Perturbation analysis of tensor $(\mathcal{B},\mathcal{C})$-inverse via Einstein product

We investigate the influence of a relatively small perturbation on various generalized inverses functions or quantities derived from a tensor $\mathcal{A}$.When a small tensor perturbation $\mathcal{E}$ is introduced, it becomes challenging to analyze generalized inverses of the perturbed tensor $\mathcal{D}=\mathcal{A}+\mathcal{E}$ and to determine how this perturbation affects a generalized inverse of $\mathcal{A}$.Our main goal is to understand the relationship between ${\mathcal{D}}^{\mathrm{⅁}}$ and ${\mathcal{A}}^{\mathrm{⅁}}$, where $(\cdot {)}^{\mathrm{⅁}}$ denotes a specific generalized inverse or a class of generalized inverses.In particular, classes of tensor inner, outer, and $(\mathcal{B},\mathcal{C})$ inverses are considered.