A cohomological index of Fuller type for parameterized set-valued maps in normed spaces

• Autorzy: Robert Skiba
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Fixed points of parameterized maps; Periodic orbit; Stationary point; Fixed point index; Fuller index

Kategorie tematyczne

• 55M20: Fixed points and coincidences
• 37C27: Periodic orbits of vector fields and flows
• 54C60: Set-valued maps
• 37C25: Fixed points, periodic points, fixed-point index theory
• 54H25: Fixed-point and coincidence theorems
• 55N05: \v Cech types

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2014
• Tom: 12
• Numer: 8
• Strony: 1164-1197

Daty

wydano

2014-08-01

online

2014-05-08

Twórcy

autor

Robert Skiba

• Nicolaus Copernicus University

Bibliografia

• [1] Chicone C., Ordinary Differential Equations with Applications, 2nd ed., Texts Appl. Math., 34, Springer, New York, 2006
• [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
• [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
• [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
• [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
• [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
• [9] Fenske C.C., A direct topological definition of the Fuller index for local semiflows, Topol. Methods Nonlinear Anal., 2003, 21(2), 195–209
• [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
• [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
• [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
• [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
• [14] Hatcher A., Algebraic Topology, Cambridge University Press, Cambridge, 2002
• [15] Kryszewski W., Homotopy Properties of Set-Valued Mappings, Nicolaus Copernicus University, Torun, 1997
• [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
• [17] Lang S., Introduction to Differentiable Manifolds, 2nd ed., Universitext, Springer, New York, 2002
• [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
• [22] Prasolov V.V., Elements of Homology Theory, Grad. Stud. Math., 81, American Mathematical Society, Providence, 2007
• [23] Spanier E.H., Algebraic Topology, McGraw-Hill, New York, 1966
• [24] Srzednicki R., Periodic orbits indices, Fund. Math., 1990, 135(3), 147–173
• [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

Identyfikatory

DOI

10.2478/s11533-014-0408-z