Several efficient iterative algorithms for solving nonlinear tensor equation \({\mathcal {X}}+{\mathcal {A}}^{T}*_N{\mathcal {X}}^{-1}*_N{\mathcal {A}}={\mathcal {I}}\) with Einstein product

In this paper, we initially introduce several theoretical results regarding the existence of solutions for the nonlinear tensor equation \(\begin{aligned} {\mathcal {X}}+{\mathcal {A}}^{T}*_N{\mathcal {X}}^{-1}*_N{\mathcal {A}}={\mathcal {I}}. \end{aligned}\) $$\begin{array}{c}\hfill \mathcal{X}+{\mathcal{A}}^T{\ast }_N{\mathcal{X}}^{-1}{\ast }_N\mathcal{A}=\mathcal{I}.\end{array}$$ Then, we present iterative algorithms without inversion to determine solutions of the nonlinear tensor equation. We demonstrate the convergence of these algorithms towards solving the nonlinear tensor equation. Finally, we provide various numerical experiments to illustrate their effectiveness and accuracy.