Generalizations of vector quasivariational inclusion problems with set-valued maps

• Authors: Pham Huu Sach;Le Anh Tuan
• Venue: Journal of Global Optimization
• Year: 2009

Existence theorems are given for the problem of finding a point (z 0,x 0) of a set E 脳 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 驴 is a relation on 2 Y (i.e., a subset of 2 Y 脳 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 驴 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.