AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation76827210.1155/2012/768272768272Research ArticleIterative Algorithms for General Multivalued Variational InequalitiesYaoYonghong1NoorMuhammad Aslam2, 3LiouYeong-Cheng4KangShin Min5NoorKhalida Inayat1Department of MathematicsTianjin Polytechnic UniversityTianjin 300387Chinatjpu.edu.cn2Mathematics DepartmentCollege of ScienceKing Saud UniversityRiyadh 1145Saudi Arabiaksu.edu.sa3Mathematics DepartmentCOMSATS Institute of Information TechnologyIslamabad 44000Pakistanciit.edu.pk4Department of Information ManagementCheng Shiu UniversityKaohsiung 833Taiwancsu.edu.tw5Department of Mathematics and RINSGyeongsang National UniversityJinju 660-701Republic of Koreagsnu.ac.kr201217112011201221102011011120112012Copyright © 2012 Yonghong Yao 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 introduce and study some new classes of variational inequalities and the Wiener-Hopf equations. Using essentially the projection technique, we establish the equivalence between these problems. This equivalence is used to suggest and analyze some iterative methods for solving the general multivalued variational in equalities in conjunction with nonexpansive mappings. We prove a strong convergence result for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general multivalued variational inequalities under some mild conditions. Several special cases are also discussed.

1. Introduction

Variational inequality problems were initially studied by Stampacchia in 1964. Variational inequalities have applications in diverse disciplines such as partial differential equations, optimal control, optimization, mathematical programming, mechanics, and finance, see  and the references therein. Variational inequalities have been extended and generalized in several directions using novel and innovative techniques. It is a common practice to study these variational inequalities in the setting of convexity. It has been observed that the optimality conditions of the differentiable convex functions can be characterized by the variational inequalities. In recent years, it has been shown that the minimum of the differentiable nonconvex functions can also be characterized by the variational inequalities. Motivated and inspired by these developments, 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 remarkable 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 these algorithms have strong convergence. In this paper, we introduce and consider a new class of variational inequalities, which is called the general multivalued variational inequality. Using essentially the projection technique, we establish the equivalence between the multivalued variational inequalities and the multivalued Wiener-Hopf equations.

Related to the variational inequalities, we have the problem of finding the fixed points of the nonexpansive mappings, which is the subject of current interest in functional analysis. It is natural to consider a unified approach to these two different problems. Noor and Huang  considered the problem of finding the common element of the set of the solutions of variational inequalities and the set of the fixed points of the nonexpansive mappings. We use the Wiener-Hopf technique to suggest and analyze some iterative methods for finding the common element the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the special general variational inequalities. We also consider the convergence criteria of the proposed algorithms under suitable conditions. Several special cases are also discussed.

2. Preliminaries

Let C be a nonempty closed convex subset of a real Hilbert space H. Let A:H2H be a multivalued mapping. Let F,g:HH be two nonlinear operators. We consider the problem of finding uC and wA(u) such thatF(u)+w,g(v)-u0,vH,g(v)C. Inequality of type (2.1) is called the general multivalued variational inequality. We will denote the set of solutions of the special general variational inequality (2.1) by SGVI(F,A,g). The general multivalued variational inequality (2.1) can be written in the following equivalent form, that is, find uC,wA(u) and g(u)C such thatρ(F(u)+w)+u-g(u),g(v)-u0,vH,g(v)C. This equivalent formulation is very important and plays a crucial role in the development of the iterative methods for solving the general multivalued variational inequalities.

We now discuss several special cases.

Special Cases

(A) If w=0, then (2.1) reduces to: find uC such that F(u),g(v)-u0,vH,g(v)C, which is called the general variational inequality, introduced and studied by Noor . It has been shown that the minimum of a class of differentiable functions can be characterized by the general variational inequality of type (2.3).

(B) If gI, the identity operator, then (2.1) reduces to find uC and wA(u) such that F(u)+w,v-u0,vC, which is known as the mildly nonlinear multivalued variational inequality and has been studied extensively.

If F and A are single-valued nonlinear operators, then problem (2.1) is equivalent to finding uC such that F(u)+A(u),v-u0,vC, which is known as the mildly nonlinear variational inequality, the origin of which can be traced back to Noor .

(C) If w=0 and gI, then (2.1) reduces to: find uC such that F(u),v-u0,vC, which is wellknown as the variational inequality, originally introduced and studied by Stampacchia  in 1964. It is clear from the above discussion that general multivalued variational inequality is quite general one. It has been shown that a wide class of problems arising in various discipline of mathematical and engineering sciences can be studied via the general multivalued variational inequalities (2.1) and its special cases.

In the sequel, we need the following well-known lemma.

Lemma 2.1.

For a given zH,uC satisfies the inequality u-z,v-u,vC, if and only if u=PCz, where PC is the projection of H into the closed convex set C.

By using Lemma 2.1, one can prove that the general multivalued variational inequality (2.1) is equivalent to the following fixed point problem.

Lemma 2.2.

uC is a solution of the special general variational inequality (2.1) if and only if uC satisfies the relation u=PC[g(u)-ρ(F(u)+w)], where ρ>0 is a constant.

Related to the general multivalued variational inequality (2.1), we consider the problem of solving the Wiener-Hopf equations. Let F,g:HH be two nonlinear operators and A:H2H be a multi-valued relaxed monotone operator. Let QC=I-gPC, where I is the identity operator. We consider the problem of finding yH:wA(y) such thatFPCy+w+ρ-1QCy=0, which is called the special general multivalued Wiener-Hopf equations. We use SGWH(F,A,g) to denote the set of solutions of the special general multivalued Wiener-Hopf equations. For different and suitable choice of the operators F,A, we can obtain various forms of the Wiener-Hopf equations, which have been studied by Noor , Shi , and others.

Using essentially the technique of Noor [17, 18] and applying Lemma 2.2, one can establish the equivalence between the Wiener-Hopf equations and the general multivalued variational inequalities (2.1). To convey an idea of the technique and for the sake of completeness, we include its proof.

Lemma 2.3.

If uSGVI(F,A,g), then yH and wA(y) satisfy the general Wiener-Hopf equations (2.10), where y=g(u)-ρ(F(u)+w),u=PCy, where ρ>0 is a constant.

Proof.

Let uSGVI(F,A,g). Then, from Lemma 2.2, we have u=PC[g(u)-ρ(F(u)+w)]. Let y=g(u)-ρ(F(u)+w),wA(u). Then, we have u=PCy. Therefore, from (2.13), we obtain y=gPCy-ρ(FPCy+w). It follows that FPCy+w+ρ-1QCy=0,wAPCy, where QC=I-gPC, which is exactly the general Wiener-Hopf equations (2.10). This completes the proof.

Remark 2.4.

Let S:CC be a nonexpansive mapping. If uF(S)SGVI(F,A,g), then one can easy to see u=Su=SPC[g(u)-ρ(F(u)+w)], which is implies that u=(1-αn)u+αnSPC[g(u)-ρ(F(u)+w)], where {αn} is a sequence in (0,1).

Using Remark 2.4 and Lemma 2.3, we can suggest the following algorithm for finding the common element of the solutions set of the variational inequalities and the set of fixed points of a nonexpansive mapping.

Algorithm 2.5.

For a given x0H  arbitrarily, let the sequence {xn} be generated by yn+1=(1-αn)xn+αn[g(xn)-ρ(F(xn)+wn)],xn=SPCyn, where {αn} is a sequence in (0,1) and ρ>0 is some constant.

Note that, if SI, then Algorithm 2.5 reduces to the following iterative method for solving the general variational inequalities.

Algorithm 2.6.

For a given x0H  arbitrarily, let the sequence{xn} be generated by yn+1=(1-αn)xn+αn[g(xn)-ρ(F(xn)+wn)],xn=PCyn, where {αn} is a sequence in (0,1) and ρ>0 is some constant.

If {wn}=0, then Algorithm 2.5 reduces to the following iterative method for solving the general variational inequalities (2.3), which was considered by Noor .

Algorithm 2.7.

For a given x0H  arbitrarily, let the sequence {xn} be generated by yn+1=(1-αn)xn+αn(g(xn)-ρF(xn)),xn=SPCyn, where {αn} is a sequence in (0,1) and ρ>0 is some constant.

If g=SI and {wn}=0, then Algorithm 2.5 reduces to the following iterative method for solving the variational inequalities (2.6).

Algorithm 2.8.

For a given x0H  arbitrarily, let the sequence {xn} be generated by yn+1=(1-αn)xn+αn(xn-ρF(xn)),xn=PCyn, where {αn} is a sequence in (0,1) and ρ>0 is some constant.

We recall the well-known concepts. The multivalued mapping A is said to be γ-Lipschitzian if there exists a constant γ>0 such thatw1-w2γu-v,w1A(u),  w2A(v). Recall that a mapping S:CC is called nonexpansive ifSx-Syx-y,x,yH. We will use F(S) to denote the set of fixed points of S.

A mapping F:CH is called α-strongly monotone if there exists a constant α>0 such thatF(x)-F(y),x-yαx-y2,x,yH, and β-Lipschitz continuous if there exists a constant β>0 such thatF(x)-F(y)βx-y,x,yH.

3. Main Results

Now we state and prove our main result.

Theorem 3.1.

Let C be a nonempty closed convex subset of a real Hilbert space H. Let F:CH be an α-strongly monotone and β-Lipschitz continuous mapping, g:CH an σ-strongly monotone and δ-Lipschitz continuous mapping and A:H2H be a γ-Lipschitz continuous mapping. Let S:CC be a nonexpansive mapping such that F(S)SGVI(F,A,g). Assume that |ρ-α-γ(1-k)β2-γ2|<(α-γ(1-k))2-(β2-γ2)k(2-k)β2-γ2,α>γ(1-k)+(β2-γ2)k(2-k),  γρ<1-k,  k<1, where k=1-2σ+δ2,n=0αn=, then the approximate solution {yn+1} obtained from Algorithm 2.5 converges strongly to ySGWH(F,A,g).

Proof.

Let x*F(S)SGVI(F,A,g). Then, from Remark 2.4, we have x*=SPCy,y=(1-αn)x*+αn[g(x*)-ρ(F(x*)+w)], where yH and wA(y) satisfy the general Wiener-Hopf equations (2.10).

From (2.19) and (3.1), we have xn+1-x*=SPCyn-SPCyyn-y. From (2.19), we have yn+1-y(1-αn)xn-x*+αng(xn)-ρ(F(xn)+wn)-[g(x*)-ρ(F(x*)+w)](1-αn)xn-x*+αnxn-x*-(g(xn)-g(x*))+αnxn-x*-ρ(F(xn)-F(x*))+ραnwn-w. Since g is an σ-strongly monotone and δ-Lipschitz continuous mapping, we have xn-x*-(g(xn)-g(x*))2=xn-x*2-2g(xn)-g(x*),xn-x*+g(xn)-g(x*)2xn-x*2-2σxn-x*2+δ2xn-x*2=(1-2σ+δ2)xn-x*2=k2xn-x*2, where k=1-2σ+δ2.

At the same time, we note that F is an α-strongly monotone and β-Lipschitz continuous mapping, so we have xn-x*-ρ(F(xn)-F(x*))2=xn-x*2-2ρF(xn)-F(x*),xn-x*+ρ2F(xn)-F(x*)2=(1-2ρα+ρ2β2)xn-x*2. From (3.5)–(3.7), we have yn+1-y(1-αn)xn-x*+αn(k+ργ+1-2ρα+ρ2β2)xn-x*=(1-αn)xn-x*+αnθxn-x*, where θ=k+γρ+1-2ρα+ρ2β2. Using (3.1), we see that θ<1. Substituting (3.4) into (3.8), we have yn+1-y[(1-αn)+(k+γρ+1-2ρα+ρ2β2)αn]yn-y=[1-(1-θ)αn]xn-x*i=0n[1-(1-θ)αi]y0-y. Since n=0αn diverges and 1-θ>0, we have i=0n[1-(1-θ)αi]=0. Consequently, the sequence {yn} converges strongly to y in H, the required result.

4. Conclusion

One of the most difficult and important problems in variational inequalities is the development of an efficient numerical methods. One of the technique is called the projection method and its variant forms. Projection method represent an important tool for finding the approximate solution of various types of variational inequalities. The projection type methods were developed in 1970s. The main idea in this techniques is to establish the equivalence between the variational inequalities and the fixed point problem using the concept of projection. These methods have been extended and modified in various ways. Shi  considered the problem of solving a system of nonlinear projections, which are called the Wiener-Hopf equations. It has been shown by Shi  that the Wiener-Hopf equations are equivalent to the variational inequalities. It turns out that this alternative formulation is more general and flexible. It has been shown that the Wiener-Hopf equations provide us a simple, natural, elegant, and convenient device to develop some efficient numerical methods for solving variational and complementarity problems. In this paper, we introduce and study some new classes of variational inequalities and Wiener-Hopf equations. Using essentially the projection technique, we establish the equivalence between these problems. This equivalence is used to suggest and analyze some iterative methods for solving the general multivalued variational inequalities in conjunction with nonexpansive mappings. We prove a strong convergence result for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general multivalued variational inequalities under some mild conditions. Several special cases are also discussed. The ideas and techniques of this paper may be a starting point for a wide range of novel and innovative applications in various fields.

Acknowledgments

The authors thank the referees for useful comments and suggestions. The research of Professor M. Aslam Noor is supported by the Visiting Professor Program of King Saud University, Riyadh, Saudi Arabia, and Research Grant: KSU.VPP.108.

CengL. C.AnsariQ. H.YaoJ. C.Relaxed extragradient iterative methods for variational inequalitiesApplied Mathematics and Computation201121831112112310.1016/j.amc.2011.01.061CengL. C.TeboulleM.YaoJ. C.Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed-point problemsJournal of Optimization Theory and Applications20101461193110.1007/s10957-010-9650-02657821ZBL1222.47091ChangS. S.LeeH. W. J.ChanC. K.LiuJ. A.A new method for solving a system of generalized nonlinear variational inequalities in Banach spacesApplied Mathematics and Computation2011217166830683710.1016/j.amc.2011.01.0212775674ZBL1215.65119CianciarusoF.MarinoG.MugliaL.YaoY.On a two-step algorithm for hierarchical fixed point problems and variational inequalitiesJournal of Inequalities and Applications200920091320869210.1155/2009/2086922551747ZBL1180.47040FacchineiF.PangJ. S.Finite-Dimensional Variational Inequalities and Complementarity Problems20031-2New York, NY, USASpringerSpringer Series in Operations ResearchGlowinskiR.Numerical Methods for Nonlinear Variational Problems1984New YorkSpringerHeB. S.A new method for a class of linear variational inequalitiesMathematical Programming199466213714410.1007/BF015811411297058ZBL0813.49009IusemA. N.SvaiterB. F.A variant of Korpelevich's method for variational inequalities with a new search strategyOptimization199742430932110.1080/023319397088443651609571ZBL0891.90135JailletP.LambertonD.LapeyreB.Variational inequalities and the pricing of American optionsActa Applicandae Mathematicae199021326328910.1007/BF000472111096582ZBL0714.90004KhobotovE. N.Modification of the extra-gradient method for solving variational inequalities and certain optimization problemsUSSR Computational Mathematics and Mathematical Physics1987275120127KorpelevichG. M.An extragradient method for finding saddle points and for other problemsEkonomika i Matematicheskie Metody19761247477560451121KumamP.PetrotN.WangkeereeR.Existence and iterative approximation of solutions of generalized mixed quasi-variational-like inequality problem in Banach spacesApplied Mathematics and Computation2011217187496750310.1016/j.amc.2011.02.0542784596LuX.XuH. K.YinX.Hybrid methods for a class of monotone variational inequalitiesNonlinear Analysis. Theory, Methods & Applications2009713-41032104110.1016/j.na.2008.11.0672527522ZBL1176.90462MarinoG.MugliaL.YaoY.Viscosity methods for common solutions of equilibrium and variational inequality problems via multi-step iterative algorithms and common fixed pointsNonlinear Analysis, Theory, Methods and Applications. In press10.1016/j.na.2011.09.019NoorM. A.On Variational Inequalities1975London, UKBrunel UniversityNoorM. A.General variational inequalitiesApplied Mathematics Letters19881211912210.1016/0893-9659(88)90054-7953368ZBL0655.49005NoorM. A.Wiener-Hopf equations and variational inequalitiesJournal of Optimization Theory and Applications199379119720610.1007/BF009418941246503ZBL0799.49010NoorM. A.Some developments in general variational inequalitiesApplied Mathematics and Computation2004152119927710.1016/S0096-3003(03)00558-72050063ZBL1134.49304NoorM. A.Differentiable non-convex functions and general variational inequalitiesApplied Mathematics and Computation2008199262363010.1016/j.amc.2007.10.0232420590ZBL1147.65047NoorM. A.Al-SaidE. A.Wiener-Hopf equations technique for quasimonotone variational inequalitiesJournal of Optimization Theory and Applications1999103370571410.1023/A:10217963268311727246ZBL0953.65050NoorM. A.HuangZ.Wiener-Hopf equation technique for variational inequalities and nonexpansive mappingsApplied Mathematics and Computation2007191250451010.1016/j.amc.2007.02.1172385552ZBL1193.49009ShiP.Equivalence of variational inequalities with Wiener-Hopf equationsProceedings of the American Mathematical Society19911112339346103722410.1090/S0002-9939-1991-1037224-3ZBL0881.35049SolodovM. V.SvaiterB. F.A new projection method for variational inequality problemsSIAM Journal on Control and Optimization199937376577610.1137/S03630129973174751675086ZBL0959.49007StampacchiaG.Formes bilineaires coercitives sur les ensembles convexesComptes Rendus de l'Academie des Sciences1964258441344160166591ZBL0124.06401VermaR. U.Projection methods, algorithms, and a new system of nonlinear variational inequalitiesComputers & Mathematics with Applications2001417-81025103110.1016/S0898-1221(00)00336-91826902ZBL0995.47042XuH. K.KimT. H.Convergence of hybrid steepest-descent methods for variational inequalitiesJournal of Optimization Theory and Applications2003119118520110.1023/B:JOTA.0000005048.79379.b62028445ZBL1045.49018YaoJ. C.Variational inequalities with generalized monotone operatorsMathematics of Operations Research199419369170510.1287/moor.19.3.6911288894ZBL0813.49010YaoY.ChenR.XuH. K.Schemes for finding minimum-norm solutions of variational inequalitiesNonlinear Analysis. Theory, Methods & Applications2010727-83447345610.1016/j.na.2009.12.0292587377ZBL1183.49012YaoY.LiouY. C.KangS. M.Two-step projection methods for a system of variational inequality problems in Banach spacesJournal of Global Optimization. In press10.1007/s10898-011-9804-0YaoY.NoorM. A.LiouY. C.Strong convergence of a modified extra-gradient method to the 4 minimum-norm solution of variational inequalitiesAbstract and Applied Analysis20122012981743610.1155/2012/817436YaoY.NoorM. A.NoorK. I.LiouY. C.YaqoobH.Modified extragradient methods for a system of variational inequalities in Banach spacesActa Applicandae Mathematicae201011031211122410.1007/s10440-009-9502-92639166ZBL1192.47065YaoY.ShahzadN.New methods with perturbations for non-expansive mappings in Hilbert 5 spacesFixed Point Theory and Applications201120117910.1186/1687-1812-2011-79YaoY.ShahzadN.Strong convergence of a proximal point algorithm with general errorsOptimization Letters. In press10.1007/s11590-011-0286-2