Unbounded Solutions of Second-Order Multipoint Boundary Value Problem on the Half-Line. (18th October 2010) - Liu, Lishan; Hao, Xinan; Wu, Yonghong
Abstract : This paper investigates the second-order multipoint boundary value problem on the half-lineu ′′ ( 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, whereα > 0, β > 0, k i ≥ 0, 0 ≤ ξ i < ∞       ( i = 1, 2, …, n ), andf : 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 thatf 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.