Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article A Note on a Beam Equation with Nonlinear Boundary Conditions | Hindawi Publishing Corporation Boundary Value Problems Volume 2011 Article ID 376782 14 pages doi 2011 376782 Research Article A Note on a Beam Equation with Nonlinear Boundary Conditions Paolamaria Pietramala Dipartimento di Matematica Universita della Calabria Arcavacata di Rende 87036 Cosenza Italy Correspondence should be addressed to Paolamaria Pietramala pietramala@ Received 14 May 2010 Revised 12 July 2010 Accepted 31 July 2010 Academic Editor Feliz Manuel Minhos Copyright 2011 Paolamaria Pietramala. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. We present new results on the existence of multiple positive solutions of a fourth-order differential equation subject to nonlocal and nonlinear boundary conditions that models a particular stationary state of an elastic beam with nonlinear controllers. Our results are based on classical fixed point index theory. We improve and complement previous results in the literature. This is illustrated in some examples. 1. Introduction The fourth-order differential equation u 4 f g t f t U ty te 0 1 arises naturally in the study of the displacement u u t of an elastic beam when we suppose that along its length a load is added to cause deformations. This classical problem has been widely studied under a variety of boundary conditions BCs that describe the controls at the ends of the beam. In particular Gupta 1 studied along other sets of local homogeneous BCs the problem u 0 0 u 0 0 u 1 0 u 1 0 that models a bar with the left end being simply supported hinged and the right end being sliding clamped. This problem and its generalizations has been studied previously by Davies and coauthors 2 Graef and Henderson 3 and Yao 4 . 2 Boundary Value Problems Multipoint versions of this problem do have a physical interpretation. For example the .
