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) + Sigma(l)(k=1) a(k)(t)y(h(k)(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) + Sigma(k is an element of 1) a(k)(t)y(t) = 0, where 1 is an element of {1, 2,..., l} is the chosen set of indices. These results are applied to analyze the stability of the nonlinear equation x(t) + Sigma(l)(k=1) a(k)(t)x(h(k)(t) = f(t, x(t), x(g(1)(t)),..., x(g(m)(t))) by the first approximation. It is to be noted that coefficients and delays are not assumed to be continuous functions. (c) 2005 Elsevier Inc. All rights reserved.