In this paper, we establish $Z$ -eigenvalue inclusion theorems for general tensors, which reveal some crucial differences between $Z$ -eigenvalues and $H$ -eigenvalues. As an application, we obtain upper bounds for the largest $Z$ -eigenvalue of a weakly symmetric nonnegative tensor, which are sharper than existing upper bounds.

In this paper, by characterizing the ratio of the smallest and largest values of a Perron vector, we propose some Ostrowski-type bound estimations for the spectral radius of weakly irreducible nonnegative tensors which improve the existing results. Based on Brualdi-type eigenvalue inclusion sets and Gershgorin eigenvalue inclusion sets, we establish some new Ky Fan-type theorems for weakly irreducible nonnegative tensors.

M-eigenvalues of fourth-order partially symmetric tensors play important roles in the nonlinear elastic material analysis and the entanglement problem of quantum physics. In this paper, we introduce M-identity tensor and establish two M-eigenvalue inclusion intervals with n parameters for fourth-order partially symmetric tensors, which are sharper than some existing results. Numerical examples are proposed to verify the efficiency of the obtained results. As applications, we provide some checkable sufficient conditions for the positive definiteness and establish bound estimations for the M-spectral radius of fourth-order partially symmetric nonnegative tensors.

In this paper, we present two S-type Z-eigenvalue inclusion sets involved with a nonempty proper subset S of N for general tensors. It is shown that the new sets are tighter than those provided by Wang et al. (Discrete Contin. Dyn. Syst., Ser. B 22(1):187-198, 2017). Furthermore, we obtain upper bounds for the spectral radius of weakly symmetric nonnegative tensors, which are sharper than existing results.

In this paper, we develop existence and stability theorems for strong vector set-valued equilibrium problems in reflexive Banach spaces. Based on asymptotic cone theory, we present the equivalent characterizations on the nonemptiness and boundedness of the solution set for strong vector set-valued equilibrium problems. Furthermore, stability results are established for strong vector set-valued equilibrium problems, when both the mapping and the constraint set are perturbed by different parameters.