Journal of the
Korean Mathematical Society JKMS

In this paper we obtain a very general theorem of $\rho$-compatibility for three multivalued mappings, one of them from the class $\frak{B}$. 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 $T\in \frak{B}(Y,X)$ 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.

Keywords: $G$-convex space, the better admissible class, fixed point, equilibrium problems

MSC numbers: 54C60, 54H25, 91B50