Asymptotic Behavior of Solutions of Linear Delay Equations

The delay differential equations of the formx′(t)=-a(t)x(t-1),t≥0 (*)are considered,where a(t)≥0 is locally integrable on [0,∞).The main result:Let 0<c(t)≤a(t)≤k(t) for large t,∫~∞ c(s)ds=+∞,and c(t)≤Mc(t′) for t,t′≥T,|t-t′|≤l with some constants l>0,M>1,T≥0.Then the condition k(t)≤3/2+αc(t),t≥T withsome constant α>0 dependent on l,M,ensures that all solutions of (*) tend to zero as t→∞.