THEOREMS OF KIGURADZE-TYPE AND BELOHOREC-TYPE REVISITED ON TIME SCALES

This article concerns the oscillation of second-order nonlinear dynamic equations. By using generalized Riccati transformations, Kiguradzetype and Belohorec-type oscillation theorems are obtained on an arbitrary time scale. Our results cover those for differential equations and difference equations, and provide new oscillation criteria for irregular time scales. Some examples are given to illustrate our results

Oscillation of Second Order Nonlinear Dynamic Equations on Time Scales

By means of Riccati transformation techniques, we establish some oscillation criteria for a second order nonlinear dynamic equation on time scales in terms of the coefficients. We give examples of dynamic equations to which previously known oscillation criteria are not applicable

Forced oscillation of second order nonlinear dynamic equations on time scales

By means of the Kartsatos technique and generalized Riccati transformation techniques, we establish some new oscillation criteria for a second order nonlinear dynamic equations with forced term on time scales in terms of the coefficients

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

Asymptotic properties of solutions of certain third-order dynamic equations

AbstractIn this paper, the well known oscillation criteria due to Hille and Nehari for second-order linear differential equations will be generalized and extended to the third-order nonlinear dynamic equation (r2(t)((r1(t)xΔ(t))Δ)γ)Δ+q(t)f(x(t))=0 on time scale T, where γ≥1 is a ratio of odd positive integers. Our results are essentially new even for third-order differential and difference equations, i.e., when T=R and T=N. Two examples of dynamic equations on different time scales are given to show the applications of our main 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

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