Asymptotic expansion of the variable eigenvalue associated to second order differential equations

Abstract

We consider a second order linear differential equation with almost constant coefficients and we give sufficient conditions to know the asymptotic behavior of the logarithmic derivative of a solution y. This allows us to investigate the asymptotic behavior of the variable eigenvalue as of the eigenvectors associated to the solution of the equation. We recover and extend Poincare and Perron's results. Moreover, we unify and generalize asymptotic results due to Levinson, Hartman-Wintner and Harris-Lutz.