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

摘要

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 o 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}+rac{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.