Generalized Prüfer angle and oscillation of half-linear differential equations

Abstract

In this paper, we introduce a new modification of the half-linear Prüfer angle. Applying this modification, we investigate the conditional oscillation of the half-linear second order differential equation (∗)tα−1r(t)Φ(x′)′+tα−1−ps(t)Φ(x)=0,Φ(x)=|x|p−1sgnx,where p>1, α≠p, and r,s are continuous functions such that r(t)>0 for large t. We present conditions on the functions r,s which guarantee that Eq. (∗) behaves like the Euler type equation [tα−1Φ(x′)]′+λtα−1−pΦ(x)=0, which is conditionally oscillatory with the oscillation constant λ0=|p−α|p∕pp.