June 2013 CHINESE JOURNAL OF ENGINEERING MATHEMATICS Vol. 30 No. 3 doi: 10.3969/j.issn.1005-3085.2013.03.017 Article ID: 1005-3085(2013)03-0467-08 Positive Solutions for a Class of Beam Equations ∗ WU Ying 1 , HAN Guo-dong 2,† (1- School of Science, Xi’an University of Science and Technology, Xi’an 710054; 2- College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062) Abstract: In engineering, the fourth-order two-point boundary value problem u (4) = f (t, u(t)),t ∈ [0, 1] is used to describe the deformation of an elastic beam un- der external vertical forces. A beam that has hinged connection at one end and roller connection in other end is called simply supported beam, and its correspond- ing equation satisfies the boundary condition u(0) = u(1) = u ′′ (0) = u ′′ (1) = 0 since its displacements and bending moments at both ends are equal to zero. In the paper, by using the descending flow invariant set method, it is proved that there exists a positive solution for a class of simply supported beam equations under the assumption that the nonlinear term f is asymptotically linear at 0 and superquadric at ∞ in u. The main result and its proof are quite different from those presented by other literature. Keywords: fourth-order boundary value problems; positive solution; nonlinear operator; critical point Classification: AMS(2000) 47H07; 34B18 CLC number: O175.8 Document code: A 1 Introduction and main results It is well known that the following fourth-order two-point boundary value problem (BVP):    u (4) = f (t, u(t)), t ∈ [0, 1], u(0) = u(1) = u ′′ (0) = u ′′ (1) = 0, (1) describes the deformation of an elastic beam, both of whose ends are simply supported at 0 and 1. In recent years, much attention has been given to BVP (1) by a number of authors, see [1–9] and references therein. Received: 11 Apr 2011. Accepted: 13 Oct 2011. Biography: Wu Ying (Born in 1977), Female, Master, Lecturer. Research field: nonlinear analysis and applications. * Foundation item: The National Natural Science Foundation of China (11101253; 10826081; 10871123); the Fundamental Research Funds for the Central Universities (GK200902046); the Scientific Research Foundation of Xi’an University of Science and Technology (200843). † Corresponding author: G. Han. E-mail address: gdhan.math@gmail.com