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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 standard nonlinearity independent of y(0) and y(1).

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