Summary: We study the scalar equation
by means of contraction mappings. Conditions are obtained to ensure that each solution
. The conditions allow
to grow as large as
, but not as large as
. This is parallel to the classical result of R. A. Smith
[Q. J. Math., Oxf. II. Ser. 12, 123–125 (1961; Zbl 0103.05604
)] for the linear equation without a delay.