In this paper we use upper and lower solutions, the monotone iterative method, and the Schauder fixed point theorem to establish a new result on the existence of bounded/periodic solutions over
R
\mathbb R
for a class of scalar delay differential equations of the form
d
u
d
t
=
f
(
t
,
u
(
t
)
,
u
(
t
−
r
1
)
,
⋯
,
u
(
t
−
r
m
)
)
\begin{equation*} \frac {du}{dt}=f(t,u(t), u(t-r_1), \cdots , u(t-r_m)) \end{equation*}
with constant delays, where
f
f
satisfies a one-sided Lipschitz condition on the state variables
u
(
t
)
u(t)
,
u
(
t
−
r
1
)
u(t-r_1)
, …,
u
(
t
−
r
m
)
u(t-r_m)
. Applications of this result to some population models are given.