Generalized Ordinary Differential Equations: Not Absolutely Continuous Solutions

208 Pages·2012·0.76 MB·English

by  Kurzweil J.

Series in Real Analysis – Vol. 11
GENERALIZED ORDINARY
DIFFERENTIAL EQUATIONS
Not Absolutely Continuous Solutions
7907.9789814324021-tp.indd 1 12/16/11 10:40 AM
SERIES IN REAL ANALYSIS
Published
Vol. 1: Lectures on the Theory of Integration
R Henstock
Vol. 2: Lanzhou Lectures on Henstock Integration
Lee Peng Yee
Vol. 3: The Theory of the Denjoy Integral & Some Applications
V G Celidze & A G Dzvarseisvili
translated by P S Bullen
Vol. 4: Linear Functional Analysis
W Orlicz
Vol. 5: Generalized ODE
S Schwabik
Vol. 6: Uniqueness & Nonuniqueness Criteria in ODE
R P Agarwal & V Lakshmikantham
Vol. 7: Henstock–Kurzweil Integration: Its Relation to Topological Vector Spaces
Jaroslav Kurzweil
Vol. 8: Integration between the Lebesgue Integral and the Henstock–Kurzweil
Integral: Its Relation to Local Convex Vector Spaces
Jaroslav Kurzweil
Vol. 9: Theories of Integration: The Integrals of Riemann, Lebesgue,
Henstock–Kurzweil, and McShane
Douglas S Kurtz & Charles W Swartz
Vol. 10: Topics in Banach Space Integration
Òtefan Schwabik & Ye Guoju
Vol. 11: Generalized Ordinary Differential Equations:
Not Absolutely Continuous Solutions
Jaroslav Kurzweil
Vol. 12: Henstock–Kurzweil Integration on Euclidean Spaces
Tuo Yeong Lee
Vol. 13: Theories of Integration: The Integrals of Riemann, Lebesgue,
Henstock–Kurzweil, and McShane (2nd Edition)
Douglas S Kurtz & Charles W Swartz
He Yue - Generalized Ordinary Diff.pmd 1 1/31/2012, 1:50 PM
Series in Real Analysis – Vol. 11
GENERALIZED ORDINARY
DIFFERENTIAL EQUATIONS
Not Absolutely Continuous Solutions
Jaroslav Kurzweil
Academy of Sciences, Czech Republic
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
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Series in Real Analysis — Vol. 11
GENERALIZED ORDINARY DIFFERENTIAL EQUATIONS
Not Absolutely Continuous Solutions
Copyright © 2012 by World Scientific Publishing Co. Pte. Ltd.
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ISBN-13 978-981-4324-02-1
ISBN-10 981-4324-02-7
Printed in Singapore.
He Yue - Generalized Ordinary Diff.pmd 2 1/31/2012, 1:50 PM
November18,2011 16:26 WorldScienti(cid:12)cBook-9inx6in jk
Preface
TheKurzweil-Henstockintegralisanonabsolutelyconvergentintegral. The
aimof this treatise isthe exploitation of this propertyin some convergence
problems in ordinary differential equations and in some situations where
solutions of infinite variation can occur. This leads to generalized differ-
ential equations the theory of which is presented in Chapters 6–18, 22–24,
27. Since the motion of Kapitza’s pendulum was a significant motivation
for the theory, the equation of motion of Kapitza’s pendulum is exposed in
Chapter2andvariousestimatesforequationsofasimilartypeareobtained
by elementary methods in Chapters 3 and 4.
The concept of the Kurzweil-Henstock integral and of the generalized
differentialequationgoesbackto1957. Thepreparationofthisbookstarted
in 2004 after W. N. Everitt had encouraged me to build a solid theory of
the generalized differential equations. The results in this direction were
reported and discussed regularly in the Seminar on Differential Equations
and Integration Theory in the Institute of Mathematics of the Academy of
Sciences of the Czech Republic in Prague.
I wish to thank the participants of the seminar for their contributions
andcomments,inparticulartoJ.Jarn´ık,B.Maslowski,I.Vrkoˇc,M.Tvrdy´
and the late Sˇ. Schwabik. The last two of them in addition read and
commentedthemanuscript. E.Ritterov´awasveryhelpfulbytypingseveral
versions of the manuscript.
Theresearchconnectedwiththepreparationofthisbookwassupported
by the grant No. IAA100190702 of the Grant Agency of the Acad. Sci. of
theCzechRepublicandbytheAcademyofSciencesoftheCzechRepublic,
Institutional Research Plan No. AV0Z10190503.
Prague, May 2011 Jaroslav Kurzweil
v
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Contents
Preface v
1. Introduction 1
2. Kapitza’s pendulum and a related problem 9
3. Elementary methods: averaging 11
4. Elementary methods: internal resonance 15
5. StrongRiemann-integrationoffunctionsofapairofcou-
pled variables 27
6. Generalized ordinary differential equations: Strong
Riemann-solutions (concepts) 37
7. Functions ψ , ψ 43
1 2
8. Strong Riemann-solutions of generalized differential
equations: a survey 47
9. Approximate solutions: boundedness 51
vii
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viii Generalized Ordinary Differential Equations
10. Approximate solutions: a Lipschitz condition 59
11. Approximate solutions: convergence 63
12. Solutions 69
13. Continuous dependence 77
14. Strong Kurzweil Henstock-integration of functions of a
pair of coupled variables 83
15. Generalized differential equations: Strong Kurzweil
Henstock-solutions 97
16. Uniqueness 101
17. Differential equations in classical form 105
18. On a class of differential equations in classical form 111
19. Integration and Strong Integration 119
20. A class of Strong Kurzweil Henstock-integrable functions 127
21. Integration by parts 135
22. A variant of Gronwall inequality 145
23. Existence of solutions of a class of generalized ordinary
differential equations 155
24. A convergence process as a source of discontinuities in
the theory of differential equations 165
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Contents ix
25. A class of Strong Riemann-integrable functions 177
26. On equality of two integrals 185
27. A class of Generalized ordinary differential equations
with a restricted right hand side 187
Appendix A. Some elementary results 189
Appendix B. Trifles from functional analysis 191
Bibliography 193
Symbols 195
Subject index 197