International Journal of Mathematics and Mathematical Sciences 
Volume 14 (1991), Issue 3, Pages 497-508
doi:10.1155/S0161171291000686

Functional differential equations with infinite delay in Banach spaces

Jin Liang1 and Tijun Xiao2

 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

Abstract

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.