Oscillation of modified Euler type half-linear differential equations via averaging technique

In this article, we analyze the oscillation behavior of half-linear differential equation $$\big( r(t) t^{p-1} \Phi(x')\big)' + \frac{s(t)}{t \log^pt} \Phi(x) = 0, \quad \Phi(x)=|x|^{p-1}\text{sgn} x, p > 1. $$ Applying modified Prufer angle and a general averaging technique over unbounded intervals, prove an criterion for studied equation. We point out that presented is new even in linear case when p=2.