Global bifurcation and multiplicity results for Sturm-Liouville problems

We prove the multiplicity theorem for the problem $$\left\{\begin{array}{l} {u''(t) + \mu u(t) + \varphi (t,u(t),u'(t)) = 0\quad \hbox{for a. e.}\, t \in (a,b)}\\ {l(u) = 0,}\end{array}\right.$$ with Sturm-Liouville boundary conditions l, and function φ satisfying Caratheodory conditions. We assume that φ(t, x, y) = Ax + o(|x| + |y|) for |x| + |y| → 0 and φ(t, x, y) = Bx + o(|x| + |y|) for |x| + |y| → +∞. We prove that the above problem has at least 2n solutions where n is the number of eigenvalues of the appropriate linear problem, laying between min(A, B) and max(A, B). Some additional remarks are following.