The existence and uniqueness of eigenvalues for monotone homogeneous mapping pairs

In this paper, the concept of eigenvalue is introduced to the monotone homogeneous mapping pairs (f,g)(f,g), and its existence and uniqueness are established successfully under the boundedness of some orbits of f,gf,g in the Hilbert semi-norm. Also, the Collatz–Wielandt min–max type property is obtained for such a class of mapping pairs. In particular, the nonlinear Perron–Frobenius property for nonnegative tensor pairs (A,B)(A,B) is obtained without involving the calculation of the tensor inversion. By particularizing the mapping gg or ff, several results can emerge as corollaries. They lead to further existence results and open problems.