Oscillation criteria for second order sublinear dynamic equations with damping term

This paper concerns the oscillation of solutions to the second order sublinear dynamic equations with damping. No sign conditions are imposed on coefficients. We illustrate the results by several examples.     

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.     

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

Oscillation criteria for higher‐order nonlinear dynamic equations with Laplacians and a deviating argument on time scales

In this paper, we study the n th‐order nonlinear dynamic equation with Laplacians and a deviating argument on an above‐unbounded time scale, where n ?2, New oscillation criteria are established for the cases when n is even and odd and when α  > γ ,α  = γ , and α  < γ , respectively, with α  = α ₁?α _{n  ? 1}. Copyright © 2017 John Wiley & Sons, Ltd.     

时间尺度上具阻尼项的二阶半线性时滞动力方程的振动准则

利用广义Riccati变换和不等式技巧,研究了一类具阻尼项的二阶半线性时滞动力方程解的振动性质,在一定条件下,建立了四个新的振动准则,其结果不仅推广和改进了已知的一些结果,而且在时间尺度上统一了具阻尼项的二阶半线性时滞微分方程和差分方程解的振动性质.     

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 Second Order Nonlinear Dynamic Equations with p-Laplacian and Damping

This paper concerns the oscillation of solutions of the second order nonlinear dynamic equation with p-Laplacian and damping

(r(t)φα(x^△(t)))^△ +p(t)φα(x△σ(t))  +q(t)f(xσ(t)) =0${\left(r\left(t\right)\phi \alpha \left(x^{△}\left(t\right)\right)\right)}^{△}\hspace{0.17em}+p\left(t\right)\phi \alpha \left(x^{△\sigma }\left(t\right)\right)\hspace{0.17em}\hspace{0.17em}+q\left(t\right)f\left(x^{\sigma }\left(t\right)\right)\hspace{0.17em}=0$

on a time scale ? which is unbounded above. Sign changes are allowed for the coefficient functions r, p and q. Several examples are given to illustrate the main results.     

Oscillation criteria for higher order nonlinear dynamic equations

In this paper, we will consider the higher‐order functional dynamic equations of the form on an above‐unbounded time scale , where and , . The function is a rd‐continuous function such that . The results extend and improve some known results in the literature on higher order nonlinear dynamic equations.     

时间尺度上具阻尼项的二阶半线性时滞动力方程振动性的新结果

本文讨论一类具阻尼项的二阶半线性时滞动力方程解的振动性质, 利用广义Riccati 变换和不等式技巧, 在一定条件下, 建立了4 个新的振动准则, 其结果改进和推广了已知的一些结果.     

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,     

时标上的二阶变时滞动力方程的振动准则

利用广义Riccati技巧及完全平方技巧,讨论了一类时标上的二阶非线性变时滞动力方程的振动性,得到了一些新的振动准则.     

Interval criteria for second-order super-half-linear functional dynamic equations with delay and advance arguments

Interval oscillation criteria are established for second-order forced super half-linear dynamic equations on time scales containing both delay and advance arguments, where the potentials and forcing term are allowed to change sign. Four discrete examples are provided to illustrate the relevance of the results. The theory can be applied to second-order dynamic equations regardless of the choice of delta or nabla derivatives.     

Necessary and sufficient conditions for oscillation of second‐order dynamic equations on time scales

In this paper, we establish necessary and sufficient conditions for oscillation of second‐order strongly superlinear and strongly sublinear dynamic equations. Our results unify and improve many known results in the literature.     

时间尺度上一类二阶具阻尼项的半线性中立型时滞动力方程的振动性

借助时间尺度的有关理论,运用Riccati变换技巧,平均函数技术及不等式技巧,研究了时间尺度上一类二阶具阻尼项的半线性中立型时滞动力方程的振动性,给出该类方程振动的几个充分条件,推广并改进了已有的某些结果.     

Oscillation in odd-order neutral delay differential equations

Consider the odd-order functional differential equation

Oscillation of second-order linear delay differential equations

The aim of this paper is to derive sufficient conditions for the linear delay differential equation (r(t)y′(t))′ + p(t)y(τ(t)) = 0 to be oscillatory by using a generalization of the Lagrange mean-value theorem, the Riccati differential inequality and the Sturm comparison theorem.      

Oscillatory and asymptotic criteria of third order nonlinear delay dynamic equations with damping term on time scales

In this paper, we are concerned with oscillatory and asymptotic behavior of third order nonlinear delay dynamic equations with damping term on time scales. By using a generalized Riccati function and inequality technique, we establish some new oscillatory and asymptotic criteria. The established results on one hand extend some known results in the literature, on the other hand unify continuous and discrete analysis as two special cases of an arbitrary time scale. We also present some applications for the established results.     

Necessary and sufficient conditions on the asymptotic behavior of second‐order neutral delay dynamic equations with positive and negative coefficients

In this paper, we establish necessary and sufficient conditions for the solutions of a second‐order nonlinear neutral delay dynamic equation with positive and negative coefficients to be oscillatory or tend to zero asymptotically. We consider three different ranges of the coefficient associated with the neutral part in one of which it is allowed to be oscillatory. Thus, our results improve and generalize the existing results in the literature to arbitrary time scales. Some examples on nontrivial time scales are also given. Copyright © 2013 John Wiley & Sons, Ltd.     

Sharp oscillation and nonoscillation tests for delay dynamic equations

In this paper, we give new sufficient conditions for both oscillation and nonoscillation of the delay dynamic equation where and satisfy τ(t) ≤ σ(t) for all large t and . As an important corollary, we obtain the time scale invariant integral condition for nonoscillation: for all large t. Also, with some examples, we show that newly presented results are sharp.