Forced oscillation of third order nonlinear dynamic equations on time scales

Baoguo Jia¹

¹Department of Mathematics Zhongshan University Guangzhou, China 510275

Annales Polonici Mathematici, Tome 99 (2010) no. 1, pp. 79-87

Résumé

Consider the third order nonlinear dynamic equation $$ x^{\Delta\Delta\Delta}(t)+p(t)f(x)=g(t),\tag{$*$} $$ on a time scale $\mathbb T$ which is unbounded above. The function $f \in C(\mathcal R,\mathcal R)$ is assumed to satisfy $xf(x)>0$ for $x\neq 0$ and be nondecreasing. We study the oscillatory behaviour of solutions of $(*)$. As an application, we find that the nonlinear difference equation $$ \Delta^3x(n)+n^{\alpha}|x|^\gamma {\rm sgn}(n)=(-1)^nn^c, $$ where $\alpha\geq -1$, $\gamma>0$, $c>3$, is oscillatory.

Détail

DOI : 10.4064/ap99-1-7

Keywords: consider third order nonlinear dynamic equation delta delta delta x tag * time scale mathbb which unbounded above function mathcal mathcal assumed satisfy neq nondecreasing study oscillatory behaviour solutions * application nonlinear difference equation delta alpha gamma sgn where alpha geq gamma oscillatory