Generalizations of vector quasivariational inclusion problems with set-valued maps

Abstract

Existence theorems are given for the problem of finding a point (z 0,x 0) of a set E × K such that $$(z_0,x_0)\in B(z_0,x_0)\times A(z_0,x_0)$$ and, for all $$\eta\in A(z_0,x_0), (F(z_0,x_0,x_0,\eta), C(z_0,x_0,x_0,\eta))\in \alpha$$ where ? is a relation on 2 Y (i.e., a subset of 2 Y × 2 Y ), $$A : E\times K\longrightarrow 2^K,$$ $$B : E\times K\longrightarrow 2^E, C : E\times K\times K\times K\longrightarrow 2^Y$$ and $$F : E\times K\times K\times K\longrightarrow 2^Y$$ are some set-valued maps, and Y is a topological vector space. Detailed discussions are devoted to special cases of ? and C which correspond to several generalized vector quasi-equilibrium problems with set-valued maps. In such special cases, existence theorems are obtained with or without pseudomonotonicity assumptions.