Existence And Uniqueness Result Of Solutions To Cauchy Problems Of Fractional Differential Equations Of Variable-order

As we all know,the initial boundary value problem of fractional differential equations has become an object of extensive research,but variable order operators belong to a more complicated category,and its order is the derivative and integration of some variable functions.Variable order extensive applications urgently call for systematic studies on the existence,uniqueness of solutions to initial value problems of these equations.In view of this,studying the existence and uniqueness of solution for the following initial value problem for differential equation of variable-order where 1<q(t)<2,0<T<+∞,D0+g(t)is about variable order q(t)Riemann-Liouville fractional derivative.The main results of this paper are as follows:1.By using the Banach compression mapping principle,studying the exis-tence and uniqueness of the solution to initial value the problem(1)for differential equation of variable-order.2.By using Schauder fixed point theorem and Leary-Schauder nonlinear choice theorem,we obtain local existence and sufficient conditions for the existence of positive solutions of the initial value problem(1)for differential equation of variable-order.3.By using upper and lower solutions and monotone iteration methods,we studies the existence and uniqueness of the solution to initial value the problem(1)for differential equation of variable-order.