The aim of this contribution is to study the role of the coefficient $r$ in the qualitative theory of the equation ${\left(r\left(t\right)\Phi \left(y^{\Delta }\right)\right)}^{\Delta }+p\left(t\right)\Phi \left(y^{\sigma }\right)=0$, where $\Phi \left(u\right)={\left|u\right|}^{\alpha -1}\mathrm{sgn}u$ with $\alpha >1$. We discuss sign and smoothness conditions posed on $r$, (non)availability of some transformations, and mainly we show how the behavior of $r$, along with the behavior of the graininess of the time scale, affect some comparison results and (non)oscillation criteria. At the same time we provide a survey of recent results acquired by sophisticated modifications of the Riccati...

We extend the classical Leighton comparison theorem to a class of quasilinear forced second order differential equations $$\left(r\left(t\right)\right|x^{\text{'}}{|}^{\alpha -2}{x}^{\text{'}}{)}^{\text{'}}+c\left(t\right){\left|x\right|}^{\beta -2}x=f\left(t\right),1<\alpha \le \beta ,t\in I=(a,b),(*)$$ where the endpoints $a$, $b$ of the interval $I$ are allowed to be singular. Some applications of this statement in the oscillation theory of (*) are suggested.

Let $𝒜$ be the real vector space of Abelian integrals$$I\left(h\right)=\int {\int }_{\{H\le h\}}R(x,y)dx\wedge dy,h\in [0,\tilde{h}]$$where $H(x,y)=(x^2+y^2)/2+...$ is a fixed real polynomial, $R(x,y)$ is an arbitrary real polynomial and $\{H\le h\}$, $h\in [0,\tilde{h}]$, is the interior of the oval of $H$ which surrounds the origin and tends to it as $h\to 0$. We prove that if $H(x,y)$ is a semiweighted homogeneous polynomial with only Morse critical points, then $𝒜$ is a free finitely generated module over the ring of real polynomials $ℝ\left[h\right]$, and compute its rank. We find the generators of $𝒜$ in the case when $H$ is an arbitrary cubic polynomial. Finally we...

The asymptotic behaviour of a Sturm-Liouville differential equation with coefficient ${\lambda }^{2}q\left(s\right),s\in [s_0,\infty )$ is investigated, where $\lambda \in ℝ$ and $q\left(s\right)$ is a nondecreasing step function tending to $\infty$ as $s\to \infty$. Let $S$ denote the set of those $\lambda$’s for which the corresponding differential equation has a solution not tending to 0. It is proved that $S$ is an additive group. Four examples are given with $S=\left\{0\right\}$, $S=\mathbb{Z}$, $S=𝔻$ (i.e. the set of dyadic numbers), and $\mathbb{Q}\subset S⫋ℝ$.

We establish a Hartman type asymptotic formula for nonoscillatory solutions of the half-linear second order differential equation $${\left(r\left(t\right)\Phi \left(y^{\text{'}}\right)\right)}^{\text{'}}+c\left(t\right)\Phi \left(y\right)=0,\Phi \left(y\right):={\left|y\right|}^{p-2}y,p>1.$$

This paper deals with the oscillatory solutions of the first order nonlinear advanced differential equation. The aim of the present paper is to obtain an oscillation condition for this equation. This result is new and improves and correlates many of the well-known oscillation criteria that were in the literature. Finally, an example is given to illustrate the main result.