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Nonoscillation theory for second order half-linear differential equations in the framework of regular variation. (English) Zbl 1047.34034

The authors study regularity properties implying nonoscillation of the solutions of the half-linear equation

$\left(|{y}^{\text{'}}{|}^{\alpha -1}{y}^{\text{'}}{\right)}^{\text{'}}+q\left(t\right){|y|}^{\alpha -1}y=0$

with $\alpha >0$ and $q$ positive and continuous on the half-axis $t\ge 0$. Some necessary and sufficient conditions for the existence of such solutions are presented. The results generalize corresponding ones for the linear equation (when $\alpha =1$) as proved in [V. Marić, Regular variation and differential equations. Lecture Notes in Mathematics 1726. Berlin: Springer-Verlag (2000; Zbl 0946.34001)].

##### MSC:
 34C11 Qualitative theory of solutions of ODE: growth, boundedness 26A12 Rate of growth of functions of one real variable, orders of infinity, slowly varying functions 34D05 Asymptotic stability of ODE 34C15 Nonlinear oscillations, coupled oscillators (ODE)