Topological Methods in Nonlinear Analysis

Total and local topological indices for maps of Hilbert and Banach manifolds

Yuri E. Gliklikh

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Total and local topological indices are constructed for various types of continuous maps of infinite-dimensional manifolds and ANR's from a broad class. In particular the construction covers locally compact maps with compact sets of fixed points (e.g. maps having a certain finite iteration compact or having compact attractor or being asymptotically compact etc.); condensing maps ($k$-set contraction) with respect to Kuratowski's or Hausdorff's measure of non-compactness on Finsler manifolds; maps, continuous with respect to the topology of weak convergence, etc.

The characteristic point is that all conditions are formulated in internal terms and the index is in fact internal while the construction is produced through transition to the enveloping space. Examples of applications are given.

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Topol. Methods Nonlinear Anal., Volume 15, Number 1 (2000), 17-31.

First available in Project Euclid: 22 August 2016

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Gliklikh, Yuri E. Total and local topological indices for maps of Hilbert and Banach manifolds. Topol. Methods Nonlinear Anal. 15 (2000), no. 1, 17--31.

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  • R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina and B. N. Sadovskiĭ, Measures of Non-Compactness and Condensing Operators, Basel-Boston-Berlin, Birkhäuser-Verlag (1992) \ref\key 2
  • Yu. G. Borisovich, Rotation of weakly continuous vector fields , Dokl. Akad. Nauk SSSR (1960, 131 ), 230–233 \ref\key 3 ––––, Rotation of weakly continuous vector fields , Proceedings of Razmadze Tbilissi Mathematical Institute of Acad. Sci. of Georgian SSR (1960, 27 ), pages 27–42 \ref\key 4 ––––, On an application of the concept of rotation of vector field. , Dokl. Akad. Nauk SSSR (1963, 153 ), 12–15 \ref\key 5
  • Yu. G. Borisovich and Yu. E. Gliklikh, The Lefschetz number of mappings of Banach manifolds and relatedness theorem , Trudy Mat. Fak. VGU (1973, 11 ), 5–13, (Russian) \ref\key 6 ––––, Fixed points of mappings of Banach manifolds and some applications , Nonlinear Anal. (1980, 4 ), 165–192 \ref\key 7 ––––, Topological theory of fixed points on infinite-dimensional manifolds , Lecture Notes Math. (1984, 1108 ), 1–23 \ref\key 8
  • F. E. Browder, Fixed point theorems on infinite-dimensional manifolds , Trans. Amer. Math. Soc. (1965, 119 ), 179–194 \ref\key 9
  • C. C. Fenske C.C. and H.-O. Peitgen, On fixed points of zero index in asymptotic fixed points theory , Pacific J. of Math. (1976, 66 ), 391–410 \ref\key 10
  • G. Fournier, Généralisations du théorème de Lefschetz pour des espases non-compacts I, II, III , Bull. Acad. Polon. Sci. Ser. Math., Astron., Phys. (1975, 23 ), 693–699, 701–706, 707–711 \ref\key 11
  • Yu. E. Gliklikh, Integral operators on manifolds , Trudy Mat. Fak. VGU (1970, 4 ), 29–35, (Russian) \ref\key 12 ––––, On shift operator along the trajectories of functional-differential equations on smooth manifolds , Trudy Nauchno-Issledovatel'skogo Instituta Matematiki VGU (1975, 17 ), 18–24, (Russian) \ref\key 13 ––––, Ordinary and Stochastic Differential Geometry as a Tool for Mathematical Physics, Dordrecht, Kluwer (1996) \ref\key 14 ––––, Global Analysis in Mathematical Physics. Geometric and Stochastic Methods, Springer-Verlag (1997) \ref\key 15
  • Yu. E. Gliklikh and V. V. Obukhovskiĭ, Differential equations of Carathéodory type on Hilbert manifolds , Trudy Mat. Fak. VGU (New Series) (1996, 1 ), 23–28, (Russian) \ref\key 16 ––––, Tubular neighbourhoods of Hilbert manifolds and differential equations of Carathéodory type on groups of diffeomorphisms , New Approaches in Nonlinear Analysis (Th. M. Rassias, ed.), Palm Harbor, Fl., Hadronic Press (1999), 109–123 \ref\key 17
  • M. A. Krasnosel'skiĭ, Topological Methods in the Theory of Nonlinear Integral Equations, Moscow, Gostekhizdat (1956) \ref\key 18
  • M. A. Krasnosel'skiĭ and P. P. Zabreĭko, Geometrical Methods of Nonlinear Analysis, Moscow, Nauka (1975) \ref\key 19
  • J.-P. Penot, Weak topology on functional manifolds , Global Analysis and its Applications, Vienna, IAEA (1974, 3 ), 75–84 \ref\key 20
  • B. N. Sadovskiĭ, Limit-compact and condensing operators , Russian Math. Surveys (1972, 27 ), 85–155