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

Jitai Liang; Ben Niu; Junjie Wei

doi:10.3934/dcdsb.2019134

Article Contents

2019, Volume 24, Issue 11: 6167-6188. Doi: 10.3934/dcdsb.2019134

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

a.
Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China
b.
School of Mathematics and Big Data, Foshan University, Foshan, Guangdong, 528000, China

^* Corresponding author: Junjie Wei
^* Corresponding author: Junjie Wei

Received: April 2018

Revised: December 2018

Early access: July 2019

Published: August 2019

This work is supported by National Natural Science Foundation of China (No. 11771109)

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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.

Keywords:

Abstract functional differential equations,
linearized stability principle,
state-dependent delay,
sectorial operator.

Mathematics Subject Classification: Primary: 34K20, 35B35; Secondary: 35K57, 35R10.

Citation:

Full Text(HTML)

Figure 1. Numerical simulations of system (37) with the initial value $ \phi ( \zeta ,y) = 0.1sin^{2}(y) $. (a) The equilibrium point $ u^{\ast } = 0 $ is asymptotically stable with the system parameters $ m = 3, $ $ D = 0.01, $ $ B = 0.1, $ $ d = 0.2. $ (b) The equilibrium point $ u^{\ast } = 0 $ is unstable with the system parameters $ m = 3, $ $ D = 0.01, $ $ B = 0.6, $ $ d = 0.2. $

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