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)nu(2n) = f (t, u, u′, . . . , (−1)[ i2 ]u(i), . . . , (−1)n−1u(2n−1)), u(2i)(0) = u(2i+1)(1) = 0(i = 0, 1, . . . , n − 1), where n ≥ 2 and f ∈ C([0, 1]×R2n + , R+)(R+ := [0,∞)). 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 byproduct, our main results are applied for establishing the existence, multiplicity and uniqueness of symmetric positive solutions for the Lidstone problem involving all derivatives.