Nonmonotone equations with large almost periodic forcing terms

We consider the scalar differential equation (u) over dot = f (u)+ch(t) where f (u) is a jumping nonlinearity and h(t) is an almost periodic function, while c is a real parameter deciding the size of the forcing term. The main result is that, if h(t) does not vanish too much in some suitable sense, then the equation admits a (unique) almost periodic solution for large values of the parameter c. The class of the h(t)'s to which the result applies is studied in detail: it includes all the nontrivial trigonometric polynomials and is generic in the Baire sense. (C) 2012 Elsevier Inc. All rights reserved.