International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 3, Pages 497-508
Functional differential equations with infinite delay in Banach spaces
1Teaching and Research Section of Mathematics, Kunming Engineering Institute, Kunming, China
2Department of Mathematics, Yunnan Teachers' University, Kunming, China
Received 17 August 1989; Revised 16 August 1990
Copyright © 1991 Jin Liang and Tijun Xiao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper, a definition of the fundamental operator for the linear autonomous functional
differential equation with infinite delay in a Banach space is given, and some sufficient and necessary
conditions of the fundamental operator being exponentially stable in abstract phase spaces which satisfy
some suitable hypotheses are obtained. Moreover, we discuss the relation between the exponential
asymptotic stability of the zero solution of nonlinear functional differential equation with infinite delay in
a Banach space and the exponential stability of the solution semigroup of the corresponding linear equation,
and find that the exponential stability problem of the zero solution for the nonlinear equation can be discussed
only in the exponentially fading memory phase space.