Leggett–Williams type theorems with applications to nonlinear differential and integral equations

The aim of this article is to prove a few Leggett–Williams type theorems, in particular for a more general class of mappings than compact ones. We examine also invariant directions which satisfy a certain additional condition formulated in terms of a given seminorm. As applications of these results we prove a few existence results concerning positive solutions to Hammerstein integral equations or to a two-point boundary value problem.