Favard theory for the adjoint equation and Fredholm Alternative

Fredholm Alternative is a classical tool of periodic linear equations, allowing to describe the existence of periodic solutions of an inhomogeneous equation in terms of the adjoint equation. A few partial extensions have been proposed in the literature for recurrent equations: our aim is to point out that they have a common root and discuss whether such a root gives rise to a general Fredholm-type Alternative. Sacker–Sell spectral theory and Favard theory are main ingredients in this discussion: a considerable effort is devoted to understand how Favard theory is affected by adjunction, at least for planar equations.