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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

1Hale, J. K.. Asymptotic behavior of dissipative system, Math Surveys and Monographs 25, (Providence, R.I.: American Mathematical Society, 1988).Google Scholar
2Krein, M. G. and Rutman, M. A.. Linear operators leaving invariant a cone in a Banach space. Trans. Amer. Math. Soc. 26 (1950), 199–325.Google Scholar
3Perron, O.. Zur Theorie der Metrizen. Math. Ann. 64 (1907), 248–263.CrossRefGoogle Scholar
4Smoller, J.. Shock waves and reaction diffusion equation, Grundlehren Math. Wiss. 258 Berlin: Springer, 1983).CrossRefGoogle Scholar