Linearized stability for abstract functional differential equations subject to state-dependent delays with applications

Abstract

In this paper, the linearized stability for a class of abstract functional differential equations (FDE) with state-dependent delays (SD) is investigated. In particular, such equations contain more general delay terms which not only cover the discrete delay and distributed delay as special cases, but also extend the SD to abstract integro-differential equation that the states belong to some infinite-dimensional space. The principle of linearized stability for such equations is established, which is nontrivial compared with ordinary differential equations with SD. Moreover, it should be stressed that such topic is untreated in the literatures up to date. Finally, we present an example to show the effectiveness of the proposed results.