Existence Of Solutions For Boundary Value Problems Of Nonlinear Fractional Difference Equations

Fractional calculus is a discipline for arbitrary order derivative and integral, it has beenthree hundred years of history. The fractional calculus was favored by many scholars almostwhen the concept of classical calculus has been introduced. Based on the study of fractionalcalculus, the study on difference equation of fractional order has entered the initial stage.Fractional difference equations also appear in rheology, self-similarity in rheological dynamicsand porous structure, popular, power network, viscoelastic, chemical physics and many otherbranches of science addition to the field of mathematics. In view of the applications offractional differential equations in various fields, the researches on the existence of solutionsfor linear and nonlinear fractional order differential equations boundary value problems hascaught attention of the domestic and foreign mathematical workers.In this paper, we consider the existence of solutions for boundary value problems ofnonlinear fractional difference equations and give some new existence theorems. Someexamples are presented to illustrate the main results respectively.In chapter1, we introduce some background materials about fractional calculus theory, thedevelopment of existence of solutions for boundary value problems of fractional differenceequations, some preliminary definitions and lemmas on fractional differential equations whichare needed in this thesis, and the arrangement for this thesis.In chapter2, we study the existence and uniqueness of solutions for two classes ofboundary value problems of fractional difference equations by contraction mapping theoremand Brouwer theorem. In first section, we establish some sufficient conditions for the existenceand uniqueness of solution to the problem with condition23by contraction mappingtheorem and brouwer theorem. We also present some examples to illustrate the main results. Insecond section, we give some sufficient conditions for the existence and uniqueness of solutionto the problem with condition N1Nby contraction mapping theorem and brouwertheorem, and we present some examples to illustrate the main results.In chapter3, we discuss the existence of solutions for two classes of boundary valueproblems of nonlinear fractional difference equations by the properties of the Green function and Guo-Krasnosel’skii fixed point theorem on cones. In first section, we study the existence ofsolution to the problem with conditions12, we give some criteria for the existence ofsolution to the problem by considering the eigenvalue intervals. We present some examples toillustrate the main results. In second section, we study the existence of solution to the problemwith conditions23, obtain the existence of solution to the problem, and present someexamples to illustrate the main results.In chapter4, we study the existence of solutions for a class of boundary value problems ofa system of fractional difference equations. We convert the problem to an equivalentsummation equation by using Green function. Then we establish some sufficient conditions forthe existence of solutions to the problem by using Guo-Krasnosel’skii fixed point theorem oncones and Schauder fixed point theorem. At last, we give some examples to illustrate the mainresults.In chapter5, we summarize the main results in this thesis, and point out the innovations ofour work. Finally, we prospect some future research work.