Coupled systems with Ambrosetti-Prodi-type differential equations

F. Minhós; F. Carapau; G. Rodrigues; F. Minhós; F. Carapau; G. Rodrigues

doi:10.3934/math.2023972

AIMS Mathematics

2023, Volume 8, Issue 8: 19049-19066. doi: 10.3934/math.2023972

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Coupled systems with Ambrosetti-Prodi-type differential equations

1.
Department of Mathematics, School of Science and Technology, Institute for Advanced Studies and Research (IIFA), University of Évora, Rua Romão Ramalho, 59, 7000-671 Évora, Portugal
2.
Center for Research in Mathematics and Applications (CIMA), Institute for Advanced Studies and Research (IIFA), University of Évora, Rua Romão Ramalho, 59, 7000-671 Évora, Portugal
3.
Coordination of Degree in Mathematics, Campus Maceió, Federal Institute of Education, Science and Technology of Alagoas (IFAL), R. Mizael Domingues, 530-Centro, Maceió-AL, 57020-600, Brazil

Received: 01 March 2023 Revised: 22 May 2023 Accepted: 24 May 2023 Published: 06 June 2023
MSC : 34B08, 34B15, 34B60

In this paper, we consider some boundary value problems composed by coupled systems of second-order differential equations with full nonlinearities and general functional boundary conditions verifying some monotone assumptions. The arguments apply the lower and upper solutions method, and defining an adequate auxiliary, homotopic, and truncated problem, it is possible to apply topological degree theory as the tool to prove the existence of solution. In short, it is proved that for the parameter values such that there are lower and upper solutions, then there is also, at least, a solution and this solution is localized in a strip bounded by lower and upper solutions. As far as we know, it is the first paper where Ambrosetti-Prodi differential equations are considered in couple systems with different parameters.
- coupled systems,
- lower and upper solutions,
- Nagumo condition,
- degree theory,
- Ambrosetti-Prodi problems
Citation: F. Minhós, F. Carapau, G. Rodrigues. Coupled systems with Ambrosetti-Prodi-type differential equations[J]. AIMS Mathematics, 2023, 8(8): 19049-19066. doi: 10.3934/math.2023972

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Abstract

In this paper, we consider some boundary value problems composed by coupled systems of second-order differential equations with full nonlinearities and general functional boundary conditions verifying some monotone assumptions. The arguments apply the lower and upper solutions method, and defining an adequate auxiliary, homotopic, and truncated problem, it is possible to apply topological degree theory as the tool to prove the existence of solution. In short, it is proved that for the parameter values such that there are lower and upper solutions, then there is also, at least, a solution and this solution is localized in a strip bounded by lower and upper solutions. As far as we know, it is the first paper where Ambrosetti-Prodi differential equations are considered in couple systems with different parameters.

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