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

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.

Keywords

• coupled systems
• lower and upper solutions
• Nagumo condition
• degree theory
• Ambrosetti–Prodi problems
• Lotka–Volterra systems