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Fixed points of multivalued maps under local Lipschitz conditions and applications

Part of: Maps and general types of spaces defined by maps General theory in ordinary differential equations Qualitative theory Nonlinear operators and their properties

Published online by Cambridge University Press:  29 January 2019

Claudio A. Gallegos  and
Hernán R. Henríquez
Claudio A. Gallegos
Affiliation:
Departamento de Matemática, Universidad de Santiago de Chile, USACH, Casilla 307, Correo 2, Santiago, Chile (claudio.gallegos@usach.cl; hernan.henriquez@usach.cl)
Hernán R. Henríquez
Affiliation:
Departamento de Matemática, Universidad de Santiago de Chile, USACH, Casilla 307, Correo 2, Santiago, Chile (claudio.gallegos@usach.cl; hernan.henriquez@usach.cl)
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Abstract

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.

Keywords

fixed pointsset-valued mapscondensing mappingsk-set contractionsdifferential inclusionsmild solutionsasymptotically almost periodic functions

MSC classification

Primary: 47H10: Fixed-point theorems 54C60: Set-valued maps
Secondary: 47H08: Measures of noncompactness and condensing mappings, $K$-set contractions, etc. 34A60: Differential inclusions 34C27: Almost and pseudo-almost periodic solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 3 , June 2020 , pp. 1467 - 1494
Copyright
Copyright © 2019 The Royal Society of Edinburgh

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