Accelerated dynamical approaches for finding the unique positive solution of K S $\mathcal {K}\mathcal {S}$ -tensor equations

A new class of tensors called $\mathcal {K}\mathcal {S}$ -tensors, which is a subset of non-singular $\mathcal {P}$ -tensors and generalization of ${\mathscr{H}}^{+}$ -tensors, is proposed. It is proved that the system of $\mathcal {K}\mathcal {S}$ -tensor equations always has a unique positive solution for any positive right-hand side by proposing a positive increasing map. Two approaches based on dynamical system are presented to find the unique positive solution. The theoretical analysis results show that the convergence of the proposed models is guaranteed, and numerical examples further illustrate that the given models are feasible and effective in finding the positive solution of $\mathcal {K}\mathcal {S}$ -tensor equations.