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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

Concavity of solutions of a 2n-th order problem with symmetry