# Generalized Systems of Variational Inequalities and Projection Methods for Inverse-Strongly Monotone Mappings

Authors: Wiyada Kumam; Prapairat Junlouchai; Poom Kumam
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

We introduce an iterative sequence for finding a common element of the set of fixed points of a nonexpansive mapping and the solutions of the variational inequality problem for three inverse-strongly monotone
mappings. Under suitable conditions, some strong convergence theorems for approximating a common element of the above two sets are obtained. Moreover, using the above theorem, we also apply to find solutions of a general system of variational inequality and a zero of a maximal monotone operator in a real Hilbert space. As applications, at the end of the paper we utilize our results to study some convergence problem
for strictly pseudocontractive mappings. Our results include the previous results as special cases extend
and improve the results of Ceng et al., (2008) and many others.


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## Body

## 1. Introduction

Variational inequalities are known to play a crucial role in mathematics as a unified framework for studying a large variety of problems arising, for instance, in structural analysis, engineering sciences and others. Roughly speaking, they can be recast as fixed-point problems, and most of the numerical methods related to this topic are based on projection methods. LetH be a real Hilbert space with inner product 〈·,·〉 and ∥·∥, and let E be a nonempty, closed, convex subset of H. A mapping A:E→H is called α-inverse-strongly monotone if there exists a positive real number α>0 such that (1.1)〈Ax-Ay,x-y〉≥α‖Ax-Ay‖2,∀x,y∈E
(see [1, 2]). It is obvious that every α-inverse-strongly monotone mapping A is monotone and Lipschitz continuous. A mapping S:E→E is called nonexpansive if (1.2)‖Sx-Sy‖≤‖x-y‖,∀x,y∈E.
We denote by F(S) the set of fixed points of S and by PE the metric projection of H onto E. Recall that the classical variational inequality, denoted by VI(A,E), is to find an x*∈E such that (1.3)〈Ax*,x-x*〉≥0,∀x∈E.
The set of solutions of VI(A,E) is denoted by Γ. The variational inequality has been widely studied in the literature; see, for example, [3–6] and the references therein.For finding an element ofF(S)∩Γ, Takahashi and Toyoda [7] introduced the following iterative scheme:(1.4)xn+1=αnxn+(1-αn)SPE(xn-λnAxn),
for every n=0,1,2,…, where x0=x∈E,{αn} is a sequence in (0,1), and {λn} is a sequence in (0,2α). On the other hand, for solving the variational inequality problem in the finite-dimensional Euclidean space Rn, Korpelevich (1976) [8] introduced the following so-called extragradient method:(1.5)x0=x∈E,yn=PE(xn-λnAxn),xn+1=PE(xn-λnAyn),
for every n=0,1,2,…, where λn∈(0,1/k). Many authors using extragradient method for approximating common fixed points and variational inequality problems (see also [9, 10]). Recently, Nadezhkina and Takahashi [11] and Zeng et al. [12] proposed some iterative schemes for finding elements in F(S)∩Γ by combining (1.4) and (1.5). Further, these iterative schemes are extended in Y. Yao and J. C. Yao [13] to develop a new iterative scheme for finding elements in F(S)∩Γ.Consider the following problem of finding(x*,y*)∈E×E such that (see cf. Ceng et al. [14]):(1.6)〈λAy*+x*-y*,x-x*〉≥0,∀x∈E,〈μBx*+y*-x*,x-y*〉≥0,∀x∈E,
which is called general system of variational inequalities (GSVI), where λ>0 and μ>0 are two constants. In particular, if A=B, then problem (1.6) reduces to finding (x*,y*)∈E×E such that(1.7)〈λAy*+x*-y*,x-x*〉≥0,∀x∈E,〈μAx*+y*-x*,x-y*〉≥0,∀x∈E,
which is defined by [15, 16], and is called the new system of variational inequalities. Further, if x*=y*, then problem (1.7) reduces to the classical variational inequalityVI(A,E), that is, find x*∈E such that 〈Ax*,x-x*〉≥0, for allx∈E.We can characteristic problem, ifx*∈F(S)∩VI(A,E), then it follows that x*=Sx*=PE[x*-ρAx*], where ρ>0 is a constant.In 2008, Ceng et al. [14] introduced a relaxed extragradient method for finding solutions of problem (1.6). Let the mappings A,B:E→H be α-inverse-strongly monotone and β-inverse-strongly monotone, respectively. Let S:E→E be a nonexpansive mapping. Suppose x1=u∈E and {xn} is generated by(1.8)yn=PE(xn-μBxn),xn+1=αnu+βnxn+γnSPE(yn-λnAyn),
where λ∈(0,2α),μ∈(0,2β), and {αn},{βn},{γn} are three sequences in [0,1] such that αn+βn+γn=1,for alln≥1. First, problem (1.6) is proven to be equivalent to a fixed point problem of a nonexpansive mapping.In this paper, motivated by what is mentioned above, we consider generalized system of variational inequalities as follows.LetE be a nonempty, closed, convex subset of a real Hilbert space H. Let A,B,C:E→H be three mappings. We consider the following problem of finding (x*,y*,z*)∈E×E×E such that(1.9)〈λAy*+x*-y*,x-x*〉≥0,∀x∈E,〈μBz*+y*-z*,x-y*〉≥0,∀x∈E,〈τCx*+z*-x*,x-z*〉≥0,∀x∈E,
which is called a general system of variational inequalities where λ>0,μ>0 and τ>0 are three constants.In particular, ifA=B=C, then problem (1.9) reduces to finding (x*,y*,z*)∈E×E×E such that(1.10)〈λAy*+x*-y*,x-x*〉≥0,∀x∈E,〈μAz*+y*-z*,x-y*〉≥0,∀x∈E,〈τAx*+z*-x*,x-z*〉≥0,∀x∈E.
Next, we consider some special classes of the GSVI problem (1.9) reduce to the following GSVI.(i)
Ifτ=0, then the GSVI problems (1.9) reduce to GSVI problem (1.6).(ii)
Ifτ=μ=0, then the GSVI problems (1.9) reduce to classical variational inequality VI(A,E) problem.The above system enters a class of more general problems which originated mainly from the Nash equilibrium points and was treated from a theoretical viewpoint in [17, 18]. Observe at the same time that, to construct a mathematical model which is as close as possible to a real complex problem, we often have to use constraints which can be expressed as one several subproblems of a general problem. These constrains can be given, for instance, by variational inequalities, by fixed point problems, or by problems of different types.This paper deals with a relaxed extragradient approximation method for solving a system of variational inequalities over the fixed-point sets of nonexpansive mapping. Under classical conditions, we prove a strong convergence theorem for this method. Moreover, the proposed algorithm can be applied for instance to solving the classical variational inequality problems.


## 2. Preliminaries

LetE be a nonempty, closed, convex subset of a real Hilbert space H. For every point x∈H, there exists a unique nearest point in E, denoted by PEx, such that (2.1)‖x-PEx‖≤‖x-y‖,∀y∈E.PE is call the metric projection of H onto E.Recall that,PEx is characterized by following properties: PEx∈E and (2.2)〈x-PEx,y-PEx〉≤0,‖x-y‖2≥‖x-PEx‖2+‖PEx-y‖2,
for all x∈H and y∈E.Lemma 2.1 (see cf. Zhang et al. [19]).
The metric projectionPE has the following properties:  (i)
PE:H→E is nonexpansive;(ii)
PE:H→E is firmly nonexpansive, that is,
(2.3)‖PEx-PEy‖2≤〈PEx-PEy,x-y〉,∀x,y∈H;(iii)
for eachx∈H,(2.4)z=PE(x)⟺〈x-z,z-y〉≥0,∀y∈E.Lemma 2.2 (see Osilike and Igbokwe [20]).
Let(E,〈·,·〉) be an inner product space. Then for all x,y,z∈E and α,β,γ∈[0,1] with α+β+γ=1, one has
(2.5)‖αx+βy+γz‖2=α‖x‖2+β‖y‖2+γ‖z‖2-αβ‖x-y‖2-αγ‖x-z‖2-βγ‖y-z‖2.Lemma 2.3 (see Suzuki [21]).
Let{xn} and {yn} be bounded sequences in a Banach space X and let {βn} be a sequence in [0,1] with 0<liminfn→∞βn≤limsupn→∞βn<1. Suppose xn+1=(1-βn)yn+βnxn for all integers n≥0 and limsupn→∞(∥yn+1-yn∥-∥xn+1-xn∥)≤0. Then, limn→∞∥yn-xn∥=0.Lemma 2.4 (see Xu [22]).
Assume{an} is a sequence of nonnegative real numbers such that
(2.6)an+1≤(1-αn)an+δn,n≥0,
where {αn} is a sequence in (0,1) and {δn} is a sequence in R such that  (i)
∑n=1∞αn=∞,(ii)
limsupn→∞(δn/αn)≤0 or ∑n=1∞|δn|<∞. Then, limn→∞an=0.Lemma 2.5 (Goebel and Kirk [23]).
Demiclosedness Principle. Assume thatT is a nonexpansive self-mapping of a nonempty, closed, convex subset E of a real Hilbert space H. If T has a fixed point, then I-T is demiclosed; that is, whenever {xn} is a sequence in E converging weakly to some x∈E (for short, xn⇀x∈E), and the sequence {(I-T)xn} converges strongly to some y (for short, (I-T)xn→y), it follows that (I-T)x=y. Here, I is the identity operator of H.The following lemma is an immediate consequence of an inner product.Lemma 2.6.
In a real Hilbert spaceH, there holds the inequality
(2.7)‖x+y‖2≤‖x‖2+2〈y,x+y〉,∀x,y∈H.Remark 2.7.
We also have that, for allu,v∈E and λ>0,
(2.8)‖(I-λA)u-(I-λA)v‖2=‖(u-v)-λ(Au-Av)‖2=‖u-v‖2-2λ〈u-v,Au-Av〉+λ2‖Au-Av‖2≤‖u-v‖2+λ(λ-2α)‖Au-Av‖2.
So, if λ≤2α, then I-λA is a nonexpansive mapping from E to H.


## 3. Main Results

In this section, we introduce an iterative precess by the relaxed extragradient approximation method for finding a common element of the set of fixed points of a nonexpansive mapping and the solution set of the variational inequality problem for three inverse-strongly monotone mappings in a real Hilbert space. We prove that the iterative sequence converges strongly to a common element of the above two sets.In order to prove our main result, the following lemmas are needed.Lemma 3.1.
For givenx*,y*,z*∈E×E×E,(x*,y*,z*) is a solution of problem (1.9) if and only if x* is a fixed point of the mapping G:E→E defined by
(3.1)G(x)=PE{PE[PE(x-τCx)-μBPE(x-τCx)]-λAPE[PE(x-τCx)-μBPE(x-τCx)]},∀x∈E,
where y*=PE(z*-μBz*) and z*=PE(x*-τCx*).Proof.
(3.2)〈λAy*+x*-y*,x-x*〉≥0,∀x∈E,〈μBz*+y*-z*,x-y*〉≥0,∀x∈E,〈τCx*+z*-x*,x-z*〉≥0,∀x∈E,⇔(3.3)〈(-y*+λAy*)+x*,x-x*〉≥0,∀x∈E,〈(-z*+μBz*)+y*,x-y*〉≥0,∀x∈E,〈(-x*+τCx*)+z*,x-z*〉≥0,∀x∈E,⇔(3.4)〈(y*-λAy*)-x*,x*-x〉≥0,∀x∈E,〈(z*-μBz*)-y*,y*-x〉≥0,∀x∈E,〈(x*-τCx*)-z*,z*-x〉≥0,∀x∈E,⇔(3.5)x*=PE(y*-λAy*),y*=PE(z*-μBz*),z*=PE(x*-τCx*),⇔x*=PE[PE(z*-μBz*)-λAPE(z*-μBz*)].
Thus,(3.6)x*=PE{PE[PE(x*-τCx*)-μBPE(x*-τCx*)]-λAPE[PE(x*-τCx*)-μBPE(x*-τCx*)]}.Lemma 3.2.
The mappingG defined by Lemma 3.1 is nonexpansive mappings.Proof.
For allx,y∈E,
(3.7)‖G(x)-G(y)‖=‖PE{PE[PE(x-τCx)-μBPE(x-τCx)]-λAPE[PE(x-τCx)-μBPE(x-τCx)]}-PE{PE[PE(y-τCy)-μBPE(y-τCy)]-λAPE[PE(y-τCy)-μBPE(y-τCy)]}≤‖[PE(x-τCx)-μBPE(x-τCx)]-λAPE[PE(x-τCx)-μBPE(x-τCx)]-[PE(y-τCy)-μBPE(y-τCy)]-λAPE[PE(y-τCy)-μBPE(y-τCy)]‖=‖(I-λA)[PE(x-τCx)-μBPE(x-τCx)]-(I-λA)[PE(y-τCy)-μBPE(y-τCy)]‖≤‖[PE(x-τCx)-μBPE(x-τCx)]-[PE(y-τCy)-μBPE(y-τCy)]‖=‖(I-μB)[PE(x-τCx)]-(I-μB)[PE(y-τCy)]‖≤‖PE(x-τCx)-PE(y-τCy)‖≤‖(x-τCx)-(y-τCy)‖=‖(I-τC)(x)-(I-τC)(y)‖≤‖x-y‖.
This shows that G:E→E is a nonexpansive mapping.Throughout this paper, the set of fixed points of the mappingG is denoted by Υ.Now, we are ready to proof our main results in this paper.Theorem 3.3.
LetE be a nonempty, closed, convex subset of a real Hilbert space H. Let the mapping A,B,C:E→H be α-inverse-strongly monotone, β-inverse-strongly monotone, and γ-inverse-strongly monotone, respectively. Let S be a nonexpansive mapping of E into itself such that F(S)∩Υ≠∅. Let f be a contraction of H into itself and given x1∈H arbitrarily and {xn} is generated by
(3.8)zn=PE(xn-τCxn),yn=PE(zn-μBzn),xn+1=αnf(xn)+βnxn+γnSPE(yn-λAyn),n≥0,
where λ∈(0,2α),μ∈(0,2β),τ∈(0,2γ), and {αn},{βn},{γn} are three sequences in [0,1] such that  (i)
αn+βn+γn=1,(ii)
limn→∞αn=0 and ∑n=1∞αn=∞,(iii)
0<liminfn→∞βn≤limsupn→∞βn<1. Then, {xn} converges strongly to x¯∈F(S)∩Γ, where x¯=PF(S)∩Γf(x¯) and (x¯,y¯,z¯) is a solution of problem (1.9), where
(3.9)y¯=PE(z¯-μBz¯),z¯=PE(x¯-τCx¯).Proof.
Letx*∈F(S)∩Γ. Then, x*=Sx* and x*=Gx*, that is,
(3.10)x*=PE{PE[PE(x*-τCx*)-μBPE(x*-τCx*)]-λAPE[PE(x*-τCx*)-μBPE(x*-τCx*)]}.
Put x*=PE(y*-λAy*) and tn=PE(yn-λAyn). Then, x*=PE[PE(z*-μBz*)-λAPE(z*-μBz*)] implies that y*=PE(z*-μBz*), where z*=PE(x*-τCx*). Since I-λA, I-μB and I-τC are nonexpansive mappings. We obtain that
(3.11)‖tn-x*‖=‖PE(yn-λAyn)-x*‖=‖PE(yn-λAyn)-PE(y*-λAy*)‖≤‖(yn-λAyn)-(y*-λAy*)‖=‖(I-λA)yn-(I-λA)y*‖≤‖yn-y*‖(3.12)=‖yn-PE(z*-μBz*)‖=‖PE(zn-μBzn)-PE(z*-μBz*)‖≤‖(I-μB)zn-(I-μB)z*‖≤‖zn-z*‖,(3.13)‖zn-z*‖=‖PE(xn-τCxn)-PE(x*-τCx*)‖≤‖(xn-τCxn)-(x*-τCx*)‖=‖(I-τC)xn-(I-τC)x*‖≤‖xn-x*‖.
Substituting (3.13) into (3.12), we have
(3.14)‖tn-x*‖≤‖xn-x*‖,
and by (3.11) we also have
(3.15)‖yn-y*‖≤‖xn-x*‖.
Since xn+1=αnf(xn)+βnxn+γnStn and by Lemma 2.2, we compute
(3.16)‖xn+1-x*‖=‖αnf(xn)+βnxn+γnStn-x*‖=‖αn(f(xn)-x*)+βn(xn-x*)+γn(Stn-x*)‖≤αn‖f(xn)-x*‖+βn‖xn-x*‖+γn‖Stn-x*‖≤αn‖f(xn)-x*‖+βn‖xn-x*‖+γn‖tn-x*‖≤αn‖f(xn)-x*‖+βn‖xn-x*‖+γn‖xn-x*‖=αn‖f(xn)-x*‖+(1-αn)‖xn-x*‖=αn‖f(xn)-f(x*)+f(x*)-x*‖+(1-αn)‖xn-x*‖≤αn‖f(xn)-f(x*)‖+αn‖f(x*)-x*‖+(1-αn)‖xn-x*‖≤αnk‖xn-x*‖+αn‖f(x*)-x*‖+(1-αn)‖xn-x*‖=(αnk+(1-αn))‖xn-x*‖+αn‖f(x*)-x*‖=(1-αn(1-k))‖xn-x*‖+αn‖f(x*)-x*‖=(1-αn(1-k))‖xn-x*‖+αn(1-k)‖f(x*)-x*‖(1-k).
By induction, we get
(3.17)‖xn+1-x*‖≤M,
where M=max{∥x0-x*∥+(1/(1-k))∥f(x*)-x*∥},n≥0. Therefore, {xn}is bounded.   Consequently, by (3.11), (3.12) and (3.13), the sequences {tn},{Stn},{yn},{Ayn},{zn},{Bzn},{Cxn}, and {f(xn)} are also bounded. Also, we observe that
(3.18)‖zn+1-zn‖=‖PE(xn+1-τCxn+1)-PE(xn-τCxn)‖≤‖(I-τC)xn+1-(I-τC)xn‖≤‖xn+1-xn‖,(3.19)‖tn+1-tn‖=‖PE(yn+1-λAyn+1)-PE(yn-λAyn)‖≤‖(yn+1-λAyn+1)-(yn-λAyn)‖=‖(I-λA)yn+1-(I-λA)yn‖≤‖yn+1-yn‖=‖PE(zn+1-μBzn+1)-PE(zn-μBzn)‖≤‖zn+1-zn‖≤‖xn+1-xn‖.
Let xn+1=(1-βn)wn+βnxn. Thus, we get
(3.20)wn=xn+1-βnxn1-βn=αnf(xn)+γnSPC(yn-λnAyn)1-βn=αnu+γnStn1-βn
It follows that
(3.21)wn+1-wn=αn+1f(xn+1)+γn+1Stn+11-βn+1-αnf(xn)+γnStn1-βn=αn+1f(xn+1)1-βn+1+γn+1Stn+11-βn+1-αn+1f(xn)1-βn+1+αn+1f(xn)1-βn+1-αnf(xn)1-βn-γnStn1-βn=αn+11-βn+1(f(xn+1)-f(xn))+(αn+11-βn+1-αn1-βn)f(xn)+γn+1Stn+11-βn+1-γnStn1-βn=αn+11-βn+1(f(xn+1)-f(xn))+(αn+11-βn+1-αn1-βn)f(xn)+γn+1Stn+11-βn+1-γn+1Stn1-βn+1+γn+1Stn1-βn+1-γnStn1-βn=αn+11-βn+1(f(xn+1)-f(xn))+(αn+11-βn+1-αn1-βn)f(xn)+γn+11-βn+1(Stn+1-Stn)+(γn+11-βn+1-γn1-βn)Stn=αn+11-βn+1(f(xn+1)-f(xn))+(αn+11-βn+1-αn1-βn)f(xn)+(αn+11-βn+1-αn1-βn)Stn+γn+11-βn+1(Stn+1-Stn)=αn+11-βn+1(f(xn+1)-f(xn))+(αn+11-βn+1-αn1-βn)(f(xn)+Stn)+γn+11-βn+1(Stn+1-Stn).
Combining (3.19) and (3.21), we obtain
(3.22)‖wn+1-wn‖-‖xn+1-xn‖≤|αn+11-βn+1|‖f(xn+1)-f(xn)‖+|αn+11-βn+1-αn1-βn|‖f(xn)+Stn‖+|γn+11-βn+1|‖Stn+1-Stn‖-‖xn+1-xn‖≤|αn+11-βn+1|k‖xn+1-xn‖+|αn+11-βn+1-αn1-βn|‖f(xn)+Stn‖+|γn+11-βn+1|‖tn+1-tn‖-‖xn+1-xn‖≤|αn+11-βn+1|k‖xn+1-xn‖+|αn+11-βn+1-αn1-βn|‖f(xn)+Stn‖+|γn+11-βn+1|‖xn+1-xn‖-‖xn+1-xn‖=|αn+11-βn+1|k‖xn+1-xn‖+|αn+11-βn+1-αn1-βn|‖f(xn)+Stn‖+|γn+1-1+βn+11-βn+1|‖xn+1-xn‖=|αn+11-βn+1|k‖xn+1-xn‖+|αn+11-βn+1-αn1-βn|‖f(xn)+Stn‖+|αn+11-βn+1|‖xn+1-xn‖.
This together with (i), (ii), and (iii) implies that
(3.23)limsup⁡n→∞(‖wn+1-wn‖-‖xn+1-xn‖)≤0.
Hence, by Lemma 2.3, we have
(3.24)lim⁡n→∞‖wn-xn‖=0.
Consequently,
(3.25)lim⁡n→∞‖xn+1-xn‖=lim⁡n→∞(1-βn)‖wn-xn‖=0.
From (3.18) and (3.19), we also have ∥zn+1-zn∥→0∥tn+1-tn∥→0 and ∥yn+1-yn∥→0 as n→∞. Since
(3.26)xn+1-xn=αnf(xn)+βnxn+γnStn-xn=αn(f(xn)-xn)+γn(Stn-xn),
it follows by (ii) and (3.25) that
(3.27)lim⁡n→∞‖xn-Stn‖=0.
Since x*∈F(S)∩Γ, from (3.15) and Lemma 2.2, we get
(3.28)‖xn+1-x*‖2=‖αnf(xn)+βnxn+γnStn-x*‖2≤αn‖f(xn)-x*‖2+βn‖xn-x*‖2+γn‖Stn-x*‖2≤αn‖f(xn)-x*‖2+βn‖xn-x*‖2+γn‖tn-x*‖2=αn‖f(xn)-x*‖2+βn‖xn-x*‖2+γn‖PE(yn-λAyn)-PE(y*-λAy*)‖2≤αn‖f(xn)-x*‖2+βn‖xn-x*‖2+γn‖(yn-λAyn)-(y*-λAy*)‖2=αn‖f(xn)-x*‖2+βn‖xn-x*‖2+γn‖(yn-y*)-λ(Ayn-Ay*)‖2=αn‖f(xn)-x*‖2+βn‖xn-x*‖2+γn[∥yn-y*∥2-2λ〈yn-y*,Ayn-Ay*〉+λ2‖Ayn-Ay*‖2]≤αn‖f(xn)-x*‖2+βn‖xn-x*‖2+γn[‖yn-y*‖2-2λα‖Ayn-Ay*‖2+λ2‖Ayn-Ay*‖2]=αn‖f(xn)-x*‖2+βn‖xn-x*‖2+γn[‖yn-y*‖2+λ(λ-2α)‖Ayn-Ay*‖2]≤αn‖f(xn)-x*‖2+βn‖xn-x*‖2+γn[‖xn-x*‖2+λ(λ-2α)‖Ayn-Ay*‖2]=αn‖f(xn)-x*‖2+βn‖xn-x*‖2+γn‖xn-x*‖2+γnλ(λ-2α)‖Ayn-Ay*‖2=αn‖f(xn)-x*‖2+(βn+γn)‖xn-x*‖2+γnλ(λ-2α)‖Ayn-Ay*‖2=αn‖f(xn)-x*‖2+(1-αn)‖xn-x*‖2+γnλ(λ-2α)‖Ayn-Ay*‖2≤αn‖f(xn)-x*‖2+‖xn-x*‖2+γnλ(λ-2α)‖Ayn-Ay*‖2.
Therefore, we have
(3.29)-γnλ(λ-2α)‖Ayn-Ay*‖2≤αn‖f(xn)-x*‖2+‖xn-x*‖2-‖xn+1-x*‖2=αn‖f(xn)-x*‖2+(‖xn-x*‖+‖xn+1-x*‖)×(‖xn-x*‖-‖xn+1-x*‖)≤αn‖f(xn)-x*‖2+(‖xn-x*‖+‖xn+1-x*‖)‖xn-xn+1‖.
From (ii), (iii), and ∥xn+1-xn∥→0, as n→∞, we get ∥Ayn-Ay*∥→0 as n→∞.
Sincex*∈F(S)∩Υ, from (3.11) and Lemma 2.2, we get
(3.30)‖xn+1-x*‖2=‖αnf(xn)+βnxn+γnStn-x*‖2≤αn‖f(xn)-x*‖2+βn‖xn-x*‖2+γn‖tn-x*‖2≤αn‖f(xn)-x*‖2+βn‖xn-x*‖2+γn‖yn-y*‖2=αn‖f(xn)-x*‖2+βn‖xn-x*‖2+γn‖PE(zn-μBzn)-PE(z*-μBz*)‖2≤αn‖f(xn)-x*‖2+βn‖xn-x*‖2+γn‖(zn-μBzn)-(z*-μBz*)‖2=αn‖f(xn)-x*‖2+βn‖xn-x*‖2+γn‖(zn-z*)-(μBzn-μBz*)‖2≤αn‖f(xn)-x*‖2+βn‖xn-x*‖2+γn[‖zn-z*‖2+μ(μ-2β)‖Bzn-Bz*‖2]≤αn‖f(xn)-x*‖2+‖xn-x*‖2+γnμ(μ-2β)‖Bzn-Bz*‖2.
Thus, we also have(3.31)-γnμ(μ-2β)‖Bzn-Bz*‖2≤αn‖f(xn)-x*‖2+‖xn-x*‖2-‖xn+1-x*‖2=αn‖f(xn)-x*‖2+(‖xn-x*‖+‖xn+1-x*‖)×(‖xn-x*‖-‖xn+1-x*‖)≤αn‖f(xn)-x*‖2+(‖xn-x*‖+‖xn+1-x*‖)‖xn-xn+1‖.
By again (ii), (iii), and (3.25), we also get ∥Bzn-Bz*∥→0 as n→∞.
Letx*∈F(S)∩Υ; again from (3.12), (3.13) and Lemma 2.2, we get
(3.32)‖xn+1-x*‖2=‖αnf(xn)+βnxn+γnStn-x*‖2≤αn‖f(xn)-x*‖2+βn‖xn-x*‖2+γn‖tn-x*‖2≤αn‖f(xn)-x*‖2+βn‖xn-x*‖2+γn‖zn-z*‖2≤αn‖f(xn)-x*‖2+βn‖xn-x*‖2+γn‖(xn-τCxn)-(x*-τCx*)‖2≤αn‖f(xn)-x*‖2+βn‖xn-x*‖2+γn[‖xn-x*‖2+τ(τ-2γ)‖Cxn-Cx*‖2]≤αn‖f(xn)-x*‖2+‖xn-x*‖2+γnτ(τ-2γ)‖Cxn-Cx*‖2.
Again, we have(3.33)-γnτ(τ-2γ)‖Cxn-Cx*‖2≤αn‖f(xn)-x*‖2+‖xn-x*‖2-‖xn+1-x*‖2=αn‖f(xn)-x*‖2+(‖xn-x*‖+‖xn+1-x*‖)×(‖xn-x*‖-‖xn+1-x*‖)≤αn‖f(xn)-x*‖2+(‖xn-x*‖+‖xn+1-x*‖)‖xn-xn+1‖.
Similarly again by (ii), (iii), and ∥xn-xn+1∥→0 as n→∞, and from (3.33), we also that ∥Cxn-Cx*∥→0.
On the other hand, we compute that(3.34)‖zn-z*‖2=‖PE(xn-τCxn)-PE(x*-τCx*)‖2≤〈(xn-τCxn)-(x*-τCx*),PE(xn-τCxn)-PE(x*-τCx*)〉=〈(xn-τCxn)-(x*-τCx*),zn-z*〉=12[‖(xn-τCxn)-(x*-τCx*)‖2+‖zn-z*‖2-‖(xn-τCxn)-(x*-τCx*)-(zn-z*)‖2]=12[‖(I-τC)xn-(I-τC)x*‖2+‖zn-z*‖2-‖(xn-τCxn)-(x*-τCx*)-(zn-z*)‖2]≤12[∥xn-x*∥2+∥zn-z*∥2-∥(xn-zn)-τ(Cxn-Cx*)-(x*-z*)∥2]=12[‖xn-x*‖2+‖zn-z*‖2-‖[(xn-zn)-(x*-z*)]-τ(Cxn-Cx*)‖2]=12[‖xn-x*‖2+‖zn-z*‖2-‖(xn-zn)-(x*-z*)‖2+2τ〈(xn-zn)-(x*-z*),Cxn-Cx*〉-τ2‖Cxn-Cx*‖2].
So, we obtain(3.35)‖zn-z*‖2≤‖xn-x*‖2-‖(xn-zn)-(x*-z*)‖2+2τ〈(xn-zn)-(x*-z*),Cxn-Cx*〉-τ2‖Cxn-Cx*‖2.
Hence, by (3.12), it follows that
(3.36)‖xn+1-x*‖2=‖αnf(xn)+βnxn+γnStn-x*‖2≤αn‖f(xn)-x*‖2+βn‖xn-x*‖2+γn‖Stn-x*‖2≤αnk‖xn-x*‖2+βn‖xn-x*‖2+γn‖tn-x*‖2≤αnk‖xn-x*‖2+βn‖xn-x*‖2+γn‖zn-z*‖2≤αnk‖xn-x*‖2+βn‖xn-x*‖2+γn‖xn-x*‖2-γn‖(xn-zn)-(x*-z*)‖2+2τγn〈(xn-zn)-(x*-z*),Cxn-Cx*〉-τ2γn‖Cxn-Cx*‖2=αnk‖xn-x*‖2+(1-αn)‖xn-x*‖2-γn‖(xn-zn)-(x*-z*)‖2+2τγn〈(xn-zn)-(x*-z*),Cxn-Cx*〉-τ2γn‖Cxn-Cx*‖2≤αnk‖xn-x*‖2+‖xn-x*‖2-γn‖(xn-zn)-(x*-z*)‖2+2τγn〈(xn-zn)-(x*-z*),Cxn-Cx*〉≤αnk‖xn-x*‖2+‖xn-x*‖2-γn‖(xn-zn)-(x*-z*)‖2+2τγn‖(xn-zn)-(x*-z*)‖‖Cxn-Cx*‖,
which implies that
(3.37)γn‖(xn-zn)-(x*-z*)‖2≤αnk‖xn-x*‖2+‖xn-x*‖2-‖xn+1-x*‖2+2τγn‖(xn-zn)-(x*-z*)‖‖Cxn-Cx*‖≤αnk‖xn+1-x*‖2+2γnτ‖(xn-zn)-(x*-z*)‖‖Cxn-Cx*‖+‖xn-xn+1‖(‖xn-x*‖+‖xn+1-x*‖).
By (ii), (iii), ∥xn-xn+1∥→0, and ∥Cxn-Cx*∥→0 as n→∞, from (3.37) and we get ∥(xn-zn)-(x*-z*)∥→0 as n→∞. Now, observe that
(3.38)‖(zn-tn)+(x*-z*)‖2=‖zn-PE(yn-λAyn)+PE(y*-λAy*)-z*‖2=‖zn-PE(yn-λAyn)+PE(y*-λAy*)-z*+μBzn-μBzn+μBz*-μBz*‖2=‖zn-μBzn-(z*-μBz*)-[PE(yn-λAyn)-PE(y*-λAy*)]+μ(Bzn-Bz*)‖2≤‖zn-μBzn-(z*-μBz*)-[PE(yn-λAyn)-PE(y*-λAy*)]‖2+2μ〈Bzn-Bz*,zn-μBzn-(z*-μBz*)-[PE(yn-λAyn)-PE(y*-λAy*)]+μ(Bzn-Bz*)〉=‖zn-μBzn-(z*-μBz*)-[PE(yn-λAyn)-PE(y*-λAy*)]‖2+2μ〈Bzn-Bz*,(zn-tn)+(x*-z*)〉≤‖zn-μBzn-(z*-μBz*)‖2-‖PE(yn-λnAyn)-PE(y*-λnAy*)‖2+2μ‖Bzn-Bz*‖‖(zn-tn)+(x*-z*)‖≤‖zn-μBzn-(z*-μBz*)‖2-‖SPE(yn-λnAyn)-SPE(y*-λnAy*)‖2+2μ‖Bzn-Bz*‖‖(zn-tn)+(x*-z*)‖=‖zn-μBzn-(z*-μBz*)‖2-‖Stn-Sx*‖2+2μ‖Bzn-Bz*‖‖(zn-tn)+(x*-z*)‖≤‖zn-μBzn-(z*-μBz*)-(Stn-x*)‖×(∥zn-μBzn-(z*-μBz*)∥+‖Stn-x*‖)+2μ‖Bzn-Bz*‖‖(zn-tn)+(x*-z*)‖.
Since ∥Stn-xn∥→0, ∥(xn-zn)-(x*-z*)∥→0, and ∥Bzn-Bz*∥→0,asn→∞, it follows that
(3.39)∥(zn-tn)+(x*-z*)∥→0,asn→∞.
Since
(3.40)‖Stn-tn‖≤‖Stn-xn‖+‖(xn-zn)-(x*-z*)‖+‖(zn-tn)+(x*-z*)‖,
we obtain
(3.41)lim⁡n→∞‖Stn-tn‖=0.
Next, we show that
(3.42)limsup⁡n→∞〈f(x¯)-x¯,xn-x¯〉≤0,
where x¯=PF(S)∩Υf(x¯).
Indeed, since{tn} and {Stn} are two bounded sequence in E, we can choose a subsequence {tni} of {tn} such that tni of tn such that tni⇀z∈E and
(3.43)limsup⁡n→∞〈f(x¯)-x¯,Stn-x¯〉=lim⁡i→∞〈f(x¯)-x¯,Stni-x¯〉.
Since limn→∞∥Stn-tn∥=0, we obtain Stni⇀z as i→∞. Now, we claim that z∈F(S)∩Υ. First by Lemma 2.5, it is easy to see that z∈F(S).
Since∥Stn-tn∥→0, ∥Stn-xn∥→0, and
(3.44)‖tn-xn‖=‖tn-Stn+Stn-xn‖≤‖tn-Stn‖+‖Stn-xn‖=‖Stn-tn‖+‖Stn-xn‖,
we conclude that∥tn-xn∥→0 as n→∞. Furthermore, by Lemma 3.2 that G is nonexpansive, then
(3.45)‖tn-G(tn)‖=‖G(xn)-G(tn)‖≤‖xn-tn‖.
Thus limn→∞∥tn-G(tn)∥=0. According to Lemma 2.5, we obtain z∈Υ. Therefore, there holds z∈F(S)∩Υ.
On the other hand, it follows from (2.2) that
(3.46)limsup⁡n→∞〈f(x¯)-x¯,xn-x¯〉=limsup⁡n→∞〈f(x¯)-x¯,Stn-x¯〉=lim⁡n→∞〈f(x¯)-x¯,Stni-x¯〉=〈f(x¯)-x¯,z-x¯〉≤0.
Finally, we show thatxn→x¯, and by (3.14) that
(3.47)‖xn+1-x¯‖2=‖αnf(xn)+βnxn+γnStn-x¯‖2≤‖βn(xn-x¯)+γn(Stn-x¯)‖2+2αn〈f(xn)-x¯,xn+1-x¯〉≤‖βn(xn-x¯)+γn(Stn-x¯)‖2+2αn〈f(xn)-f(x¯),xn+1-x¯〉+2αn〈f(x¯)-x¯,xn+1-x¯〉≤[βn‖xn-x¯‖2+γn‖Stn-x¯‖2]+2αn‖f(xn)-f(x¯)‖‖xn+1-x¯‖+2αn〈f(x¯)-x¯,xn+1-x¯〉≤[βn‖xn-x¯‖2+γn‖tn-x¯‖2]+2αnα‖xn-x¯‖‖xn+1-x¯‖+2αn〈f(x¯)-x¯,xn+1-x¯〉≤(1-αn)2‖xn-x¯‖2+αnα(‖xn-x¯‖2+‖xn+1-x¯‖2)+2αn〈f(x¯)-x¯,xn+1-x¯〉
which implies that
(3.48)‖xn+1-x¯‖2≤(1-2(1-α)αn1-ααn)‖xn-x¯‖2+αn21-ααn‖xn-x¯‖2+2αn1-ααn〈f(x¯)-x¯,xn+1-x¯〉∶=(1-σn)‖xn-x¯‖2+δn,n≥0,
where σn=(2(1-α)αn)/1-ααn and δn=αn2/(1-ααn)∥xn-x¯∥2+2αn/(1-ααn)〈f(x¯)-x¯,xn+1-x¯〉. Therefore, by (3.46) and Lemma 2.4, we get that {xn} converges to x¯, where x¯=PF(S)∩Υf(x¯). This completes the proof.SettingA=B=C, we obtain the following corollary.Corollary 3.4.
LetE be a nonempty, closed, convex subset of a real Hilbert space H. Let the mapping A:E→H be α-inverse-strongly monotone. Let S be a nonexpansive mapping of E into itself such that F(S)∩Υ≠∅. Let f be a contraction of H into itself and given x0∈H arbitrarily and {xn} is generated by
(3.49)zn=PE(xn-τAxn),yn=PE(zn-μAzn),xn+1=αnf(xn)+βnxn+γnSPE(yn-λAyn),n≥1,
where λ,μ,τ∈(0,2α) and {αn},{βn},{γn} are three sequences in [0,1] such that  (i)
αn+βn+γn=1,(ii)
limn→∞αn=0 and ∑n=1∞αn=∞,(iii)
0<liminfn→∞βn≤limsupn→∞βn<1. Then, {xn} converges strongly to x¯∈F(S)∩Υ, where x¯=PF(S)∩Υf(x¯) and (x¯,y¯,z¯) is a solution of problem (1.10), where
(3.50)y¯=PE(z¯-μAz¯),z¯=PE(x¯-τAx¯).SettingA≡B≡0 (the zero operators), we obtain the following corollary for solving the foxed points problem and the classical variational inequality problems.Corollary 3.5.
LetE be a nonempty, closed, convex subset of a real Hilbert space H. Let the mapping A:E→H be α-inverse-strongly monotone. Let S be a nonexpansive mapping of E into itself such that F(S)∩VI(A,E)≠∅. Let f be a contraction of H into itself and given x0∈H arbitrarily and {xn} is generated by
(3.51)xn+1=αnf(xn)+βnxn+γnSPE(xn-λAxn),n≥1,
where λ∈(0,2α) and {αn},{βn},{γn} are three sequences in [0,1] such that  (i)
αn+βn+γn=1,(ii)
limn→∞αn=0 and ∑n=1∞αn=∞,(iii)
0<liminfn→∞βn≤limsupn→∞βn<1.Then,{xn} converges strongly to x¯∈F(S)∩VI(A,E), where x¯=PF(S)∩VI(A,E)f(x¯).


## 4. Some Applications

We recall that a mappingT:E→E is called strictly pseudocontractive if there exists some k with 0≤k<1 such that (4.1)‖Tx-Ty‖2≤‖x-y‖2+k‖(I-T)x-(I-T)y‖2,∀x,y∈E.
For recent convergence result for strictly pseudocontractive mappings, put A=I-T. Then, we have (4.2)‖(I-A)x-(I-A)y‖2≤‖x-y‖2+k‖Ax-Ay‖2.
On the other hand, (4.3)‖(I-A)x-(I-A)y‖2≤‖x-y‖2+‖Ax-Ay‖2-2〈x-y,Ax-Ay〉.
Hence, we have (4.4)〈x-y,Ax-Ay〉≥1-k2‖Ax-Ay‖2.
Consequently, if T:E→E is a strictly pseudocontractive mapping with constant k, then the mapping A=I-T is (1-k)/2-inverse-strongly monotone.SettingA=I-T, B=I-V, and C=I-W, we obtain the following corollary.Theorem 4.1.
LetE be a nonempty, closed, convex subset of a real Hilbert space H. Let T,V,W be strictly pseudocontractive mappings with constant k of C into itself, and let S be a nonexpansive mapping of E into itself such that F(S)∩Υ≠∅. Let f be a contraction of H into itself and given x0∈H arbitrarily and {xn} is generated by
(4.5)zn=(I-τ)xn+τWxn,yn=(I-μ)zn+μVzn,xn+1=αnf(xn)+βnxn+γnS((1-λ)yn+λTyn),n≥1,
where λ∈(0,2α),μ∈(0,2β),andτ∈(0,2γ) and {αn},{βn},{γn} are three sequences in [0,1] such that  (i)
αn+βn+γn=1,(ii)
limn→∞αn=0 and ∑n=1∞αn=∞,(iii)
0<liminfn→∞βn≤limsupn→∞βn<1. Then {xn} converges strongly to x¯∈F(S)∩Υ, where x¯=PF(S)∩Υf(x¯) and (x¯,y¯,z¯) is a solution of problem (1.9), where
(4.6)y¯=PE(z¯-μBz¯),z¯=PE(x¯-τCx¯).Proof.
SinceA=I-T, B=I-V, and C=I-W, we have
(4.7)PE(xn-τCxn)=(I-τ)xn+τWxn,PE(yn-λAyn)=(I-λ)yn+λTyn,PE(zn-μBzn)=(I-μ)zn+μVzn.
Thus, the conclusion follows immediately from Theorem 3.3.Iff(x)=x0,forallx∈E, and T=V=W in Theorem 4.1, we obtain the following corollary.Corollary 4.2.
LetE be a nonempty, closed, convex subset of a real Hilbert space H. Let T be strictly pseudocontractive mappings with constant k of C into itself, and let S be a nonexpansive mapping of E into itself such that F(S)∩Υ≠∅. Given x0∈H arbitrarily and {xn} is generated by
(4.8)zn=(I-τ)xn+τTxn,yn=(I-μ)zn+μTzn,xn+1=αnx0+βnxn+γnS((1-λ)yn+λTyn),n≥1,
where λ∈(0,2α),μ∈(0,2β),andτ∈(0,2γ) and {αn},{βn},{γn} are three sequences in [0,1] such that
(4.9)αn+βn+γn=1,lim⁡n→∞⁡αn=0and∑n=1∞αn=∞,0<liminf⁡n→∞⁡βn≤limsup⁡n→∞⁡βn<1.
Then, {xn} converges strongly to x¯∈F(S)∩Υ, where x¯=PF(S)∩Υx¯ and (x¯,y¯,z¯) is a solution of problem (1.10), where
(4.10)y¯=PE(z¯-μAz¯)z¯=PE(x¯-τAx¯).



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*Source: 976505-2011-07-14.xml*