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Báo cáo sinh học: " Research Article Nodal Solutions for a Class of Fourth-Order Two-Point Boundary Value Problems"

Tuyển tập các báo cáo nghiên cứu về sinh học được đăng trên tạp chí sinh học Journal of Biology đề tài: Research Article Nodal Solutions for a Class of Fourth-Order Two-Point Boundary Value Problems | Hindawi Publishing Corporation Boundary Value Problems Volume 2010 Article ID 570932 8 pages doi 10.1155 2010 570932 Research Article Nodal Solutions for a Class of Fourth-Order Two-Point Boundary Value Problems Jia Xu1 2 and XiaoLing Han1 1 Department of Mathematics Northwest Normal University Lanzhou 730070 China 2 College of Physical Education Northwest Normal University Lanzhou 730070 China Correspondence should be addressed to Jia Xu xujia@nwnu.edu.cn Received 18 February 2010 Accepted 27 April 2010 Academic Editor Irena Rachunkova Copyright 2010 J. Xu and X. Han. 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 consider the fourth-order two-point boundary value problem u Mu Xh f f u 0 t 1 u 0 u 1 u 0 u 1 0 where A e R is a parameter M e -K4 n4 64 is given constant h e C 0 1 0 x with h t 0 on any subinterval of 0 1 f e C R R satisfies f ù u 0 for all u 0 and lim - f u u 0 lim ot f u u f OT lim 0f u u f0 for some f OT fa e 0 x . By using disconjugate operator theory and bifurcation techniques we establish existence and multiplicity results of nodal solutions for the above problem. 1. Introduction The deformations of an elastic beam in equilibrium state with fixed both endpoints can be described by the fourth-order ordinary differential equation boundary value problem u Xh t f Ù 0 t 1 L L L _ . V1 u 0 u 1 u 0 u 1 0 where f R R is continuous A e R is a parameter. Since the problem 1.1 cannot transform into a system of second-order equation the treatment method of second-order system does not apply to the problem 1.1 . Thus existing literature on the problem 1.1 is limited. In 1984 Agarwal and chow 1 firstly investigated the existence of the solutions of the problem 1.1 by contraction mapping and iterative methods subsequently Ma and Wu 2 and Yao 3 4 studied the existence of positive solutions of this