Nonoscillation theory for second order half-linear differential equations in the framework of regular variation.

*(English)*Zbl 1047.34034The authors study regularity properties implying nonoscillation of the solutions of the half-linear equation

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

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)].

Reviewer: Vojislav Marić (Novi Sad)

##### 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) |

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