Further study on tensor absolute value equations

In this paper, we consider the tensor absolute value equations (TAVEs), which is a newly introduced problem in the context of multilinear systems. Although the system of the TAVEs is an interesting generalization of matrix absolute value equations (AVEs), the well-developed theory and algorithms for the AVEs are not directly applicable to the TAVEs due to the nonlinearity (or multilinearity) of the problem under consideration. Therefore, we first study the solutions existence of some classes of the TAVEs with the help of degree theory, in addition to showing, by fixed point theory, that the system of the TAVEs has at least one solution under some checkable conditions. Then, we give a bound of solutions of the TAVEs for some special cases. To find a solution to the TAVEs, we employ the generalized Newton method and report some preliminary results.