Continuity of the generalized spectral radius in max algebra

sLet parallel to (.) parallel to be an induced matrix norm associated with a monotone norm on R-n and beta be the collection of all nonempty closed and bounded subsets of n x n nonnegative matrices under this matrix norm. For Psi, Phi is an element of beta, the Hausdorff metric H between Psi and Phi is given by H(Psi, Phi) = max{sup(A is an element of Psi) inf (B is an element of Phi)parallel to A-B parallel to,sup(B is an element of Phi) inf(A is an element of Psi)parallel to A - B parallel to}. The max algebra system consists of the set of nonnegative numbers with sum a circle times b = max{a, b} and the standard product ab for a,b >= 0. For n x n non-negative matrices A, B their product is denoted by A circle times B, where vertical bar A circle times B vertical bar(ij) = max(1 <= k <= n) a(ik)b(kj).For each Psi is an element of beta, the max algebra version of the generalized spectral radius of Psi is mu(Psi) = limsup(m ->infinity)[sup(A is an element of Psi circle times)(m)mu(A)](1/m). where Psi(m)(circle times) = {A(1) circle times A(2) circle times ... circle times A(m) : A(i) is an element of Psi}. Here mu(A) is the maximum circuit geometric mean. In this paper we prove that the max algebra version of the generalized spectral radius is continuous on the Hausdorff metric space (beta, H). The notion of the max algebra version of simultaneous nilpotence of matrices is also proposed. Necessary and sufficient conditions for the max algebra version of simultaneous nilpotence of matrices are presented as well. (C) 2008 Elsevier Inc. All rights reserved.