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Keywords:
neutral type difference equation; nonoscillatory solution; asymptotic behavior; oscillation; third order linear difference equations
neutral type difference equation; nonoscillatory solution; asymptotic behavior; oscillation; third order linear difference equations
Summary:
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: \[ \BOF\align \lbrace 0<p(n)&\le 1, \ \sigma (n)=n+k,\ \tau (n)=n+l\rbrace , \lbrace p(n)&>1, \ \sigma (n)=n-k,\ \tau (n)=n-l\rbrace , \BOF\endalign \] where $k$, $l$ are positive integers and we obtain sufficient conditions under which all solutions of the above equations are oscillatory.
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