Weighted Variational Inequalities In Normed Spaces

powerful tool to investigate and study a wide class of problems arising in industrial, physical, pure and applied sciences in a unified and general framework. The ideas and techniques of variational inequalities are being applied in diverse areas and are proving to be productive and innovative. Variational inequalities have been extended and generalized in several directions using novel and new techniques.Vector-valued variational inequalities were first introduced and studied by Giannessi [8] in a finite-dimensional Euclidean space in 1980. Later Chen et al. [11-14] and Yang [10,15] have intensively studied some kinds of vector variational inequalities, vector quasi-variational inequalities, and vector complementarity problems in Banach spaces.It is known that the theory of variational inequalities is a powerful tool to study a wide class of problems such as optimization and equilibrium problems (see, e.g., [4]). Vector variational inequality problems were related to vector network equilibrium problems [5] and vector optimization problems [10]. Over the last two decades, vector variational inequality problems have been intensively and extensively studied (see, e.g. , [4] for details). Recently, using systems of vector variational inequalities, Ansari, Schaible and Yao [3] studied vector Nash equilibrium problems in topological vector spaces ordered by pointed, closed and convex cones with nonempty interiors . Most recently, Ansari, Khan and Siddiqi [2] investigated the problems of weighted variational inequalities in finite dimensional Euclidean spaces ordered by their respective nonnegative orthants over product of sets and established a number of existence results for their solutions. Clearly, this study is helpful to the study of Nash equilibrium problems in finite dimensional Euclidean spaces ordered by their respective nonnegative orthants. However, it should be noted that the study of weighted variational inequalities over product of sets in normed spaces ordered by general nontrivial, closed and convex cones with nonempty interiors will be helpful to the study of vector variational inequalities over product of sets in normed spaces, and in turn, should be helpful to the study of vector Nash equilibrium problems in normed spaces . In this paper, we introduce weighted variational inequalities over product of sets and systems of weighted variational inequalities in normed spaces ordered by nontrivial, closed and convex cones with nonempty interiors. We extend most of the results of [2] to weighted variational inequalities over product of sets in normed spaces ordered by nontrivial, closed and convex cones with nonempty interiors. We also modified the settings of problems (SGVVI) and ( )SGVVI w in Section 4 of [2] and present more precise conclusions of Lemmas 4.2 and 4.3, Theorems 4.1, 4.2 and 4.3, and Corollary 4.2 in [2].