Favard theory and Fredholm alternative for disconjugate recurrent second order equations (Q523843)

The authors consider a boundary value problem for a class of scalar equations of the type \[ x''+\lambda(t)x'+a(t)x=g(t) \] in the recurrent framework. It is assumed that \(a(t)\) and \(\lambda(t)\) are jointly recurrent in the sense that their joint hull \(\mathrm{cls}\{(a_\tau,\lambda_\tau):\tau\in\mathbb{R}\}\) is compact minimal, where subscripts stand for translation factors and closure is taken with respect to the compact-open topology. From authors' abstract: The existence of Fredholm-type alternative is discussed for the recurrent 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.