Existence Of Solutions For Two Classes Boundary Value Problems Of Fractional Q-differences Equation With Perturbation

Fractional calculus is a discipline for arbitrary order derivative and integral,it has been three hundred years of history.The fractional calculus was favored by many scholars at home and abroad.q-difference theory is an important part of discrete mathematics,similar to fractional calculus.With the development of mathematics,The theory of q-difference is applied more and more to natural science and engineering.Driven by the boundary value problem of fractional differential equation,more and more scholars apply the methods of fractional differential equation theory to q-difference theory.Therefore,the theory of fractional difference boundary value problem has obtained many research results.In this paper,two kinds of boundary value problems for fractional q-difference equations with perturbation terms are studied.The first kind of equation is two different signs higher order fractional q-difference equations with perturbation terms.First,the expression of the solution of the equation is obtained,which is related to the Green function and properties of Green functions.And then our used two fixed point theorems of mixed monotone operators with perturbation for fractional q-difference equations.Our results can not only guarantee the existence of a unique positive solution,but also be applied to construct an iterative scheme for approximating it.Some examples is given to illustrate the main result.The second kind of equation is concerned with the existence and uniqueness of positive solutions for the following fractional boundary value problem:The process is similar to the first kind of equation.First,the expression of the solution of the equation is obtained,which is related to the Green function and properties of Green functions.And then our used three fixed point theorems of mixed monotone operators with perturbation for fractional q-difference equations.Our results can not only guarantee the existence of a unique positive solution,but also be applied to construct an iterative scheme for approximating it.Some examples is given to illustrate the main result.