Existence and multiplicity of solutions for fourth-order boundary value problems with parameters

In this paper we study the existence and multiplicity of the solutions for the fourth-order boundary value problem (BVP) u(4)(t) + ηu (t) − ζu(t) = λf (t, u(t)), 0 < t < 1, u(0) = u(1) = u (0) = u (1) = 0, where f : [0, 1] × R→R is continuous, ζ,η ∈ R and λ ∈ R+ are parameters. By means of the idea of the decomposition of operators shown by Chen [W.Y. Chen, A decomposition problem for operators, Xuebao of Dongbei Renmin University 1 (1957) 95–98], see also [M. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations, Gostehizdat, Moscow, 1956], and the critical point theory, we obtain that if the pair (η, ζ ) is on the curve ζ =−η2/4 satisfying η <2π2, then the above BVP has at least one, two, three, and infinitely many solutions for λ being in different interval, respectively. © 2006 Elsevier Inc. All rights reserved.