Existence Of Solutions For Two Classes Boundary Value Problems Of Nonlinear Fractional Q,?-Differences Equation

The q-difference theory has been put forward for more than 100 years.The theoretical study of the q-difference boundary value problem has also attracted extensive attention of scholars at home and abroad.Similar to the emergence of q-difference theory,the emergence of q,?-difference theory is also widely used in other disciplines other than mathematics,such as engineering science,quantum mechanics,biological engineering,especially in the fields of power systems,mathematical and physical problems and quantum physical models,playing an extremely important role.Driven by the boundary value problem of fractional differential equation,a large number of scholars have applied the methods used in the theory of fractional differential equation to the fractional q,?-difference theory.The fractional q,?-difference theory has obtained a lot of research results.In this paper,we study two kinds of boundary value problems for nonlinear fractional difference equations.First,we study a class of boundary value problems for mixed q,?-difference equations including the fractional derivatives of RiemannLiouville and Caputo.The existence and uniqueness of the solutions to the above problems are obtained by using Expansion and contraction theorem of cone,Banach compressed image theory and Selection theorem,and the effectiveness of the results is illustrated by examples.Secondly,we study the boundary value problem of a kind of q,?-difference equations with integral boundary conditions.The Green function and its properties of the problem are derived.By using these properties and the fixed point theory on the cone,the existence of three positive solutions is obtained.Finally,the effectiveness of the main results in this paper is illustrated by examples.