Absztrakt: | The paper is concerned with stability properties of functional-differential
equations. The technique of the proofs is based on the method
of Lyapunov functionals. This method is combined with the annulus
argument -- the method which can detect that a curve in $\bbfR^n$
crosses an annulus around the origin infinitely many times. The
author modifies this method and gives annulus arguments not requiring
the boundedness of the derivatives of the functions involved.
Using these results, the author establishes several Lyapunov
type theorems for the attractivity, asymptotic $D$-stability
and partial stability of the zero solution of a nonautonomous
functional-differential equation $x'(t)=f(t,x_t)$. The right-hand
side of this equation is not assumed to be bounded with respect
to $t$. The results are applied to the scalar equation with one
constant delay $$ x'(t)=-c(t)x(t)+b(t)x(t-h) \qquad (c(t)\ge
0), $$ to the scalar equation with several constant delays $$
x'(t)=-c(t)x(t)+\sum _{i=1}^{n} b_i(t)x(t-h_i) \qquad (c(t)\ge
0) $$ and to the system $$ x'(t)=B(t)x(t-h)-C(t)x(t), $$ where
$B(t)$ and $C(t)$ are continuous matrix functions. The results
generalize and improve several results of {\it S. N. Busenberg}
and {\it K. L. Cooke} [Q. Appl. Math. 42, 295-306 (1984; Zbl
0558.34059)].}
\RV{J.Kalas (Brno) |