Boundary Value Problems For Fractional Differential Equations With Derivatives In Nonlinear Terms

Fractional differential equation is an important branch of differential equation,because it can be widely used in the viscoelastic mechanics,information processing,biomedicine and system control,and other fields,and be able to more accurately and objectively describe the nonlinear phenomenon or condition,scholars at home and abroad have done a lot of research on the structure and properties of solutions of fractional differential equations,especially in the area of boundary value problems of fractional differential equations.In this thesis,fixed point theorems with monotone nonlinear operators,monotone iterative techniques and some classical fixed point theorems are used.The properties of solutions for multi-point boundary value problems of fractional differential equations with Riemann-Liouville(R-L)type derivatives and Caputo type derivatives in nonlinear terms are discussed,and the existence,uniqueness and multiplicity of solutions are obtained.The research contents are as follows:Firstly,the existence and uniqueness of positive solutions for multi-point boundary value problems with R-L type derivatives in nonlinear terms is studied.Considering that the derivatives contained in the nonlinear term and the boundary value condition are of the same type(R-L)with different orders,and then the Banach space which is consistent with the R-L type derivatives of nonlinear term is defined.Using the fixed point theory of “sum type”operators,a sufficient condition for the existence and uniqueness of the solution of the equation is obtained,and two iterative sequences are constructed to approximate the unique solution.In the last,two examples are given to verify the conclusion.Secondly,the existence of multiple solutions for a four-point boundary value problem with R-L type derivatives and -Laplacian boundary conditions are studied.By using the fixed point theorem of cone expansion-compression and Legget-Williams fixed point theorems,we prove that the sufficient conditions for the existence of one solution and three solutions respectively,and the feasibility of the conclusion is illustrated by two examples.Finally,the existence and uniqueness of the solutions for the four-point boundary value problem with Caputo type derivatives in the nonlinear term is studied.According to the characteristics of Caputo type derivatives in the nonlinear term,Banach space is defined and Green function is calculated.Using Schauder fixed point theorem and Banach contraction mapping principle,the properties of the solution of the equation are obtained.In the last,two examples are given to verify the conclusion.