Qualitative Analysis Of Several Kinds Of Fractional Difference Equation

q-calculus(or quantum calculus)has been an important bridge between mathematics and physics since its birth,and it plays an important role in quantum physics,material mechanics,spectral analysis and so on.Fractional q-difference theory contains the advances of fractional calculus and discrete mathematics,so it has rich research values.In recent years,fractional q-difference theory has attracted the attention of scholars at home and abroad,and a series of excellent results have been obtained.With the in-depth study of q-calculus,inspired by its applications,many researchers have developed the theory of(p,q)-quantum calculus(also known as post-quantum calculus)based on two parameters.In recent years,with the extensive research and application of(p,q)-difference in quantum model,signal processing and other mathematical and physical problems,people combine(p,q)-difference equation with fractional calculus to begin the study of fractional(p,q)-difference equation,which greatly enriches the background of the application of difference equation theory.Compared with q-difference equation,the(p,q)-difference equation has two truly independent quantitative parameter sums,which is more widely applicable in mathematical models such as quantum mechanics and fluid mechanics.Therefore,the fractional(p,q)-difference equation has richer theoretical research significance and application value.At present,the qualitative theory of fractional q-difference equation has been widely studied by scholars,and the qualitative theory of fractional(p,q)-difference equation has gradually attracted the attention of scholars,especially for its existence and stability,which is not only the requirement of the development of its theory,but also the need of social production and life.It is expected that it can play a corresponding role in practical application.This paper study the solvability and stability of initial or boundary value problems for several classes of fractional q-difference equations and fractional(p,q)-difference equations,including singular equations,impulsive equations,Schr(?)dinger equations and beam equations.It involves the existence,multiplicity,uniqueness and stability of solutions or positive solutions.And some new results are obtained.In Chapter 1,the history of development and application prospect of fractional calculus,fractional q-difference equations and fractional(p,q)-difference equation are introduced.Some basic definitions of fractional q-difference equation and fractional(p,q)-difference equation,some related lemmas and main tools used in this paper are listed.The main content of this paper is presented.In Chapter 2,the solvability of two classes of fractional q-difference boundary value problems is studied.By using Banach fixed point theorem,Avery-Peterson theorem,Krasnosel’skii fixed point theorem and upper and lower solution method combined with monotone iterative technique,sufficient conditions for the existence,uniqueness and multiplicity of solutions for two kinds of fractional q-difference boundary value problems are obtained.In Chapter 3,two classes of boundary value problems of fractional stationary Schr(?)dinger difference equations are studied.By using Schauder fixed point theorem,monotone iterative method,Banach fixed point theorem and fixed point theorem on cone,some sufficient conditions for the existence of solutions are obtained.In Chapter 4,the initial value problem for a class of singular fractional order(p,q)-difference equations is studied.In the first section,the existence criterion of the solution of the initial value problem is established under the assumption that the nonlinear term is point singular.In the second section,the stability result of the solution of the initial value problem is obtained by establishing a(p,q)-Gronwall inequality,and the stability result means that the uniqueness of the solution holds.In Chapter 5,the boundary value problem for a class of fractional(p,q)-difference equations is studied.By using the generalized Banach fixed point theorem and the fixed point theorem on the cone,some sufficient conditions for the existence of positive solutions and the uniqueness of solutions are presented.In Chapter 6,the work of this paper is summarized and the future work is prospected.