P. Nesterov, "On the dynamics of certain higher-order scalar difference equation: asymptotics, oscillation, stability," Turkish Journal of Mathematics, vol 44, pp. 1612-1639 (2020)

http://dx.doi.org/10.3906/mat-1912-89

Abstract: We construct the asymptotics for solutions of the higher-order scalar difference equation that is equivalent to the linear delay difference equation ∆y(n) = −g(n)y(n − k) . We assume that the coefficient of this equation oscillates at the certain level and the oscillation amplitude decreases as n → ∞. Both the ideas of the centre manifold theory and the averaging method are used to construct the asymptotic formulae. The obtained results are applied to the oscillation and stability problems for the solutions of the considered equation.