The Existence Of Solutions Of Boundary Value Problems For Two Classes Of Fractional Order Difference Equations

This dissertation is mainly concerned with the boundary value problems of fractional order difference equations with order number between 3 and 4. The main contents of this dissertation are summarized as follows.Firstly, the background, significance and current situation of the research on the fractional order difference equations are summarized, and then the necessary knowledge and preliminaries are given.The third chapter is concerned with boundary value problems of parameterized fractional differential equations with order number between 3 and 4. Green function is technically constructed for linearly homogeneous fractional order difference equations, and the positivity of Green function is obtained. Under the suitable conditions, the existence and uniqueness of solutions of boundary value problems for nonlinearly fractional order difference equations is proved by using of Banach contraction mapping principle and Brouwer fixed point theorem. In particular, the difference between fractional order difference equations and integer order difference equations is found, which is significant on research of fractional order difference equations.In the fourth chapter, the boundary value problems for fractional difference equations with order number between 3 and 4 are discussed. Green function is technically constructed for linearly homogeneous fractional order difference equations, and the positivity of Green function is obtained. Under the suitable conditions, the existence of positive solutions of boundary value problems of nonlinearly fractional order difference equations is proved by using of Guo-Krasnoselskii fixed point theorem and the fixed point index theorem. In particular, the existence of two and three positive solutions at least is obtained.