In this article we apply an extension of a Leggett-Williams type fixed point theorem to a two-point boundary value problem for a 2n-th order ordinary differential equation. The fixed point theorem employs concave and convex functionals defined on a cone in a Banach space. Inequalities that extend the notion of concavity to 2n-th order differential inequalities are derived and employed to provide the necessary estimates. Symmetry is employed in the construction of the appropriate Banach space.
Altwaty, Abdulmalik and Eloe, Paul W., "Concavity of solutions of a 2n-th order problem with symmetry" (2013). Mathematics Faculty Publications. 116.