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) + etau"(t) - zetau(t) = lambdaf(t, u(t)), 0 < t < 1, u(0) = u(1) = u"(0) = u"(1) = 0, where f (t, u): [0, 1] x [0, +infinity) --> [0, +infinity) is continuous and zeta, eta and lambda are parameters. We show that there exists a lambda* > (0) over dot such that the above boundary value problem (BVP) has at least two, one and no positive solutions for 0 < lambda < lambda*, lambda = lambda* and lambda > lambda*, 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 u(lambda)(t) of the BVP depends continuously on the parameter lambda, i.e., u(lambda)(t) is nondecreasing in lambda, lim(lambda-->0+) parallel tou(lambda)(t)parallel to = 0 and lim(lambda-->+infinity) parallel tou(lambda)(t)parallel to = +infinity for any t is an element of [0, 1]. (C) 2004 Elsevier Inc. All rights reserved.