Differential Inequalities at Resonance

Abstract

It is well known that the theory of differential inequalities for the initial value problems has been very useful in the theory of differential equations [3,8]. Recently, such types of differential inequalities were developed for boundary value problems [1,6] and were used in proving the existence of solutions. It is natural to expect that differential inequalities for problems at resonance will be 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 [2,4,5]. In this paper following [7] we develop differential inequalities for boundary value problems at resonance for first and second order differential equations. As a simple application, we prove existence of multiple solutions as limits of monotone iterates for first and second order periodic boundary value problems.

Research partially supported by U.S. Army Research Grant #DAAG29-80-C-0060.