The Initial Value Problems Of Two Kinds Of Caputo Type Fractional Difference Equations

Many mathematical models in physics,space science,life sciences,and engi-neering etc.can be described by fractional difference equations.Therefore,the research on fractional difference equations has very important theoretical value and practical significance.This dissertation is concerned with the initial problems for two classes of non-linear Caputo type fractional difference equations.The main contents are as follows.In Chapter 1,the history and current situation of fractional differential equar tions and fractional difference equations are summarized.Some preliminaries are given,and the main results are introduced.In the second chapter,the initial value problems for three Caputo type frac-tional difference equations with lower bounds of 0 are studied-respectively.The initial problems of fractional difference equations with order between 0 and 1,1 and 2 are discussed,respectively.By using the properties of fractional difference and fractional summation,Volterra summation equation is constructed.Under the conditions that the right-hand function is bounded and satisfies the Lipschitz con-ditions,the existence and uniqueness of initial value problems are proved by using the progressive approximation method and the fractional Gronwall inequality.Es-pecially,when the order is between 1 and 2,an example is given to illustrate the significance of conclusions.Continuous dependence of solutions on the initial datum is also investigated by use of fractional Gronwall inequality.Finally,the relevant conclusions are extended to initial value problems with each order greater than zero.In the chapter 3,the conclusions in the second chapter are respectively ex-tended to three initial value problems of Caputo type fractional difference equations with nonzero lower bound by use of similar methods to ones in the second chapter.