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

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.

Automated summary

This paper investigates two types of second-order differential equation systems with parameters: coupled systems with Sturm-Liouville type boundary conditions and classical systems with Dirichlet boundary conditions. We discuss the Ambosetti-Prodi alternative for each system. We provide sufficient conditions for the existence and non-existence of solutions for both types of systems using the lower and upper solutions method and the properties of the Leary-Schauder topological degree theory. This study is the first to obtain the Ambrosetti-Prodi alternative for such systems with different parameters.