AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation94914110.1155/2012/949141949141Research ArticleIterative Algorithms Approach to Variational Inequalities and Fixed Point ProblemsLiouYeong-Cheng1YaoYonghong2TsengChun-Wei1LinHui-To1YangPei-Xia2NoorKhalida Inayat1Department of Information ManagementCheng Shiu UniversityKaohsiung 833Taiwancsu.edu.tw2Department of MathematicsTianjin Polytechnic UniversityTianjin 300387Chinatjpu.edu.cn2012161201220122109201116112011171120112012Copyright © 2012 Yeong-Cheng Liou et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider a general variational inequality and fixed point problem, which is to find a point x* with the property that (GVF): x*GVI(C,A) and g(x*)Fix(S) where GVI(C,A) is the solution set of some variational inequality Fix(S) is the fixed points set of nonexpansive mapping S, and g is a nonlinear operator. Assume the solution set Ω of (GVF) is nonempty. For solving (GVF), we suggest the following method g(xn+1)=βg(xn)+(1-β)SPC[αnF(xn)+(1-αn)(g(xn)-λAxn)], n0. It is shown that the sequence {xn} converges strongly to x*Ω which is the unique solution of the variational inequality F(x*)-g(x*),g(x)-g(x*)0, for all xΩ.

1. Introduction

Let A:CH and g:CC be two nonlinear mappings. We concern the following generalized variational inequality of finding uC,  g(u)C such thatg(v)-g(u),Au0,g(v)C. The solution set of (1.1) is denoted by GVI(C,A,g). It has been shown that a large class of unrelated odd-order and nonsymmetric obstacle, unilateral, contact, free, moving, and equilibrium problems arising in regional, physical, mathematical, engineering, and applied sciences can be studied in the unified and general framework of the general variational inequalities (1.1), see  and the references therein. Noor  has introduced a new type of variational inequality involving two nonlinear operators, which is called the general variational inequality. It is worth mentioning that this general variational inequality is remarkably different from the so-called general variational inequality which was introduced by Noor  in 1988. Noor  proved that the general variational inequalities are equivalent to nonlinear projection equations and the Wiener-Hopf equations by using the projection technique. Using this equivalent formulation, Noor  suggested and analyzed some iterative algorithms for solving the special general variational inequalities and further proved that these algorithms have strong convergence.

For g=I, where I is the identity operator, problem (1.1) is equivalent to finding uC such thatv-u,Au0,vC, which is known as the classical variational inequality introduced and studied by Stampacchia  in 1964. This field has been extensively studied due to a wide range of applications in industry, finance, economics, social, pure and applied sciences. For related works, please see . Our main purposes in the present paper is devoted to study this topic.

Motivated and inspired by the works in this field, in this paper, we consider a general variational inequality and fixed point problem, which is to find a point x* with the property thatx*GVI(C,A),g(x*)Fix(S), where Fix(S) is the fixed points set of nonexpansive mapping S. Assume the solution set Ω of (GVF) is nonempty. For solving (GVF), we suggest the following methodg(xn+1)=βg(xn)+(1-β)SPC[αnF(xn)+(1-αn)(g(xn)-λAxn)],n0. It is shown that the sequence {xn} converges strongly to x*Ω which is the unique solution of the following variational inequalityF(x*)-g(x*),g(x)-g(x*)0,xΩ. Our results contain some interesting results as special cases.

2. Preliminaries

Let H be a real Hilbert space with inner product ·,· and norm ·, respectively. Let C be a nonempty closed convex subset of H. Recall that a mapping S:CC is said to be nonexpansive ifSx-Syx-y, for all x,yC. We denote by Fix(S) the set of fixed points of S. A mapping F:CH is said to be L-Lipschitz continuous, if there exists a constant L>0 such that F(x)-F(y)Lx-y for all x,yC. A mapping A:CH is said to be α-inverse strongly g-monotone if and only ifAx-Ay,g(x)-g(y)αAx-Ay2, for some α>0 and for all x,yC. A mapping g:CC is said to be strongly monotone if there exists a constant γ>0 such thatg(x)-g(y),x-yγx-y2, for all x,yC.

Let B be a mapping of H into 2H. The effective domain of B is denoted by dom(B), that is, dom(B)={xH:Bx}. A multivalued mapping B is said to be a monotone operator on H if and only ifx-y,u-v0, for all x,ydom(B),  uBx, and vBy. A monotone operator B on H is said to be maximal if and only if its graph is not strictly contained in the graph of any other monotone operator on H. Let B be a maximal monotone operator on H and let B-10={xH:0Bx}.

It is well known that, for any uH, there exists a unique u0C such thatu-u0=inf{u-x:xC}. We denote u0 by PCu, where PC is called the metric projection of H onto C. The metric projection PC of H onto C has the following basic properties:

PCx-PCyx-y for all x,yH;

x-y,PCx-PCyPCx-PCy2 for every x,  yH;

x-PCx,y-PCx0 for all xH, yC.

It is easy to see that the following is true:uGVI(C,A,g)g(u)=PC(g(u)-λA(u)),λ>0.

We use the following notation:

xnx stands for the weak convergence of (xn) to x;

xnx stands for the strong convergence of (xn) to x.

We need the following lemmas for the next section.

Lemma 2.1.

Let C be a nonempty closed convex subset of a real Hilbert space H. Let G:CC be a nonlinear mapping and let the mapping A:CH be α-inverse strongly g-monotone. Then, for any λ>0, one has PC[g(x)-λAx]-PC[g(y)-λAy]2g(x)-g(y)2+λ(λ-2α)Ax-Ay2,x,yC.

Proof.

Consider the following:PC[g(x)-λAx]-PC[g(y)-λAy]2g(x)-g(y)-λ(Ax-Ay)2=g(x)-g(y)2-2λAx-Ay,g(x)-g(y)+λ2Ax-Ay2g(x)-g(y)2-2λαAx-Ay2+λ2Ax-Ay2g(x)-g(y)2+λ(λ-2α)Ax-Ay2. If λ[0,2α], we have PC[g(x)-λAx]-PC[g(y)-λAy]g(x)-g(y)-λ(Ax-Ay)g(x)-g(y).

Lemma 2.2 (see [<xref ref-type="bibr" rid="B32">36</xref>]).

Let C be a closed convex subset of a Hilbert space H. Let S:CC be a nonexpansive mapping. Then Fix(S) is a closed convex subset of C and the mapping I-S is demiclosed at 0, that is, whenever {xn}C is such that xnx and (I-S)xn0, then (I-S)x=0.

Lemma 2.3 (see [<xref ref-type="bibr" rid="B34">37</xref>]).

Let {xn} and {yn} be bounded sequences in a Banach space X and let {βn} be a sequence in [0,1] with 0<liminfnβnlimsupnβn<1. Suppose xn+1=(1-βn)yn+βnxn for all n0 and limsupn(yn+1-yn-xn+1-xn)0. Then, limnyn-xn=0.

Lemma 2.4 (see [<xref ref-type="bibr" rid="B35">38</xref>]).

Assume {an} is a sequence of nonnegative real numbers such that an+1(1-γn)an+δnγn, where {γn} is a sequence in (0,1) and {δn} is a sequence such that

n=1γn=;

limsupnδn0 or n=1|δnγn|<.

Then limnan=0.

3. Main Results

In this section, we will prove our main results.

Theorem 3.1.

Let C be a nonempty closed and convex subset of a real Hilbert space H. Let F:CH be an L-Lipschitz continuous mapping, g:CC be a weakly continuous and γ-strongly monotone mapping such that R(g)=C. Let A:CH be an α-inverse strongly g-monotone mapping and let S:CC be a nonexpansive mapping. Suppose that Ω. Let β(0,1) and γ(L,2α). For given x0C, let {xn}C be a sequence generated by g(xn+1)=βg(xn)+(1-β)SPC[αnF(xn)+(1-αn)(g(xn)-λAxn)],n0, where {αn}(0,1) satisfies (C1):  limnαn=0 and (C2):  nαn=. Then the sequence {xn} generated by (3.1) converges strongly to x*Ω which is the unique solution of the following variational inequality: F(x*)-g(x*),g(x)-g(x*)0,xΩ.

Proof.

First, we show the solution set of variational inequality (3.2) is singleton. Assume x̃Ω also solves (3.2). Then, we have F(x*)-g(x*),g(x̃)-g(x*)0,F(x̃)-g(x̃),g(x*)-g(x̃)0. It follows that F(x̃)-g(x̃)-F(x*)+g(x*),g(x*)-g(x̃)0g(x*)-g(x̃)2F(x*)-F(x̃),g(x*)-g(x̃)g(x*)-g(x̃)2F(x*)-F(x̃),g(x*)-g(x̃)F(x*)-F(x̃)g(x*)-g(x̃)g(x*)-g(x̃)F(x*)-F(x̃). Since g is γ-strongly monotone, we have γx-y2g(x)-g(y),x-yg(x)-g(y)x-y,x,yC. Hence, γx-yg(x)-g(y),x,yC. In particular, γx*-x̃g(x*)-g(x̃). By (3.4), we deduce γx*-x̃g(x*)-g(x̃)F(x*)-F(x̃)Lx*-x̃, which implies that x̃=x* because of L<γ by the assumption. Therefore, the solution of variational inequality (3.2) is unique.

Pick up any uΩ. It is obvious that uGVI(C,A,g) and g(u)Fix(S). Set un=PC[αnF(xn)+(1-αn)(g(xn)-λAxn)],  n0. From (2.6), we know g(u)=PC[g(u)-μAu] for any μ>0. Hence, we have g(u)=PC[g(u)-(1-αn)λAu]=PC[αng(u)+(1-αn)(g(u)-λAu)],n0. From (3.6), (3.8), and Lemma 2.1, we get un-g(u)=PC[αnF(xn)+(1-αn)(g(xn)-λAxn)]-PC[αng(u)+(1-αn)(g(u)-λAu)]αnF(xn)-g(u)+(1-αn)(g(xn)-λAxn)-(g(u)-λAu)αnF(xn)-F(u)+αnF(u)-g(u)+(1-αn)g(xn)-g(u)αnLxn-u+αnF(u)-g(u)+(1-αn)g(xn)-g(u)αnLγg(xn)-g(u)+αnF(u)-g(u)+(1-αn)g(xn)-g(u)=[1-(1-Lγ)αn]g(xn)-g(u)+αnF(u)-g(u). It follows from (3.1) that g(xn+1)-g(u)βg(xn)-g(u)+(1-β)Sun-Sg(u)βg(xn)-g(u)+(1-β)un-g(u)βg(xn)-g(u)+(1-β)[1-(1-Lγ)αn]g(xn)-g(u)+(1-β)αnF(u)-g(u)=[1-(1-Lγ)(1-β)αn]g(xn)-g(u)+(1-Lγ)(1-β)αnF(u)-g(u)1-L/γ. This indicates by induction that g(xn+1)-g(u)max{g(xn)-g(u),F(u)-g(u)1-L/γ}. Hence, {g(xn)} is bounded. By (3.6), we have xn-u(1/γ)g(xn)-g(u). This implies that {xn} is bounded. Consequently, {F(xn)},{Axn},{un}, and {Sun} are all bounded.

Note that we can rewrite (3.1) as g(xn+1)=βg(xn)+(1-β)Sun for all n. Next, we will use Lemma 2.3 to prove that xn+1-xn0. In fact, we firstly have Sun-Sun-1=SPC[αnF(xn)+(1-αn)(g(xn)-λAxn)]-SPC[αn-1F(xn-1)+(1-αn-1)(g(xn-1)-λAxn-1)][αnF(xn)+(1-αn)(g(xn)-λAxn)]-[αn-1F(xn-1)+(1-αn-1)(g(xn-1)-λAxn-1)]αnF(xn)-F(xn-1)+|αn-αn-1|F(xn-1)+(1-αn)g(xn)-λAxn-(g(xn-1)-λAxn-1)+|αn-αn-1|g(xn-1)-λAxn-1αnLxn-xn-1+(1-αn)g(xn)-g(xn-1)+|αn-αn-1|(F(xn-1)+g(xn-1)-λAxn-1)αn(Lγ)g(xn)-g(xn-1)+(1-αn)g(xn)-g(xn-1)+|αn-αn-1|(F(xn-1)+g(xn-1)-λAxn-1)=[1-(1-Lγ)αn]g(xn)-g(xn-1)+|αn-αn-1|(F(xn-1)+g(xn-1)-λAxn-1). It follows that Sun-Sun-1-g(xn)-g(xn-1)|αn-αn-1|(F(xn-1)+g(xn-1)-λAxn-1). Since αn0 and the sequences {F(xn)},{g(xn)}, and {Axn} are bounded, we have limsupn(Sun-Sun-1-g(xn)-g(xn-1))0. By Lemma 2.3, we obtain limnSun-g(xn)=0. Hence, limng(xn+1)-g(xn)=limn(1-β)Sun-g(xn)=0. This together with (3.6) imply that limnxn+1-xn=0. By the convexity of the norm and (3.9), we have g(xn+1)-g(u)2=β(g(xn)-g(u))+(1-β)(Sun-Sg(u))2βg(xn)-g(u)2+(1-β)Sun-Sg(u)2βg(xn)-g(u)2+(1-β)un-g(u)2βg(xn)-g(u)2+(1-β)[αnF(xn)-g(u)+(1-αn)(g(xn)-λAxn)-(g(u)-λAu)]2βg(xn)-g(u)2+(1-β)[αnF(xn)-g(u)2+(1-αn)(g(xn)-λAxn)-(g(u)-λAu)2]. From Lemma 2.1, we derive (g(xn)-λAxn)-(g(u)-λAu)2g(xn)-g(u)2+λ(λ-2α)Axn-Au2. Thus, g(xn+1)-g(u)2βg(xn)-g(u)2+(1-β)[αnF(xn)-g(u)2+(1-αn)(g(xn)-g(u)2+λ(λ-2α)Axn-Au2)]=(1-β)αnF(xn)-g(u)2+[1-(1-β)αn]g(xn)-g(u)2+(1-β)(1-αn)λ(λ-2α)Axn-Au2. So, (1-β)(1-αn)λ(2α-λ)Axn-Au2(1-β)αnF(xn)-g(u)2+g(xn)-g(u)2-g(xn+1)-g(u)2(1-β)αnF(xn)-g(u)2+(g(xn)-g(u)+g(xn+1)-g(u))g(xn+1)-g(xn). Since αn0,  g(xn+1)-g(xn)0 and liminfn(1-β)(1-αn)λ(2α-λ)>0, we obtain limnAxn-Au=0. Set yn=g(xn)-λAxn-(g(u)-λAu) for all n. By using the property of projection, we get un-g(u)2=PC[αnF(xn)+(1-αn)(g(xn)-λAxn)]-PC[αng(u)+(1-αn)(g(u)-λAu)]2αn(F(xn)-g(u))+(1-αn)yn,un-g(u)=12{αn(F(xn)-g(u))+(1-αn)yn2+un-g(u)2-αn(F(xn)-g(u))+(1-αn)yn-un+g(u)2}12{αnF(xn)-g(u)2+(1-αn)g(xn)-g(u)2+un-g(u)2-αn(F(xn)-g(u)-yn)+g(xn)-un-λ(Axn-Au)2}=12{αnF(xn)-g(u)2+(1-αn)g(xn)-g(u)2+un-g(u)2-g(xn)-un2-λ2Axn-Au-αn2F(xn)-g(u)-yn2+2λαnAxn-Au,F(xn)-g(u)-yn+2λg(xn)-un,Axn-Au-2αng(xn)-un,F(xn)-g(u)-yn(1-αn)g(xn)-g(u)2}. It follows that un-g(u)2αnF(xn)-g(u)2+(1-αn)g(xn)-g(u)2-g(xn)-un2+2λαnAxn-AuF(xn)-g(u)-yn+2λg(xn)-unAxn-Au+2αng(xn)-unF(xn)-g(u)-yn. From (3.18) and (3.24), we have g(xn+1)-g(u)2βg(xn)-g(u)2+(1-β)un-g(u)2βg(xn)-g(u)2+(1-β)αnF(xn)-g(u)2+(1-αn)(1-β)g(xn)-g(u)2-(1-β)g(xn)-un2+2λ(1-β)αnAxn-AuF(xn)-g(u)-yn+2λ(1-β)g(xn)-unAxn-Au+2(1-β)αng(xn)-unF(xn)-g(u)-yng(xn)-g(u)2+αnF(xn)-g(u)2-(1-β)g(xn)-un2+2λαnAxn-AuF(xn)-g(u)-yn+2λg(xn)-unAxn-Au+2αng(xn)-unF(xn)-g(u)-yn. Then, we obtain (1-β)g(xn)-un2(g(xn)-g(u)+g(xn+1)-g(u))g(xn+1)-g(xn)+αnF(xn)-g(u)2+2λαnAxn-AuF(xn)-g(u)-yn+2λg(xn)-unAxn-Au+2αng(xn)-unF(xn)-g(u)-yn. Since limnαn=0, limng(xn+1)-g(xn)=0 and limnAxn-Au=0, we have limng(xn)  -un=0. Next, we prove limsupnF(x*)-g(x*),un-g(x*)0 where x* is the unique solution of (3.2). We take a subsequence {uni} of {un} such that limsupnF(x*)-g(x*),un-g(x*)=limiF(x*)-g(x*),uni-g(x*)=limiF(x*)-g(x*),g(xni)-g(x*). Since {xni} is bounded, there exists a subsequence {xnij} of {xni} which converges weakly to some point zC. Without loss of generality, we may assume that xniz. This implies that g(xni)g(z) due to the weak continuity of g. Now, we show zΩ. First, we note that from (3.15) and (3.27) that g(xn)-Sg(xn)0. Hence, limig(xni)-Sg(xni)=0. By the demiclosedness principle of the nonexpansive mapping (see Lemma 2.2), we deduce g(z)Fix(S). Next, we only need to prove zGVI(C,A,g). Set Tv={Av+NC(v),    vC,,vC. By , we know that T is maximal g-monotone. Let (v,w)G(T). Since w-AvNC(v) and xnC, we have g(v)-g(xn),w-Av0. From un=PC[αnF(xn)+(1-αn)(g(xn)-λAxn)], we get g(v)-un,un-[αnF(xn)+(1-αn)(g(xn)-λAxn)]0. It follows that g(v)-un,un-g(xn)λ+Axn-αnλ(F(xn)-g(xn)+λAxn)0. Then, g(v)-g(xni),wg(v)-g(xni),Avg(v)-g(xni),Av-g(v)-uni,uni-g(xni)λ-g(v)-uni,Axni+αniλg(v)-uni,F(xni)-g(xni)+λAxni=g(v)-g(xni),Av-Axni+g(v)-g(xni),-Axni-g(v)-uni,uni-g(xni)λ-g(v)-uni,Axni+αniλg(v)-uni,F(xni)-g(xni)+λAxni-g(v)-uni,uni-g(xni)λ-g(xni)-uni,Axni+αniλg(v)-uni,F(xni)-g(xni)+λAxni. Since g(xni)-uni0 and g(xni)g(z), we deduce that g(v)-g(z),w0 by taking i in (3.33). Thus, zT-10 by the maximal g-monotonicity of T. Hence, zGVI(C,A,g). Therefore, zΩ. From (3.28), we obtain limsupnF(x*)-g(x*),un-g(x*)=limiF(x*)-g(x*),g(xni)-g(x*)=F(x*)-g(x*),g(z)-g(x*)0. We take u=x* in (3.23) to get un-g(x*)2αnF(xn)-g(x*),un-g(x*)+(1-αn)g(xn)-λAxn-(g(x*)-λAx*),un-g(x*)αnF(xn)-F(x*),un-g(x*)+αnF(x*)-g(x*),un-g(x*)+(1-αn)g(xn)-λAxn-(g(x*)-λAx*)un-g(x*)αnLxn-x*un-g(x*)+αnF(x*)-g(x*),un-g(x*)+(1-αn)g(xn)-g(x*)un-g(x*)αn(Lγ)g(xn)-g(x*)un-g(x*)+αnF(x*)-g(x*),un-g(x*)+(1-αn)g(xn)-g(x*)un-g(x*)=[1-(1-Lγ)αn]g(xn)-g(x*)un-g(x*)+αnF(x*)-g(x*),un-g(x*)=1-(1-L/γ)αn2g(xn)-g(x*)2+12un-g(x*)2+αnF(x*)-g(x*),un-g(u). It follows that un-g(x*)2[1-(1-Lγ)αn]g(xn)-g(x*)2+2αnF(x*)-g(x*),un-g(x*). Therefore, g(xn+1)-g(x*)2βg(xn)-g(x*)2+(1-β)un-g(x*)2βg(xn)-g(x*)2+(1-β)[1-(1-Lγ)αn]g(xn)-g(x*)2+2(1-β)αnF(x*)-g(x*),un-g(x*)=[1-(1-Lγ)(1-β)αn]g(xn)-g(x*)2+2(1-β)αnF(x*)-g(x*),un-g(x*)=[1-(1-Lγ)(1-β)αn]g(xn)-g(x*)2+(1-Lγ)(1-β)αn(21-L/γF(x*)-g(x*),un-g(x*))=(1-γn)g(xn)-g(x*)2+δnγn, where γn=(1-L/γ)(1-β)αn and δn=(2/(1-L/γ))F(x*)-g(x*),un-g(x*). From condition (C2), we have nγn=. By (3.34), we have limsupnδn0. We can therefore apply Lemma 2.4 to conclude that g(xn)g(x*) and xnx*. This completes the proof.

Corollary 3.2.

Let C be a nonempty closed and convex subset of a real Hilbert space H. Let F:CH be an L-contraction. Let A:CH be an α-inverse strongly monotone mapping and let S:CC be a nonexpansive mapping. Suppose that Ω. Let β(0,1) and γ(L,2α). For given x0C, let {xn}C be a sequence generated by xn+1=βxn+(1-β)SPC[αnF(xn)+(1-αn)(xn-λAxn)],n0, where {αn}(0,1) satisfies (C1):  limnαn=0 and (C2):  nαn=. Then the sequence {xn} generated by (3.38) converges strongly to x*Ω which is the unique solution of the following variational inequality: F(x*)-x*,x-x*0,xΩ.

Acknowledgments

The authors are very grateful to the referees for their comments and suggestions which improved the presentation of this paper. The first author was partially supported by the Program TH-1-3, Optimization Lean Cycle, of Subprojects TH-1 of Spindle Plan Four in Excellence Teaching and Learning Plan of Cheng Shiu University and was supported in part by NSC 100-2221-E-230-012. The second author was supported in part by Colleges and Universities Science and Technology Development Foundation (20091003) of Tianjin, NSFC 11071279 and NSFC 71161001-G0105.

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