Generalized Prüfer angle and oscillation of half-linear differential equations

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 equation(∗∗)tα−1r(t)Φ(x′)′+tα−1−ps(t)Φ(x)=0,Φ(x)=|x|p−1sgnx,where p>1p>1, α≠pα≠p, and r,sr,s are continuous functions such that r(t)>0r(t)>0 for large tt. We present conditions on the functions r,sr,s which guarantee that Eq. (∗∗) behaves like the Euler type equation [tα−1Φ(x′)]′+λtα−1−pΦ(x)=0,[tα−1Φ(x′)]′+λtα−1−pΦ(x)=0, which is conditionally oscillatory with the oscillation constant λ0=|p−α|p∕ppλ0=|p−α|p∕pp.