Constant-Sign Solutions of Systems of Integral Equations

651 Pages·2013·4.17 MB·English

by  Ravi P. Agarwal

Ravi P. Agarwal · Donal O’Regan
Patricia J.Y. Wong
Constant-Sign
Solutions
of Systems
of Integral
Equations
Constant-Sign Solutions of Systems of Integral
Equations

Ravi P. Agarwal • Donal O’Regan
Patricia J.Y. Wong
Constant-Sign Solutions
of Systems of Integral
Equations
123
RaviP.Agarwal DonalO’Regan
DepartmentofMathematics SchoolofMathematics,Statistics
TexasA&MUniversity–Kingsville andAppliedMathematics
Kingsville,TX,USA NationalUniversityofIreland,Galway
Galway,Ireland
PatriciaJ.Y.Wong
SchoolofElectrical&Electronic
Engineering
NanyangTechnologicalUniversity
Singapore
ISBN978-3-319-01254-4 ISBN978-3-319-01255-1(eBook)
DOI10.1007/978-3-319-01255-1
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013291023
MathematicsSubjectClassification:45-02
©SpringerInternationalPublishingSwitzerland2013
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection
with reviews or scholarly analysis or material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of
this publication or parts thereof is permitted only under the provisions of the Copyright Law of the
Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.
PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations
areliabletoprosecutionundertherespectiveCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
While the advice and information in this book are believed to be true and accurate at the date of
publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor
anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with
respecttothematerialcontainedherein.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
RPA: Tomy wifeSadhnaAgarwal
DOR: Tomy daughterNiamh EllieO’Regan
PJYW: Tomy motherWong-PhuahLiewHong

Preface
Themainpurposeofthisbookistodevelopindetailexistencetheoryandprovide
some insights into systems of integral equations. The intention is to present a
systematic investigationand a comprehensivetreatmentof Volterra and Fredholm
equations.
Thebookconsistsof20chapters.Thenecessaryfixedpointtheory,compactness
criteria,andbasicpreliminarieswillbepresentedinChap.1.Throughoutthebook
wewilldiscusstwosystemsofFredholmequations,oneisonthefiniteinterval
Z
1
u .t/D g .t;s/f .s;u .s/;u .s/;(cid:2)(cid:2)(cid:2) ;u .s//ds; t 2Œ0;1(cid:2); 1(cid:3)i (cid:3)n
i i i 1 2 n
0
andtheotherisonthehalf-lineŒ0;1/
Z
1
u .t/D g .t;s/f .s;u .s/;u .s/;(cid:2)(cid:2)(cid:2) ;u .s//ds; t 2Œ0;1/; 1(cid:3)i (cid:3)n
i i i 1 2 n
0
andthesolutionuD.u ;u ;(cid:2)(cid:2)(cid:2) ;u /willbesoughtinappropriatefunctionspaces,
1 2 n
andthesystemofVolterraintegralequationswestudyis
Z
t
u .t/D g .t;s/f .s;u .s/;u .s/;(cid:2)(cid:2)(cid:2) ;u .s//ds; t 2Œ0;T(cid:2); 1(cid:3)i (cid:3)n
i i i 1 2 n
0
whereT > 0isfixed.Oneofthemainaimsofthisbookistodiscussforintegral
equationsthenaturalfunctionspaceswhereasolutionliesandtopresentproperties
and behavior of solutions. The arguments throughout the book will be based on
fixedpointarguments.Inparticular(butnotexclusively)wewillpresentarguments
basedontheSchauderfixedpointtheorem,Krasnosel’skiifixedpointtheorem,the
Leggett–Williams fixed point theorem, and the Leray–Schauder alternative. The
functionspaceswhicharise naturally,forexample,inFredholmintegralequations
on the finite interval in this book will be .CŒ0;1(cid:2)/n D CŒ0;1(cid:2) (cid:4) (cid:2)(cid:2)(cid:2) (cid:4) CŒ0;1(cid:2)
vii
viii Preface
or .LpŒ0;1(cid:2)/n D LpŒ0;1(cid:2) (cid:4) (cid:2)(cid:2)(cid:2) (cid:4) LpŒ0;1(cid:2) or more generally .L Œ0;1(cid:2)/n D
(cid:3)
L Œ0;1(cid:2)(cid:4)(cid:2)(cid:2)(cid:2)(cid:4)L Œ0;1(cid:2) where L Œ0;1(cid:2) is an Orlicz space. Throughoutthe book
(cid:3) (cid:3) (cid:3)
many examples are included to illustrate the theory. In Chap.2 we will present
existencetheory(singleandmultipleconstant-signsolutions)forFredholmintegral
equations both on the finite interval and on the half-line. The set of eigenvalues
anditspropertiesforvariousFredholmintegralequationsonthefiniteintervaland
on the half-line are discussed in Chap.3. Multiplicity of solutions of Fredholm
integral equations is investigated in Chap.4. In particular the existence of triple
constant-signsolutionsis studiedthere. Solutionsin Chaps.2–4onfinite intervals
are sought in .CŒ0;1(cid:2)/n D CŒ0;1(cid:2)(cid:4)(cid:2)(cid:2)(cid:2)(cid:4)CŒ0;1(cid:2). In Chap.5 solutions on finite
intervalsare soughtin .LpŒ0;1(cid:2)/n D LpŒ0;1(cid:2)(cid:4)(cid:2)(cid:2)(cid:2)(cid:4)LpŒ0;1(cid:2). VariousFredholm
integral equations both on the finite interval and on the half-line are discussed in
thischapter.InparticularinChap.5theresultsobtainedassumeapolynomialtype
restriction (in u) on the nonlinearity f.t;u/. The semipositone case for various
Fredholmintegralequationsbothonthefiniteintervalandonthehalf-lineistackled
in Chap.6. Semipositone problems arise naturally in chemical reactor theory and
existence will be established in Chap.6 provided the Thiele modulus is suitably
chosen.InChaps.7–9singularitiesofspecialtype,namelyintegrablesingularities,
areinvestigatedforintegralequations.Inparticularthenonlinearityf .t;u ;(cid:2)(cid:2)(cid:2) ;u /
i 1 n
may be singular at t D 0 and u D 0 where j 2 f1;(cid:2)(cid:2)(cid:2) ;ng. Our theory is
j
motivatedfromproblems,forexample,incommunications,boundarylayertheory,
andHomannflow.Thesingularcase(withothertypesofsingularities)forvarious
Fredholm and Volterra integral equations is studied in detail in Chaps.10–12.
Periodicandalmostperiodicsolutionsforvariousintegralequationsarediscussed
inChaps.13–15.Inparticulartheintegralequation
Z
t
u .t/D g .t;s/f .s;u .s/;u .s/;(cid:2)(cid:2)(cid:2) ;u .s//ds; t 2.(cid:5)1;1/; 1(cid:3)i (cid:3)n
i i i 1 2 n
t(cid:2)(cid:4)
whichdescribesthespreadofntypesofepidemicswhichareindependentofeach
other is tackled in Chap.13, whereas a system of Hill’s equation is examined in
Chap.14. Solutions in Orlicz spaces for various Fredholm and Volterra integral
equations will be presented in Chaps.16 and 17, and a solution to a Fredholm
integralequation on the finite interval will be soughtin .L Œ0;1(cid:2)/n D L Œ0;1(cid:2)(cid:4)
(cid:3) (cid:3)
(cid:2)(cid:2)(cid:2) (cid:4) L Œ0;1(cid:2). In particular in these chapters using Orlicz spaces other than Lp
(cid:3)
will lead to restrictions that are not of polynomialtype (in u) on the nonlinearity
f.t;u/ and hence will allow us to consider new classes of equations. In Chap.18
theUrysohnintegralequation
Z
1
u .t/D g .t;s;u .s/;u .s/;(cid:2)(cid:2)(cid:2) ;u .s//ds; t 2Œ0;1(cid:2); 1(cid:3)i (cid:3)n
i i 1 2 n
0
will be discussed and a solution will be sought in .CŒ0;1(cid:2)/n D CŒ0;1(cid:2) (cid:4) (cid:2)(cid:2)(cid:2)(cid:4)
CŒ0;1(cid:2). In addition the set of eigenvalues and its properties for Urysohn integral
Preface ix
equationswill be studied in Chap.18. Finally Brezis–Browdertype argumentsfor
various Fredholm and Volterra integral equations will be presented in Chaps.19
and20.
We wish to express our gratitude to our colleague Dr. Fengmin Chen for
typesettingsomepartsofparticularchapters.
Kingsville,TX,USA RaviP.Agarwal
Galway,Ireland DonalO’Regan
Singapore PatriciaJ.Y.Wong