RSME Springer Series Volume 6
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RSME Springer Series Volume 6 Editors-in-Chief Joaquín Pérez, Departamento de Geometría y Topología, University of Granada, Granada, Spain Series Editors Nicolas Andruskiewitsch, FaMAF - CIEM (CONICET), Universidad Nacional de Córdoba, Córdoba, Argentina María Emilia Caballero, Instituto de Matemáticas, Universidad Nacional Autónoma de México, México, Mexico Pablo Mira, Departamento de Matematica Aplicada y Estadistica, Universidad Politécnica de Cartagena, Cartagena, Spain Timothy G. Myers, Centre de Recerca Matemàtica, Barcelona, Spain Marta Sanz-Solé, Department of Mathematics and Informatics, Barcelona Graduate School of Mathematics (BGSMath), Universitat de Barcelona, Barcelona, Spain Karl Schwede, Department of Mathematics, University of Utah, Salt Lake City, UT, USA
As of 2015, RSME - Real Sociedad Matemática Española - and Springer cooperate in order to publish works by authors and volume editors under the auspices of a co-branded series of publications including advanced textbooks, Lecture Notes, collections of surveys resulting from international workshops and Summer Schools, SpringerBriefs, monographs as well as contributed volumes and conference pro- ceedings. The works in the series are written in English only, aiming to offer high level research results in the fields of pure and applied mathematics to a global readership of students, researchers, professionals, and policymakers. More information about this series at http://www.springer.com/series/13759
Rubén Figueroa Sestelo • Rodrigo López Pouso • Jorge Rodríguez López Degree Theory for Discontinuous Operators Applications to Discontinuous Differential Equations
Rubén Figueroa Sestelo Rodrigo López Pouso
IES Pedras Rubias Department of Statistics, Mathematical
Salceda de Caselas Analysis and Optimization, Faculty of
Pontevedra, Spain Mathematics
University of Santiago de Compostela
Santiago de Compostela
La Coruña, Spain
Jorge Rodríguez López
Department of Statistics, Mathematical
Analysis and Optimization, Faculty of
Mathematics
University of Santiago de Compostela
Santiago de Compostela
La Coruña, Spain
ISSN 2509-8888 ISSN 2509-8896 (electronic)
RSME Springer Series
ISBN 978-3-030-81603-2 ISBN 978-3-030-81604-9 (eBook)
https://doi.org/10.1007/978-3-030-81604-9
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland
AG 2021
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The registered company address is: Gewerbestrasse 11, 6330 Cham, SwitzerlandPreface
This book brings together in an unified extended form the work carried out by the
authors in the last few years on degree theory techniques applied to discontinu-
ous differential equations. Besides many new existence results for discontinuous
differential equations, the core contribution in this book is an extension of the
Leray–Schauder’s degree theory which underlies our proofs. Alternative potential
uses of such theory are vast: any theorem proven by means of the classical
“continuous” degree theory can have its generalization through the “discontinuous”
theory introduced here. Readers may get the best account of possible applications
from Mawhin’s expository paper “Leray–Schauder degree: a half century of exten-
sions and applications,” included in the list of references.
For readability, we have split the monograph into two parts. A first part devoted
to degree theory, and a second part concerning applications to discontinuous
differential equations. Our new degree theory yields interesting results for discon-
tinuous differential equations because it allows us to replace some typical stringent
assumptions. For example, we can find in the literature many existence results for
discontinuous equations with monotone right-hand sides deduced by means of fixed
point theorems for monotone operators, such as Tarski’s, or Heikkilä’s well-order
chains of iterations. Here we prove new fixed point theorems for discontinuous
operators which need not be monotone and whose application to differential
equations yield novel existence results. In this sense, this monograph provides
researchers interested in Carathéodory solutions for discontinuous ODEs with a
new approach, complementary to classical ones which can be found in the books
by Filippov or Carl, Heikkilä, and Lakshmikantham.
Santiago de Compostela, Spain Rubén Figueroa Sestelo
Santiago de Compostela, Spain Rodrigo López Pouso
Santiago de Compostela, Spain Jorge Rodríguez López
April 2021
vContents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1
2 Degree Theory for a Class of Discontinuous Operators . . . . . . . . . . . . . . . . . 5
2.1 A Topological Degree for Discontinuous Operators . . . . . . . . . . . . . . . . . . 6
2.2 Basic Properties of the Degree . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10
2.3 Fixed Point Index for Discontinuous Operators . . . . . . . . . . . . . . . . . . . . . . . 17
3 Fixed Point Theorems for Some Discontinuous Operators .. . . . . . . . . . . . . 25
3.1 Schauder Type Fixed Point Theorems . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 26
3.2 Krasnosel’skiı̆’s Compression–Expansion Type Fixed Point
Theorems in Cones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 27
3.3 Krasnosel’skiı̆ Type Fixed Point Theorems in Cones
for Monotone Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 31
3.4 A Generalization of Leggett–Williams’ Three-Solutions
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 34
3.5 A Vectorial Version of Krasnosel’skiı̆’s Fixed Point Theorem .. . . . . . . 39
4 First Order Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 45
4.1 Existence Result for First Order Scalar Problems
with Functional Initial Conditions . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 47
4.2 Existence Results for Non-autonomous Systems . .. . . . . . . . . . . . . . . . . . . . 57
4.3 Discontinuous First-Order Functional Boundary Value Problems .. . . 72
4.3.1 An Application to Second-Order Problems
with Functional Boundary Conditions .. . . .. . . . . . . . . . . . . . . . . . . . 77
5 Second Order Problems and Lower and Upper Solutions . . . . . . . . . . . . . . 83
5.1 Existence Results on Bounded Domains . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 84
5.1.1 Existence Results via Well-Ordered Lower
and Upper Solutions . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 86
5.1.2 Existence of Extremal Solutions Between the Lower
and Upper Solutions . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 103
viiviii Contents
5.1.3 Existence Results via Non-ordered Lower
and Upper Solutions . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 108
5.1.4 Multiplicity Results . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 113
5.2 Existence Results on Unbounded Domains . . . . . . . .. . . . . . . . . . . . . . . . . . . . 118
5.2.1 Existence Results on the Half Line . . . . . . . .. . . . . . . . . . . . . . . . . . . . 119
5.2.2 Extremal Solutions Between the Lower and Upper
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 131
6 Positive Solutions for Second and Higher Order Problems . . . . . . . . . . . . . 135
6.1 Second Order Problems with Sturm–Liouville Boundary
Conditions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 136
6.2 Second Order Systems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 149
6.3 Multiplicity Result to a Three-Point Problem . . . . .. . . . . . . . . . . . . . . . . . . . 160
6.4 Positive Solutions to a One Dimensional Beam Equation . . . . . . . . . . . . 167
6.4.1 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 168
6.4.2 A Multiplicity Result . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 177
A Degree Theory for Multivalued Operators .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 181
References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 185You can also read
