On periodic Ambrosetti-Prodi-type problems

Feliz Minhós; Nuno Oliveira; Feliz Minhós; Nuno Oliveira

doi:10.3934/math.2023654

AIMS Mathematics

2023, Volume 8, Issue 6: 12986-12999. doi: 10.3934/math.2023654

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On periodic Ambrosetti-Prodi-type problems

Feliz Minhós ^{1
,
,},
Nuno Oliveira ²

1.
Departament of Mathematics, School of Sciences and Technology, Research Centre in Mathematics and Applications (CIMA), Institute for Research and Advanced Training (IIFA), University of Évora. Rua Romão Ramalho, 59, 7000-671 Évora, Portugal
2.
Research Centre in Mathematics and Applications (CIMA), Institute for Research and Advanced Training (IIFA), University of Évora. Rua Romão Ramalho, 59, 7000-671 Évora, Portugal

Received: 23 December 2022 Revised: 09 March 2023 Accepted: 17 March 2023 Published: 31 March 2023
MSC : 34B15, 34B18, 34L30

This work presents a discussion of Ambrosetti-Prodi-type second-order periodic problems. In short, the existence, non-existence, and multiplicity of solutions will be discussed on the parameter $ \lambda $. The arguments rely on a Nagumo condition, to guarantee an apriori bound on the first derivative, lower and upper-solutions method, and the Leray-Schauder's topological degree theory. There are two types of new results based on the parameter's variation: An existence and non-existence theorem and a multiplicity theorem, proving the existence of a bifurcation point. An application to a damped and forced pendulum is studied, suggesting a method to estimate the critical values of the parameter.
- Nagumo-type condition,
- lower and upper solutions,
- Leray-Schauder degree,
- Ambrosetti-Prodi problems
Citation: Feliz Minhós, Nuno Oliveira. On periodic Ambrosetti-Prodi-type problems[J]. AIMS Mathematics, 2023, 8(6): 12986-12999. doi: 10.3934/math.2023654

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Abstract

This work presents a discussion of Ambrosetti-Prodi-type second-order periodic problems. In short, the existence, non-existence, and multiplicity of solutions will be discussed on the parameter $ \lambda $. The arguments rely on a Nagumo condition, to guarantee an apriori bound on the first derivative, lower and upper-solutions method, and the Leray-Schauder's topological degree theory. There are two types of new results based on the parameter's variation: An existence and non-existence theorem and a multiplicity theorem, proving the existence of a bifurcation point. An application to a damped and forced pendulum is studied, suggesting a method to estimate the critical values of the parameter.

References

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