Oscillation criteria for quasi-linear functional dynamic equations on time scales

This paper is concerned with oscillation of the second-order quasilinear functional dynamic equation

$$(r(t)(x^\Delta (t))^\gamma )^\Delta + p(t)x^\beta (\tau (t)) = 0,$$

on a time scale \(\mathbb{T}\) where γ and β are quotient of odd positive integers, r, p, and τ are positive rd-continuous functions defined on \(\mathbb{T},\tau :\mathbb{T} \to \mathbb{T}\) and \(\mathop {\lim }\limits_{t \to \infty } \tau (t) = \infty \). We establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. Our results improve the oscillation results in the literature when γ = β, and τ(t) ≤ t and when τ(t) > t the results are essentially new. Some examples are considered to illustrate the main results.

     

Oscillation criteria for nonlinear neutral functional dynamic equations on time scales

In this paper, we establish some new sufficient conditions for oscillation of the second-order neutral functional dynamic equation $$\left[ {r\left( t \right)\left[ {m\left( t \right)y\left( t \right) + p\left( t \right)y\left( {\tau \left( t \right)} \right)} \right]^\Delta } \right]^\Delta + q\left( t \right)f\left( {y\left( {\delta \left( t \right)} \right)} \right) = 0$$ on a time scale $\mathbb{T}$ which is unbounded above, where m, p, q, r, T and δ are real valued rd-continuous positive functions defined on $\mathbb{T}$ . The main investigation of the results depends on the Riccati substitutions and the analysis of the associated Riccati dynamic inequality. The results complement the oscillation results for neutral delay dynamic equations and improve some oscillation results for neutral delay differential and difference equations. Some examples illustrating our main results are given.     

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 dynamic equations on time scales

Erbe’s and Hassan’s contributions regarding oscillation criteria are interesting in the development of oscillation theory of dynamic equations on time scales. The objective of this paper is to amend these results.     

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 criteria for a class of second-order Emden-Fowler delay dynamic equations on time scales

By means of Riccati transformation technique, we establish some new oscillation criteria for the second-order Emden-Fowler delay dynamic equations

xΔΔ(t)+p(t)xγ(τ(t))=0     

Oscillation criteria of second order half linear delay dynamic equations on time scales

In this paper, we establish some new oscillation criteria for a non autonomous second order delay dynamic equation

$${\left( {r\left( t \right)g\left( {{x^\Delta }\left( t \right)} \right)} \right)^\Delta } + p\left( t \right)f\left( {x\left( {\tau \left( t \right)} \right)} \right) = 0,$$

on a time scale T. Oscillation behavior of this equation is not studied before. Our results not only apply on differential equations when T=?, difference equations when T=? but can be applied on different types of time scales such as when T=q^? for q > 1 and also improve most previous results. Finally, we give some examples to illustrate our main results.

     

On the oscillation for third-order nonlinear neutral delay dynamic equations on time scales

In this paper, we consider the third-order nonlinear neutral delay dynamic equations

$$\begin{aligned} \left( b(t)\left( \left( ((x(t)-p(t)x(\tau (t)))^\Delta )^{\alpha _1}\right) ^\Delta \right) ^{\alpha _2}\right) ^\Delta +f(t,x(\delta (t)))=0 \end{aligned}$$

on a time scale \(\mathbb {T}\), where \(\alpha _i\) are quotients of positive odd integers, \(i=1\), 2, \(|f(t,u)|\ge q(t)|u|\), \(b,\ p\) and q are real-valued positive rd-continuous functions defined on \(\mathbb {T}\). By using the Riccati transformation technique and integral averaging technique, some new sufficient conditions which ensure that every solution oscillates or tends to zero are established. Our results are new for third-order nonlinear neutral delay dynamic equations and extend many known results for oscillation of third order dynamic equations. Some examples are given here to illustrate our main results.

     

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 nonlinear perturbed dynamic equations of second-order on time scales

In this paper, by using the Riccati transformation technique we establish some new oscillation criteria for second-order nonlinear perturbed dynamic equation on time scales. An example illustrating our main results is also given.     

Oscillation of even order nonlinear delay dynamic equations on time scales

One of the important methods for studying the oscillation of higher order differential equations is to make a comparison with second order differential equations. The method involves using Taylor’s Formula. In this paper we show how such a method can be used for a class of even order delay dynamic equations on time scales via comparison with second order dynamic inequalities. In particular, it is shown that nonexistence of an eventually positive solution of a certain second order delay dynamic inequality is sufficient for oscillation of even order dynamic equations on time scales. The arguments are based on Taylor monomials on time scales.     

Oscillation of second-order half-linear delay dynamic equations with damping on time scales

By using the generalized Riccati transformation and the inequality technique, we establish a few new oscillation criterions for certain second-order half-linear delay dynamic equations with damping on a time scale. Our results extend and improve some known results, but also unify the oscillation of the second-order half-linear delay differential equation with damping and the second-order half-linear delay difference equation with damping.     

Oscillation of second-order nonlinear delay dynamic equations with damping on time scales

By using the generalized Riccati transformation and the inequality technique, we establish some new oscillation criteria for the second-order nonlinear delay dynamic equations with damping on a time scale. Our results extend and improve some known theorems, but also unify the oscillation of the second-order nonlinear delay differential equation with damping and the second-order nonlinear delay difference equation with damping.     

时间测度链上一类二阶动力方程的振动准则

研究了时间测度链上的一类二阶非线性中立型时滞动力方程的振动性,利用时间测度链上的理论和一些分析技巧,通过引入参数函数和Riccati变换,得到了该方程振动的几个充分条件,推广和改进了现有文献中的有关结果,并给出了一些例子用以说明文中的主要结论.     

Oscillation of delay differential equations on time scales

Consider the following equation: , where t is in a measure chain. We apply the theory of measure chains to investigate the oscillation and nonoscillation of the above equation on the basis of some well-known results. And in some sense, we show a method to unify the delay differential equation and delay difference equation.