Unbounded Solutions of Second-Order Multipoint Boundary Value Problem on the Half-Line. (18th October 2010)

• Record Type: Journal Article
• Authors: Liu Liu, Lishan Lishan; Hao Hao, Xinan Xinan; Wu Wu, Yonghong Yonghong
• Other Names: Radulescu Radulescu Vicentiu Vicentiu Academic Editor.
• Is Part Of: Boundary value problems. Volume 2010(2010)
• Publication Date: 2010-10-18
• Subjects: Boundary value problems -- Periodicals; Boundary value problems; Electronic journals; Periodicals; 515.35
• DOI: 10.1155/2010/236560
• Languages: English
• ISSNs: 1687-2762

Abstract

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 +, α u ( 0 ) - β u ' ( 0 ) - ∑ i = 1 n k i u ( ξ i ) = a ≥ 0, lim t → + ∞ u ' ( t ) = b > 0, where α > 0, β > 0, k i ≥ 0, 0 ≤ ξ i < ∞ ( 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.