Maximum principles for the Hill's equation

246 Pages·2018·1.762 MB·English

by  Cabada, Alberto, Cid, José, López-Somoza, Lucía et al.

MAXIMUM PRINCIPLES
FOR THE HILL’S
EQUATION

MAXIMUM PRINCIPLES
FOR THE HILL’S
EQUATION
AlbertoCabada
UniversidadedeSantiagodeCompostela,
InstitutodeMatemáticas,
FacultadedeMatemáticas,
SantiagodeCompostela,Galicia,Spain
JoséÁngelCid
UniversidadedeVigo,
DepartamentodeMatemáticas,
Ourense,Galicia,Spain
LucíaLópez-Somoza
UniversidadedeSantiagodeCompostela,
InstitutodeMatemáticas,
FacultadedeMatemáticas,
SantiagodeCompostela,Galicia,Spain
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This book is dedicated to my brother and sisters José Luis, Marina, Mercedes,
Nieves and Victoria.
Alberto Cabada
This book is dedicated to my wife Natalia and our children, Gael and Noa.
José Ángel Cid
This book is dedicated to my parents, Deme and Aurora, and my brother Pablo.
Lucía López-Somoza
ABOUT THE AUTHORS
Alberto CabadaisProfessorattheUniversityofSantiagodeCompostela.
His line of research is devoted to nonlinear differential equations. He has
obtained some results of existence and multiplicity of solution in differ-
ential equations, both ordinary and partial, as well as difference equations
and fractional ones. The techniques used are mainly based on topological
methodsanditerativetechniques.Animportantpartofhisresearchfocuses
on the study of, both quantitative and qualitative, properties of the so-
calledGreen’sfunctions.Heistheauthorofmorethanonehundredthirty
research articles indexed in the Citation Index Report and has authored
two monographs. He has supervisedseveralMasterand Ph.D. studentsand
has been the lead of different academic institutions as the Department of
Mathematical Analysis and the Institute of Mathematics of the University
of Santiago de Compostela.
José Ángel Cid is Associate Professor at the University of Vigo. His re-
searchisfocusedinthefieldofordinarydifferentialequations.Hehasdealt
mainlywithqualitativepropertieslikeexistence,uniquenessandmultiplic-
ity, obtained by means of topological and variational methods, fixed point
theory and monotone iterative techniques. He has authored more than
fortyresearchpapersincludedintheCitationIndexReport.Hehastaught
at the universities of Santiago de Compostela, Jaén and Vigo.
Lucía López-Somoza is a Ph.D. student at University of Santiago de
Compostela.Herlineofresearchisthestudyofnonlinearfunctionaldiffer-
ential equations. She studies Hill’s equation and, in particular, the relations
between the solutions of this equation under different types of boundary
conditions. Atpresentsheis a researchfellowship at Universityof Santiago
de Compostela.
ix
PREFACE
This book is devoted to the study of basic properties of the Hill’s equa-
tion, both in homogeneous and non homogeneous cases. As regards the
homogeneous problem, the spectral problem will be treated, along with
the oscillation of the solutions and their stability. Concerning the non
homogeneous problem, we will consider comparison principles for the
Hill’s equation. More concisely it will be delivered to the properties of the
Green’sfunctionsrelatedtosuchequationcoupledwithdifferentboundary
value conditions. We will establish its relationship with the spectral the-
ory developed for the homogeneous case. So stability and constant sign
solutions of the equation will be considered. Classical and recent results
obtained by us and another authors will be presented.
Theexistenceofsolutionsofnonlinearboundaryvalueproblemswillbe
also studied. The used techniques will mainly consist on the construction
ofintegraloperatorsdefinedonabstractspaceswhosefixedpointscoincide
with the solutions of the nonlinear problems that we are considering. So,
thefundamentalconstructionoftheclassicalLeray–Schauderdegreewillbe
shown, and classical fixed point theorems will be deduced. We will make
specialemphasisonoperatorsdefinedonconeswhich,aswewillsee,allow
ustofindconstantsignsolutions.Moreover,thetheoryoflowerandupper
solutions and the monotone iterative techniques will be also developed in
ageneralframeworkandappliedtononlinearproblemsrelatedtotheHill’s
equation.
Thisbookisdirectedtoawiderangeofmathematicians,includingboth
theoreticalandappliedorientedones,workingonthesubjectofdifferential
equations. The book also could be used for a Ph. D course addressed to
graduate students.
Theaudiencewillbenefitofashortbookprovidingbothcompleteand
accessible information of classical results and recent developments related
to the subject.
Alberto Cabada, José Ángel Cid, Lucía López-Somoza
Ourense and Santiago de Compostela
September 2017
xi
ACKNOWLEDGMENT
We thank the editorial team at Elsevier, specially Mr. Graham Nisbet, Se-
nior Acquisitions Editor, and Ms. Susan Ikeda, Editorial Project Manager,
for guidance throughout the publishing process.
We also thank to Pr. F. Adrián F. Tojo for his interesting suggestions in
the preparation of this manuscript.
ThisbookwassupportedbyMinisteriodeEconomíayCompetitividad,
Spain,andFEDER,projectMTM2013-43014-P,AgenciaEstataldeInves-
tigación (AEI) of Spain under grant MTM2016-75140-P, co-financed by
theEuropeanCommunityfundFEDER,andbyXuntadeGalicia(Spain),
project EM2014/032.
Alberto Cabada, José Ángel Cid, Lucía López-Somoza
Ourense and Santiago de Compostela
September 2017
xiii
CHAPTER 1
Introduction
Contents
1.1 Hill’sEquation 1
1.2 StabilityintheSenseofLyapunov 4
1.3 Floquet’sTheoremfortheHill’sEquation 8
References 18
1.1 HILL’SEQUATION
The Hill’s equation,
u(cid:2)(cid:2)(t)+a(t)u(t)=0, (1.1)
hasnumerousapplicationsinengineeringandphysics.Amongthemwecan
findsomeproblemsinmechanics,astronomy,circuits,electricconductivity
ofmetalsandcyclotrons.Hill’sequationisnamedafterthepioneeringwork
ofthemathematicalastronomerGeorgeWilliamHill(1838–1914),see[6].
He also made contributions to the three and the four body problems.
Moreover, the theory related to the Hill’s equation can be extended to
every differential equation in the form
u(cid:2)(cid:2)(t)+p(t)u(cid:2)(t)+q(t)u(t)=0, (1.2)
such that the coefficients p and q have enough regularity. This is due to
the fact that, with a suitable change of variable, the previous equation
transforms in one of the type of (1.1) (see the details in Section 2.2 of
Chapter 2).
As a first example we could consider a mass-spring system, that is, a
springwithamass m hangingfromit.Itisverywell-knownthat,denoting
by u(t) the position of the mass at the instant t and assuming absence of
friction, the previous model can be expressed as
k
u(cid:2)(cid:2)(t)+ u(t)=0,
m
with k>0 the elastic constant of the spring.
However, in a real physical system, there exists a friction force which
opposesthemovementandisproportionaltotheobject’sspeed.Inthiscase
MaximumPrinciplesfortheHill’sEquation.
DOI:http://dx.doi.org/10.1016/B978-0-12-804117-8.00001-1 1
Copyright©2018ElsevierInc.Allrightsreserved.