Existence Of Solutions Of Boundary Value Problems For Some Fourth-order Differential Equations With Impulsive Effects

The theory of boundary value problems of differential equations is an important branch of differential equation theory and is widely used.Much more researches on the boundary value problems of second-order,third-order linear or nonlinear differential equations have been done,but far less studies on fourth-order which mainly results from physical,biological and chemical phenomena of various types,and,in particular,enjoys a wide application in the theory of elastic beams and its stability.It is well known that the most buildings are framed by beams sloped or deformed by their own gravity and external forces.The nonlinear term f varies in the boundary value problems of fourth-order differential equation,and the different boundary value conditions represents the elastic beams deformation of different types.In this paper,the existence,multiplicity and Ulam stability of solutions for boundary value problems of fourth order impulsive differential equations are studied by using critical point theory and topological degree theory,and some new results are obtained.This paper is divided into six chapters,and the specific contents are as follows:Chapter one is the introduction.Firstly,it briefly introduces the research background and significance of boundary value problems of fourth order differential equations and impulsive differential equations.Then,it emphatically introduces the research status of several kinds of boundary value problems of fourth order impulsive differential equations that are directly related to this paper.Finally,the main work and innovation of this paper are summarized.In chapter two,the existence and multiple solutions of the fourth order SturmLiouville impulsive differential equation with simply supported beam boundary value problem in the framework of spectrum are studied.Firstly,the generalized Lazer-Leach theorem for boundary value problems of Sturm-Liouville impulsive differential equations of the fourth order is obtained by using Prüfer transformation and topological degree theory under the condition that the nonlinear term does not cross resonance.Secondly,the existence and multiplicity of solutions for boundary value problems of sturm-Liouville impulsive differential equations of the fourth order are obtained by using the variational method and Liapunov-Schmit process under the condition that the nonlinear terms cross resonance.In chapter three,the solvability of boundary value problems for Kirchhoff type impulsive differential equations of the fourth order is studied on infinite intervals.Firstly,the existence of the forced solution and the mountain path solution are obtained by using the minimum principle and mountain path theorem respectively.Secondly,the multiplicity of the nontrivial weak solution of the problem is obtained by using the properties of the genus.Finally,the existence result of the nontrivial ground state solution of the problem is obtained by using Nehari manifold method.Due to the lack of upper compactness of infinite interval,it is necessary to construct a new embedding theorem,which brings some difficulties to the discussion of this problem.In chapter four,the existence and Ulam stability of the cantilever beam-type boundary value problem of the fourth order P-Laplacian impulsive differential equation with nonlinear term and damping term are studied.Firstly,the existence and uniqueness of the solution are obtained by using the variational method and iterative technique.Secondly,the Ulam stability theorem is established by analyzing the Ulam stability of the boundary value problem.There are damping terms in the equation,which brings additional difficulties to the establishment of the variational structure,so the variational method and iterative method are combined to demonstrate.In chapter five,the existence of solutions for the periodic boundary value problem of the fourth order variable exponential p(t)-Laplacian impulsive equation is discussed.In the framework of degree theory,the continuation theorem of the fourth order variable exponential p(t)-laplacian operator with impulsive effect under periodic boundary value condition is constructed.Then,by using the continuation theorem,we obtain the existence of solutions for a class of periodic boundary value problems of the fourth order variable exponential p(t)-Laplacian impulsive equation.Since the problem discussed in this chapter is variable exponential operator with impulse effect,it brings great difficulty to the construction of continuation theorem.The last part is the main summary of the work done in this paper and makes a plan for the follow-up discussion.