Existence Of Nontrivial Solutions For Boundary Value Problems Of Some Differential Equations

Boundary value problem of differential equation is an important branch of differential equation theory,which is widely used in natural science and engineering technology.Nonlinear functional analysis is an important branch of modern mathematics and plays an important role in many fields.Especially in differential equations,the theory of topological degree in nonlinear functional analysis used to study the boundary value problems of differential equations,it has always been a subject with lasting vitality.The multipoint boundary value problems of differential equation have attracted much attention,and the existence,uniqueness and multiplicity of their solutions is a hot research topic.Therefore,on the basis of the previous theory,"Boundary value problems of fourth order ordinary differential equations","Multipoint boundary value problems of second order ordinary differential equations" and "A system of fractional differential equations with two parameters" are discussed and studied respectively by using the fixed point theorem and the properties of Green's function.This paper is divided into four chapters:In Chapter 1,the research background of boundary value problems of differential equations and the main contents of this paper are briefly introduced.In Chapter 2,by using the fixed point theorem with lattice structure,we study the boundary value problems of fourth-order ordinary differential equation.By proving that the operator is a complete continuous operator and a quasi-additive operator,we discuss the problem under sublinear conditions,asymptotically linear conditions and superlinear conditions respectively.Under sublinear conditions,we obtain that the boundary value problems of differential equation have at least three nontrivial solutions,among them are one positive solution,one negative solution and one sign-changing solution.Under asymptotic linear conditions,by using the boundedness of operators,we obtain at least one nontrivial solution and at least three nontrivial solutions in another case.Under superlinear conditions,we discuss the existence of solutions according to Krein-Rutmann theorem and operator satisfying H condition.In Chapter 3,by using the fixed point theorem,the multipoint boundary value problems for a class of second-order ordinary differential equations are considered.By proving the e-continuity of corresponding nonlinear operators in a certain region and the properties of Green's function,it is concluded that there are at least two positive solutions,two negative solutions and one sign-changing solution.In Chapter 4,the existence of positive solutions for a class of fractional differential equations with two parameters is considered.In this chapter,we adopt a conformable fractional derivative,by the fixed point theorem of cone tension and compression,the existence of positive solutions is obtained.