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