Nodal solutions of boundary value problems of fourth-order ordinary differential equations

We study the existence of nodal solutions of the fourth-order two-point boundary value problem y +β(t)y = a(t)f (y), 0 < t <1, y(0) = y(1) = y (0) = y (1) = 0, where β ∈ C[0, 1] with β(t) < π2 on [0, 1], a ∈ C[0, 1] with a 0 on [0, 1] and a(t) ≡ 0 on any subinterval of [0, 1], f ∈ C(R) satisfies f (u)u > 0 for all u = 0. We give conditions on the ratio f (s)/s at infinity and zero that guarantee the existence of nodal solutions. The proof of our main results is based upon bifurcation techniques. © 2005 Elsevier Inc. All rights reserved