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

Abstract:

Consider the functional equations of neutral type (1) $(d/dt)D(t,{x_t}) = f(t,{x_t})$ 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 ${\lim _{t \to \infty }}\pi (t) = 0$.