A note on periodic solutions of the delay differential equation 𝑥’(𝑡)=-𝑓(𝑥(𝑡-1))

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.