In this paper, we prove that the general quasi variational inequalities are equivalent to the fixed point problem using the projection technique. We use this equivalent formulation to study the sensitivity analysis of the general quasi variational inequality. Our approach is very simple. Several special cases are also discussed

Khalida Inayat Noor

Muhammad Aslam Noor

Themistocles M. Rassias

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MATHEMATICAL COMMUNICATIONS 205Math. Commun., Vol. 15, No. 1, pp. 205-212 (2010)Parametric general quasi variational inequalitiesMuhammad Aslam Noor1,∗, Khalida Inayat Noor2 and Themistocles M.Rassias31 Mathematics Department, College of Science, King Saud University, Riyadh 11 451,Saudi Arabia2 Mathematics Department, COMSATS Institute of Information Technology, Islamabad,Pakistan3 Mathematics Department, National Technical University of Athens, Zografou Campus,Athens 15 780, GreeceReceived June 27, 2009; accepted January 4, 2010Abstract. In this paper, we prove that the general quasi variational inequalities are equiv-alent to the fixed point problem using the projection technique. We use this equivalentformulation to study the sensitivity analysis of the general quasi variational inequality. Ourapproach is very simple. Several special cases are also discussed.AMS subject classifications: 49J40, 90C33Key words: variational inequalities, sensitivity analysis, fixed-point problem1. IntroductionThe ideas and techniques of variational inequalities are being applied in a varietyof diverse areas of pure and applied sciences, and proved to be innovative and pro-ductive. It has been shown that variational inequalities provide a novel, natural,simple, unified and efficient framework for a general treatment of a wide class ofunrelated linear and nonlinear problems. This theory combines theoretical and al-gorithmic advances with a new and novel domain of applications. Analysis of theseproblems requires a blend of techniques from convex analysis, functional analysisand numerical analysis, see [1-35] and the references therein.In recent years, variational inequalities have been generalized and extended inseveral directions using new and novel techniques. Noor [25] has introduced andconsidered a new class of variational inequalities which is called the general quasivariational inequality. In this paper, we study sensitivity analysis of general quasivariational inequalities, that is, examining how the solutions of such problems changewhen the data of the problems are changed. This is an important problem for severalreasons. We would like to mention that sensitivity analysis provides useful informa-tion for designing or planning various equilibrium systems. Sensitivity analysis canprovide a new insight and can stimulate new ideas and techniques for problem solv-ing. Dafermos [4] studied sensitivity analysis of variational inequalities using the∗Corresponding author. Email addresses: noormaslam@hotmail.com (M.A.Noor),khalidanoor@hotmail.com (K. I. Noor), trassias@math.ntua.gr (Th.M. Rassias)http://www.mathos.hr/mc c©2010 Department of Mathematics, University of Osijek206 M.A.Noor, K. I. Noor and Th.M.Rassiasfixed point technique. This technique has been modified and extended by many au-thors, see Noor-Noor and Rassias [27]. This approach has strong geometrical flavour.We extend this for general quasi variational inequalities. We show that general quasivariational inequalities are equivalent to the fixed point problem. This alternativeequivalent form is used to consider sensitivity analysis of general quasi variationalinequalities without assuming the differentiability of the given data. Our analysisis in the spirit of Dafermos [5] and Noor [18,19]. Since general quasi variational in-equalities include general variational inequalities, quasi variational inequalities andcomplementarity problems as special cases, our result continues to hold for theseproblems.2. PreliminariesLet H be a real Hilbert space whose inner product and norm are denoted by 〈·, ·〉and ‖.‖, respectively. Let K : H −→ H be a point to set mapping, which is closedand convex valued. In other words, for every u ∈ H, the set K(u) is closed andconvex.For given nonlinear operators T, g : H → H, consider the problem of findingu ∈ H : g(u) ∈ K(u) such that〈ρTu + u− g(u), g(v)− u〉 ≥ 0, ∀v ∈ H : g(v) ∈ K(u), (1)where ρ > 0 is a constant. Inequality of type (1) is called a general quasi variationalinequality involving two operators, which was introduced and studied by Noor [25].This class of quasi variational inequalities is a quite general and unified one.If K(u) ≡ K, that is, the convex set K(u) is independent of the solution u, thengeneral quasi variational inequalities (1) are equivalent to finding u ∈ H : g(u) ∈K(u) such that〈Tu + u− g(u), g(v)− u〉 ≥ 0, ∀v ∈ H : g(v) ∈ K, (2)which is called a general variational inequality involving two operators and wasintroduced and studied by Noor [22]. It has been shown [22] that the minimum of anonconvex differentiable function on a nonconvex set K in H can be characterizedby a general variational inequality (2).Noor [14] considered and studied the following quasi variational inequality offinding u ∈ H : g(u) ∈ K(u) such that〈ρTu + g(u)− g(u), v − g(u)〉 ≥ 0, ∀v ∈ K(u). (3)Obviously problems (1) and (3) are quite different and have applications in pure andapplied sciences.For g = I, the identity operator, the general quasi variational inequality (1) isequivalent to finding u ∈ K(u) such that〈Tu, v − u〉 ≥ 0, ∀v ∈ K(u), (4)which is known as a classical quasi variational inequality, introduced and studied byBensoussan and Lions [3] in the study of impulse control theory. See also [1-32].Parametric quasi variational inequalities 207If K(u) ≡ K, and g = I, the identity operator, then problem (1) is equivalent tofinding u ∈ K such that〈Tu, v − u〉 ≥ 0, ∀v ∈ K, (5)which is known as a classical variational inequality which was introduced in 1964 byStampacchia [32]. For recent applications, numerical methods, sensitivity analysis,dynamical systems and formulation of variational inequalities, see [1-35] and thereferences therein. In brief, we conclude that the general quasi variational inequality(1) is quite general and includes several classes of variational inequalities and relatedoptimization problems as special cases.We also need the following standard and classical result.Lemma 1 (see [13]). Let K(u) be a closed and convex set in H. Then, for a givenz ∈ H, u ∈ K(u) satisfies the inequality〈u− z, v − u〉 ≥ 0, ∀v ∈ K(u),if and only ifu = PK(u)z,where PK(u) is the projection of H onto the closed convex set K(u) in H.We would like to point out that the implicit projection operator PK(u) is notnonexpansive. We shall assume that the implicit projection operator PK(u) satisfiesthe Lipschitz type continuity, which plays an important and fundamental role in theexistence theory and in developing numerical methods for solving quasi variationalinequalities.Assumption 1. The implicit projection operator PK(u) satisfies the condition‖PK(u)w − PK(v)w|‖ ≤ ν‖u− v||, ∀u, v, w ∈ H,where ν > 0 is a positive constant.Assumption 1 has been used to prove the existence of a solution of quasi varia-tional inequalities as well as in analyzing convergence of the iterative methods, see[18, 19, 20].In many important applications [2, 3, 19] the convex-valued set K(u) can bewritten asK(u) = m(u) + K, (6)where m(u) is a point-point mapping and K is a convex set. In this case, we havePK(u)w = Pm(u)+K(w) = m(u) + PK [w −m(u)], ∀u, v ∈ H.We note that if K(u) is defined by (6) and m(u) is a Lipschitz continuous mappingwith constant γ > 0, then‖PK(u)w − PK(v)w‖ = ‖m(u)−m(v) + PK [w −m(u)]− PK [w −m(v)‖≤ 2‖m(u)−m(v)‖ ≤ 2γ‖u− v‖, ∀u, v, w ∈ H,208 M.A.Noor, K. I. Noor and Th.M.Rassiaswhich shows that Assumption 1 holds with ν = 2γ.We now consider parametric versions of problem (1). To formulate the problem,let M be an open subset of H in which the parameter λ takes values. Let T (u, λ)be the given operator defined on H ×H ×M and take value in H ×H. From nowonward, we denote Tλ(.) ≡ T (., λ) unless otherwise specified.The parametric general variational inequality problem is to find (u, λ) ∈ H ×Msuch that〈ρTλu + u− g(u), g(v)− u〉 ≥ 0, ∀v ∈ H : g(v) ∈ K. (7)We also assume that for some λ ∈ M , problem (7) has a unique solution u.From Lemma 1, we see that parametric general quasi variational inequalities areequivalent to the fixed point problem:Fλ(u) ≡ u = PK [g(u)− ρTλu], ∀(u, λ) ∈ X ×M. (8)We use this equivalence to study sensitivity analysis of general quasi variationalinequalities. We assume that for some λ ∈ M , problem (7) has a solution u and Xis a closure of a ball in H centered at u. We want to investigate those conditionsunder which, for each λ in a neighborhood of λ, problem (7) has a unique solutionz(λ) near u and the function u(λ) is (Lipschitz) continuous and differentiable.Definition 1. Let Tλ(.) be an operator on X ×M . Then, the operator Tλ(.) is saidto be:(a) Locally strongly monotone, if there exists a constant α > 0 such that〈Tλ(u)− Tλ(v), u− v〉 ≥ α‖u− v‖2, ∀λ ∈ M, u, v ∈ X.(b) Locally Lipschitz continuous, if there exists a constant β > 0 such that‖Tλ(u)− Tλ(v)‖ ≤ β‖u− v‖, ∀λ ∈ M, u, v ∈ X.3. Main resultsWe consider the case when the solutions of the parametric general quasi variationalinequality (8) lie in the interior of X. Following the ideas of Dafermos [5] and Noor[18,19], we consider the map Fλ(u) as defined by (8). We have to show that themap Fλ(u) has a fixed point, which is a solution of the parametric general quasivariational inequality (7). First of all, we prove that the map Fλ(u), defined by (8),is a contraction map with respect to z uniformly in λ ∈ M .Lemma 2. Let Tλ(.) be a locally strongly monotone with constant α > 0 and locallyLipschitz continuous with constant β > 0. Let the operator g be strongly monotonewith constants σ > 0 and Lipschitz continuous with constants δ > 0, respectively. IfAssumption 1 holds and for all u1, u2 ∈ X and λ ∈ M , we have‖Fλ(u1)− Fλ(u2)‖ ≤ θ‖u1 − u2‖,Parametric quasi variational inequalities 209whereθ ={√1− 2σ + δ2 + ν +√1− 2αρ + β2ρ2}(9)for|ρ− αβ2| <√α2 − β2k(2− k)β2, α > β√k(2− k), k < 1, (10)wherek =√1− 2σ + δ2 + ν. (11)Proof. In order to prove the existence of a solution of (7), it is enough to show thatthe mapping Fλ(u), defined by (8), is a contraction mapping.For u1 6= u2 ∈ H, and using Assumption 1, we have‖Fλ(u1)− Fλ(u2)‖ = ‖PK(u1)[g(u1)− ρTλu1]− PK(u2)[g(u2)− ρTλu2]‖≤ ‖PK(u1)[g(u1)− ρTλu1]− PK(u2)[g(u1)− ρTλu1]‖+‖PK(u2)[g(u1)− ρTλu1]− PK(u2)[g(u2)− ρTλu2]‖≤ ν‖u1 − u2‖+ ‖g(u1)− g(u2)− ρ(Tλu1 − Tλu2)‖= ν‖u1 − u2‖+‖ − (u1− u2− (g(u1)− g(u2)))+ u1 − u2 − ρ(Tλu1 − Tλu2)‖≤ ν‖u1 − u2‖+ ‖u1 − u2 − (g(u1)− g(u2))‖+‖u1 − u2 − ρ(Tλu1 − Tλu2)‖. (12)Since the operator Tλ is a locally strongly monotone with constant α > 0 and locallyLipschitz continuous with constant β > 0, it follows that||u1 − u2 − ρ(Tλu1 − Tλu2)||2 ≤ ||u1 − u2||2 − 2ρ〈Tλu1 − Tλu2, u1 − u2〉+ρ2||Tλu1 − Tλu2||2≤ (1− 2ρα + ρ2β2)||u1 − u2||2. (13)In a similar way, we have||u1 − u2 − (g(u1)− g(u2))||2 ≤ (1− 2σ + δ2)||u1 − u2||2, (14)where we have used the fact that g is strongly monotone with constant σ > 0 andLipschitz continuous with constant δ > 0.From (9), (11), (13) and (14), we have‖Fλ(u1)− Fλ(u2)‖ ≤{ν +√(1− 2σ + δ2) +√(1− 2ρα + ρ2β2)}‖u1 − u2‖={k +√(1− 2ρα + ρ2β2)}‖u1 − u2‖= θ‖u1 − u2‖,210 M.A.Noor, K. I. Noor and Th.M.Rassiaswhereθ = k +√(1− 2ρα + ρ2β2). (15)From (10), it follows that θ < 1. Thus it follows that the mapping Fλ(u), defined by(8), is a contraction mapping and consequently it has a fixed point, which belongs toK(u) satisfying the general quasi variational inequality (7), the required result.Remark 1. From Lemma 2, we see that the map Fλ(z) defined by (8) has a uniquefixed point u(λ), that is, u(λ) = Fλ(u). Also, by assumption, the function u, forλ = λ is a solution of the parametric general quasi variational inequality (7). Againusing Lemma 2, we see that u, for λ = λ, is a fixed point of Fλ(u) and it is also afixed point of Fλ(u). Consequently, we conclude thatu(λ) = u = Fλ(u(λ)).Using Lemma 2, we can prove the continuity of the solution u(λ) of the parametricgeneral quasi variational inequality (7) using the technique of Noor [13,14]. However,for the sake of completeness and to convey an idea of the techniques involved, wegive its proof.Lemma 3. Assume that the operator Tλ(.) is locally Lipschitz continuous with re-spect to the parameter λ. If the operator Tλ(.) is locally Lipschitz continuous andthe map λ → PKλu is continuous (or Lipschitz continuous), then the function u(λ)satisfying (8) is (Lipschitz) continuous at λ = λ.Proof. For all λ ∈ M, invoking Lemma 2 and the triangle inequality, we have‖u(λ)− u(λ̄)‖ ≤ ‖Fλ(u(λ))− Fλ̄(u(λ̄)‖+ ‖Fλ(u(λ̄))− Fλ̄(u(λ̄))‖≤ θ‖u(λ)− u(λ̄)‖+ ‖Fλ(u(λ̄))− Fλ̄(u(λ̄))‖. (16)From (8) and the fact that the operator Tλ is Lipschitz continuous with respect tothe parameter λ, we have‖Fλ(u(λ̄))− Fλ̄(u(λ̄))‖ = ‖u(λ̄)− u(λ̄) + ρ(Tλ(u(λ̄), u(λ̄))− Tλ̄(u(λ̄), u(λ̄)))‖≤ ρµ‖λ− λ̄‖. (17)Combining (16) and (17), we obtain‖u(λ)− u(λ̄)‖ ≤ ρµ1− θ‖λ− λ̄‖, for all λ, λ̄ ∈ M,from which the required result follows.We now state and prove the main result of this paper which is the motivation ofour next result.Theorem 1. Let u be the solution of the parametric general quasi variational in-equality (7) for λ = λ. Let Tλ(u) be a locally strongly monotone Lipschitz continuousoperator for all u, v ∈ X. If the map λ → PK is (Lipschitz) continuous at λ = λ,then there exists a neighborhood N ⊂ M of λ such that for λ ∈ N , the parametricgeneral quasi variational inequality (7) has a unique solution u(λ) in the interior ofX,u(λ) = u and u(λ) is (Lipschitz) continuous at λ = λ.Parametric quasi variational inequalities 211Proof. Its proof follows from Lemmas 2, 3 and Remark 1.AcknowledgementThe authors would like to thank Dr. S. M. Junaid Zaidi, Rector, CIIT, for providingexcellent research facilities.References[1] R.P.Agarwal, J. J. Cho, N. J.Huang, Sensitivity analysis for strongly nonlinearquasi variational inclusions, Appl. Math. Letters 13(2000), 19–24.[2] C.Baiocchi, A.Capelo, Variational and Quasi Variational Inequalities, J. Wiley &Sons, London, 1984.[3] A.Bejancu, Geometry of CR-Submanifolds, Kluwer Academic Publishers, Dortrecht,1986.[4] A.Bensoussan, J. L. Lions, Applications des Inequations Variationelles en Controlet en Stochastiques, Dunod, Paris, 1978.[5] S.Dafermos, Sensitivity analysis in variational inequalities, Math. Opers. Res.13(1988), 421–434.[6] F.Giannessi, A.Maugeri, Variational Inequalities and Network Equilibrium Prob-lems, Plenum Press, New York, 1995.[7] D.Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities andTheir Applications, Academic Press, London, 1980.[8] A. S.Kravchuk, P. J.Neittaanmaki, Variational and Quasi Variational Inequali-ties in Mechanics, Springer, Dordrecht, 2007.[9] J.Kyparisis, Sensitivity analysis framework for variational inequalities, Math. Progr.38(1987), 203–213.[10] J.Kyparisis, Sensitivity analysis for variational inequalities and nonlinear comple-mentarity problems, Annals. Opers. Res. 27(1990), 143–174.[11] J. Liu, Sensitivity analysis in nonlinear programs and variational inequalities via con-tinuous selections, SIAM J. Control Optim. 33(1995), 1040–1068.[12] A.Moudafi, M.Aslam Noor, Sensitivity analysis for variational inclusions byWiener-Hopf equations technique, J. Appl. Math. Stoch. Anal. 12(1999), 223–232.[13] M.A.Noor, An iterative scheme for a class of quasi variational inequalities, J. Math.Anal. Appl. 110(1985), 463–468.[14] M.A.Noor, General variational inequalities, Appl. Math. Lett. 1(1988), 119–121.[15] M.A.Noor, Quasi variational inequalities, Appl. Math. Lett. 1(1988), 367–370.[16] M.A.Noor, Some recent advances in variational inequalities, Part. I, basic concepts,New Zealand J. Math. 26(1997), 53–80.[17] M.A.Noor, Some recent advances in variational inequalities, Part II, other concepts,New Zealand J. Math. 26(1997), 229–255.[18] M.A.Noor, Generalized multivalued quasi variational inequalities (II), Comput.Math. Appl. 35(1998), 63–78.[19] M.A.Noor, Sensitivity analysis for quasi variational inequalities, J. Optim. TheoryAppl. 95(1997), 399–407.[20] M.A.Noor, Some developments in general variational inequalities, Appl. Math.Comput. 152(2004), 199–277.[21] M.A.Noor, On merit functions for quasi variational inequalities, J. Math. Inequal.1(2007), 259–268.212 M.A.Noor, K. I. Noor and Th.M.Rassias[22] M.A.Noor, Existence results for quasi variational inequalities, Banach J. Math.Anal. 1(2007), 186–194.[23] M.A.Noor, Differentiable non-convex functions and general variational inequalities,Appl. Math. Comput. 199(2008), 623–630.[24] M.A.Noor, Implicit Wiener-Hopf equations and quasi variational inequalities2(2008), 15–25.[25] M.A.Noor, Principles of Variational Inequalities, Lap-Lambert Academic Publish-ing Ag& Co, Saarbrücken, 2009.[26] M.A.Noor, K. Inayat Noor, Sensitivity analysis for quasi variational inclusions,J. Math. Anal. Appl. 236(1999), 290–299.[27] M.A.Noor, K. Inayat Noor, Th.M.Rassias, Some aspects of variational inequal-ities, J. Comput. Appl. Math. 47(1993), 285–312.[28] M.A.Noor, K. Inayat Noor, Th.M.Rassias, Set-valued resolvent equations andmixed variational inequalities, J. Math. Anal. Appl. 220(1998), 741–759.[29] M.A.Noor, Th.M.Rassias, Resolvent equations for set-valued mixed variationalinequalities, Nonlinear Anal. 42(2000), 71–83.[30] M.A.Noor, Th.M.Rassias, A class of projection methods for general variationalinequalities, J. Math. Anal. Appl. 268(2002), 334–343.[31] Y.Qiu, T. L.Magnanti, Sensitivity analysis for variational inequalities defined onpolyhedral sets, Math. Oper. Res. 14(1989), 410–432.[32] G. Stampacchia, Formes bilineaires coercivities sur les ensembles convexes, C. R.Acad. Sci. Paris 258(1966), 4413–4416.[33] R.L.Tobin, Sensitivity analysis for variational inequalities, J. Optim. Theory Appl.48(1986), 191–204.[34] N.D.Yen, Holder continuity of solutions to a parametric variational inequality, Appl.Math. Optim. 31(1995), 245–255.[35] N.D.Yen, G.M. Lee, Solution sensitivity of a class of variational inequalities, J.Math. Anal. 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Parametric general quasi variational inequalities

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MATHEMATICAL COMMUNICATIONS 205Math. Commun., Vol. 15, No. 1, pp. 205-212 (2010)Parametric general quasi variational inequalitiesMuhammad Aslam Noor1,∗, Khalida Inayat Noor2 and Themistocles M.Rassias31 Mathematics Department, College of Science, King Saud University, Riyadh 11 451,Saudi Arabia2 Mathematics Department, COMSATS Institute of Information Technology, Islamabad,Pakistan3 Mathematics Department, National Technical University of Athens, Zografou Campus,Athens 15 780, GreeceReceived June 27, 2009; accepted January 4, 2010Abstract. In this paper, we prove that the general quasi variational inequalities are equiv-alent to the fixed point problem using the projection technique. We use this equivalentformulation to study the sensitivity analysis of the general quasi variational inequality. Ourapproach is very simple. Several special cases are also discussed.AMS subject classifications: 49J40, 90C33Key words: variational inequalities, sensitivity analysis, fixed-point problem1. IntroductionThe ideas and techniques of variational inequalities are being applied in a varietyof diverse areas of pure and applied sciences, and proved to be innovative and pro-ductive. It has been shown that variational inequalities provide a novel, natural,simple, unified and efficient framework for a general treatment of a wide class ofunrelated linear and nonlinear problems. This theory combines theoretical and al-gorithmic advances with a new and novel domain of applications. Analysis of theseproblems requires a blend of techniques from convex analysis, functional analysisand numerical analysis, see [1-35] and the references therein.In recent years, variational inequalities have been generalized and extended inseveral directions using new and novel techniques. Noor [25] has introduced andconsidered a new class of variational inequalities which is called the general quasivariational inequality. In this paper, we study sensitivity analysis of general quasivariational inequalities, that is, examining how the solutions of such problems changewhen the data of the problems are changed. This is an important problem for severalreasons. We would like to mention that sensitivity analysis provides useful informa-tion for designing or planning various equilibrium systems. Sensitivity analysis canprovide a new insight and can stimulate new ideas and techniques for problem solv-ing. Dafermos [4] studied sensitivity analysis of variational inequalities using the∗Corresponding author. Email addresses: noormaslam@hotmail.com (M.A.Noor),khalidanoor@hotmail.com (K. I. Noor), trassias@math.ntua.gr (Th.M.Rassias)http://www.mathos.hr/mc c©2010 Department of Mathematics, University of Osijek206 M.A.Noor, K. I. Noor and Th.M.Rassiasfixed point technique. This technique has been modified and extended by many au-thors, see Noor-Noor and Rassias [27]. This approach has strong geometrical flavour.We extend this for general quasi variational inequalities. We show that general quasivariational inequalities are equivalent to the fixed point problem. This alternativeequivalent form is used to consider sensitivity analysis of general quasi variationalinequalities without assuming the differentiability of the given data. Our analysisis in the spirit of Dafermos [5] and Noor [18,19]. Since general quasi variational in-equalities include general variational inequalities, quasi variational inequalities andcomplementarity problems as special cases, our result continues to hold for theseproblems.2. PreliminariesLet H be a real Hilbert space whose inner product and norm are denoted by 〈·, ·〉and ‖.‖, respectively. Let K : H −→ H be a point to set mapping, which is closedand convex valued. In other words, for every u ∈ H, the set K(u) is closed andconvex.For given nonlinear operators T, g : H → H, consider the problem of findingu ∈ H : g(u) ∈ K(u) such that〈ρTu+ u− g(u), g(v)− u〉 ≥ 0, ∀v ∈ H : g(v) ∈ K(u), (1)where ρ > 0 is a constant. Inequality of type (1) is called a general quasi variationalinequality involving two operators, which was introduced and studied by Noor [25].This class of quasi variational inequalities is a quite general and unified one.If K(u) ≡ K, that is, the convex set K(u) is independent of the solution u, thengeneral quasi variational inequalities (1) are equivalent to finding u ∈ H : g(u) ∈K(u) such that〈Tu+ u− g(u), g(v)− u〉 ≥ 0, ∀v ∈ H : g(v) ∈ K, (2)which is called a general variational inequality involving two operators and wasintroduced and studied by Noor [22]. It has been shown [22] that the minimum of anonconvex differentiable function on a nonconvex set K in H can be characterizedby a general variational inequality (2).Noor [14] considered and studied the following quasi variational inequality offinding u ∈ H : g(u) ∈ K(u) such that〈ρTu+ g(u)− g(u), v − g(u)〉 ≥ 0, ∀v ∈ K(u). (3)Obviously problems (1) and (3) are quite different and have applications in pure andapplied sciences.For g = I, the identity operator, the general quasi variational inequality (1) isequivalent to finding u ∈ K(u) such that〈Tu, v − u〉 ≥ 0, ∀v ∈ K(u), (4)which is known as a classical quasi variational inequality, introduced and studied byBensoussan and Lions [3] in the study of impulse control theory. See also [1-32].Parametric quasi variational inequalities 207If K(u) ≡ K, and g = I, the identity operator, then problem (1) is equivalent tofinding u ∈ K such that〈Tu, v − u〉 ≥ 0, ∀v ∈ K, (5)which is known as a classical variational inequality which was introduced in 1964 byStampacchia [32]. For recent applications, numerical methods, sensitivity analysis,dynamical systems and formulation of variational inequalities, see [1-35] and thereferences therein. In brief, we conclude that the general quasi variational inequality(1) is quite general and includes several classes of variational inequalities and relatedoptimization problems as special cases.We also need the following standard and classical result.Lemma 1 (see [13]). Let K(u) be a closed and convex set in H. Then, for a givenz ∈ H, u ∈ K(u) satisfies the inequality〈u− z, v − u〉 ≥ 0, ∀v ∈ K(u),if and only ifu = PK(u)z,where PK(u) is the projection of H onto the closed convex set K(u) in H.We would like to point out that the implicit projection operator PK(u) is notnonexpansive. We shall assume that the implicit projection operator PK(u) satisfiesthe Lipschitz type continuity, which plays an important and fundamental role in theexistence theory and in developing numerical methods for solving quasi variationalinequalities.Assumption 1. The implicit projection operator PK(u) satisfies the condition‖PK(u)w − PK(v)w|‖ ≤ ν‖u− v||, ∀u, v, w ∈ H,where ν > 0 is a positive constant.Assumption 1 has been used to prove the existence of a solution of quasi varia-tional inequalities as well as in analyzing convergence of the iterative methods, see[18, 19, 20].In many important applications [2, 3, 19] the convex-valued set K(u) can bewritten asK(u) = m(u) +K, (6)where m(u) is a point-point mapping and K is a convex set. In this case, we havePK(u)w = Pm(u)+K(w) = m(u) + PK [w −m(u)], ∀u, v ∈ H.We note that if K(u) is defined by (6) and m(u) is a Lipschitz continuous mappingwith constant γ > 0, then‖PK(u)w − PK(v)w‖ = ‖m(u)−m(v) + PK [w −m(u)]− PK [w −m(v)‖≤ 2‖m(u)−m(v)‖ ≤ 2γ‖u− v‖, ∀u, v, w ∈ H,208 M.A.Noor, K. I. Noor and Th.M.Rassiaswhich shows that Assumption 1 holds with ν = 2γ.We now consider parametric versions of problem (1). To formulate the problem,let M be an open subset of H in which the parameter λ takes values. Let T (u, λ)be the given operator defined on H ×H ×M and take value in H ×H. From nowonward, we denote Tλ(.) ≡ T (., λ) unless otherwise specified.The parametric general variational inequality problem is to find (u, λ) ∈ H ×Msuch that〈ρTλu+ u− g(u), g(v)− u〉 ≥ 0, ∀v ∈ H : g(v) ∈ K. (7)We also assume that for some λ ∈M , problem (7) has a unique solution u.From Lemma 1, we see that parametric general quasi variational inequalities areequivalent to the fixed point problem:Fλ(u) ≡ u = PK [g(u)− ρTλu], ∀(u, λ) ∈ X ×M. (8)We use this equivalence to study sensitivity analysis of general quasi variationalinequalities. We assume that for some λ ∈ M , problem (7) has a solution u and Xis a closure of a ball in H centered at u. We want to investigate those conditionsunder which, for each λ in a neighborhood of λ, problem (7) has a unique solutionz(λ) near u and the function u(λ) is (Lipschitz) continuous and differentiable.Definition 1. Let Tλ(.) be an operator on X ×M . Then, the operator Tλ(.) is saidto be:(a) Locally strongly monotone, if there exists a constant α > 0 such that〈Tλ(u)− Tλ(v), u− v〉 ≥ α‖u− v‖2, ∀λ ∈M,u, v ∈ X.(b) Locally Lipschitz continuous, if there exists a constant β > 0 such that‖Tλ(u)− Tλ(v)‖ ≤ β‖u− v‖, ∀λ ∈M,u, v ∈ X.3. Main resultsWe consider the case when the solutions of the parametric general quasi variationalinequality (8) lie in the interior of X. Following the ideas of Dafermos [5] and Noor[18,19], we consider the map Fλ(u) as defined by (8). We have to show that themap Fλ(u) has a fixed point, which is a solution of the parametric general quasivariational inequality (7). First of all, we prove that the map Fλ(u), defined by (8),is a contraction map with respect to z uniformly in λ ∈M .Lemma 2. Let Tλ(.) be a locally strongly monotone with constant α > 0 and locallyLipschitz continuous with constant β > 0. Let the operator g be strongly monotonewith constants σ > 0 and Lipschitz continuous with constants δ > 0, respectively. IfAssumption 1 holds and for all u1, u2 ∈ X and λ ∈M , we have‖Fλ(u1)− Fλ(u2)‖ ≤ θ‖u1 − u2‖,Parametric quasi variational inequalities 209whereθ ={√1− 2σ + δ2 + ν +√1− 2αρ+ β2ρ2}(9)for|ρ− αβ2| <√α2 − β2k(2− k)β2, α > β√k(2− k), k < 1, (10)wherek =√1− 2σ + δ2 + ν. (11)Proof. In order to prove the existence of a solution of (7), it is enough to show thatthe mapping Fλ(u), defined by (8), is a contraction mapping.For u1 6= u2 ∈ H, and using Assumption 1, we have‖Fλ(u1)− Fλ(u2)‖ = ‖PK(u1)[g(u1)− ρTλu1]− PK(u2)[g(u2)− ρTλu2]‖≤ ‖PK(u1)[g(u1)− ρTλu1]− PK(u2)[g(u1)− ρTλu1]‖+‖PK(u2)[g(u1)− ρTλu1]− PK(u2)[g(u2)− ρTλu2]‖≤ ν‖u1 − u2‖+ ‖g(u1)− g(u2)− ρ(Tλu1 − Tλu2)‖= ν‖u1 − u2‖+‖ − (u1− u2− (g(u1)− g(u2)))+ u1 − u2 − ρ(Tλu1 − Tλu2)‖≤ ν‖u1 − u2‖+ ‖u1 − u2 − (g(u1)− g(u2))‖+‖u1 − u2 − ρ(Tλu1 − Tλu2)‖. (12)Since the operator Tλ is a locally strongly monotone with constant α > 0 and locallyLipschitz continuous with constant β > 0, it follows that||u1 − u2 − ρ(Tλu1 − Tλu2)||2 ≤ ||u1 − u2||2 − 2ρ〈Tλu1 − Tλu2, u1 − u2〉+ρ2||Tλu1 − Tλu2||2≤ (1− 2ρα+ ρ2β2)||u1 − u2||2. (13)In a similar way, we have||u1 − u2 − (g(u1)− g(u2))||2 ≤ (1− 2σ + δ2)||u1 − u2||2, (14)where we have used the fact that g is strongly monotone with constant σ > 0 andLipschitz continuous with constant δ > 0.From (9), (11), (13) and (14), we have‖Fλ(u1)− Fλ(u2)‖ ≤{ν +√(1− 2σ + δ2) +√(1− 2ρα+ ρ2β2)}‖u1 − u2‖={k +√(1− 2ρα+ ρ2β2)}‖u1 − u2‖= θ‖u1 − u2‖,210 M.A.Noor, K. I. Noor and Th.M.Rassiaswhereθ = k +√(1− 2ρα+ ρ2β2). (15)From (10), it follows that θ < 1. Thus it follows that the mapping Fλ(u), defined by(8), is a contraction mapping and consequently it has a fixed point, which belongs toK(u) satisfying the general quasi variational inequality (7), the required result.Remark 1. From Lemma 2, we see that the map Fλ(z) defined by (8) has a uniquefixed point u(λ), that is, u(λ) = Fλ(u). Also, by assumption, the function u, forλ = λ is a solution of the parametric general quasi variational inequality (7). Againusing Lemma 2, we see that u, for λ = λ, is a fixed point of Fλ(u) and it is also afixed point of Fλ(u). Consequently, we conclude thatu(λ) = u = Fλ(u(λ)).Using Lemma 2, we can prove the continuity of the solution u(λ) of the parametricgeneral quasi variational inequality (7) using the technique of Noor [13,14]. However,for the sake of completeness and to convey an idea of the techniques involved, wegive its proof.Lemma 3. Assume that the operator Tλ(.) is locally Lipschitz continuous with re-spect to the parameter λ. If the operator Tλ(.) is locally Lipschitz continuous andthe map λ→ PKλu is continuous (or Lipschitz continuous), then the function u(λ)satisfying (8) is (Lipschitz) continuous at λ = λ.Proof. For all λ ∈M, invoking Lemma 2 and the triangle inequality, we have‖u(λ)− u(λ¯)‖ ≤ ‖Fλ(u(λ))− Fλ¯(u(λ¯)‖+ ‖Fλ(u(λ¯))− Fλ¯(u(λ¯))‖≤ θ‖u(λ)− u(λ¯)‖+ ‖Fλ(u(λ¯))− Fλ¯(u(λ¯))‖. (16)From (8) and the fact that the operator Tλ is Lipschitz continuous with respect tothe parameter λ, we have‖Fλ(u(λ¯))− Fλ¯(u(λ¯))‖ = ‖u(λ¯)− u(λ¯) + ρ(Tλ(u(λ¯), u(λ¯))− Tλ¯(u(λ¯), u(λ¯)))‖≤ ρµ‖λ− λ¯‖. (17)Combining (16) and (17), we obtain‖u(λ)− u(λ¯)‖ ≤ ρµ1− θ‖λ− λ¯‖, for all λ, λ¯ ∈M,from which the required result follows.We now state and prove the main result of this paper which is the motivation ofour next result.Theorem 1. Let u be the solution of the parametric general quasi variational in-equality (7) for λ = λ. Let Tλ(u) be a locally strongly monotone Lipschitz continuousoperator for all u, v ∈ X. 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Parametric general quasi variational inequalities

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