The Application Of Variational Methods And Measure Of Noncompactness In Some Differential Equations

With the birth and development of calculus,the research on nonlinear dif-ferential equations has been enriched in practical application,and the related theories of nonlinear functional analysis have become a powerful supporting tool in the application of most disciplines.At the same time,many phenomena in nature and many problems in real life can be transformed into corresponding differential equations or systems through mathematical modeling,so as to apply relevant theories in nonlinear analysis for qualitative analysis.With the rapid development of science,technology,engineering and other fields,differential equation model has been increasingly widely used.Therefore,the research on the existence of solutions of nonlinear differential equations and other related properties has always been a hot topic in nonlinear science.In this paper,we apply the variational method theory and the measure of noncompactness theory to study the boundary value problems of some nonlinear differential equations or systems,and obtain the conclusions of existence and multiple solutions.This paper is divided into five chapters.Chapter 1 is the introduction,which introduces the research background of differential equation system,impulse differential equation and fractional dif-ferential equation.At the same time,it briefly introduces the research methods applied in this paper,the development status and main content of this research,and finally describes the difficulties overcome in this paper.Chapter 2 introduces the basic knowledge of this dissertation,including the basic definitions,theorems and related operations in fractional calculus.In chapter 3,we study a class of fourth-order boundary value problems for linear and nonlinear differential equations with non-instantaneous impulse per-turbations.We apply the variational method to obtain the existence of solutions of the corresponding boundary value problems by Lax-Milgram theorem and critical point theory,respectively.In chapter 4,we study the existence of solutions for a class of fractional differential boundary value problems.By using the measure of noncompactness theory and Darbo's fixed point theorem,we obtain the existence of solutions for fractional differential boundary value systems in abstract Spaces.Chapter 5 is summary and prospect.We summarize the work of this dis-sertation and look forward to what we can do in the future.