On the oscillation of solutions of third order linear difference equations of neutral type

Abstract: In this note we consider the third order linear difference equations of neutral type
\label{E} \Delta^{3}[x(n)-p(n)x(\sigma(n))]+\delta q(n)x(\tau(n))=0, \quad n \in N(n_0), \tag{$ E$}
where $\delta=\pm1$, $p,q N(n_0)\rightarrow\bb R_+;$ $\sigma,\tau N(n_0)\rightarrow\bb N$, $\lim_{n \rightarrow\infty}\sigma(n)= \lim\limits_{n \rightarrow\infty}\tau(n)= \infty.$ We examine the following two cases:
\align\{0<p(n)&\leq1, \sigma(n)=n+k, \tau(n)=n+l\},
\{p(n)&>1, \sigma(n)=n-k, \tau(n)=n-l\},
where $k$, $l$ are positive integers and we obtain sufficient conditions under which all solutions of the above equations are oscillatory.

Keywords: neutral type difference equation, nonoscillatory solution, asymptotic behavior

Classification (MSC2000): 39A11

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