Boundary and periodic value problems for systems of differential equations under Bernstein-Nagumo growth condition (Q1894165)

The author establishes several existence results for systems of second- order differential equations (1) \(x'' = f(t,x,x')\) subjected on the interval \(0,1\) to various boundary conditions (e.g. nonhomogeneous Dirichlet, Neumann, Sturm-Liouville conditions, periodic conditions). It is assumed that \(f : 0,1 \times \mathbb{R} 2n \to \mathbb{R} n\) fulfils the Carathéodory conditions and some (Bernstein-type or Nagumo-type) growth conditions. Proofs are obtained via the theory of topological transversality for continuous compact operators in the case of the Bernstein-type conditions and for upper semi-continuous, compact, multivalued operators in the case of the Nagumo-type conditions. On the contrary to the previously published results concerning the subject, Hartman's condition is replaced by another condition which is automatically satisfied in the scalar case.