On the Uniform Asymptotic Stability of Functional Differential Equations of the Neutral Type

Abstract. Consider the functional equations of neutral type (1) (d/dt)D(t, x,) -/(*, *,) and (2) (d/dt) [D(t, x,) -G(t, *,)} =/(/, *,) + F(t, Xt) where D,f are bounded linear operators from C[a, b] into R" or C" for each fixed t in [0, °° ), F= Fi+Fh G = Gi+Gh | F¡(t, <*>) \ ¿v(t)\\, \Gx(t,4>)\ ¿Tr(t)\ 0, there exists 8(e) >0 such that | F2(t, 0)| á«|*|, |Gs(/,0)| ê«||,<^0, 1*1 <4(t). The authors prove that if (1) is uniformly asymptotically stable, then there is a f o, 0 0,0 0, Af0>0, so>0, such that if ir(f)0, then the solution x=Q of (2) is uniformly asymptotically stable. The result generalizes previous results which consider only terms of the form F¡, Gi or Fi, Gï but not both simultaneously, and the stronger hypothesis lim(^»ir(/) =0.