On the oscillation of solutions of third order linear difference equations of neutral type. (English) Zbl 1110.39002

Andruch-Sobiło, Anna; Migda, Małgorzata

Math. Bohem. 130, No. 1, 19-33 (2005).

Summary: We consider the third order linear difference equations of neutral type \(\Delta ^{3}[x(n)-p(n)x(\sigma (n))]+\delta q(n)x(\tau (n))=0\), \(n \in N(n_0),\) where \(\delta =\pm 1\), \(p,q\: N(n_0)\rightarrow \mathbb R_+;\) \(\sigma ,\tau \: N(n_0)\rightarrow \mathbb N\), \(\lim _{n \rightarrow \infty }\sigma (n)= \lim \limits _{n \rightarrow \infty }\tau (n)= \infty .\) We examine the following two cases: \[ \begin{aligned} \{0<p(n)&\leq 1, \;\sigma (n)=n+k,\;\tau (n)=n+l\},\\ \{p(n)&>1, \;\sigma (n)=n-k,\;\tau (n)=n-l\}, \end{aligned} \] where \(k\), \(l\) are positive integers and we obtain sufficient conditions under which all solutions of the above equations are oscillatory.

MSC:

39A11

Stability of difference equations (MSC2000)

Keywords:

neutral type difference equation; nonoscillatory solution; asymptotic behavior; oscillation; third order linear difference equations