Mild solutions, variation of constants formula, and linearized stability for delay differential equations

Nishiguchi Junya: Mild solutions, variation of constants formula, and linearized stability for delay differential equations. (2023)

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Abstract

The method and the formula of variation of constants for ordinary differential equations (ODEs) is a fundamental tool to analyze the dynamics of an ODE near an equilibrium. It is natural to expect that such a formula works for delay differential equations (DDEs), however, it is well-known that there is a conceptual difficulty in the formula for DDEs. Here we discuss the variation of constants formula for DDEs by introducing the notion of a mild solution, which is a solution under an initial condition having a discontinuous history function. Then the principal fundamental matrix solution is defined as a matrix-valued mild solution, and we obtain the variation of constants formula with this function. This is also obtained in the framework of a Volterra convolution integral equation, but the treatment here gives an understanding in its own right. We also apply the formula to show the principle of linearized stability and the Poincaré–Lyapunov theorem for DDEs, where we do not need to assume the uniqueness of a solution.

Item Type: Journal
Publication full: Electronic journal of qualitative theory of differential equations
Date: 2023
Number: 32
ISSN: 1417-3875
Number of Pages: 77
Language: English
Place of Publication: Szeged
DOI: 10.14232/ejqtde.2023.1.32
Uncontrolled Keywords: Differenciálegyenlet - késleltetett
Additional Information: Bibliogr.: p. 74-77. ; összefoglalás angol nyelven
Date Deposited: 2023. Nov. 16. 13:50
Last Modified: 2023. Nov. 16. 13:50
URI: http://acta.bibl.u-szeged.hu/id/eprint/82282

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