Existence of positive solutions for the cantilever beam equations with fully nonlinear terms. (February 2016) - Li, Yongxiang
Abstract: 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 ∈ [ 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 ] × R + 3 × 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.,