Strong vector variational like inequality problems with properly quasimonotone bifunctions

In this paper, we introduce two types of proper quasimonotone maps over cones for a vector-valued bifunction and discuss their relations with generalized monotone maps, namely cone pseudomonotone and cone quasimonotone maps. Strong vector variational like inequality problems of the Stampacchia and the Minty type have been defined. These problems are a generalization of the classical Stampacchia and Minty problems and encompass many problems studied in the literature. A generalization of celebrated Minty lemma, relating the solutions of the two problems, has been proved. Existence results for strong Stampacchia and Minty type vector variational like inequality problems have been established using the notions of proper quasimonotone maps over cones. Gap functions have also been proposed for both problems.