INCLUSION AND INTERSECTION THEOREMS WITH APPLICATIONS IN EQUILIBRIUM THEORY IN G-CONVEX SPACES
Mircea Balaj, Donal O’Regan · Journal of the Korean Mathematical Society · 2010
In this paper we obtain a very general theorem of $\rho$ -compatibility for three multivalued mappings, one of them from the class $\mathfrak{B}$ . More exactly, we show that given a G-convex space Y, two topological spaces X and Z, a (binary) relation $\rho$ on $2^Z$ and three mappings P : X $\multimap$ Z, Q : Y $\multimap$ Z and $T\;{\in}\;\mathfrak{B}$ (Y,X) satisfying a set of conditions we can find ( $\widetilde{x},\;\widetilde{y}$ ) ${\in}$ $X\;{\times}\;Y$ such that $\widetilde{x}\;{\in}\;T(\widetilde{y})$ and $P(\widetilde{x}){\rho}\;Q(\widetilde{y})$ . Two particular cases of this general result will be then used to establish existence theorems for the solutions of some general equilibrium problems.