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Ambrosetti–Prodi Alternative for Coupled and Independent Systems of Second-Order Differential Equations

Author

Listed:
  • Feliz Minhós

    (Department of Mathematics, School of Science and Technology, University of Évora, Rua Romão Ramalho, 59, 7000-671 Évora, Portugal
    Center for Research in Mathematics and Applications, Institute for Advanced Studies and Research, University of Évora, Rua Romão Ramalho, 59, 7000-671 Évora, Portugal)

  • Gracino Rodrigues

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

Abstract

This paper deals with two types of systems of second-order differential equations with parameters: coupled systems with the boundary conditions of the Sturm–Liouville type and classical systems with Dirichlet boundary conditions. We discuss an Ambosetti–Prodi alternative for each system. For the first type of system, we present sufficient conditions for the existence and non-existence of its solutions, and for the second type of system, we present sufficient conditions for the existence and non-existence of a multiplicity of its solutions. Our arguments apply the lower and upper solutions method together with the properties of the Leary–Schauder topological degree theory. To the best of our knowledge, the present study is the first time that the Ambrosetti–Prodi alternative has been obtained for such systems with different parameters.

Suggested Citation

  • Feliz Minhós & Gracino Rodrigues, 2023. "Ambrosetti–Prodi Alternative for Coupled and Independent Systems of Second-Order Differential Equations," Mathematics, MDPI, vol. 11(17), pages 1-19, August.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:17:p:3645-:d:1223538
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