Local existence theorems for ordinary differential equations of fractional order

Abstract

In this paper, we prove two local existence theorems, by using both the Picard method and the Schauder fixed-point theorem, for the following initial-value problem:

$$g(\alpha )(x) = f(x,g(x))(almost all x\varepsilon [a,a + h])$$

with (A)

$$g(\alpha - 1)(a) = b\Gamma (\alpha ),0 < \alpha \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ < } 1,$$

where g^(α) denotes the derivative of order α of a real-valued function g; γ(α) is the Gamma function where α > 0; b is a real number, and under suitable conditions on the function f.

If α = 1 in the initial-value problem (A), then the existence theorems corresponding to this problem are known (sometimes) as the Carathéodory theorem, see Coddington and Levinson (1955) and Hale (1969). Finally, we prove a local existence theorem of the maximum and the minimum solutions for the initial-value problem (A) above; when α = 1, this theorem reduces to Theorem 1,2 of Coddington and Levinson (1955).