Periodic solutions of nonlinear boundary value problems

Abstract

ONE OF the well known techniques in the theory of nonlinear boundary value problems (BVP) is the method of differential inequalities or the method of upper and lower solutions. The method of “Alternative Problems”, a global variant of the Lyapunov-Schmidt method, has been used in the study of problems at resonance. The investigation of periodic BVP’s forms an important subclass of problems at resonance. Our aim is to combine the two approaches to discuss nonlinear problems at resonance. We restrict ourselves in this paper to the discussion of periodic boundary value problems. In Section 1, we shall indicate the method of upper and lower solutions. The results for systems are given in a general way so as to include known results and also offer new directions. Section 2 deals with the abstract existence results at resonance in the desired framework. In Section 3, we consider the scalar periodic BVP’s by combining the two techniques namely alternative methods and the method of differential inequalities.