Cauchy Problem for Fractional ODEs

Abstract

In this chapter, we discuss the existence, uniqueness and regularity of solutions to fractional ordinary differential equations (odes). The existence and uniqueness can be analyzed using the method of successive approximations and fixed point argument. The former, pioneered by Cauchy, Lipschitz, Peano and Picard for classical odes in various settings, shows the existence by constructing suitable approximations, and lends itself to a constructive algorithm, although not necessarily efficient. The latter is an abstraction of the former and converts the existence issue into the existence of a fixed point for a certain (nonlinear) map, often derived via suitable integral transform.