A Neumann problem for a system depending on the unknown boundary values of the solution

Abstract

A semilinear system of second order ODEs under Neumann conditions is studied. The system has the particularity that its nonlinear term depends on the (unknown) Dirichlet values $y(0)$ and $y(1)$ of the solution. Asymptotic and non-asymptotic sufficient conditions of Landesman-Lazer type for existence of solutions are given. We generalize our previous results for a scalar equation, and a well known result by Nirenberg for a nonlinearity independent of $y(0)$ and $y(1)$.

Keywords

• two-ion electro-diffusion models
• landesman-lazer conditions
• nonlinear systems
• topological degree