Oscillation constants for second-order nonlinear dynamic equations of Euler type on time scales

journal article in Web of Science

en

We are concerned with the oscillation problem for second-order nonlinear dynamic equations on time scales of the form $x^{\Delta \Delta} + f(x)/(t \sigma(t)) = 0$, where $f(x)$ satisfies $x f(x) > 0$ if $x \neq 0$. By means of Riccati technique and phase plane analysis of a system, (non)oscillation criteria are established. A necessary and sufficient condition for all nontrivial solutions of the Euler-Cauchy dynamic equation $y^{\Delta \Delta} +\lambda/(t \sigma(t))\, y = 0$ to be oscillatory plays a crucial role in proving our results.

We are concerned with the oscillation problem for second-order nonlinear dynamic equations on time scales of the form $x^{\Delta \Delta} + f(x)/(t \sigma(t)) = 0$, where $f(x)$ satisfies $x f(x) > 0$ if $x \neq 0$. By means of Riccati technique and phase plane analysis of a system, (non)oscillation criteria are established. A necessary and sufficient condition for all nontrivial solutions of the Euler-Cauchy dynamic equation $y^{\Delta \Delta} +\lambda/(t \sigma(t))\, y = 0$ to be oscillatory plays a crucial role in proving our results.

Oscillation constant; Dynamic equations on time scales; Euler-Cauchy equation; Riccati technique; Phase plane analysis; Schauder fixed point theorem

07.09.2017

Taylor and Francis

1563-5120

23

11

1884–1900

17