Applications Of Critical Point Theory And Bifurcation Theory In Several Kinds Of Differential Equations

Differential equations have a wide range of applications.The problems in the biological,chemical,economic,physical and technical field can be trans-formed into solving differential equations.On one hand,the introduction of nonlinear terms and boundary value conditions makes the research of solu-tions of differential equations more complex,so the theory of well-posedness is applied to solve this problem.On the other hand,the research of differential ecosystems implements the strategy of sustainable development.Natural envi-ronment regulation and artificial intervention make the study of the behavioral trajectory of the solution of differential systems difficult.The qualitative and stability theory of the differential equation came into being.The well-posedness theory,qualitative and stability theory of differential equations have always been hot topics in the field of mathematics.This dissertation focuses on these two research hotspots and further discussion.In this dissertation,the existence and multiplicity of solutions of the bound-ary value problems of nonlinear impulsive differential equations and differential inclusion are studied by using smooth critical point theory,non-smooth critical point theory,fixed point theorem on Banach space,space decomposition theory and variational inequalities.Besides,the existence,stability of the equilibrium state and bifurcation problem of differential models are studied by using the eigenvalue theory and bifurcation theory.The full text is divided into seven chapters to discuss.The first chapter is the Introduction.On one hand,the proposition,applica-tion and research methods of boundary value problems of nonlinear impulsive differential equations are introduced.And the variational method,the devel-opment history and research status of critical point theory are given.On the other hand,the research methods and the development process of qualitative and stability of differential models,particularly,the background and signifi-cance,research history and research methods of the predator-prey model are shown.At the same time,the main research work of this paper is given.The second chapter is the Preliminaries.The definitions,lemmas,inequal-ities,and theorems used in this paper are given to prepare for following chapters.In chapter 3,the existence of solutions of fourth-order anti-periodic bound-ary value problems of differential equations with impulsive effects are studied by using critical point theory.The research content of this chapter is divided into two parts.In the first part,the solution space is decomposed into orthogo-nal space based on eigenvalues,moreover,the saddle point theorem is applied to prove the existence of solutions of fourth-order antiperiodic boundary value problems of differential equation with impulsive effects.Compared with the existing literatures,the differential model studied is more general and practical.So we promote the existing conclusions.In the second part,the Banach space is defined and the fixed point theorem is applied to obtain the existence of the solution of the auxiliary problem.The existence and the properties of solutions of the fourth-order antiperiodic boundary value problem of differential equation with impulsive effects are given by the critical point theory and the relationship between the auxiliary problem and the studied problem.Any minimizing se-quence on the closed convex subset of the solution space is bounded,which is more conducive to the application of the extreme value theorem.In addition,it provides a new proof that the critical point of energy functional is the classical solution of the studied problem.In chapter 4,the multiplicity of solutions of fourth-order periodic bound-ary value problems of differential equations with impulsive effects are studied by using critical point theory.In the first part,the existence of the solution of the linear problem is given by Lax-Milgram theorem.And the multiplicity of solution of the fourth-order periodic boundary value problem of differen-tial equation with impulsive effects is given by the mountain path theorem and variational methods.In the second part,the convergence and the existence of infinitely many solutions of the fourth-order periodic boundary value problem of differential equation with oscillatory nonlinear terms and impulsive effects are obtained by the maximum minimum theory.The main method is construct-ing a auxiliary problem,and then we obtain the existence of infinitely many solutions and the convergence of the auxiliary problem.The transformation is used to make the solution of the auxiliary problem be equal to the studied problem.The impulsive effect is not considered in the previous literatures and the restrictions of nonlinear terms in the study are weakened.So the existing research work has been expanded.In chapter 5,the existence and multiplicity of solutions of impulsive dif-ferential inclusion with relativistic operators are studied by using non-smooth critical point theory.In the first part,the existence of nonnegative solution is obtained by non-smooth critical point theorem and restriction of nonlinear terms and impulsive terms.The range of the critical point makes singular problems and non-singular problems equivalent.In the second part,the existence of in-finitely many solutions of the impulsive differential inclusions with oscillating nonlinear terms is shown by the non-smooth critical point theorem.Besides,the convergence of the solutions transforms the singular and non-singular problems into each other.Compared with the existing literatures involving relativistic op-erators,the impulsive effect is added to the differential inclusion and we adopt a new method to make the solution of singular and non-singular systems equiv-alent.Besides,a new method is used to judge the nonnegative and the norm convergence of the solution.Furthermore,new conclusions are obtained.In chapter 6,the stability of the differential model is studied by eigen-value theory and bifurcation theory.In the first part,the stability of equilib-rium state of the improved predator-prey model is analyzed by the eigenvalue theory.Moreover,the type,stability and normative type of bifurcation of the model are obtained by the bifurcation theory.In the second part,the existence,multiplicity,local stability and global stability of the equilibrium state of the differential model with cross terms isolated from the infectious disease model are studied.Furthermore,numerical simulations are given to verify the correct-ness of theoretical analysis.The time-delay and piecewise constant arguments are simultaneously introduced into the predator-prey model invasively in this chapter.And the results of the coexistence of the two bifurcations are differ-ent from the previous literatures.In addition,there are cross terms between the variables in studied differential model.So the research of the model is more general.Direct analysis of eigenfunctions is more complicated.Eigenfunction is simplified by power drop,which is more conducive to analysis eigenvalues.Research work in this chapter proves the stability of such model in theory and enriches the existing work.In chapter 7,the research content of this paper is summarized and the follow-up research questions are expected.