S. H. Saker: On Oscillation of a Certain Class of Third-Order Nonlinear Functional Dynamic Equations on Time Scales, p.365-389

In this paper, we establish some new sufficient conditions for oscillation of the third order nonlinear functional dynamic equation

$\begin{equation*} \left[ p(t)\left[ (r(t)x^{\Delta }(t))^{\Delta }\right] ^{\gam... ...t)))=0,\text{ \ for }t\in \lbrack t_{0},\infty )_{% \mathbb{T}}, \end{equation*}$

on a time scale ${\mathbb{T}}$ , where $\gamma >0$ is the quotient of odd positive integers,

and $\tau$ are positive rd-continuous functions defined on ${\mathbb{T}}$ and $f\in C({\mathbb{R}},{\mathbb{R}}% \mathbf{)}$ ,

and $f(u)/u^{\gamma }\geqslant K>0,$ for $u\neq 0$ . The results provided substantial improvement over those obtained by Yu and Wang [J. Comp. Appl. Math. 225 (2009), 531-540] and Hassan [Oscillation of third order nonlinear delay dynamic equations on time scales, Math. Comp. Modelling 49 (2009), 1573-1586], in the sense that our results can be applied when $0<\gamma <1,$ $(\tau \circ \sigma )(t)\neq (\sigma \circ \tau )(t),$ and do not require that $\int_{t_{0}}^{\infty }q(t)\Delta t=\infty$ . Some examples illustrating the main results are given.

S. H. Saker: On Oscillation of a Certain Class of Third-Order Nonlinear Functional Dynamic Equations on Time Scales, p.365-389

Abstract: