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 - A, 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.  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.   Google Scholar

[3]

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

[4]

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

[5]

B. Cornet, Euler characteristic and fixed-point theorems,, Positivity, 6 (2002), 243.  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.   Google Scholar

[7]

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

[8]

A. Ćwiszewski and W. Kryszewski, Homotopy invariants for tangent vector fields on closed sets,, Nonlinear Anal., 65 (2006), 175.  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.  doi: 10.1515/crll.1984.350.1.  Google Scholar

[10]

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

[11]

R. Engelking, "General Topology,", Sigma Series in Pure Mathematics, (1989).   Google Scholar

[12]

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

[13]

A. Granas and J. Dugundji, "Fixed Point Theory,", Springer-Verlag, (2003).   Google Scholar

[14]

M. W. Hirsch, "Differential Topology,", Graduate Texts in Mathematics, 33 (1976).   Google Scholar

[15]

S. T. Hu, "Theory of Retracts,", Wayne State Univ. Press, (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.  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, (1992).   Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (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.   Google Scholar

[25]

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

[26]

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

[27]

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

[28]

R. Srzednicki, K. Wójcik and P. Zgliczyński, Fixed point results based on the Ważewski method,, in, (2005), 905.  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.   Google Scholar

[30]

J. H. C. Whitehead, Note on a theorem due to Borsuk,, Bull. Amer. Math. Soc., 54 (1948), 1125.  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.  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.   Google Scholar

[3]

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

[4]

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

[5]

B. Cornet, Euler characteristic and fixed-point theorems,, Positivity, 6 (2002), 243.  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.   Google Scholar

[7]

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

[8]

A. Ćwiszewski and W. Kryszewski, Homotopy invariants for tangent vector fields on closed sets,, Nonlinear Anal., 65 (2006), 175.  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.  doi: 10.1515/crll.1984.350.1.  Google Scholar

[10]

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

[11]

R. Engelking, "General Topology,", Sigma Series in Pure Mathematics, (1989).   Google Scholar

[12]

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

[13]

A. Granas and J. Dugundji, "Fixed Point Theory,", Springer-Verlag, (2003).   Google Scholar

[14]

M. W. Hirsch, "Differential Topology,", Graduate Texts in Mathematics, 33 (1976).   Google Scholar

[15]

S. T. Hu, "Theory of Retracts,", Wayne State Univ. Press, (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.  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, (1992).   Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (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.   Google Scholar

[25]

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

[26]

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

[27]

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

[28]

R. Srzednicki, K. Wójcik and P. Zgliczyński, Fixed point results based on the Ważewski method,, in, (2005), 905.  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.   Google Scholar

[30]

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

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