L~P-Solutions of Fractional Differential Equations

摘要

We study fractional differential equations of Caputo type ~cD~q x(t) = u(t,x(t)), 0 < q < 1, of both linear and nonlinear type. That equation is inverted as an integral equation with kernel C(t — s) := (1/Γ(q))(t- s)~(q-1). We then transform the integral equation into one with kernel R(t - s) so that 0 < R(t)≤ C(t) and ∫_0~∞R(s)ds = 1. A variety of techniques are introduced by which we are able to show that solutions are in L~P [0, ∞) for appropriate p≥ 1.