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

DOI:
https://doi.org/10.1090/S0002-9939-1971-0274900-1

MathSciNet review:
0274900

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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 ${\lim _{t \to \infty }}\pi (t) = 0$.

- J. K. Hale and M. A. Cruz,
*Asymptotic behavior of neutral functional differential equations*, Arch. Rational Mech. Anal.**34**(1969), 331–353. MR**249760**, DOI https://doi.org/10.1007/BF00281436 - J. K. Hale and M. A. Cruz,
*Existence, uniqueness and continuous dependence for hereditary systems*, Ann. Mat. Pura Appl. (4)**85**(1970), 63–81. MR**262633**, DOI https://doi.org/10.1007/BF02413530 - Jack K. Hale and Kenneth R. Meyer,
*A class of functional equations of neutral type*, Memoirs of the American Mathematical Society, No. 76, American Mathematical Society, Providence, R.I., 1967. MR**0223842**

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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