This work establishes multiple novel representations for the m-weak core inverse, accompanied by proofs of their validity. Furthermore, we derive perturbation bounds and analyze continuity properties for this generalized inverse. By utilizing the m-weak core inverse, we characterize the unique solution to a constrained minimization problem in the Frobenius norm framework: min ǁ M^{m+1} X- M^2m (M^m) †B ǁF², subject to the range constraint R (X) ⊆R (M^k), where m∈ N, B ∈ C ^{n×q}, M∈ C^{n×n} and ind (M) =k.
Li et al. (Wed,) studied this question.