Comparison Results for First and Second Order Boundary Value Problems at Resonance

Abstract

**Please note that the full text is embargoed** ABSTRACT: It is well known that the comparison principle for the initial value problems has been very useful in the theory of differential equations [1, 2,5]. Recently, such types of comparison results were developed for boundary value problems [3,4] and were used in proving the existence of solutions. It is natural to expect that comparison results for problems at resonance will he useful in proving, for example, existence results for periodic boundary value problems. Recently, existence of periodic solutions for first and second order differential equations have been considered by utilizing the method of upper and lower solutions and Lyapunov-Schmidt method, where certain simple comparison theorems have been proved [6,7]. In this paper, we develop systematically general comparison results of various types for boundary value problems at resonance for first and second order differential equations. We do hope that our general comparison theorems will play an important role in the existence theory of boundary value resonance.