Boundary Value Problems
Volume 2010 (2010), Article ID 236560, 15 pages
doi:10.1155/2010/236560
Research Article

Unbounded Solutions of Second-Order Multipoint Boundary Value Problem on the Half-Line

Lishan Liu,1,2 Xinan Hao,1 and Yonghong Wu2

1School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China
2Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia

Received 14 May 2010; Revised 4 September 2010; Accepted 11 October 2010

Academic Editor: Vicentiu Radulescu

Copyright © 2010 Lishan Liu et al. 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 investigates the second-order multipoint boundary value problem on the half-line u′′(t)+f(t,u(t),u'(t))=0, t∈ℝ+, αu(0)-βu'(0)-∑i=1nkiu(ξi)=a≥0, lim⁡t→+∞u'(t)=b>0, where α>0, β>0, ki≥0, 0≤ξi<∞    (i=1,2,…,n), and f:ℝ+×ℝ×ℝ→ℝ is continuous. We establish sufficient conditions to guarantee the existence of unbounded solution in a special function space by using nonlinear alternative of Leray-Schauder type. Under the condition that f is nonnegative, the existence and uniqueness of unbounded positive solution are obtained based upon the fixed point index theory and Banach contraction mapping principle. Examples are also given to illustrate the main results.