Existence of positive solutions for the cantilever beam equations with fully nonlinear terms

In this paper we discuss the existence of positive solutions of the fully fourth-order boundary value problem {u((4)) = f(t, u, u', u '', u '''), t is an element of [0, 1], u(0) = u'(0) = u ''(1) = u '''(1) = 0, which models a statically elastic beam fixed at the left and freed at the right, and it is called cantilever beam in mechanics, where f : [0, 1] x R-+(3) x R- -> R+ is continuous. Some inequality conditions on f guaranteeing the existence of positive solutions are presented. Our conditions allow that f(t, x(0), x(1), x(2), x(3)) is superlinear or sublinear growth on x(0), x(1), x(2), x(3). For the superlinear case, a Nagumo-type condition is presented to restrict the growth of f on x(2) and x(3). Our discussion is based on the fixed point index theory in cones. (C) 2015 Elsevier Ltd. All rights reserved.