Geodesic
Asymptotic behaviour of solutions of third order nonlinear difference equations of neutral type
Mathematica Bohemica, Tome 133 (2008) no. 3, pp. 247-258

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In the paper we consider the difference equation of neutral type $$ \Delta ^{3}[x(n)-p(n)x(\sigma (n))] + q(n)f(x(\tau (n)))=0, \quad n \in \Bbb N (n_0), $$ where $p,q\colon\Bbb N(n_0)\rightarrow \Bbb R_+$; $\sigma , \tau \colon\Bbb N\rightarrow \Bbb Z$, $\sigma $ is strictly increasing and $\lim \limits _{n \rightarrow \infty }\sigma (n)=\infty ;$ $\tau $ is nondecreasing and $\lim \limits _{n \rightarrow \infty }\tau (n)=\infty $, $f\colon\Bbb R\rightarrow {\Bbb R}$, $xf(x)>0$. We examine the following two cases: \[ 0
In the paper we consider the difference equation of neutral type $$ \Delta ^{3}[x(n)-p(n)x(\sigma (n))] + q(n)f(x(\tau (n)))=0, \quad n \in \Bbb N (n_0), $$ where $p,q\colon\Bbb N(n_0)\rightarrow \Bbb R_+$; $\sigma , \tau \colon\Bbb N\rightarrow \Bbb Z$, $\sigma $ is strictly increasing and $\lim \limits _{n \rightarrow \infty }\sigma (n)=\infty ;$ $\tau $ is nondecreasing and $\lim \limits _{n \rightarrow \infty }\tau (n)=\infty $, $f\colon\Bbb R\rightarrow {\Bbb R}$, $xf(x)>0$. We examine the following two cases: \[ 0

(n)\leq \lambda ^* 1,\quad \sigma (n)=n-k,\quad \tau (n)=n-l, \] and \[1\lambda _*\leq p(n),\quad \sigma (n)=n+k,\quad \tau (n)=n+l,\] where $k$, $l$ are positive integers. We obtain sufficient conditions under which all nonoscillatory solutions of the above equation tend to zero as $n\rightarrow \infty $ with a weaker assumption on $q$ than the usual assumption $\sum \limits _{i=n_0}^{\infty }q(i)=\infty $ that is used in literature.
DOI : 10.21136/MB.2008.140615
Classification : 34K40, 39A10, 39A12, 39A21, 39A22
Keywords: neutral type difference equation; third order difference equation; nonoscillatory solutions; asymptotic behavior 
Andruch-Sobiło, Anna; Drozdowicz, Andrzej. Asymptotic behaviour of solutions of third order nonlinear difference equations of neutral type. Mathematica Bohemica, Tome 133 (2008) no. 3, pp. 247-258. doi: 10.21136/MB.2008.140615
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