An extension of the MPD and MP weak group inverses

Combining the Moore-Penrose (MP) inverse with m -weak group inverse (m-WGI) in an appropriate way, we solve certain systems of equations and establish a novel class of generalized inverses, which is called the Moore-Penrose m-WGI (MP-m-WGI). Since the weak group inverse (WGI) and Drazin inverse are particular instances of the m-WGI family, it clearly follows that MP weak group (MPWG) inverse and MPD inverse are subclasses of the MP-m-WGI. We present a number of effective representations and characterizations for the MP-m-WGI. As corollaries, we propose new generalized inverses and validate some known properties and representations of the MPD inverse. The MP-2-WGI is considered as one important kind of the MP-m-WGI. Continuity of the MP-m-WGI is studied too. Solvability of a few linear equations is proved and general solutions are expressed as expressions which include the MP-m-WGI.