Asymptotic behaviour of solutions of third order nonlinear difference equations of neutral type

Andruch-Sobiło, Anna ; Drozdowicz, Andrzej

Mathematica Bohemica, Tome 133 (2008) no. 3, pp. 247-258.

Résumé

In the paper we consider the difference equation of neutral type $$ \Delta ^{3}[x(n)-p(n)x(\sigma (n))] + q(n)f(x(\tau (n)))=0, \quad n \in \Bbb N (n_0), $$ where $p,q:\mathbb{N}(n_0)\to {\mathbb{R}}_{+}$; $\sigma ,\tau :\mathbb{N}\to \mathbb{Z}$, $\sigma$ is strictly increasing and $\underset{n\to \mathrm{\infty }}{lim}\sigma (n)=\mathrm{\infty };$ $\tau$ is nondecreasing and $\underset{n\to \mathrm{\infty }}{lim}\tau (n)=\mathrm{\infty }$, $f:\mathbb{R}\to \mathbb{R}$, $xf(x)>0$. We examine the following two cases: \[ 0(n)\leq \lambda ^* 1,\quad \sigma (n)=n-k,\quad \tau (n)=n-l, \] and \[1\lambda _*\leq p(n),\quad \sigma (n)=n+k,\quad \tau (n)=n+l,\] where $k$, $l$ are positive integers. We obtain sufficient conditions under which all nonoscillatory solutions of the above equation tend to zero as $n\to \mathrm{\infty }$ with a weaker assumption on $q$ than the usual assumption $\sum _{i=n_0}^{\mathrm{\infty }}q(i)=\mathrm{\infty }$ that is used in literature.

Détail

DOI : 10.21136/MB.2008.140615

Classification : 34K40, 39A10, 39A12, 39A21, 39A22
Mots-clés : neutral type difference equation; third order difference equation; nonoscillatory solutions; asymptotic behavior