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The Perron-Frobenius theorem for homogeneous, monotone functions. (English) Zbl 1067.47064
The classical Perron-Frobenius theorem is extended to nonlinear maps. We state the main result of the paper. Let \(\mathbb{R}^+\) be the set of positive reals. A map \(f:(\mathbb{R}^+)^n \to(\mathbb{R}^+)^n\) of the positive cone in the \(n\)-dimensional real vector space is called homogeneous if \(f(\lambda x)=\lambda f(x)\) for every \(\lambda\in \mathbb{R}^+\) and for every \(x\in(\mathbb{R}^+)^n\), and monotone if \(x\leq y\), \(x,y\in(\mathbb{R}^+)^n\) implies \(f(x)\leq f(y)\) (the partial order \(\leq\) in \((\mathbb{R}^+)^n\) is the product ordering). If \(f\) is any homogeneous monotone function, the associate graph \(G(f)\) is defined as a directed graph on the vertices \(\{1,2,\dots,n\}\), and with edge from \(i\) to \(j\) if and only if \[ \lim_{u\to\infty} f_i(u_{\{j\}}) =\infty, \] where \(f_i\) is the \(i\)th component of \(f\) and \(u_{\{j\}}\in (\mathbb{R}^+)^n\) has the \(j\)th component \(u\) and all other components 1.
Theorem. If \(f\) is a homogeneous monotone map, and if \(G(f)\) is strongly connected, i.e., there exists a directed path between any two distinct vertices, then \(f\) has an eigenvector, i.e., there exist \(x\in(\mathbb{R}^+)^n\) and \(\lambda\in\mathbb{R}^+\) such that \(f(x)=\lambda x\). In contrast to the case when \(f\) is linear, the vector \(x\) need not be unique (up to a scalar multiple).
The proof is based on the techniques of the Hilbert projective metric. In particular, the property that a homogeneous monotone function \(f\) has an eigenvector if and only if some orbit (and hence all orbits) of \(f\) are bounded in the Hilbert projective metric, plays a key role.

MSC:
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
15B48 Positive matrices and their generalizations; cones of matrices
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