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Variational inequalities with generalized monotone operators. (English) Zbl 0813.49010

The purpose of this paper is to derive some more existence results for pseudomonotone operators $T$ for the problem: Find $\overline{x}\in K$ such that

$\left(x-\overline{x},T\overline{x}\right)\ge 0\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}x\in K,$

where $T$ is an operator from a closed convex subset $K$ of $B$ into ${B}^{*}$, $B$ is a real Banach space with norm $\parallel ·\parallel$, ${B}^{*}$ is its topological conjugate space endowed with weak * topology and $\left(u,\nu \right)$ is the paring between $u\in {B}^{*}$ and $\nu \in {B}^{*}$. In a final section same existence and uniqueness results for minimization problems with pseudoconvex functions in Banach spaces are obtained.

##### MSC:
 49J40 Variational methods including variational inequalities 90C48 Programming in abstract spaces 90C33 Complementarity and equilibrium problems; variational inequalities (finite dimensions) 49J27 Optimal control problems in abstract spaces (existence)