Multiplicity theorems involving functions with non-convex range

Here is a sample of the results proved in this paper: Let f : R -> R be a continuous function, let rho > 0 and let omega : [0, rho[-> [0, +infinity[ be a continuous increasing function such that [GRAPHICS] Then, the following assertions are equivalent: [root degrees root degrees] (a) the restriction of f to - 2 , is not constant; 2 (b) for every convex set S subset of C0([0, 1]) x C0([0, 1]) dense in C0([0, 1]) x C0([0, 1]), there exists (alpha, beta) is an element of S such that the problem [GRAPHICS] has at least two classical solutions.