# American Institute of Mathematical Sciences

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
 [1] Yu-Xia Liang, Ze-Hua Zhou. Supercyclic translation $C_0$-semigroup on complex sectors. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 361-370. doi: 10.3934/dcds.2016.36.361 [2] Jiří Neustupa. On $L^2$-Boundedness of a $C_0$-Semigroup generated by the perturbed oseen-type operator arising from flow around a rotating body. Conference Publications, 2007, 2007 (Special) : 758-767. doi: 10.3934/proc.2007.2007.758 [3] Jacek Banasiak, Marcin Moszyński. Hypercyclicity and chaoticity spaces of $C_0$ semigroups. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 577-587. doi: 10.3934/dcds.2008.20.577 [4] Todd Young. A result in global bifurcation theory using the Conley index. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 387-396. doi: 10.3934/dcds.1996.2.387 [5] M. C. Carbinatto, K. Mischaikow. Horseshoes and the Conley index spectrum - II: the theorem is sharp. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 599-616. doi: 10.3934/dcds.1999.5.599 [6] José A. Conejero, Alfredo Peris. Hypercyclic translation $C_0$-semigroups on complex sectors. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1195-1208. doi: 10.3934/dcds.2009.25.1195 [7] Marius Mitrea. On Bojarski's index formula for nonsmooth interfaces. Electronic Research Announcements, 1999, 5: 40-46. [8] Marian Gidea. Leray functor and orbital Conley index for non-invariant sets. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 617-630. doi: 10.3934/dcds.1999.5.617 [9] Jintao Wang, Desheng Li, Jinqiao Duan. On the shape Conley index theory of semiflows on complete metric spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1629-1647. doi: 10.3934/dcds.2016.36.1629 [10] Anna Go??biewska, S?awomir Rybicki. Equivariant Conley index versus degree for equivariant gradient maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 985-997. doi: 10.3934/dcdss.2013.6.985 [11] Ketty A. De Rezende, Mariana G. Villapouca. Discrete conley index theory for zero dimensional basic sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1359-1387. doi: 10.3934/dcds.2017056 [12] Katsutoshi Shinohara. On the index problem of $C^1$-generic wild homoclinic classes in dimension three. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 913-940. doi: 10.3934/dcds.2011.31.913 [13] Cleon S. Barroso. The approximate fixed point property in Hausdorff topological vector spaces and applications. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 467-479. doi: 10.3934/dcds.2009.25.467 [14] Grzegorz Graff, Piotr Nowak-Przygodzki. Fixed point indices of iterations of $C^1$ maps in $R^3$. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 843-856. doi: 10.3934/dcds.2006.16.843 [15] Sergiu Aizicovici, Nikolaos S. Papageorgiou, V. Staicu. The spectrum and an index formula for the Neumann $p-$Laplacian and multiple solutions for problems with a crossing nonlinearity. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 431-456. doi: 10.3934/dcds.2009.25.431 [16] Tiphaine Jézéquel, Patrick Bernard, Eric Lombardi. Homoclinic orbits with many loops near a $0^2 i\omega$ resonant fixed point of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3153-3225. doi: 10.3934/dcds.2016.36.3153 [17] Litismita Jena, Sabyasachi Pani. Index-range monotonicity and index-proper splittings of matrices. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 379-388. doi: 10.3934/naco.2013.3.379 [18] Elisa Gorla, Maike Massierer. Index calculus in the trace zero variety. Advances in Mathematics of Communications, 2015, 9 (4) : 515-539. doi: 10.3934/amc.2015.9.515 [19] Ha Pham, Plamen Stefanov. Weyl asymptotics of the transmission eigenvalues for a constant index of refraction. Inverse Problems & Imaging, 2014, 8 (3) : 795-810. doi: 10.3934/ipi.2014.8.795 [20] Alexander Krasnosel'skii, Jean Mawhin. The index at infinity for some vector fields with oscillating nonlinearities. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 165-174. doi: 10.3934/dcds.2000.6.165

2018 Impact Factor: 1.143