Asymptotic behaviour of solutions of third order nonlinear difference equations of neutral type. (English) Zbl 1199.39022

Andruch-Sobiło, Anna; Drozdowicz, Andrzej

Math. Bohem. 133, No. 3, 247-258 (2008).

Summary: We consider the difference equation of neutral type \[ \Delta ^{\!3}[x(n)-p(n)x(\sigma (n))] + q(n)f(x(\tau (n)))=0, \quad n \in \mathbb N (n_0), \] where \(p,q\:\mathbb N(n_0)\rightarrow \mathbb R_+\); \(\sigma , \tau \:\mathbb N\rightarrow \mathbb Z\), \(\sigma \) is strictly increasing and \(\lim \limits _{n \rightarrow \infty }\sigma (n)=\infty \); \(\tau \) is nondecreasing and \(\lim \limits _{n \rightarrow \infty }\tau (n)=\infty \), \(f\:\mathbb R\rightarrow {\mathbb R}\), \(xf(x)>0\). We examine the following two cases: \[ 0<p(n)\leq \lambda ^*< 1,\quad \sigma (n)=n-k,\quad \tau (n)=n-l, \] 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 \(n\rightarrow \infty \) with a weaker assumption on \(q\) than the usual assumption \(\sum \limits _{i=n_0}^{\infty }q(i)=\infty \) that is used in the literature.

MSC:

39A22	Growth, boundedness, comparison of solutions to difference equations
39A10	Additive difference equations
39A12	Discrete version of topics in analysis
34K40	Neutral functional-differential equations
39A21	Oscillation theory for difference equations

Keywords:

neutral type difference equation; third order difference equation; nonoscillatory solutions; asymptotic behavior