  The theory of ion-acoustic solitons in nonrelativistic fully degenerate
plasmas and nonrelativistic and ultra-relativistic degenerate plasmas at low
temperatures is known. We consider a multi-component relativistic degenerate
electron-positron-ion plasma at finite temperatures. Specifically, we focus on
the intermediate region where the particle's Fermi energy $(k_BT_F)$ slightly
differs from the thermal energy $(k_BT)$ and the rest mass energy $(mc^2)$,
i.e., $k_BT_F>k_BT\sim mc^2$ and the normalized chemical energy
($\xi=\mu/k_BT$) is positive and finite. Two different parameter regimes with
$\beta\equiv k_BT/mc^2<1$ and $\beta>1$, relevant for astrophysical plasmas,
are defined, and the existence of small amplitude ion-acoustic solitons in
these regimes are studied, including the critical cases where the known KdV
(Korteweg-de Vries) theory fails. We show that while the solitons with both the
positive (compressive) and negative (rarefactive) potentials coexist in the
case of $\beta<1$, only compressive solitons can exist in the other regime
$(\beta>1)$. Furthermore, while the rarefactive solitons within the parameter
domains of $\beta$ and $\xi$ can evolve with increasing amplitude and hence
increasing energy, the energy of compressive solitons reaches a steady state.
