Existence And Stability Of Solutions To The Fractional Q-difference Equations

The q-difference calculus was first proposed by Jackson,which plays an important role in quantum calculus.It usually used to describes some physical processes imposed on the time scale set Tq in quantum dynamics and stochastic analysis.With extensive applications in various fields including quantum mechanics,hypergeometric series,particle physics and complex analysis and so on.It more convenient and accurately describe practical problems.And about fractional q-difference equations have been increasingly studied.This paper studies the boundary value problem of three types about fractional q-difference equations,and it discusses the existence and stability of the solutions.This article is composed as follows:Chapter 1,the background and current situation of the fractional q-difference equation and the main works of this paper are given.Chapter 2,some notations,definitions and lemmas are introduced about this paper.Chapter 3,the existence of positive solutions to the Caputo fractional q-difference equation is investigated.This part mainly describes the existence of positive solutions of ?=0 by iterative method.Secondly,by using the properties of Green function and cone stretching and cone compression fixed point theorem.We obtain the existence conditions and interval of positive solutions of boundary value problems with the range of parameter ?>0.Finally it illustrate the applicability of the results.Chapter 4,it first studies the existence and stability of solutions of fractional coupled time delay q-difference systems by the fixed point theorem.Secondly,we consider the results of solutions on the generalized Banach space.Finally,Two examples verifies the validity of the results.Chapter 5,we study the existence and uniqueness of solutions about the Hilfer fractional qdifference equation by topological degree theory and fixed point theorem.Then we discuss the existence of extremal solutions in ordered Banach space by a monotone iterative method.Then we consider the results about Ulam stability of equation.Finally,two examples are given to illustrate the feasibility of these conclusions.Chapter 6,the summary of this article is given,meanwhile we make a prospect for the future work.