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

Abstract

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