JEDNOSTKA NAUKOWA KATEGORII A+

# Wydawnictwa / Czasopisma IMPAN / Annales Polonici Mathematici / Wszystkie zeszyty

## Forced oscillation of third order nonlinear dynamic equations on time scales

### Tom 99 / 2010

Annales Polonici Mathematici 99 (2010), 79-87 MSC: 34K11, 39A10, 39A99. DOI: 10.4064/ap99-1-7

#### Streszczenie

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.