Opuscula Math. 33, no. 4 (2013), 603-613
Concavity of solutions of a 2n-th order problem with symmetry
Abstract. 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.
Keywords: Fixed-point theorems, concave and convex functionals, differential inequalities, symmetry.
Mathematics Subject Classification: 34B15, 34B27, 47H10.