Asymptotic Integration of Differential and Difference Equations by Sigrun Bodine & Donald A. Lutz

Author:Sigrun Bodine & Donald A. Lutz
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(5.11)

(5.12)

Suppose that for i ≠ j and k ≠ i holds that

(5.13)

where Also, let the eigenvalues μ k (n) of Λ(n) + V (n) be nowhere zero for n ≥ n 0 and satisfy the discrete Levinson dichotomy conditions ( 3.8 ), (3.​9). Finally, suppose that the d × d matrix R(n) satisfies that

(5.14)

Then

(5.15)

has for n sufficiently large d solutions satisfying

(5.16)

Proof

Writing the diagonal matrix of eigenvalues of Λ(n) + V (n) as

the beginning of the proof follows exactly the proof of Theorem 4.​8. In particular, one utilizes the implicit function theorem to show the existence of a matrix Q(n) satisfying diag Q(n) ≡ 0 and

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