American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021159
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Periodic solutions for a class of second-order differential delay equations

 School of Mathematics and Information Science, Guangzhou University, Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou, Guangdong 510006, China

* Corresponding author

Received  June 2021 Revised  August 2021 Early access September 2021

Fund Project: This project is supported by National Natural Science Foundation of China (No. 11871171)

In this paper, we study the existence of periodic solutions of the following differential delay equations
 $$$z^{\prime\prime}(t) = \sum\limits_{k = 1}^{M-1}(-1)^kf(z(t-k)), \notag$$$
where
 $f\in C(\mathbf{R}^N, \mathbf{R}^N)$
,
 $M,N\in \mathbf{N}$
and
 $M$
is odd. By making use of
 $S^1$
-geometrical index theory, we obtain an estimation about the number of periodic solutions in term of the difference between eigenvalues of asymptotically linear matrices at the origin and at infinity.
Citation: Xuan Wu, Huafeng Xiao. Periodic solutions for a class of second-order differential delay equations. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021159
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