A Riccati technique for proving oscillation of a half-linear equation.
The aim of this contribution is to study the role of the coefficient in the qualitative theory of the equation , where with . We discuss sign and smoothness conditions posed on , (non)availability of some transformations, and mainly we show how the behavior of , 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 where the endpoints , of the interval are allowed to be singular. Some applications of this statement in the oscillation theory of (*) are suggested.
Let be the real vector space of Abelian integralswhere is a fixed real polynomial, is an arbitrary real polynomial and , , is the interior of the oval of which surrounds the origin and tends to it as . We prove that if is a semiweighted homogeneous polynomial with only Morse critical points, then is a free finitely generated module over the ring of real polynomials , and compute its rank. We find the generators of in the case when is an arbitrary cubic polynomial. Finally we...
The asymptotic behaviour of a Sturm-Liouville differential equation with coefficient is investigated, where and is a nondecreasing step function tending to as . Let denote the set of those ’s for which the corresponding differential equation has a solution not tending to 0. It is proved that is an additive group. Four examples are given with , , (i.e. the set of dyadic numbers), and .
We establish a Hartman type asymptotic formula for nonoscillatory solutions of the half-linear second order differential equation
Our aim in this paper is to obtain a new oscillation criterion for equation with a nonnegative coefficients which extends and improves some oscillation criteria for this equation. In the special case of equation (*), namely, for equation , our results solve the open question of .