Existence Of Solutions For Boundary Value Problems Of Two Fractional Differential Equations

Integer differential equations are the basic mathematical tools we are familiar with,but in actual production and life,integer differential equations have obvious narrowness.In comparison,fractional differential equations are more intuitive and flexible in reflecting the answers and nature of the questions,so they are used very frequently and ingeniously by scholars and math enthusiasts.In the first chapter of this paper,in the context of the field of fractional differential equations,we briefly describe the history and current situation of fractional differential equations,and briefly describe the process of fractional differential equations in China and other countries for decades.In the second chapter of this paper,some theories,theorems and methods for proving the existence of solutions of fractional differential equations are enumerated and introduced in detail.In the third chapter of this paper,we take the fractional differential equation(1.1)with higher derivatives as an example to study the existence of positive solutions.And give two practical examples to illustrate its application.Among them,x[2](t)=x(x(t)),??(1,2].After fully utilizing the cone stretching and cone compression theorem and the principle of compression mapping,the Schauder fixed point theorem and other proof methods are used to compare the methods and results of similar problems in the literature,and different methods are used to prove the solution.In the fourth chapter of this paper,we study the following(1.2)fractional differential equations for a class of nonlinear boundary value problems with integral boundary conditions,and verify the existence and multiplicity of positive solutions by solving them.And give two practical examples to illustrate its application.Among them,n-1<?<n,0<?<n-1,we use nested methods,Banach space compression mapping principle,Guo-Krasnoselskii fixed point theorem to get the existence and uniqueness of understanding.In the fifth chapter of this paper,based on the basis and method used to prove the two main problems of this paper,summarize the conclusions and some techniques to guide and help other types of similar problems.effect.