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 and 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.

[2]

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

[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.

[4]

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

[5]

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

[6]

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

[7]

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

[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.

[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.

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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.

[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.

[18]

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

[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.

[20]

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

[21]

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

[22]

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

[23]

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

[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.

[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.

[26]

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

[27]

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

[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.

[29]

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

[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.

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.

[2]

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

[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.

[4]

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

[5]

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

[6]

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

[7]

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

[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.

[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.

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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.

[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.

[18]

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

[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.

[20]

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

[21]

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

[22]

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

[23]

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

[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.

[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.

[26]

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

[27]

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

[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.

[29]

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

[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.

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