tailieunhanh - báo cáo hóa học:" Research Article Existence of Solutions to a Nonlocal Boundary Value Problem with Nonlinear Growth"

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 Existence of Solutions to a Nonlocal Boundary Value Problem with Nonlinear Growth | Hindawi Publishing Corporation Boundary Value Problems Volume 2011 Article ID 416416 15 pages doi 2011 416416 Research Article Existence of Solutions to a Nonlocal Boundary Value Problem with Nonlinear Growth Xiaojie Lin School of Mathematical Sciences Xuzhou Normal University Xuzhou Jiangsu 221116 China Correspondence should be addressed to Xiaojie Lin linxiaojie1973@ Received 17 July 2010 Accepted 17 October 2010 Academic Editor Feliz Manuel Minhos Copyright 2011 Xiaojie Lin. 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. This paper deals with the existence of solutions for the following differential equation x t f t x t x t t e 0 1 subject to the boundary conditions x 0 ax ị x 1 J 0 x s dg s where a 0 0 ị 1 f 0 1 X R2 R is a continuous function g 0 1 0 to is a nondecreasing function with g 0 0. Under the resonance condition g 1 1 some existence results are given for the boundary value problems. Our method is based upon the coincidence degree theory of Mawhin. We also give an example to illustrate our results. 1. Introduction In this paper we consider the following second-order differential equation x t f t x t x 0 t e 0 1 subject to the boundary conditions x 0 axjif x 1 f x s dg s 0 where a 0 0 ị 1 f 0 1 X R2 R is a continuous function g 0 1 0 to is a nondecreasing function with g 0 0. In boundary conditions the integral is meant in the Riemann-Stieltjes sense. 2 Boundary Value Problems We say that BVP is a problem at resonance if the linear equation x t 0 t e 0 1 with the boundary condition has nontrivial solutions. Otherwise we call them a problem at nonresonance. Nonlocal boundary value problems were first considered by Bicadze and Samarskii 1 and later by Il pin and Moiseev 2 3 . In a recent paper 4 Karakostas and Tsamatos studied the following nonlocal .