The system of nonlinear tensor equations with Einstein product \(\begin{aligned} \begin{array}{l} \mathcal {X}+\mathcal {A}^{T}*_N\mathcal {Y}^{-1}*_N\mathcal {A}=\mathcal {I},\\ \mathcal {Y}+\mathcal {B}^{T}*_N\mathcal {X}^{-1}*_N\mathcal {B}=\mathcal {I}, \end{array} \end{aligned}\) which is the generalized form of a system of nonlinear matrix equations studied in the literature, occurs in many applications such as network systems, filtering, control theory, and optimal control. In this study, first, we present a family of iterative methods avoid tensor inversion to solve the system of nonlinear matrix equations based on fixed point iteration. We show that the proposed method converges to the solution of this system of nonlinear matrix equations under situations. Then, by using the method, we present an extended iterative method with the Einstein product to solve the system of nonlinear tensor equations. Finally, various numerical examples are presented to demonstrate the effectiveness and accuracy of the proposed method.
