The Initial Value Problem Of A Class Of Nonlinear Fractional Differential Equations

Fractional calculus is a important branch of mathematical analysis which dealswith the investigation and applications of integrals and derivatives of arbitrary order.In recent decades, as the field of fractional calculus has attracted many researchersin several areas including physics, chemistry, engineering and even finance and socialsciences, definite problem of fractional diferential equations become a internationalhot research subject, especially for initial value problem of the following nonlinearfractional diferential equationD~su=f (x, u),(1)where0<s <1, Dsis fractional derivative. Since the nonlinear fractional diferen-tial equation (1) has been considered for the first time in1996by D. Delbosco andL. Rodino[1], its initial value problems have received considerable attention[1–10].In this paper, we are concerned with the nonlinear fractional diferential equa-tion become a typical sequential fractional diferential equation. Let α1=α2=1, thenthe equation (2) become a one dimensional p-Laplacian.The main purpose of this paper is to investigate the existence and uniquenessof solutions to the initial value problem (2),(3). We fist transform the problem(2),(3) into an equivalent integral equation, and then we are going to prove the ex-istence and uniqueness of this integral equation. According to Schauder fixed pointtheorem, we get the existence of local solution, furthermore, under some condition,the existence interval of local solution can extend to [0,∞). As for uniqueness ofsolution to the problem (2),(3), we discuss it by contraction mapping principle andemploying mathematical induction in a local interval. Finally, we extend the exis-tence interval of local unique solution to [0,∞).