A global stability criterion in nonautonomous delay differential equations

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.