  We are concerned with the tensor equation with an M-tensor or Z-tensor, which
we call the M- tensor equation or Z-tensor equation respectively. We derive a
necessary and sufficient condition for a Z (or M)-tensor equation to have
nonnegative solutions. We then develop a monotone iterative method to find a
nonnegative solution to an M-tensor equation. The method can be regarded as an
approximation to Newton's method for solving the equation. At each iteration,
we solve a system of linear equations. An advantage of the proposed method is
that the coefficient matrices of the linear systems are independent of the
iteration. We show that if the initial point is appropriately chosen, then the
sequence of iterates generated by the method converges to a nonnegative
solution of the M- tensor equation monotonically and linearly. At last, we do
numerical experiments to test the proposed methods. The results show the
efficiency of the proposed methods.
