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Existence of solutions of generalized variational inequalities in reflexive Banach spaces. (English) Zbl 1163.47307

Summary: In this work, we study the following Generalized Variational Inequality Problem (for short, GVIP): Given a closed convex set K in a reflexive Banach space E with the dual E * , a multifunction T:K2 E * , and a vector bE * , find x ¯K such that there exists u ¯T(x ¯) satisfying

u ¯-b,y-x ¯0,forallyK·

By using generalized projection and Ky Fan’s well-known KKM theorem, we prove existence results for solutions of GVIP. Our results extend some recent results from the literature.

MSC:
47J20Inequalities involving nonlinear operators
46N10Applications of functional analysis in optimization and programming
47N10Applications of operator theory in optimization, convex analysis, programming, economics
49J40Variational methods including variational inequalities
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