### Oscillation theorems for second order damped nonlinear difference equations

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Some new oscillation and nonoscillation criteria for the second order neutral delay difference equation $$\Delta \left({c}_{n}\Delta ({y}_{n}+{p}_{n}{y}_{n-k})\right)+{q}_{n}{y}_{n+1-m}^{\beta}=0,\phantom{\rule{1.0em}{0ex}}n\ge {n}_{0}$$ where $k$, $m$ are positive integers and $\beta $ is a ratio of odd positive integers are established, under the condition ${\sum}_{n={n}_{0}}^{\infty}\frac{1}{{c}_{n}}<\infty .$

The aim of this paper is to study asymptotic properties of the third-order quasi-linear neutral functional differential equation $${\left[a\left(t\right){\left({[x\left(t\right)+p\left(t\right)x\left(\delta \left(t\right)\right)]}^{\text{'}\text{'}}\right)}^{\alpha}\right]}^{\text{'}}+q\left(t\right){x}^{\alpha}\left(\tau \left(t\right)\right)=0\phantom{\rule{0.166667em}{0ex}},E$$ where $\alpha >0$, $0\le p\left(t\right)\le {p}_{0}<\infty $ and $\delta \left(t\right)\le t$. By using Riccati transformation, we establish some sufficient conditions which ensure that every solution of () is either oscillatory or converges to zero. These results improve some known results in the literature. Two examples are given to illustrate the main results.

This paper deals with oscillatory and asymptotic behaviour of solutions of second order quasilinear difference equation of the form $$\Delta ({a}_{n-1}|\Delta {y}_{n-1}{|}^{\alpha -1}\Delta {y}_{n-1})+F(n,{y}_{n})=G(n,{y}_{n},\Delta {y}_{n}),\phantom{\rule{1.0em}{0ex}}n\in N\left({n}_{0}\right)\phantom{\rule{2.0em}{0ex}}\left(\mathrm{E}\right)$$ where $\alpha >0$. Some sufficient conditions for all solutions of (E) to be oscillatory are obtained. Asymptotic behaviour of nonoscillatory solutions of (E) are also considered.

The asymptotic and oscillatory behavior of solutions of mth order damped nonlinear difference equation of the form $$\Delta \left({a}_{n}{\Delta}^{m-1}{y}_{n}\right)+{p}_{n}{\Delta}^{m-1}{y}_{n}+{q}_{n}f\left({y}_{\sigma (n+m-1)}\right)=0$$ where $m$ is even, is studied. Examples are included to illustrate the results.

We shall investigate the properties of solutions of second order linear difference equations defined over a discrete Hardy field via canonical valuations.

In this paper the authors give necessary and sufficient conditions for the oscillation of solutions of nonlinear delay difference equations of Emden– Fowler type in the form ${\Delta}^{2}{y}_{n-1}+{q}_{n}{y}_{\sigma \left(n\right)}^{\gamma}={g}_{n}$, where $\gamma $ is a quotient of odd positive integers, in the superlinear case $(\gamma >1)$ and in the sublinear case $(\gamma <1)$.

Some new criteria for the oscillation of third order nonlinear neutral difference equations of the form $$\Delta \left({a}_{n}{\left({\Delta}^{2}({x}_{n}+{b}_{n}{x}_{n-\delta})\right)}^{\alpha}\right)+{q}_{n}{x}_{n+1-\tau}^{\alpha}=0$$ and $$\Delta \left({a}_{n}{\left({\Delta}^{2}({x}_{n}-{b}_{n}{x}_{n-\delta})\right)}^{\alpha}\right)+{q}_{n}{x}_{n+1-\tau}^{\alpha}=0$$ are established. Some examples are presented to illustrate the main results.

The asymptotic and oscillatory behavior of solutions of Volterra summation equations $${y}_{n}={p}_{n}\pm \sum _{s=0}^{n-1}K(n,s)f(s,{y}_{s}),\phantom{\rule{4pt}{0ex}}n\in \mathbb{N}$$ where $\mathbb{N}=\{0,1,2,\cdots \}$, are studied. Examples are included to illustrate the results.

The aim of this paper is to investigate the oscillatory and asymptotic behavior of solutions of a third-order delay difference equation. By using comparison theorems, we deduce oscillation of the difference equation from its relation to certain associated first-order delay difference equations or inequalities. Examples are given to illustrate the main results.

In this paper, we present several sufficient conditions for the existence of nonoscillatory solutions to the following third order neutral type difference equation $${\Delta}^{3}({x}_{n}+{a}_{n}{x}_{n-l}+{b}_{n}{x}_{n+m})+{p}_{n}{x}_{n-k}-{q}_{n}{x}_{n+r}=0,\phantom{\rule{1.0em}{0ex}}n\ge {n}_{0}$$ via Banach contraction principle. Examples are provided to illustrate the main results. The results obtained in this paper extend and complement some of the existing results.

We obtain some new sufficient conditions for the oscillation of the solutions of the second-order quasilinear difference equations with delay and advanced neutral terms. The results established in this paper are applicable to equations whose neutral coefficients are unbounded. Thus, the results obtained here are new and complement some known results reported in the literature. Examples are also given to illustrate the applicability and strength of the obtained conditions over the known ones.

Sufficient conditions are obtained for the third order nonlinear delay difference equation of the form $$\Delta \left({a}_{n}\left(\Delta \left({b}_{n}{\left(\Delta {y}_{n}\right)}^{\alpha}\right)\right)\right)+{q}_{n}f\left({y}_{\sigma \left(n\right)}\right)=0$$ to have property $\left(\mathrm{A}\right)$ or to be oscillatory. These conditions improve and complement many known results reported in the literature. Examples are provided to illustrate the importance of the main results.

We study the oscillatory behavior of the second-order quasi-linear retarded difference equation $$\Delta \left(p\left(n\right){\left(\Delta y\left(n\right)\right)}^{\alpha}\right)+\eta \left(n\right){y}^{\beta}(n-k)=0$$ under the condition ${\sum}_{n={n}_{0}}^{\infty}{p}^{-\frac{1}{\alpha}}\left(n\right)<\infty $ (i.e., the noncanonical form). Unlike most existing results, the oscillatory behavior of this equation is attained by transforming it into an equation in the canonical form. Examples are provided to show the importance of our main results.

The authors consider the difference equation $${\Delta}^{m}[{y}_{n}-{p}_{n}{y}_{n-k}]+\delta {q}_{n}{y}_{\sigma (n+m-1)}=0\phantom{\rule{2.0em}{0ex}}(*)$$ where $m\ge 2$, $\delta =\pm 1$, $k\in {N}_{0}=\{0,1,2,\cdots \}$, $\Delta {y}_{n}={y}_{n+1}-{y}_{n}$, ${q}_{n}>0$, and $\left\{\sigma \right(n\left)\right\}$ is a sequence of integers with $\sigma \left(n\right)\le n$ and ${lim}_{n\to \infty}\sigma \left(n\right)=\infty $. They obtain results on the classification of the set of nonoscillatory solutions of ($*$) and use a fixed point method to show the existence of solutions having certain types of asymptotic behavior. Examples illustrating the results are included.

2000 Mathematics Subject Classification: 34C15

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