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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 $T:K\to {2}^{{E}^{*}}$, and a vector $b\in {E}^{*}$, find $\overline{x}\in K$ such that there exists $\overline{u}\in T\left(\overline{x}\right)$ satisfying

$〈\overline{u}-b,y-\overline{x}〉\ge 0,\phantom{\rule{2.em}{0ex}}\text{for}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}y\in K·$

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:
 47J20 Inequalities involving nonlinear operators 46N10 Applications of functional analysis in optimization and programming 47N10 Applications of operator theory in optimization, convex analysis, programming, economics 49J40 Variational methods including variational inequalities
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