Asymptotics of perturbed discrete Euler equations in the critical case

Abstract

We consider the difference equation Δ2y(k)+p(k)y(k+1)=0 under the condition limk→∞⁡k2p(k)=1/4; such a setting can cover various perturbations of the critical (double root) case in Euler type difference equations. We establish precise asymptotic formulae for all nonoscillatory solutions. An important role is played by the concept of the refined discrete regular variation, transformations of dependent and independent variable, and asymptotic theory of dynamic equations on time scales.