On Bounded Solutions of One Difference Equation of the Second Order
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ON BOUNDED SOLUTIONS OF ONE DIFFERENCE EQUATION OF THE SECOND ORDER M. F. Gorodnii1;2 and V. P. Kravets’3
UDC 517.929.2
We study the problem of existence of a unique bounded (on Z/ solution of a linear difference equation of the second order with a jump of the operator coefficient in the finite-dimensional Banach space.
N Also let I and Let X be an m-dimensional complex Banach space with the norm k � k and the null element 0: O be the identity and null operators in X and let A and B be fixed linear operators in X: Consider a difference equation xnC1 � 2xn C xn�1 D Fn xn C yn ;
n 2 Z;
(1)
where fyn ; n 2 Zg is a given sequence and fxn ; n 2 Zg is the required sequence of elements of the space X; Fn D A; n � 1; Fn D B; n 0: The aim of the present work is to establish necessary and sufficient conditions for the operators A and B under which the following condition is satisfied: Boundedness Condition For any sequence fyn ; n 2 Zg bounded in X; Eq. (1) possesses a unique bounded solution fxn ; n 2 Zg in the space X: A similar problem for a difference equation of the first order with a jump of the operator coefficient was studied in [1]. The case where the matrices of the operators A and B are reduced to the diagonal form was considered in [2]. As for the difference equations of the second order with constant operator coefficients and their applications, see [3, p. 17], [4] and the references therein. Auxiliary Assertions We set !ˇ Ω ˇ ˇ .1/ .2/ X D xD ˇ x ;x 2 X : x .2/ ˇ 2
º
x .1/
Let X 2 be a 2m-dimensional complex Banach space with coordinatewise addition and multiplication by a scalar and with the following norm:
1 Shevchenko
� � � � kxk⇤ D �x .1/ � C �x .2/ �;
xD
x .1/ x .2/
!
2 X 2:
Kyiv National University, Volodumyrs’ka Str., 64/13, Kyiv, 01033, Ukraine; e-mail: [email protected]. author. 3 Shevchenko Kyiv National University, Volodumyrs’ka Str., 64/13, Kyiv, 01033, Ukraine; e-mail: [email protected]. 2 Corresponding
Translated from Neliniini Kolyvannya, Vol. 22, No. 2, pp. 196–201, April–June, 2019. Original article submitted March 26, 2019. 1072-3374/20/2494–0601
c 2020 �
Springer Science+Business Media, LLC
601
M. F. G ORODNII AND V. P. K RAVETS ’
602
If E; F; G; and H are linear operators in X; then, as in the case of numerical matrices, ✓
E F T D G H
◆
specifies a linear operator in X 2 according to the rule
Tx D
Ex .1/ C F x .2/ Gx .1/
C
H x .2/
!
;
x .1/
xD
x .2/
!
2 X 2:
Let TA D
A C 2I
�I
I
O
!
TB D
;
B C 2I
�I
I
O
!
;
and let �.TA / be a collection of eigenvalues of the operator TA ; S D fz 2 C j jzj D 1g : In what follows, we use the following assertions: Lemma 1. The operator TA�1
D
O
I
�I
A C 2I
!
inverse to the operator TA exists. Lemma 2. A number � ¤ 0 is an eigenvalue of TA corresponding to an eigenvector �C
1 � 2 is an eigenvalue of A corresponding to the eigenvector v: �
✓
�v v
◆
if and only if
✓ ◆ 1 �v Lemma 3. If � 2 �.TA / is associated with an eigenvector ; then � ¤ 0; 2 � .TA /; and it corresponds v � ✓ ◆ v : to the e
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