Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On the uniform asymptotic stability of functional differential equations of the neutral type

Authors: J. K. Hale and A. F. Izé
Journal: Proc. Amer. Math. Soc. 28 (1971), 100-106
MSC: Primary 34.75
MathSciNet review: 0274900
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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},\vert{F_1}(t,\phi )\vert \leqq v(t)\vert\phi \vert,\vert{G_1}(t,\phi )\vert \leqq \pi (t)\vert\phi \vert,\pi (t)$, bounded and for any $ \varepsilon > 0$, there exists $ \delta (\varepsilon) > 0$ such that $ \vert{F_2}(t,\phi )\vert \leqq \varepsilon\vert\phi \vert,\vert{G_2}(t,\phi )\... ...eqq \varepsilon\vert\phi \vert,t \geqq 0,\vert\phi \vert < \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$.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 34.75

Retrieve articles in all journals with MSC: 34.75

Additional Information

PII: S 0002-9939(1971)0274900-1
Keywords: Functional differential equations of neutral type, Banach space, topology of uniform convergence, uniform asymptotic stability, bounded operator, bounded variation, uniformly nonatomic, trajectory, variation of constants formula, uniform stability, exponential asymptotic stability
Article copyright: © Copyright 1971 American Mathematical Society