Fixed Points of a Family of Set-Valued Mappings Under Generalized Caristi-Type Conditions

Abstract

We investigate the existence and uniqueness of common fixed points for a family of set-valued mappings {F_r}_r\in[0,1] defined on a complete metric space, under generalized Caristi-type conditions. By introducing a unified inequality framework involving lower semicontinuous control functions and weakly orbitally continuous selections, we establish fixed point results that generalize and subsume several known theorems for single valued and multivalued operators.  The main results provide a systematic extension of Caristis theorem to parametrized families of set-valued maps, with applications illustrated through carefully constructed examples.