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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 $$(\vert y'\vert^{\alpha- 1}y')'+ q(t)\vert y\vert^{\alpha- 1}y= 0$$ 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 [{\it V. Marić}, Regular variation and differential equations. Lecture Notes in Mathematics 1726. Berlin: Springer-Verlag (2000; Zbl 0946.34001)].

MSC:
34C11Qualitative theory of solutions of ODE: growth, boundedness
26A12Rate of growth of functions of one real variable, orders of infinity, slowly varying functions
34D05Asymptotic stability of ODE
34C15Nonlinear oscillations, coupled oscillators (ODE)
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References:
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