Remarks on the range properties of certain semilinear problems of Landesman-Lazer type

In the article we consider the resonant Dirichlet problem, u " (t) + u(t) + g(u ' (t)) = f(t), t is an element of [0, pi], u(0) = u(pi)= 0, where g is a continuous function having finite limits at plus and minus inf inity: g(-infinity) < g(+<infinity>). Imposing g(-infinity) < g(s) < g(+inf inity) for any s is an element of R we formulate a necessary condition. Our main result provides a characteristic of the set of functions f is an elem ent of C[0, T], such that (1) has a solution. However, to prove this we nee d that g is odd, increasing, and satisfies a certain asymptotic condition a t infinity, in addition. The main tools used ape Lyapunov-Schmidt reduction and certain asymptotical methods. (C) 2001 Academic Press.