## 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)].

### 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

Zbl 0946.34001
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### References:

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