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The index of Lyapunov stable fixed points in two dimensions. (English) Zbl 0811.34018
Let \(U \subset R^ 2\) be an open and connected set, \(0 \in U\), \(f:U \to R^ 2\) be a continuous, one-to-one and orientation-preserving mapping, \(f(0) = 0\). If 0 is stable and isolated as a fixed point of \(f\), then the fixed point index is equal to one. Applications to differential equations are given.
Reviewer: M.Bartušek (Brno)

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
47H11 Degree theory for nonlinear operators
34D20 Stability of solutions to ordinary differential equations
Full Text: DOI
[1] Amann, H. (1982). A note on degree theory for gradient mappings.Proc. Am. Math. Soc. 85, 591–595. · Zbl 0501.58012
[2] Brown, M. (1984). A new proof of Brouwer’s lemma on translation arcs.Houston Math. J. 10, 35–41. · Zbl 0551.57005
[3] Brown, M. (1985). Homeomorphisms of two-dimensional manifolds.Houston Math. J. 11, 455–469. · Zbl 0605.57005
[4] Erle, D. (1993). Stable equilibria and vector field index.Topol. Appl. 49, 231–235. · Zbl 0777.58032
[5] Krasnoselskii, M. A. (1968).Translation Along Trajectories of Differential Equations, Am. Math. Soc., Providence, RI.
[6] Krasnoselskii, M. A., and Zabreiko, P. P. (1984).Geometrical Methods of Nonlinear Analysis, Springer-Verlag, Berlin.
[7] Krasnoselskii, M. A., Perov, A. I., Povolotskii, A. I., and Zabreiko, P. P. (1966).Plane Vector Fields, Academic Press, New York.
[8] Levi-Civita, T. (1901). Sopra alcuni criteri di instabilita.Ann. Mat. V, 221–307. · JFM 32.0720.01
[9] Massera, J. L. (1950). The existence of periodic solutions of systems of differential equations.Duke Math. J 17, 457–475. · Zbl 0038.25002
[10] Mawhin, J. (1985).Points Fixes, Points Critiques et Problèmes aux Limites, Les Presses de l’Université de Montréal, Montréal. · Zbl 0561.34001
[11] Ortega, R. (1990). Topological degree and stability of periodic solutions for certain differential equations.J. London Math. Soc. 42, 505–516. · Zbl 0677.34042
[12] Pliss, V. A. (1966).Nonlocal Problems of the Theory of Oscillations, Academic Press, New York. · Zbl 0151.12104
[13] Siegel, C. L., and Moser, J. K. (1971).Lectures on Celestial Mechanics, Springer-Verlag, Berlin. · Zbl 0312.70017
[14] Spanier, E. H. (1989).Algebraic Topology, Springer-Verlag, Berlin. · Zbl 0145.43303
[15] Thews, K. (1989). Der Abbildungsgrad von Vektorfeldern zu stabilen Ruhelagen.Arch. Math. 52, 71–74. · Zbl 0633.34027
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