On the structure of the solution set of u '' = f(t, u), u(0) = u(1)=0

The solution set of a Dirichlet problem x " = f(t, x), x(0) = x(1) = 0, on a Banach space E and with f satisfying a Lipschitz condition, is homeomorph ic to a closed subset of E. We prove that to any closed subset C of E there is a function f with Lipschitz constant arbitrarily close to pi(2), such t hat the solution set of the corresponding Dirichlet problem is homeomorphic to C.