  To search for new candidates of the true and simultaneous two-proton ($2p$)
radioactivity, the $2p$ decay energies (\textit{Q}$_{2p}$) are extracted by the
Weizs\"{a}cker-Skyrme-4 (WS4) model, the finite-range droplet model (FRDM), the
Kourra-Tachibaba-Uno-Yamada (KTUY) model and the Hartree-Fock-Bogoliubov
mean-field model with the BSk29 Skyrme interaction (HFB29). Then, the $2p$
radioactivity half-lives are calculated within the generalized liquid drop
model (GLDM) by inputting the four types of \textit{Q}$_{2p}$ values. By the
energy and half-life constraints, it is found that the probable $2p$ decay
candidates are the nuclei beyond the proton-drip line in the region of
\textit{Z} $<$50 or \textit{Z}$\leq 50$ based on each nuclear mass model. In
the region beyond \textit{Z}=50, the $2p$-decaying candidates are predicted
only using the HFB29 mass model. Finally, the competition between the true $2p$
radioactivity and $\alpha$-decay for the nuclei above the
\textit{N}=\textit{Z}=50 shell closures is discussed. It is shown that
$^{101}$Te, $^{111}$Ba and $^{114}$Ce prefer to $2p$ radioactivity and the
dominant decay mode of $^{107}$Xe and $^{116}$Ce is $\alpha$-decay.
