We construct a cohomological index of the Fuller type for set-valued flows in normed linear spaces satisfying the properties of existence, excision, additivity, homotopy and topological invariance. In particular, the constructed index detects periodic orbits and stationary points of set-valued dynamical systems, i.e., those generated by differential inclusions. The basic methods to calculate the index are also presented.
@article{bwmeta1.element.doi-10_2478_s11533-014-0408-z,
author = {Robert Skiba},
title = {A cohomological index of Fuller type for parameterized set-valued maps in normed spaces},
journal = {Open Mathematics},
volume = {12},
year = {2014},
pages = {1164-1197},
zbl = {1295.55004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-014-0408-z}
}
Robert Skiba. A cohomological index of Fuller type for parameterized set-valued maps in normed spaces. Open Mathematics, Tome 12 (2014) pp. 1164-1197. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-014-0408-z/
[1] Chicone C., Ordinary Differential Equations with Applications, 2nd ed., Texts Appl. Math., 34, Springer, New York, 2006 | Zbl 1120.34001
[2] Chow S.N., Mallet-Paret J., The Fuller index and global Hopf bifurcation, J. Differential Equations, 1978, 29(1), 66–84 http://dx.doi.org/10.1016/0022-0396(78)90041-4 | Zbl 0369.34020
[3] Crabb M.C., Potter A.J.B., The Fuller index, In: Invitations to Geometry and Topology, Oxf. Grad. Texts Math., 7, Oxford University Press, 2002, 92–125 | Zbl 0996.54506
[4] Dold A., Lectures on Algebraic Topology, Grundlehren Math. Wiss., 200, Springer, New York-Berlin, 1972 http://dx.doi.org/10.1007/978-3-662-00756-3
[5] Dold A., The fixed point index of fibre-preserving maps, Invent. Math., 1974, 25(3–4), 281–297 http://dx.doi.org/10.1007/BF01389731 | Zbl 0284.55007
[6] Dold A., The fixed point transfer of fibre-preserving maps, Math. Z., 1976, 148(3), 215–244 http://dx.doi.org/10.1007/BF01214520 | Zbl 0329.55007
[7] Fenske C.C., A simple-minded approach to the index of periodic orbits, J. Math. Anal. Appl., 1988, 129(2), 517–532 http://dx.doi.org/10.1016/0022-247X(88)90269-7
[8] Fenske C.C., An index for periodic orbits of functional-differential equations, Math. Ann., 1989, 285(3), 381–392 http://dx.doi.org/10.1007/BF01455063 | Zbl 0663.34057
[9] Fenske C.C., A direct topological definition of the Fuller index for local semiflows, Topol. Methods Nonlinear Anal., 2003, 21(2), 195–209 | Zbl 1035.37016
[10] Franzosa R.D., An homology index generalizing Fuller’s index for periodic orbits, J. Differential Equations, 1990, 84(1), 1–14 http://dx.doi.org/10.1016/0022-0396(90)90124-8 | Zbl 0706.58055
[11] Fuller F.B., An index of fixed point type for periodic orbits, Amer. J. Math., 1967, 89, 133–148 http://dx.doi.org/10.2307/2373103 | Zbl 0152.40204
[12] Górniewicz L., Topological Fixed Point Theory of Multivalued Mappings, Math. Appl., 495, Kluwer, Dordrecht, 1999 http://dx.doi.org/10.1007/978-94-015-9195-9 | Zbl 0937.55001
[13] Granas A., Dugundji J., Fixed Point Theory, Springer Monogr. Math., Springer, New York, 2003 http://dx.doi.org/10.1007/978-0-387-21593-8 | Zbl 1025.47002
[14] Hatcher A., Algebraic Topology, Cambridge University Press, Cambridge, 2002
[15] Kryszewski W., Homotopy Properties of Set-Valued Mappings, Nicolaus Copernicus University, Torun, 1997 | Zbl 1250.54022
[16] Kryszewski W., Skiba R., A cohomological index of Fuller type for set-valued dynamical systems, Nonlinear Anal., 2012, 75(2), 684–716 http://dx.doi.org/10.1016/j.na.2011.09.002 | Zbl 1232.54023
[17] Lang S., Introduction to Differentiable Manifolds, 2nd ed., Universitext, Springer, New York, 2002 | Zbl 1008.57001
[18] Lee J.M., Introduction to Smooth Manifolds, Grad. Texts in Math., 218, Springer, New York, 2003
[19] Massey W.S., Homology and Cohomology Theory, Monogr. Textbooks Pure Appl. Math., 46, Marcel Dekker, New York-Basel, 1978
[20] Potter A.J.B., On a generalization of the Fuller index, In: Nonlinear Functional Analysis and its Applications, 2, Proc. Sympos. Pure Math., 45, American Mathematical Society, Providence, 1986, 283–286 http://dx.doi.org/10.1090/pspum/045.2/843615
[21] Potter A.J.B., Approximation methods and the generalised Fuller index for semiflows in Banach spaces, Proc. Edinburgh Math. Soc., 1986, 29(3), 299–308 http://dx.doi.org/10.1017/S0013091500017740 | Zbl 0606.34043
[22] Prasolov V.V., Elements of Homology Theory, Grad. Stud. Math., 81, American Mathematical Society, Providence, 2007 | Zbl 1120.55001
[23] Spanier E.H., Algebraic Topology, McGraw-Hill, New York, 1966
[24] Srzednicki R., Periodic orbits indices, Fund. Math., 1990, 135(3), 147–173 | Zbl 0715.55005
[25] Srzednicki R., The fixed point homomorphism of parametrized mappings of ANR’s and the modified fuller index, Ruhr-Universität Bochum, Preprint No. 143/1990
[26] Srzednicki R., Fixed point homomorphisms for parameterized maps, J. Fixed Point Theory Appl., 2013, 13(2), 489–518 http://dx.doi.org/10.1007/s11784-013-0131-6 | Zbl 1285.55002