Consider the delay differential equation
x
′
(
t
)
=
−
f
(
x
(
t
−
1
)
)
x’(t)=-f(x(t-1))
, where
f
∈
C
(
R
,
R
)
f\in C(\mathbb {R}, \mathbb {R})
is odd and satisfies
x
f
(
x
)
>
0
xf(x)>0
for
x
≠
0
x\ne 0
. When
α
=
lim
x
→
0
f
(
x
)
x
\alpha =\lim _{x\to 0}\frac {f(x)}{x}
and
β
=
lim
x
→
∞
f
(
x
)
x
\beta =\lim _{x\to \infty }\frac {f(x)}{x}
exist, there is at least one Kaplan-Yorke periodic solution with period
4
4
if
min
{
α
,
β
}
>
π
2
>
max
{
α
,
β
}
\min \{\alpha ,\beta \}>\frac {\pi }{2}>\max \{\alpha ,\beta \}
. When this condition is not satisfied, we present several sufficient conditions on the existence/nonexistence of such periodic solutions. It is worthy of mention that some results are on the existence of at least two Kaplan-Yorke periodic solutions with period
4
4
and in some cases we do not require the limits
α
\alpha
and/or
β
\beta
to exist. Hence our results not only greatly improve but also complement existing ones. Moreover, some of the theoretical results are illustrated with examples.