Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations

By: R.P. Agarwal, Said R. Grace, Donal O'Regan

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Hardcover | 31 July 2002

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In this monograph, the authors present a compact, thorough, systematic, and self-contained oscillation theory for linear, half-linear, superlinear, and sublinear second-order ordinary differential equations. An important feature of this monograph is the illustration of several results with examples of current interest. This book will stimulate further research into oscillation theory.

This book is written at a graduate level, and is intended for university libraries, graduate students, and researchers working in the field of ordinary differential equations.

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"The monograph under review is a very successful attempt to collect and systematically present numerous oscillation or nonoscillation criteria for various classes of linear and nonlinear differential equations and related results on asymptotic behavior of solutions ... . The book is nicely written and the selection of the material is very interesting ... . The book is a welcome addition to the existing literature on oscillation and shall be useful for researchers and graduate students in differential equations ... ." (Yuri V. Rogovchenko, Zentralblatt MATH, Vol. 1073 (24), 2005)

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Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Functional Analysis & Transforms

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Published: 22nd January 2011

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Preface	p. xi
Preliminaries
Introduction	p. 1
Continuability, Boundedness and Convergence to Zero	p. 2
Some Useful Results from Analysis and Fixed Point Theorems	p. 10
Notes and General Discussions	p. 12
References	p. 12
Oscillation and Nonoscillation of Linear Ordinary Differential Equations
Introduction	p. 13
Sturm Comparison Theorem	p. 13
Nonoscillatory Characterizations and Comparison Theorems	p. 15
Nonoscillation Criteria	p. 25
Oscillation Criteria	p. 35
Oscillation Criteria-Integral Averaging	p. 40
Oscillation Criteria-Integrable Coefficients	p. 70
Forced Oscillations	p. 82
Notes and General Discussions	p. 90
References	p. 91
Oscillation and Nonoscillation of Half-Linear Differential Equations
Introduction	p. 93
Preliminaries	p. 94
Sturm-Type Comparison Theorems	p. 94
Levin-Type Comparison Theorems	p. 100
Liapunov's Inequality	p. 105
An Oscillation Criterion for Almost-Periodic Sturm-Liouville Equations	p. 107
On Zeros of Solutions of a Singular Half-linear Differential Equation	p. 113
Nonoscillation Criteria	p. 140
Nonoscillatory Characterization	p. 140
Comparison Theorems	p. 147
Sufficient Conditions for Nonoscillation of Equation (3.1.1)	p. 148
Further Results on Nonoscillation of Equation (3.1.1)	p. 154
Oscillation Criteria	p. 163
Oscillation Criteria-Integral Averaging	p. 168
Oscillation Theorems for Equation (3.1.1)	p. 168
Interval Criteria for the Oscillation of Equation (3.1.1)	p. 179
The Weighted-Average Oscillation Criteria	p. 188
Oscillation Criteria-Integrable Coefficients	p. 196
Oscillation of Damped and Forced Equations	p. 205
Oscillation of Equation (3.11.1)	p. 206
Oscillation of Equation (3.11.2)	p. 207
Distance of Zeros of Oscillatory Solutions	p. 210
Oscillation and Nonoscillation of Half-linear Equations with Deviating Arguments	p. 218
Some Useful Lemmas	p. 219
Oscillation of Equation (3.13.1)	p. 223
Some Extensions and More Oscillation Criteria	p. 232
Classification of Nonoscillatory Solutions	p. 237
A Comparison Theorem	p. 251
Oscillation of Equation (3.13.2)	p. 253
Nonoscillation of Solutions of Equation (3.13.127)	p. 259
Notes and General Discussions	p. 267
References	p. 268
Oscillation Theory for Superlinear Differential Equations
Introduction	p. 273
Superlinear Oscillation Criteria	p. 273
Further Results on Superlinear Oscillations	p. 296
Existence of Nonoscillatory Solutions and Comparison Results	p. 328
A Necessary Condition for Oscillation	p. 328
Comparison of Nonlinear Equations of the Same Form	p. 332
Linearization of Oscillation Theorems	p. 339
Oscillation Criteria for Certain Nonlinear Differential Equations	p. 341
Superlinear Forced Oscillations	p. 358
Oscillation of Nonlinear Differential Equations with Nonlinear Damping Term	p. 362
Damping with Nonlinear Growth-Oscillation Criteria	p. 362
Damping with Nonlinear Growth-A Nonoscillation Theorem	p. 367
Notes and General Discussions	p. 371
References	p. 372
Oscillation Theory for Sublinear Differential Equations
Introduction	p. 376
Sublinear Oscillation Criteria	p. 376
Further Sublinear Oscillation Criteria	p. 387
Linearization of Sublinear Oscillation Theorems	p. 429
Nonoscillation Theorems for Sublinear Differential Equations	p. 433
Oscillation Criteria for Certain Nonlinear Differential Equations	p. 444
Notes and General Discussions	p. 453
References	p. 453
Further Results on the Oscillation of Differential Equations
Introduction	p. 457
Oscillation Criteria of Olech-Opial-Wazewski Type	p. 457
Some Definitions and Notation	p. 458
Oscillation Criteria	p. 459
Further Results	p. 463
Oscillation Criteria for Half-linear Differential Equations	p. 466
Preliminary Results	p. 467
Oscillation Criteria	p. 469
Oscillation Criteria via Liapunov's Second Method	p. 476
Preliminaries	p. 477
Oscillation Criteria	p. 479
Notes and General Discussions	p. 489
References	p. 490
Oscillation Results for Differential Systems
Introduction	p. 491
Oscillation of Nonlinear Two-dimensional Differential Systems	p. 491
Superlinear Case	p. 493
Sublinear Case	p. 498
Further Oscillation Criteria	p. 500
Oscillation Theorems for Linear Differential Systems	p. 503
Oscillation Theorems for Differential Systems with Functionally Commutative Matrix Coefficients	p. 533
Comparison Theorems for Operator-Valued Linear Differential Equations	p. 540
Oscillation Results for Differential Systems with Forcing Terms	p. 549
Notes and General Discussions	p. 552
References	p. 554
Asymptotic Behavior of Solutions of Certain Differential Equations
Introduction	p. 558
Asymptotic Behavior of Solutions of Nonlinear Differential Equations	p. 558
Asymptotic Behavior of Nonoscillatory Solutions	p. 559
Bounded Asymptotically Linear Solutions	p. 562
Unbounded Asymptotically Linear Solutions	p. 567
Further Extensions and Improvements	p. 574
Asymptotic Behavior of Positive Solutions of Singular Emden-Fowler-Type Equations	p. 584
Positive Solutions of Equation (8.2.1)	p. 585
Positive Decaying Solutions of Equation (8.2.2)	p. 595
Positive Solutions of Singular Emden-Fowler-Type Systems	p. 614
Notes and General Discussions	p. 622
References	p. 624
Miscellaneous Topics
Introduction	p. 625
Comparison and Nonoscillation Results	p. 625
Nonoscillatory Solutions of Forced Differential Equations	p. 632
Limit Circle Criteria and Related Properties for Nonlinear Equations	p. 638
Properties of Certain Differential Equations	p. 650
Notes and General Discussions	p. 654
References	p. 654
Nonoscillation Theory for Multivalued Differential Equations
Introduction	p. 656
Preliminaries	p. 656
Differential Inclusions	p. 657
Neutral Inclusions	p. 664
Notes and General Discussions	p. 668
References	p. 668
Subject Index	p. 670
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Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations

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