Selecting families and their applications

Deguire and Lassonde [P. Deguire, M. Lassonde, Familles sélectantes, Topol. Methods Nonlinear Anal. 5 (1995) 261–269] extend the concept of continuous selection and introduce the notion of selecting family for a family of set-valued mappings. In this paper, we first establish a new existence theorem of selecting families. The existence of the selecting families will be then used in order to obtain several fixed component theorems of Fan–Browder type, an intersection result, a maximal element theorem for a family of set-valued mappings and a minimax inequality.