Representation of Bounded Solutions of Linear Discrete Equations
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REPRESENTATION OF BOUNDED SOLUTIONS OF LINEAR DISCRETE EQUATIONS V. Yu. Slyusarchuk
UDC 517.988.6
A representation of bounded solutions of linear discrete equations is obtained.
1. Statement of the Main Problem Let G be an arbitrary countable set and let Eg ; g 2 G; be finite-dimensional Banach spaces over the field K of real or complex numbers with the norms k � kEg ; g 2 G; and null elements 0g ; g 2 G; respectively. By M we [ denote the set of mappings xW G ! Eg for each of which x.g/ 2 Eg ; g 2 G; and g2G
sup kx.g/kEg < C1:
g2G
For any two elements x; y 2 M and a number k 2 K; we define the sum x C y of the elements x and y and the product kx of the element x by the number k by the equalities .x C y/.g/ D x.g/ C y.g/
and
.kx/.g/ D kx.g/;
g 2 G:
It is clear that x C y 2 M and kx 2 M for all x; y 2 M and k 2 K: The set M with the introduced operations of addition and multiplication by a number is a linear space [1]. This space is also normalized with a norm k � kM specified by the equality kxkM D sup kx.g/kEg : g2G
In view of the completeness of the spaces Eg ; g 2 G; the space M is also a complete and, hence, Banach space. In the case where the Banach spaces Eg ; g 2 G; coincide with a certain space E; we denote the space M by l1 .G; E/: Let X and Y be arbitrary Banach spaces with the norms k � kX and k � kY ; respectively, and let L.X; Y / be a Banach space of linearly continuous operators A acting from the space X into the space Y with the norm kAkL.X;Y / D
sup kAxkY :
kxkX D1
Consider an equation X
˛2G
A.g; ˛/x.˛/ D y.g/;
g 2 G;
(1)
National University of Water Management and Utilization of Natural Resources, Soborna Str., 11, Rivne, 33000, Ukraine; e-mail: [email protected]. Translated from Neliniini Kolyvannya, Vol. 22, No. 2, pp. 262–279, April–June, 2019. Original article submitted July 2, 2017. 1072-3374/20/2494–0673
c 2020 �
Springer Science+Business Media, LLC
673
V. Y U. S LYUSARCHUK
674
where A.g; ˛/ 2 L.E˛ ; Eg /; g; ˛ 2 G; sup
X
g2G ˛2G
kA.g; ˛/kL.E˛ ;Eg / < C1;
(2)
and y 2 M: This equation can be represented in the form Ax D y; where AW M ! M is a linear and continuous [in view of (2)] operator given by the formula .Az/.g/ D
X
A.g; ˛/z.˛/;
˛2G
g 2 G;
(3)
and z is an arbitrary element of the space M: Assume that the following condition is satisfied: Condition A. For any y 2 M; Eq. (1) possesses a unique solution x 2 M: In the present paper, we study the representations of solutions of Eq. (1). It is clear that countable systems of linear algebraic equations and linear difference equations in finitedimensional spaces are special cases of Eq. (1) (see, e.g., [2, 3]). Note that the problem of representation of the bounded solutions of Eq. (1) in the case where Eg D E for all g 2 G; where E is a finite-dimensional space and G is a countable additive group, was solved in [4]. For the difference equation xn C An�1 xn�1 D yn ;
n 2 Z;
where An�1 2 L.En�1 ; En /; Z is the set of all integers, and En ; n 2 Z; are infinite-dimensional spaces, a similar problem wa
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