Multiple periodic solutions of nonlinear second order differential equations

Keqiang Li; Shangjiu Wang; Keqiang Li; Shangjiu Wang

doi:10.3934/math.2023570

AIMS Mathematics

2023, Volume 8, Issue 5: 11259-11269. doi: 10.3934/math.2023570

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Research article

Multiple periodic solutions of nonlinear second order differential equations

Keqiang Li ¹,
Shangjiu Wang ^{2,3
,
,}

1.
College of Digital Technology and Engineering, Ningbo University of Finance and Economics, Ningbo 315175, China
2.
School of Mathematics and Statistics, Shaoguan University, Shaoguan 512005, China
3.
School of Economics and Statistics, Guangzhou University, Guangzhou 510006, China

Received: 29 November 2022 Revised: 16 February 2023 Accepted: 01 March 2023 Published: 10 March 2023
MSC : 34C25, 47J30, 58E05

In this paper, we are interested in the existence of multiple nontrivial $ T $-periodic solutions of the nonlinear second ordinary differential equation $ \ddot{x}+V_x(t, x) = 0 $ in $ N(\geq 1) $ dimensions. Using homological linking and morse theory, we get at least two critical points of the functional corresponding to our problem. And, we also prove that two critical points are different by critical groups. Then, we obtain there are at least two nontrivial $ T $-periodic solutions of the problem.
- multiple periodic solutions,
- morse theory,
- homological linking,
- critical groups,
- nontrivial periodic solutions
Citation: Keqiang Li, Shangjiu Wang. Multiple periodic solutions of nonlinear second order differential equations[J]. AIMS Mathematics, 2023, 8(5): 11259-11269. doi: 10.3934/math.2023570

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Abstract

In this paper, we are interested in the existence of multiple nontrivial $ T $-periodic solutions of the nonlinear second ordinary differential equation $ \ddot{x}+V_x(t, x) = 0 $ in $ N(\geq 1) $ dimensions. Using homological linking and morse theory, we get at least two critical points of the functional corresponding to our problem. And, we also prove that two critical points are different by critical groups. Then, we obtain there are at least two nontrivial $ T $-periodic solutions of the problem.

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