Uniqueness of positive solutions of a class of ODE with nonlinear boundary conditions
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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 < b < ∞, B1 and B2 ∈ C 1 (R), a ∈ C[0, ∞) with a ≤ 0 on [0, ∞) and f ∈ C[0, ∞) ∩ C 1 (0, ∞) 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,
B1 u(0) − u (0) = 0,
t ∈ (0,b)
B2 u(b) + u (b) = 0,
(1.1) (1.2)
where 0 < b < ∞. We make the following assumptions: (C1) f ∈ C[0, ∞) ∩ C 1 (0, ∞) with f (0) = 0, f (u) > 0,
u f (u) < f (u),
Copyright © 2006 Hindawi Publishing Corporation Boundary Value Problems 2005:3 (2005) 289–298 DOI: 10.1155/BVP.2005.289
for u > 0;
(1.3)
290
Uniqueness of positive solutions of a class of ODE
(C2) a ∈ C[0, ∞) with a(t) ≤ 0 for t ≥ 0; (C3) Bi ∈ C 1 [0, ∞) satisfies Bi (0) = 0, Bi (x) > 0 for x > 0, Bi (x) is nondecreasing on (0, ∞) (i = 1,2). Remark 1.1. Condition (C3) implies that Bi (x) ≥ 0 for x ≥ 0 (i = 1,2). In fact, we have from Bi (0) = 0 and Bi (x) > 0 for x > 0 that Bi (0) ≥ 0.
(1.4)
This together with the assumption Bi (x) is nondecreasing on (0, ∞) implies that Bi (x) ≥ 0 for x ≥ 0. The main result of this paper is the following. Theorem 1.2. Let (C1)–(C3) hold. Then problem (1.1), (1.2) has at most one positive solution. Here we say u(t) is a positive solution of (1.1), (1.2), if that u(t) > 0 on [0,b] and satisfies the differential equation (1.1) as well as the boundary conditions (1.2). Remark 1.3. As an application of Theorem 1.2, we consider the nonlinear problem
u + a(t)u + u p = 0,
k
u(0) − u (0) = 0,
t ∈ (0,b), l
u(b) + u (b) = 0,
(1.5)
where p ∈ (0,1), k, l ∈ (1, ∞) are given, a ∈ C[0, ∞) with a ≤ 0 on [0, ∞). Clearly all of the conditions of Theorem 1.2 are satisfied. Therefore by Theorem 1.2, (1.5) has at most a positive for any b ∈ (0, ∞). The proof of the main result is motivated by the work of Erbe and Tang [3, 4, 5] and is based on the s
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