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### On asymptotic behaviour of solutions of some difference equation

Mathematica Slovaca

### Asymptotic behaviour of solutions of third order nonlinear difference equations of neutral type

Mathematica Bohemica

In the paper we consider the difference equation of neutral type ${\Delta }^{3}\left[x\left(n\right)-p\left(n\right)x\left(\sigma \left(n\right)\right)\right]+q\left(n\right)f\left(x\left(\tau \left(n\right)\right)\right)=0,\phantom{\rule{1.0em}{0ex}}n\in ℕ\left({n}_{0}\right),$ where $p,q:ℕ\left({n}_{0}\right)\to {ℝ}_{+}$; $\sigma ,\tau :ℕ\to ℤ$, $\sigma$ is strictly increasing and $\underset{n\to \infty }{lim}\sigma \left(n\right)=\infty ;$ $\tau$ is nondecreasing and $\underset{n\to \infty }{lim}\tau \left(n\right)=\infty$, $f:ℝ\to ℝ$, $xf\left(x\right)>0$. We examine the following two cases: $0 and $1<{\lambda }_{*}\le p\left(n\right),\phantom{\rule{1.0em}{0ex}}\sigma \left(n\right)=n+k,\phantom{\rule{1.0em}{0ex}}\tau \left(n\right)=n+l,$ where $k$, $l$ are positive integers. We obtain sufficient conditions under which all nonoscillatory solutions of the above equation tend to zero as $n\to \infty$ with a weaker assumption on $q$ than the usual assumption $\sum _{i={n}_{0}}^{\infty }q\left(i\right)=\infty$ that is used in literature.

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