On stability of some linear and nonlinear delay differential equations

New explicit conditions of exponential stability are obtained for the nonautonomous equation with several delays ˙y(t) + l k=1 ak(t)y hk(t) = 0 by the following method: several delays in the left-hand side are chosen and the solution is estimated using an auxiliary ordinary differential equation ˙y(t) + k∈I ak(t)y(t) = 0, where I ∈ {1, 2, . . . , l} is the chosen set of indices. These results are applied to analyze the stability of the nonlinear equation ˙x(t) + l k=1 ak(t)x hk(t) = f t,x(t),x g1(t) , . . . , x gm(t) by the first approximation. It is to be noted that coefficients and delays are not assumed to be continuous functions.  2005 Elsevier Inc. All rights reserved.