# Viscosity Approximation Methods for Equilibrium Problems, Variational Inequality Problems of Infinitely Strict Pseudocontractions in Hilbert Spaces

Authors: Aihong Wang
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of the solutions of the equilibrium problem and the set of fixed points of infinitely strict pseudocontractive mappings. Strong convergence theorems are established in Hilbert spaces. Our results improve and extend the corresponding results announced by many others recently.


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## Body

## 1. Introduction

LetH be a real Hilbert space and let C be a nonempty convex subset of H.A mappingS of C is said to be a κ-strict pseudocontraction if there exists a constant κ∈[0,1) such that
(1.1)∥Sx-Sy∥2≤∥x-y∥2+κ∥(I-S)x-(I-S)y∥2,
for all x,y∈C; see [1]. We denote the set of fixed points S by F(S) (i.e., F(S)={x∈C:Sx=x}).Note that the class of strict pseudocontraction strictly includes the class of nonexpansive mappings which are mappingsS on C such that
(1.2)∥Sx-Sy∥≤∥x-y∥,
for all x,y∈C. That is, S is nonexpansive if and only if S is a 0-strict pseudocontraction. Let Φ be a bifunction from C×C to ℝ, where ℝ is the set of real numbers. The equilibrium problem for Φ:C×C→ℝ is to find x∈C such that
(1.3)Φ(x,y)≥0,∀y∈C.
The set of solutions of (1.3) is denoted by EP(Φ). Given a mapping B:C→H, let Φ(x,y)=〈Bx,y-x〉 for all x,y∈C. Then the classical variational inequality problem is to find x∈C such that 〈Bx,y-x〉≥0. We denote the solution of the variational inequality by VI(C,B); that is
(1.4)VI(C,B)={x∈C:〈Bx,y-x〉≥0}.
Let A be a strongly positive linear-bounded operator on H if there is a constant γ->0 with property
(1.5)〈Ax,x〉≥γ-∥x∥2,∀x∈H.
A typical problem is to minimize a quadratic function over the set of the fixed points a nonexpansive mapping on a real Hilbert space H:
(1.6)minx∈E12〈Ax,x〉-〈x,b〉,
where A is a linear-bounded operator, E is the fixed point set of a nonexpansive mapping S on H, and b is a given point in H. The problem (1.3) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others; see [1–11]. In particular, Combettes and Hirstoaga [4] proposed several methods for solving the equilibrium problem. On the other hand, Mann [6], Shimoji and Takahashi [8] considered iterative schemes for finding a fixed point of a nonexpansive mapping. Further, Acedo and Xu [12] projected new iterative methods for finding a fixed point of strict pseudocontractions.In 2006, Marino and Xu [7] introduced the general iterative method: for x1=x∈C,
(1.7)xn+1=αnγf(xn)+(I-αnA)Txn,n≥1.
They proved that the sequence {xn} of parameters satisfies appropriate condition and that the sequence {xn} generated by (1.7) converges strongly to the unique solution of the variational inequality 〈(γf-A)q,p-q〉≤0,p∈F(T). Recently, Liu [5] considered a general iterative method for equilibrium problems and strict pseudocontractions:
(1.8)Φ(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,yn=βnun+(1-βn)Sun,xn+1=εnγf(xn)+(I-εnA)un,∀n≥1,
where S is a k-strict pseudocondition mapping and {εn}, {βn} are sequences in (0,1). They proved that under certain appropriate conditions over {εn}, {βn}, and {rn}, the sequences {xn} and {un} both converge strongly to some q∈F(S)⋂‍EP(Φ), which solves some variational inequality problems. Tian [10] proposed a new general iterative algorithm: for nonexpansive mapping T:H→H with F(T)≠ϕ,
(1.9)xn+1=αnγf(xn)+(I-μαnF)Txn,∀n≥1,
where F is a k-Lipschitzian and η-strong monotone operator. He obtained that the sequence xn generated by (1.9) converges to a point q in F(T), which is the unique solution of the variational inequality 〈(γf-A)q,p-q〉≤0,p∈F(T). Very recently, Wang [13] considered a general composite iterative method for infinite family strict pseudocontractions: for x1=x∈C,
(1.10)yn=βnxn+(1-βn)Wnxn,xn+1=αnγf(xn)+(I-μαnF)yn,∀n≥1,
where Wn is a mapping defined by (2.5), F is a k-Lipschitzian, and η-strongly monotone operator. With some appropriate condition, the sequence {xn} generated by (1.10) converges strongly to a common element of the fixed point of an infinite family of λi-strictly pseudocontractive mapping, which is a unique solution of the variational inequality 〈(γf-A)q,p-q〉≤0,p∈F(T). Kumam proposed many algorithms for the equilibrium and the fixed point problems with k-strict pseudoconditions; see [14–16]. In particular, in 2011, Kumam and Jaiboon [14] considered a system of mixed equilibrium problems, variational inequality problems, and strict pseudocontractive mappings:
(1.11)x1∈E,un∈E,vn∈E,un=Trϕ1,φxn,vn=Tsϕ2,φxn,zn=PE(un-μnCun),yn=PE(vn-λnCvn),kn=anSkxn+bnyn+cnzn,xn+1=εnγf(xn)+βnxn+((1-βn)I-εnA)kn,∀n≥1,
where S is a k-strict pseudocondition mapping. They proved that under certain appropriate conditions over {εn}, {βn}, {rn}, {an}, {bn}, {cn}, {λn}, and {μn}, the sequence {xn} converges strongly to a point q∈Θ which is the unique solution of the variational inequality 〈(A-γf)q,x-q〉≥0. Inprasit [17] proposed a viscosity approximation methods to solving the generalized equilibrium and fixed point problems of finite family of nonexpansive mapping in Hilbert spaces.In this paper, motivated by the above facts, we use the viscosity approximation method to find a common element of the set of solutions of the equilibrium problemVI(C,B) and the set of fixed points of a infinite family of strict pseudocontractions.


## 2. Preliminaries

Throughout this paper, we always write⇀ for weak convergence and → for strong convergence. We need some facts and tools in a real Hilbert space H which are listed as below.Lemma 2.1.
LetH be a real Hilbert space. There hold the following identities:(i)
∥x-y∥2=∥x∥2-∥y∥2-2〈x-y,y〉,∀x,y∈H,(ii)
∥tx+(1-t)y∥2=t∥x∥2+(1-t)∥y∥2-t(1-t)∥x-y∥2,∀t∈[0,1],∀x,y∈H.Lemma 2.2 (see [18]).
Assume that{αn} is a sequence of nonnegative real numbers such that
(2.1)αn+1≤(1-ρn)αn+σn,
where {ρn} is a sequence in (0,1) and {σn} is a sequence such that(i)
∑n=1∞‍ρn=∞,(ii)
limsupn→∞(σn/ρn)≤0 or ∑n=1∞‍|σn|<∞.
Thenlimn→∞an=0.Recall that given a nonempty closed convex subsetC of a real Hilbert space H, for any x∈H, there exists a unique nearest point in C, denoted by PCx, such that
(2.2)∥x-PCx∥≤∥x-y∥,
for all y∈C. Such a PC is called the metric (or the nearest point) projection of H onto C. As we all know, y=PCx if and only if there holds the relation:
(2.3)〈x-y,y-z〉≥0∀z∈C.Lemma 2.3 (see [13]).
LetA:H→H be an L-Lipschitzian and η-strongly monotone operator on a Hilbert space H with L>0,η>0,0<μ<2η/L2, and 0<t<1. Then S=(I-tμA):H→H is a contraction with contractive coefficient 1-tτ and τ=(1/2)μ(2η-μL2).Lemma 2.4 (see [1]).
LetS:C→C be a κ-strict pseudocontraction. Define T:C→C by Tx=λx+(1-λ)Sx for each x∈C. Then, as λ∈[κ,1), T is a nonexpansive mapping such that F(T)=F(S).Lemma 2.5 (see [10]).
LetH be a Hilbert space and f:H→H a contraction with coefficient 0<α<1, and A:H→H an L-Lipschitzian continuous operator and η-strongly monotone with L>0,η>0. Then for 0<γ<μη/α,
(2.4)〈x-y,(μA-γf)x-(μA-γf)y〉≥(μη-γα)∥x-y∥2,x,y∈H.
That is, μA-γf is strongly monotone with coefficient μη-γα.Let{Sn} be a sequence of κn-strict pseudo-concontractions. Define Sn′=θnI+(1-θn)Sn,θn∈[κn,1). Then, by Lemma 2.4, Sn′ is nonexpansive. In this paper, we consider the mapping Wn defined by
(2.5)Un,n+1=I,Un,n=tnSn′Un,n+1+(1-tn)I,Un,n-1=tn-1Sn-1′Un,n+(1-tn-1)I,⋯,Un,i=tiSi′Un,i+1+(1-ti)I,⋯,Un,2=t2S2′Un,3+(1-t2)I,Wn=Un,1=t1S1′Un,2+(1-t1)I.Lemma 2.6 (see [8]).
LetC be a nonempty closed convex subset of a strictly convex Banach space E, let S1′,S2′,… be nonexpansive mappings of C into itself such that ∩i=1∞F(Si′)≠∅, and let t1,t2,… be real numbers such that 0<ti≤b<1, for every i=1,2,…. Then, for any x∈C and k∈N, the limit limn→∞Un,kx exists.Using Lemma2.6, one can define the mapping W of C into itself as follows:
(2.6)Wx:=limn→∞Wnx=limn→∞Un,1x,x∈C.Lemma 2.7 (see [8]).
LetC be a nonempty closed convex subset of a strictly convex Banach space E. Let S1′,S2′,… be nonexpansive mappings of C into itself such that ∩i=1∞F(Si′)≠∅, and let t1,t2,… be real numbers such that 0<ti≤b<1, for all i≥1. If K is any bounded subset of C, then
(2.7)limn→∞supx∈K∥Wx-Wnx∥=0.Lemma 2.8 (see [3]).
LetC be a nonempty closed convex subset of a Hilbert space H,let{Si′:C→C} be a family of infinite nonexpansive mappings with ∩i=1∞F(Si′)≠∅, and let t1,t2,… be real numbers such that 0<ti≤b<1, for every i=1,2,…. Then F(W)=∩i=1∞F(Si′).For solving the equilibrium problem, let us assume that the bifunctionΦ satisfies the following conditions:(A1)
Φ(x,x)=0 for all x∈C;(A2)
Φ is monotone; that is Φ(x,y)+Φ(y,x)≤0 for any x,y∈C;(A3)
for eachx,y,z∈C,limsupt→0Φ(tz+(1-t)x,y)≤Φ(x,y);(A4)
Φ(x,·) is convex and lower semicontinuous for each x∈C.We recall some lemmas which will be needed in the rest of this paper.Lemma 2.9 (see [2]).
LetC be a nonempty closed convex subset of H, let Φ be bifunction from C×C to ℝ satisfying (A1)–(A4), and let r>0 and x∈H. Then there exists  z∈C such that
(2.8)Φ(z,y)+1r〈y-z,z-x〉≥0,∀y∈C.Lemma 2.10 (see [4]).
Letϕ be a bifunction from C×C into ℝ satisfying (A1)–(A4). Then, for any r>0 and x∈H, there  exists  z∈C such that
(2.9)Φ(z,y)+1r〈y-z,z-x〉≥0,∀y∈C.
Further, if Trx={z∈C;Φ(z,y)+(1/r)〈y-z,z-x〉≥0,∀y∈C}, then the following hold:(1)
Tr is single-valued;(2)
Tr is firmly nonexpansive;(3)
F(Tr)=EP(ϕ);(4)
EP(ϕ) is closed and convex.Lemma 2.11 (see [9]).
Let{xn} and {zn} be bounded sequences in a Banach space, and let {βn} be a sequence of real numbers such that 0<liminfn→∞βn≤limsupn→∞βn<1 for all n=0,1,2,…. Suppose that xn+1=(1-βn)zn+βnxn for all n=0,1,2,…. and limsupn→∞∥zn+1-zn∥-∥xn+1-xn∥≤0. Then limn→∞∥zn-xn∥=0.Lemma 2.12 (see [11]).
LetC,H,F, and Trx be as in Lemma 2.9. Then the following holds:
(2.10)∥Tsx-Ttx∥2≤〈Tsx-Ttx,Tsx-x〉,
for all s,t>0, and x∈H.Lemma 2.13 (see [13]).
LetH be a Hilbert space, and let C be a nonempty closed convex subset of H, and T:C→C a nonexpansive mapping with F(T)≠∅. If {xn} is a sequence in C weakly converging to x and if {(I-T)xn} converges strongly to y, then (I-T)x=y.


## 3. Main Results

Now we start and prove our main result of this paper.Theorem 3.1.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let ϕ be a bifunction from C×C→ℝ satisfying (A1)–(A4). Let Si:C→C be a family κi-strict pseudocontractions for some 0≤κi<1. Assume the set Ω=VI(C,B)∩i=1∞F(Si)∩EP(ϕ)≠∅. Let f be a contraction of H into itself with α∈(0,1), and let A be a strongly positive linear bounded operator on H with coefficient γ>0 and 0<γ<γ-/γ. Let B:C→H be an ξ-inverse strongly monotone mapping. Let Wn be the mapping generated by Si′ and ti as in (2.5). Let {xn} be a sequence generated by the following algorithm:
(3.1)ϕ(zn,y)+1λn〈y-zn,zn-xn〉≥0,yn=PC(I-μnB)zn,Kn=αnxn+(1-αn)Wnyn,xn+1=εnγf(xn)+βnxn+((1-βn)I-εnA)Kn,
where {εn}, {βn}, {αn}, and {λn} are sequences in (0,1). Assume that the control sequences satisfy the following restrictions:(i)
limn→∞εn=0 and Σn=1∞εn=∞;(ii)
0<liminfn→∞βn≤limsupn→∞βn<1;(iii)
0<limn→∞(λn/λn+1)=1;(iv)
limn→∞|αn+1-αn|=0;(v)
0<μn≤2ξ;(vi)
limn→∞αn=a.
Then{xn} converges strongly to q∈Ω which is the unique solution of the variational inequality
(3.2)〈(A-γf)q,x-q〉≥0,∀x∈Ω,
or equivalent q=PΩ(I-A+γf)(q), where P is a metric projection mapping form H onto Ω.Proof.
Sinceεn→0, as n→∞, we may assume, without loss of generality, that εn≤(1-βn)∥A∥-1 for all n∈ℕ. By Lemma 2.3, we know that if 0≤ρ≤∥A∥-1, then ∥I-ρA∥≤1-ργ-. We will assume that ∥I-A∥≤1-γ-. Since A is a strongly positive bounded linear operator on H, we have
(3.3)∥A∥=sup{|〈Ax,x〉|:x⊆H,∥x∥=1}.
Observe that
(3.4)〈((1-βn)I-εnA)x,x〉=(1-βn)∥x∥2-εn〈Ax,x〉≥(1-βn)∥x∥2-εn∥A∥≥0.
So this shows that (1-βn)I-εnA is positive. It follows that
(3.5)∥(1-βn)I-εnA∥=sup{|〈((1-βn)I-εnA)x,x〉|:x⊆H,∥x∥=1}=sup{1-βn-εn〈Ax,x〉x⊆H,∥x∥=1}≤1-βn-εnγ-.
Step1. We claim that the mapping PΩ(I-A+γf) where Ω=⋂i=1∞‍F(Si)⋂‍EP(Φ) has a unique fixed point. Let  f be a contraction of H into itself with α∈(0,1). Then, we have
(3.6)∥PC(I-A+γf)(x)-PC(I-A+γf)(y)∥≤∥(I-A+γf)(x)-(I-A+γf)(y)∥≤∥I-A∥∥x-y∥+γ∥f(x)-f(y)∥≤(1-γ-)∥x-y∥+γα∥x-y∥=(1-(γ--γα))∥x-y∥,
for all x,y∈H. Since 0<1-(γ--γα)<1, it follows that PΩ(I-A+γf) is a contraction of H into itself. Therefore the Banach contraction mapping principle implies that there exists a unique element q∈H such that q=PΩ(I-A+γf)(q).
Step2. We shall show that (I-μnB) is nonexpansive. Let x,y∈C. Since B is ξ-inverse strongly monotone and λn<2α for all n∈ℕ, we obtain
(3.7)∥(I-μnB)x-(I-μnB)y∥2=∥x-y-μn(Bx-By)∥2=∥x-y∥2-2μn〈x-y,Bx-By〉+μn2∥Bx-By∥2≤∥x-y∥2-2ξμn∥Bx-By∥2+μn2∥Bx-By∥2=∥x-y∥2+μn(μn-2ξ)∥Bx-By∥2≤∥x-y∥2,
where μn≤2ξ, for all n∈N. So we have that the mapping (I-λnA) is nonexpansive.
Step 3. We claim that {xn} is bounded.
Letp∈Ω; from Lemma 2.10, we have
(3.8)p=PC(p-μnBp)=Tλnp,∥zn-p∥=∥Tλnxn-Tλnp∥≤∥xn-p∥.
Note that
(3.9)∥yn-p∥=∥PC(I-μnB)zn-p∥=∥PC(I-μnB)zn-PC(I-μnB)p∥≤∥(zn-μnBzn)-(p-μnBp)∥=∥(I-μnB)(zn-p)∥≤∥zn-p∥≤∥xn-p∥,∥Kn-p∥=∥αnxn+(1-αn)Wnyn-p∥=∥αn(xn-p)+(1-αn)(Wnyn-p)∥≤αn∥xn-p∥+(1-αn)∥Wnyn-p∥≤αn∥xn-p∥+(1-αn)∥yn-p∥≤∥xn-p∥.
It follows that
(3.10)∥xn+1-p∥=∥εnγf(xn)+βnxn+((1-βn)I-εnA)Kn-p∥=∥εn(γf(xn)-Ap)+βn(xn-p)+((1-βn)I-εnA)(Kn-p)∥≤εn∥γf(xn)-Ap∥+βn∥xn-p∥+(1-βn-εnγ-)∥Kn-p∥≤εn∥γf(xn)-Ap∥+βn∥xn-p∥+(1-βn-εnγ-)∥xn-p∥≤(1-εnγ-)∥xn-p∥+εnγ∥f(xn)-f(p)∥+εn∥γf(p)-Ap∥≤(1-εnγ-)∥xn-p∥+εnγα∥xn-p∥+εn∥γf(p)-Ap∥=(1-(γ--αγ)εn)∥xn-p∥+(γ--αγ)εn∥γf(p)-Ap∥γ--αγ≤max{∥xn-p∥,∥γf(p)-Ap∥γ--αγ}.
By simple induction, we have
(3.11)∥xn-p∥≤max{∥x1-p∥,∥γf(p)-Ap∥γ--αγ},∀n∈ℕ.
Hence {xn} is bounded. This implies that {Kn}, {f(xn)} are also bounded.
Step 4. Show that limsupn→∞∥xn+1-xn∥=0.
Observing thatzn=Tλnxn and zn+1=Tλn+1, we get
(3.12)∥zn+1-zn∥=∥Tλn+1xn+1-Tλnxn∥=∥Tλn+1xn+1-Tλn+1xn+Tλn+1xn-Tλnxn∥≤∥xn+1-xn∥+∥Tλn+1xn-Tλnxn∥.
By Lemma 2.10, we obtain
(3.13)Φ(Tλnxn,y)+1λn〈y-Tλnxn,Tλnun-xn〉≥0,∀y∈C,Φ(Tλn+1xn,y)+1λn+1〈y-Tλn+1xn,Tλn+1xn-xn〉≥0,∀y∈C.
In particular, we have
(3.14)Φ(Tλnxn,Tλn+1xn)+1λn〈Tλn+1xn-Tλnxn,Tλnxn-xn〉≥0,Φ(Tλn+1xn,Tλnxn)+1λn+1〈Tλnxn-Tλn+1xn,Tλn+1xn-xn〉≥0.
Summing up (3.14) and using (A2),  we obtain(3.15)1λn+1〈Tλnxn-Tλn+1xn,Tλn+1xn-xn〉+1λn〈Tλn+1xn-Tλnxn,Tλnxn-xn〉≥0,
for all y∈C. It follows that
(3.16)〈Tλnxn-Tλn+1xn,Tλn+1xn-xnλn+1-Tλnxn-xnλn〉≥0.
This implies
(3.17)0≤〈Tλn+1xn-Tλnxn,Tλnxn-xn-λnλn+1(Tλn+1xn-xn)〉=〈Tλn+1xn-Tλnxn,Tλnxn-Tλn+1xn+(1-λnλn+1)(Tλn+1xn-xn)〉.
It follows that
(3.18)∥Tλn+1xn-Tλnxn∥2≤|1-λnλn+1|∥Tλn+1xn-Tλnxn∥(∥Tλn+1xn∥+∥xn∥).
Hence, we obtain
(3.19)∥Tλn+1xn-Tλnxn∥2≤|1-λnλn+1|L,
where L=sup{∥xn∥+∥Tλn+1xn∥:n∈ℕ}. By (3.12) and (3.19), we obtain
(3.20)∥zn+1-zn∥≤∥un+1-un∥+∥Tλn+1un-Tλnun∥≤∥xn+1-xn∥+|1-λnλn+1|L,∥yn+1-yn∥=∥PC(I-μnB)zn+1-PC(I-μnB)zn∥≤∥(I-μnB)(zn+1-zn)∥≤∥zn+1-zn∥≤∥xn+1-xn∥+|1-λnλn+1|L.
From (2.5), we have
(3.21)∥Wn+1yn-Wnyn∥=∥t1S1′Un+1,2yn-t1S1′Un,2yn∥≤t1∥Un+1,2yn-Un,2∥≤t1∥t2S2′Un+1,3yn-t2S2′Un,3yn∥≤t1t2∥Un+1,3yn-Un,3yn∥≤⋯≤M1∏i=1n‍ti,
where M1=supn{∥Un+1,n+1yn-Un,n+1yn∥}.
Note that(3.22)∥Kn+1-Kn∥=∥αn+1xn+1+(1-αn+1)Wn+1yn+1-αnxn-(1-αn)Wnyn∥=∥αn+1(xn+1-xn)+αn+1xn+(1-αn+1)(Wn+1yn+1-Wnyn)-αnxn+(1-αn+1)Wnyn-(1-αn)Wnyn∥=∥αn+1(xn+1-xn)+(αn+1-αn)xn(1-αn+1)(Wn+1yn+1-Wnyn)+(αn-αn+1)Wnyn∥≤αn+1∥xn+1-xn∥+|αn+1-αn|∥xn∥+(1-αn+1)∥Wn+1yn+1-Wnyn∥+|αn-αn+1|∥Wnyn∥≤αn+1∥xn+1-xn∥+|αn+1-αn|∥xn∥+(1-αn+1)[∥Wn+1yn+1-Wn+1yn∥+∥Wn+1yn-Wnyn∥]+|αn-αn+1|∥Wnyn∥≤αn+1∥xn+1-xn∥+|αn+1-αn|∥xn∥+(1-αn+1)[∥xn+1-xn∥+(1-αn+1)|1-λnλn+1|L+M1∏i=1n‍ti]+|αn-αn+1|∥Wnyn∥≤∥xn+1-xn∥+2|αn+1-αn|M2′+(1-αn+1)(M1∏i=1n‍ti+|1-λnλn+1|L),
where M2′=sup{∥xn∥,∥Wnyn∥}.
Supposexn+1=βnxn+(1-βn)ln, then ln=(xn+1-βnxn)/(1-βn) =(εnγf(xn)+((1-βn)I-εnF)Kn)/(1-βn).
Hence, we have(3.23)ln+1-ln=εn+1γf(xn+1)+((1-βn+1)I-εnA)Kn+11-βn+1-εnγf(xn)+((1-βn)I-εnA)Kn1-βn=εn+11-βn+1γf(xn+1)-εn1-βnγf(xn)+Kn+1-Kn+εn1-βnAKn-εn+11-βn+1AKn+1=εn+11-βn+1(γf(xn+1-AKn+1))+εn1-βn(AKn-γf(xn))+Kn+1-Kn.
Then
(3.24)∥ln+1-ln∥-∥xn+1-xn∥≤εn+11-βn+1(∥γf(xn+1)∥+∥AKn+1∥)+εn1-βn(∥AKn∥+∥γf(xn)∥)+∥Kn+1-Kn∥-∥xn+1-xn∥≤εn+11-βn+1(∥γf(xn+1)∥+∥AKn+1∥)+εn1-βn(∥AKn∥+∥γf(xn)∥)+2|αn+1-αn|M2′+(1-αn+1)(|1-λnλn+1|L+M1∏i=1n‍ti).
Combining with (i), (iii), and (iv), we have
(3.25)limsupn→∞(∥ln+1-ln∥-∥xn+1-xn∥)≤0.
Hence, by Lemma 2.11, we obtain ∥ln-xn∥→0 as n→∞. It follows that
(3.26)limn→∞∥xn+1-xn∥=limn→∞(1-βn)∥ln-xn∥=0.
We also know that
(3.27)xn+1-xn=εnγf(xn)+βnxn+[(1-βn)I-εnA]Kn-xn=εn(γf(xn)-Axn)+[(1-βn)I-εnA](Kn-xn).
So
(3.28)limn→∞∥Kn-xn∥=0.
Step 5. We claim that ∥xn-Wxn∥→0.
Observe that(3.29)∥xn-Wnyn∥=∥xn-xn+1+xn+1-yn+Kn-Wnyn∥≤∥xn-xn+1∥+∥xn+1-yn∥+αn∥xn-Wnyn∥.
From (A1), (A3), and (3.28), using step 2, we have
(3.30)(1-αn)∥xn-Wnyn∥≤∥xn-xn+1∥+∥εnγf(xn)+βnxn-βnKn-εnAKn∥≤∥xn-xn+1∥+εn∥γf(xn)-AKn∥+βn∥xn-Kn∥.
This implies that
(3.31)∥xn-Wnyn∥→0(asn→∞).
Next we want to show limn→∞∥xn-yn∥=0.
Letp∈⋂i=1∞‍F(Si)⋂‍EP(Φ); we have
(3.32)∥zn-p∥2=∥Tλnxn-Tλnp∥2≤〈Tλnxn-Tλnp,xn-p〉=〈zn-p,xn-p〉=12(∥zn-p∥2+∥xn-p∥2-∥xn-zn∥2).
Therefore
(3.33)∥zn-p∥2≤∥xn-p∥2-∥xn-zn∥2.
Note that
(3.34)∥Kn-p∥=∥αnxn+(1-αn)Wnyn-p∥2=∥αn(xn-p)+(1-αn)(Wnyn-p)∥2≤αn∥xn-p∥2+(1-αn)∥yn-p∥2≤αn∥xn-p∥2+(1-αn)∥zn-p∥2≤αn∥xn-p∥2+(1-αn)[∥xn-p∥2-∥xn-zn∥2]=∥xn-p∥2-(1-αn)∥xn-zn∥2.
From (3.34), we have
(3.35)∥xn+1-p∥2=∥εnγf(xn)+βnxn+((1-βn)I-εnA)Kn-p∥2=∥εn(γf(xn)-Ap)∥2+βn∥xn-p∥2+((1-βn)I-εnA)∥Kn-p∥2≤εn∥γf(xn)-Ap∥2+βn∥xn-p∥2+(1-βn-εnγ-)∥Kn-p∥2≤εn∥γf(xn)-Ap∥2+βn∥xn-p∥2+(1-βn-εnγ-)(∥xn-p∥2-(1-αn)∥xn-zn∥2)=εn∥γf(xn)-Ap∥2+(1-εnγ-)∥xn-p∥2-(1-βn-εnγ-)(1-αn),∥xn-zn∥2≤εn∥γf(xn)-Ap∥2+∥xn-p∥2+(1-βn-εnγ-)(1-αn)∥xn-zn∥2.
It follows that
(3.36)(1-βn-εnγ-)(1-αn)∥xn-zn∥2≤εn∥γf(xn)-Ap∥2+∥xn-p∥2-∥xn+1-p∥2=εn∥γf(xn)-Ap∥2+(∥xn-p∥+∥xn+1-p∥)×∥xn-xn+1∥.
From conditions (i), (vi) and (3.26), we have
(3.37)∥xn-zn∥→0.
We also compute
(3.38)∥yn-p∥2=∥PC(I-μnB)zn-PC(I-μnB)p∥≤∥(zn-μnBzn)-(p-μnBp)∥2=∥zn-p∥2-2μn〈zn-p,Bzn-Bp〉+μn2∥Bzn-Bp∥2≤∥xn-p∥2-μn(2ξ-μn)∥Bzn-Bp∥2,(3.39)∥Kn-p∥2=∥αnxn+(1-αn)Wnyn-p∥2=∥αn(xn-p)-(1-αn)(Wnyn-p)∥2≤αn∥xn-p∥2+(1-αn)∥yn-p∥2=αn∥xn-p∥2+(1-αn){∥xn-p∥2-μn(2ξ-μn)∥Bzn-Bp∥2}=∥xn-p∥2-μn(1-αn)(2ξ-μn)∥Bzn-Bp∥2.
So, from (3.39), we get
(3.40)∥xn+1-p∥2=∥εnγf(xn)+βnxn+((1-βn)I-εnA)Kn-p∥2=∥εn(γf(xn)-Ap)∥2+βn(xn-p)+((1-βn)I-εnA)∥Kn-p∥2≤εn∥γf(xn)-Ap∥2+βn∥xn-p∥2+(1-βn-εnγ-)∥Kn-p∥2≤εn∥γf(xn)-Ap∥2+βn∥xn-p∥2+(1-βn-εnγ-)×{∥xn-p∥2-μn(1-αn)(2ξ-μn)∥Bzn-Bp∥2}≤εn∥γf(xn)-Ap∥2+∥xn-p∥2-(1-βn-εnγ-)μn(1-αn)(2ξ-μn)∥Bzn-Bp∥2.
It follows that(3.41)(1-βn-εnγ-)μn(1-αn)(2ξ-μn)∥Bzn-Bp∥2≤εn∥γf(xn)-Ap∥2+∥xn-xn+1∥(∥xn-p∥+∥xn+1-p∥).
So
(3.42)∥Bzn-Bp∥→0.
On the other hand, we also know that
(3.43)∥yn-p∥2=∥PC(I-μnB)zn-PC(I-μnB)p∥2≤〈(I-μnB)zn-(I-μnB)p,yn-p〉=12{∥(I-μnB)zn-(I-μnB)p∥2+∥yn-p∥2-∥(I-μnB)zn-(I-μnB)p-(yn-p)∥2}≤12{∥xn-p∥2+∥yn-p∥2-∥zn-yn∥2-μn∥Bzn-Bp∥2+2μn〈zn-yn,Bzn-Bp〉∥yn-p∥2},
and hence
(3.44)∥yn-p∥2≤∥xn-p∥2-∥zn-yn∥2+2μn∥zn-yn∥2∥Bzn-Bp∥.
So
(3.45)∥Kn-p∥2=∥αnxn+(1-αn)Wnyn-p∥2≤αn∥xn-p∥2+(1-αn){∥xn-p∥2-∥zn-yn∥2}+2μn∥zn-yn∥∥Bzn-Bp∥=∥xn-p∥2-(1-αn)∥zn-yn∥2+2(1-αn)μn∥zn-yn∥∥Bzn-Bp∥,(3.46)∥xn+1-p∥2=∥εnγf(xn)+βnxn+((1-βn)I-εnA)Kn-p∥2≤εn∥γf(xn)-Ap∥2+βn∥xn-p∥2+(1-βn-εnγ-)∥Kn-p∥2≤εn∥γf(xn)-Ap∥2+∥xn-p∥2-(1-βn-εnγ-)(1-αn)∥zn-yn∥2+2(1-αn)μn(1-βn-εnγ-)∥zn-yn∥∥Bzn-Bp∥.
Hence
(3.47)(1-βn-εnγ-)(1-αn)∥zn-yn∥2≤εn∥γf(xn)-Ap∥2+2(1-αn)μn(1-βn-εnγ-)∥zn-yn∥∥Bzn-Bp∥+∥xn-p∥2-∥xn+1-p∥2≤εn∥γf(xn)-Ap∥2+2(1-αn)μn(1-βn-εnγ-)∥zn-yn∥∥Bzn-Bp∥+∥xn-xn+1∥(∥xn-p∥+∥xn+1-p∥).
From (i), (3.42), and (3.26), we know that
(3.48)∥zn-yn∥→0.
From (3.37) and (3.42), we can get
(3.49)∥xn-yn∥→0.
On the other hand, we have
(3.50)∥xn-Wxn∥≤∥xn-Wnyn∥+∥Wnyn-Wxn∥≤∥xn-Wnyn∥+sup∥Wnyn-Wxn∥.
By (3.30), (3.49), and using Lemma 2.7, we have
(3.51)∥xn-Wxn∥→0.
Step 6. We claim that limsupn→∞〈(A-γf)q,q-xn〉≤0, where q=PΩ(I-A+γf)(q) is the unique solution of 〈(A-γf)q,x-q〉≥0, for all x∈Ω.
Indeed, take a subsequence{xnj} of {xn} such that
(3.52)limsupn→∞〈(A-γf)q,q-xn〉=limsupn→∞〈(A-γf)q,q-xnj〉.
Since {xnj} is bounded, there exists a subsequence {xnjk} of {xnj}, which converges weakly to p; without loss of generality, we can assume xnj⇀p and Wxnj⇀p, we arrive at
(3.53)limsupn→∞〈(A-γf)q,q-xnj〉=〈(A-γf)q,q-p〉≤0.
Step 7. We show that xn→q.
Since(3.54)〈(A-γf)q,q-xn+1〉=〈(A-γf)q,xn-xn+1〉+〈(A-γf)q,q-xn〉≤∥(A-γf)q∥·∥xn-xn+1∥+〈(A-γf)q,q-xn〉,
so
(3.55)limsupn→∞〈(A-γf)q,q-xn+1〉≤0.
Note that
(3.56)∥xn+1-q∥2=∥εnγf(xn)+βnxn+((1-βn)I-εnA)Kn-q∥2=∥εn(γf(xn)-Aq)+βn(xn-q)+((1-βn)I-εnA)(Kn-q)∥2≤∥(1-βn)(1-βn)I-εnA1-βn(Kn-q)+βn(xn-q)∥2+2εn〈(A-γf)q,xn+1-q〉≤(1-βn)∥(1-βn)I-εnA1-βn(Kn-q)∥2+βn∥xn-q∥2+2εnγα×〈f(xn)-f(q),xn+1-q〉+2εn〈γf(q)-Aq,xn+1-q〉≤(1-βn)∥(1-βn)I-εnA1-βn(Kn-q)∥2+βn∥xn-q∥2+2εnγα×∥xn-q∥·∥xn+1-q∥+2εn〈γf(q)-Aq,xn+1-q〉≤∥(1-βn)I-εnA∥21-βn∥Kn-q∥2+βn∥xn-q∥2+εnγα×(∥xn-q∥2+∥xn+1-q∥2)+2εn〈γf(q)-Aq,xn+1-q〉≤(((1-βn)-γ-εn)21-βn+βn+εnγα)∥xn-q∥2+εnγα∥xn+1-q∥2+2εn〈γf(xn)-Aq,xn+1-q〉,
which implies that
(3.57)∥xn+1-q∥2≤(1-2(γ--αγ)εn1-αγεn)∥xn-q∥2+εn1-αγεn{γ-2εn21-βn∥xn-q∥2+2εn〈γf(xn)-Aq,xn+1-q〉}.
Let
(3.58)σn=εn1-αγεn{γ-2εn21-βn∥xn-q∥2+2εn〈γf(xn)-Aq,xn+1-q〉},ρn=2(γ--αγ)εn1-αγεn,
then we have
(3.59)limsupn→∞σnρn≤0.
Applying Lemma 2.2, we can conclude that {xn} converges strongly to q in norm. This completes the proof.As direct consequences of Theorem3.1, we obtain the following corollary.Corollary 3.2.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C×C→ℝ satisfying (A1)–(A4). Let Si:C→C be a family κi-strict pseudocontractions for some 0≤κi<1. Assume the set Ω=∩i=1∞F(Si)∩EP(F)≠∅. Let f be a contraction of H into itself with α∈(0,1) and let A be an α-inverse strongly monotone mapping. Let F be a strongly positive linear-bounded operator on H with coefficient γ>0 and 0<γ<γ-/α and τ<1. Let Wn be the mapping generated by Si′ and ti, where Si:C→C is a nonexpansive mapping with a fixed point. Let {xn} and {un} be sequences generated by the following algorithm:
(3.60)F(zn,y)+1λn〈y-zn,zn-xn〉≥0,Kn=αnxn+(1-αn)Wnzn,xn+1=εnγf(xn)+βnxn+((1-βn)I-εnA)Kn,
where{εn}, {βn}, {αn}, and {λn} are sequences in (0,1). Assume that the control sequences satisfy the following restrictions:  (i)
limn→∞εn=0 and Σn=1∞εn=∞;(ii)
0<liminfn→∞βn≤limsupn→∞βn<1;(iii)
0<liminfn→∞λn≤limsupn→∞λn<1;(iv)
limn→∞|λn+1-λn|=limn→∞|αn+1-αn|=0;(v)
0<tn≤b<1;(vi)
λn<2α.
Then{xn} converges strongly to w∈Ω where w=PΩ(I-A+γf)w.



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*Source: 150145-2012-09-26.xml*