On stability of some linear and nonlinear delay differential equations

New explicit conditions of exponential stability are obtained for the nonautonomous equation with several delaysy˙(t)+∑k=1lak(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 equationy˙(t)+∑k∈Iak(t)y(t)=0, where I∈{1,2,…,l}I∈{1,2,…,l} is the chosen set of indices.These results are applied to analyze the stability of the nonlinear equationx˙(t)+∑k=1lak(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.