Existence Of Solutions To Boundary Value Problems For Nonlinear Mixed Fractional Q-Difference Equations

This paper mainly studies the existence and uniqueness of solutions of mixed fractional order q-difference equations,which is divided into three chapters.The first chapter is the introduction,which mainly describes the background significance and research status of fractional order q-difference equations.In the second chapter,we discuss the existence of positive solutions of boundary value problems for a class of fractional q-difference equations with integral boundary conditions:(?)where 0<q<1,2<α≤3,0≤β≤1,Dqα,Dqβ,Dqη are the fractional Riemann-Liouville q-derivative of order α,β and η.A is a function of bounded variation.∫01 Dqβu(t)dA(t)denotes the Riemann-Stieltjes integral.f:[0,1]×[0,∞)×[0,∞)→[0,∞),g:[0,1]×[0,∞)→[0,∞)are continuous functions.a(t),b(t)are allowed to be singular at t=0 or t=1.By using the fixed point theorem with mixed monotone operator,we obtain the existence and uniqueness of positive solution.In the third chapter,we study the unique iterative positive solutions of boundary value problems for mixed fractional q-difference equations:(?) where 0<q<1,2<α,β≤3,0<η,κ≤1,Dqα,Dqβ,Dqη,Dqκ are the fractional Riemann-Liouville q-derivative of order α,β,η and κ.By applying the monotone iterative technique,the existence and uniqueness of positive solution for the fractional q-difference equation(3.1.1)can be gained.