The Application Of Critical Point Theory And Topological Degree Theory In Some Differential Equations

Over the past few decades,with the development of nonlinear science,the existence of solutions to nonlinear differential equations has been the core of nonlinear science.Along with the rapid development of research in many fields of science,technology and engineering,a large number of real phenomena can be attributed to the mathematical model of nonlinear differential equations.In this dissertation,the existence results of solutions for several kinds of nonlin?ear differential equations are obtained by critical point theory and topological degree theory.This dissertation consists of six chapters.Chapter 1 is the introduction,introducing the application background of impulsive differential equations,impulsive differential systems and differential inclusions.Meanwhile,the methods and related theories to be used are also introduced.Finally,the framework and research content are pointed out.Chapter 2 presents some of the basic knowledge required in this disser-tation,including basic definitions,theorems and basic calculation formulas on fractional differential equations.In Chapter 3,the existence of solutions to two kinds of boundary value problems of impulsive differential equations is studied.The existence and mul-tiplicity of solutions for two kinds of fourth-order impulsive differential equa-tions are obtained by critical point theory.In Chapter 4,the existence of solutions to a boundary value problem of fractional differential inclusion is studied.By a nonsmooth critical point theo-rem,the fractional differential inclusion shows infinitely many solutions with the nonlinear term oscillating at the zero and infinity,respectively.In Chapter 5,the existence and asymptotic behavior of solutions for a per-turbed impulsive differential system are studied.Using topological degree the-ory,the existence conditions of periodic solutions for perturbed impulsive dif-ferential systems are established.Chapter 6 summarizes this dissertation and takes several views to other possible research areas.