On the oscillation of third-order quasi-linear neutral functional differential equations

summary:The aim of this paper is to study asymptotic properties of the third-order quasi-linear neutral functional differential equation \begin{equation*} \big [a(t)\big ([x(t)+p(t)x(\delta (t))]^{\prime \prime }\big )^\alpha \big ]^{\prime }+q(t)x^\alpha (\tau (t))=0\,, E \end{equation*} where $\alpha >0$, $0\le p(t)\le p_0<\infty$ and $\delta (t)\le t$. 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

On the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations

By using comparison principles, we analyze the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations. Due to less restrictive assumptions on the coefficients of the equation and on the deviating argument, our criteria improve a number of related results reported in the literature.publishedVersio

Analysis and explicit solvability of degenerate tensorial problems

We study a two-dimensional boundary value problem described by a tensorial equation in a bounded domain. Once its more general definition is given, we conclude that its analysis is linked to the resolution of an overdetermined hyperbolic problem and hence present some discussions and considerations. Secondly, for a simplified version of the original formulation, which leads to a degenerate problem on a rectangle, we prove the existence and uniqueness of a solution under proper assumptions on the data

Oscillation Theorems for Second-Order Nonlinear Neutral Delay Differential Equations

We analyze the oscillatory behavior of solutions to a class of second-order nonlinear neutral delay differential equations. Our theorems improve a number of related results reported in the literature

On the oscillation of third order half-linear neutral type difference equations

In this paper, the authors study the oscillatory properties of third order quasilinear neutral difference equation of the form $$\Delta(a_{n}(\Delta^{2}(x_{n} + p_{n}x_{n-\delta}))^{\alpha}) + q_{n} {x^{\alpha}_{n-\tau}} = 0,\quad n\geq 0, \tag{E}$$ where $\alpha > 0, 0\leq p_{n}\leq p < \infty.$ By using Riccati transformation we estabilish some new sufficient conditions which ensure that every solution of equation (E) is either oscillatory or converges to zero. These results improve some known results in the literature. Examples are provided to illustrate the main results

Oscillation theorems for second order neutral differential equations

In this paper new oscillation criteria for the second order neutral differential equations of the form \begin{equation*} \left(r(t)\left[x(t)+p(t)x(\tau(t))\right]'\right)'+q(t)x(\sigma(t))+v(t)x(\eta(t))=0 \tag{$E$}\end{equation*} are presented. Gained results are based on the new comparison theorems, that enable us to reduce the problem of the oscillation of the second order equation to the oscillation of the first order equation. Obtained comparison principles essentially simplify the examination of the studied equations. We cover all possible cases when arguments are delayed, advanced or mixed

Oscillation Theorems for Second-Order Quasilinear Neutral Functional Differential Equations

New oscillation criteria are established for the second-order nonlinear neutral functional differential equations of the form (r(t)|z′(t)|α−1z′(t))’+f(t,x[σ(t)])=0, t≥t0, where z(t)=x(t)+p(t)x(τ(t)), p∈C1([t0,∞),[0,∞)), and α≥1. Our results improve and extend some known results in the literature. Some examples are also provided to show the importance of these results