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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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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DOI: https://doi.org/10.1007/s00025-021-01401-w
Keywords
- Periodic solution
- resonance and non-resonance
- Landesman–Lazer conditions
- rotation number
- Poincaré–Bohl theorem