Existence of Positive Solutions of Nonlinear Second-Order M-Point Boundary Value Problem

Under suitable conditions on f t u , ( ) , the nonlinear second-order m-point boundary value problem ′′ + ( ) = u t f t u t ( ) , ( ) 0 , 0 1 < < t u( ) 0 0 = , u a u i i i m ( ) ( ) 1 1 2 = = − ∑ x has at least one positive solution. In this paper, the authors examine the positive solutions of nonlinear second-order m-point boundary value problem. Ma, 2001; Sun, 2005) for some recent results of nonlinear multi-point boundary value problems. In this paper, we investigate the existence of positive solutions to nonlinear second-order multi-point boundary value problems: ′′ + ( ) = u t f t u t ( ) , ( ) 0 , 0 1 < < t (1.1) u( ) 0 0 = , u a u i i i m ( ) ( ) 1 1 2 = = − ∑ x (1.2) DOI: 10.4018/jalr.2011010103 International Journal of Artificial Life Research, 2(1), 28-33, January-March 2011 29 Copyright © 2011, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. where a i 3 0 for i m = − 1 2 3 , ,..., and a m− > 2 0 , xi satisfy 0 1 1 2 2 < < < < < − x x x ... m and a u i i i m ( ) x = − ∑ < 1 2 1 . We also make the following assumptions. (A 1 ) f C ∈ × ∞ ∞ ( ) [ , ] [ , );[ , ) 0 1 0 0 (A 2 ) There are real numbers H 1 , H 2 satisfying f s u M H ( , )£ 1 1 on [ , ] [ , ] 0 1 0 1 ́ H and f s u M H ( , )3 2 2 on [ , ] [ , ) x m H − × ∞ 2 2 1 Γ , where M a i i i m 1 1 2 2 1 = ⋅ − ( ) =− ∑ x and