State-Dependent Impulses: Boundary Value Problems on Compact Interval

194 Pages·2015·2.515 MB·English

by  Irena Rachůnková, Jan Tomeček (auth.)

Atlantis Briefs in Differential Equations
Series Editors: Zuzana Dosla · Sarka Necasova · Milan Pokorny
Irena Rachůnková
Jan Tomeček
State-Dependent
Impulses
Boundary Value Problems on Compact
Interval
Atlantis Briefs in Differential Equations
Volume 6
Series editors
Zuzana Dosla, Brno, Czech Republic
Sarka Necasova, Prague 1, Czech Republic
Milan Pokorny, Praha 8, Czech Republic
About this Series
Theaimoftheseriesisrapiddisseminationofnewresultsandoriginalmethodsin
the theory of Differential Equations, including topics not yet covered by standard
monographs. The series features compact volumes of 75–200 pages, written in a
concise,clearwayandgoingdirectlytothepoint;theintroductorymaterialshould
be restricted to a minimum or covered by suitable references.
For more information on this series and our other book series, please visit our
website at: www.atlantis-press.com/publications/books
AMSTERDAM—PARIS—BEIJING
ATLANTIS PRESS
Atlantis Press
29, avenue Laumière
75019 Paris, France
More information about this series at http://www.springer.com/series/13609
ů á č
Irena Rach nkov Jan Tome ek
(cid:129)
State-Dependent Impulses
Boundary Value Problems on Compact
Interval
IrenaRachůnková Jan Tomeček
Faculty of Science Faculty of Science
Palacký University Palacký University
Olomouc Olomouc
Czech Republic Czech Republic
ISSN 2405-6405 ISSN 2405-6413 (electronic)
Atlantis Briefs in Differential Equations
ISBN978-94-6239-126-0 ISBN978-94-6239-127-7 (eBook)
DOI 10.2991/978-94-6239-127-7
LibraryofCongressControlNumber:2015948780
©AtlantisPressandtheauthor(s)2015
Thisbook,oranypartsthereof,maynotbereproducedforcommercialpurposesinanyformorbyany
means, electronic or mechanical, including photocopying, recording or any information storage and
retrievalsystemknownortobeinvented,withoutpriorpermissionfromthePublisher.
Printedonacid-freepaper
Preface
By writing this book we wanted to present a new approach to impulsive boundary
value problems, which is applicable both to the state-dependent and the fixed-time
impulses. Existence results for boundary value problems with state-dependent
impulsesareratherrareintheliteratureandaregivenmostlyforperiodicproblems.
Our main goal here isto provide general existence principles which can serve as a
tool for the investigation of solvability of impulsive boundary value problems, in
particularwithimpulsesoccurringatmomentsdependingonthestatevariables.To
demonstrate it, we bring new existence theorems and examples for such problems.
Ontheotherhand,thoughthereisvastliteraturedealingwithfixed-timeimpulsive
boundary value problems, we decided to include existence results for several
important problems of this type, for example, problems with time or space
singularities.
The book is organized as follows:
Part I, which consists of Chaps. 2–5, is devoted to fixed-time impulsive
boundaryvalueproblems.InChap.2wediscussthesolvabilityofthesecondorder
problemwithnonlinearboundaryconditionsandshowtheapplicationofthelower
andupperfunctionsmethod.Chapters3and4investigatethesecondorderDirichlet
problems, time singularities being studied in Chap. 3 and space singularities in
Chap. 4. Chapter 5 deals with higher order differential equations and with systems
thataresubjecttogenerallinearboundaryconditions.ThemainpurposeofChap.5
is, besides the existence results, to find an operator representation of the boundary
conditions in the space of regulated functions, which is used in Part II.
All chapters ofPart IIaredevoted tostate-dependent impulsive boundary value
problems.InChap.6wedevelopanewapproachgivingeffectiveconditionsforthe
solvability of the second order Dirichlet boundary value problem with one
state-dependentimpulseconditionimposedonthefirstderivativeofthesolution.In
Chap.7,aproblemmoregeneralthantheprecedingoneisinvestigated.Itisshown
that our approach can be easily extended to problems with a finite number of
state-dependent impulses mentioned above. In Chap. 8, the Sturm–Liouville
boundary value problem is investigated for a more general right-hand side of a
v
vi Preface
differentialequationdependingnotonlyonasolutionbutalsoonitsfirstderivative.
Moreover, state-dependent impulse conditions are imposed both on a solution and
onitsfirstderivative.Furthergeneralizationtohigherorderdifferentialequationsor
differential systems subject to general linear boundary conditions is presented in
Chaps. 9 and 10.
Olomouc Irena Rachůnková
May 2015 Jan Tomeček
Acknowledgment
The authors would like to express thanks to the Grant Agency of the Czech
Republic. The writing of this book was supported by Grant No. 14-06958S.
vii
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
1
1.1 Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2 Impulsive Ordinary Differential Equations . . . . . . . . . . . . . . .
2
1.2.1 Impulses at Fixed Times . . . . . . . . . . . . . . . . . . . . . .
3
1.2.2 State-Dependent Impulses . . . . . . . . . . . . . . . . . . . . .
4
1.2.3 Discontinuous Dynamical Systems . . . . . . . . . . . . . . .
5
1.2.4 Other Approaches to IODEs. . . . . . . . . . . . . . . . . . . . 6
1.3 Impulsive Boundary Value Problems. . . . . . . . . . . . . . . . . . . 6
1.3.1 Impulses at Fixed Times . . . . . . . . . . . . . . . . . . . . . .
6
1.3.2 State-Dependent Impulses . . . . . . . . . . . . . . . . . . . . .
8
1.3.3 Main Differences Between Impulses at Fixed Moments
and State-Dependent Impulses . . . . . . . . . . . . . . . . . .
8
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
14
Part I Problems with Impulses at Fixed Times
2 Second Order Problem with Nonlinear Boundary Conditions . . ..
21
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.2 Auxiliary Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
38
3 Dirichlet Problem with Time Singularities. . . . . . . . . . . . . . . . . .. 41
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Existence Principle for Singular Problem . . . . . . . . . . . . . . . .
43
3.3 Regular Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.4 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
61
ix
x Contents
4 Dirichlet Problem with Space Singularities . . . . . . . . . . . . . . . . ..
63
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
4.2 Main Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
4.3 A Priori Estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
4.4 Uniform Integrability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.5 Existence Principle for Regular Impulsive BVPs. . . . . . . . . . . 79
4.6 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
86
5 Systems of Differential Equations and Higher-Order Differential
Equations with General Linear Boundary Conditions . . . . . . . . ..
89
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
5.2 Operator Representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.3 Application to nth Order Differential Equations. . . . . . . . . . . . 100
5.4 Fredholm-Type Existence Theorems . . . . . . . . . . . . . . . . . . .
107
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
108
Part II Problems with State-Dependent Impulses
6 Dirichlet Problem with One Impulse Condition. . . . . . . . . . . . . ..
113
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.3 Main Result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
120
6.4 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
123
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
125
7 Dirichlet Problem via Lower and Upper Functions. . . . . . . . . . ..
127
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
127
7.2 Operators and Auxiliary Problem . . . . . . . . . . . . . . . . . . . . . 129
7.3 Main Result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.4 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
138
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
141
8 Sturm–Liouville Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
143
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
143
8.2 Existence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
146
8.3 Unbounded Data Functions . . . . . . . . . . . . . . . . . . . . . . . . . 150
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 154
9 Higher Order Equation with General Linear
Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
157
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
158
9.2 Transversality Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . .
159
9.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
162
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 168