The Unique Solution Of Two Kinds Of Q-Fractional Difference Boundary Value Problems

In this paper,we mainly study the unique solutions of two kinds of q-fractional difference equations.The main content of this paper is divided into two parts:The first part mainly describes the existence and uniqueness of non-trivial solutions for a class of nonlinear q-fractional-order boundary value problems of difference equations.Firstly,a new concept is defined on a new set.Then,by using the definition of normal cone,the existence of unique fixed points for two mixed monotone operators is established.Green functions for linear fractional-order boundary value problems are obtained,and the upper and lower bounds of Green functions are estimated.Secondly,by using the abstract theorem,we discuss the non-linear terms which meet the conditions of the theorem,establish the existence of the unique solution of the above problem,and obtain the iteration sequence approaching the unique solution.Furthermore,the existence and uniqueness of nontrivial solutions for boundary value problems of fractional difference equations are proved.Finally,we give an example to illustrate the validity of the main theorems and results.The second part focuses on the unique solution of a class of nonlinear q-fractional difference equations.The content of the second part can be regarded as a supplement to the first part.The main results obtained in this paper include the existence of unique solutions of the equations.The specific contents are arranged as follows:The first chapter expounds the research achievements of q-fractional differential equation and its historical background and significance,the research status at home and abroad,and briefly introduces the main research contents of this paper.In the second chapter,the first main content of this paper is introduced,that is,for a class of nonlinear q-fractional-order boundary value problems,the existence results of unique solutions and their corresponding properties can be obtained under proper assumptions.In the third chapter,the unique solution of a class of nonlinear q-fractional difference equations is given.The main result has the existence of a unique solution.