A new family of G_B-MAJORIZED mappings from a topological space into a finite continuous topological spaces (in short, FC-space) involving a better admissible set-valued mapping is introduced. Some existence theorems of maximal elements for the family of G_B-majorized mappings are proved under noncompact setting of product FC-spaces. Some applications to fixed point and system of minimax inequalities are given in product FC-spaces. These theorems improve, unify and generalize many important results in recent literature.