Page 631 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts

DYNAMICAL SYSTEMS AND ORDINARY DIFFERENTIAL EQUATIONS AND
APPLICATIONS

same dimension as the kernel and cokernel of the boundary-value problem. The condition for
the sequence of characteristic matrices to converge is found. We obtain a constructive criterion
under which the solutions to these problems depend continuously on the small parameter ε at
ε = 0, and find the degree of convergence of the solutions. Also applications of these results to
multipoint boundary-value problems are obtained.

References

[1] O. M. Atlasiuk, V. A. Mikhailets, Fredholm one-dimensional boundary-value problems in
Sobolev spaces. Ukr. Math. J. 70 (2019), no. 10, 1526 – 1537.

[2] O. M. Atlasiuk, V. A. Mikhailets, Fredholm one-dimensional boundary-value problems
with parameter in Sobolev spaces. Ukr. Math. J. 70 (2019), no. 11, 1677 – 1687.

[3] O. M. Atlasiuk, Limit theorems for solutions of multipoint boundary-value problems in
Sobolev spaces. Journal of Mathematical Sciences 247 (2020), no. 2, 238 – 247.

Exponential dichotomy conditions for difference equations with perturbed
coefficients

Anna Bondar, anna.alex.bondar@gmail.com
Novosibirsk State University, Sobolev Institute of Mathematics, Russian Federation

A system of linear difference equations with periodic coefficients is considered

yn+1 = (A(n) + B(n))yn, n ∈ Z, (1)

where A(n) are non-degenerate matrices of size m × m and the matrix sequence {A(n)} is
N -periodic, i.e. A(n + N ) = A(n), n ∈ Z. The sequence {B(n)} is an N -periodic sequence
of perturbations. We assume that the system

xn+1 = A(n)xn, n ∈ Z, (2)

is exponentially dichotomous. As shown in[ 1], this is equivalent to the fact that there are Her-
mitian matrices H(0), H(1), . . . , H(N − 1) and a matrix P satisfying the following boundary
value problem

 −1 P ∗Ul∗UlP Ul−1
 H(l) − A∗(l)H(l + 1)A(l) = Ul∗


 −1

 (I − P )∗Ul∗Ul(I − P )Ul−1,

 − Ul∗ l = 0, 1, . . . , N − 1,















H(0) = H(N ) > 0, (3)







 H (0) = P ∗H(0)P + (I − P )∗H(0)(I − P ),












 P2 = P, P UN = UN P,


where Ul is the Cauchy matrix of (2). This criterion is analogous to the criterion of M. G. Krein
for the exponential dichotomy of difference equations with constant coefficients [2].

629