Abstract :
Consider the following nonautonomous nonlinear delay differential equation:
⎧⎪
⎨⎪
⎩
dy(t)
dt =−a(t)y(t) −
m
i=0
ai (t)gi y τi (t) , t t0,
y(t) = φ(t), t t0,
where we assume that there is a strictly monotone increasing function f (x) on (−∞,+∞) such that
⎧⎪
⎨⎪
⎩
f (0) = 0, 0 <
gi(x)
f (x)
1, x = 0, 0 i m, and
if f (x) ≡ x, then lim
x→−∞
f (x) or lim
x→+∞
f (x) is finite.
In this paper, to the above nonautonomous nonlinear delay differential equation, we establish conditions of
global asymptotic stability for the zero solution. In particular, for a special wide class of f (x) which contains
two cases f (x) = ex −1 and f (x) = x, we give more explicit conditions which are some extension of
the “3/2-type criterion.” Applying these to discrete models of nonautonomous delay differential equations,
we also obtain new sufficient conditions of the global asymptotic stability of the zero solution.
© 2006 Elsevier Inc. All rights reserved.
