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Non-resonance and Double Resonance for a Planar System via Rotation Numbers

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Abstract

We consider a planar system \(z'=f(t,z)\) under non-resonance or double resonance conditions and obtain the existence of \(2\uppi \)-periodic solutions by combining a rotation number approach together with Poincaré-Bohl theorem. Firstly, we allow that the angular velocity of solutions of \(z'=f(t,z)\) is controlled by the angular velocity of solutions of two positively homogeneous system \(z'=L_i(t,z),i=1,2\), whose rotation numbers satisfy \(\rho (L_1)>n\) and \(\rho (L_2)<n+1\), namely, nonresonance occurs in the sense of the rotation number. Secondly, we prove the existence of \(2\uppi \)-periodic solutions when the nonlinearity is allowed to interact with two positively homogeneous system \(z'=L_i(t,z),i=1,2\), with \(\rho (L_1)\ge n\) and \(\rho (L_2)\le n+1\), which gives rise to double resonance, and some kind of Landesman–Lazer conditions are assumed at both sides.

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Acknowledgements

The authors are grateful to an anonimous referee for a careful reading of a first version of this paper. This work is supported by the National Natural Science Foundation of China (No. 12071327), Spanish ERDF project MTM2017-82348-C2-1-P and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 19KJD100004).

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Authors and Affiliations

  1. School of Mathematical Sciences, Soochow University, Suzhou, 215006, People’s Republic of China

    Chunlian Liu & Dingbian Qian

  2. Departamento de Matematica Aplicada, Universidad de Granada, 18071, Granada, Spain

    Pedro J. Torres

  3. College of Xinglin, Nantong University, Nantong, 226008, People’s Republic of China

    Chunlian Liu

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  1. Chunlian Liu
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  2. Dingbian Qian
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  3. Pedro J. Torres
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Correspondence to Pedro J. Torres.

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Liu, C., Qian, D. & Torres, P.J. Non-resonance and Double Resonance for a Planar System via Rotation Numbers. Results Math 76, 91 (2021). https://doi.org/10.1007/s00025-021-01401-w

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