Boundary value problems and homoclinic solutions for systems of nonlinear second-order ordinary differential equations

In this paper, we introduce some extensions of the classical Hartman type conditions for nonlinear second-order systems of ordinary differential equations [Formula: see text] submitted to the classical Dirichlet, mixed, Neumann and periodic boundary conditions. We also give a new proof of an estimate for the derivatives of solutions due to Hartman. As a consequence, we obtain a priori bounds for the solutions. The obtained results are applied to prove the existence of solutions to the following ‘homoclinic’ problem: [Formula: see text] [Formula: see text], [Formula: see text].