On the existence of two solutions with a prescribed number of zeros for a superlinear two-point boundary value problem

where the function f has superlinear growth at infinity and p grows at most linearly in u and u′. Our method is based on a continuation theorem for a coincidence equation of the form Lu = N(u, λ), with the parameter λ varying in the unit interval I. In [1] and the preceding papers [4], [9, §5.5] we introduced a continuous functional φ(u, λ) which takes integer values on large solutions and, under certain conditions, we were able to show that there are solutions of Lu = N(u, 1) with φ(u, 1) equal to some positive integer. In the applications to scalar ordinary differential equations φ was related to the number of zeros of the solutions and thus we could prove that the boundary value problems we considered have at least one solution with k zeros in [a, b] for each sufficiently large k.