Admissible characterization in delay equations and robustness of exponential dichotomies against small-delay perturbations

Abstract

Robustness of exponential dichotomies against small-delay perturbations presents a fundamental obstacle: the lack of an effective admissible characterization in nonautonomous delay equations. Considerable efforts have been devoted to obtaining such a characterization for differential equations in Banach spaces. However, even unlike ordinary differential equations, the variation of constants formula for delay equations requires extending the phase space to a space of discontinuous functions, and the dependence on the past states leads to another challenge in establishing exponential dichotomy via admissibility, i.e., constructing a suitable admissible pair and then estimating the growth/decay rates along the unstable/stable subspaces via the admissibility property. In this paper, we provide an admissible characterization based on two pairs of Banach spaces that yields explicit dichotomy exponents. Through this characterization and an operator perturbation method, we prove that small-delay perturbations preserve exponential dichotomies.