Fixed Points of Multivalued Maps Under Local Lipschitz Conditions and Applications - Henriquez-Miranda, H; Gallegos-Castro, C (Cambridge University Press, 2020)
In this work we are concerned with the existence of fixed points for multivalued maps defined on Banach spaces. Using the Banach spaces scale concept, we establish the existence of a fixed point of a multivalued map in a vector subspace where the map is only locally Lipschitz continuous. We apply our results to the existence of mild solutions and asymptotically almost periodic solutions of an abstract Cauchy problem governed by a first-order differential inclusion. Our results are obtained by using fixed point theory for the measure of noncompactness. © 2019 The Royal Society of Edinburgh.
Keywords: fixed points; set-valued maps; condensing mappings; k-set contractions; differential inclusions; mild solutions; asymptotically almost periodic functions