Two-parameter nonresonance condition for the existence of fourth-order boundary value problems ✩

In this paper, we discuss the existence of the fourth-order boundary value problem u(4) = f (t,u,u ), 0 < t <1, u(0) = u(1) = u (0) = u (1) = 0, where f : [0, 1]×R×R→R is continuous, and partly solve the Del Pino and Manasevich’s conjecture on the nonresonance condition involving the two-parameter linear eigenvalue problem. We also present a two-parameter nonresonance condition described by circle.  2004 Elsevier Inc. All rights reserved