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This volume provides a comprehensive review of the developments which have taken place during the last thirty years concerning the asymptotic properties of solutions of nonautonomous ordinary differential equations. The conditions of oscillation of solutions are established, and some general theorems on the classification of equations according to their oscillatory properties are proved. In addition, the conditions are found under which nonlinear equations do not have singular, proper, oscillatory and monotone solutions. The book has five chapters: Chapter I deals with linear differential equations; Chapter II with quasilinear equations; Chapter III with general nonlinear differential equations; and Chapter IV and V deal, respectively, with higher-order and second-order differential equations of the Emden-Fowler type. Each section contains problems, including some which presently remain unsolved. The volume concludes with an extensive list of references. For researchers and graduate students interested in the qualitative theory of differential equations.
Series Editor's Preface. Preface. Basic Notation. I: Linear Differential Equations. 1. Equations having Properties A and £Ii£. 2. Oscillatory and Nonoscillatory Equations. 3. Oscillation Properties of Solutions Vanishing at Infinity. 4. The Subspace of Solutions Vanishing at Infinity. 5. Bounded and Unbounded Solutions. 6. Asymptotic Formulas. II: Quasilinear Differential Equations. 7. Statement of the Problem; Auxiliary Assertions. 8. The Family of Lh Type Solutions of the Equation. 9. L0h, LINFINITYh and Lh Type Equations. III: General Nonlinear Differential Equations. 10. Theorems on the Classification of Equations with respect to their Oscillation Properties. 11. Singular Solutions. 12. Fast Growing Solutions. 13. Kneser Solutions. 14. Proper Oscillatory Solutions. IV: Higher Order Differential Equations of Emden-Fowler Type. 15. Oscillatory Solutions. 16. Nonoscillatory Solutions. V: Second Order Differential Equations of Emden-Fowler Type. 17. Existence Theorems for Proper and Singular Solutions. 18. Oscillation and Nonoscillation Criteria for Proper Solutions. 19. Unbounded and Bounded Solutions; Solutions Vanishing at Infinity. 20. Asymptotic Formulas. References. Author Index. Subject Index.