Positive solutions of a 2nth-order boundary value problem involving all derivatives via the order reduction method

This paper is mainly concerned with the existence, multiplicity and uniqueness of positive solutions for the 2nth-order boundary value problem {(-1)(n)u((2n)) = f (t, u, u, u' ,..., (-1)(vertical bar 1/2 vertical bar)u(i) ,..., (-1)(n-1)u((2n-1))), u((2i))(0) = u((2i+1)) (1) = 0(i = 0(i = 0(i = 0, 1 ,..., n-1), where n >= 2 andf epsilon C([0,1] xR(+)(2n,) R(+))(R(+:) = [0, infinity))We first use the method of order reduction to transform the above problem into an equivalent initial value problem for a first-order integro-differential equation and then use the fixed point index theory to prove the existence, multiplicity, and uniqueness of positive solutions for the resulting problem, based on a priori estimates achieved by developing spectral properties of associated parameterized linear integral operators. Finally, as a by product, our main results are applied for establishing the existence, multiplicity and uniqueness of symmetric positive solutions for the Lidstone problem involving all derivatives. (C) 2010 Elsevier Ltd. All rights reserved.