A basic problem in both pure and applied mathematics is solving various kinds of equations. In recent years, multilinear systems of equations have received a great deal of attention and active research. However, most studies are focused on those multilinear systems with the tensors involved being strong $\mathcal{M}$-tensors. It is well known that many tensors from real problems are non-negative, including strong completely positive tensors. In this paper, we consider the multilinear system of equations whose coefficient tensors are strong completely positive tensors of even order. First we show that this multilinear system has a unique real solution. Secondly, we propose a homotopy approximation method for solving the multilinear system, which has global convergence, and it possesses locally superlinear convergence under certain conditions. Preliminary numerical experiment results show that the proposed algorithm is effective.

纯数学和应用数学中的一个基本问题是求解各种方程。近年来，多线性方程组受到了广泛的关注和积极的研究。然而，大多数研究都集中在那些涉及强张量的多线性系统上$\mathcal{米}$-张量。众所周知，实际问题中的许多张量都是非负的，包括强完全正张量。在本文中，我们考虑系数张量是偶数阶的强完全正张量的多重线性方程组。首先我们证明这个多线性系统有一个唯一的实数解。其次，我们提出了一种求解多线性系统的同伦近似方法，该方法具有全局收敛性，并且在一定条件下具有局部超线性收敛性。初步的数值实验结果表明该算法是有效的。