Existence of positive solutions for the cantilever beam equations with fully nonlinear terms. (February 2016)

• Record Type: Journal Article
• Authors: Li, Yongxiang
• Is Part Of: Nonlinear analysis. Volume 27(2016)
• Page Start: 221
• Page End: 237
• Publication Date: 2016-02
• Subjects: Fully fourth-order boundary value problem -- Cantilever beam equation -- Positive solution -- Cone -- Fixed point index; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2015.07.016
• Languages: English
• ISSNs: 1468-1218

Abstract

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.