# A Viscosity Hybrid Steepest Descent Method for Equilibrium Problems, Variational Inequality Problems, and Fixed Point Problems of Infinite Family of Strictly Pseudocontractive Mappings and Nonexpansive Semigroup

Authors: Haitao Che; Xintian Pan
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

In this paper, modifying the set of variational inequality and extending the nonexpansive mapping of hybrid steepest descent method to nonexpansive semigroups, we introduce a new iterative scheme by using the viscosity hybrid steepest descent method for finding a common element of the set of solutions of a system of equilibrium problems, the set of fixed points of an infinite family of strictly pseudocontractive mappings, the set of solutions of fixed points for nonexpansive semigroups, and the sets of solutions of variational inequality problems with relaxed cocoercive mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above sets under some mild conditions. The results shown in this paper improve and extend the recent ones announced by many others.


---

## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 and induced norm ∥·∥. Let C be a nonempty closed convex subset of H and let F:C×C→R be a bifunction. We consider the following equilibrium problem (EP) which is to find x*∈C such that
(1)EP:F(x*,y)≥0,∀y∈C.
The set of solutions of EP is denoted by EP(F).Let{Fi,i=1,2,…,N} be a finite family of bifunctions from C×C into R, where R is the set of real numbers. The system of equilibrium problems for {F1,F2,…,FN} is to find a common element x*∈C such that
(2)F1(x*,y)≥0,∀y∈C,F2(x*,y)≥0,∀y∈C,⋮FN(x*,y)≥0,∀y∈C.
We denote the set of solutions of (2) by ∩k=1NSEP(Fk), where SEP(Fk) is the set of solutions to the equilibrium problems, that is,(3)Fk(x*,y)≥0,∀y∈C.IfN=1, then the problem (2) is reduced to the equilibrium problems.IfN=1 and F(x*,y)=〈Tx*,y-x*〉, then the problem (2) is reduced to the variational inequality problems of finding x*∈C such that
(4)〈Tx*,y-x*〉≥0,∀y∈C.
The set of solutions of (4) is denoted by VI(C,T).The equilibrium problem is very general in the sense that it includes, as special cases, fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems in noncooperative games, and numerous problems in physics, economics, and others. Some methods have been proposed to solveVI(C,T), EP(F), and SEP(Fk); see, for example, [1–29] and references therein. Formulations (2) extend this formulism to such problems, covering in particular various forms of feasibility problems [30, 31].Definition 1.
One-parameter family mappingΓ={T(t):t∈R+} from C into itself is said to be a nonexpansive semigroup on C if it satisfies the following conditions:(i)
T(0)x=x for all x∈C,(ii)
T(s+t)=T(s)T(t) for all s,t∈R+,(iii)
for eachx∈C, the mapping T(t)x is continuous,(iv)
∥T(t)x-T(t)y∥≤∥x-y∥ for all x,y∈C and t∈R+.Remark 2.
We denote byF(Γ) the set of all common fixed points of Γ, that is, F(Γ):=∩t∈R+F(T(t))={x∈C:T(t)x=x}.LetB:C→H be a nonlinear mapping. Now, we recall the following definitions.(1)
B is said to be monotone if
(5)〈Bx-By,x-y〉≥0,∀x,y∈C.(2)
B:C→C is called ω-Lipschitzian if there exists a positive constant ω such that
(6)∥Bx-By∥≤ω∥x-y∥,∀x,y∈C.(3)
B is said to be η-strongly monotone if there exists a positive constant η such that
(7)〈Bx-By,x-y〉≥η∥x-y∥2,∀x,y∈C.(4)
B is said to be nonexpansive if
(8)∥Bx-By∥≤∥x-y∥,∀x,y∈C.AndF(B) denotes the set of fixed points of the mapping B, that is, F(B)={x∈C:Bx=x}.(5)
B:C→C is said to be k-strictly pseudocontractive mapping if there exists a constant 0≤k<1 such that
(9)∥Bx-By∥2≤∥x-y∥2+k∥(I-B)x-(I-B)y∥2,∀x,y∈C.(6)
B is said to be α-inverse-strongly monotone if there exists a constant α>0 such that
(10)〈Bx-By,x-y〉≥α∥Bx-By∥2,∀x,y∈C.(7)
B is said to be relaxed (u,v)-cocoercive if there exist positive real numbers u,v such that
(11)〈Bx-By,x-y〉≥(-u)∥Bx-By∥2+v∥x-y∥2,∀x,y∈C.(8)
A linear bounded operatorB is strong positive if there exists a constant γ¯>0 with the property
(12)〈Bx,x〉≥γ¯∥x∥2,∀x∈C.(9)
A set-valued mappingQ:H→2H is called monotone if for all x,y∈H, f∈Qx and g∈Qy imply 〈x-y,f-g〉≥0.(10)
A monotone mappingQ:H→2H is called maximal if the graph G(Q) of Q is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping Q is maximal if and only if for (x,f)∈H×H, 〈x-y,f-g〉≥0 for every (y,g)∈G(Q) implies f∈Qx.Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. Convex minimization problems have a great impact and influence on the development of almost all branches of pure and applied sciences. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert spaceH:
(13)θ(x)=12〈Ax,x〉-〈x,b〉,∀x∈F(S),
where A is a linear bounded operator, F(S) is the fixed point set of a nonexpansive mapping S, and b is a given point in H [16].For finding a common element of the set of fixed points of nonexpansive mappings and the set of the variational inequalities, in 2006, Marino and Xu [16] introduced the general iterative method and proved that for a given x0∈H, the sequence {xn} generated by the algorithm
(14)xn+1=αnγf(xn)+(I-αnB)Txn,
where T is a self-nonexpansive mapping on H, f is an α-contraction of H into itself (i.e., ∥f(x)-f(y)∥≤α∥x-y∥,∀x,y∈H and α∈(0,1)), {αn}⊂(0,1) satisfies certain conditions, and B is strongly positive bounded linear operator on H and converges strongly to fixed point x* of T which is the unique solution to the following variational inequality:
(15)〈(γf-B)x*,x*-x〉≤0,∀x∈F(T),
which is the optimality condition for the minimization problem
(16)minx∈F(S)∩EP12〈Bx,x〉-h(x),
where h is a potential function for rf(i.e., h′(x)=rf(x) for z∈H).Takahashi and Toyoda [32] introduced the following iterative scheme:
(17)x0∈C,xn+1=γnxn+(1-γn)SPC(xn-αnBxn),
where B is a ξ-inverse-strongly monotone mapping, {γn} is a sequence in (0,1), and {αn} is a sequence in (0,2ξ). They showed that if F(S)∩VI(C,B)≠∅, then the sequence {xn} generated by (17) converges weakly to some z∈F(S)∩VI(C,B).Yamada [33] introduced the following iterative scheme called the hybrid steepest descent method:
(18)xn+1=Sxn+αnμBSxn,n∈N,
where x1=x∈H, {αn}⊂(0,1), and let B:H→H be a strongly monotone and Lipschitz continuous mapping and μ is a positive real number. He proved that the sequence {xn} generated by (18) converges strongly to the unique solution of F(S)∩VI(C,B).LetC be a nonempty closed convex subset of H. Given r>0 the operators JrF:H→C defined by
(19)JrFx={z∈C:F(z,y)+1r(y-z,z-x)≥0,∀y∈C}
are called the resolvent of F (see [19]). It is shown in [19] that under suitable hypotheses on F (to be stated precisely in Section 2), JrF:H→C is single-valued and firmly nonexpansive and satisfied F(JrF)=EP(F),∀r>0.For finding a common element ofEP(F)∩F(S), S. Takahashi and W. Takahashi [23] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solution (1) and the set of fixed points of a nonexpansive mapping in a Hilbert space. Let S:C→H be a nonexpansive mapping. Starting with arbitrary initial point x1∈H, define sequences {xn} and {un} recursively by
(20)F(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnf(xn)+(1-αn)Sun,∀n∈N.
They proved that under certain appropriate conditions imposed on {αn} and {rn}, the sequences {xn} and {un} converge strongly to z∈F(S)∩EP(F), where z=PF(S)∩EP(F)f(z).In 2012, Chamnarnpan and Kumam [34] introduced the following explicit viscosity scheme with respect to W-mappings for an infinite family of nonexpansive mappings
(21)xn+1=εnrf(xn)+βnxn+((1-βn)I-εnA)WnJrFxn.
They prove that sequence {xn} and JrnF converge strongly to z∈(∩n=1∞F(Tn))∩EP(F), where z is an equilibrium point for F and is the unique solution of the variational inequality
(22)〈(rf-A)z,x-z〉≤0,∀x∈⋂n=1∞F(Tn)∩EP(F).In 2012, Kangtunyakarn [35] modify the set of variational inequality to construct a new iterative scheme for finding a common element of the set of fixed point problems of infinite family of ki pseudocontractive mappings and the set of equilibrium problem and two sets of variational inequality problems. Let
(23)F∶=(⋂i=1∞{F(Ti)})∩(⋂k=1MSEP(Fk))∩VI(C,A)∩VI(C,B).
Starting with arbitrary initial point x1∈C, define sequences {xn} and {un} recursively by
(24)F(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnu+(1-αn)SnPC(I-γ(aA+(1-a)B)un),∀n∈N,
where {Sn:C→C} is the sequence defined by (37), A,B is α and β-inverse-strongly monotone mapping, respectively, a∈(0,1), 0<r<min{2α,2β} and {rn}⊂[a,b]⊂(0,min{2α,2β}). Under certain appropriate conditions they proved that the sequences {xn} and {un} converge strongly to z∈F, where z=PFu.LetAi:C→H be a mapping, for i=1,2,…,N. By modification of (4), for δi∈(0,1), we have
(25)VI(C,∑i=1N‍δiAi)={x*∈C:〈y-x*,∑i=1N‍δiAix*〉≥0,∀y∈C,∑i=1N‍δi=1}.In this paper, motivated by the above results, we extend the nonexpansive mapping of hybrid steepest descent method (18) to nonexpansive semigroups and introduce a new iterative scheme for finding a common element of the set of solutions of a system of equilibrium problems, the set of fixed points of an infinite family of strictly pseudocontractive mappings, the set of solutions of fixed points for nonexpansive semigroups, and the set of solutions of variational inequality problems for relaxed cocoercive mapping in a real Hilbert space by the hybrid steepest descent method. The results shown in this paper improve and extend the recent ones announced by many others.


## 2. Preliminaries

Throughout this paper, we always assume thatC is a nonempty closed convex subset of a Hilbert space H. We write xn⇀x to indicate that the sequence {xn} converges weakly to x. xn→x implies that {xn} converges strongly to x. We denote by N and R the sets of positive integers and real numbers, respectively. For any x∈H, there exists a unique nearest point in C, denoted by PCx, such that
(26)∥x-PCx∥≤∥x-y∥,∀y∈C.
Such a PC is called the metric projection of H onto C. It is known that PC is nonexpansive. Furthermore, for x∈H and u∈C,
(27)u=PCx⟺〈x-u,u-y〉≥0,∀y∈C.
It is widely known that H satisfies Opial’s condition [8], that is, for any sequence {xn} with xn⇀x, the inequality
(28)limn→∞inf∥xn-x∥<limn→∞inf∥xn-y∥
holds for every y∈H with y≠x.In order to solve the equilibrium problem for a bifunctionF:C×C→R, we assume that F satisfies the following conditions:(A1)
F(x,x)=0,∀x∈C,(A2)
F is monotone, that is, F(x,y)+F(y,x)≤0,∀x,y∈C,(A3)
limt↓0F(tz+(1-t)x,y)≤F(x,y),∀x,y,z∈C,(A4)
For eachx∈C,y↦F(x,y) is convex and lower semicontinuous.Let us recall the following lemmas which will be useful for our paper.Lemma 3 (see [19]).
LetF be a bifunction from C×C into R satisfying (A1), (A2), (A3), and (A4). Then, for any r>0 and x∈H, there exists z∈C such that
(29)F(z,y)+1r(y-z,z-x)≥0,∀y∈C.
Furthermore, if JrFx={z∈C:F(z,y)+(1/r)(y-z,z-x)≥0,∀y∈C}, then the following hold:(1)
JrF is single-valued,(2)
JrF is firmly nonexpansive, that is,
(30)∥JrFx-JrFy∥2≤〈JrFx-JrFy,x-y〉,∀x,y∈H,(3)
F(JrF)=EP(F),(4)
EP(F) is closed and convex.Lemma 4 (see [12]).
LetC be a nonempty bounded closed and convex subset of a real Hilbert space H. Let Γ={T(s):s∈R+} from C be a nonexpansive semigroup on C, then for all h>0,
(31)limt→∞supx∈C∥1t∫0t‍T(s)xds-T(h)(1t∫0t‍T(s)xds)∥=0.Lemma 5 (see [13]).
LetC be a nonempty bounded closed and convex subset of a real Hilbert space H, let {xn} be a sequence, and let Γ={T(s):s∈R+} from C be a nonexpansive semigroup on C, if the following conditions are satisfied:(i)
xn⇀z,(ii)
limsups→∞limsupn→∞∥T(s)xn-xn∥=0,
then,z∈F(Γ).Lemma 6 (see [36]).
In a Hilbert spaceH, there holds the inequality
(32)∥x+y∥2≤∥x∥2+2〈y,x+y〉,∀x,y∈H.Lemma 7 (see [16]).
AssumeA be a strongly positive linear bounded operator on H with coefficient γ¯>0 and 0≤ρ≥∥A∥-1, then ∥I-ρA∥≤1-ργ¯.Lemma 8 (see [37]).
LetB be a monotone mapping of C into H and let NCω1 be the normal cone to C at ω1∈C, that is,
(33)NCω1={ω∈H:〈ω1-ω2,ω〉≥0,∀ω2∈C},
and define a mapping Q on C by
(34)Qω1={Bω1+NCω1,ω1∈C,∅,ω1∉C.
Then Q is maximal monotone and 0∈Qω1 if and only if, 〈Bω1,ω1-ω2〉≥0 for all ω2∈C.Lemma 9 (see [27]).
Let{xn} and {yn} be bounded sequences in a Banach space E and {γn} be a sequence in [0,1] satisfying the following condition:
(35)0<limn→∞infγn≤limn→∞supγn<1.
Suppose that xn+1=γnxn+(1-γn)yn,n≥0 and limn→∞sup(∥yn+1-yn∥-∥xn+1-xn∥)≤0. Then limn→∞∥yn-xn∥=0.Lemma 10 (see [28]).
Assume that{an} is a sequence of nonnegative real numbers such that
(36)an+1≤(1-bn)an+cn,n≥0,
where {bn} is a sequence in (0,1) and {δn} is a sequence in R, such that(i)
∑i=1∞bi=∞,(ii)
limn→∞sup(cn/bn)≤0 or ∑i=1∞|cn|<∞.
Then,limn→∞an=0.LetC be a nonempty closed convex subset of a Hilbert space H. Let {Ti}i=1∞ be mapping of C into self. For all j=1,2,…, let ρj=(α1j,α2j,α2j)∈I×I×I where I=[0,1] and α1j+α2j+α2j=1. For every n∈N, we define the mapping Sn:C→C as follows:
(37)Un,n+1∶=I,Un,n:=α1nTnUn,n+1+α2nUn,n+1+α3nI,⋮Un,k∶=α1kTkUn,k+1+α2kUn,k+1+α3kI,⋮Un,2∶=α12T2Un,3+α22Un,3+α32I,Sn=Un,1:=α11T1Un,2+α21Un,2+α31I.
This mapping is called S-mapping generated by T1,…,Tn and ρ1,…,ρn.Lemma 11 (see [38]).
LetC be a nonempty closed convex subset of a Hilbert space H. Let {Ti}i=1∞ be a ki-strict pseudocontractive mapping of C into self with κ=supiki and let ρj=(α1j,α2j,α2j)∈I×I×I where I=[0,1], α1j+α2j+α2j=1, α1j+α2j≤b<1, and α1j,α2j,α2j∈(κ,1) for all j=1,2,…. For every n∈N, let Sn and S-mapping generated by T1,T2,…,Tn and ρ1,ρ2,…,ρn and T1,T2,…, and ρ1,ρ2,…, respectively. Then, for every x∈C and k∈N, the limit limn→∞Un,kx exists.In view of the previous lemma, we will define the mappingS:C→C as follows:
(38)Sx∶=limn→∞Snx=limn→∞Un,1x,x∈C.Remark 12 (see [38]).
For eachn∈N, Sn is nonexpansive and limn→∞supx∈D∥Snx-Sx∥=0 for every bounded subset D of C.Lemma 13 (see [38]).
LetC be a nonempty closed convex subset of a Hilbert space H. Let {Ti}i=1∞ be a ki-strict pseudocontractive mapping of C into self such that ∩i=1∞F(Ti)≠∅ with κ=supiki and let ρj=(α1j,α2j,α2j)∈I×I×I where I=[0,1], α1j+α2j+α2j=1, α1j+α2j≤b<1, and α1j,α2j,α2j∈(κ,1) for all j=1,2,…. For every n∈N, let Sn and S-mapping generated by T1,T2,…,Tn and ρ1,ρ2,…,ρn, respectively. Then, F(S)=∩i=1∞F(Ti)≠∅.


## 3. Main Results

In this section, we will present our main results. To establish our results, we need the following technical lemmas.Lemma 14.
LetC be a nonempty closed convex subset of a Hilbert space H and let Ai:C→H be ωi-Lipschitz continuous and relaxed (ui,vi)-cocoercive mappings with vi-uiωi2>0, for i=1,2,…,N. If ∩i=1NVI(C,Ai)≠∅, then, for δi∈(0,1) and ∑i=1Nδi=1,
(39)⋂i=1NVI(C,Ai)=VI(C,∑i=1N‍δiAi).Proof.
The proof is by induction. This holds forN=2. In fact, for a∈(0,1), it is obvious that
(40)VI(C,A1)∩VI(C,A2)⊆VI(C,∑i=1N‍δiAi).
Next, we will show that VI(C,∑i=1NδiAi)⊆VI(C,A1)∩VI(C,A2).
Let(41)x0∈VI(C,∑i=1N‍δiAi),x*∈VI(C,A1)∩VI(C,A2).
It follows that
(42)〈y-x*,A1x*〉≥0,∀y∈C,(43)〈y-x*,A2x*〉≥0,∀y∈C.
Then, for every a∈(0,1), one has
(44)〈y-x*,aA1x*〉≥0,∀y∈C,〈y-x*,(1-a)A2x*〉≥0,∀y∈C.
From x0∈VI(C,∑i=1NδiAi) and (43), one has
(45)〈x*-x0,aA1x0〉=〈x*-x0,(∑i=1N‍δiAi)x0〉-〈x*-x0,(1-a)A2x0〉≥(1-a)〈x0-x*,A2x0〉=(1-a)〈x0-x*,A2x0-A2x*〉+(1-a)〈x0-x*,A2x*〉≥(1-a)(v2-u2ω22)∥x0-x*∥2≥0,
which means
(46)〈x*-x0,A1x0〉≥0.
On the other hand, from x*∈VI(C,A1), we have
(47)〈x*-x0,A1x0〉=〈x*-x0,A1x0-A1x*〉+〈x*-x0,A1x*〉≤〈x*-x0,A1x0-A1x*〉≤u1∥A1x0-A1x*∥2-v1∥x0-x*∥2≤u1ω12∥x0-x*∥2-v1∥x0-x*∥2=(u1ω12-v1)∥x0-x*∥2≤0.
This together with (46) leads to
(48)A1x*=A1x0.
Furthermore, for every y∈C, from (46) and (48), we obtain
(49)〈y-x0,A1x0〉=〈y-x*,A1x0〉+〈x*-x0,A1x0〉≥〈y-x*,A1x0〉=〈y-x*,A1x*〉≥0,
which implies
(50)x0∈VI(C,A1).
It follows from (45) and (42) that
(51)〈x*-x0,(1-a)A2x0〉≥〈x0-x*,aA1x0〉=a〈x0-x*,A1(x0-x*)〉+a〈x0-x*,A1x*〉≥a〈x0-x*,A1(x0-x*)〉≥a(v1-u1ω12)∥x0-x*∥2≥0.
It yields that
(52)〈x*-x0,A2x0〉≥0.
From x*∈VI(C,A2) and (52), one has
(53)0≤〈x*-x0,A2x0〉=〈x*-x0,A2x0-A2x*〉+〈x*-x0,A2x*〉≤〈x*-x0,A2x0-A2x*〉≤u2∥A2x0-A2x*∥2-v2∥x0-x*∥2≤u2ω22∥x0-x*∥2-v1∥x0-x*∥2=(u2ω22-v2)∥x0-x*∥2≤0.
That is,
(54)A2x*=A2x0.
Therefore, for every y∈C, from (52) and (54), we obtain
(55)〈y-x0,A2x0〉=〈y-x*,A2x0〉+〈x*-x0,A2x0〉≥〈y-x*,A2x0〉=〈y-x*,A2x*〉≥0,
which means
(56)x0∈VI(C,A2).
And hence,
(57)x0∈VI(C,A1)∩VI(C,A2).
Thus, we have
(58)VI(C,∑i=1N‍δiAi)⊆VI(C,A1)∩VI(C,A2).
Thus,
(59)VI(C,∑i=1N‍δiAi)=VI(C,A1)∩VI(C,A2).
Assume now that∩i=1kVI(C,Ai)=VI(C,∑i=1kδiAi) is true for some k, and we show that it continues to hold for k+1. For δi∈(0,1) and ∑i=1k+1δi=1, we have
(60)VI(C,∑i=1k+1‍δiAi)=VI(C,δ1A1+∑i=2k+1‍δiAi)=VI(C,δ1A1+(1-δ1)∑i=2k+1‍δi1-δ1Ai)=VI(C,δ1A1)∩VI(C,(1-δ1)∑i=2k+1‍δi1-δ1Ai)=VI(C,A1)∩VI(C,∑i=2k+1‍δi1-δ1Ai)=VI(C,A1)∩(⋂i=2k+1VI(C,Ai))=⋂i=1k+1VI(C,Ai).
By induction, ∩i=1kVI(C,Ai)=VI(C,∑i=1kδiAi) holds for k=1,2,…,N and this completes the proof.Lemma 15.
LetC be a nonempty closed convex subset of a Hilbert space H, let Γ={T(s):s∈R+} from C be a nonexpansive semigroup on C, and let Ai:C→H be ωi-Lipschitz continuous and relaxed (μi,vi)-cocoercive mappings with vi-μiωi2>0,  for i=1,2,…,N. Assume that D=∑i=1NδiAi, for δi∈(0,1) and ∑i=1Nδi=1. If Kn(x)=(1/tn)∫0tn‍T(s)Snxds, where {Sn:C→C} is the sequence defined by (37) with 0≤αn≤(2∑i=1Nδi(vi-μiωi2))/(∑i=1Nδiωi2), then Kn-αnDKn is a nonexpansive mapping in H. Furthermore, I-αnD is a nonexpansive mapping in H.Proof.
Since0≤αn≤(2∑i=1Nδi(vi-μiωi2))/(∑i=1Nδiωi2), for every x,y∈C, we have
(61)∥(Kn-αnDKn)x-(Kn-αnDKn)y∥2=∥(Knx-Kny)-αn(DKnx-DKny)∥2=∥Knx-Kny∥2-2αn〈Knx-Kny,DKnx-DKny〉+αn2∥DKnx-DKny∥2≤∥Knx-Kny∥2-2αn∑i=1N‍δi(vi-μiωi2)×∥Knx-Kny∥2+αn2∑i=1N‍δiωi2∥Knx-Kny∥2=(1-2αn∑i=1N‍δi(vi-μiωi2)+αn2∑i=1N‍δiωi2)×∥Knx-Kny∥2≤∥x-y∥2.
Thus, we obtain that Kn-αnDKn is a nonexpansive mapping.
Similarly, we can obtain thatI-αnD is a nonexpansive mapping in H and this completes the proof.The following main results follow from Lemmas14 and 15.Theorem 16.
LetC be a nonempty closed convex subset of a real Hilbert space H, and let Fk,k∈{1,2,…,M} be a bifunction from C×C→R satisfying (A1)–(A4). Let Γ={T(s):0≤s<∞} be a nonexpansive semigroup on C and let tn be a positive real divergent sequence. Let {Ti}i=1∞ be ki-strict pseudo-contractive mappings of C into self with κ=supiki and let ρj=(α1j,α2j,α2j)∈I×I×I, where I=[0,1], α1j+α2j+α2j=1,α1j+α2j≤b<1, and α1j,α2j,α2j∈(κ,1) for all j=1,2,…. For every n∈N, let Sn and S-mapping generated by T1,T2,…,Tn and ρ1,ρ2,…,ρn with Ti(F(Γ))⊂F(Γ). Let Ai:C→H be ωi-Lipschitz continuous and relaxed (μi,vi)-cocoercive mappings with vi-μiωi2>0,  for i=1,2,…,N, let f be a contraction of H into itself with η∈(0,1), and let A be is a strongly positive linear bounded self-adjoint operator with the coefficients γ¯>0 and 0<r<γ¯/η. Assume that
(62)Θ∶=F(Γ)∩(⋂i=1∞{F(Ti)})∩(⋂k=1MSEP(Fk))∩(⋂i=1NVI(C,Ai)).
Let {xn} be a sequence generated by x1∈C and
(63)un=JrM,nFMJrM-1,nFM-1⋯Jr2,nF2Jr1,nF1xn,zn=PC(1tn∫0tn‍T(s)Snunds-αn(∑i=1N‍δiAi)×1tn∫0tn‍T(s)Snunds(∑i=1N‍δiAi)),yn=εnrf(xn)+βnxn+((1-βn)I-εnA)zn,xn+1=γnxn+(1-γn)yn,∀n∈N,
where {Sn:C→C} is the sequence defined by (37) and δi∈(0,1), ∑i=1Nδi=1. If {εn}, {βn} are two sequences in (0,1) and {γn}⊂[c1,c2]⊂(0,1) and {rk,n}, for k∈{1,2,…,M} is a real sequence in (0,∞) satisfy the following conditions:(i)
limn→∞εn=0,
∑i=1∞εn=∞,(ii)
0<limn→∞infβn≤limn→∞supβn<1 and limn→∞|βn+1-βn|=0,(iii)
limn→∞infrk,n>0 and limn→∞|rk,n+1-rk,n|=0, for k∈{1,2,…,M},(iv)
{αn}⊂[g1,g2]⊂(0,(2∑i=1Nδi(vi-μiωi2))/(∑i=1Nδiωi2)) and limn→∞|αn+1-αn|=0,(v)
limn→∞|tn-tn-1|/tn=0.
Then{xn} converges strongly to z∈Θ, where z is the unique solution of variational inequality
(64)limn→∞sup〈(rf-A)z,p-z〉≤0,∀p∈Θ,
which is the optimality condition for the minimization problem
(65)minz∈Θ12〈Az,z〉-h(z),
where h is a potential function for rf(i.e., h′(z)=rf(z) for z∈H).Proof.
From the restrictions on control sequences, we may assume, without loss of generality, thatεn≤(1-βn)∥A∥-1 for all n≥1. From Lemma 7, 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 linear bounded self-adjoint operator on H, we have
(66)∥A∥=sup{|〈Ax,x〉|:x∈H,∥x∥=1}.
Note that
(67)〈((1-βn)I-εnA)x,x〉=1-βn-εn〈Ax,x〉≥1-βn-εn∥A∥≥0.
That is, (1-βn)I-εnA is positive. Furthermore,
(68)∥(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¯.
Next, We divide the proof of Theorem into five steps.
Step 1. We show that {xn} is bounded.
Takep∈Θ. Let ℑnk=Jrk,nFkJrk-1,nFk-1⋯Jr2,nF2Jr1,nF1, for k∈{1,2,…M} and ℑn0=I, for any n∈N. Since Jrk,nFk is nonexpansive for each k=1,2,…M and un=ℑnMxn, we have
(69)∥un-p∥=∥ℑnMxn-ℑnMp∥≤∥xn-p∥.
From Lemma 15 and (69), one has
(70)∥zn-p∥=∥PC(1tn∫0tn‍T(s)Snunds-αn(∑i=1N‍δiAi)×1tn∫0tn‍T(s)Snunds)-p(∑i=1N‍δiAi)∥=∥PC(1tn∫0tn‍T(s)Snunds-αn(∑i=1N‍δiAi)×1tn∫0tn‍T(s)Snunds-αn(∑i=1N‍δiAi))-PC(1tn∫0tn‍T(s)Snpds-αn(∑i=1N‍δiAi)×1tn∫0tn‍T(s)Snpds(∑i=1N‍δiAi))∥≤∥(1tn∫0tn‍T(s)Snunds-αn(∑i=1N‍δiAi)×1tn∫0tn‍T(s)Snunds(∑i=1N‍δiAi))-(1tn∫0tn‍T(s)Snpds-αn(∑i=1N‍δiAi)×1tn∫0tn‍T(s)Snpds(1tn∫0tn‍T(s)Snunds-αn(∑i=1N‍δiAi))∥≤∥un-p∥≤∥xn-p∥.
It follows that
(71)∥yn-p∥=∥εnγf(xn)+βnxn+((1-βn)I-εnA)zn-p∥=∥εn(γf(xn)-Ap)+βn(xn-p)+((1-βn)I-εnA)(zn-p)∥≤εn∥γ(f(xn)-f(p))+γf(p)-Ap∥+βn∥xn-p∥+(1-βn-εnγ¯)∥zn-p∥≤εnγη∥xn-p∥+εn∥γf(p)-Ap∥+(1-εnγ¯)∥zn-p∥≤(1-εn(γ¯-γη))∥xn-p∥+εn∥γf(p)-Ap∥.
Furthermore,
(72)∥xn+1-p∥=∥γnxn+(1-γn)yn-p∥≤γn∥xn-p∥+(1-γn)∥yn-p∥≤(1-εn(1-γn)(γ¯-γη))∥xn-p∥+(1-γn)εn∥γf(p)-Ap∥≤max{∥xn-p∥,∥γf(p)-Ap∥γ¯-γη}.
By induction, we have
(73)∥xn-p∥≤max{∥x1-p∥,∥γf(p)-Ap∥γ¯-γη},n≥1.
Hence, {xn} is bounded and we also obtain that {un},{zn},{yn},{(∑i=1NδiAi)(1/tn)∫0tn‍T(s)Snunds} and {f(xn)} are all bounded.
Step 2. We claim that  limn→∞∥xn-xn+1∥=0.
From the definition ofzn and Lemma 15, for p∈Θ, we have
(74)∥zn+1-zn∥=∥PC(1tn+1∫0tn+1‍T(s)Sn+1un+1ds-αn+1(∑i=1N‍δiAi)×1tn+1∫0tn+1‍T(s)Sn+1un+1ds(∑i=1N‍δiAi))-PC(1tn∫0tn‍T(s)Snunds-αn(∑i=1N‍δiAi)×1tn∫0tn‍T(s)Snunds)(1tn+1∫0tn+1‍T(s)Sn+1un+1ds∥≤∥(1tn+1∫0tn+1‍T(s)Sn+1un+1ds-αn+1(∑i=1N‍δiAi)×1tn+1∫0tn+1‍T(s)Sn+1un+1ds(∑i=1N‍δiAi))-(1tn∫0tn‍T(s)Snunds-αn(∑i=1N‍δiAi)×1tn∫0tn‍T(s)Snunds(∑i=1N‍δiAi))∥≤∥(I-αn+1(∑i=1N‍δiAi))×(1tn+1∫0tn+1‍T(s)Sn+1un+1ds-1tn∫0tn‍T(s)Snunds)(I-αn+1(∑i=1N‍δiAi))∥+∥(αn-αn+1)(∑i=1N‍δiAi)1tn∫0tn‍T(s)Snunds∥≤∥1tn+1∫0tn+1‍T(s)Sn+1un+1ds-1tn∫0tn‍T(s)Snunds∥+|αn-αn+1|∥(∑i=1N‍δiAi)1tn∫0tn‍T(s)Snunds∥≤∥1tn+1∫0tn+1‍T(s)(Sn+1un+1-Snun)ds∥+∥(1tn+1-1tn)∫0tn‍T(s)Snunds+1tn+1∫tntn+1‍T(s)Snunds∥+|αn-αn+1|∥(∑i=1N‍δiAi)1tn∫0tn‍T(s)Snunds∥≤∥1tn+1∫0tn+1‍T(s)(Sn+1un+1-Snun)ds∥+∥(1tn+1-1tn)∫0tn‍T(s)(Snun-Snp)ds+1tn+1∫tntn+1‍T(s)(Snun-Snp)ds∥+|αn-αn+1|∥(∑i=1N‍δiAi)1tn∫0tn‍T(s)Snunds∥≤∥Sn+1un+1-Snun∥+2|tn+1-tn|tn+1∥un-p∥+|αn-αn+1|∥(∑i=1N‍δiAi)1tn∫0tn‍T(s)Snunds∥≤∥un+1-un∥+∥Sn+1un-Snun∥+2|tn+1-tn|tn+1∥un-p∥+|αn-αn+1|∥(∑i=1N‍δiAi)1tn∫0tn‍T(s)Snunds∥.
First, we will show that if{xn} is bounded, then
(75)limn→∞∥ℑnkxn-ℑn+1kxn∥=0
for k∈{1,2,…,M}.
From Step 2 of the proof in [4], we have for k∈{1,2,…,M}(76)limn→∞∥Jrk,n+1Fkxn-Jrk,nFkxn∥=0.
For k∈{1,2,…,M}, notice that
(77)ℑnk=Jrk,nFkJrk-1,nFk-1⋯Jr2,nF2Jr1,nF1=Jrk,nFkℑnk-1.
It follows that
(78)∥ℑnkxn-ℑn+1kxn∥=∥Jrk,nFkℑnk-1xn-Jrk,n+1Fkℑn+1k-1xn∥≤∥Jrk,nFkℑnk-1xn-Jrk,n+1Fkℑnk-1xn∥+∥Jrk,n+1Fkℑnk-1xn-Jrk,n+1Fkℑn+1k-1xn∥≤∥Jrk,nFkℑnk-1xn-Jrk,n+1Fkℑnk-1xn∥+∥Jrk-1,nFk-1ℑnk-2xn-Jrk-1,n+1Fk-1ℑnk-2xn∥+∥ℑnk-2xn-ℑn+1k-2xn∥≤∥Jrk,nFkℑnk-1xn-Jrk,n+1Fkℑnk-1xn∥+∥Jrk-1,nFk-1ℑnk-2xn-Jrk-1,n+1Fk-1ℑnk-2xn∥+⋯+∥Jr2,nF2ℑn1xn-Jr2,n+1F2ℑn1xn∥+∥Jr1,nF1xn-Jr1,n+1F1xn∥.
Therefore, from (76), we conclude (75).
Second, we estimate∥un+1-un∥. From un+1=ℑn+1Mxn+1 and un=ℑnMxn=JrM,nFMℑnM-1xn, we obtain
(79)FM(un+1,y)+1rM,n+1〈y-un+1,un+1-ℑn+1M-1xn+1〉≥0,∀y∈C,(80)FM(un,y)+1rM,n〈y-un,un-ℑnM-1xn〉≥0,∀y∈C.
Taking y=un in (79) and y=un+1 in (80), we have
(81)FM(un+1,un)+1rM,n+1〈un-un+1,un+1-ℑn+1M-1xn+1〉≥0,FM(un,un+1)+1rM,n〈un+1-un,un-ℑnM-1xn〉≥0.
So, from (A2), one has
(82)〈un+1-un,un-ℑnM-1xnrM,n-un+1-ℑn+1M-1xn+1rM,n+1〉≥0.
Furthermore,
(83)〈(1-rM,nrM,n+1)un+1-un,un-ℑnM-1xn-(un+1-ℑn+1M-1xn+1)+(1-rM,nrM,n+1)(un+1-ℑn+1M-1xn+1)〉≥0.
Since limn→∞rk,n>0, we assume that there exists a real number such that rk,n>a>0 for all n∈N. Thus, we obtain
(84)∥un+1-un∥≤∥ℑnM-1xn-ℑn+1M-1xn+1∥+|1-rM,nrM,n+1|∥un+1-ℑn+1M-1xn+1∥≤∥ℑnM-1xn-ℑn+1M-1xn∥+∥ℑn+1M-1xn-ℑn+1M-1xn+1∥+1a|rM,n+1-rM,n|∥un+1-ℑn+1M-1xn+1∥≤∥ℑnM-1xn-ℑn+1M-1xn∥+∥xn-xn+1∥+1a|rM,n+1-rM,n|∥un+1-ℑn+1M-1xn+1∥.
Third, we estimate ∥Sn+1un-Snun∥. It follows from (37) that
(85)∥Sn+1un-Snun∥2=∥Un+1,1un-Un,1un∥2=∥α11T1Un+1,2un+α21Un+1,2un+α31un-(α11T1Un,2un+α21Un,2un+α31un)∥2=∥α11(T1Un+1,2-T1Un,2)un+α21(Un+1,2-Un,2)un∥2≤α11∥(T1Un+1,2-T1Un,2)un∥2+α21∥(Un+1,2-Un,2)un∥2-α11α21∥(T1Un+1,2-T1Un,2)un-(Un+1,2-Un,2)un∥2≤α11(∥(I-T1)Un+1,2un-(I-T1)Un,2un∥2∥Un+1,2un-Un,2un∥2+κ∥(I-T1)Un+1,2un-(I-T1)Un,2un∥2)+α12∥Un+1,2un-Un,2un∥2-α11α21∥(I-T1)Un+1,2un-(I-T1)Un,2un∥2≤(1-α13)∥Un+1,2un-Un,2un∥2⋮≤∏i=1n‍(1-α3i)∥Un+1,n+1un-Un,n+1un∥2,
which means that
(86)∥Sn+1un-Snun∥≤L1∏i=1n‍(1-α3i),
where L1≥0 is a constant such that ∥Un+1,n+1un-Un,n+1un∥≤L1, for all n∈N.
Next, we estimate∥yn+1-yn∥. Substituting (84) and (86) into (74), one has
(87)∥zn+1-zn∥≤∥xn-xn+1∥+∥ℑnM-1xn-ℑn+1M-1xn∥+1a|rM,n+1-rM,n|∥un+1-ℑn+1M-1xn+1∥+L1∏i=1n‍(1-α3i)+|αn-αn+1|∥(∑i=1N‍δiAi)1tn∫0tn‍T(s)Snunds∥+2|tn+1-tn|tn+1∥un-p∥.
From (61), we have
(88)∥yn+1-yn∥=∥εn+1γ(f(xn+1)-f(xn))+(εn+1-εn)(γf(xn)-Azn)+βn+1(xn+1-xn)+(βn+1-βn)(xn-zn)+((1-βn+1)I-εn+1A)(zn+1-zn)∥≤εn+1γη∥xn+1-xn∥+|εn+1-εn|×∥γf(xn)-Azn∥+βn+1∥xn+1-xn∥+|βn+1-βn|∥xn-zn∥+(1-βn+1-εn+1γ¯)∥zn+1-zn∥.
Substitution (87) into (88) yields that
(89)∥yn+1-yn∥≤(1-εn+1(γ¯-γη))∥xn+1-xn∥+L2(|εn+1-εn|+|βn+1-βn|+2|tn+1-tn|tn+1+|rM,n+1-rM,n|+|αn-αn+1|2|tn+1-tn|tn+1)+∥ℑnM-1xn-ℑn+1M-1xn∥+L1∏i=1n‍(1-α3i),
where L2 is an appropriate constant such that
(90)L2=max{supn≥1{∥(∑i=1N‍δiAi)1tn∫0tn‍T(s)Snunds∥},supn≥1{1a∥un+1-ℑn+1M-1xn+1∥},supn≥1{∥γf(xn)-Azn∥},supn≥1{∥xn-zn∥},supn≥1{∥un-p∥}(∑i=1N‍δiAi)}.
It follows from (89) that
(91)∥yn+1-yn∥-∥xn+1-xn∥≤L2(|εn+1-εn|+|βn+1-βn|+2|tn+1-tn|tn+1+|rM,n+1-rM,n|+|αn-αn+1|2|tn+1-tn|tn+1)+∥ℑnM-1xn-ℑn+1M-1xn∥+L1∏i=1n‍(1-α3i).
Consequently, from (75) and the conditions in Theorem 16, we obtain
(92)limn→∞sup(∥yn+1-yn∥-∥xn+1-xn∥)≤0.
Hence, by Lemma 9, one has
(93)limn→∞∥yn-xn∥=0.
Since xn+1=γnxn+(1-γn)yn, this shows that
(94)limn→∞∥xn+1-xn∥=limn→∞(1-γn)∥yn-xn∥=0.
Step 3. We claim that limn→∞∥(1/tn)∫0tn‍T(s)Snunds-un∥=0.
Observingyn=εnγf(xn)+βnxn+((1-βn)I-εnA)zn, we obtain
(95)∥xn-zn∥≤∥xn-yn∥+∥yn-zn∥≤∥xn-yn∥+εn∥γf(xn)-Azn∥+βn∥xn-zn∥,
which means that
(96)∥xn-zn∥≤11-βn∥xn-yn∥+εn1-βn∥γf(xn)-Azn∥.
This together with the conditions (i) and (ii) imply that
(97)limn→∞∥xn-zn∥=0.
From (93) and (97), one has
(98)limn→∞∥yn-zn∥≤limn→∞(∥yn-xn∥+∥xn-zn∥)=0.
Forp∈Θ, we see that
(99)∥zn-p∥2=∥PC(1tn∫0tn‍T(s)Snunds-αn(∑i=1N‍δiAi)×1tn∫0tn‍T(s)Snunds)-p(∑i=1N‍δiAi)∥2=∥PC(1tn∫0tn‍T(s)Snunds-αn(∑i=1N‍δiAi)×1tn∫0tn‍T(s)Snunds(∑i=1N‍δiAi))-PC(1tn∫0tn‍T(s)Snpds-αn(∑i=1N‍δiAi)×1tn∫0tn‍T(s)Snpds(∑i=1N‍δiAi))∥2≤∥(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds)-αn(∑i=1N‍δiAi)×(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds)∥2=∥(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds)∥2-2αn〈(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds),(∑i=1N‍δiAi)(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds)〉+αn2∥(∑i=1N‍δiAi)×((1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds))(∑i=1N‍δiAi)∥2≤∥xn-p∥2-αn∑i=1N‍δi(2viωi2-2μi-αn)×∥Ai(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds)∥2.
It follows from (42) that(100)∥yn-p∥2=∥εnrf(xn)+βnxn+((1-βn)I-εnA)zn-p∥2=∥((1-βn)I-εnA)(zn-p)+βn(xn-p)+εn(rf(xn)-Ap)∥2=∥((1-βn)I-εnA)(zn-p)+βn(xn-p)∥2+εn2∥rf(xn)-Ap∥2+2εn〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap((1-βn)I-εnA)〉+2βnεn〈xn-p,rf(xn)-Ap〉≤((1-βn-εnγ¯)∥zn-p∥+βn∥xn-p∥)2+εn2∥rf(xn)-Ap∥2+2εn〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap((1-βn)I-εnA)〉+2βnεn〈xn-p,rf(xn)-Ap〉=(1-βn-εnγ¯)2∥zn-p∥2+βn2∥xn-p∥2+2βn(1-βn-εnγ¯)×∥zn-p∥∥xn-p∥+εn2∥rf(xn)-Ap∥2+2εn×〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap((1-βn)I-εnA)〉+2βnεn〈xn-p,rf(xn)-Ap〉≤(1-βn-εnγ¯)2∥zn-p∥2+βn2∥xn-p∥2+βn(1-βn-εnγ¯)×(∥zn-p∥2+∥xn-p∥2)+εn2∥rf(xn)-Ap∥2+2εn×〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap((1-βn)I-εnA)〉+2βnεn〈xn-p,rf(xn)-Ap〉=(1-εnγ¯)(1-βn-εnγ¯)∥zn-p∥2+(1-εnγ¯)βn∥xn-p∥2+εn2∥rf(xn)-Ap∥2+2εn×〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap((1-βn)I-εnA)〉+2βnεn〈xn-p,rf(xn)-Ap〉.
Substituting (99) into (100) yields that
(101)∥yn-p∥2≤(1-εnγ¯)(1-βn-εnγ¯)×{∥xn-p∥2-aαn(2v1ω12-2μ1+αn)×∥A1(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds)∥2-(1-a)αn(2v2ω22-2μ2+αn)×∥A2(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds)∥2}+(1-εnγ¯)βn∥xn-p∥2+εn2∥rf(xn)-Ap∥2+2εn〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap((1-βn)I-εnA)〉+2βnεn〈xn-p,rf(xn)-Ap〉=(1-εnγ¯)2∥xn-p∥2+(1-εnγ¯)(1-βn-εnγ¯)×{-1tn∫0tn‍T(s)Snpds)∥22-αn∑i=1N‍δi(2viωi2-2μi-αn)×∥Ai(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds)-1tn∫0tn‍T(s)Snpds)∥2αn∑i=1N‍δi(2viωi2-2μi-αn)}+2εn〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap((1-βn)I-εnA)〉+2βnεn〈xn-p,rf(xn)-Ap〉+εn2∥rf(xn)-Ap∥2≤∥xn-p∥2+(1-εnγ¯)(1-βn-εnγ¯)×{-1tn∫0tn‍T(s)Snpds)∥2-αn∑i=1N‍δi(2viωi2-2μi-αn)×∥Ai(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds)∥2∑i=1N‍}+2εn〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap((1-βn)I-εnA)〉+2βnεn〈xn-p,rf(xn)-Ap〉+εn2∥rf(xn)-Ap∥2.
Furthermore,
(102)∥xn+1-p∥2=∥γnxn+(1-γn)yn-p∥2≤γn∥xn-p∥2+(1-γn)∥yn-p∥2≤∥xn-p∥2+(1-γn)(1-εnγ¯)×(1-βn-εnγ¯)×{-αn∑i=1N‍δi(2viωi2-2μi-αn)Ai×∥(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds)∥2∑i=1N‍δi(2viωi2-2μi-αn)}+(1-γn)εn2∥rf(xn)-Ap∥2+2(1-γn)εn×〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap((1-βn)I-εnA)〉+2(1-γn)βnεn〈xn-p,rf(xn)-Ap〉.
It follows that
(103)(1-e1)(1-εnγ¯)(1-βn-εnγ¯)×{∑i=1N‍δi(2g1viωi2-2g2μi-g22)×∥Ai(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds)∥2∑i=1N‍}≤(1-γn)(1-εnγ¯)(1-βn-εnγ¯)×{-αn∑i=1N‍δi(2viωi2-2μi-αn)×∥Ai(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds)∥2∑i=1N‍δi(2viωi2-2μi-αn)}≤∥xn-p∥2-∥xn+1-p∥2+(1-γn)εn2∥rf(xn)-Ap∥2+2(1-γn)εn×〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap〉+2(1-γn)βnεn〈xn-p,rf(xn)-Ap〉.
From (94) and the condition (i), for i=1,2,…,N, we have
(104)limn→∞∥Ai(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds)∥=0.
Then, for δi∈(0,1) and ∑i=1Nδi=1,
(105)limn→∞∥(∑i=1N‍δiAi)(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds)(∑i=1N‍δiAi)∥=0.
On the other hand, one has(106)∥zn-p∥2=∥PC((∑i=1N‍δiAi)1tn1tn∫0tn‍T(s)Snunds-αn(∑i=1N‍δiAi)1tn∫0tn‍T(s)Snunds)-p∥2≤〈(∑i=1N‍δiAi)1tn∫0tn‍T(s)Snunds-αn(∑i=1N‍δiAi)1tn∫0tn‍T(s)Snunds-(1tn∫0tn‍T(s)Snpds-αn(∑i=1N‍δiAi)×1tn∫0tn‍T(s)Snpds(∑i=1N‍δiAi)),zn-p〉=12{∥(∑i=1N‍δiAi)1tn∫0tn‍T(s)Snunds-αn(∑i=1N‍δiAi)1tn∫0tn‍T(s)Snunds-((∑i=1N‍δiAi)1tn∫0tn‍T(s)Snpds-αn(∑i=1N‍δiAi)1tn∫0tn‍T(s)Snpds)∥2+∥zn-p∥2-∥1tn∫0tn‍T(s)Snunds-αn(∑i=1N‍δiAi)1tn∫0tn‍T(s)Snunds-(1tn∫0tn‍T(s)Snpds-αn(∑i=1N‍δiAi)×1tn∫0tn‍T(s)Snpds)-(zn-p)∥2}≤12{∥(1tn∫0tn‍T(s)Snunds-zn)∥un-p∥2+∥zn-p∥2-∥(1tn∫0tn‍T(s)Snunds-zn)-αn(∑i=1N‍δiAi)×(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds)∥2}≤12{〈(∑i=1N‍δiAi)∥xn-p∥2+∥zn-p∥2-∥1tn∫0tn‍T(s)Snunds-zn∥2-αn2∥(∑i=1N‍δiAi)×(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds)(∑i=1N‍δiAi)∥2+2αn〈(∑i=1N‍δiAi)×(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds),1tn∫0tn‍T(s)Snunds-zn(∑i=1N‍δiAi)〉},which means that
(107)∥zn-p∥2≤∥xn-p∥2-∥1tn∫0tn‍T(s)Snunds-zn∥2+2αn∥(∑i=1N‍δiAi)×(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds)(∑i=1N‍δiAi)∥×∥1tn∫0tn‍T(s)Snunds-zn∥.
It follows that
(108)∥yn-p∥2≤(1-εnγ¯)(1-βn-εnγ¯)∥zn-p∥2+(1-εnγ¯)βn∥xn-p∥2+εn2∥rf(xn)-Ap∥2+2εn〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap((1-βn)I-εnA)〉+2βnεn〈xn-p,rf(xn)-Ap〉≤(1-εnγ¯)(1-βn-εnγ¯)×{∥xn-p∥2-∥1tn∫0tn‍T(s)Snunds-zn∥2+2αn∥(∑i=1N‍δiAi)×(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds)(∑i=1N‍δiAi)∥×∥Snun-zn∥∥1tn∫0tn‍T(s)Snunds-zn∥2}+(1-εnγ¯)βn∥xn-p∥2+εn2∥rf(xn)-Ap∥2+2εn〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap〉+2βnεn〈xn-p,rf(xn)-Ap〉=(1-εnγ¯)2∥xn-p∥2-(1-εnγ¯)×(1-βn-εnγ¯)∥1tn∫0tn‍T(s)Snunds-zn∥2+2αn(1-εnγ¯)(1-βn-εnγ¯)×∥(∑i=1N‍δiAi)×(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds)(∑i=1N‍δiAi)∥∥Snun-zn∥+εn2∥rf(xn)-Ap∥2+2εn〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap((1-βn)I-εnA)〉+2βnεn〈xn-p,rf(xn)-Ap〉≤∥xn-p∥2-(1-εnγ¯)(1-βn-εnγ¯)×∥1tn∫0tn‍T(s)Snunds-zn∥2+2αn(1-εnγ¯)(1-βn-εnγ¯)×∥(∑i=1N‍δiAi)(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds)(∑i=1N‍δiAi)∥×∥Snun-zn∥+εn2∥rf(xn)-Ap∥2+2εn×〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap〉+2βnεn〈xn-p,rf(xn)-Ap〉.
Therefore, from (108) and (102), one has
(109)∥xn+1-p∥2≤γn∥xn-p∥2+(1-γn)∥yn-p∥2≤γn∥xn-p∥2+(1-γn)×{∥1tn∫0tn‍T(s)Snunds-zn∥2∥xn-p∥2-(1-εnγ¯)×(1-βn-εnγ¯)×∥1tn∫0tn‍T(s)Snunds-zn∥2}+2αn(1-εnγ¯)(1-βn-εnγ¯)×{∥(∑i=1N‍δiAi)×(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds)(∑i=1N‍δiAi)∥×∥Snun-zn∥+εn2∥rf(xn)-Ap∥2+2εn〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap((1-βn)I-εnA)〉+2βnεn〈xn-p,rf(xn)-Ap〉∥(∑i=1N‍δiAi)}=∥xn-p∥2-(1-γn)(1-εnγ¯)×(1-βn-εnγ¯)∥1tn∫0tn‍T(s)Snunds-zn∥2+2αn(1-γn)(1-εnγ¯)(1-βn-εnγ¯)×{∥(∑i=1N‍δiAi)×(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds)(∑i=1N‍δiAi)∥×∥Snun-zn∥+(1-γn)εn2×∥rf(xn)-Ap∥2+2(1-γn)εn×〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap((1-βn)I-εnA)〉+2(1-γn)βnεn×〈xn-p,rf(xn)-Ap〉(∑i=1N‍δiAi)}.
Then,
(110)(1-γn)(1-εnγ¯)(1-βn-εnγ¯)×∥1tn∫0tn‍T(s)Snunds-zn∥2≤∥xn-p∥2-∥xn+1-p∥2+2αn(1-γn)(1-εnγ¯)(1-βn-εnγ¯)×{∥(∑i=1N‍δiAi)(1tn∫0tn‍T(s)Snunds-1tn∫0tn‍T(s)Snpds)(∑i=1N‍δiAi)∥×∥Snun-zn∥+(1-γn)εn2×∥rf(xn)-Ap∥2+2(1-γn)εn×〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap〉+2(1-γn)βnεn×〈xn-p,rf(xn)-Ap〉∥(∑i=1N‍δiAi)(1tn∫0tn‍T(s)Snunds}.
From (94), (105), and condition (i), one has
(111)limn→∞∥1tn∫0tn‍T(s)Snunds-zn∥=0.
Letp∈Θ and k∈{1,2,…,M}. Since Jrk,nFk is firmly nonexpansive, we obtain
(112)∥ℑnkxn-p∥2=∥Jrk,nFkℑnk-1xn-Jrk,nFkp∥2=〈Jrk,nFkℑnk-1xn-p,ℑnk-1xn-p〉=12(∥ℑnkxn-p∥2+∥ℑnk-1xn-p∥2-∥ℑnkxn-ℑnk-1xn∥2).
It follows that
(113)∥ℑnkxn-p∥2≤∥xn-p∥2-∥ℑnkxn-ℑnk-1xn∥2.
Consequently, from (108), one has
(114)∥yn-p∥2≤(1-εnγ¯)(1-βn-εnγ¯)∥zn-p∥2+(1-εnγ¯)βn∥xn-p∥2+εn2∥rf(xn)-Ap∥2+2εn×〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap〉+2βnεn〈xn-p,rf(xn)-Ap〉≤(1-εnγ¯)(1-βn-εnγ¯)×∥un-p∥2+(1-εnγ¯)βn∥xn-p∥2+εn2∥rf(xn)-Ap∥2+2εn×〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap((1-βn)I-εnA)〉+2βnεn〈xn-p,rf(xn)-Ap〉=(1-εnγ¯)(1-βn-εnγ¯)×∥ℑnkxn-p∥2+(1-εnγ¯)βn×∥xn-p∥2+εn2∥rf(xn)-Ap∥2+2εn〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap((1-βn)I-εnA)〉+2βnεn〈xn-p,rf(xn)-Ap〉≤(1-εnγ¯)(1-βn-εnγ¯)×(∥xn-p∥2-∥ℑnkxn-ℑnk-1xn∥2)+(1-εnγ¯)βn∥xn-p∥2+εn2∥rf(xn)-Ap∥2+2εn〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap((1-βn)I-εnA)〉+2βnεn〈xn-p,rf(xn)-Ap〉≤∥xn-p∥2-(1-εnγ¯)(1-βn-εnγ¯)×∥ℑnkxn-ℑnk-1xn∥2+εn2∥rf(xn)-Ap∥2+2εn〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap((1-βn)I-εnA)〉+2βnεn〈xn-p,rf(xn)-Ap〉.
Then,
(115)∥xn+1-p∥2=∥γnxn+(1-γn)yn-p∥2≤γn∥xn-p∥2+(1-γn)∥yn-p∥2≤∥xn-p∥2+(1-γn)×{∥ℑnkxn-ℑnk-1xn∥2-(1-εnγ¯)(1-βn-εnγ¯)×∥ℑnkxn-ℑnk-1xn∥2+εn2∥rf(xn)-Ap∥2+2εn〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap((1-βn)I-εnA)〉+2βnεn〈xn-p,rf(xn)-Ap〉∥ℑnkxn-ℑnk-1xn∥2}.
That is,
(116)(1-γn)-(1-εnγ¯)(1-βn-εnγ¯)×∥ℑnkxn-ℑnk-1xn∥2≤∥xn-p∥2-∥xn+1-p∥2+(1-γn)εn2∥rf(xn)-Ap∥2+2(1-γn)εn×〈((1-βn)I-εnA)(zn-p),rf(xn)-Ap〉+2(1-γn)βnεn〈xn-p,rf(xn)-Ap〉.
By condition (i) and (94), for k∈{1,2,…,M}, we obtain
(117)limn→∞∥ℑnkxn-ℑnk-1xn∥=0.
Therefore, we have
(118)∥un-xn∥=∥ℑnkxn-ℑn0xn∥≤∥ℑnkxn-ℑnk-1xn∥+∥ℑnk-1xn-ℑnk-2xn∥+⋯+∥ℑn1xn-ℑn0xn∥.
From (117), one has
(119)limn→∞∥un-xn∥=0.Notice that
(120)∥un-yn∥≤∥un-xn∥+∥xn-yn∥,
Applying (119) and (93), we have
(121)limn→∞∥un-yn∥=0.
Since
(122)∥un-zn∥≤∥un-yn∥+∥yn-zn∥,
this together with (94) yields that
(123)limn→∞∥un-zn∥=0.
Consequently, we obtain
(124)limn→∞∥1tn∫0tn‍T(s)Snunds-un∥=0.
Step
4. Letting z=PΘ(I-A+rf)z, we show
(125)limn→∞sup〈(rf-A)z,xn-z〉≤0.
We know that PΘ(I-A+rf) is a contraction. Indeed, for any x,y∈H, we have
(126)∥PΘ(I-A+rf)x-PΘ(I-A+rf)y∥≤∥(I-A+rf)x-(I-A+rf)y∥≤(1-(γ¯-rη))∥x-y∥,
and hence PΘ(I-A+rf) is a contraction due to (1-(γ¯-rη))∈(0,1). Thus, Banach’s Contraction Mapping Principle guarantees that PΘ(I-A+rf) has a unique fixed point, which implies z=PΘ(I-A+rf)z.
We claim thatz∈F(Γ). Since {uni}⊂{un} is bounded in C, without loss of generality, we can assume that {uni}⇀z. Since C is closed and convex, C is weakly closed. Thus we have z∈C. For 0≤s<∞, notice that
(127)∥uni-T(h)uni∥≤∥uni-1tni∫0tni‍T(s)Sniunids∥+∥1tni∫0tni‍T(s)Sniunids-T(h)1tni∫0tni‍T(s)Sniunids∥+∥T(h)1tni∫0tni‍T(s)Sniunids-T(h)uni∥≤2∥uni-1tni∫0tni‍T(s)Sniunids∥+∥1tni∫0tni‍T(s)Sniunids-T(h)1tni∫0tni‍T(s)Sniunids∥.
It follows from (124) and Lemma 4 that
(128)limn→∞∥uni-T(h)uni∥=0.
Thus, (128) and Lemma 5 assert that z∈F(Γ). Since {xni}⊂{xn} is bounded in C, without loss of generality, we can assume that {xni}⇀ω. It follows from (94) that zni⇀ω. Since C is closed and convex, C is weakly closed. Thus we have ω∈C.
Let us showω∈F(S). For the sake of contradiction, suppose that ω∉F(S), that is, Sω≠ω. Since z∈F(Γ), by our assumption, we have Tiω∈F(Γ) and then Snω∈F(Γ). Hence (1/tn)∫0tn‍T(s)Snωds=Snω. Therefore, by (124) and Opial condition, we have
(129)limn→∞inf∥uni-ω∥<limn→∞inf∥uni-Sω∥≤limn→∞inf{∥uni-1tni∫0tniT(s)Sniunids∥+∥1tni∫0tniT(s)Sniunids-1tni∫0tniT(s)Sniωds∥+∥Sniω-Sω∥∥uni-1tni∫0tni‍T(s)Sniunids∥}≤limn→∞inf∥uni-ω∥,
which derives a contradiction. Thus, we obtain ω∈F(S)=∩i=1∞F(Ti).
Next, we claim thatω∈∩i=1MSEP(Fi). Since un=ℑnkxn for k=1,2,…,M, we obtain
(130)Fk(ℑnkxn,y)+1rn〈y-ℑnkxn,ℑnkxn-ℑnk-1xn〉≥0,∀y∈C.From (A2), one has
(131)1rn〈y-ℑnkxn,ℑnkxn-ℑnk-1xn〉≥F(y,ℑnkxn).
Replacing n by ni, we have
(132)〈y-ℑnikxni,1rni(ℑnikxni-ℑnik-1xni)〉≥Fk(y,ℑnikxni).
It follows from (1/rni)(ℑnikxni-ℑnik-1xni)→0 and ℑnikxni⇀ω that
(133)Fk(y,ω)≤0,y∈C,
for k=1,2,…,M.
Putzt=ty+(1-t)ω for all t∈(0,1] and y∈C. Then, we have zt∈C and then F(zt,ω)≤0. Hence, from (A1) and (A4), we have
(134)0=Fk(zt,zt)≤tFk(zt,y)+(1-t)Fk(zt,y)≤tFk(zt,y),
which means Fk(zt,y)≥0. From (A3), we obtain Fk(ω,y)≥0 for y∈C and then ω∈SEP(Fk) for k=1,2,…,M, that is, ω∈∩i=1MSEP(Fk).
Finally, we claim thatω∈∩i=1NVI(C,Ai).
We define the maximal monotone operator(135)Qq1={(∑i=1N‍δiAi)q1+NCq1,ω1∈C,∅,ω1∉C.
SinceAi is relaxed (μi,vi)-cocoercive for i=1,2, we have
(136)〈(∑i=1N‍δiAi)x-(∑i=1N‍δiAi)y,x-y〉=∑i=1N‍δi〈Aix-Aiy,x-y〉≥∑i=1N‍δi(-μi∥Aix-Aiy∥2+vi∥x-y∥2)≥∑i=1N‍δi(vi-μiω1i)∥x-y∥2≥0,
which yields that ∑i=1NδiAi is monotone. Thus, Q is maximal monotone. Let (q1,q2)∈G(Q). Since q2-(∑i=1NδiAi)q1∈NCq1 and zn∈C, we have
(137)〈q1-zn,q2-(∑i=1N‍δiAi)q1〉≥0.
On the other hand, it follows from zn=PC((1/tn)∫0tn‍T(s)Snunds-αn(∑i=1NδiAi)(1/tn)∫0tn‍T(s)Snunds) that
(138)〈(∑i=1N‍δiAi)q1-zn,zn-(1tn∫0tnT(s)Snunds-αn(∑i=1N‍δiAi)×1tn∫0tn‍T(s)Snunds)(∑i=1N‍δiAi)〉≥0,
and hence
(139)〈q1-zn,zn-(1/tn)∫0tnT(s)Snundsαn+(∑i=1N‍δiAi)1tn∫0tn‍T(s)Snundszn-(1/tn)∫0tnT(s)Snundsαn〉≥0.
It follows that
(140)〈q1-zni,q2〉≥〈q1-zni,(∑i=1N‍δiAi)q1〉≥〈q1-zni,(∑i=1N‍δiAi)q1〉-〈q1-zni,zni-Sniuniαni+(∑i=1N‍δiAi)Sniuni〉=〈q1-zni,(∑i=1N‍δiAi)×(q1-1tni∫0tni‍T(s)Sniunids)-zni-(1/tni)∫0tni‍T(s)Sniunidsαni〉=〈q1-zni,(∑i=1N‍δiAi)(q1-zni)〉+〈q1-zni,(∑i=1N‍δiAi)(zni-Sniuni)〉-〈q1-zni,zni-Sniuniαni〉≥〈q1-zni,(∑i=1N‍δiAi)×(zni-1tni∫0tni‍T(s)Sniunids)〉-〈q1-zni,zni-(1/tni)∫0tni‍T(s)Sniunidsαni〉,
which implies that
(141)〈q1-ω,q2〉≥0.
Since Q is maximal monotone, we obtain that ω∈Q-10. From Lemma 8, we obtain ω∈VI(C,∑i=1NδiAi), that is, ω∈(∩i=1NVI(C,Ai)). Thus, ω∈Θ.
Sincez=PΘ(I-A+rf)z, one has
(142)limn→∞sup〈(rf-A)z,xn-z〉=limi→∞〈(rf-A)z,xni-z〉=〈(rf-A)z,ω-z〉≤0.
Furthermore,
(143)〈(rf-A)z,yn-z〉=〈(rf-A)z,yn-xn〉+〈(rf-A)z,xn-z〉.
From (93) and (142), we have
(144)limn→∞sup〈(rf-A)z,yn-z〉≤0.
Step
5. Finally, we show that xn converges strongly to z=PΘ(I-A+rf)z. Indeed, from (61) and (70), we obtain
(145)∥yn-z∥2=∥((1-βn)I-εnA)εnrf(xn)+βnxn+((1-βn)I-εnA)zn-z∥2=∥((1-βn)I-εnA)(zn-z)+βn(xn-z)+εn(rf(xn)-Az)((1-βn)I-εnA)∥2≤∥((1-βn)I-εnA)(zn-z)+βn(xn-p)∥2+2εn〈((1-βn)I-εnA)(zn-z)+βn(xn-z)+εn(rf(xn)-Az),rf(xn)-Ap((1-βn)I-εnA)〉=∥((1-βn)I-εnA)(zn-z)+βn(xn-z)∥2+2εn〈(yn-z,rf(xn)-Az〉≤(1-βn)∥((1-βn)I-εnA)1-βn(zn-z)∥2+βn∥xn-z∥2+2rεn〈yn-z,f(xn)-f(z)〉+2εn〈yn-z,f(z)-Az〉≤(1-βn-εnγ¯)21-βn∥xn-z∥2+βn∥xn-z∥2+rηεn(∥xn-z∥2+∥yn-z∥2)+2εn〈yn-z,f(z)-Az〉=(1-(2γ¯-rη)+(εnγ¯)21-βn)∥xn-z∥2+rηεn∥yn-z∥2+2εn〈yn-z,f(z)-Az〉,
which implies that
(146)∥yn-z∥2≤(1-2(γ¯-rη)εn1-rηεn)∥xn-z∥2+εn1-rηεn×{γ¯2εn1-βn∥xn-z∥2+2〈yn-z,f(z)-Az〉γ¯2εn1-βn}.
It follows from (146) that
(147)∥xn+1-z∥2=∥γnxn+(1-γn)yn-p∥2≤γn∥xn-z∥2+(1-γn)∥yn-z∥2≤∥xn-z∥2+(1-γn)×{(1-2(γ¯-rη)εn1-rηεn)∥xn-z∥2+εn1-rηεn×(γ¯2εn1-βn∥xn-z∥2+2〈yn-z,f(z)-Az〉γ¯2εn1-βn)}=(1-2(1-γn)(γ¯-rη)εn1-rηεn)∥xn-z∥2+(1-γn)εn1-rηεn×(γ¯2εn1-βn∥xn-z∥2+2〈yn-z,f(z)-Az〉γ¯2εn1-βn).
From condition (i) and (142), we know that
(148)∑i=1∞‍2(1-γn)(γ¯-rη)εn1-rηεn=∞,limn→∞sup(1-γn)εn1-rηεn(γ¯2εn1-βn∥xn-z∥2+2〈yn-z,f(z)-Az〉γ¯2εn1-βn)≤0.
we can conclude from Lemma 10 that xn→z as n→∞. This completes the proof of Theorem 16.Theorem 17.
LetC be a nonempty closed convex subset of a real Hilbert space H, and let Fk,k∈{1,2,…,M} be bifunction from C×C→R satisfying (A1)–(A4). Let {Ti}i=1∞ be ki-strict pseudocontractive mappings of C into self with κ=supiki and let ρj=(α1j,α2j,α2j)∈I×I×I, where I=[0,1], α1j+α2j+α2j=1, α1j+α2j≤b<1, and α1j,α2j,α2j∈(κ,1) for all j=1,2,…. For every n∈N, let Sn and S be S-mapping generated by Tn,Tn-1,…,T1 and ρn,ρn-1,…,ρ1 and Tn,Tn-1,…, and ρn,ρn-1,…, respectively. Let B:C→H be ω-Lipschitz continuous and relaxed (μ,v)-cocoercive mappings with v-μω2>0, let f be a contraction of H into itself with η∈(0,1), and let A be a strongly positive linear bounded self-adjoint operator with the coefficients γ¯>0 and 0<r<γ¯/η. Assume that
(149)Θ∶=(⋂i=1∞{F(Ti)})∩(⋂k=1MSEP(Fk))∩VI(C,B).
Let {xn} be a sequence generated by x1∈C and
(150)un=JrM,nFMJrM-1,nFM-1⋯Jr2,nF2Jr1,nF1xn,zn=PC(Snun-αnBSnun),yn=εnrf(xn)+βnxn+((1-βn)I-εnA)zn,xn+1=γnxn+(1-γn)yn,∀n∈N,
where {Sn:C→C} is the sequence defined by (37). If {εn}, {βn} are two sequences in (0,1) and {γn}⊂[c1,c2]⊂(0,1) and {rk,n}, for k∈{1,2,…,M} is a real sequence in (0,∞) satisfing the following conditions:(i)
limn→∞εn=0,
∑i=1∞εn=∞,(ii)
0<limn→∞infβn≤limn→∞supβn<1 and limn→∞|βn+1-βn|=0,(iii)
limn→∞infrk,n>0 and limn→∞|rk,n+1-rk,n|=0, for k∈{1,2,…,M},(iv)
{αn}⊂[g1,g2]⊂(0,(2(v-μω2))/ω2) and limn→∞|αn+1-αn|=0.
Then{xn} converges strongly to z∈Θ, where z is the unique solution of variational inequality
(151)limn→∞sup〈(rf-A)z,p-z〉≤0,∀p∈Θ,
which is the optimality condition for the minimization problem
(152)minz∈Θ12〈Az,z〉-h(z),
where h is a potential function for rf(i.e., h′(z)=rf(z) for z∈H).Proof.
By Theorem16, for i=1,2,…,N, letting Ai=B, we can obtain Theorem 17.Theorem 18.
LetC be a nonempty closed convex subset of a real Hilbert space H, and let Fk,k∈{1,2,…,M} be a bifunction from C×C→R satisfying (A1)–(A4). Let {Ti}i=1∞ be ki-strict pseudo-contractive mappings of C into self with κ=supiki and let ρj=(α1j,α2j,α2j)∈I×I×I, where I=[0,1], α1j+α2j+α2j=1, α1j+α2j≤b<1, and α1j,α2j,α2j∈(κ,1) for all j=1,2,…. For every n∈N, let Sn and S be S-mapping generated by Tn,Tn-1,…,T1 and ρn,ρn-1,…,ρ1 and Tn,Tn-1,…, and ρn,ρn-1,⋯, respectively. Let Ai:C→H be ωi-Lipschitz continuous and relaxed (μi,vi)-cocoercive mappings with vi-μiωi2>0, for i=1,2,…,N, let f be a contraction of H into itself with η∈(0,1), and let A be is a strongly positive linear bounded self-adjoint operator with the coefficients γ¯>0 and 0<r<γ¯/η. Assume that
(153)Θ:=(⋂i=1∞{F(Ti)})∩EP(F)∩(⋂i=1NVI(C,Ai)).
Let {xn} be a sequence generated by x1∈C and
(154)F(un,y)+1r(y-un,un-xn)≥0,zn=PC(Snun-αn(∑i=1N‍δiAi)Snun),yn=εnrf(xn)+βnxn+((1-βn)I-εnA)zn,xn+1=γnxn+(1-γn)yn,∀n∈N,
where {Sn:C→C} is the sequence defined by (37) and δi∈(0,1), ∑i=1Nδi=1. If {εn}, {βn} are two sequences in (0,1) and {γn}⊂[c1,c2]⊂(0,1) and {rk,n}, for k∈{1,2,…,M} is a real sequence in (0,∞) satisfing the following conditions:(i)
limn→∞εn=0,
∑i=1∞εn=∞,(ii)
0<limn→∞infβn≤limn→∞supβn<1 and limn→∞|βn+1-βn|=0,(iii)
limn→∞infrn>0 and limn→∞|rn+1-rn|=0,(iv)
{αn}⊂[g1,g2]⊂(0,(2∑i=1Nδi(vi-μiωi2))/(∑i=1Nδiωi2)) and limn→∞|αn+1-αn|=0.
Then{xn} converges strongly to z∈Θ, where z is the unique solution of variational inequality
(155)limn→∞sup〈(rf-A)z,p-z〉≤0,∀p∈Θ,
which is the optimality condition for the minimization problem
(156)minz∈Θ12〈Az,z〉-h(z),
where h is a potential function for rf(i.e., h′(z)=rf(z) for z∈H).Proof.
By Theorem16, letting M=1 for all n≥1, we can obtain Theorem 19.Theorem 19.
LetC be a nonempty closed convex subset of a real Hilbert space H, and let Fk,k∈{1,2,…,M} be a bifunction from C×C→R satisfying (A1)–(A4). Let B:C→H be ω-Lipschitz continuous and relaxed (μ,v)-cocoercive mappings with v-μω2>0, and let f be a contraction of H into itself with η∈(0,1), and let A be is a strongly positive linear bounded self-adjoint operator with the coefficients γ¯>0 and 0<r<γ¯/η. Assume that
(157)Θ:=(⋂k=1MSEP(Fk))∩VI(C,B).
Let {xn} be a sequence generated by x1∈C and
(158)un=JrM,nFMJrM-1,nFM-1⋯Jr2,nF2Jr1,nF1xn,zn=PC(un-αnBun),yn=εnrf(xn)+βnxn+((1-βn)I-εnA)zn,xn+1=γnxn+(1-γn)yn,∀n∈N.
If {εn}, {βn} are two sequences in (0,1) and {γn}⊂[c1,c2]⊂(0,1) and {rk,n}, for k∈{1,2,…,M} is a real sequence in (0,∞) satisfing the following conditions:(i)
limn→∞εn=0,
∑i=1∞εn=∞,(ii)
0<limn→∞infβn≤limn→∞supβn<1 and limn→∞|βn+1-βn|=0,(iii)
limn→∞infrk,n>0 and limn→∞|rk,n+1-rk,n|=0, for k∈{1,2,…,M}.(iv)
{αn}⊂[g1,g2]⊂(0,(2(v-μω2))/ω2) and limn→∞|αn+1-αn|=0.Then {xn} converges strongly to z∈Θ, where z is the unique solution of variational inequality
(159)limn→∞sup〈(rf-A)z,p-z〉≤0,∀p∈Θ,
which is the optimality condition for the minimization problem
(160)minz∈Θ12〈Az,z〉-h(z),
where h is a potential function for rf(i.e., h′(z)=rf(z) for z∈H).Proof.
By Theorem17, letting Tn=I for all n≤1, we can obtain Theorem 19.



---

*Source: 204948-2013-07-30.xml*