Periodic solutions for a class of second-order differential delay equations

Xuan Wu; Huafeng Xiao

doi:10.3934/cpaa.2021159

Article Contents

2021, Volume 20, Issue 12: 4253-4269. Doi: 10.3934/cpaa.2021159

This issue Previous Article Asymptotic expansion of the ground state energy for nonlinear Schrödinger system with three wave interaction Next Article Partial regularity result for non-autonomous elliptic systems with general growth

Periodic solutions for a class of second-order differential delay equations

Xuan Wu and
Huafeng Xiao^,

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

^* Corresponding author
^* Corresponding author

Received: June 2021

Revised: August 2021

Early access: September 22, 2021

Published online: September 22, 2021

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

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Abstract

In this paper, we study the existence of periodic solutions of the following differential delay equations

$ \begin{equation} z^{\prime\prime}(t) = \sum\limits_{k = 1}^{M-1}(-1)^kf(z(t-k)), \notag \end{equation} $

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.

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Mathematics Subject Classification: Primary: 34K13; Secondary: 58E05.

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