Resonant nonlinear boundary value problems with almost periodic nonlinearity

In this paper we give a qualitative and quantitative description of the set of continuous functions h for which the resonant boundary value problem -u"(x) - u (x) +g(u(x)) = h(x), x is an element of [0,pi], u(0) = u(pi) 0, has solution. Here, g is a continuous and bounded function (not identically zero), with primitive G satisfying the following hypothesis: there exist sequences {x(n)} --> +infinity such that G(x(n)) --> sup {G(t) : t greater than or equal to 0}, G(y(n)) --> inf {G(t) : t greater than or equal to 0}. In particular, this is the case if g is continuous and bounded and G is an almost periodic function. A noteworthy example, from the point of view of the applications to some problems in Menchanics, is when the function g is of the form g =Sigma(i=l)(n) g(i), where each function g(i) is a continuous periodic function with period T-i and with zero mean value, i.e., integral(0)(Ti) g(i)(t) dt = 0, 1 less than or equal to i less than or equal to n. In the proofs we use the Liapunov-Schmidt 0 reduction, the shooting method and a detailed study of the oscillatory properties of the integral expressions associated to the bifurcation equation.