Oscillation constant for modified Euler type half-linear equations

Abstract

Applying the modified half-linear Prufer angle, we study oscillation properties of the half-linear differential equation $$ [ r(t) t^{p-1} \Phi(x')]' + \frac{s(t)}{t \log^pt} \Phi(x) = 0, \quad \Phi(x)=|x|^{p-1}\hbox{sgn} x. $$ We show that this equation is conditionally oscillatory in a very general case. Moreover, we identify the critical oscillation constant (the borderline depending on the functions r and s which separates the oscillatory and non-oscillatory equations). Note that the used method is different from the standard method based on the half-linear Prufer angle.

Keywords

• Half-linear equations
• Prufer angle
• oscillation theory
• conditional oscillation
• oscillation constant