Initial Value Problems For Several Kinds Of Fractional Difference Equations

With the rapid development of science and technology,the fractional difference equations are widely applied in many fields such as physics,engineering and ect.Due t.o its challenge and broad applications in many fields,the fra.ct.ional difference equations have attracte.d wide attention of many scholars.In order to enric:h and develop the t.heory of t,he fractional difference equat.ions,some real work is done in this paper.In the first chapter,the relevant background and research status of the fractional difference equations are int,roduced briefly.The symbols,not,ion and preliminaries are presented in the second chapter.The third chapter is concerned with the properties of fractional differences and fractional summations.Firstly,the proofs of two basic theorems[7]are given by using of the new method different form t,he method used in[7].Then,one of the theorems is generalized using mathematical induction.In the fourth chapter,R-L fractional-order difference equations initial value prob-lem are studied.Volterra summation equation with parameters is constructed which is equivalent to the above Cauchy problems.The existence and uniqueness of solu-tions is obtained by successive iteration and fractional Gronwall inequality.Finally,an example is given to illustrate the main results.The fifth chapter is concerned for Caputo fractional-order difference equations initial value problem.Volterra summation equation with parameters is constructed which is equivalent to the above Cauchy problems.The existence and uniqueness of solutions is gained by successive iteration and fractional Gronwall inequality.Finally,an example is given to illustrate the main results.