  In this paper the fractional Cox-Ingersoll-Ross process on $\mathbb{R}_+$ for
$H<1/2$ is defined as a square of a pointwise limit of the processes
$Y_{\varepsilon}$, satisfying the SDE of the form $d Y_{\varepsilon}(t)=(
\frac{k}{ Y_{\varepsilon}(t)\mathbb{1}_{\{
Y_{\varepsilon}(t)>0\}}+\varepsilon}-a Y_{\varepsilon}(t))dt+\sigma dB^H(t)$,
as $\varepsilon\downarrow0$. Properties of such limit process are considered.
SDE for both the limit process and the fractional Cox-Ingersoll-Ross process
are obtained.
