New Concepts Of Fractional Hahn's Q,w-Derivative And Applications

Quantum calculus plays an important role in the development of mathematics.Its research not only has important theoretical significance,but also plays an import.ant role in the research of dynamic system?conformal quantum mechanics?high energy physics?signal analysis and so on.There are many different types of quantum dif-ference operators such as h-difference operator,q-difference operator,Hahn difference operator,the forward difference operator,the backward difference operator,and so on.These operators are found in many applications of mathematical areas such as orthogonal polynomials,basic hypergeometric functions,combinatorics,the calculus of variations and the theory of relativity.The theory and application of defferential equation is one of the hot topics studied by scholars at home and abroad.At present,the theory of integer Hahn differential equations has achieved some significant results.However,there are few research results on fractional Hahn calculus.So this paper studies the fractional Hahn q,?-calculus and Hahn differential equations,and studies the existence of solutions of fractional Hahn differential equation with fixed point theorem.This paper is divided into theoretical part and application part.The concepts of q,?-exponential function,q,?-gamma function and q,?-Laplace transform is intro-duced,and its related properties are studied.Next,the definition of fractional Hahn q,?-derivative of Riemann-Liouville type and Caputo type is given,some properties and their relations are discussed.In application,we consider the existence of solutions of two kinds of fractional differential equations with boundary conditions.whereg ?(0,1),1<??2,0<??2,0<?<1.and ?>0 is parameter,f:[a,b]×R+?R+is continuous.whereq ?(0,1),2<?:<3,f:[a,b]×[0,+?).By using Krasnoselskii fixed point theorem,monotone iterative technique and lower-upper solution method,an existence result of positive solutions to the boundary value problem is enunciated.