# De la Vallée Poussin type inequality and eigenvalue problem for generalized half-linear differential equation

## Libor Báňa and Ondřej Došlý

Address: Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, CZ-611 37 Brno, Czech Republic

E-mail:

xbanal@math.muni.cz

dosly@math.muni.cz

Abstract: We study the generalized half-linear second order differential equation via the associated Riccati type differential equation and Prüfer transformation. We establish a de la Vallée Poussin type inequality for the distance of consecutive zeros of a nontrivial solution and this result we apply to the “classical” half-linear differential equation regarded as a perturbation of the half-linear Euler differential equation with the so-called critical oscillation constant. In the second part of the paper we study a Dirichlet eigenvalue problem associated with the investigated half-linear equation.

AMSclassification: primary 34C10.

Keywords: generalized half-linear differential equation, de la Vallée Poussin inequality, half-linear Euler differential equation, Dirichlet eigenvalue problem.

DOI: 10.5817/AM2014-4-193