# A New Iterative Method for Finding Common  Solutions of a System of Equilibrium Problems, Fixed-Point Problems, and Variational Inequalities

Authors: Jian-Wen Peng; Soon-Yi Wu; Jen-Chih Yao
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

We introduce a new iterative scheme based on extragradient method and viscosity approximation method for finding a common element of the solutions set of a system of equilibrium problems, fixed point sets of an infinite family of nonexpansive mappings, and the solution set of a variational inequality for a relaxed cocoercive mapping in a Hilbert space. We prove strong convergence theorem. The results in this paper unify and generalize
some well-known results in the literature.


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## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 and induced norm ∥·∥. Let C be a nonempty, closed, and convex subset of H. Let {Fk}k∈Γ be a countable family of bifunctions from C×C to ℝ, where ℝ is the set of real numbers. Combettes and Hirstoaga [1] considered the following system of equilibrium problems: (1.1)Findx∈Csuchthat(∀k∈Γ),(∀y∈C),Fk(x,y)≥0.IfΓ is a singleton, problem (1.1) becomes the following equilibrium problem: (1.2)Findingx∈CsuchthatF(x,y)≥0,∀y∈C.The solutions set of (1.2) is denoted by EP(F). And clearly the solutions set of problem (1.1) can be written as ⋂k∈ΓEP(Fk).Problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games, and others; see for instance, [1–4].Recall that a mappingS of a closed and convex subset C into itself is nonexpansive if (1.3)∥Sx-Sy∥≤∥x-y∥∀x,y∈C.
We denote fixed-points set of S by Fix(S). A mapping f:C→C is called contraction if there exists a constant α∈(0,1) such that (1.4)∥fx-fy∥≤α∥x-y∥,∀x,y∈C.
A bounded linear operator B on H is strongly positive, if there is a constant γ¯>0 such that 〈Bx,x〉≥γ¯∥x∥2 for all x∈H.Combettes and Hirstoaga [1] introduced an iterative scheme for finding a common element of the solutions set of problem (1.1) in a Hilbert space and obtained a weak convergence theorem. Peng and Yao [2] introduced a new viscosity approximation scheme based on the extragradient method for finding a common element in the solutions set of the problem (1.1), fixed-points set of an infinite family of nonexpansive mappings and the solutions set of the variational inequality for a monotone and Lipschitz continuous mapping in a Hilbert space and obtained a strong convergence theorem. Colao et al. [3] introduced an implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed-points of infinite family of nonexpansive mappings in a Hilbert space and obtained a strong convergence theorem. Saeidi [4] introduced some iterative algorithms for finding a common element of the solutions set of a system of equilibrium problems and of fixed-points set of a finite family and a left amenable semigroup of nonexpansive mappings in a Hilbert space and obtained some strong convergence theorems.Several algorithms for problem (1.2) have been proposed (see [5–20]). S. Takahashi and W. Takahashi [5] introduced and studied the following iterative scheme by the viscosity approximation method for finding a common element of the solutions set of problem (1.2) and fixed-points set of a nonexpansive mapping in a Hilbert space. Let an arbitrary x1∈H define sequences {xn} and {un} by(1.5)F(un,y)+1βn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnf(xn)+(1-αn)Sun,∀n∈N.
Shang et al. [6] introduced the following iterative scheme by the viscosity approximation method for finding a common element of the solutions set of problem (1.2) and fixed-points set of a nonexpansive mapping in a Hilbert space. Let an arbitrary x1∈H, define sequences {xn} and {un} by(1.6)F(un,y)+1βn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnγf(xn)+(I-αnB)Sun,∀n∈N.
They proved that under certain appropriate conditions imposed on {αn} and {βn}, the sequences {xn} and {un} generated by (1.6) converge strongly to the unique solution of the variational inequality (1.7)〈(B-γf)x*,x-x*〉≥0,∀x∈Fix(S)∩EP(F),
which is the optimality condition for the minimization problem(1.8)minx∈Fix(S)∩EP(F)12〈Bx,x〉-h(x),
where h is a potential function for γf (i.e., h′(x)=γf(x) for x∈H). If C=H, the algorithm (1.6) was also studied by Plubtieng and Punpaeng [7].LetA:C→H be a monotone mapping. The variational inequality problem is to find a point x∈C such that (1.9)〈Ax,y-x〉≥0
for all y∈C. The solutions set of the variational inequality problem is denoted by VI(C,A). Qin et al. [8] introduced the following general iterative scheme for finding a common element of the solutions set of problem (1.2), the solutions set of a variational inequality and fixed-points set of a nonexpansive mapping in a Hilbert space. Let an arbitrary x1∈H, define sequences {xn} and {un} by(1.10)F(un,y)+1βn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnγf(xn)+(I-αnB)SPC(I-snA)un,∀n∈N.
They proved that under certain appropriate conditions imposed on {αn}, {sn} and {βn}, the sequences {xn} and {un} generated by (1.10) converge strongly to the unique solution of the variational inequality (1.11)〈(B-γf)x*,x-x*〉≥0,∀x∈Fix(S)∩VI(C,A)∩EP(F).Qin et al. [9] introduced the following general iterative scheme for finding a common element of the solutions set of problem (1.2) and fixed-points set of a finite family of nonexpansive mappings in a Hilbert space. Let an arbitrary x1∈H, define sequences {xn} and {un} by(1.12)F(un,y)+1βn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnγf(Wnxn)+(1-αnB)WnPC(I-snA)un,∀n∈N,
where Wn is the W-mapping generated by T1,T2,…,TN and λn1,λn2,…,λnN. They proved that under certain appropriate conditions imposed on {αn}, {sn} and {βn}, the sequences {xn} and {un} generated by (1.12) converge strongly to the unique solution of the variational inequality (1.13)〈(B-γf)x*,x-x*〉≥0,∀x∈⋂i=1NFix(Ti)∩VI(C,A)∩EP(F).A typical problem is to minimize a quadratic function over the fixed-points set of a nonexpansive mappingS on a real Hilbert space H, that is, (1.14)minx∈Fix(S)12〈Bx,x〉-〈x,b〉,
where b is a given point in H. In 2003, Xu [21] proved that the sequence {xn} defined by the iterative method below, with the initial point x0∈H, chosen arbitrarily: (1.15)xn+1=(I-αnB)Sxn+αnu,n≥0,
converges strongly to the unique solution of the minimization problem (1.15) provided the sequence {αn} satisfies certain conditions. Marino and Xu [22] combine the iterative method (1.15) with the viscosity approximation in [23] and consider the following general iterative method: with the initial point x0∈H, chosen arbitrarily: (1.16)xn+1=(1-αnB)Sxn+αnγf(xn),n≥0.They proved that if the sequence{αn} satisfies appropriate conditions, then the sequence {xn} generated by (1.16) converges strongly to the unique solution of the variational inequality (1.17)〈(B-γf)x*,x-x*〉≥0,x∈Fix(S)
which is the optimality condition for the minimization problem(1.18)minx∈Fix(S)12〈Bx,x〉-h(x),
where h is a potential function for γf.Recently, Qin et al. [24] introduced the following general iterative process: with the initial point x1∈C, chosen arbitrarily:(1.19)yn=PC(I-snA)xn,xn+1=αnγf(Wnxn)+(I-αnB)WnPC(I-rnA)yn,∀n∈N,
where Wn is the W-mapping generated by T1,T2,…,TN and λn1,λn2,…,λnN. They proved that if the sequences of parameters {αn}, {rn} and {sn} satisfies appropriate conditions, then the sequence {xn}, {yn} generated by (1.19) converge strongly to a point x* which is the unique solution of the variational inequality (1.20)〈(B-γf)x*,x-x*〉≥0,∀x∈⋂i=1NF(Ti)∩VI(C,A).Inspired and motivated by above works, we introduce a new iterative scheme based on extragradient method and viscosity approximation method for finding a common element of the solutions set of a system of equilibrium problems, fixed-points set of a family of infinitely nonexpansive mappings and the solutions set of a variational inequality for a relaxed cocoercive mapping in a Hilbert space. We prove strong convergence theorem. The results in this paper unify, generalize and extend some well-known results in [6–9, 21, 22, 24].


## 2. Preliminaries

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥. Let C be a nonempty, closed, and convex subset of H. Let symbols → and ⇀ denote strong and weak convergence, respectively. It is well 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∈[0,1].For anyx∈H, there exists a unique nearest point in C, denoted by PC(x), such that ∥x-PC(x)∥≤∥x-y∥ for all y∈C. The mapping PC is called the metric projection of H onto C. We know that PC is a nonexpansive mapping from H onto C, PC(x)∈C and (2.2)〈x-PC(x),PC(x)-y〉≥0
for all x,y∈H.It is easy to see that (2.2) is equivalent to (2.3)∥x-y∥2≥∥x-PC(x)∥2+∥y-PC(x)∥2
for all x,y∈H. It is also known that PC has the following firmly nonexpansive property: (2.4)〈x-y,PCx-PCy〉≥∥PCx-PCy∥2
for all x,y∈H.Recall also that a mappingA of C into H is called monotone if (2.5)〈Ax-Ay,x-y〉≥0,
for all x,y∈C.A is said to be μ-cocoercive, if for each x,y∈C, we have (2.6)〈Ax-Ay,x-y〉≥μ∥Ax-Ay∥2,
for a constant μ>0.A is said to be relaxed (u,v)-cocoercive, if there exist two constants u,v>0 such that (2.7)〈Ax-Ay,x-y〉≥(-u)∥Ax-Ay∥2+v∥x-y∥2,∀x,y∈C.
Let A be a monotone mapping of C into H. In the context of the variational inequality problem the characterization of projection (2.2) implies the following:(2.8)u∈VI(C,A)⇒u=PC(u-λAu),λ>0,u=PC(u-λAu)forsomeλ>0⇒u∈VI(C,A).It is also known thatH satisfies the Opial's condition [25], that is, for any sequence {xn}⊂H with xn⇀x, the inequality(2.9)lim infn→∞∥xn-x∥<lim infn→∞∥xn-y∥
holds for every y∈H with x≠y.A set-valued mappingT:H→2H is called monotone if for all x,y∈H, f∈Tx and g∈Ty imply 〈x-y,f-g〉≥0. A monotone mapping T:H→2H is maximal if its graph G(T) of T is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if for (x,f)∈H×H,〈x-y,f-g〉≥0 for every (y,g)∈G(T) implies f∈Tx. Let A be a monotone and k-Lipschitz-continuous mapping of C into H and let NCv be normal cone to C at v∈C, that is, NCv={w∈H:〈v-u,w〉≥0,forallu∈C}. Define(2.10)Tv={Av+NCvifv∈C,∅ifv∉C.
Then T is maximal monotone and 0∈Tv if and only if v∈VI(C,A) (see [26]).For solving the problem (1.1), let us assume that the bifunction F satisfies the following condition:(A1)
F(x,x)=0 for all x∈C;(A2)
F is monotone, that is, F(x,y)+F(y,x)≤0 for any x,y∈C;(A3)
for eachx,y,z∈C,
(2.11)limt↓0F(tz+(1-t)x,y)≤F(x,y);(A4)
for eachx∈C,y↦F(x,y) is convex;(A5)
for eachx∈C,y↦F(x,y) is lower semicontinuous.We recall some lemmas needed later.Lemma 2.1 (see [1, 10]).
LetC be a nonempty, closed, and convex subset of H, and let F be a bifunction from C×C to ℝ which satisfies conditions (A1)–(A5). For β>0 and x∈H, define the mapping TβF:H→C as follows:
(2.12)TβF(x)={z∈C:F(z,y)+1β〈y-z,z-x〉≥0,∀y∈C}
for all x∈H. Then, the following statements hold: (1)
TβF(x)≠∅;(2)
TβF is single-valued;(3)
TβF is firmly nonexpansive, that is, for any x,y∈H,(2.13)∥TβF(x)-TβF(y)∥2≤〈TβF(x)-TβF(y),x-y〉;(4)
Fix(TβF)=EP(F);(5)
EP(F) is closed and convex.Lemma 2.2 (see [27]).
Assume that{sn} is a sequence of nonnegative real numbers such that
(2.14)sn+1≤(1-αn)sn+αnβn+δn,n≥1,
where {αn}, {βn} and {δn} are sequences of numbers which satisfy the conditions:(i)
{αn}⊂[0,1],∑n=1∞αn=∞, or equivalently, ∏i=1∞(1-αn)=0;(ii)
limsupn→∞βn≤0;(iii)
δn≥0(n≥1), ∑n=1∞δn<∞;
Then,limn→∞sn=0.Lemma 2.3.
In a real Hilbert spaceH, the following inequality holds:
(2.15)∥x+y∥2≤∥x∥2+2〈y,x+y〉
for all x,y∈H.Lemma 2.4 (see [22]).
Assume thatA is a strongly positive linear bounded operator on a Hilbert space H with coefficient γ̃>0 and 0<ρ≤∥A∥-1. Then ∥I-ρA∥≤1-ργ̃.LetS1,S2,… be a family of infinitely nonexpansive mappings of C into itself and let ξ1,ξ2,… be real numbers such that 0≤ξi≤1 for every i∈N. For any n∈N, define a mapping Wn of C into C as follows:(2.16)Un,n+1=I,Un,n=ξnSnUn,n+1+(1-ξn)I,Un,n-1=ξn-1Sn-1Un,n+(1-ξn-1)I,⋮Un,k=ξkSkUn,k+1+(1-ξk)I,Un,k-1=ξk-1Sk-1Un,k+(1-ξk-1)I,⋮Un,2=ξ2S2Un,3+(1-ξ2)I,Wn=Un,1=ξ1S1Un,2+(1-ξ1)I.Such a mappingWn is called the W-mapping generated by Sn,Sn-1,…,S1 and ξn,ξn-1,…,ξ1; see [28, 29].Lemma 2.5 (see [28]).
LetC be a nonempty, closed, and convex subset of a Banach space E. Let S1,S2,... be a family of infinitely nonexpansive mappings of C into itself such that ⋂i=1∞Fix(Si) is nonempty, and let ξ1,ξ2,… be real numbers such that 0<ξi≤d<1 for every i∈N. For any n∈N, let Wn be the W-mapping of C into itself generated by Sn,Sn-1,…,S1 and ξn,ξn-1,…,ξ1. Then Wn is asymptotically regular and nonexpansive. Further, if E is strict convex, then F(Wn)=⋂i=1nFix(Si).Lemma 2.6 (see [29]).
LetC be a nonempty, closed, and convex subset of a strictly convex Banach space E. Let S1,S2,… be a family of infinitely nonexpansive mappings of C into itself such that ⋂i=1∞Fix(Si) is nonempty, and let ξ1,ξ2,… be real numbers such that 0<ξi≤d<1 for every i∈N. Then for every x∈C and k∈N, the limit limn→∞Un,kx exists.Remark 2.7.
Using Lemma2.6, one can define mappings U∞,k and W of C into itself as follows:
(2.17)U∞,kx=limn→∞Un,kx,
and Wx=limn→∞Wnx=limn→∞Un,1x for every x∈C. Such a mapping W is called the W-mapping generated by S1,S2,… and ξ1,ξ2,…. Since Wn is nonexpansive, W:C→C is also nonexpansive. Indeed, observe that for each x,y∈C(2.18)∥Wx-Wy∥=limn→∞∥Wnx-Wny∥≤∥x-y∥.If{xn} is a bounded sequence in C, then we have (2.19)limn→∞∥Wx-Wnx∥=0.Lemma 2.8 (see [29]).
LetC be a nonempty, closed and convex subset of a strictly convex Banach space E. Let S1,S2,… be an infinite family of nonexpansive mappings of C into itself such that ⋂i=1∞Fix(Si) is nonempty, and let ξ1,ξ2,… be real numbers such that 0<ξi≤d<1 for every i∈N. Then Fix(W)=⋂n=1∞Fix(Sn).


## 3. Strong Convergence Theorem

In this section, we prove strong convergence theorem which solve the problem of finding a common element of the solutions set of a system of equilibrium problems, fixed-points set of a family of infinitely nonexpansive mappings, and the solutions set of a variational inequality for a relaxed cocoercive mapping in Hilbert space.Theorem 3.1.
LetC be a nonempty, closed, and convex subset of H. Let F1,F2,…,Fm be bifunctions from C×C to ℝ which satisfies conditions (A1)–(A5). Let A:C→H be relaxed (u,v)-cocoercive and μ-Lipschitz continuous and B a strongly positive linear bounded operator on H with coefficient γ̃>0. Assume that 0<γ<γ̃/α. Let S1,S2,… be a family of infinitely nonexpansive mappings of C into itself such that Ω=⋂i=1∞Fix(Si)∩VI(C,A)∩⋂k=1mEP(Fk)≠∅ and let ξ1,ξ2,… be real numbers such that 0<ξi≤δ<1 for every i∈N, and let Wn be the W-mapping of C into itself generated by Sn,Sn-1,…,S1 and ξn,ξn-1,…,ξ1. Let f:C→C be a contraction with coefficient α(0<α<1) and {xn}, {un}, and {yn} be sequences generated by
(3.1)x1=x∈H,un=TβnFmTβnFm-1⋯TβnF2TβnF1xn,yn=PC(I-snA)un,xn+1=αnγf(Wnxn)+(I-αnB)WnPC(I-rnA)yn
for every n=1,2,…, where {αn}, {βn}, {rn}, and {sn} are sequences of numbers which satisfy the conditions:(C1)
{αn}⊂[0,1] with limn→∞αn=0, ∑n=1∞αn=∞, and ∑n=1∞|αn+1-αn|<∞;(C2)
{rn}⊂[a,b] and {sn}⊂[a,b] for some a,b with 0≤a≤b≤2(υ-uμ2)/μ2, ∑n=1∞|rn+1-rn|<∞, and ∑n=1∞|sn+1-sn|<∞;(C3)
lim infn→∞βn>0 and ∑n=1∞|βn+1-βn|<∞.
Then,{xn}, {yn}, and {un} converge strongly to a point q∈Ω which solves the following variational inequality:(3.2)〈γfq-Bq,p-q〉≤0,∀p∈Ω.
Equivalently, one has q=PΩ(γf+(I-B))(q).Proof.
Sinceαn→0 from condition (C1), we may assume, with no loss of generality, that αn≤∥B∥-1 for all n. Lemma 2.4 implies ∥I-αnB∥≤1-αnγ̃. Next, we will assume that ∥I-B∥≤1-γ̃. Now, we show that the mappings I-snA and I-rnA are nonexpansive. Indeed, from the relaxed (u,v)-cocoercivity and μ-Lipschitz continuity of A and condition (C2), we have
(3.3)∥(I-snA)x-(I-snA)y∥2=∥(x-y)-sn(Ax-Ay)∥2=∥x-y∥2-2sn〈x-y,Ax-Ay〉+sn2∥Ax-Ay∥2≤∥x-y∥2-2sn[-u∥Ax-Ay∥2+υ∥x-y∥2]+sn2∥Ax-Ay∥2≤∥x-y∥2+2snμ2u∥x-y∥2-2snυ∥x-y∥2+μ2sn2∥x-y∥2=(1+2snμ2u-2snυ+μ2sn2)∥x-y∥2≤∥x-y∥2,
which implies the mapping I-snA is nonexpansive, so is I-rnA.
Fork∈{0,1,2,…,m}, and for any positive integer number n, we define the operator Θβnk:H→C as follows:(3.4)Θβn0x=x,Θβnkx=TβnFkTβnFk-1⋯TβnF2TβnF1x,k=1,2,…,m.
Next, we show that the sequence{xn} is bounded. Let p∈Ω. Then from Lemma 2.1(3), we know that for k∈{1,2,…,m}, TβnFk is nonexpansive and p=TβnFkp, and(3.5)∥un-p∥=∥Θβnmxn-p∥=∥Θβnmxn-Θβnmp∥≤∥xn-p∥
for all n=1,2,…. By p=PC(I-snA)p and (3.5), we have
(3.6)∥yn-p∥=∥PC(I-snA)un-PC(I-snA)p∥≤∥(I-snA)un-(I-snA)p∥≤∥un-p∥≤∥xn-p∥.
Sincexn+1=αnγf(Wnxn)+(I-αnB)WnPC(I-rnA)yn and p=Wnp, we have(3.7)∥xn+1-p∥=∥αn(γf(Wnxn)-Bp)+(I-αnB)(WnPC(I-rnA)yn-p)∥≤αn∥γf(Wnxn)-Bp∥+(1-αnγ̃)∥PC(I-rnA)yn-p∥≤αnγ∥f(Wnxn)-f(p)∥+αn∥γf(p)-Bp∥+(1-αnγ̃)∥yn-p∥≤[1-αn(γ̃-αγ)]∥xn-p∥+αn∥γf(p)-Bp∥.
By inductions, we have
(3.8)∥xn-p∥≤max{∥x0-p∥,∥γf(p)-Bp∥γ̃-αγ},
which proves that the sequence {xn} is bounded. It follows from (3.5) and (3.6) that {yn} and {un} are also bounded.
SinceΘβnkxn=TβnFkΘβnk-1xn and Θβn+1kxn+1=Tβn+1FkΘβn+1k-1xn+1 for each k=1,2,…,m, by Lemma 2.1, we have(3.9)Fk(Θβnkxn,y)+1βn〈y-Θβnkxn,Θβnkxn-Θβnk-1xn〉≥0∀y∈C,(3.10)Fk(Θβn+1kxn+1,y)+1βn+1〈y-Θβn+1kxn+1,Θβn+1kxn+1-Θβn+1k-1xn+1〉≥0∀y∈C,
Settingy=Θβn+1kxn+1 in (3.9) and y=Θβnkxn in (3.10), we have(3.11)Fk(Θβnkxn,Θβn+1kxn+1)+1βn〈Θβn+1kxn+1-Θβnkxn,Θβnkxn-Θβnk-1xn〉≥0,Fk(Θβn+1kxn+1,Θβnkxn)+1βn+1〈Θβnkxn-Θβn+1kxn+1,Θβn+1kxn+1-Θβn+1k-1xn+1〉≥0.
Adding the two inequalities and from the monotonicity ofF, we get(3.12)〈Θβn+1kxn+1-Θβnkxn,Θβnkxn-Θβnk-1xnβn-Θβn+1kxn+1-Θβn+1k-1xn+1βn+1〉≥0
and hence(3.13)∥Θβn+1kxn+1-Θβnkxn∥2≤〈Θβn+1kxn+1-Θβnkxn,(Θβn+1k-1xn+1-Θβnk-1xn)+(1-βnβn+1)(Θβn+1kxn+1-Θβn+1k-1xn+1)〉.
Without loss of generality, let us assume that there exists a real numberd such that βn>d>0 for all n=1,2,…. Hence, for each k=1,2,…,m we have(3.14)∥Θβn+1kxn+1-Θβnkxn∥≤∥Θβn+1k-1xn+1-Θβnk-1xn∥+1βn+1|βn+1-βn|∥Θβn+1kxn+1-Θβn+1k-1xn+1∥≤∥Θβn+1k-1xn+1-Θβnk-1xn∥+1d|βn+1-βn|M0,
where M0 is an approximate constant such that
(3.15)M0≥max{supn≥1{∥Θβn+1kxn+1-Θβn+1k-1xn+1∥},k=1,2,…,m}.
It follows from (3.14) that(3.16)∥un+1-un∥=∥Θβn+1mxn+1-Θβnmxn∥≤∥xn+1-xn∥+md|βn+1-βn|M0.
Putρn=PC(I-rnA)yn. We have(3.17)∥yn-yn+1∥=∥PC(I-snA)un-PC(I-sn+1A)un+1∥≤∥(I-snA)un-(I-sn+1A)un+1∥=∥(un-snAun)-(un+1-snAun+1)+(sn+1-sn)Aun+1∥≤∥un-un+1∥+|sn+1-sn|M1,
where M1 is an approximate constant such that M1≥max{supn≥1{∥Aun∥},M0}.
Substituting (3.16) into (3.17), we have(3.18)∥yn-yn+1∥≤∥xn+1-xn∥+[md|βn+1-βn|+|sn+1-sn|]M1.
It follows from (3.18) that
(3.19)∥ρn-ρn+1∥=∥PC(I-rnA)yn-PC(I-rn+1A)yn+1∥≤∥(I-rnA)yn-(I-rn+1A)yn+1∥=∥(yn-rnAyn)-(yn+1-rnAyn+1)+(rn+1-rn)Ayn+1∥≤∥yn-yn+1∥+|rn+1-rn|M2≤∥xn-xn+1∥+[md|βn+1-βn|+|sn+1-sn|+|rn+1-rn|]M2,
where M2 is an approximate constant such that M2≥max{M1,supn≥1{∥Ayn+1∥}}.
Observe that(3.20)xn+1=αnγf(Wnxn)+(I-αnB)Wnρn,xn+2=αn+1γf(Wn+1xn+1)+(I-αn+1B)Wn+1ρn+1,
we have
(3.21)xn+2-xn+1=αn+1γ[f(Wn+1xn+1)-f(Wnxn)]+(I-αn+1B)(Wn+1ρn+1-Wnρn)+(αn+1-αn)[γf(Wnxn)-BWnρn].
It follows that
(3.22)∥xn+2-xn+1∥≤αn+1γα∥Wn+1xn+1-Wnxn∥+(1-αn+1γ̃)∥Wn+1ρn+1-Wnρn∥+|αn+1-αn|∥γf(Wnxn)-BWnρn∥≤αn+1γα(∥xn+1-xn∥+∥Wn+1xn-Wnxn∥)+(1-αn+1γ̃)(∥ρn+1-ρn∥+∥Wn+1ρn-Wnρn∥)+|αn+1-αn|∥γf(Wnxn)-BWnρn∥.
Next we estimate∥Wn+1xn-Wnxn∥ and ∥Wn+1ρn-Wnρn∥. It follows from the definition of Wn and nonexpansiveness of Si that(3.23)∥Wn+1xn-Wnxn∥=∥Un+1,1xn-Un,1xn∥=∥ξ1S1Un+1,2xn+(1-ξ1)xn-{ξ1S1Un,2xn+(1-ξ1)xn}∥=ξ1∥S1Un+1,2xn-S1Un,2xn∥≤ξ1∥Un+1,2xn-Un,2xn∥=ξ1∥ξ2S2Un+1,3xn+(1-ξ2)xn-{ξ2S2Un,3xn+(1-ξ2)xn}∥=ξ1ξ2∥S2Un+1,3xn-S2Un,3xn∥≤ξ1ξ2∥Un+1,3xn-Un,3xn∥⋮≤Πi=1nξi∥Un+1,n+1xn-Un,n+1xn∥=Πi=1nξi∥ξn+1Sn+1xn+(1-ξn+1)xn-xn∥=Πi=1n+1ξi∥Sn+1xn-xn∥≤Πi=1n+1ξiM3,
where M3 is an approximate constant such that
(3.24)M3≥max{M2,supn≥1{∥Sn+1xn-xn∥},supn≥1{∥Sn+1ρn-ρn∥}}.
Similarly, we have(3.25)∥Wn+1ρn-Wnρn∥≤Πi=1n+1ξiM3.
Substituting (3.19), (3.23), and (3.25) into (3.22) yields that(3.26)∥xn+2-xn+1∥≤αn+1γα(∥xn+1-xn∥+Πi=1n+1ξiM3)+(1-αn+1γ̃)(∥xn+1-xn∥+[md|βn+1-βn|+|sn+1-sn|+|rn+1-rn|]M2+Πi=1n+1ξiM3)+|αn+1-αn|∥γf(Wnxn)-BWnρn∥≤[1-αn+1(γ̃-γα)]∥xn+1-xn∥+M4(md|βn+1-βn|+|sn+1-sn|+|rn+1-rn|+|αn+1-αn|),+Πi=1n+1ξi
where M4 is an approximate constant such that
(3.27)M4≥max{M3,supn≥1{∥γf(Wnxn)-BWnρn∥}}.
It follows from conditions (C1)–(C3) and Πi=1n+1ξi≤δn+1 and Lemma 2.2 that
(3.28)∥xn+1-xn∥→0.
Observe that(3.29)xn+1-Wnρn=αn(γf(Wnxn)-BWnρn),
it follows from (C1) that
(3.30)limn→∞∥Wnρn-xn+1∥=0.
For p∈Ω, we have
(3.31)∥yn-p∥2=∥PC(I-snA)un-PC(I-snA)p∥2≤∥(un-p)-sn(Aun-Ap)∥2=∥un-p∥2-2sn〈un-p,Aun-Ap〉+sn2∥Aun-Ap∥2≤∥xn-p∥2-2sn[-u∥Aun-Ap∥2+v∥un-p∥2]+sn2∥Aun-Ap∥2≤∥xn-p∥2+(2snu+sn2-2snvμ2)∥Aun-Ap∥2.
Similarly, we have
(3.32)∥ρn-p∥2≤∥xn-p∥2+(2rnu+rn2-2rnvμ2)∥Ayn-Ap∥2.
On the other hand, we have
(3.33)∥xn+1-p∥2=∥αn(γf(Wnxn)-Bp)+(I-αnB)(Wnρn-p)∥2≤(αn∥γf(Wnxn)-Bp∥+(1-αnγ̃)∥ρn-p∥)2≤αn∥γf(Wnxn)-Bp∥2+∥ρn-p∥2+2αn∥ρn-p∥∥γf(Wnxn)-Bp∥.
Substituting (3.32) into (3.33), we have
(3.34)∥xn+1-p∥2≤αn∥γf(Wnxn)-Bp∥2+∥xn-p∥2+(2rnu+rn2-2rnvμ2)∥Ayn-Ap∥2+2αn∥ρn-p∥∥γf(Wnxn)-Bp∥.
It follows from condition (C2) that
(3.35)(2avμ2-2bu-b2)∥Ayn-Ap∥2≤αn∥γf(Wnxn)-Bp∥2+∥xn-p∥2-∥xn+1-p∥2+2αn∥ρn-p∥∥γf(Wnxn)-Bp∥≤αn∥γf(Wnxn)-Bp∥2+(∥xn-p∥+∥xn+1-p∥)∥xn+1-xn∥+2αn∥ρn-p∥∥γf(Wnxn)-Bp∥.
As ∥xn+1-xn∥→0 and limn→∞αn=0, we have
(3.36)limn→∞∥Ayn-Ap∥=0
It is easy to see that∥ρn-p∥≤∥yn-p∥. Using (3.33) again, we have(3.37)∥xn+1-p∥2≤αn∥γf(Wnxn)-Bp∥2+∥yn-p∥2+2αn∥ρn-p∥∥γf(Wnxn)-Bp∥.
Substituting (3.31) into (3.37), we can obtain(3.38)∥xn+1-p∥2≤αn∥γf(Wnxn)-Bp∥2+∥xn-p∥2+(2snu+sn2-2snvμ2)∥Aun-Ap∥2+2αn∥ρn-p∥∥γf(Wnxn)-Bp∥.
It follows from (C2) that
(3.39)(2avμ2-2bv-b2)∥Aun-Ap∥2≤αn∥γf(Wnxn)-Bp∥2+∥xn-p∥2-∥xn+1-p∥2+2αn∥ρn-p∥∥γf(Wnxn)-Bp∥≤αn∥γf(Wnxn)-Bp∥2+(∥xn-p∥-∥xn+1-p∥)∥xn+1-xn∥+2αn∥ρn-p∥∥γf(Wnxn)-Bp∥.
As ∥xn+1-xn∥→0 and limn→∞αn=0, we have
(3.40)limn→∞∥Aun-Ap∥=0.
Observe that
(3.41)∥ρn-p∥2=∥PC(I-rnA)yn-PC(I-rnA)p∥2≤〈(I-rnA)yn-(I-rnA)p,ρn-p〉=12{∥(I-rnA)yn-(I-rnA)p∥2+∥ρn-p∥2-∥(I-rnA)yn-(I-rnA)p-(ρn-p)∥2}≤12{∥yn-p∥2+∥ρn-p∥2-∥(yn-ρn)-rn(Ayn-Ap)∥2}≤12{∥xn-p∥2+∥ρn-p∥2-∥yn-ρn∥2-rn2∥Ayn-Ap∥2+2rn〈yn-ρn,Ayn-Ap〉∥xn-p∥2},
which yields that
(3.42)∥ρn-p∥2≤∥xn-p∥2-∥yn-ρn∥2+2rn∥yn-ρn∥∥Ayn-Ap∥.
Substituting (3.42) into (3.33) we have
(3.43)∥xn+1-p∥2≤αn∥γf(Wnxn)-Bp∥2+∥xn-p∥2-∥yn-ρn∥2+2rn∥yn-ρn∥∥Ayn-Ap∥+2αn∥ρn-p∥∥γf(Wnxn)-Bp∥,
which implies that
(3.44)∥yn-ρn∥2≤αn∥γf(Wnxn)-Bp∥2+∥xn-p∥2-∥xn+1-p∥2+2rn∥yn-ρn∥∥Ayn-Ap∥+2αn∥ρn-p∥∥γf(Wnxn)-Bp∥≤αn∥γf(Wnxn)-Bp∥2+(∥xn-p∥+∥xn+1-p∥)∥xn+1-xn∥+2rn∥yn-ρn∥∥Ayn-Ap∥+2αn∥ρn-p∥∥γf(Wnxn)-Bp∥.
It follows from (C1), ∥xn+1-xn∥→0, and ∥Ayn-Ap∥→0 that limn→∞∥yn-ρn∥=0.
Forp∈Ω, we have(3.45)∥yn-p∥2=∥PC(I-snA)un-PC(I-snA)p∥2≤〈PC(I-snA)un-PC(I-snA)p,(I-snA)un-(I-snA)p〉=〈yn-p,(I-snA)un-(I-snA)p〉=12(∥yn-p∥2+∥(I-snA)un-(I-snA)p∥2-∥(yn-p)-[un-p-sn(Aun-Ap)]∥2)≤12(∥yn-p∥2+∥un-p∥2-∥yn-p-[un-p-sn(Aun-Ap)]∥2)=12(∥yn-p∥2+∥un-p∥2-∥yn-un∥2+2sn〈yn-un,Aun-Ap〉-sn2∥Aun-Ap∥2).
This implies that
(3.46)∥yn-p∥2≤∥un-p∥2-∥yn-un∥2+2sn〈yn-un,Aun-Ap〉-sn2∥Aun-Ap∥2≤∥un-p∥2-∥yn-un∥2+2sn∥yn-un∥∥Aun-Ap∥.
By (3.46), (3.37), and (3.5), we obtain(3.47)∥xn+1-p∥2≤αn∥γf(Wnxn)-Bp∥2+∥un-p∥2-∥yn-un∥2+2sn∥yn-un∥∥Aun-Ap∥+2αn∥ρn-p∥∥γf(Wnxn)-Bp∥≤αn∥γf(Wnxn)-Bp∥2+∥xn-p∥2-∥yn-un∥2+2sn∥yn-un∥∥Aun-Ap∥+2αn∥ρn-p∥∥γf(Wnxn)-Bp∥.
It follows that
(3.48)∥yn-un∥2≤αn∥γf(Wnxn)-Bp∥2+∥xn-p∥2-∥xn+1-p∥2+2sn∥yn-un∥∥Aun-Ap∥+2αn∥ρn-p∥∥γf(Wnxn)-Bp∥≤αn∥γf(Wnxn)-Bp∥2+(∥xn-p∥+∥xn+1-p∥)(∥xn-xn+1∥)+2sn∥yn-un∥∥Aun-Ap∥+2αn∥ρn-p∥∥γf(Wnxn)-Bp∥.
It follows from (C1),∥Aun-Ap∥→0, and ∥xn+1-xn∥→0 that ∥yn-un∥→0. It follows from ∥ρn-un∥≤∥ρn-yn∥+∥yn-un∥ that limn→∞∥un-ρn∥=0.
We now show that(3.49)limn→∞∥Θβnkxn-Θβnk-1xn∥=0,k=1,2,…,m.
Indeed, letp∈Ω, it follows from the firmly nonexpansiveness of TβnFk, we have for each k∈{1,2,…,m},(3.50)∥Θβnkxn-p∥2=∥TβnFkΘβnk-1xn-TβnFkp∥2≤〈Θβnkxn-p,Θβnk-1xn-p〉=12(∥Θβnkxn-p∥2+∥Θβnk-1xn-p∥2-∥Θβnkxn-Θβnk-1xn∥2).
Thus, we get
(3.51)∥Θβnkxn-p∥2≤∥Θβnk-1xn-p∥2-∥Θβnkxn-Θβnk-1xn∥2,k=1,2,…,m.
This implies that for each k∈{1,2,…,m},
(3.52)∥Θβnkxn-p∥2≤∥Θβn0xn-p∥2-∥Θβnkxn-Θβnk-1xn∥2-∥Θβnk-1xn-Θβnk-2xn∥2-⋯-∥Θβn2xn-Θβn1xn∥2-∥Θβn1xn-Θβn0xn∥2.
It follows fromun=Θβnmxn that for each k=1,2,…,m(3.53)∥un-p∥2≤∥xn-p∥2-∥Θβnkxn-Θβnk-1xn∥2.
By (3.37), (3.6), and (3.53), we have that for each k=1,2,…,m(3.54)∥xn+1-p∥2≤αn∥γf(Wnxn)-Bp∥2+∥un-p∥2+2αn∥ρn-p∥∥γf(Wnxn)-Bp∥≤αn∥γf(Wnxn)-Bp∥2+∥xn-p∥2-∥Θβnkxn-Θβnk-1xn∥2+2αn∥ρn-p∥∥γf(Wnxn)-Bp∥.
Thus, we have that for each k=1,2,…,m(3.55)∥Θβnkxn-Θβnk-1xn∥2≤αn∥γf(Wnxn)-Bp∥2+∥xn-p∥2-∥xn+1-p∥2+2αn∥ρn-p∥∥γf(Wnxn)-Bp∥≤αn∥γf(Wnxn)-Bp∥2+(∥xn-p∥+∥xn+1-p∥)∥xn-xn+1∥+2αn∥ρn-p∥∥γf(Wnxn)-Bp∥.
It follows from (C1) and ∥xn+1-xn∥→0 that for each k=1,2,…,m(3.56)∥Θβnkxn-Θβnk-1xn∥→0.
Since(3.57)∥Wnρn-ρn∥≤∥Wnρn-xn+1∥+∥xn+1-xn∥+∥xn-Θβn1xn∥+∥Θβn1xn-Θβn2xn∥+⋯+∥Θβnm-1xn-Θβnmxn∥+∥un-yn∥+∥yn-ρn∥.
It follows from (3.56) that
(3.58)limn→∞∥Wnρn-ρn∥=0.
Observe that
(3.59)∥Wρn-ρn∥≤∥Wρn-Wnρn∥+∥Wnρn-ρn∥.
It follows from Remark2.7 that(3.60)limn→∞∥Wρn-ρn∥=0.
We show thatPΩ(γf+(I-B)) is a contraction. Indeed, for all x,y∈H, we have(3.61)∥PΩ(γf+(I-B))(x)-PΩ(γf+(I-B))(y)∥≤∥(γf+(I-B))(x)-(γf+(I-B))(y)∥≤γ∥f(x)-f(y)∥+∥I-B∥∥x-y∥≤γα∥x-y∥+(1-γ̃)∥x-y∥=(γα+1-γ̃)∥x-y∥.
The Banach's Contraction Mapping Principle guarantees thatPΩ(γf+(I-B)) has a unique fixed point, say q∈H. That is, q=PΩ(γf+(I-B))(q).
Next, we show that(3.62)lim supn→∞〈γf(q)-Bq,xn-q〉≤0.
To show that, we choose a subsequence {xni} of xn such that
(3.63)lim supn→∞〈γf(q)-Bq,xn-q〉=limi→∞〈γf(q)-Bq,xni-q〉.
As{xni} is bounded, we know that there is a subsequence {xnij} of {xni} which converges weakly to p. We may assume, without loss of generality, that xni⇀p. From ∥Θβnkxn-Θβnk-1xn∥→0 for each k=1,2,…,m, we obtain that Θβnikxni⇀p for k=1,2,…,m. From ∥un-ρn∥→0, we also obtain that ρni⇀p. Since {uni}⊂C and C is closed and convex, we obtain p∈C.
Now we show thatp∈Ω. Indeed, let us first show that p∈VI(C,A). Put(3.64)Tw1={Aw1+NCw1ifw1∈C,∅ifw1∉C.
Since A is relaxed (u,v)-cocoercive, we have
(3.65)〈Ax-Ay,x-y〉≥(-u)∥Ax-Ay∥2+v∥x-y∥2≥(v-uμ2)∥x-y∥2≥0,
which yields that A is monotone. Thus T is maximal monotone. Let (w1,w2)∈G(T). Since w2-Aw1∈NCw1 and ρn∈C, we have
(3.66)〈w1-ρn,w2-Aw1〉≥0.
On the other hand, from ρn=PC(I-rnA)yn, we have
(3.67)〈w1-ρn,ρn-(I-rnA)yn〉≥0
and hence
(3.68)〈w1-ρn,ρn-ynrn+Ayn〉≥0.
It follows that
(3.69)〈w1-ρni,w2〉≥〈w1-ρni,Aw1〉≥〈w1-ρni,Aw1〉-〈w1-ρni,ρni-ynirni+Ayni〉≥〈w1-ρni,Aw1-ρni-ynirni-Ayni〉=〈w1-ρni,Aw1-Aρni〉+〈w1-ρni,Aρni-Ayni〉-〈w1-ρni,ρni-ynirni〉≥〈w1-ρni,Aρni-Ayni〉-〈w1-ρni,ρni-ynirni〉,
which implies that 〈w1-p,w2〉≥0. We have p∈T-10 and hence p∈VI(C,A).
We next show thatp∈⋂k=1mEP(Fk). Indeed, by Lemma 2.1, we have that for each k=1,2,…,m,(3.70)Fk(Θβnkxn,y)+1βn〈y-Θβnkxn,Θβnkxn-Θβnk-1xn〉≥0,∀y∈C.
It follows from (A2) that(3.71)1βn〈y-Θβnkxn,Θβnkxn-Θβnk-1xn〉≥Fk(y,Θβnkxn),∀y∈C.
Hence,(3.72)〈y-Θβnikxni,Θβnikxni-Θβnik-1xniβni〉≥Fk(y,Θβnikxni),∀y∈C.
It follows from (A4), (A5),(Θβnikxni-Θβnik-1xni)/βni→0, and Θβnikxni⇀p that for each k=1,2,…,m,(3.73)Fk(y,p)≤0,∀y∈C.
Fort with 0<t≤1 and y∈C, let yt=ty+(1-t)p. Since y∈C and p∈C, we obtain yt∈C and hence Fk(yt,p)≤0. So by (A4), we have(3.74)0=Fk(yt,yt)≤tFk(yt,y)+(1-t)Fk(yt,p)≤tFk(yt,y).
Dividing byt, we get that for each k=1,2,…,m,(3.75)Fk(yt,y)≥0.
Letting t→0, it follows from (A3) that for each k=1,2,…,m,
(3.76)Fk(p,y)≥0
for all y∈C and hence p∈EP(Fk) for k=1,2,…,m. That is, p∈⋂k=1mEP(Fk).
We now show thatp∈Fix(W). Assume that p∉Fix(W). Since ρni⇀p and p≠Wp, from (3.60) and the Opial condition we have(3.77)lim infi→∞∥ρni-p∥<lim infi→∞∥ρni-Wp∥≤lim infi→∞{∥ρni-Wρni∥+∥Wρni-Wp∥}≤lim infi→∞∥ρni-p∥,
which is a contradiction. So, we get p∈Fix(W)=⋂i=1∞Fix(Si). This implies that p∈Ω.
Sinceq=PΩ(γf+(I-B))(q), we have(3.78)lim supn→∞〈γf(q)-Bq,xn-q〉=limi→∞〈γf(q)-Bq,xni-q〉=〈γf(q)-Bq,p-q〉≤0.
That is, (3.62) holds. Next, we consider
(3.79)∥xn+1-q∥2=∥αn(γf(Wnxn)-Bq)+(I-αnB)(Wnρn-q)∥2≤(1-αnγ̃)2∥Wnρn-q∥2+2αn〈γf(Wnxn)-Bq,xn+1-q〉≤(1-αnγ̃)2∥xn-q∥2+2αnγ〈f(Wnxn)-f(q),xn+1-q〉+2αn〈γf(q)-Bq,xn+1-q〉≤(1-αnγ̃)2∥xn-q∥2+αnγα(∥xn-q∥2+∥xn+1-q∥2)+2αn〈γf(q)-Bq,xn+1-q〉
So, we can obtain
(3.80)∥xn+1-q∥2≤(1-αnγ̃)2+αnγα1-αnγα∥xn-q∥2+2αn1-αnγα〈γf(q)-Bq,xn+1-q〉=1-2αnγ̃+αnγα1-αnγα∥xn-q∥2+αn2γ̃21-αnγα∥xn-q∥2+2αn1-αnγα〈γf(q)-Bq,xn+1-q〉≤[1-2αn(γ̃-αγ)1-αnγα]∥xn-q∥2+2αn(γ̃-αγ)1-αnγα[1γ̃-αγ〈γf(q)-Bq,xn+1-q〉+αnγ̃22(γ̃-αγ)M],
where M is an approximate constant such that M≥supn≥1{∥xn-q∥2}}.
Putln=2αn(γ̃-αγ)/(1-αnγα) and tn=(1/(γ̃-αγ))〈γf(q)-Bq,xn+1-q〉+(αnγ̃2/2(γ̃-αγ))M. That is,(3.81)∥xn+1-q∥2≤(1-ln)∥xn-q∥+lntn.
From condition (C1) and Lemma 2.2, we concluded that xn→q∈Ω. It is easy to see that un→q and yn→q. This completes the proof.Corollary 3.2.
LetC be a nonempty, closed and convex subset of H. Let F be a bifunction from C×C to ℝ satisfies conditions (A1)–(A5). Let A:C→H be relaxed (u,v)-cocoercive and μ-Lipschitz continuous and B a strongly positive linear bounded operator on H with coefficient γ̃>0. Assume that 0<γ<γ̃/α. Let S1,S2,… be a family of infinitely nonexpansive mappings of C into itself such that Γ=⋂i=1∞Fix(Si)∩VI(C,A)∩EP(F)≠∅, let ξ1,ξ2,… be real numbers such that 0<ξi≤δ<1 for every i∈N and Wn be the W-mapping of C into itself generated by Sn,Sn-1,…,S1 and ξn,ξn-1,…,ξ1. Let f:C→C be a contraction with coefficient α(0<α<1) and {xn}, {un} and {yn} be sequences generated by
(3.82)x1=x∈H,F(un,y)+1βn〈y-un,un-xn〉≥0,∀y∈C,yn=PC(I-snA)un,xn+1=αnγf(Wnxn)+(I-αnB)WnPC(I-rnA)yn
for every n=1,2,…, where {αn}, {βn}, {rn} and {sn} are sequences of numbers satisfying the conditions:(C1)
{αn}⊂[0,1] with limn→∞αn=0, ∑n=1∞αn=∞, and ∑n=1∞|αn+1-αn|<∞;(C2)
{rn}⊂[a,b] and {sn}⊂[a,b] for some a,b with 0≤a≤b≤2(υ-uμ2)/μ2, ∑n=1∞|rn+1-rn|<∞, and ∑n=1∞|sn+1-sn|<∞;(C3)
lim infn→∞βn>0 and ∑n=1∞|βn+1-βn|<∞.
Then,{xn} and {yn} converge strongly to q∈Γ, which solves the following variational inequality:(3.83)〈γfq-Bq,p-q〉≤0,∀p∈Γ.Proof.
Letm=1, by Theorem 3.1, we obtain the desired result.Corollary 3.3.
LetC be a nonempty, closed, and convex subset of H. Let A:C→H be relaxed (u,v)-cocoercive and μ-Lipschitz continuous and let B be a strongly positive linear bounded operator on H with coefficient γ̃>0. Assume that 0<γ<γ̃/α. Let S1,S2,… be a family of infinitely nonexpansive mappings of C into itself such that Δ=⋂i=1∞Fix(Si)∩VI(C,A)≠∅, let ξ1,ξ2,… be real numbers such that 0<ξi≤δ<1 for every i∈N, and let Wn be the W-mapping of C into itself generated by Sn,Sn-1,…,S1 and ξn,ξn-1,…,ξ1. Let f:C→C be a contraction with coefficient α(0<α<1) and {xn}, {un}, and {yn} be sequences generated by
(3.84)x1=x∈C,yn=PC(I-snA)xn,xn+1=αnγf(Wnxn)+(I-αnB)WnPC(I-rnA)yn
for every n=1,2,…, where {αn}, {βn}, {rn}, and {sn} are sequences of numbers satisfying the conditions:(C1)
{αn}⊂[0,1] with limn→∞αn=0, ∑n=1∞αn=∞, and ∑n=1∞|αn+1-αn|<∞;(C2)
{rn}⊂[a,b] and {sn}⊂[a,b] for some a,b with 0≤a≤b≤2(υ-uμ2)/μ2, ∑n=1∞|rn+1-rn|<∞ and ∑n=1∞|sn+1-sn|<∞.
Then,{xn} and {yn} converge strongly to q∈Δ, which solves the following variational inequality:(3.85)〈γfq-Bq,p-q〉≤0,∀p∈Δ.Proof.
LetF(x,y)=0 for x,y∈C, by Corollary 3.2 we obtain the desired result.Corollary 3.4.
LetC be a nonempty, closed and convex subset of H. Let F1,F2,…,Fm be bifunctions from C×C to ℝ satisfies conditions (A1)–(A5). Let A:C→H be relaxed (u,v)-cocoercive and μ-Lipschitz continuous and B a strongly positive linear bounded operator on H with coefficient γ̃>0 such that Ξ=⋂k=1mEP(Fk)∩VI(C,A)≠∅. Let f:C→C be a contraction with coefficient α(0<α<1) and {xn}, {un} and {yn} be sequences generated by
(3.86)x1=x∈H,un=TβnFmTβnFm-1⋯TβnF2TβnF1xn,yn=PC(I-snA)un,xn+1=αnγf(xn)+(I-αnB)PC(I-rnA)yn
for every n=1,2,…, where {αn}, {βn}, {rn} and {sn} are sequences of numbers satisfying the conditions:(C1)
{αn}⊂[0,1] with limn→∞αn=0, ∑n=1∞αn=∞, and ∑n=1∞|αn+1-αn|<∞;(C2)
{rn}⊂[a,b] and {sn}⊂[a,b] for some a,b with 0≤a≤b≤2(υ-uμ2)/μ2, ∑n=1∞|rn+1-rn|<∞, and ∑n=1∞|sn+1-sn|<∞;(C3)
lim infn→∞βn>0 and ∑n=1∞|βn+1-βn|<∞.
Then,{xn}, {yn} and {un} converge strongly to q∈Ω, which solves the following variational inequality:(3.87)〈γfq-Bq,p-q〉≤0,∀p∈Ξ.Remark 3.5.
(i) Ifsn=0 for all n≥0, by Corollary 3.2, we get Theorem 2.1 in [9]. If sn=0 and Si=I for all n≥0, by Corollary 3.2, we get Theorem 2.1 in [8] with S=I. If sn=0, rn=0 and Si=I for all n≥0, by Corollary 3.2, we get Theorem 3.1 in [6] with S=I and Theorem 3.3 in [7] with S=I and C=H.
(ii) Corollary3.3 extends, generalizes and improves the main results in [21, 22, 24].
(iii) It is easy to see that Theorem3.1 is different from the main results in [1–4].



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*Source: 428293-2010-08-08.xml*