Page 632 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts

AMICAL SYSTEMS AND ORDINARY DIFFERENTIAL EQUATIONS AND
APPLICATIONS

Using the fact that the solution of the boundary value problem (3) is represented as

−1 ∞ k N +l−1
H(l) = Ul∗ UN∗ P ∗
Ui∗Ui P UNk Ul−1
k=0
i=l
−1 ∞ k Ul−1 = H−(l) + H+(l),
+ Ul∗ UN∗ (I − P )∗ N +l−1

k=1 Ui∗Ui (I − P )UNk

i=l

we can obtain conditions for perturbations {B(n)} under which the system (1) is also exponen-
tially dichotomous.

Theorem. Let det(A(n)) = 0 and the matrix sequence of perturbations {B(n)} satisfy the
condition

max{ B(0) , . . . , B(N − 1) }

< h− 1 − 1 + 1 h− H(0) + h+ 1 −1
h− 1+ +1
h+ H(0) ,
h+

where

h− = max{ H−(0) , H−(1) , . . . , H−(N − 1) },
h+ = max{ H+(0) , H+(1) , . . . , H+(N − 1) },

then the perturbed system (1) is exponentially dichotomous .

This paper is a continuation of [1, 3–5].
This work was supported by the Russian Foundation for Basic Research (project No. 19-
01-00754).

References

[1] Demidenko G.V., Bondar A.A. Exponential dichotomy of systems of linear difference
equations with periodic coefficients // Sib. Math. J. 2016. V. 57, No. 6. P. 969–980.

[2] Daleckii Ju.L., Krein M.G. Solutions to Differential Equations in Banach Space. Amer.
Math. Soc., Providence, 1974.

[3] Aydin K., Bulgak H., Demidenko G.V. Numeric characteristics for asymptotic stability
of solutions to linear difference equations with periodic coefficients // Sib. Math. J. 2000.
V. 41, No. 6. P. 1005–1014.

[4] Demidenko G. V. Stability of solutions to difference equations with periodic coefficients in
linear terms // J. Comp. Math. Optim. 2010. Vol. 6, No. 1. P. 1–12.

[5] Demidenko G. V. On conditions for exponential dichotomy of systems of linear differential
equations with periodic coefficients // Int. J. Dyn. Syst. Differ. Equ. 2016. Vol. 6, No. 1.
P. 63–74.

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