Non-oscillation of periodic half-linear equations in the critical case

Abstract

Recently, it was shown that the Euler type half-linear differential equations $$ [ r (t) t^{p-1}\Phi(x')]' + \frac{s (t)}{ t \log^p t} \Phi(x) = 0 $$ with periodic coefficients r and s are conditionally oscillatory and the critical oscillation constant was found. Nevertheless, the critical case remains unsolved. The objective of this article is to study the critical case. Thus, we consider the critical value of the coefficients and we prove that any considered equation is non-oscillatory. Moreover, we analyze the situation when the periods of coefficients r and s do not need to coincide.

Keywords

• Half-linear equations
• Pr\"{u}fer angle
• oscillation theory
• conditional oscillation
• oscillation constant