Maximal elements and generalized games involving condensing mappings in locally FC-uniform spaces and applications(Ⅰ)

First,the notions of the measure of noncompactnees and condensing set- valued mappings are introduced in locally FC-uniform spaces without convexity struc- ture.A new existence theorem of maximal elements of a family of set-valued mappings involving condensing mappings is proved in locally FC-uniform spaces.As applications, some new equilibrium existence theorems of generalized game involving condensing map- pings are established in locally FC-uniform spaces.These results improve and generalize some known results in literature to locally FC-uniform spaces.Some further applications of our results to the systems of generalized vector quasi-equilibrium problems will be given in a follow-up paper.