Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

On the uniform asymptotic stability of functional differential equations of the neutral type
HTML articles powered by AMS MathViewer

by J. K. Hale and A. F. Izé PDF
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 ${\lim _{t \to \infty }}\pi (t) = 0$.
References
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
  • © Copyright 1971 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 28 (1971), 100-106
  • MSC: Primary 34.75
  • DOI: https://doi.org/10.1090/S0002-9939-1971-0274900-1
  • MathSciNet review: 0274900