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Primordial black holes (PBHs) has been considered to form a part of darkmatter for a long time but the possibility has been poorly constrained over awide mass range, including the stellar mass range ($1-100~M_{\odot}$). However,due to the discovery of merger events of black hole binaries by LIGO-Virgogravitational wave observatories, the interest for PBHs in the stellar masswindow has been aroused again. Fast radio bursts (FRBs) are bright radiotransients with millisecond duration and very high all-sky occurrence rate.Lensing effect of these bursts has been proposed as one of the optimal probesfor constraining the abundance of PBHs in the stellar mass range. In thispaper, we first investigate constraints on the abundance of PBHs from thelatest $593$ FRB observations for both the monochromatic mass distribution andthree other popular extended mass distributions related to different formationmechanisms of PBHs. It is found that constraints from currently public FRBobservations are relatively weaker than those from existing gravitational wavedetections. Furthermore, we forecast constraining power of future FRBobservations on the abundance of PBHs with different mass distributions of PBHsand different redshift distributions of FRBs taken into account. Finally, Wefind that constraints of parameter space on extended mass distributions from$\sim10^5$ FRBs with $\overline{\Delta t}\leq1 ~\rm ms$ would be comparablewith what can be constrained from gravitational wave events. It is foreseenthat upcoming complementary multi-messenger observations will yieldconsiderable constraints on the possibilities of PBHs in this intriguing masswindow.

Constraints on the abundance of primordial black holes with different mass distributions from lensing of fast radio bursts