Sharp results for oscillation of second-order neutral delay differential equations

The aim of the present paper is to continue earlier works by the authors on the oscillation problem of second-order half-linear neutral delay differential equations. By revising the set method, we present new oscillation criteria which essentially improve a number of related ones from the literature. A couple of examples illustrate the value of the results obtained

Necessary and sufficient conditions for oscillation of second-order differential equations with nonpositive neutral coefficients

summary:In this work, we present necessary and sufficient conditions for oscillation of all solutions of a second-order functional differential equation of type $$(r(t)(z'(t))^\gamma )' +\sum _{i=1}^m q_i(t)x^{\alpha _i}(\sigma _i(t))=0, \quad t\geq t_0,$$ where $z(t)=x(t)+p(t)x(\tau (t))$. Under the assumption $\int ^{\infty }(r(\eta ))^{-1/\gamma } {\rm d}\eta =\infty$, 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

Oscillation of Noncanonical Second-Order Advanced Differential Equations via Canonical Transform

In this paper, we develop a new technique to deduce oscillation of a second-order noncanonical advanced differential equation by using established criteria for second-order canonical advanced differential equations. We illustrate our results by presenting two examples

Asymptotic proximity to higher order nonlinear differential equations

Abstract The existence of unbounded solutions and their asymptotic behavior is studied for higher order differential equations considered as perturbations of certain linear differential equations. In particular, the existence of solutions with polynomial-like or noninteger power-law asymptotic behavior is proved. These results give a relation between solutions to nonlinear and corresponding linear equations, which can be interpreted, roughly speaking, as an asymptotic proximity between the linear case and the nonlinear one. Our approach is based on the induction method, an iterative process and suitable estimates for solutions to the linear equation

OSCILLATION of SECOND-ORDER HALF-LINEAR NEUTRAL NONCANONICAL DYNAMIC EQUATIONS

In This Paper, We Shall Establish Some New Criteria for the Oscillation of Certain Second-Order Noncanonical Dynamic Equations with a Sublinear Neutral Term. This Task is Accomplished by Reducing the Involved Nonlinear Dynamic Equation to a Second-Order Linear Dynamic Inequality. We Also Establish Some New Oscillation Theorems Involving Certain Integral Conditions. Three Examples, Illustrating Our Results, Are Presented. Our Results Generalize Results for Corresponding Differential and Difference Equations

Asymptotic Behavior of Even-Order Damped Differential Equations with p-Laplacian like Operators and Deviating Arguments

We study the asymptotic properties of the solutions of a class of even-order damped differential equations with p-Laplacian like operators, delayed and advanced arguments. We present new theorems that improve and complement related contributions reported in the literature. Several examples are provided to illustrate the practicability, maneuverability, and efficiency of the results obtained. An open problem is proposed

Improved results for testing the oscillation of functional differential equations with multiple delays

In this article, we test whether solutions of second-order delay functional differential equations oscillate. The considered equation is a general case of several important equations, such as the linear, half-linear, and Emden-Fowler equations. We can construct strict criteria by inferring new qualities from the positive solutions to the problem under study. Furthermore, we can incrementally enhance these characteristics. We can use the criteria more than once if they are unsuccessful the first time thanks to their iterative nature. Sharp criteria were obtained with only one condition that guarantees the oscillation of the equation in the canonical and noncanonical forms. Our oscillation results effectively extend, complete, and simplify several related ones in the literature. An example was given to show the significance of the main results

Oscillation Criteria for Fourth Order Nonlinear Positive Delay Differential Equations with a Middle Term

In this article, we establish some new criteria for the oscillation of fourth order nonlinear delay differential equations of the form (Equation presented) provided that the second order equation (Equation presented) is nonoscillatiory or oscillatory. This equation with g(t) = t is considered in [8] and some oscillation criteria for this equation via certain energy functions are established. Here, we continue the study on the oscillatory behavior of this equation via some inequalities

On the oscillatory behavior of even order neutral delay dynamic equations on time-scales

We establish some new criteria for the oscillation of the even order neutral dynamic equation \begin{equation*} \left( a(t)\left( \left( x(t)-p(t)x(\tau (t))\right) ^{\Delta^{n-1}}\right) ^{\alpha }\right) ^{\Delta }+q(t)\left( x^{\sigma}(g(t))\right) ^{\lambda }=0 \end{equation*} on a time scale $\mathbb{T}$, where $n \geq 2$ is even, $\alpha$ and $\lambda$ are ratios of odd positive integers, $a$, $p$ and $q$ are real valued positive rd-continuous functions defined on $\mathbb{T}$, and $g$ and $\tau$ are real valued rd-continuous functions on $\mathbb{T}$. Examples illustrating the results are included