On first-order neutral differential equations of Euler form with unbounded delays

In this paper, we investigate the first-order neutral differential equation of Euler form with variable unbounded delay (∗)ddtx(t)-cx(αt)+1t∑i=1npix(βit)=0,t⩾t0>0,where 0⩽c<1,0<α<1,0<βi<1 and pi>0 are constants, i=1,2,…,n. Some necessary and sufficient as well as explicit sufficient conditions for the oscillation of all solutions are established. Several relevant results in the literature are improved.