Existence And Uniqueness Of Solutions For BVP Of Nonlinear Multi-order Fractional Differential Equation

The research object of fractional calculus is fractional differential and fractional integral.Nowadays,there are many researches on fractional calculus in special func-tions,fractional physics,viscoelastic mechanics and so on.With the development of fractional calculus theory and the promotion of its application,it is a natural subject to extend the research results of integer order boundary value problems to fractional boundary value problems.In this paper,we study the existence and uniqueness of solutions to the following boundary value problem of nonlinear multi-order fractional differential equation(?)where L(D)=(?),2<??3,1<?1<…?l<2,0<?1<…<?m<1,?i?C[0,1](i=1,2,…,l),?j?C[0,1](j=1,2,…,m),??C[0,1],f?C([0,1]× R,R),??1,CD0+?,CD0+?i,CD0+?j denote the standard Caputo fractional derivatives.In the first chapter,we mainly introduce some basic knowledge(definitions,lemmas,theorems)related to the theory of fractional differential equations,which will be the main theoretical basis for our subsequent research work.In the second chapter,based on two lemmas,the solution of the boundary value problem of nonlinear fractional differential equation is transformed into the solvabil-ity of its equivalent integral equation.Existence and uniqueness of solution for the above problem is established by imposing some suitable conditions on nonlinear ter-m and applying the fixed point theorem and Banach contraction mapping theory.And we have two corresponding examples to illustrate the results.