Geodesic
On the oscillation of solutions of third order linear difference equations of neutral type
Mathematica Bohemica, Tome 130 (2005) no. 1, pp. 19-33.

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In this note we consider the third order linear difference equations of neutral type \[ \Delta ^{3}[x(n)-p(n)x(\sigma (n))]+\delta q(n)x(\tau (n))=0, \quad n \in N(n_0), \qquad \mathrm{({\mathrm E})}\] 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: \[ \BOF\align \lbrace 0(n)\le 1, \ \sigma (n)=n+k,\ \tau (n)=n+l\rbrace , \lbrace p(n)>1, \ \sigma (n)=n-k,\ \tau (n)=n-l\rbrace , \BOF\endalign \] where $k$, $l$ are positive integers and we obtain sufficient conditions under which all solutions of the above equations are oscillatory.
DOI : 10.21136/MB.2005.134217
Classification : 39A11
Keywords: neutral type difference equation; nonoscillatory solution; asymptotic behavior; oscillation; third order linear difference equations 
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Andruch-Sobiło, Anna; Migda, Małgorzata. On the oscillation of solutions of third order linear difference equations of neutral type. Mathematica Bohemica, Tome 130 (2005) no. 1, pp. 19-33. doi : 10.21136/MB.2005.134217. http://geodesic.mathdoc.fr/articles/10.21136/MB.2005.134217/

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