tailieunhanh - Báo cáo toán học: " Sturm-Liouville BVP in Banach space"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học được đăng trên tạp chí toán học quốc tế đề tài: Sturm-Liouville BVP in Banach space | Su et al. Advances in Difference Equations 2011 2011 65 http content 2011 1 65 o Advances in Difference Equations a SpringerOpen Journal RESEARCH Open Access Sturm-Liouville BVP in Banach space Hua Su1 Lishan Liu2 and Xinjun Wang3 Correspondence jnsuhua@163. com 1School of Mathematic and Quantitative Economics Shandong University of Finance and Economics Jinan Shandong 250014 China Full list of author information is available at the end of the article Springer Abstract We consider the existence of single and multiple positive solutions for fourth-order Sturm-Liouville boundary value problem in Banach space. The sufficient condition for the existence of single and multiple positive solutions is obtained by fixed theorem of strict set contraction operator in the frame of the ODE technique. Our results significantly extend and improve many known results including singular and nonsingular cases. 1 Introduction The boundary value problems BVPs for ordinary differential equations play a very important role in both theory and application. They are used to describe a large number of physical biological and chemical phenomena. In this article we will study the existence of positive solutions for the following fourth-order nonlinear Sturm-Liouville BVP in a real Banach space E P1ỉ p t u t f u ti 0 1 1 a1 u 0 - b1Uz 0 0 Y1U 1 51UZ 1 0 1 1 a 2 u 0 - 2 lim p t u t 0 V2U 1 2 iim p t u t 0 t 1- where a b ỏ gi 0 i 1 2 are constants such that p 1 b1g1 a1 71 a 1Ỏ1 0 B t s ft Ệ P2 b2g2 a2g2 B 0 1 a2ỏ2 0 and p e C1 0 1 0 . P T Moreover p may be singular at t 0 and or 1. BVP is often referred to as the deformation of an elastic beam under a variety of boundary conditions for detail see 1-17 . For example BVP subject to Lidstone boundary value conditions u 0 u 1 u 0 u 1 0 are used to model such phenomena as the deflection of elastic beam simply supported at the endpoints see 1 3 5 7-11 13 14 . We notice that the above articles use the .