Difference Equations & Inequalities

982 Pages·2003·31.47 MB·English

by  Ravi P. Agarwal

PURE AND APPLIED MATHEMATICS
A Program of Monographs, Textbooks, and Lecture Notes
EXECUTIVE EDITORS
Earl J. Taft Zuhair Nashed
Rutgers University University o f Delaware
New Brunswick, New Jersey Newark, Delaware
EDITOFUAL BOARD
M. S. Baouendi Ani1 Nerode
University of California, Cornell University
San Diego
Donald Passman
Jane Cronin University of Wisconsin,
Rutgers University Madison
Jack K. Hale Fred S. Roberts
Georgia Institute of Technology Rutgers University
S. Kobayashi David L. Russell
University of California, Virginia Polytechnic Institute
Berkeley and State University
Marvin Marcus Walter Schernpp
University of California, Universitat Siegen
Santa Barbara
Mark Teply
W. S. Massey University of Wisconsin,
Yale University Milwaukee
DIFFERENCE EQUATIONS
AND INEQUALITIES
Theory, Methods, and Applications
Second Edition, Revised and Eqpanded
Ravi P. Agarwal
National University of Singapore
Kent Ridge, Singapore
MARCEL -
MARCELD EKKEIRN,C. NEWY ORK BASEL
DEKKER
Library of Congress Cataloging-in-Publication Data
Aganval, Ravi P.
Difference equations and inequalities :t heory, methods, and applications l
-
Ravi P. Aganval 2nd ed., rev. and expanded
p. cm. -(M onographs and textbooks in pure and applied mathematics;
228)
Includes bibliographical references and indexes.
ISBN 0-8247-9007-3(a cid-kee paper)
1. Difference equations. 2. Inequalities (Mathematics) I. Title. 11. Series.
QA431 .A43 2000
515'.6254c21 99-058166
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Preface to the Second Edition
Since its publication in 1992, Difference Equations and Inequalities has
been received very positively by the international scientific community. Its
success prompted a request from the publisher for an updated edition.
In this edition, besides a new chapter on the qualitative properties of
solutions of neutral difference equations, new material has been added in all the
existing chapters of the first edition. This includes a variety of interesting
examples from real world applications, new theorems, over 200 additional
problems and 400 further references.
The theory of difference equations has grown at an accelerated pace in the
past decade. It now occupies a central position in applicable analysis and will
no doubt continue to play an important role in mathematics as a whole. It is
hoped that this new edition will be a timely and welcomed reference.
It is a pleasure to thank all those who have helped in the preparation of this
edition. I would especially like to thank Ms. Maria Allegra of Marcel Dekker,
Inc., whose help was instrumental in the successful completion of this project.
Ravi P. Agarwal
Preface to the First Edition
Examples of discrete phenomena in nature abound and yet somehow the
continuous version has commandeered all our attention -p erhaps owing to that
special mechanism in human nature that permits us to notice only what we have
been conditioned to. Although difference equations manifest themselves as
mathematical models describing real life situations in probability theory,
queuing problems, statistical problems, stochastic time series, combinatorial
analysis, number theory, geometry, electrical networks, quanta in radiation,
genetics in biology, economics, psychology, sociology, etc., unfortunately, these
are only considered as the discrete analogs of differential equations. It is an
indisputable fact that difference equations appeared much earlier than
V
vi Preface to the First Edition
differential equations and were instrumental in paving the way for the
development of the latter. It is only recently that difference equations have
started receiving the attention they deserve. Perhaps this is largely due to the
advent of computers, where differential equations are solved by using their
approximate difference equation formulations. This self-contained monograph
is an in-depth and up-to-date coverage of more than 400 recent publications and
may be of interest to practically every user of mathematics in almost every
discipline.
It is impossible to acknowledge individually colleagues and friends to
whom I am indebted for assistance, inspiration and criticism in writing this
monograph. I must, however, express my appreciation and thanks to
Ms.
Rubiah Tukimin for her excellent and careful typing of the manuscript.
Ravi P. Agarwal
Contents
Preface to the Second Edition V
Preface to the First Edition V
Chapter 1 Preliminaries
1.1. Notations 1
1.2. Difference Equations 2
1.3. Initial Value Problems 4
1.4. Some Examples: Initial Value Problems 6
1.5. Boundary Value Problems 11
1.6. Some Examples: Boundary Value Problems 13
1.7. Some Examples: Real World Phenomena 22
1.8. Finite Difference Calculus 26
1.9. Problems 34
1 .lo. Notes 44
1. ll . References 45
Chapter 2 Linear Initial Value Problems
2.1. Introduction 49
2.2. Preliminary Results from Algebra 49
2.3. Linear Dependence and Independence 54
2.4. Matrix Linear Systems 55
2.5. Variation of Constants Formula 58
2.6. Green’s Matrix 59
2.7. Adjoint Systems 60
2.8. Systems with Constant Coefficients 62
2.9. Periodic Linear Systems 69
2.10. Almost Periodic Linear Systems 72
2.11. Higher Order Linear Equations 74
2.12. Method of Generating Functions 80
2.13. Bernoulli’s Method 85
2.14. Poincark’s and Perron’s Theorems 87
2.15. Regular and Singular Perturbations 91
2.16. Problems 97
2.17. Notes 113
2.18. References 114
vii
viii Contents
Chapter 3 Miscellaneous Difference Equations
3.1. Clairaut’s Equation 117
3.2. Euler’s Equation 118
3.3. Riccati’s Equation 120
3.4. Bernoulli’s Equation 125
3.5. Verhulst’s Equation 126
3.6. Best Discrete Approximations:
Harmonic Oscillator Equation 128
3.7. Duffing’s Equation 130
3.8. van der Pol’s Equation 135
3.9. Hill’s Equation 142
3.10. Mathieu’s Equation 147
3.11. Weierstrass’ Elliptic Equations 148
3.12. Volterra’s Equations 150
3.13. Elementary Partial Difference Equations:
Riccati’s Extended Form 152
3.14. Wave Equation 158
3.15. FitzHugh-Nagumo’s Equation 159
3.16. Korteweg-de Vries’ Equation 160
3.17. Modified KdV Equation 161
3.18. Lagrange’s Equations 162
3.19. Problems 167
3.20. Notes 180
3.21. References 181
Chapter 4 Difference Inequalities
4.1. Gronwall Inequalities 184
4.2. Nonlinear Inequalities 193
4.3. Inequalities Involving Differences 200
4.4. Finite Systems of Inequalities 205
4.5. Opial Type Inequalities 209
4.6. Wirtinger Type Inequalities 214
4.7. Problems 219
4.8. Notes 229
4.9. References 230
Chapter 5 Qualitative Properties of Solutions of
Difference Systems
5.1. Dependence on Initial Conditions and Parameters 234
5.2. Asymptotic Behavior of Linear Systems 238
5.3. Asymptotic Behavior of Nonlinear Systems 247
Contents ix
5.4. Concepts of Stability 250
5.5. Stability of Linear Systems 255
5.6. Stability of Nonlinear Systems 262
5.7. Nonlinear Variation of Constants 269
5.8. Dichotomies 272
5.9. Lyapunov’s Direct Method for Autonomous Systems 281
5.10. Lyapunov’s Direct Method for Non-Autonomous Systems 289
5.11. Stability of Discrete Models in Population Dynamics 292
5.12. Converse Theorems 298
5.13. Total Stability 303
5.14. Practical Stability 305
5.15. Mutual Stability 306
5.16. Problems 308
5.17. Notes 326
5.18. References 328
Chapter 6 Qualitative Properties of Solutions of
Higher Order Difference Equations
6.1. General Properties of Solutions of
+ +
(6.1.1) p(k)~(k1 ) p(k -l )u(k- 1 ) = q(k)u(k) 335
6.2. Boundedness of Solutions of (6.1.1) 337
6.3. Recessive and Dominant Solutions of (6.1.1) 342
6.4. Oscillation and Nonoscillation for (6.1.1) 348
6.5. Riccati Type Transformations for (6.1.1) 349
6.6. Riccati Type Transformations for
+ +
(6.6.1) A(p(k)Au(k)) ~(k)u(k 1) = 0 355
6.7. Olver Type Comparison Results 362
6.8. Sturm Type Comparison Results 365
6.9. Variety of Properties of Solutions of
+ +
(6.9.1) p(k)z(k 1) p(k -l )z(k- l )
= q(k)z(k)+ 4k) 367
6.10. Variety of Properties of Solutions of
+
(6.10.1) A2u(k- 1 ) p(k)ur(k)= 0 370
6.11. Oscilla,tion and Nonoscillation for
+
(6.11.1) A(r(k)Au(k)) f(k)F(u(k))= 0 378
6.12. Asymptotic Behavior of Solutions of
+
(6.12.1) A(r(k)Au(k)) f(k)F(u(k))= g(k) 384
6.13. l2 and c0 Solutions of
+
(6.13.1) A2u(k) f(k,u (k))= 0 387
6.14. Oscillation and Nonoscillation for
(6.14.1) Aiu(k)= f(k,u(k),A p(k)) 391
X Contents
6.15. Oscillation and Nonoscillation for
+
(6.15.1) A(~(k)Au(k)f)( k)F(k,u(k)A, u(k))
= g(ku(k),W k)) 393
6.16. Variety of Properties of Solutions of
(6.16.1) A4u(k -2 ) = p(k)u(k) 398
6.17. Asymptotic Behavior of Solutions of
(6.17.1) Anu(k)+ f(k,~ ( kA) u, (k),. . . , A n-'u(k)) = 0 403
6.18. Asymptotic Behavior, Oscillation and
Nonoscillation for
(6.18.1) Anu(k)+ h(k)F(k~, ( k A) ,u (k),. . . , A n-lu(k))
= g(k,u(k)A, u(k),. . . , A n-'u(k)) 408
6.19. Oscillation and Nonoscillation for
+
(6.19.1) An.(k) CLl fi(k)Fi(~(kA)u, (k),. . . ,
An-' u(lc))= 0
414
6.20. Oscillation and Nonoscillation for
+ +
(6.20.1) ~ ( k1) -~ ( k )p (k)u(k- m )= 0 420
6.21. Oscillation and Nonoscillation for
(6.21.1)6 A,u(k) + bC,"=f,z (k)Fz(u(gz(k)=)) 0 425
6.22. Oscillation and Nonoscillation for
+ + +
(6.22.1) A(p(k)(A~(k))q~(k) l)f(u(k I)) = 0 427
6.23. Oscillation and Nonoscillation for
+
(6.23.1) ~ ( t-)~ (-tT ) p(t)u(t- 0 )= 0 436
6.24. Problems 443
6.25. Notes 473
6.26. References 474
Chapter 7 Qualitative Properties of Solutions of
Neutral Difference Equations
7.1. Oscillation and Nonoscillation for
+ +
(7.1.1) A(u(k) p ( k- T )) q(k)u(k- o ) = 0 485
7.2. Existence and Asymptotic Behavior
of Nonoscillatory Solutions of (7.1.1) 493
7.3. Oscillation and Comparison Theorems for (7.1. l) 509
7.4. Global Asymptotic Stability Criterion for (7.1.1) 512
7.5. Oscillation and Nonoscillation for
(7.5.1) A (U(k)+pu(k-T))+q(k)zl(k-ol)"h(k).1L(lc-aa) = 0 517
7.6. Oscillation and Nonoscillation for
+ +
(7.6.1) A (u(k) p ( k- T )) q(k)u(k- o )= F(k) 519
7.7. Oscillation and Nonoscillation for