Favard theory for the adjoint equation and Fredholm Alternative

Abstract

Abstract. 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.

Recurrent equations with sign and Fredholm alternative

“…The aim of the present paper is to test the result of [1] in some concrete situations, showing that it gives new insights even in cases already studied in the literature. The first concrete result concerns a matrix A which has a sign, say for instance it is nonnegative in the classical sense:…”

“…x = A(t)x + f (t) (1) where the vector f is bounded continuous on time and the matrix A is recurrent.…”

“…is compact minimal with respect to the compact-open topology and the flow induced by translations (Aτ )(t) = A(t + τ ). A necessary condition in order (1) to admit bounded solutions can be easily obtained integrating by parts, namely:…”

“…In [1] we discussed whether or not this condition gives rise to a Fredholm Alternative in the recurrent framework, as implicitly suggested in [14]. To be more precise the question in [1] is to decide if, under the additional assumption that f is jointly recurrent with A, condition (2) is also sufficient for equation (1) to admit a solution x which is not only bounded, but even recurrent and with the same recurrence properties of A and f . This can be expressed by saying that a continuous flow epimorphism exits:…”

“…If for instance A and f are almost periodic, then this restriction implies that x is almost periodic too. The main result of [1] is that (2) is sufficient to get such x satisfying (4), as soon as A and f are jointly recurrent and A fulfills the additional condition:…”