Critical oscillation constant for Euler-type dynamic equations on time scales

摘要

In this paper we study the second-order dynamic equation on the time scale T of the form(r(t)yΔ)Δ+γq(t)tσ(t)yσ=0,where r,q are positive rd-continuous periodic functions with inf{r(t),t∈T}>0 and γ is an arbitrary real constant. This equation corresponds to Euler-type differential (resp. Euler-type difference) equation for continuous (resp. discrete) case. Our aim is to prove that this equation is conditionally oscillatory, i.e., there exists a constant Γ>0 such that studied equation is oscillatory for γ>Γ and non-oscillatory for γ<Γ.