If
f
:
R
→
R
f:{\mathbf {R}} \to {\mathbf {R}}
is a continuous odd function satisfying
x
f
(
x
)
>
0
xf(x) > 0
,
x
≠
0
x \ne 0
, and
f
(
x
)
=
o
(
x
−
2
)
f(x) = o({x^{ - 2}})
as
x
→
∞
x \to \infty
, then so-called periodic solutions of long period seem to play a prominent role in the dynamics of
(
∗
)
({\ast })
\[
x
′
(
t
)
=
−
α
f
(
x
(
t
−
1
)
)
,
α
>
0.
x’(t) = - \alpha f(x(t - 1)),\qquad \alpha > 0.
\]
In this paper we prove the existence of long-period periodic solutions of
(
∗
)
({\ast })
for a class of nonodd functions that decay "rapidly" to
0
0
at infinity and satisfy
x
f
(
x
)
⩾
0
xf(x) \geqslant 0
. These solutions have quite different qualitative features than in the odd case.