Linear Noetherian Boundary-Value Problem for a Matrix Difference Equation
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LINEAR NOETHERIAN BOUNDARY-VALUE PROBLEM FOR A MATRIX DIFFERENCE EQUATION A. A. Boichuk,1 S. M. Chuiko,2,3 and Ya. V. Kalinichenko4
UDC 517.9
We establish constructive conditions for the solvability of a linear Noetherian boundary-value problem for a matrix difference equation and propose a procedure of construction of its solutions. We suggest an original scheme of regularization of a linear Noetherian boundary-value problem for a linear degenerate system of difference equations.
1. Statement of the Problem We study the problem of determination of bounded solutions [1, 2] Z(k) 2 R↵⇥β , k 2 ⌦ := {0, 1, 2, . . . , !} of a linear Noetherian (↵ 6= β 6= λ 6= µ) boundary-value problem [3] Z(k + 1) = AZ(k) + Z(k)B + F (k),
LZ(·) = A.
(1)
The components Z (i,j) (k), F (i,j) (k) : ⌦ ! R1 of the matrices Z(k) and F (k) 2 R↵⇥β are assumed to be bounded functions on the set ⌦. Here, A 2 R↵⇥↵ , B 2 Rβ⇥β , and A 2 Rλ⇥µ are constant matrices and LZ(·) is a linear bounded matrix functional: LZ(·) :
�
Z(k) : ⌦ ! R↵⇥β ! Rλ⇥µ .
Constructive conditions for the solvability of the general Noetherian boundary-value problem and the structure of its solution were obtained in [1]. Conditions of solvability and the structure of the Green operator for the Noetherian boundary-value problem (1) for a conventional (β = µ = 1) difference equation were established in [3] as a result of generalization of the classical results for systems of difference equations [4]. In turn, the conditions of solvability and the structure of periodic solutions for systems of matrix differential equations were obtained in [2] by using the procedure of generalized inversion of matrices and operators described in [5]. The general solution of the semihomogeneous Cauchy problem Z(k + 1) = AZ(k) + Z(k)B + F (k),
Z(0) = ⇥,
(2)
can be represented in the form [6]
1
⇥ ⇤ Z(k) = W (k, ⇥) + K F (s) (k),
Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine; e-mail: [email protected]. Donbass State Pedagogic University, Slavyansk, Ukraine; e-mail: [email protected]. 3 Corresponding author. 4 Donbass State Pedagogic University, Slavyansk, Ukraine. 2
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 3, pp. 340–354, March, 2020. Original article submitted January 6, 2019. 386
0041-5995/20/7203–0386
© 2020
Springer Science+Business Media, LLC
L INEAR N OETHERIAN B OUNDARY-VALUE P ROBLEM FOR A M ATRIX D IFFERENCE E QUATION
387
where W (k, ⇥) :=
k X
Ckk−j Ak−j ⇥ B j
j=0
is the general solution of the homogeneous part of the matrix difference equation (1) and ⇥
⇤
K F (s) (k) :=
k−1 X j=0
⇥ ⇤ W j, F (k − 1 − j)
is the generalized Green operator of the Cauchy problem (2). Lemma 1. The general solution of the linear semihomogeneous Cauchy problem (2) ⇥ ⇤ Z(k) = W (k, ⇥) + K F (s) (k),
⇥ 2 R↵⇥β ,
is determined by the generalized Green operator of the Cauchy problem (2). To check this assertion, it suffices to substitute the general solution in the matrix difference equation (1). Substituting the general solutio
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