$L^p$-Solutions of Fractional Differential Equations

Abstract

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 $C(t-s) =(1/\Gamma (q))(t-s)^{q-1}$.  We then transform the integral equation into one with kernel $R(t-s)$ so that $0<R(t) \leq C(t)$ and $\int_0^{\infty} R(s)ds=1.  A variety of techniques are introduced by which we are able to show that solutions are in $L^p[0,\infty)$ for appropriate $p \geq 1$.