Oscillation of first‐order dynamic equations with nonmonotone delay

This paper discusses higher order nonlinear neutral mixed type difference equations of the form Δ^{m}[x(n)+p(n)h(x(σ(n)))]+q(n)f(x(τ(n)))=0, n=0,1,2,…, where (p(n)), (q(n)) are sequences of nonnegative real numbers, h, f:R→R are continuous and nondecreasing with uh(u)>0, uf(u)>0 for all u≠0, and (σ(n)) and (τ(n)) are sequences of integers such that lim_{n→∞}τ(n)=lim_{n→∞}σ(n)=∞. In general, we will examine the oscillatory behavior of the solutions for the above equation. Especially, when m is even, the result obtained here complements studies related to the oscillation of the above equation. In addition, examples showing the accuracy of the results are given.

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“…then all solutions of (1.5) are oscillatory, while, in 2020, the present author [16] proved that, if…”

Section: Introductionmentioning

confidence: 62%

Oscillatory Behavior of Higher Order Nonlinear Mixed Type Difference Equations With a Nonlinear Neutral Term

Çakır

Öcalan

Yıldız

2023

JAM

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“…where ϑ(ζ) is defined by (1.4), then all solutions of (1.3) oscillate. In 2020, Öcalan [17] obtained the following criteria. If −r ∈ R + and…”

Section: Introductionmentioning

confidence: 99%

Oscillation condition for first order linear dynamic equations on time scales

Öcalan

Kilic

2023

Malaya J. Mat.

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In this paper, we deal with the first-order dynamic equations withnonmonotone arguments\begin{equation*}y^{\Delta }(t)+\underset{i=1}{\overset{m}{\sum }}p_{i}(t)y\left( \tau_{i}(t)\right) =0,\text{ }t\in \lbrack t_{0},\infty )_{\mathbb{T}}\end{equation*}where $p_{i}\in C_{rd}\left( [t_{0},\infty )_{\mathbb{T}},%\mathbb{R}^{+}\right) ,$ $\tau _{i}\in C_{rd}\left( [t_{0},\infty )_{\mathbb{%T}},\mathbb{T}\right) $ and $\tau _{i}(t)\leq t,\ \lim_{t\rightarrow \infty}\tau _{i}(t)=\infty $ for $1\leq i\leq m$. Also, we present a new sufficient condition for theoscillation of delay dynamic equations on time scales. Finally, we give anexample illustrating the result.

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“…In recent years, there has been much research activity concerning the oscillation and non-oscillation of solutions of delay differential and difference equations. For these oscillatory and non-oscillatory results, we refer, for instance [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

“…In [16], Öcalan proved that if is not necessarily monotone and then all solutions of (1.1) oscillate. It can be seen immediately that if is non-decreasing, then condition (1.20) returns to condition (1.15).…”

Section: Introductionmentioning

confidence: 99%

Upper and lower limit oscillation conditions for first-order difference equations

Öcalan¹

2022

Proceedings of the Edinburgh Mathematical Society

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In this work, we consider the first-order difference equation with general argument \[ \varDelta x(n)+p(n)x\left( \tau (n)\right) =0,\quad n\geq 0, \] where $(p(n))$ is a sequence of non-negative real numbers, $(\tau (n))$ is a sequence of integers such that $\tau (n)< n$ for $\ n\in \mathbf {N},\,$ and $\ \lim _{n\rightarrow \infty }\tau (n)=\infty.$ Under the assumption that the deviating argument is not necessarily monotone, we obtain some new oscillation conditions and improve the all known results for the above equation in the literature, involving only upper and only lower limit conditions. Two examples illustrating the results are also given.

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Oscillation of first‐order dynamic equations with nonmonotone delay

Cited by 4 publications

References 28 publications

Oscillatory Behavior of Higher Order Nonlinear Mixed Type Difference Equations With a Nonlinear Neutral Term

Oscillatory Behavior of Higher Order Nonlinear Mixed Type Difference Equations With a Nonlinear Neutral Term

Oscillation condition for first order linear dynamic equations on time scales

Upper and lower limit oscillation conditions for first-order difference equations

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