The nonlinear tensor equation \(\begin{aligned} {\mathcal {X}}- {\mathcal {B}}*_n({\mathcal {X}}^{-1}+{\mathcal {A}})^{-1}*_n{\mathcal {B}}^{T}={\mathcal {I}}, \end{aligned}\) plays a significant role in various areas of mathematics, physics, and engineering. It finds applications in problems involving signal processing, machine learning, network theory, and the modeling of multidimensional systems. In this work, we propose a novel estimation approach for approximating the positive definite solution of the tensor equation. By utilizing this estimate, we construct a fixed-point iterative method that circumvents the need for direct tensor inversion. This not only enhances computational efficiency but also ensures numerical stability, particularly for large-scale problems where direct inversion is impractical. Then, the convergence analysis of the proposed iterative method is rigorously established, demonstrating its effectiveness in finding the unique positive definite solution. The method’s flexibility allows it to be adapted to a wide range of tensor structures and operations, including the Einstein product and other tensor contractions. Finally, to validate the practical utility of the proposed method, we present a series of numerical experiments. These examples include applications in solving tensor equations arising in multi-dimensional dynamic systems, image processing, and optimization problems. The results illustrate the accuracy, robustness, and computational advantages of the proposed method compared to existing approaches. This study not only contributes a new computational tool for solving nonlinear tensor equations but also opens avenues for further research in extending the method to more complex tensor formulations and higher-order systems.
