# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## On the uniform asymptotic stability of functional differential equations of the neutral typeHTML articles powered by AMS MathViewer

by J. K. Hale and A. F. Izé
Proc. Amer. Math. Soc. 28 (1971), 100-106 Request permission

## Abstract:

Consider the functional equations of neutral type (1) $(d/dt)D(t,{x_t}) = f(t,{x_t})$ and (2) $(d/dt)[D(t,{x_t}) - G(t,{x_t})] = f(t,{x_t}) + F(t,{x_t})$ where $D,f$ are bounded linear operators from $C[a,b]$ into ${R^n}$ or ${C^n}$ for each fixed $t$ in $[0,\infty ),F = {F_1} + {F_2},G = {G_1} + {G_2},|{F_1}(t,\phi )| \leqq v(t)|\phi |,|{G_1}(t,\phi )| \leqq \pi (t)|\phi |,\pi (t)$, bounded and for any $\varepsilon > 0$, there exists $\delta (\varepsilon ) > 0$ such that $|{F_2}(t,\phi )| \leqq \varepsilon |\phi |,|{G_2}(t,\phi )| \leqq \varepsilon |\phi |,t \geqq 0,|\phi | < \delta (\varepsilon )$. The authors prove that if (1) is uniformly asymptotically stable, then there is a ${\zeta _0},0 < {\zeta _0} < 1$, such that for any $p > 0,0 < \zeta < {\zeta _0}$, there are constants ${v_0} > 0,{M_0} > 0,{s_0} > 0$, such that if $\pi (t) < {M_0},t \geqq {s_0},(1/p)\int _t^{t + p} {v(s)ds < \zeta {v_0}} ,t > 0$, then the solution $x = 0$ of (2) is uniformly asymptotically stable. The result generalizes previous results which consider only terms of the form ${F_1},{G_1}$ or ${F_{2,}}{G_2}$ but not both simultaneously, and the stronger hypothesis ${\lim _{t \to \infty }}\pi (t) = 0$.
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