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

Electronic Journal of Qualitative Theory of Differential Equations [electronic only]

### 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...

### Bounded nonoscillatory solutions of neutral type difference systems.

Electronic Journal of Qualitative Theory of Differential Equations [electronic only]

### Criterion of $p$-criticality for one term $2n$-order difference operators

Archivum Mathematicum

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.

### Dynamics for nonlinear difference equation ${x}_{n+1}=\left(\alpha {x}_{n-k}\right)/\left(\beta +\gamma {x}_{n-l}^{p}\right)$.

Advances in Difference Equations [electronic only]

### Existence of non-oscillatory solutions for a higher-order nonlinear neutral difference equation.

Electronic Journal of Differential Equations (EJDE) [electronic only]

### Expressions of solutions for a class of differential equations.

Analele Ştiinţifice ale Universităţii “Ovidius" Constanţa. Seria: Matematică

### Forced oscillation of third order nonlinear dynamic equations on time scales

Annales Polonici Mathematici

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 $\Delta ³x\left(n\right)+{n}^{\alpha }{|x|}^{\gamma }sgn\left(n\right)=\left(-1\right)ⁿ{n}^{c}$, where α ≥ -1, γ > 0, c > 3, is oscillatory.

### Global behavior of a third order rational difference equation

Mathematica Bohemica

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

### Global behavior of the difference equation ${x}_{n+1}=\frac{a{x}_{n-3}}{b+c{x}_{n-1}{x}_{n-3}}$

Archivum Mathematicum

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.

### Global stability and oscillation of a discrete annual plants model.

Abstract and Applied Analysis

### Iterated oscillation criteria for delay dynamic equations of first order.

Advances in Difference Equations [electronic only]

### Iterated oscillation criteria for delay partial difference equations

Mathematica Bohemica

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,...

### Linearized Oscillation of Nonlinear Difference Equations with Advanced Arguments

Archivum Mathematicum

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 ${p}_{i}\in \left(-\infty ,0\right)$ and ${k}_{i}\in \left\{\cdots ,-2,-1\right\}$ 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 ${k}_{i}\in \left\{\cdots ,-2,-1\right\}$ for $i=1,2,\cdots ,m$.

### Necessary and sufficient conditions for the oscillation of forced nonlinear second order delay difference equation

Kybernetika

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 $\left(\gamma >1\right)$ and in the sublinear case $\left(\gamma <1\right)$.

### Non-oscillation of second order linear self-adjoint nonhomogeneous difference equations

Mathematica Bohemica

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 $¬\equiv 0\right)$ for large $n$. Further, these results are used to obtain...

### On oscillation and nonoscillation properties of Emden-Fowler difference equations

Open Mathematics

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.

### On the oscillation of second order nonlinear neutral delay difference equations.

Electronic Journal of Qualitative Theory of Differential Equations [electronic only]

### On the oscillatory behavior for a certain class of third order nonlinear delay difference equations.

Electronic Journal of Qualitative Theory of Differential Equations [electronic only]

### Oscillation conditions for difference equations with several variable arguments

Mathematica Bohemica

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 $\Delta x\left(n\right)=x\left(n+1\right)-x\left(n\right)$ [$\nabla x\left(n\right)=x\left(n\right)-x\left(n-1\right)$]. 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)$$\left[\sigma ...$

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