Massimo Tarallo

The method of upper and lower solutions is a main tool to prove the existence of periodic solutions to periodic differential equations. It is known that, in general, the method does not extend to the almost periodic case. The aim of the present paper is to show that, however, something interesting survives: if the classical assumptions of the method are satisfied, then the expected existence result holds generically in the limit periodic framework.

We discuss the existence of a Fredholm–type Alternative for a recurrent second order linear equation, which is disconjugate in a strong sense. The basic result is about bounded solutions of equations with bounded coefficients: it depends on kinematic similarities that allow to reduce the problem to a pair of very simple normal forms. Then the result is specialized to recurrent equations, by means of Favard theory.

We prove that a Fredholm--type Alternative holds for recurrent equations with sign, extending a previous result by Cieutat and Haraux in [3]. Moreover, we show that this can be seen a particular case of [1] and we provide a solution to an interesting question raised by Hale in [6]. Finally we characterize the existence of exponential dichotomies also in the nonrecurrent case.