### Asymptotic and oscillatory behavior of second order neutral quantum equations with maxima.

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In the paper we consider the difference equation of neutral type $${\Delta}^{3}[x\left(n\right)-p\left(n\right)x\left(\sigma \left(n\right)\right)]+q\left(n\right)f\left(x\left(\tau \left(n\right)\right)\right)=0,\phantom{\rule{1.0em}{0ex}}n\in \mathbb{N}\left({n}_{0}\right),$$ where $p,q:\mathbb{N}\left({n}_{0}\right)\to {\mathbb{R}}_{+}$; $\sigma ,\tau :\mathbb{N}\to \mathbb{Z}$, $\sigma $ is strictly increasing and $\underset{n\to \infty}{lim}\sigma \left(n\right)=\infty ;$$\tau $ is nondecreasing and $\underset{n\to \infty}{lim}\tau \left(n\right)=\infty $, $f:\mathbb{R}\to \mathbb{R}$, $xf\left(x\right)>0$. We examine the following two cases: $$0<p\left(n\right)\le {\lambda}^{*}<1,\phantom{\rule{1.0em}{0ex}}\sigma \left(n\right)=n-k,\phantom{\rule{1.0em}{0ex}}\tau \left(n\right)=n-l,$$ and $$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...

We investigate the criticality of the one term $2n$-order difference operators $l{\left(y\right)}_{k}={\Delta}^{n}\left({r}_{k}{\Delta}^{n}{y}_{k}\right)$. We explicitly determine the recessive and the dominant system of solutions of the equation $l{\left(y\right)}_{k}=0$. Using their structure we prove a criticality criterion.

Consider the third order nonlinear dynamic equation ${x}^{\Delta \Delta \Delta}\left(t\right)+p\left(t\right)f\left(x\right)=g\left(t\right)$, (*) on a time scale which is unbounded above. The function f ∈ C(,) is assumed to satisfy xf(x) > 0 for x ≠ 0 and be nondecreasing. We study the oscillatory behaviour of solutions of (*). As an application, we find that the nonlinear difference equation $\Delta \xb3x\left(n\right)+{n}^{\alpha}{\left|x\right|}^{\gamma}sgn\left(n\right)=(-1)\u207f{n}^{c}$, where α ≥ -1, γ > 0, c > 3, is oscillatory.

In this paper, we determine the forbidden set and give an explicit formula for the solutions of the difference equation $${x}_{n+1}=\frac{a{x}_{n}{x}_{n-1}}{-b{x}_{n}+c{x}_{n-2}},\phantom{\rule{1.0em}{0ex}}n\in {\mathbb{N}}_{0}$$ where $a$, $b$, $c$ are positive real numbers and the initial conditions ${x}_{-2}$, ${x}_{-1}$, ${x}_{0}$ are real numbers. We show that every admissible solution of that equation converges to zero if either $a<c$ or $a>c$ with $(a-c)/b<1$. When $a>c$ with $(a-c)/b>1$, we prove that every admissible solution is unbounded. Finally, when $a=c$, we prove that every admissible solution converges to zero.

In this paper, we introduce an explicit formula and discuss the global behavior of solutions of the difference equation $${x}_{n+1}=\frac{a{x}_{n-3}}{b+c{x}_{n-1}{x}_{n-3}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.0em}{0ex}}n=0,1,\cdots $$ where $a,b,c$ are positive real numbers and the initial conditions ${x}_{-3}$, ${x}_{-2}$, ${x}_{-1}$, ${x}_{0}$ are real numbers.

In this paper, by using an iterative scheme, we advance the main oscillation result of Zhang and Liu (1997). We not only extend this important result but also drop a superfluous condition even in the noniterated case. Moreover, we present some illustrative examples for which the previous results cannot deliver answers for the oscillation of solutions but with our new efficient test, we can give affirmative answers for the oscillatory behaviour of solutions. For a visual explanation of the examples,...

This paper is concerned with the nonlinear advanced difference equation with constant coefficients $${x}_{n+1}-{x}_{n}+\sum _{i=1}^{m}{p}_{i}{f}_{i}\left({x}_{n-{k}_{i}}\right)=0\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.0em}{0ex}}n=0,1,\cdots $$ where ${p}_{i}\in (-\infty ,0)$ and ${k}_{i}\in \{\cdots ,-2,-1\}$ for $i=1,2,\cdots ,m$. We obtain sufficient conditions and also necessary and sufficient conditions for the oscillation of all solutions of the difference equation above by comparing with the associated linearized difference equation. Furthermore, oscillation criteria are established for the nonlinear advanced difference equation with variable coefficients $${x}_{n+1}-{x}_{n}+\sum _{i=1}^{m}{p}_{in}{f}_{i}\left({x}_{n-{k}_{i}}\right)=0\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.0em}{0ex}}n=0,1,\cdots $$ where ${p}_{in}\le 0$ and ${k}_{i}\in \{\cdots ,-2,-1\}$ for $i=1,2,\cdots ,m$.

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 $(\gamma <1)$.

In the paper, conditions are obtained, in terms of coefficient functions, which are necessary as well as sufficient for non-oscillation/oscillation of all solutions of self-adjoint linear homogeneous equations of the form $$\Delta \left({p}_{n-1}\Delta {y}_{n-1}\right)+q{y}_{n}=0,\phantom{\rule{1.0em}{0ex}}n\ge 1,$$ where $q$ is a constant. Sufficient conditions, in terms of coefficient functions, are obtained for non-oscillation of all solutions of nonlinear non-homogeneous equations of the type $$\Delta \left({p}_{n-1}\Delta {y}_{n-1}\right)+{q}_{n}g\left({y}_{n}\right)={f}_{n-1},\phantom{\rule{1.0em}{0ex}}n\ge 1,$$ where, unlike earlier works, ${f}_{n}\ge 0$ or $\le 0$ (but $\neg \equiv 0)$ for large $n$. Further, these results are used to obtain...

A characterization of oscillation and nonoscillation of the Emden-Fowler difference equation $$\Delta \left({a}_{n}{\left|\Delta {x}_{n}\right|}^{\alpha}sgn\Delta {x}_{n}\right)+{b}_{n}{\left|{x}_{n+1}\right|}^{\beta}sgn{x}_{n+1}=0$$ is given, jointly with some asymptotic properties. The problem of the coexistence of all possible types of nonoscillatory solutions is also considered and a comparison with recent analogous results, stated in the half-linear case, is made.

Consider the difference equation $$\Delta x\left(n\right)+\sum _{i=1}^{m}{p}_{i}\left(n\right)x\left({\tau}_{i}\left(n\right)\right)=0,\phantom{\rule{1.0em}{0ex}}n\ge 0\phantom{\rule{1.0em}{0ex}}\left[\nabla x\left(n\right)-\sum _{i=1}^{m}{p}_{i}\left(n\right)x\left({\sigma}_{i}\left(n\right)\right)=0,\phantom{\rule{1.0em}{0ex}}n\ge 1\right],$$ where $\left({p}_{i}\left(n\right)\right)$, $1\le i\le m$ are sequences of nonnegative real numbers, ${\tau}_{i}\left(n\right)$ [${\sigma}_{i}\left(n\right)$], $1\le i\le m$ are general retarded (advanced) arguments and $\Delta $ [$\nabla $] denotes the forward (backward) difference operator $\Delta x\left(n\right)=x(n+1)-x\left(n\right)$ [$\nabla x\left(n\right)=x\left(n\right)-x(n-1)$]. New oscillation criteria are established when the well-known oscillation conditions $$\underset{n\to \infty}{lim\; sup}\sum _{i=1}^{m}\sum _{j=\tau \left(n\right)}^{n}{p}_{i}\left(j\right)>1\phantom{\rule{1.0em}{0ex}}\left[\underset{n\to \infty}{lim\; sup}\sum _{i=1}^{m}\sum _{j=n}^{\sigma \left(n\right)}{p}_{i}\left(j\right)>1\right]$$ and $$\underset{n\to \infty}{lim\; inf}\sum _{i=1}^{m}\sum _{j={\tau}_{i}\left(n\right)}^{n-1}{p}_{i}\left(j\right)>\frac{1}{\mathrm{e}}\phantom{\rule{1.0em}{0ex}}\left[\underset{n\to \infty}{lim\; inf}\sum _{i=1}^{m}\sum _{j=n+1}^{{\sigma}_{i}\left(n\right)}{p}_{i}\left(j\right)>\frac{1}{\mathrm{e}}\right]$$ are not satisfied. Here $\tau \left(n\right)={max}_{1\le i\le m}{\tau}_{i}\left(n\right)$$[\sigma ...$