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Fixed points of multivalued maps under local Lipschitz conditions and applications

Part of: Qualitative theory Maps and general types of spaces defined by maps General theory in ordinary differential equations 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 ([email protected]; [email protected])
Hernán R. Henríquez
Affiliation:
Departamento de Matemática, Universidad de Santiago de Chile, USACH, Casilla 307, Correo 2, Santiago, Chile ([email protected]; [email protected])
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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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References

1Abada, N., Benchohra, M. and Hammouche, H.. Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions. J. Diff. Equ. 246 (2009), 3834–3863.CrossRefGoogle Scholar
2Abbas, S., Benchohra, M. and N'Guérékata, G. M.. Topics in fractional differential equations (New York: Springer Science, 2012).CrossRefGoogle Scholar
3Akhmerov, R. R., Kamenskiĭ, M. I., Potapov, A. S., Rodkina, A. E. and Sadovskiĭ, B. N.. Measures of noncompactness and condensing operators (Basel: Birkhäuser Verlag, 1992).CrossRefGoogle Scholar
4Arendt, W., Batty, C., Hieber, M. and Neubrander, F.. Vector-valued laplace transforms and Cauchy problems (Basel: Birkhäuser, 2001).CrossRefGoogle Scholar
5Banas, J. and Goebel, K.. Measures of noncompactness in Banach spaces. Lect. Notes Pure Appl. Math. 60 (New York: Marcel Dekker, 1980).Google Scholar
6Benchohra, M., Hendersson, J. and Ntouyas, S.. Impulsive differential equations and inclusions, New York: Hindawi Publishing Corporation, 2006).CrossRefGoogle Scholar
7Chuong, N. M. and Ke, T. D.. Generalized Cauchy problems involving nonlocal and impulsive conditions. J. Evol. Equ. 12 (2012), 367–392.CrossRefGoogle Scholar
8Deimling, K.. Multivalued differential equations, De Gruyter series in nonlinear analysis and applications 1 (Berlin: Walter de Gruyter & Co., 1992).Google Scholar
9Diagana, T.. Almost automorphic type and almost periodic type functions in abstract spaces (Switzerland: Springer, 2013).CrossRefGoogle Scholar
10Diestel, J., Ruess, W. M. and Schachermayer, W.. Weak compactness in L 1(μ, X). Proc. Amer. Math. Soc. 118 (1993), 447–453.Google Scholar
11Engel, K-J. and Nagel, R.. One-parameter semigroups for linear evolution equations (New York: Springer-Verlag, 2000).Google Scholar
12Gatsori, E. P., Górniewicz, L., Ntouyas, S. K. and Sficas, G. Y.. Existence results for semilinear functional differential inclusions with infinite delay. Fixed Point Theory 6 (2005), 47–58.Google Scholar
13Górniewicz, L.. Topological fixed point theory of multivalued mappings, 2nd edn (Dordrecht: Springer, 2006).Google Scholar
14Granas, A and Dugundji, J.. Fixed point theory, New York: Springer-Verlag, 2003).CrossRefGoogle Scholar
15Heinz, H-P.. On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions. Nonlinear Anal. Theory, Methods & Appl. 7 (1983), 1351–1371.CrossRefGoogle Scholar
16Henríquez, H. R.. Asymptotically almost-periodic solutions of abstract differential equations. J. Math. Anal. Appl. 160 (1991), 157–175.CrossRefGoogle Scholar
17Henríquez, H. R.. Introducción a la Integración Vectorial, Editorial Académica Española, ISBN: 978-3-659-04096-2, 2012.Google Scholar
18Henríquez, H. R., Hernández, M. E. and dos Santos, J. C.. Asymptotically almost periodic and almost periodic solutions for partial neutral integrodifferential equations. Zeitschrift für Anal. und ihre Anwendungen 26 (2007), 363–375.CrossRefGoogle Scholar
19Henríquez, H. R., Poblete, V. and Pozo, J. C.. Mild solutions of non-autonomous second order problems with nonlocal initial conditions. J. Math. Anal. Appl. 412 (2014), 1064–1083.CrossRefGoogle Scholar
20Kamenskii, M., Obukhovskii, V. and Zecca, P.. Condensing multivalued maps and semilinear differential inclusions in Banach spaces, De Gruyter series in nonlinear analysis and applications 7 (Berlin: Walter de Gruyter & Co., 2001).Google Scholar
21Kavitha, V., Mallika Arjunan, M. and Ravichandran, C.. Existence results for a second order impulsive neutral functional integrodifferential inclusions in Banach spaces with infinite delay. J. Nonlinear Sci. Appl. 5 (2012), 321–333.CrossRefGoogle Scholar
22Marle, C-M.. Mesures et Probabilités (Paris: Hermann, 1974).Google Scholar
23Martin, R. H.. Nonlinear operators and differential equations in Banach spaces (New York: John Wiley & Sons, 1976).Google Scholar
24Obukhovskii, V. and Yao, J.-C.. On impulsive functional differential inclusions with Hille-Yosida operators in Banach spaces. Nonlinear Anal. Theory, Methods & Appl. 73 (2010), 1715–1728.CrossRefGoogle Scholar
25Pazy, A.. Semigroups of linear operators and applications to partial differential equations (New York: Springer-Verlag, 1983).CrossRefGoogle Scholar
26Samoilenko, A. M. and Perestyuk, N. A.. Impulsive differential equations (Singapore: World Scientific, 1995).CrossRefGoogle Scholar
27Smirnov, G. V.. Introduction to the theory of differential inclusions (Providence: Amer. Math. Soc., 2002).Google Scholar
28Tolstonogov, A. A.. Solutions of evolution inclusions I. Siberian Math. Journal 33 (1992), 500–511.CrossRefGoogle Scholar
29Tolstonogov, A.. Differential inclusions in a Banach space (Dordrecht: Springer, 2000).CrossRefGoogle Scholar
30Tolstonogov, A. A.. Properties of the set of extreme solutions for a class of nonlinear second-order evolution inclusions. Set-Valued Anal. 10 (2002), 53–77.CrossRefGoogle Scholar
31Tolstonogov, A. A. and Umanskii, Ya. I.. On solutions of evolution inclusions II. Siberian Math. J. 33 (1992), 693–702.CrossRefGoogle Scholar
32Vrabie, I. I.. Compactnes methods for nonlinear evolutioins (Essex: Longman, 1995).Google Scholar
33Zaidman, S.. Almost-periodic functions in abstract spaces (London: Pitman Advanced Publ. Program, 1985).Google Scholar