Interval Oscillation Criteria for Forced Second-Order Nonlinear Delay Dynamic Equations with Damping and Oscillatory Potential on Time Scales

We are concerned with the interval oscillation of general type of forced second-order nonlinear dynamic equation with oscillatory potential of the form rtg1xt,xΔtΔ+p(t)g2(x(t),xΔ(t))xΔ(t)+q(t)f(x(τ(t)))=e(t), on a time scale T. We will use a unified approach on time scales and employ the Riccati technique to establish some oscillation criteria for this type of equations. Our results are more general and extend the oscillation criteria of Erbe et al. (2010). Also our results unify the oscillation of the forced second-order nonlinear delay differential equation and the forced second-order nonlinear delay difference equation. Finally, we give some examples to illustrate our results

Qualitative analysis of dynamic equations on time scales

In this article, we establish the Picard-Lindelof theorem and approximating results for dynamic equations on time scale. We present a simple proof for the existence and uniqueness of the solution. The proof is produced by using convergence and Weierstrass M-test. Furthermore, we show that the Lispchitz condition is not necessary for uniqueness. The existence of epsilon-approximate solution is established under suitable assumptions. Moreover, we study the approximate solution of the dynamic equation with delay by studying the solution of the corresponding dynamic equation with piecewise constant argument. We show that the exponential stability is preserved in such approximations.Comment: 13 page

Oscillation theorems for certain third order nonlinear delay dynamic equations on time scales

In this paper, we establish some new oscillation criteria for the third order nonlinear delay dynamic equations $$\left(b(t)\left([a(t)(x^\Delta(t))^{\alpha_1}]^\Delta\right)^{\alpha_2}\right)^\Delta+q(t)x^{\alpha_3}(\tau(t))=0$$ on a time scale $\mathbb{T}$, where $\alpha_i$ are ratios of positive odd integers, $i=1,\ 2,\ 3,$ $b,\ a$ and $q$ are positive real-valued rd-continuous functions defined on $\mathbb{T}$, and the so-called delay function $\tau:\mathbb{T}\rightarrow \mathbb{T}$ is a strictly increasing function such that $\tau(t)\leq t$ for $t\in \mathbb{T}$ and $\tau(t)\rightarrow\infty$ as $t\rightarrow\infty.$ By using the Riccati transformation technique and integral averaging technique, some new sufficient conditions which insure that every solution oscillates or tends to zero are established. Our results are new for third order nonlinear delay dynamic equations and complement the results established by Yu and Wang in J. Comput. Appl. Math., 2009, and Erbe, Peterson and Saker in J. Comput. Appl. Math., 2005. Some examples are given here to illustrate our main results

OSCILLATION OF SOLUTION TO SECOND-ORDER HALF-LINEAR DELAY DYNAMIC EQUATIONS ON TIME SCALES

This article concerns the oscillation of solutions to second-order half-linear dynamic equations with a variable delay. By using integral averaging techniques and generalized Riccati transformations, new oscillation criteria are obtained. Our results extend Kamenev-type, Philos-type and Li-type oscillation criteria. Several examples are given to illustrate our results

Oscillation of Second-Order Nonlinear Delay Dynamic Equations with Damping on Time Scales

We use the generalized Riccati transformation and the inequality technique to establish some new oscillation criteria for the second-order nonlinear delay dynamic equation with damping on a time scale T(r(t)g(x(t), xΔ(t)))Δ+p(t)g(x(t), xΔ(t))  + q(t)f(x(τ(t)))=0, where r(t), p(t), and q(t) are positive right dense continuous (rd-continuous) functions on T. Our results improve and extend some results established by Zhang et al., 2011. Also, our results unify the oscillation of the second-order nonlinear delay differential equation with damping and the second-order nonlinear delay difference equation with damping. Finally, we give some examples to illustrate our main results

On the oscillatory behavior of even order neutral delay dynamic equations on time-scales

We establish some new criteria for the oscillation of the even order neutral dynamic equation \begin{equation*} \left( a(t)\left( \left( x(t)-p(t)x(\tau (t))\right) ^{\Delta^{n-1}}\right) ^{\alpha }\right) ^{\Delta }+q(t)\left( x^{\sigma}(g(t))\right) ^{\lambda }=0 \end{equation*} on a time scale $\mathbb{T}$, where $n \geq 2$ is even, $\alpha$ and $\lambda$ are ratios of odd positive integers, $a$, $p$ and $q$ are real valued positive rd-continuous functions defined on $\mathbb{T}$, and $g$ and $\tau$ are real valued rd-continuous functions on $\mathbb{T}$. Examples illustrating the results are included

Oscillation of Second-Order Nonlinear Delay Dynamic Equations on Time Scales

In this work, we use the generalized Riccati transformation and the inequality technique to establish some new oscillation criteria for the second-order nonlinear delay dynamic equation (p(t)(xΔ(t))γ)Δ+q(t)f(x(τ(t)))=0, on a time scale , where γ is the quotient of odd positive integers and p(t) and q(t) are positive right-dense continuous (rd-continuous) functions on 𝕋. Our results improve and extend some results established by Sun et al. 2009. Also our results unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation. Finally, we give some examples to illustrate our main results