m-Weak MPD and DMP inverses

By using adequate powers of a given matrix and its minimal rank weak Drazin inverse and minimal rank right weak Drazin inverse, we propose two new systems of matrix equations as generalizations of the systems for introducing weak MPD and weak DMP inverses. The solutions of these two new systems are called [Formula: see text]-weak MPD and [Formula: see text]-weak DMP inverses, respectively, and they encompass a large class of matrix generalized inverses. We present various characterizations and formulae for the [Formula: see text]-weak MPD and [Formula: see text]-weak DMP inverses, establish properties of some significant particular types of [Formula: see text]-weak MPD and [Formula: see text]-weak DMP inverses, and apply the [Formula: see text]-weak MPD and [Formula: see text]-weak DMP inverses to solve certain linear equations.