Nonlinear oscillation of second-order dynamic equations on time scales

Interval oscillation criteria are established for a second-order nonlinear dynamic equation on time scales by utilizing a generalized Riccati technique and the Young inequality. The theory can be applied to second-order dynamic equations regardless of the choice of delta or nabla derivatives.     

Oscillation for nonlinear second order dynamic equations on a time scale

We obtain some oscillation criteria for solutions to the nonlinear dynamic equation

xΔΔ+q(t)xΔσ+p(t)(f○xσ)=0,     

Oscillation of second-order nonlinear neutral dynamic equations on time scales

This paper is concerned with the oscillation of second-order nonlinear neutral dynamic equations of the form

(r(t)((y(t)+p(t)y(τ(t)))^Δ)^γ)^Δ+f(t,y(δ(t)))=0,     

Kamenev-type oscillation criteria for second order nonlinear dynamic equations on time scales

The purpose of this paper to establish oscillation criteria for second order nonlinear dynamic equation

(r(t)(x^Δ(t))^γ)^Δ+f(t,x(g(t)))=0,     

二阶非线性矩阵微分方程的振动性

Some new oscillation criteria are established for the second-order matrix differential system(r(t)Z′(t))′ p(t)Z′(t) Q(t)F(Z′(t))G(Z(t)) = 0, t ≥ to ＞ 0,are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [t0, ∞), rather than on the whole half-line. The results weaken the condition of Q(t) and generalize some well-known results of Wong (1999) to nonlinear matrix differential equation.     

Oscillation criteria for second-order nonlinear delay dynamic equations

In this paper, we consider the second-order nonlinear delay dynamic equation

(r(t)xΔ(t)Δ⁾+p(t)f(x(τ(t)))=0,     

Oscillation criteria for half-linear dynamic equations on time scales

This paper is concerned with oscillation of the second-order half-linear dynamic equation

(r(t)(xΔγ⁾Δ⁾+p(t)xγ(t)=0,     

Oscillation of second-order forced functional dynamic equations with oscillatory potentials

Oscillation criteria are established for a second-order forced dynamic equation on time scales containing both delay and advance arguments. Moreover, the potentials are allowed to change sign. Several nontrivial examples from difference equations are provided to illustrate the easy application of the results. The theory can be applied to second-order dynamic equations regardless of the choice of delta or nabla derivatives.     

A note on oscillation criteria for second-order neutral dynamic equations on isolated time scales

The principal goal of this paper is to amend oscillation results obtained in the recent paper by Saker and O’Regan (2011) [9].     

Li type oscillation theorem for delay dynamic equations

In this paper, we give a new sufficient condition for oscillation of first‐order delay dynamic equations on time scales, which generalize the main results of the papers [Proc. Amer. Math. Soc. 124 (1996), no. 12, 3729–3737] by Li and [Comput. Math. Appl. 37 (1999), no. 7, 11–20] by Tang and Yu. To emphasize the significance of the new result, an example for which all the results fail is also given on a nonstandard time scale. Copyright © 2013 John Wiley & Sons, Ltd.     

Wong's comparison theorem for second order linear dynamic equations on time scales

We obtain Wong-type comparison theorems for second order linear dynamic equations on a time scale. The results obtained extend and are motivated by Wong's comparison theorems. As a particular application of our results, we show that the difference equation

     

Some new oscillation results for a nonlinear dynamic system on time scales

We study the oscillation of a system of two first order nonlinear equations on time scales. This form includes the classical Emden-Fowler differential equation and many of its extensions. We generalize some well-known results of Atkinson, Bohner, Erbe, Peterson and others. We illustrate the results by several examples, including a superlinear Emden-Fowler dynamic system.     

Oscillation theorems for second-order nonlinear neutral delay dynamic equations on time scales

By employing the generalized Riccati transformation technique, we will establish some new oscillation criteria and study the asymptotic behavior of the nonoscillatory solutions of the second-order nonlinear neutral delay dynamic equation

Hille and Nehari type criteria for third-order dynamic equations

In this paper, we extend the oscillation criteria that have been established by Hille [E. Hille, Non-oscillation theorems, Trans. Amer. Math. Soc. 64 (1948) 234-252] and Nehari [Z. Nehari, Oscillation criteria for second-order linear differential equations, Trans. Amer. Math. Soc. 85 (1957) 428-445] for second-order differential equations to third-order dynamic equations on an arbitrary time scale T, which is unbounded above. Our results are essentially new even for third-order differential and difference equations, i.e., when T=R and T=N. We consider several examples to illustrate our results.     

Kamenev-type and interval oscillation criteria for second-order linear differential equations on a measure chain

We establish Kamenev-type criteria and interval criteria for oscillation of the second-order scalar differential equation (p(t)x^Δ(t))^Δ+q(t)x(σ(t))=0 on a measure chain. Our results cover those for differential equations and provide new oscillation criteria for difference equations. Several examples are given to show the significance of the results.     

Adapted Riccati technique and non-oscillation of linear and half-linear equations

The aim of this paper is to mention a generalization of the adapted Riccati equation and, using this method, to prove a non-oscillatory result concerning half-linear differential equations with coefficients having mean values. Note that this result is new even for linear equations.     

Some oscillation results for second-order nonlinear delay dynamic equations

This paper is concerned with oscillatory behavior of a class of second-order delay dynamic equations on a time scale. Two new oscillation criteria are presented that improve some known results in the literature. The results obtained are sharp even for the second-order ordinary differential equations.