Exponential stability for abstract linear autonomous functional differential equations with infinite delay

• Main Authors: Jin Liang, Falun Huang, Tijun Xiao
• Format: Article
• Language: English
• Published: Hindawi Limited 1998-01-01
• Series: International Journal of Mathematics and Mathematical Sciences
• Subjects: Exponentially stable; solution semigroup; abstract linear autonomous functional differential equation; fundamental operator; fading memory space.
• Online Access: http://dx.doi.org/10.1155/S0161171298000362
• ISSN: 0161-1712; 1687-0425

Based on our preceding paper, this note is concerned with the exponential stability of the solution semigroup for the abstract linear autonomous functional differential equation x˙(t)=L(xt)                              (∗) where L is a continuous linear operator on some abstract phase space B into a Banach space E. We prove that the solution semigroup of (∗) is exponentially stable if and only if the fundamental operator (∗) is exponentially stable and the phase space B is an exponentially fading memory space.