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A dynamical systems proof of the Krein–Rutman Theorem and an extension of the Perron Theorem

Published online by Cambridge University Press:  14 November 2011

Nicholas D. Alikakos  and
Giorgio Fusco
Nicholas D. Alikakos
Affiliation:
Mathematics Department, The University of Tennessee, Knoxville, TN 37996-1300, U.S.A.
Giorgio Fusco
Affiliation:
Mathematics Department, II University of Rome, Rome, Italy
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Synopsis

In this paper we establish Perron and Krein–Rutman-like theorems for an operator mapping a cone into the interior of the cone, by considering the discrete dynamical system for the induced operator on the projective space (= sphere). Existence of a positive eigenvector reduces to showing that the ω-limit set of the induced operator consists of a single equilibrium. A special feature of our approach is that the convexity of the cone is needed only for establishing the non-emptiness of the w-limit set. This allows us in finite dimensions to establish an abstract Perron Theorem for non-convex cones.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 117 , Issue 3-4 , 1991 , pp. 209 - 214
Copyright
Copyright © Royal Society of Edinburgh 1991

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References

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