On a fractional differential equation with infinitely many solutions

We present a set of restrictions on the fractional differential equation $x^{(\alpha )}(t)=g(x(t))$, $t\ge 0$, where $\alpha \in (0,1)$ and $g(0)=0$, that leads to the existence of an infinity of solutions starting from $x(0)=0$. The operator $x^{(\alpha )}$ is the Caputo differential operator.