Resonance with Landesman-Lazer conditions for parameter-dependent equations: a multiplicity result via the Poincaré-Birkhoff theorem

Chunlian Liu; Shuang Wang; Fanfan Chen; Chunlian Liu; Shuang Wang; Fanfan Chen

doi:10.3934/math.20241374

AIMS Mathematics

2024, Volume 9, Issue 10: 28320-28340. doi: 10.3934/math.20241374

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Resonance with Landesman-Lazer conditions for parameter-dependent equations: a multiplicity result via the Poincaré-Birkhoff theorem

1.
School of Mathematics and Statistics, Nantong University, Nantong 226019, China
2.
School of Mathematics and Statistics, Yancheng Teachers University, Yancheng 224051, China
3.
School of Science, Zhejiang Sci-Tech University, Hangzhou 310018, China

Received: 08 August 2024 Revised: 21 September 2024 Accepted: 26 September 2024 Published: 30 September 2024
MSC : 34B15, 34C15, 34C25

We investigate the resonance problem and prove the existence of multiple periodic solutions to a second order parameter-dependent equation $ x''+f(t, x) = sp(t) $. We weaken the usual requirement on the sublinearity of perturbations when $ |x| $ becomes large; and develop a more general method to investigate the rotational characterizations of the Landesman-Lazer conditions. Furthermore, $ f $ does not satisfy the common sign condition, and even the global existence of the solution is not guaranteed.
- periodic solutions,
- Poincaré-Birkhoff theorem,
- resonance,
- Landesman-Lazer conditions,
- parameter-dependent
Citation: Chunlian Liu, Shuang Wang, Fanfan Chen. Resonance with Landesman-Lazer conditions for parameter-dependent equations: a multiplicity result via the Poincaré-Birkhoff theorem[J]. AIMS Mathematics, 2024, 9(10): 28320-28340. doi: 10.3934/math.20241374

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Abstract

We investigate the resonance problem and prove the existence of multiple periodic solutions to a second order parameter-dependent equation $ x''+f(t, x) = sp(t) $. We weaken the usual requirement on the sublinearity of perturbations when $ |x| $ becomes large; and develop a more general method to investigate the rotational characterizations of the Landesman-Lazer conditions. Furthermore, $ f $ does not satisfy the common sign condition, and even the global existence of the solution is not guaranteed.

References

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