On exponential dichotomy for abstract differential equations with delayed argument
Abstract
UDC 517.9
We consider linear differential equations of the first order with delayed arguments in a Banach space. We establish conditions for the operator coefficients necessary for the existence of exponential dichotomy on the real axis. It is proved that the analyzed differential equation is equivalent to a difference equation in a certain space. It is shown that, under the conditions of existence and uniqueness of a solution bounded on the entire real axis, the condition of exponential dichotomy is also satisfied for any bounded known function. The explicit formula for projectors, which form the dichotomy, is found for the case of a single delay.
References
A. V. Chaikovs'kyi, On solutions defined on an axis for differential equations with shifts of the argument, Ukr. Math. J., 63, № 9, 1470–1477 (2012).
A. V. Chaikovs'kyi, Investigation of one linear differential equation by using generalized functions with values in a Banach space, Ukr. Math. J., 53, № 5, 796–803 (2001).
F. Riss, B. Sekefalvi-Nad, Lectures on functional analysis, Ungar Publ., New York (1955).
J. K. Hale, Theory of functional differential equations, Springer, New York (1977).
Jack K. Hale, Weinian Zhang, On uniformity of exponential dichotomies for delay equations, J. Different. Equat., 204, 1–4 (2004).
A. A. Boichuk, A. M. Samoilenko, Generalized inverse operators and Fredholm boundary-value problems, VSP, Utrecht, Boston (2004).
A. A. Boichuk, A. M. Samoilenko, Generalized inverse operators and Fredholm boundary-value problems, De Gruyter, Berlin (2016).
A. A. Boichuk, A. A. Pokutnyi, Exponential dichotomy and bounded solutions of differential equations in the Frechet space, Ukr. Math. J., 66, № 12, 1781–1792 (2015).
A. A. Boichuk, V. F. Zhuravlev, Dichotomy on semiaxes and the solutions of linear systems with delay bounded on the entire axis, J. Math. Sci., 220, № 4, 377–393 (2017).
A. M. Gomilko, M. F. Gorodnii, O. A. Lagoda, On the boundedness of a recurrence sequence in a Banach space, Ukr. Math. J., 55, № 10, 1699–1708 (2003).
M. F. Horodnii, O. A. Lahoda, Bounded solutions for some classes of difference equations with operator coefficients, Ukr. Math. J., 53, № 11, 1817–1824 (2001).
A. Chaikovs’kyi, O. A. Lagoda, Bounded solutions of difference equations in a banach space with input data from subspaces, Ukr. Math. J., 73, № 12, 1810–1824 (2022).
Copyright (c) 2023 Andriy Chaikovs'kyi, Оксана Лагода
This work is licensed under a Creative Commons Attribution 4.0 International License.
