The Existence,Uniqueness And Multiple Solutions Of Three-point Boundary Value Problems

In recent years,in the field of differential equations,the three-point boundary value problem has been widely used in physics,chemistry,biology and other fields.In the era of rapid development of science and technology,the application of the three-point boundary value problem is more and more common.With the increasing demand for three-point boundary value problems of differential equations,many scholars have gradually deepened their research on the positive solutions of three-point boundary value problems.However,we find that there are very few studies on multiple solutions,infinite solutions and sign-changing solutions of three-point boundary value problems.In this thesis,we will study the solutions of three-point boundary value problems.Firstly,we will study the positive solution of special three-point boundary value problems.Secondly,for more general three-point boundary value problems,we will obtain the existence of its multiple solutions.In the Chapter 1,we introduce the background of three-point boundary value problems and the main work of this paper,and we give some preliminary knowledge.In the Chapter 2,we consider the three boundary value problems(?) where K ? C[0,1],0<a<1,0<?<1,and ? is a positive parameter.In this chapter,we establish the upper and lower solution theorem for the three-point boundary value problems,and combine this theorem with the knowledge of comparison principle,eigenvalues and corresponding eigenfunctions,and Arzela-Ascoli lemma.When K,a,? and ? are in different ranges,we obtain the existence,uniqueness,and dependence of the solutions on the three-point boundary value problem with singular nonlinear terms.In Chapter 3,we study the following problem(?)In this chapter,we make a study on the existence of multiple solutions to the three-point boundary value problem.Firstly,we define a generalized definition of the upper and lower solutions for this problem.And secondly,to establish the connection between the upper and lower solution method and degree theory,we give the definition of the strictly upper and lower solutions.In the presence of the upper and lower solutions,the existence of three solutions is given by using the degree theory.In addition,we combine the upper and lower solution method with the variational method.In space W1 1,2((0,1)),we transform the above problem into an energy functional.By finding the critical point of the functional,we obtain the existence conclusion of the solution of the three-point boundary value problems.We use this method to obtain the existence of four solutions,five solutions,and sign-changing solutions for the three-point boundary value problem by changing the conditions.In addition,when the right term f(t,u(t))with special forms,we obtain that the problem has infinitely many solutions.