  We shall firstly study the existence of one positive solution of a model
problem for one equation via the classical Krasnosel'ski\u\i{} fixed-point
theorem. Secondly we investigate how to handle this problem via the fixed point
index theory for compact maps. Thirdly we illustrate how this approach can be
tailored in order to deal with non-trivial solutions for systems of ODEs
subject to local BCs. The case of nonlocal and nonlinear BCs will also be
investigated. Finally we present some applications to the existence of radial
solutions of some systems of elliptic PDEs.
