Opuscula Math. 33, no. 4 (2013), 603-613
http://dx.doi.org/10.7494/OpMath.2013.33.4.603

Opuscula Mathematica

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

Abdulmalik Al Twaty
Paul W. Eloe

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.

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• Abdulmalik Al Twaty
• University of Benghazi, Faculty of Arts & Sciences / Al Kufra, Department of Mathematics, Al Kufra, Libya
• Paul W. Eloe
• University of Dayton, Department of Mathematics, Dayton, Ohio 45469-2316 USA
• Revised: 2013-04-13.
• Accepted: 2013-04-13.