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"" + beta(t)y" = a(t)f(y), 0 < t < 1, y(0) = y(1) = y"(0) = y"(1) = 0, where beta is an element of C [0, 1] with beta(t) < pi(2) on [0, 1], a is an element of C [0, 1] with a >= 0 on [0, 1] and a(t) not equivalent to 0 on any subinterval of [0, 1], f is an element of C (R) satisfies f (u) u > 0 for all u not equal 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. (c) 2005 Elsevier Inc. All rights reserved.