### On the existence of periodic solutions for a class of non-autonomous differential delay equations

#### Abstract

This paper considers the existence of periodic solutions for a

class of non-autonomous differential delay equations

$$

x'(t)=-\sum_{i=1}^{n-1}f(t,x(t-i\tau)), \leqno{(*)}

$$

where $\tau> 0$ is a given

constant. It is shown that under some conditions on $f$ and by

using symplectic transformations, Floquet theory and some results

in critical point theory, the existence of single periodic

solution of the differential delay equation $(*)$ is obtained.

These results generalize previous results on the cases that the

equations are autonomous.

class of non-autonomous differential delay equations

$$

x'(t)=-\sum_{i=1}^{n-1}f(t,x(t-i\tau)), \leqno{(*)}

$$

where $\tau> 0$ is a given

constant. It is shown that under some conditions on $f$ and by

using symplectic transformations, Floquet theory and some results

in critical point theory, the existence of single periodic

solution of the differential delay equation $(*)$ is obtained.

These results generalize previous results on the cases that the

equations are autonomous.

#### Keywords

Hamiltonian system; Floquet theory; symplectic transformation; periodic solution; delay equation; critical point theory

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