Existence Of Solutions For Boundary Value Problems With Two Coupled Systems Of Fractional Q-Differences

Fractional calculus is the extension of the calculus with integer order.It is the theory of researching characteristics and applications of differential in arbitrary order(real order or complex order)and integral non-standard operators.In recent years,as fractional differential equations has been used widely in nature sciences and en-gineering,boundary value problems of fractional differential equations have become an important research field,in which attracts broad attentions of numerous scholars at home and abroad.q-difference theory is one of significant branches of discrete mathematics,it has evolved a multidisciplinary body which used increasingly in ques-tions of mathematical physics,dynamic system,quantum models and economics so far with the rapid development of scientific technology.Under the prompts of re-search boundary value problems of fractional differential equations,multitudinous re-searchers introduced fractional calculus theory into fractional q-difference and studied relative theories of fractional q-difference equations.Due to theories about boundary value problems of fractional g-difference equation has obtained numerous achievements,therefore,we predict that many applications of the fractional q-difference will appear later.This dissertation consists of six chapter.Chapter one is the introduction part,first of all,we sketch the historical background?relevance of significance and research status for the g-difference equation,followed by an overview of the main contents of this paper.the second chapter is the preliminary part,it mainly introduces some relevant definitions,properties and lemmas used in the dissertation.The third chapter mainly studies the existence and uniqueness of solutions for the sequential fractional q-differences system with two-term sum.Firstly,using q-exponential,a representation for the solution to this equation is given.then,the existence theorem of solutions are proven by using Leray-Schauder alternative theorem,Krasnoselskii fixed point theorem and Banach contraction mapping principle.The fourth chapter mainly studies the existence of solutions for the sequential fractional q-differences system with three-term sum.Firstly,using q-exponential,a representation for the solution to this equation is given.Then the existence theorem of solutions are proven by using Perov's fixed point theorem,Schauder fixed point theorem.The fifth chapter mainly studies the Existence of multiple positions solutions for boundary value problems with a coupled system of fractional q-differences on the half-line.Firstly,we analyze some properties of the Green function.The second,the existence theorem of multiple positive solutions of the equation are proved by applying Krasnoselskii fixed point theorem,Leggett-Williams fixed point theorem,and a new extension of Krasnoselskii fixed point theorem.The last chapter,it is the conclusion and references.