Existence theorems for solutions of differential equations of non-integral order

where </>(x, 3/) is a known function, y(x) is an unknown function, and D^y is the Riemann-Liouvillef generalized derivative of order a of the function y(x). For ce = l the equation (1.1) is an ordinary differential equation of the first order and the restrictions on c/>(x, y) for non-integral a are found to be quite similar to those imposed on the function in the integral case. In establishing the fundamental existence theorem we first prove (§2) a theorem of the kind considered by Birkhoff and Kellogg. J Our proof rests on three lemmas which are contained in §3 along with the definition of the generalized derivative. In §4 we establish the existence of a unique solution in the small for 0 < c e < l . The extension of this solution throughout the region of definition of cj>(x, y) and the case a>l are considered in §§5 and 6 respectively.