Research On The Properties Of Solutions For Fractional Differential Equations And Difference Equations

Fractional calculus is the theory of arbitrary order differential and integral,which is the generalization of integer order calculus.Compared with integer order differential equations,mathematical models established by fractional differential equations are closer to reality.Relatively speaking,fractional difference equations are studied later.In recent years,it has made great development because of its wide application in medicine,physics and other fields.In this paper,we will study the existence and uniqueness of solutions for several classes of fractional differential equations and fractional difference equations.The paper is mainly divided into four chapters.In Chapter 1,we summarize the related theories of fractional differential equations and fractional difference equations,along with some definitions used in this paper.In Chapter 2,we consider the fractional boundary value problem with nabla difference equation.The eigenvalues and eigenfunctions of linear nabla difference equation are obtained by using Z transformation.Combining topological theory and eigenvalues,the sufficient conditions for the existence of solutions of the corresponding nonlinear fractional boundary value problem are obtained.In Chapter 3,we discuss the existence and uniqueness of solutions and Lyapunovtype inequality for mixed fractional boundary value problem.According to the properties of fractional calculus,the differential equation involving right Riemann-Liouville and left Caputo fractional derivatives is transformed into integral equation.The Green's function is obtained and Lyapunov-type inequality of linear fractional differential equation is established.Finally,using the fixed point theorem for ?-?-contractive mapping and Banach contractive mapping principle,the existence and uniqueness of solutions for nonlinear equation are studied.In Chapter 4,we study the properties of solutions for Atangana-Baleanu fractional differential equation with infinite point boundary value conditions.Using the definition of Atangana-Baleanu fractional differential and integral,we get the corresponding integral equation.And the existence of the solutions is transformed into a fixed point problem by constructing an operator.Then the existence and uniqueness of solutions are obtained by using the fixed point theory.Finally,the Hyers-Ulam stability of the problem is analyzed.