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Keywords:
projectively bounded and invariant sets; generalized Perron–Frobenius conditions; nonlinear eigenvalue; Collatz–Wielandt relations
projectively bounded and invariant sets; generalized Perron–Frobenius conditions; nonlinear eigenvalue; Collatz–Wielandt relations
Summary:
This work is concerned with the eigenvalue problem for a monotone and homogenous self-mapping $f$ of a finite dimensional positive cone. Paralleling the classical analysis of the (linear) Perron–Frobenius theorem, a verifiable communication condition is formulated in terms of the successive compositions of $f$, and under such a condition it is shown that the upper eigenspaces of $f$ are bounded in the projective sense, a property that yields the existence of a nonlinear eigenvalue as well as the projective boundedness of the corresponding eigenspace. The relation of the communication property studied in this note with the idea of indecomposability is briefly discussed.
References:
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