Asymptotic approximations for second-order linear difference equations in Banach algebras, I

Abstract

An asymptotic approximation theory is developed for some classes of linear second-order difference equations in Banach algebras, subject to "finite moments perturbations." The special case of linear matrix difference equations (or, equivalently, of second-order systems) is included. Rigorous and explicitly computable bounds for the error terms are obtained.

Asymptotic approximations for second-order linear difference equations in Banach algebras, II

“…since V n decreases with n. The last term in (14) does not depend on n and is summable being V ν 1 < 1. Hence, the series in (12) converges uniformly with respect to n for n ν 1 .…”

“…Hence, the series in (12) converges uniformly with respect to n for n ν 1 . From (12) and (14) we obtain promptly the estimate…”

“…As for the question concerning having a basis of all solutions to Eq. (1), we proceed as in Section 2 for the analogous case of A > 0 and in [14], that is we establish the invertibility of the Casorati matrix. Now we obtain…”

“…A pair of linearly independent generators for (1) is called "a basis" of the module of solutions. In the first part, paper [14], the case of a coefficient Q n as above but with A = 0 was considered. For either A = 0 or A = 0, it was shown in [14, §2] that Eq.…”

Canonical forms and discrete Liouville–Green asymptotics for second-order linear difference equations

“…4. It is possible to proceed similarly in case of matrix difference equations or, equivalently, for systems, but we will not do it in this paper; see, e.g., [4,11,12]. Finally, we summarize the high points of the paper in the short concluding Sect.…”