摘要 : We consider the nonautonomous linear delay differential equation x'(t) + p(t)x(t - 1) = 0, t >= t(0), where p: [t(0), infinity) -> R is a continuous function such that p(t) >= a > 1/e for t >= t(0). It is shown that the smallest p... 展开 We consider the nonautonomous linear delay differential equation x'(t) + p(t)x(t - 1) = 0, t >= t(0), where p: [t(0), infinity) -> R is a continuous function such that p(t) >= a > 1/e for t >= t(0). It is shown that the smallest positive constant omega = omega(a) such that every solution has at least one zero in [t(0) - 1, t(0) + omega] coincides with the first positive root zeta(a) of the fundamental solution of the equation with constant coefficient y'(t) + ay(t - 1) = 0. A detailed analysis of the function zeta: (1/epsilon, infinity) -> (1, infinity) is given with the emphasis on the asymptotic description of zeta(a) as a -> 1/e(+). As corollaries new oscillation criteria are obtained which improve some previous results in the literature. With appropriate modifications the method applies to a more general class of linear differential equations with time-dependent delay. (C) 2021 The Author(s). Published by Elsevier Inc. 收起