A generalized eigenvalue problem in the max algebra

We consider the generalized eigenvalue problemA⊗x=λB⊗x,x⩾0,x≠0,where A and B are (entrywise) nonnegative n × n matrices, and the “max” product ⊗ satisfies(A⊗x)i≔maxm=1naimxm.The case B = I has been studied by several authors, and for irreducible (e.g., positive) A there is exactly one eigenvalue λ in the above “max” sense.The generalized problem is different, and for example neither existence nor uniqueness of eigenvalues is guaranteed, even for 2 × 2 positive matrices A and B, which can be analysed by graphical methods. For general n, existence and uniqueness are discussed by different methods including degree theory.