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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
\[
(| y'|^{\alpha- 1}y')'+ q(t)| y|^{\alpha- 1}y= 0
\]
with \(\alpha> 0\) and \(q\) positive and continuous on the half-axis \(t\geq 0\). 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 | Growth and boundedness of solutions to ordinary differential equations |

26A12 | Rate of growth of functions, orders of infinity, slowly varying functions |

34D05 | Asymptotic properties of solutions to ordinary differential equations |

34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |

### Citations:

Zbl 0946.34001
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\textit{J. Jaroš} et al., Result. Math. 43, No. 1--2, 129--149 (2003; Zbl 1047.34034)

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### References:

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