Nonoscillation Criteria for Nonlinear Delay Dynamic Systems on Time Scales

Oscillation and Nonoscillation Criteria for Nonlinear Dynamic Systems on Time Scales

Oscillation and nonoscillation criteria for linear dynamic systems on time scales

We consider the nonlinear dynamic system x Δ t atgyt, y Δ t −ft, x σ t. We establish some necessary and sufficient conditions for the existence of oscillatory and nonoscillatory solutions with special asymptotic properties for the system. We generalize the known results in the literature. Some examples are included to illustrate the results.

Oscillation and Nonoscillation Criteria for Two-dimensional Time-scale Systems of First-order Nonlinear Dynamic Equations

Oscillation criteria for two-dimensional difference and differential systems of first-order linear difference equations are generalized and extended to nonlinear dynamic equations on arbitrary time scales. This unifies and extends under one theory previous linear results from discrete and continuous systems. An example is given illustrating that a key theorem is sharp on all time scales. 1. pre...

A Nonlinear Sturm–Picone Comparison Theorem for Dynamic Equations on Time Scales

The authors derive an analog of the well-known Picone identity but for nonlinear dynamic equations on time scales. As a consequence, they obtain a nonlinear comparison theorem in the spirit of the classical Sturm–Picone comparison theorem. Comparison results yielding the nonoscillation of all solutions of nonlinear equations are also obtained. AMS subject classification: 34C10, 34C15, 39A11.

Oscillation Criteria for Second-Order Quasilinear Neutral Delay Dynamic Equations on Time Scales

We establish some new oscillation criteria for the second-order quasilinear neutral delay dynamic equations r t zΔ t γ Δ q1 t x τ1 t q2 t x τ2 t 0 on a time scale T, where z t x t p t x τ0 t , 0 < α < γ < β. Our results generalize and improve some known results for oscillation of second-order nonlinear delay dynamic equations on time scales. Some examples are considered to illustrate our main r...