Positive solutions for (n−1, 1) three-point boundary value problems with coefficient that changes sign

In this paper, we establish existence results for positive solutions for the (n − 1, 1) three-point boundary value problems consisting of the equation u(n) +λa(t)f u(t ) = 0, t∈ (0, 1), with one of the following boundary value conditions: u(0) = αu(η), u(1) = βu(η), u(i)(0) =0 fori = 1, 2, . . . ,n− 2, and u(n−2)(0) = αu(n−2)(η), u(n−2)(1) = βu(n−2)(η), u(i)(0) =0 fori = 0, 1, . . . ,n− 3, where η ∈ (0, 1), α 0, β 0, and a : (0, 1)→R may change sign and R = (−∞,+∞). f (0) > 0, λ > 0 is a parameter. Our approach is based on the Leray–Schauder degree theory. This paper is motivated by Eloe and Henderson (Nonlinear Anal. 28 (1997) 1669–1680).  2003 Elsevier Inc. All rights reserved