Asymptotic summation of perturbed linear difference systems in critical case

On the dynamics of certain higher-order scalar difference equation: asymptotics, oscillation, stability

“…It can be shown (see [12]) that the dynamics of the solutions of system (2.2) that lie on W(n) for sufficiently large n is described by the following two-dimensional system of difference equations:…”

“…Finally, we obtain linear matrix difference equations that are uniquely solvable in the considered class of matrices. We can now formulate the main theoretical results concerning the used asymptotic summation method (see [12] is a certain (k + 1) × 2 matrix from ℓ 1 whose columns belong to the subspace Q .…”

“…Thus, we have the eigenvalue λ 1,2 = 1 of the algebraic multiplicity 2 and all the other eigenvalues lie inside the unit circle |λ| < 1 in C. This is a critical case in the asymptotic summation problem. The method for constructing the asymptotics for solutions of systems having form (2.2) in this situation was proposed in [12]. We will follow it here.…”