On the fractional differential equations

摘要

In this paper we are concerned with the semilinear differential equation dαx(t)dtα=f(t,x(t)), t>;0 where α is any positive real number. In [3] the author has proved the existence, uniqueness, and some properties of the solution of this equation when 0<α<1. Here we mainly study (besides the other properties) the continuation of the solution of this equation to the solution of the corresponding initial value problem when [α]=k, k=1,2,3,⋯ Applications of singular integro-differential equations are considered.