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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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