Journal of Optimization Theory and Applications

, Volume 127, Issue 2, pp 263–283 | Cite as

Weighted Variational Inequalities

  • Q. H. Ansari
  • Z. Khan
  • A. H. Siddiqi


In this paper, we introduce weighted variational inequalities over product of sets and system of weighted variational inequalities. It is noted that the weighted variational inequality problem over product of sets and the problem of system of weighted variational inequalities are equivalent. We give a relationship between system of weighted variational inequalities and systems of vector variational inequalities. We define several kinds of weighted monotonicities and establish several existence results for the solution of the above-mentioned problems under these weighted monotonicities. We introduce also the weighted generalized variational inequalities over product of sets, that is, weighted variational inequalities for multivalued maps and systems of weighted generalized variational inequalities. Extensions of weighted monotonicities for multivalued maps are also considered. The existence of a solution of weighted generalized variational inequalities over product of sets is also studied. The existence results for a solution of weighted generalized variational inequality problem give also the existence of solutions of systems of generalized vector variational inequalities.


Weighted variational inequalities system of vector variational inequalities weighted generalized variational inequalities system of generalized vector variational inequalities Nash equilibrium problems 


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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Q. H. Ansari
    • 1
    • 2
  • Z. Khan
    • 2
  • A. H. Siddiqi
    • 1
  1. 1.Department of Mathematical SciencesKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia
  2. 2.Department of MathematicsAligarh Muslim UniversityAligarhIndia

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