Nonmonotone equations with large almost periodic forcing terms

We consider the scalar differential equation 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)ʼs to which the result applies is studied in detail: it includes all the nontrivial trigonometric polynomials and is generic in the Baire sense.