Journal of Mathematical Inequalities
In this article we apply an extension of an Avery type fixed point theorem to a family of boundary value problems for higher order ordinary differential equations. The theorem employs concave and convex functionals defined on a cone in a Banach space. We begin by extending a known application to a right focal boundary value problem for a second order problem to a conjugate boundary value problem for a second order problem. We then extend inductively to a two point boundary value problem for a higher order equation. Concavity of differentiable functions plays a key role in the application to second order equations. A concept of generalized concavity plays the same key role in the application to the higher order equation.
Altwaty, Abdulmalik A. and Eloe, Paul W., "The role of concavity in applications of Avery type fixed point theorems to higher order differential equations" (2012). Mathematics Faculty Publications. 113.