Continuous Dependence Of Solutions On Initial Data For Two Classes Of Fractional Difference Equations

With the progress of science and technology, the wide application background and the popularity of computing technology, the difference equations have stronger applicability and challenging, fractional order differential equations more and more get the attention of people in particular. In order to better solve the problems of theory and applications of encounter, the basic theory of fractional order difference equations needs to be enriched and developed for guiding and supporting the applications of the fractional order differential equations.This dissertation focuses on the continuous dependence on initial data for two classes of fractional differential equations. The full text is divided into four main parts and the contents are as follows.The chapter 1 introduces the development background of the difference equations and fractional order differential equations. Also, the situation of study and development of the fractional order difference equations are summarized, the main work of this dissertation is briefly described.In the chapter 2, the terms, definitions and symbols, etc., are given in order to facilitate the needs of the following text. For completeness, the brief proofs of some lemmas are also given.Chapter 3 is concerned with the continuous dependence of solutions on initial data for a class of fractional difference equations of Riemann-Liouville type. Under the suitable conditions, the existence of solutions is proved by using of classical methods. Then the uniqueness of solutions is obtained by means of generalized discrete fractional Gronwall inequality, which this method is different from one of existing literature. Finally, the generalized discrete fractional Gronwall inequality technique is applied to obtain the continuous dependence of solutions on the initial data, which means that our main results are extended to any fractional order difference equations.In the chapter 4, the continuous dependence of solutions on initial data is considered for a class of fractional order difference equations of Caputo type with zero for lower limit. At first, under suitable conditions, the existence and uniqueness of solutions is proved for general case of fractional order. Then, using the technique of generalized discrete fractional Gronwall inequality combined with discrete Mittag-Leffler function, the continuous dependence of solutions on initial data is gained according to the two cases of 0<α<1 and α>0, respectively. To our knowledge, the main results are new.The conclusion part summarizes the main results of this dissertation, and the some research directions are also given for further study in the related field of fractional order difference equations.