tailieunhanh - UNIQUENESS OF POSITIVE SOLUTIONS OF A CLASS OF ODE WITH NONLINEAR BOUNDARY CONDITIONS RUYUN MA AND

UNIQUENESS OF POSITIVE SOLUTIONS OF A CLASS OF ODE WITH NONLINEAR BOUNDARY CONDITIONS RUYUN MA AND YULIAN AN Received 19 August 2004 and in revised form 27 January 2005 We study the uniqueness of positive solutions of the boundary value problem u + a(t)u + f (u) = 0, t ∈ (0,b), B1 (u(0)) − u (0) = 0, B2 (u(b)) + u (b) = 0, where 0 | UNIQUENESS OF POSITIVE SOLUTIONS OF A CLASS OF ODE WITH NONLINEAR BOUNDARY CONDITIONS RUYUN MA AND YULIAN AN Received 19 August 2004 and in revised form 27 January 2005 We study the uniqueness of positive solutions of the boundary value problem u a t u f u 0 t G 0 b B1 u 0 - u 0 0 B2 u b u b 0 where 0 b TO B1 and B2 G C1 R a G C 0 to with a 0 on 0 to and f G C 0 to n C1 0 to satisfy suitable conditions. The proof of our main result is based upon the shooting method and the Sturm comparison theorem. 1. Introduction The existence of positive solutions of second order ordinary differential equations ODEs with linear boundary conditions has been extensively studied in the literature see Coffman 1 Henderson and Wang 7 Lan and Webb 8 and the references therein. Also the existence of positive solutions of second order ODEs with nonlinear boundary conditions has been studied by several authors see Dunninger and Wang 2 Wang 11 and Wang and Jiang 12 for some references along this line. However for the uniqueness problem of second order ODEs even in the linear boundary conditions case very little was known see Ni and Nussbaum 9 Fu and Lin 6 and Peletier and Serrin 10 . To the best of our knowledge no uniqueness results of positive solutions were established for second order ODEs subject to nonlinear boundary conditions. In this paper we attempt to prove some uniqueness results in this direction. More precisely we consider the uniqueness of positive solutions of the boundary value problem u a t u f u 0 t G 0 b B1 u 0 - u 0 0 B2 u b u b 0 where 0 b TO. We make the following assumptions C1 f G C 0 to n C1 0 to with f 0 0 f u 0 uf u f u for u 0 Copyright 2006 Hindawi Publishing Corporation BoundaryValue Problems 2005 3 2005 289-298 DOI 290 Uniqueness of positive solutions of a class of ODE C2 a G C 0 o with a t 0 for t 0 C3 Bị G C1 0 o satisfies Bi 0 0 Bi x 0 for x 0 B i x is nondecreasing on 0 o i 1 2 . Remark . Condition C3 implies that B i x