Collective fixed points and maximal elements with applications to abstract economies

In this paper, we first establish collective fixed points theorems for a family of multivalued maps with or without assuming that the product of these multivalued maps is Φ-condensing. As an application of our collective fixed points theorem, we derive the coincidence theorem for two families of multivalued maps defined on product spaces. Then we give some existence results for maximal elements for a family of L_S-majorized multivalued maps whose product is Φ-condensing. We also prove some existence results for maximal elements for a family of multivalued maps which are not L_S-majorized but their product is Φ-condensing. As applications of our results, some existence results for equilibria of abstract economies are also derived. The results of this paper are more general than those given in the literature.