Complementary equations: a fractional differential equation and a Volterra integral equation

Abstract

It is shown that a continuous, absolutely integrable function satisfies the initial value problem \[ D^{q}x(t) = f(t,x(t)), \qquad \lim_{t \to 0^+} t^{1-q}x(t) = x^{0} \qquad (0 < q < 1) \] on an interval $(0, T]$ if and only if it satisfies the Volterra integral equation \[ x(t) = x^{0}t^{q-1}+\frac{1}{\Gamma (q)}\int_{0}^{t}(t-s)^{q-1}f(s, x(s))\,ds \] on this same interval. In contradistinction to established existence theorems for these equations, no Lipschitz condition is imposed on $f(t,x)$. Examples with closed-form solutions illustrate this result.

Keywords

• fractional differential equations
• riemann-liouville operators
• singular kernels
• volterra integral equations