Variational inequalities with generalized monotone operators.

*(English)*Zbl 0813.49010The 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

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

where $T$ is an operator from a closed convex subset $K$ of $B$ into ${B}^{*}$, $B$ is a real Banach space with norm $\parallel \xb7\parallel $, ${B}^{*}$ is its topological conjugate space endowed with weak * topology and $(u,\nu )$ 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.

Reviewer: H.Benker (Merseburg)

##### 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) |