ISSN ONLINE(23198753)PRINT(23476710)
Khushbu^{1}, Zubair Khan^{2}

Related article at Pubmed, Scholar Google 
Visit for more related articles at International Journal of Innovative Research in Science, Engineering and Technology
In this paper, we introduce and study the system of general variational inequalities which is equivalent to the general variational inequality problem over the product of sets. The usual concept of monotonicity has been extended here. We establish existence results for the solution of general variational inequality problem over the product of sets in the setting of real Hausdroff topological vector space.
Keywords 
General variational inequalities, Monotonicity, Multivalued mapping, Convexity, Reflexive Banach space 
INTRODUCTION 
Variational inequality theory has emerged as a powerful tool for a wide class of unrelated problems arising in various branches of physical, engineering, pure and applied sciences in unified and general framework. Variational inequalities have been extended and generalized in different directions by using novel and innovative techniques and ideas, both for their own sake and for their application. 
In the last three decades, the Nash equilibrium problem [1] has been studied by many authors either by using Ky Fan minimax inequality [2] or by using fixed point technique. In 2000, Ansari and Yao [3] gave a new direction to solve the Nash equilibrium problem for nondifferentiable functions. They introduced a system of optimization problems which includes the Nash equilibrium problem as a special case. They proved that every solution of system of generalized variational inequalities is also a solution of system of optimization problems for nondifferentiable functions and also for nondifferentiable and nonconvex functions. 
It is mentioned by J. P. Aubin in his book [4] that the Nash equilibrium problem [1, 5] for differentiable functions can be formulated in the form of a variational inequality problem over product of sets (for short, VIPPS). Not only the Nash equilibrium problem but also various equilibriumtype problems, like, traffic equilibrium, spatial equilibrium, and general equilibrium programming problems, from operations research, economics, game theory, mathematical physics, and other areas, can also be uniformly modeled as a VIPPS, see for example [6,7,8] and the references therein. Pang [8] decomposed the original variational inequality problem defined on the product of sets into a system of variational inequalities (for short, SVI), which is easy to solve, to establish some solution methods for VIPPS. Later, it was found that these two problems VIPPS and SVI are equivalent. In the recent past VIPPS or SVI has been considered and studied by many authors, see for example [3, 6, 7, 9, 10, 11] and references therein. Konnov [10] extended the concept of (pseudo) monotonicity and established the existence results for a solution of VIPPS. Recently, Ansari and Yao [12] introduced the system of generalized implicit variational inequalities and proved the existence of its solution. They derived the existence results for a solution of system of generalized variational inequalities and used their results as tools to establish the existence of a solution of system of optimization problems, which include the Nash equilibrium problem as a special case, for nondifferentiable functions. Later Ansari and Khan [13, 14] also generalized these existing results. 
Motivated by the works given in [13,14], in this paper we formulate the problem of system of general variational inequalities and general variational inequality problem over the product of sets. It is noticed that every solution of general variational inequality problem over the product of sets (GVIPPS) is a solution of system of general variational inequalities (SGVI) and viceversa. We extend various kinds of monotonicities defined by Ansari and Yao [12], Ansari and Zubair [13,14]. We adopt the technique of Yang and Yao [15] to establish the existence results for a solution of general variational inequality problem over the product of sets (GVIPPS) and hence a solution of system of general variational inequality problem (SGVIP). 
PRELIMINARIES AND FORMULATION OF PROBLEM 
References 
[1] J. F. Nash, Ã¢ÂÂEquilibrium Point in nPersons GameÃ¢ÂÂ, Proc. Nat. Acad Sci., U.S.A., 36, pp.4849, 1950. [2] K. Fan, Ã¢ÂÂFixed Point and Minimax Theorems in Locally Convex Topological Linear SpacesÃ¢ÂÂ, Proc. Nat. Acad. Sci., U.S.A, 38, pp.121126, 1952. [3] Q. H. Ansari and J. C. Yao, Ã¢ÂÂA Fixed Point Theorem and its Applications to the System of Variational InequalitiesÃ¢ÂÂ, Bull. Austral. Math. Soc, 59, pp.433442, 1999. [4] J. P. Aubin, Mathematical Methods of Game Theory and Economic, NorthHolland, Amsterdam, 1982. [5] J. F. Nash, Noncooperative games, Ann. Math., 54, pp.286295, 1951. [6] M. Ferris and J. S. Pang, Engineering and economic applications of complementarity problems, SIAM Rev. , 39, pp.669713, 1997. [7] A. Nagurney, Network Economics: A Variational Inequality Approach, Kluwer Academic Publishers, DordrechtBostonLondon, 1993. [8] J. S. Pang, Asymmetric Variational Inequalities over product of sets: Applications and iterative methods, Math. Prog. , 31, pp.206219, 1985. [9] C. Cohen and F. Chaplais, Nested monotony for variational inequalities over product of spaces and convergence of iterative algorithms, J. Optim. Theory Appl., 59, pp.360390, 1988. [10] I. V. Konnov, Relatively monotone variational inequalities over product sets, Oper. Res. Lett. , 28, pp.2126, 2001 [11] S. MaklerScheimberg, V. H. Nguyen and J. J. Strodiot, Family of perturbation methods for variational inequalities, J. Optim. Theory Appl.,89, pp.423452, 1996. [12] Q. H. Ansari and J. C. Yao, Ã¢ÂÂSystem of generalized variational inequalities and their applicationsÃ¢ÂÂ, Appl. Anal., 76 (34), pp.203217, 2000. [13] Q. H. Ansari and Z. Khan, Ã¢ÂÂDensely relative pseudomonotone variational inequalities over product of setsÃ¢ÂÂ, Journal of Nonlinear and Convex Analysis, 7, pp.179188, 2011. [14] Q. H. Ansari and Z. Khan, Ã¢ÂÂRelatively Bpseudomonotone variational inequalities over product of setsÃ¢ÂÂ, Journal of Inequalities in Pure and Applied Mathematics, 4, pp.149163, 2003. [15] X. Q. Yang and J. C. Yao, Ã¢ÂÂGap functions and existence of setvalued vector variational inequalitiesÃ¢ÂÂ, J. Optim. Theory Appl., 115(2) (2002), 407417. [16] D. T. Luc, Ã¢ÂÂExistence results for densely pseudomonotone variational inequalitiesÃ¢ÂÂ, J. Math. Annal. Appl.,254, pp.291308, 2001 [17] H. Brezis,Ã¢ÂÂEquations et inequations non lineaires dans les espaces vectoriels en dualiteÃ¢ÂÂ, Ann. Inst. Fourier (Grenoble) 18, pp.115175, 1968. [18] D. Repovs and P. V. Semenov, Ã¢ÂÂContinuous Selections of Multivalued MappingsÃ¢ÂÂ, Kluwer Academic Publishers, DordrechtBostonLondon, 1998. 