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.

This paper is concerned with the delay partial difference equation (1) $A_{m+1,n}+A_{m,n+1}-A_{m,n}+{\Sigma }_{i=1}^u{p}_i{A}_{m-k_i,n-l_i}=0$ where $p_i$ are real numbers, $k_i$ and $l_i$ are nonnegative integers, u is a positive integer. Sufficient and necessary conditions for all solutions of (1) to be oscillatory are obtained.

In this paper, we present several sufficient conditions for the existence of nonoscillatory solutions to the following third order neutral type difference equation $${\Delta }^3(x_n+a_n{x}_{n-l}+b_n{x}_{n+m})+p_n{x}_{n-k}-q_n{x}_{n+r}=0,n\ge n_0$$ 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.

In the paper we consider the difference equation of neutral type $${\Delta }^3[x\left(n\right)-p\left(n\right)x\left(\sigma \left(n\right)\right)]+q\left(n\right)f\left(x\left(\tau \left(n\right)\right)\right)=0,n\in \mathbb{N}\left(n_0\right),$$ where $p,q:\mathbb{N}\left(n_0\right)\to {ℝ}_{+}$; $\sigma ,\tau :\mathbb{N}\to \mathbb{Z}$, $\sigma$ is strictly increasing and $\underset{n\to \infty }{lim}\sigma \left(n\right)=\infty ;$ $\tau$ is nondecreasing and $\underset{n\to \infty }{lim}\tau \left(n\right)=\infty$, $f:ℝ\to ℝ$, $xf\left(x\right)>0$. We examine the following two cases: $$0<p\left(n\right)\le {\lambda }^{*}<1,\sigma \left(n\right)=n-k,\tau \left(n\right)=n-l,$$ and $$1<{\lambda }_{*}\le p\left(n\right),\sigma \left(n\right)=n+k,\tau \left(n\right)=n+l,$$ 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.

In this paper we study the boundedness of solutions of some third-order delay differential equation in which $h\left(x\right)$ is not necessarily differentiable but satisfy a Routh–Hurwitz condition in a closed interval $[\delta ,kab]\subset (0,ab)$.

By means of Riccati transformation technique, we establish some new oscillation criteria for third-order nonlinear delay dynamic equations ${\left({\left(x^{\Delta \Delta }\left(t\right)\right)}^{\gamma }\right)}^{\Delta }+p\left(t\right)x^{\gamma }\left(\tau \left(t\right)\right)=0$ on a time scale ; here γ > 0 is a quotient of odd positive integers and p a real-valued positive rd-continuous function defined on . Our results not only extend and improve the results of T. S. Hassan [Math. Comput. Modelling 49 (2009)] but also unify the results on oscillation of third-order delay differential equations and third-order delay...

Si dànno condizioni sufficienti perché l’equazione $$L_{n}x(t)+{(-1)}^{n+1}a(t)x(g(t))=0$$ abbia soluzione positiva limitata.

We study the oscillatory properties of the solutions of the third-order nonlinear semi-noncanonical delay difference equation $$D_{3}y\left(n\right)+f\left(n\right)y^{\beta }\left(\sigma \left(n\right)\right)=0,$$ where $D_{3}y\left(n\right)=\Delta \left(b\left(n\right)\Delta \left(a\left(n\right){\left(\Delta y\left(n\right)\right)}^{\alpha }\right)\right)$ is studied. The main idea is to transform the semi-noncanonical operator into canonical form. Then we obtain new oscillation theorems for the studied equation. Examples are provided to illustrate the importance of the main results.

This paper is devoted to the investigation on the stability for two characteristic functions $f_1\left(z\right)=z^2+p{e}^{-z\tau }+q$ and $f_2\left(z\right)=z^2+pz{e}^{-z\tau }+q$, where $p$ and $q$ are real numbers and $\tau >0$. The obtained theorems describe the explicit stability dependence on the changing delay $\tau$. Our results are applied to some special cases of a linear differential system with delay in the diagonal terms and delay-dependent stability conditions are obtained.

In this work, we present necessary and sufficient conditions for oscillation of all solutions of a second-order functional differential equation of type $${\left(r\left(t\right){\left(z^{\text{'}}\left(t\right)\right)}^{\gamma }\right)}^{\text{'}}+\sum _{i=1}^m{q}_i\left(t\right)x^{{\alpha }_i}\left({\sigma }_i\left(t\right)\right)=0,t\ge t_0,$$ where $z\left(t\right)=x\left(t\right)+p\left(t\right)x\left(\tau \right(t\left)\right)$. Under the assumption ${\int }^{\infty }{\left(r\left(\eta \right)\right)}^{-1/\gamma }\mathrm{d}\eta =\infty$, we consider two cases when $\gamma >{\alpha }_i$ and $\gamma <{\alpha }_i$. Our main tool is Lebesgue’s dominated convergence theorem. Finally, we provide examples illustrating our results and state an open problem.