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

Yao, Jen-Chih

Math. Oper. Res. 19, No. 3, 691-705 (1994).

The purpose of this paper is to derive some more existence results for pseudomonotone operators \(T\) for the problem: Find \(\bar x\in K\) such that \[ (x- \bar x, T\bar x)\geq 0\quad\text{for all } x\in K, \] where \(T\) is an operator from a closed convex subset \(K\) of \(B\) into \(B^*\), \(B\) is a real Banach space with norm \(\|.\|\), \(B^*\) 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 inequalities
90C48	Programming in abstract spaces
90C33	Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
49J27	Existence theories for problems in abstract spaces

Keywords:

variational inequalities; pseudomonotone operators; existence; uniqueness; minimization problems; pseudoconvex functions; Banach spaces