Equivalence of Volterra equations and Grunwald–Letnikov equations for fractional order below one

Let $G_K(x)$ be continuous on the relevant interval, let $a<t$, and let $0<\alpha<1$. The Volterra integral equation

$$x(t)=\frac{1}{\Gamma(\alpha)}\int_a^t\frac{G_K(x(\tau))\,d\tau}{(t-\tau)^{1-\alpha}}$$

is compared with the Grunwald–Letnikov fractional differential equation

$${}_aD_t^{\alpha}x(t)=G_K(x(t)).$$

Volterra–Grunwald–Letnikov equivalence conjecture. Theorem 5, which asserts equivalence of these equations under the stated continuity and zero-initial-condition hypotheses, is valid for $0<\alpha<1$.

The preceding theorem establishes the equivalence for $\alpha\geq 1$, while the source says that the methods used there do not prove the needed result in the negative-order range and formulates this extension as a conjecture. The exact role of the negative-order discussion is not fully clear from the supplied excerpt.