p∥+λnαn∥p∥]≤∥xn-p∥2+2λnαn∥p∥[∥xn-p∥+λnαn∥p∥].
Then, from the last inequality we conclude that
∥xn+1-p∥2=∥βnxn+(1-βn)S(ln)-p∥2≤βn∥xn-p∥2+(1-βn)∥S(ln)-p∥2-βn(1-βn)∥xn-S(ln)∥2≤βn∥xn-p∥2+(1-βn)∥ln-p∥2-βn(1-βn)∥xn-S(ln)∥2≤βn∥xn-p∥2+(1-βn)[(αn+∥A∥2)2[λn2(αn+∥A∥2)2-1]∥xn-p∥2+2λnαn∥p∥∥p-yn∥+(1-βn)m+[λn2(αn+∥A∥2)2-1]∥xn-yn∥2]-βn(1-βn)∥xn-S(ln)∥2≤∥xn-p∥2+2λnαn∥p∥∥p-yn∥+(1-βn)(λn2(αn+∥A∥2)2-1)∥xn-yn∥2-βn(1-βn)∥xn-S(ln)∥2≤∥xn-p∥2+αn(λn2∥p∥2+∥p-yn∥2)+(1-βn)(λn2(αn+∥A∥2)2-1)∥xn-yn∥2-βn(1-βn)∥xn-S(ln)∥2≤∥xn-p∥2+αn[λn2∥p∥2+(∥xn-p∥+λnαn∥p∥)2]+(1-βn)(λn2(αn+∥A∥2)2-1)∥xn-yn∥2-βn(1-βn)∥xn-S(ln)∥2≤∥xn-p∥2+αn[λn2∥p∥2+2∥xn-p∥2+2λn2αn2∥p∥2]+(1-βn)(λn2(αn+∥A∥2)2-1)∥xn-yn∥2-βn(1-βn)∥xn-S(ln)∥2=(1+2αn)∥xn-p∥2+αnλn2∥p∥2(1+2αn2)+(1-βn)(λn2(αn+∥A∥2)2-1)∥xn-yn∥2-βn(1-βn)∥xn-S(ln)∥2≤(1+2αn)∥xn-p∥2+αnλn2∥p∥2(1+2αn2)=(1+δn)∥xn-p∥2+bn,
where δn=2αn and bn=αnλn2∥p∥2(1+2αn2). Since ∑n=0∞‍αn<∞ and {λn}⊂[a,b] for some a,b∈(0,1/∥A∥2), we conclude that ∑n=0∞‍δn<∞ and ∑n=0∞‍bn<∞. Therefore, by Lemma 7, we note that limn→∞∥xn-p∥ exists for each p∈F(S)∩Γ∩Ω and hence the sequences {xn},{ln},{yn},{zn}, and {wn} are bounded. From (22), we also obtain
(23)(1-d)(1-b2(αn+∥A∥2)2)∥xn-yn∥2+c(1-d)∥xn-S(ln)∥2≤(1-βn)(1-λn2(αn+∥A∥2)2)∥xn-yn∥2+βn(1-βn)∥xn-S(ln)∥2≤(1+2αn)∥xn-p∥2-∥xn+1-p∥2+αnλn2∥p∥2(1+2αn2),
where {λn}⊂[a,b] and {βn}⊂[c,d]. Since limn→∞∥xn-p∥ exists and αn→0, it follows that
(24)limn→∞∥xn-yn∥=limn→∞∥xn-S(ln)∥≤limn→∞[∥xn-p∥+∥p-S(ln)∥]≤limn→∞[∥xn-p∥+∥p-ln∥]=0.
Similarly, from inequality (22), we have
(25)(1-d)(1-b2(αn+∥A∥2)2)∥xn-zn∥2+c(1-d)∥xn-S(ln)∥2≤(1-βn)(1-λn2(αn+∥A∥2)2)∥xn-zn∥2+βn(1-βn)∥xn-S(ln)∥2≤(1+αn)∥xn-p∥2-∥xn+1-p∥2+αnλn2∥p∥2,
where {λn}⊂[a,b] and {βn}⊂[c,d]. Since limn→∞∥xn-p∥ exists and αn→0, we obtain
(26)limn→∞∥xn-zn∥=limn→∞∥xn-S(ln)∥=0.
Moreover, we note that
(27)∥yn-ln∥=∥PC(zn-λn∇fαnzn)-PC(xn-λn∇fαn(yn))∥≤∥(zn-λn∇fαn(zn))-(xn-λn∇fαn(yn))∥=∥zn-xn-(λn∇fαn(zn)-λn∇fαn(yn))∥≤∥zn-xn∥+∥λn∇fαn(zn)-λn∇fαn(yn)∥≤∥zn-xn∥+λn[∥∇fαn(zn)-∇fαn(xn)∥∥zn-xn∥+λnm+∥∇fαn(xn)-∇fαn(yn)∥]≤∥zn-xn∥+λn[(αn+∥A∥2)∥xn-yn∥∥zn-xn∥+λnm+∥∇fαn(zn)-∇fαn(xn)∥].
From (26) and αn→0, it is implied that
(28)limn→∞∥yn-ln∥=0.
Note that ∥ln-S(ln)∥≤∥ln-yn∥+∥yn-xn∥+∥xn-S(ln)∥. This together with (24) and (28) implies that limn→∞∥ln-S(ln)∥=0. Also, from ∥xn-ln∥≤∥xn-yn∥+∥yn-ln∥, it follows that limn→∞∥xn-ln∥=0. Since ∇f=A*(I-PC)A is a Lipschitz condition, where A* is the adjoint of A, we have limn→∞∥∇f(yn)-∇f(ln)∥=0. Since {xn} is a bounded sequence, there exists a subsequence {xnj} of {xn} that converges weakly to some x^.
Next, we show thatx^∈Γ. Since ∥xn-ln∥→0 and ∥yn-ln∥→0, it is known that lni⇀x^ and yni⇀x^. Let T:H→2H be a set value mappings defined by
(29)Tv={∇f(v)+NCvifv∈C,∅ifv∉C,
where NCv={w∈H1:〈v-u,w〉≥0,forallu∈C}. Hence, by [34], T is maximal monotone and 0∈Tv if and only if v∈VI(C,∇f). Let (v,w)∈G(T). Then we have w∈Tv=∇f(v)+NCv and hence w-∇f(v)∈NCv. So, we obtain 〈v-u,w-∇f(v)〉≥0,forallu∈C. On the other hand, from ln=PC(xn-λn∇fαn(yn)) and v∈C, we have
(30)〈xn-λn∇fαn(yn)-ln,ln-v〉≥0,
and hence
(31)〈v-ln,ln-xnλn+∇fαn(yn)〉≥0.
Therefore, from w-∇f(v)∈NCv and lni∈C, we get
(32)〈v-lni,w〉≥〈v-lni,∇f(v)〉≥〈v-lni,∇f(v)〉-〈v-lni,lni-xniλni+∇fαni(yni)〉=〈v-lni,∇f(v)〉-〈v-lni,lni-xniλni+∇f(yni)〉-αni〈v-lni,yni〉=〈v-lni,∇f(v)-∇f(lni)〉+〈v-lni,∇f(lni)-∇f(yni)〉-〈v-lni,lni-xniλni〉-αni〈v-lni,yni〉≥〈v-lni,∇f(lni)-∇f(yni)〉-〈v-lni,lni-xniλni〉-αni〈v-lni,yni〉.
By taking i→∞, we obtain 〈v-x^,w〉≥0. Since T is maximal monotone, it follows that x^∈T-10 and hence x^∈VI(C,∇f). Therefore, by Lemma 8, x^∈Γ.
Next, we show thatx^∈F(S). Since lni⇀x^ and ∥lni-S(lni)∥→0, it follows by the demiclosed principle that x^∈F(S). Hence, we have x^∈F(S)∩Γ.
Next, we show thatx^∈Ω. Let G be a mapping which is defined as in Lemma 6, thus we have
(33)∥zn-G(zn)∥=∥T1,μT2,μxn-G(zn)∥=∥G(xn)-G(zn)∥≤∥xn-zn∥,
and hence
(34)∥xn-G(xn)∥≤∥xn-zn∥+∥zn-G(zn)∥+∥G(zn)-G(xn)∥≤∥xn-zn∥+∥xn-zn∥+∥zn-xn∥=3∥xn-zn∥.
By taking n→∞, we have ∥xn-G(xn)∥→0. From limn→∞∥xn-zn∥=0 and znj⇀x^, we obtain xnj⇀x^. According to demiclosedness and Lemma 6, we have x^∈Ω. Therefore, we have x^∈F(S)∩Γ∩Ω. Let {xnj} be another subsequence of {xn} such that xnj⇀x^. We show that x^=x~, suppose that x^=x~. Since limn→∞∥xn-x^∥ exists for all x^∈F(S)∩Γ∩Ω, it follows by the Opial's condition that
(35)limn→∞∥xn-x^∥=liminfi→∞∥xni-x^∥<liminfi→∞∥xni-x~∥=limn→∞∥xn-x~∥=liminfj→∞∥xnj-x~∥<liminfj→∞∥xnj-x^∥=limn→∞∥xnjxn-x^∥.
It is a contradiction. Thus, we have x^=x~ and so xn⇀x~∈F(S)∩Γ∩Ω. Further, from ∥xn-yn∥→0, it follows that yn⇀x~ and hence zn, wn. This completes the proof.Theorem9 extends the extragradient method according to Nadezhkina and Takahashi [35].Corollary 10.
LetC be a nonempty closed convex subset in a real Hilbert space H. Let F1 and F2 be two bifunctions from C×C→ℝ satisfying (A1)–(A4). Let μ>0 and let T1,μ and T2,μ be defined as in Lemma 5 associated to F1 and F2, respectively. Let S be a quasi-nonexpansive mapping of C into itself such that F(S)∩Γ∩Ω≠∅. Let {xn}, {yn}, {zn}, and {wn} be the sequence in C generated by the following extragradient algorithm:
(36)x0=x∈C,wn∈C;F2(wn,z)+1μ〈z-wn,wn-xn〉≥0,∀z∈C,zn∈C;F1(zn,z)+1μ〈z-zn,zn-wn〉≥0,∀z∈C,yn=PC(I-λn∇f)zn,xn+1=βnxn+(1-βn)SPC(xn-λn∇f(yn)),
where ∑n=0∞‍αn<∞, {λn}⊂[a,b] for some a,b∈(0,1/∥A∥2) and {βn}⊂[c,d] for some c,d∈(0,1). Then, the sequences {xn} and {yn} converge weakly to an element x^∈F(S)∩Γ∩Ω.Corollary 11.
LetC be a nonempty closed convex subset in a real Hilbert space H. Let F1 and F2 be two bifunctions from C×C→ℝ satisfying (A1)–(A4). Let μ>0 and let T1,μ and T2,μ be defined as in Lemma 5 associated to F1 and F2, respectively. Let S be a quasi-nonexpansive mapping of C into itself such that F(S)∩Γ∩Ω≠∅. Let {xn}, {yn}, and {zn} be the sequence in C generated by the following extragradient algorithm:
(37)x0=x∈C,zn∈C;F1(zn,z)+φ(z)-φ(zn)+1μ〈z-zn,zn-xn〉≥0,∀z∈C,yn=PC(I-λn∇fαn)zn,xn+1=βnxn+(1-βn)SPC(xn-λn∇fαn(yn)),
where ∑n=0∞‍αn<∞, {λn}⊂[a,b] for some a,b∈(0,1/∥A∥2) and {βn}⊂[c,d] for some c,d∈(0,1). Then, the sequences {xn} and {yn} converge weakly to an element x^∈F(S)∩Γ∩Ω.Proof.
SettingF2=0 in Theorem 9, we obtain the desired result.Corollary 12.
LetC be a nonempty closed convex subset in a real Hilbert space H. Let S be a quasi-nonexpansive mapping of C into itself such that F(S)∩Γ≠∅. Let {xn}, {yn}, and {zn} be the sequence in C generated by the following extragradient algorithm:
(38)x0=x∈C,yn=PC(I-λn∇fαn)xn,xn+1=βnxn+(1-βn)SPC(xn-λn∇fαn(yn)),
where ∑n=0∞‍αn<∞, {λn}⊂[a,b] for some a,b∈(0,1/∥A∥2) and {βn}⊂[c,d] for some c,d∈(0,1). Then, the sequences {xn} and {yn} converge weakly to an element x^∈F(S)∩Γ.Proof.
SettingF1=F2=0 in Theorem 9, we obtain the desired result.



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*Source: 430409-2013-02-26.xml*