A note on bounded solutions of second order differential equations at resonance

Abstract

In the paper we study the existence of bounded solutions for differential equations of the form: $ x''-Ax= f(t,x)$, where $A\in L(H)$, $f: {\mathbb R}\times H \to H$ ($H$ -- a Hilbert space) is a continuous mapping. Using a perturbation of the equation, the Leray-Schauder topological degree and fixed point theory, we overcome the difficulty that the linear problem is non-Fredholm in any resonable Banach space.