Boundary Value Problems
Volume 2011 (2011), Article ID 416416, 15 pages
doi:10.1155/2011/416416
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

Received 17 July 2010; Accepted 17 October 2010

Academic Editor: Feliz Manuel Minhós

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.

Abstract

This paper deals with the existence of solutions for the following differential equation: 𝑥 ( 𝑡 ) = 𝑓 ( 𝑡 , 𝑥 ( 𝑡 ) , 𝑥 ( 𝑡 ) ) , 𝑡 ∈ ( 0 , 1 ) , subject to the boundary conditions: 𝑥 ( 0 ) = 𝛼 𝑥 ( 𝜉 ) , 𝑥 ∫ ( 1 ) = 1 0 𝑥 ( 𝑠 ) 𝑑 𝑔 ( 𝑠 ) , where 𝛼 ≥ 0 , 0 < 𝜉 < 1 , 𝑓 ∶ [ 0 , 1 ] × 𝑅 2 → 𝑅 is a continuous function, 𝑔 ∶ [ 0 , 1 ] → [ 0 , ∞ ) 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.