Exponentially Dichotomous Difference Equations with Piecewise Constant Operator Coefficients
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EXPONENTIALLY DICHOTOMOUS DIFFERENCE EQUATIONS WITH PIECEWISE CONSTANT OPERATOR COEFFICIENTS V. Yu. Slyusarchuk
UDC 517.988.6
We establish necessary and sufficient conditions for the exponential dichotomy of the solutions of linear difference equations with piecewise constant operator coefficients.
1. Statement of the Main Problem Let En , n 2 Z, be Banach spaces, let k · kEn be the norm in En , let 0n be the null element of the space En , and let M be a Banach space of bounded two-sided sequences x = (xn ) for each of which xn 2 En , n 2 Z, with the norm kxkM = sup kxn kEn n2Z
and the null element 0 = (0n ). By l1 (Z, E) we denote a Banach space of bounded two-sided sequences x = (xn ) for each of which xn is an element of the Banach space E, n 2 Z, with the norm kxkl1 (Z,E) = sup kxn kE . n2Z
It is clear that l1 (Z, E) = M if En = E for all n 2 Z. Let L(Ek , El ) be a Banach space of linear continuous operators A : Ek ! El with the norm kAkL(Ek ,El ) =
sup kAxkEl .
kxkEk =1
We consider operators An 2 L(En , En+1 ), n 2 Z, for which supn2Z kAn kL(En ,En+1 ) < +1 and linear difference equations xn = An−1 xn−1 , xn = An−1 xn−1 + fn ,
n 2 Z, n 2 Z,
(1) (2)
where xn 2 En for all n 2 Z and f = (fn ) 2 M. For the theory of difference equations, it is important to know the existence and uniqueness conditions for the solutions of Eq. (2) for each sequence f 2 M, i.e., the conditions of invertibility of the operator National University of Water Management and Utilization of Natural Resources, Rivne, Ukraine; e-mail: [email protected]. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 6, pp. 822–841, June, 2020. Ukrainian DOI: 10.37863/umzh.v72i6.1052. Original article submitted August 29, 2019. 0041-5995/20/7206–0953
c 2020
Springer Science+Business Media, LLC
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V. Y U. S LYUSARCHUK
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(Ay)n = yn − An−1 yn−1 ,
n 2 Z,
acting in the space M,. This problem was solved in [1] by using the exponential dichotomy of the solutions of Eq. (1). Definition 1. For Eq. (1), the property of exponential dichotomy is realized on Z (the equation is e-dichotomous) if, for each m 2 Z, the space Em can be represented in the form of direct sum of its closed subspaces + ⊕ E − and the following conditions are satisfied: Em = Em m + and P − onto the subspaces E + and E − are uniformly bounded, i.e., (a) the projectors Pm m m m
� + � − sup kPm kL(Em ,Em ) + kPm kL(Em ,Em ) < +1;
(3)
m2Z
+ , the solution y of the problem (b) for every z 2 Em n
yn+1 = An yn ,
n ≥ m,
(4)
ym = z
satisfies the inequality kyn kEn N1 (q1 )n−m kzkEm for all n ≥ m with some N1 > 0 and q1 2 (0, 1) independent of n and m; − , the solution y of the problem (c) for every z 2 Em n
yn+1 = An yn ,
n < m,
(5)
ym = z
satisfies the inequality kyn kEn N2 (q2 )m−n kzkEm for all n m with some N2 > 0 and q2 2 (0, 1) independent of n and m.
We study Eqs. (1) and (2) and special cases of these equations in the general case under the assumption that + and E − is a null space. If this assumption is not true, then all solutio
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