Said R. Grace

Oscillation Theory for Difference and Functional Differential Equations

Preface. 1. Oscillation of Difference Equations. 1.1. Introduction. 1.2. Oscillation of Scalar Difference Equations. 1.3. Oscillation of Orthogonal Polynomials. 1.4. Oscillation of Functions Recurrence Equations. 1.5. Oscillation in Ordered Sets. 1.6. Oscillation in Linear Spaces. 1.7. Oscillation in Archimedean Spaces. 1.8. Oscillation of Partial Recurrence Equations. 1.9. Oscillation of System of Equations. 1.10. Oscillation Between Sets. 1.11. Oscillation of Continuous-Discrete Recurrence Equations. 1.12. Second Order Quasilinear Difference Equations. 1.13. Oscillation of Even Order Difference Equations. 1.14. Oscillation of Odd Order Difference Equations. 1.15. Oscillation of Neutral Difference Equations. 1.16. Oscillation of Mixed Difference Equations. 1.17. Difference Equations Involving Quasi-differences. 1.18. Difference Equations with Distributed Deviating Arguments. 1.19. Oscillation of Systems of Higher Order Difference Equations. 1.20. Partial Difference Equations with Continuous Variables. 2. Oscillation of Functional Differential Equations. 2.1. Introduction. 2.2. Definitions, Notations and Preliminaries. 2.3. Ordinary Difference Equations. 2.4. Functional Difference Equations. 2.5. Comparison of Equations of the Same Form. 2.6. Comparison of Equations with Others of Lower Order. 2.7. Further Comparison Results. 2.8. Equations with Middle Term of Order (n - 1). 2.9. Forced Differential Equations. 2.10.Forced Equations with Middle Term of Order (n - 1). 2.11. Superlinear Forced Equations. 2.12. Sublinear Forced Equations. 2.13. Perturbed Functional Equations. 2.14. Comparison of Neutral Equations with Nonneutral Equations. 2.15 Comparison of Neutral Equations with Equations of the Same Form. 2.16. Neutral Differential Equations of Mixed Type. 2.17. Functional Differential Equations Involving Quasi-derivatives. 2.18. Neutral and Damped Functional Differential Equations Involving Quasi-derivatives. 2.19. Forced Functional Differential Equations Involving Quasi-derivatives. 2.20. Systems of Higher Order Functional Differential Equations. References. Subject Index.

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

Preface. 1. Preliminaries. 2. Oscillation and Nonoscillation of Linear Ordinary Differential Equations. 3. Oscillation and Nonoscillation of Half-Linear Differential Equations. 4. Oscillation Theory for Superlinear Differential Equations. 5. Oscillation Theory for Sublinear Differential Equations. 6. Further Results on the Oscillation of Differential Equations. 7. Oscillation Results for Differential Systems. 8. Asymptotic Behavior of Solutions of Certain Differential Equations. 9. Miscellaneous Topics. 10. Nonoscillation Theory for Multivalued Differential Equations. Subject Index.

Discrete Oscillation Theory

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Oscillation theory for second order dynamic equations

Preliminaries. Introduction. Initial Value Problem, Oscillation and Nonoscillation. Continuability and Boundedness. Some Basic Results for Second Order Linear Ordinary Differential Equations. Some Useful Criteria for First Order. Some Useful Results from Analysis and Fixed Point Theorems. Notes and General Discussions. References. Oscillations of Differential Equations with Deviating Arguments. Oscillation Theorems (I). Oscillation Theorems (II). Comparison Theorems for Second Order Functional Differential Equations. Oscillation of Functional Equations with a Damping Term. Oscillation of Second Order Linear Delay Differential Equations. Oscillation of Forced Functional Differential Equations. Oscillation of Functional Equations with Damping and Forcing Terms. Necessary and Sufficient Conditions for the Oscillation of Forced Equations. Oscillation for Perturbed Differential Equations. Asymptotic Behavior of Oscillatory Solutions of Functional Equations. Notes and General Discussions. References. Oscillation of Neutral Functional Differential Equations. Oscillation of Nonlinear Neutral Equations. Oscillation of Neutral Equations with Damping. Oscillation of Forced Neutral Equations. Oscillation of Neutral Equations with Mixed Type. Necessary and Sufficient Conditions for Oscillations of Neutral Equations with Deviating Arguments. Comparison and Linearized Oscillation Theorems for Neutral Equations. Existence of Nonoscillatory Solutions of Neutral Delay Differential Equations. Asymptotic Behavior of Nonoscillatory Solutions of Neutral Nonlinear Delay Differential Equations. Notes and General Discussions. References. Conjugacy and Nonoscillation for Second Order Differential Equations. Conjugacy of Linear Second Order Ordinary Differential Equations. Nonoscillation Theorems. Integral Conditions and Nonoscillations. Notes and General Discussions. References. Oscillation of Impulsive Differential Equations. Oscillation Criteria for Impulsive Delay Differential Equations. Oscillation of Second Order Linear Differential Equations with Impulses. Notes and General Discussions. References. Subject Index with Deviating Arguments. Oscillation of Neutral Functional Differential Equations. Conjugacy and Nonoscillation for Second Order Differential Equations. Oscillation of Impulsive Differential Equations.

Oscillation theorems for nonlinear differential equations of second order

Abstract In this paper, we present some new criteria for the oscillation of the differential equation ( a ( t ) ψ ( x ( t )) x . ( t )) . + p ( t ) x . ( t ) + q ( t ) f ( x ( t )) = 0. The obtained results extend, improve, and correlate a number of existing criteria.

On the oscillation of certain third order nonlinear functional differential equations

Abstract We offer some sufficient conditions for the oscillation of all solutions of third order nonlinear functional differential equations of the form d d t a ( t ) d 2 d t 2 x ( t ) α + q ( t ) f ( x [ g ( t ) ] ) = 0 and d d t a ( t ) d 2 d t 2 x ( t ) α = q ( t ) f ( x [ g ( t ) ] ) + p ( t ) h ( x [ σ ( t ) ] ) , when ∫ ∞ a - 1 / α ( s ) d s ∞ . The case when ∫ ∞ a - 1 / α ( s ) d s = ∞ is also included.

On the oscillation of second-order half-linear dynamic equations1

We obtain some oscillation criteria for solutions to the second-order half-linear dynamic equation when or . These criteria unify and extend known criteria for corresponding half-linear differential and difference equations. Some of our results are new even in the continuous and the discrete cases.

On the oscillation of fractional differential equations

In this paper we initiate the oscillation theory for fractional differential equations. Oscillation criteria are obtained for a class of nonlinear fractional differential equations of the form

Asymptotic stability of certain neutral differential equations

D_a^q x + f_1 (t,x) = v(t) + f_2 (t,x),\mathop {\lim }\limits_{t \to a} J_a^{1 - q} x(t) = b_1

Oscillation theorems for certain neutral functional differential equations

, where Daq denotes the Riemann-Liouville differential operator of order q, 0 < q ≤ 1. The results are also stated when the Riemann-Liouville differential operator is replaced by Caputo’s differential operator.