Uniqueness of positive solutions of nonlinear second-order equations

Abstract

In this paper we study the uniqueness question of positive solutions of the two-point boundary value problem: u ( t ) + f ( | t | , u ( t ) ) = 0 , − R > t > R , u ( ± R ) = 0 u(t) + f(|t|,u(t)) = 0, - R > t > R,u( \pm R) = 0 where R > 0 R > 0 is fixed and f : [ 0 , R ] × [ 0 , ∞ ) → R f:[0,R] \times [0,\infty ) \to \mathbb {R} is in C 1 ( [ 0 , R ] × [ 0 , ∞ ) ) {C^1}([0,R] \times [0,\infty )) . A uniqueness result is proved when f satisfies some appropriate conditions. Some examples illustrating our theorem are also given.