July  2011, 29(3): 953-978. doi: 10.3934/dcds.2011.29.953

On a generalized Poincaré-Hopf formula in infinite dimensions

1. 

Faculty of Mathematics and Computer Science, Nicholas Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland, Poland

Received  January 2010 Revised  June 2010 Published  November 2010

We prove a formula relating the fixed point index of rest points of a completely continuous semiflow defined on a (not necessarily locally compact) metric space in the interior of an isolating block $B$ to the Euler characteristic of the pair $(B,B^-)$, where $B^-$ is the exit set. The proof relies on a general concept of an approximate neighborhood extension space and a full fixed point index theory for self-maps of such spaces. As a consequence, a generalized Poincaré-Hopf type formula for the differential equation determined by a perturbation of the generator of a compact $C_0$ semigroup is obtained.
Citation: Aleksander Ćwiszewski, Wojciech Kryszewski. On a generalized Poincaré-Hopf formula in infinite dimensions. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 953-978. doi: 10.3934/dcds.2011.29.953
References:
[1]

R. Agarwal and D. O'Regan, A note on the Lefschetz fixed point theorem for admissible spaces, Bull. Korean Math. Soc., 42 (2005), 307-313. doi: 10.4134/BKMS.2005.42.2.307.  Google Scholar

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T. Bartsch and N. Dancer, Poincaré-Hopf formulas on convex sets of Banach spaces, Topol. Math. Nonl. Anal., 34 (2009), 213-230.  Google Scholar

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H. Ben-El-Mechaiekh, The coincidence problem for compositions of set-valued maps, Bull. Austral. Math. Soc., 41 (1990), 421-434. doi: 10.1017/S000497270001830X.  Google Scholar

[4]

M. Clapp, On a generalization of absolute neighborhood retracts, Fund. Math., 70 (1971), 117-130.  Google Scholar

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B. Cornet, Euler characteristic and fixed-point theorems, Positivity, 6 (2002), 243-260. doi: 10.1023/A:1020242731195.  Google Scholar

[6]

B. Cornet and M.-O. Czarnecki, Existence of (generalized) equilibria: Necessary and sufficient conditions, Communications on Applied Nonlinear Analysis, 7 (2000), 21-53.  Google Scholar

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A. Ćwiszewski, Topological degree methods for perturbations of operators generating compact $C_0$ semigroups, J. Diff. Eq., 220 (2006), 434-477.  Google Scholar

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A. Ćwiszewski and W. Kryszewski, Homotopy invariants for tangent vector fields on closed sets, Nonlinear Anal., 65 (2006), 175-209. doi: 10.1016/j.na.2005.09.010.  Google Scholar

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E. N. Dancer, Degenerate critical points, homotopy indices and Morse inequalities, J. Reine Angew. Math., 350 (1984), 1-22. doi: 10.1515/crll.1984.350.1.  Google Scholar

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S. Eilenberg and N. Steenrod, "Foundations of Alegebraic Topology," Princeton Univ. Press, Princeton, 1952.  Google Scholar

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R. Engelking, "General Topology," Sigma Series in Pure Mathematics, Heldermann Verlag, Berlin, 1989.  Google Scholar

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J. Girolo, Approximating compact sets in normed linear spaces, Pacific J. Math., 98 (1982), 81-89.  Google Scholar

[13]

A. Granas and J. Dugundji, "Fixed Point Theory," Springer-Verlag, New York, Berlin, 2003.  Google Scholar

[14]

M. W. Hirsch, "Differential Topology," Graduate Texts in Mathematics, 33, Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

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S. T. Hu, "Theory of Retracts," Wayne State Univ. Press, Detroit, 1965.  Google Scholar

[16]

M. Kamenskii and M. Quincampoix, Existence of fixed points on compact epilipschitz sets without invariance conditions, Fixed Point Theory and Appl., 3 (2005), 267-279. doi: 10.1155/FPTA.2005.267.  Google Scholar

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A. G. Kartsatos, Recent results involving compact perturbations and compact resolvents of accretive operators in Banach spaces, in "Proceedings of the First World Congress of Nonlinear Analysts, Tampa, Florida, 1992," Walter de Gruyter, New York, 1996.  Google Scholar

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R. Knill, A general setting for local fixed point theory, J. Math. Pures et Appl., 54 (1975), 389-428.  Google Scholar

[19]

Ch. K. McCord, On the Hopf index and the Conley index, Trans. Amer. Math. Soc., 313 (1989), 853-860. doi: 10.1090/S0002-9947-1989-0961594-0.  Google Scholar

[20]

J. W. Milnor, "Topology from the Differentiable Viewpoint," Princeton University Press, Princeton, NJ, 1997.  Google Scholar

[21]

M. Mrozek, The fixed point index of a translation operator of a semiflow, Univ. Iagel. Acta Math., 27 (1988), 13-22.  Google Scholar

[22]

H. Noguchi, A generalization of absolute neighborhood retracts, Ködai Math. Sem. Rep., 1 (1953), 20-22. doi: 10.2996/kmj/1138843296.  Google Scholar

[23]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.  Google Scholar

[24]

K. P. Rybakowski, On a relation between the Brouwer degree and the Conley index for gradient flows, Bull. Soc. Math. Belg. Ser. B, 37 (1985), 87-96.  Google Scholar

[25]

K. P. Rybakowski, On the homotopy index for infinite-dimensional semiflows, Trans. Amer. Math. Soc., 269 (1982), 351-382. doi: 10.1090/S0002-9947-1982-0637695-7.  Google Scholar

[26]

K. P. Rybakowski, "Homotopy Index and Partial Differential Equations," Springer-Verlag, Berlin, 1987.  Google Scholar

[27]

R. Srzedncki, On rest points of dynamical systems, Fund. Math., 126 (1985), 69-81.  Google Scholar

[28]

R. Srzednicki, K. Wójcik and P. Zgliczyński, Fixed point results based on the Ważewski method, in "Handbook of Topological Fixed Point Theory," Springer, Dordrecht, (2005), 905-943. doi: 10.1007/1-4020-3222-6_23.  Google Scholar

[29]

M. Styborski, Conley index in Hilbert spaces and the Leray-Schauder degree, Topol. Meth. Nonl. Anal., 33 (2009), 131-148.  Google Scholar

[30]

J. H. C. Whitehead, Note on a theorem due to Borsuk, Bull. Amer. Math. Soc., 54 (1948), 1125-1132. doi: 10.1090/S0002-9904-1948-09138-8.  Google Scholar

show all references

References:
[1]

R. Agarwal and D. O'Regan, A note on the Lefschetz fixed point theorem for admissible spaces, Bull. Korean Math. Soc., 42 (2005), 307-313. doi: 10.4134/BKMS.2005.42.2.307.  Google Scholar

[2]

T. Bartsch and N. Dancer, Poincaré-Hopf formulas on convex sets of Banach spaces, Topol. Math. Nonl. Anal., 34 (2009), 213-230.  Google Scholar

[3]

H. Ben-El-Mechaiekh, The coincidence problem for compositions of set-valued maps, Bull. Austral. Math. Soc., 41 (1990), 421-434. doi: 10.1017/S000497270001830X.  Google Scholar

[4]

M. Clapp, On a generalization of absolute neighborhood retracts, Fund. Math., 70 (1971), 117-130.  Google Scholar

[5]

B. Cornet, Euler characteristic and fixed-point theorems, Positivity, 6 (2002), 243-260. doi: 10.1023/A:1020242731195.  Google Scholar

[6]

B. Cornet and M.-O. Czarnecki, Existence of (generalized) equilibria: Necessary and sufficient conditions, Communications on Applied Nonlinear Analysis, 7 (2000), 21-53.  Google Scholar

[7]

A. Ćwiszewski, Topological degree methods for perturbations of operators generating compact $C_0$ semigroups, J. Diff. Eq., 220 (2006), 434-477.  Google Scholar

[8]

A. Ćwiszewski and W. Kryszewski, Homotopy invariants for tangent vector fields on closed sets, Nonlinear Anal., 65 (2006), 175-209. doi: 10.1016/j.na.2005.09.010.  Google Scholar

[9]

E. N. Dancer, Degenerate critical points, homotopy indices and Morse inequalities, J. Reine Angew. Math., 350 (1984), 1-22. doi: 10.1515/crll.1984.350.1.  Google Scholar

[10]

S. Eilenberg and N. Steenrod, "Foundations of Alegebraic Topology," Princeton Univ. Press, Princeton, 1952.  Google Scholar

[11]

R. Engelking, "General Topology," Sigma Series in Pure Mathematics, Heldermann Verlag, Berlin, 1989.  Google Scholar

[12]

J. Girolo, Approximating compact sets in normed linear spaces, Pacific J. Math., 98 (1982), 81-89.  Google Scholar

[13]

A. Granas and J. Dugundji, "Fixed Point Theory," Springer-Verlag, New York, Berlin, 2003.  Google Scholar

[14]

M. W. Hirsch, "Differential Topology," Graduate Texts in Mathematics, 33, Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

[15]

S. T. Hu, "Theory of Retracts," Wayne State Univ. Press, Detroit, 1965.  Google Scholar

[16]

M. Kamenskii and M. Quincampoix, Existence of fixed points on compact epilipschitz sets without invariance conditions, Fixed Point Theory and Appl., 3 (2005), 267-279. doi: 10.1155/FPTA.2005.267.  Google Scholar

[17]

A. G. Kartsatos, Recent results involving compact perturbations and compact resolvents of accretive operators in Banach spaces, in "Proceedings of the First World Congress of Nonlinear Analysts, Tampa, Florida, 1992," Walter de Gruyter, New York, 1996.  Google Scholar

[18]

R. Knill, A general setting for local fixed point theory, J. Math. Pures et Appl., 54 (1975), 389-428.  Google Scholar

[19]

Ch. K. McCord, On the Hopf index and the Conley index, Trans. Amer. Math. Soc., 313 (1989), 853-860. doi: 10.1090/S0002-9947-1989-0961594-0.  Google Scholar

[20]

J. W. Milnor, "Topology from the Differentiable Viewpoint," Princeton University Press, Princeton, NJ, 1997.  Google Scholar

[21]

M. Mrozek, The fixed point index of a translation operator of a semiflow, Univ. Iagel. Acta Math., 27 (1988), 13-22.  Google Scholar

[22]

H. Noguchi, A generalization of absolute neighborhood retracts, Ködai Math. Sem. Rep., 1 (1953), 20-22. doi: 10.2996/kmj/1138843296.  Google Scholar

[23]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.  Google Scholar

[24]

K. P. Rybakowski, On a relation between the Brouwer degree and the Conley index for gradient flows, Bull. Soc. Math. Belg. Ser. B, 37 (1985), 87-96.  Google Scholar

[25]

K. P. Rybakowski, On the homotopy index for infinite-dimensional semiflows, Trans. Amer. Math. Soc., 269 (1982), 351-382. doi: 10.1090/S0002-9947-1982-0637695-7.  Google Scholar

[26]

K. P. Rybakowski, "Homotopy Index and Partial Differential Equations," Springer-Verlag, Berlin, 1987.  Google Scholar

[27]

R. Srzedncki, On rest points of dynamical systems, Fund. Math., 126 (1985), 69-81.  Google Scholar

[28]

R. Srzednicki, K. Wójcik and P. Zgliczyński, Fixed point results based on the Ważewski method, in "Handbook of Topological Fixed Point Theory," Springer, Dordrecht, (2005), 905-943. doi: 10.1007/1-4020-3222-6_23.  Google Scholar

[29]

M. Styborski, Conley index in Hilbert spaces and the Leray-Schauder degree, Topol. Meth. Nonl. Anal., 33 (2009), 131-148.  Google Scholar

[30]

J. H. C. Whitehead, Note on a theorem due to Borsuk, Bull. Amer. Math. Soc., 54 (1948), 1125-1132. doi: 10.1090/S0002-9904-1948-09138-8.  Google Scholar

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