Regular Variation and Differential Equations

✍ Scribed by Vojislav Maric

• Publisher: Springer
• Year: 2000
• Tongue: English
• Leaves: 143
• Series: Lecture Notes in Mathematics, 1726
• Category: Library

✦ Synopsis

This is the first book offering an application of regular variation to the qualitative theory of differential equations. The notion of regular variation, introduced by Karamata (1930), extended by several scientists, most significantly de Haan (1970), is a powerful tool in studying asymptotics in various branches of analysis and in probability theory. Here, some asymptotic properties (including non-oscillation) of solutions of second order linear and of some non-linear equations are proved by means of a new method that the well-developed theory of regular variation has yielded. A good graduate course both in real analysis and in differential equations suffices for understanding the book.

Content Level » Research

Keywords » Boundary layer equation - Differential equations of Thomas-Fermi type - Second order linear differential equations

Related subjects » Dynamical Systems & Differential Equations

✦ Table of Contents

Cover......Page 1
Lecture Notes in Mathematics 1726......Page 2
Regular Variation and Differential Equations......Page 4
ISBN 3-540-67160-9......Page 5
Dedicated To the memory of Vojislav G. Avakitmovié.......Page 6
Preface......Page 8
Contents......Page 10
0 Introduction......Page 12
1.1. Preliminaries.......Page 20
1.2. The case f(x) < 0.......Page 23
1.3. \PI — and \Gamma— varying solutions......Page 32
1.4. The case of f(x) of arbitrary sign.......Page 37
1.5. Regular boundedness of solutions.......Page 51
1.6. Generalizations.......Page 52
1.7. Examples.......Page 55
1.8. Comments.......Page 57
2.1. Slowly varying solutions.......Page 60
2.2. Regularly varying solutions.......Page 68
2.3. On zeros of oscillatory solutions.......Page 73
2.4. Examples......Page 76
2.5. Comments......Page 81
3.1. Introduction and preliminaries.......Page 82
3.2. The case of regularly varying f and \phi......Page 86
3.3. Examples......Page 100
3.4. The case of rapidly varying f or \phi......Page 101
3.5. Examples......Page 110
3.6. A more general case......Page 113
4.1. Introduction......Page 116
4.2. Existence and uniqueness.......Page 117
4.3. Estimates and asymptotic behaviour of solutions.......Page 120
4.4. Comments.......Page 125
Appendix: Properties of regularly varying and related functions......Page 126
References......Page 130
Index......Page 136
List of Publications......Page 140