Positive solutions of the nonlinear fourth-order beam equation with three parameters

This paper is concerned with the existence and nonexistence of positive solutions of the nonlinear fourth-order beam equation 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 (t,u): [0, 1] × [0,+∞) → [0,+∞) is continuous and ζ , η and λ are parameters. We show that there exists a λ∗ > ˙0 such that the above boundary value problem (BVP) has at least two, one and no positive solutions for 0 < λ < λ∗, λ = λ∗ and λ > λ∗, respectively. Furthermore, by using the semiorder method on cones of Banach space, we establish a uniqueness criterion for positive solution of the BVP. In particular such a positive solution uλ(t ) of the BVP depends continuously on the parameter λ, i.e., uλ(t ) is nondecreasing in λ, limλ→0+ uλ(t ) = 0 and limλ→+∞ uλ(t ) =+∞for any t ∈ [0, 1].  2004 Elsevier Inc. All rights reserved.