The Existence Of Solutions For Several Fractional Difference Equations

Fractional difference equations are widely used in physics,engineering mechanics and so on,which have attracted the attention of many scholars.There are a large number of literature and results on fractional difference equations.Especially,because of the rapid development of modern computer technology,the importance of fractional difference equations has been concerned by many scholars.In order to enrich and develop the theory of fractional difference equations,and explore the further applications of fractional difference equations,some practical work is done in this dissertation.In this dissertation,the existence and uniqueness of solutions for a class of fractional difference equations has been discussed and the undetermined coefficient method is used to solve fractional difference equation with constant coefficients.The paper is divided into five parts as follows.In Chapter 1,the history and current situation of fractional differential equations and fractional difference equations are summarized.Some preliminaries are given,and the main results are introduced.The Chapter 2 is concerned with the initial value problem for a class of Riemann-Liouville fractional difference equations.Firstly,the Volterra summation equation with parameters which is equivalent to the solution of the initial value problem is built,and then the parameters is solved by the given initial conditions,and then the existence and uniqueness of solutions of the initial problems is proved by using iterative method and fractional Gronwall inequality,finally,an example is given to find the unique solution of the initial value problem.The boundary value problem for a class of Riemann-Liouville fractional difference equations with nonlocal conditions is studied in Chapter 3.Firstly,the Volterra summation equation with parameters is built,and parameters are solved according to given nonlocal conditions.Secondly,the existence and uniqueness of solutions of the boundary value problem is proved by usingContraction Mapping Principle.Finally,under suitable conditions,the existence of solutions of the boundary problem value is proved by use of Brouwer Theorem.Chapter 4 is concerned with the following fractional difference equation of(k,q)order(?)A new method for solving the fractional order difference equation of(?,q)order is given by constructing a special function A(-?,?)and using undetermined coefficient method.Finally,the research contents and main results are summarized,and the future research of fractional difference equations is pointed out.