Differential/Difference Equations

The study of oscillatory phenomena is an important part of the theory of differential equations. Oscillations naturally occur in virtually every area of applied science including, e.g., mechanics, electrical, radio engineering, and vibrotechnics. This Special Issue includes 19 high-quality papers with original research results in theoretical research, and recent progress in the study of applied problems in science and technology. This Special Issue brought together mathematicians with physicists, engineers, as well as other scientists. Topics covered in this issue: Oscillation theory; Differential/difference equations; Partial differential equations; Dynamical systems; Fractional calculus; Delays; Mathematical modeling and oscillations

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 fourth-order quasi-linear delay differential equations

In this paper, we deal with the asymptotic and oscillatory behavior of quasi-linear delay differential equations of fourth order. We first find new properties for a class of positive solutions of the studied equation, $\mathcal{N}_{a}$. As an extension of the approach taken in [1], we establish a new criterion that guarantees that $\mathcal{N}_{a} = \emptyset$. Then, we create a new oscillation criterion

Oscillation and Asymptotic Behavior of Higher-Order Nonlinear Differential Equations

The aim of this paper is to offer a generalization of the Philos and Staikos lemma. As a possible application of the lemma in the oscillation theory, we study the asymptotic properties and oscillation of the nth order delay differential equations (E)(r(t)[x(n−1)(t)]γ)′+q(t)xγ(τ(t))=0. The results obtained utilize also the comparison theorems

On Eventually Positive Solutions of Quasilinear Second-Order Neutral Differential Equations

We study the second-order neutral delay differential equation [r(t)Φγ(z′(t))]′+q(t)Φβ(x(σ(t)))=0, where Φα(t)=|t|α-1t, α≥1 and z(t)=x(t)+p(t)x(τ(t)). Based on the conversion into a certain first-order delay differential equation we provide sufficient conditions for nonexistence of eventually positive solutions of two different types. We cover both cases of convergent and divergent integral ∫∞r-1/γ(t)dt. A suitable combination of our results yields new oscillation criteria for this equation. Examples are shown to exhibit that our results improve related results published recently by several authors. The results are new even in the linear case

Oscillation Criteria for Certain Even Order Neutral Delay Differential Equations with Mixed Nonlinearities

We establish some oscillation criteria for the following certain even order neutral delay differential equations with mixed nonlinearities: rtzn-1tα-1zn-1t'+q0(t)(xτ0tα-1x(τ0(t))+q1t(x(τ1(t))β-1x(τ1(t))+q2t(x(τ2(t))γ-1x(τ2(t))=0, t≥t0, where z(t)=x(t)+p(t)x(σ(t)),n is even integer, and γ>α>β>0. Our results generalize and improve some known results for oscillation of certain even order neutral delay differential equations with mixed nonlinearities