International Journal of Mathematics and Mathematical Sciences 
Volume 21 (1998), Issue 2, Pages 255-259
doi:10.1155/S0161171298000362

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

Jin Liang,1 Falun Huang,2 and Tijun Xiao3

 1Teaching and Research Section of Mathematics, Kunming University of Science and Technology, Kunming 650093, China
2Department of Mathematics, Sichuan University, Chengdu 610064, China
3Department of Mathematics, Yunnan Normal University, Kunming 650092, China
Received 5 May 1994; Revised 27 September 1996

Abstract

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.