Positive Solution Of Non-linear Second-order Periodic Boundary Value Problems With Parameters

The periodic boundary value problem of ordinary differential equation is an important type of mathematical problems in the theory of differential equations. The study started many years ago. Because of the universality of cyclic phenomena, the research on this issue has great theoretical and practical significance. In the past, with the continuous development and improvement of nonlinear functional theory and method, people research the existence of the solutions by the fixed point theory and gradually developed theory of differential equations in Banach space as a new mathematical branch. We can see that foreign scholars and our scholars have done a lot of pioneering work in the research on the existence and multiplicity of the solutions by using the fixed point the fixed point theorem, which are the theoretical foundation of theory of differential equations in Banach space.At present, the study of second -order periodic boundary value problems are focused on ordinary differential equations with one parameter, and the papers constructed the research on second-order ordinary differential equations with more uncommon parameters. The main purpose of this paper is to research second -order non-linear periodic boundary value problems by using the cone theory, fixed point theorem and so on.Full-paper is divided into three chapters, the main contents are as follows:The first chapter is divided into two sections. Section I Introduction, describes a number of related issues and the question. Section II gives some basic definitions, theorems.The second chapter is divided into five sections. Major study is the conditions of non-linear second-order periodic boundary value problems with two parameters: .The estimate of upper and lower boundary and the Green function are established. Under the parametersα,βthe nonlinearity f satisfying the appropriate conditions, the conclusion on the existence of positive solutions of the boundary problem is obtained, by using Krasnosellskii fixed-point theorem, which enriched further the literature results. The section II has been published in the "Journal of Cheng Du University". The sectionsⅤobtains the same conclusion on the condition of variable parameters.The third chapter is divided into four sections. Major study is the conditions of non-linear second-order periodic boundary value problems with three parameters: u (t )∈C 0,2π∩C20,2π. The sections II obtains Green functional expression, and the upper and lower bound estimation. Section III, under the parametersα,β,λthe nonlinearity f satisfying the appropriate conditions, the conclusion on existence of positive solutions of the boundary problem is obtained by using the upper-lower solution method and cone fixed point theorem.Finally, all research content is summarized.