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: \[ \BOF\align \lbrace 01, \ \sigma (n)=n-k,\ \tau (n)=n-l\rbrace , \BOF\endalign \] where $k$, $l$ are positive integers and we obtain sufficient conditions under which all solutions of the above equations are oscillatory.
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 
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
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