Existence results for a class of even-order boundary value problems.

Abstract

This thesis focuses on establishing the existence of positive solutions to even-order boundary value problems... Similar problems have been considered by other authors. What distinguishes this work is the method employed: Beginning with a transformation of the problem into a system of second-order differential equations satisfying homogeneous boundary conditions, the work culminates in successive applications of the Guo-Krasnosel'skii Fixed Point Theorem giving at least three positive solutions. This method is based on the work of Marcos, Lorca, and Ubilla, who developed the technique to establish existence results for a class of fourth-order boundary value problems; however, the general framework under which this thesis operates is more accurately attributed to Hopkins, who extended the method to even-order problems on both continuous and discrete domains. Future work could seek to further demonstrate the broad applicability of the method by considering more challenging boundary conditions, by generalizing the results to time scales, or by utilizing different fixed point theorems in order to characterize solutions further.