A class of abstract delay differential equations in the light of suns and stars. II

TL;DR

This paper develops a generalized framework for analyzing abstract delay differential equations using admissible ranges and perturbations, extending existing theories to non-sun-reflexive Banach spaces and facilitating the study of bifurcations.

Contribution

It introduces the notion of admissible ranges and perturbations for non-sun-reflexive semigroups, and extends center manifold theory to this broader setting.

Findings

Established relationships between admissible ranges and subspaces of dual spaces.

Provided conditions for spectral decompositions without eventual compactness.

Generalized center manifold theorem for abstract DDEs in non-sun-reflexive spaces.

Abstract

Recent work in arXiv:1901.11526 by the author about a class of abstract delay differential equations (DDEs), as well as earlier work by Diekmann and Gyllenberg on other classes of delay equations, motivates the introduction of the general notion of an admissible range and an admissible perturbation for a given $\mathcal{C}_0$-semigroup $T_0$ on a Banach space $X$ that is not assumed to be sun-reflexive with respect to $T_0$. We investigate the relationship between admissible ranges for $T_0$ and the subspace $X^{\odot\times}$ of $X^{\odot\star}$ introduced by Van Neerven. We answer two questions about robustness of admissibility with respect to bounded linear perturbations and we use these answers to study the semilinear problem and its linearization. Partly as an application of the material developed up to that point, and partly as a justification of existing work on local bifurcations in models taking the form of abstract DDEs, we compare the construction of center manifolds in the non-sun-reflexive case with known results by Diekmann and Van Gils for the sun-reflexive case. We show that a systematic use of the space $X^{\odot\times}$ facilitates a generalization of the existing results with relatively little effort. In this context we also give sufficient conditions for the existence of appropriate spectral decompositions of $X$ and $X^{\odot\times}$ without assuming that the linearized semiflow is eventually compact. A center manifold theorem for the motivating class of abstract DDEs then follows as a particular case.