Conditions for the structural existence of an eigenvalue of a bipartite \((\min,\max,+)\)-system. (Q1870583)

The paper considers bipartite \((\min,\max,+)\)-systems of the form \[ x(k+1)= A\otimes y(k),\quad y(k+1)= B\otimes'x(k), \] where \(x\), \(y\) are vectors, \(A\), \(B\) are matrices of adequate sizes, and \(\otimes\), \(\otimes'\) denote the matrix multiplication in the sense of the \((\max,+)\)- and \((\min,+)\)-algebra, respectively. Under the mild assumption that every row in \(A\) and \(B\) contains at least a finite entry, the pair \((A,B)\) being irreducible represents a necessary and sufficient condition for the structural existence of a finite eigenvalue and a corresponding finite eigenvector. This result is also illustrated by an example.