Topological Methods in Nonlinear Analysis

Applications of weighted maps to periodic problems of autonomous differential equations

Robert Skiba

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Abstract

In this paper we present a new approach for solving the problem of the existence of closed trajectories for autonomous differential equations without the uniqueness property. To this aim, we are using a special class of set-valued maps, called weighted carriers or weighted maps.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 42, Number 1 (2013), 9-49.

Dates
First available in Project Euclid: 21 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461247291

Mathematical Reviews number (MathSciNet)
MR3155613

Zentralblatt MATH identifier
1319.34070

Citation

Skiba, Robert. Applications of weighted maps to periodic problems of autonomous differential equations. Topol. Methods Nonlinear Anal. 42 (2013), no. 1, 9--49. https://projecteuclid.org/euclid.tmna/1461247291


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References

  • J. Andres, G. Gabor and L. Górniewicz, Topological structure of solution sets to multivalued asymptotic problems , Z. Anal. Anwendungen, 19 (2000), 35–60 \ref\key 2
  • N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems, Recent Advances, North-Holland Mathematical Library, 52 , Amsterdam, North-Holland (1994) \ref\key 3
  • J.P. Aubin, Viability Theory, Birkhäuser, Boston (1991) \ref\key 4
  • J.P. Aubin and A. Cellina, Differential Inclusions, Springer–Verlag, Berlin (1984) \ref \key 5
  • R. Bader and W. Kryszewski, On the solution sets of constrained differential inclusions with applications , Set-Valued Anal., 9 (2001), 289–313 \ref\key 6
  • H. Ben-El-Mechaiekh and W. Kryszewski, Equilibria of set-valued maps on nonconvex domains , Trans. Amer. Math. Soc. (10), 349 (1997), 4159–4179 \ref\key 7
  • K. Borsuk, Theory of Retracts, Monografie Matematyczne, PWN, Warszawa (1967) \ref\key 8
  • R. Brown, A Topological Introduction to Nonlinear Analysis, Birkhäuser (1993) \ref\key 9
  • C.I. Byrnes, Topological methods for nonlinear oscillations , Notices Amer. Math. Soc. (9), 57 (2010), 1080–1091 \ref\key 10
  • F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley-Interscience, New York (1983) \ref\key 11
  • G. Conti and J. Pejsachowicz, Fixed point theorems for multivalued weighted maps , Ann. Mat. Pura Appl. (4), 126 (1980), 319–341 \ref\key 12
  • A. Ćwiszewski and W. Kryszewski, Equilibria of set-valued maps: variational approach , Nonlinear Anal., 9 (2001), 289–313 \ref\key 13
  • G. Darbo, Teoria dell'omologia in una categoria di mappe plurivalenti ponderate , Rend. Sem. Mat. Univ. Padova, 28 (1958), 188–220 \ref\key 14 ––––, Estensione alle mappe ponderate del teorema di Lefschetz sui punti fissi , Rend. Sem. Mat. Univ. Padova, 31 (1961), 46–57 \ref\key 15
  • K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin (1992) \ref\key 16
  • A. Dold, Lectures on Algebraic Topology, Die Grundlehren der mathematischen Wissenschaften, Band 200, Springer–Verlag, New York (1972) \ref\key 17
  • F.B. Fuller, Note on trajectories in a solid torus , Ann. of Math., 58 (1952), 438–439 \ref\key 18
  • L. Górniewicz, Topological structure of solution sets: current results , Arch. Math. (Brno), 36 (2000), 343–382 \ref\key 21 ––––, Topological Fixed Point Theory of Multivalued Mappings, second edition, Topological Fixed Point Theory and Its Applications, 4 , Kluwer Academic Publishers, Dordrecht (2006) \ref\key 22
  • A. Granas and J. Dugundji, Fixed Point Theory, Springer Monographs in Mathematics, Springer–Verlag, New York (2003) \ref\key 23
  • D.M. Hyman, On decreasing sequences of compact absolute retracts , Fund. Math., 64 (1969), 91–97 \ref\key 24
  • F. von Haeseler, H.-O. Peitgen and G. Skordev, Lefschetz fixed point theorem for acyclic maps with multiplicity , Topol. Methods Nonlinear Anal., 19 (2002), 339–374 \ref\key 25
  • F. von Haeseler and G. Skordev, Borsuk–Ulam theorem, fixed point index and chain approximations for maps with multiplicity , Pacific J. Math., 153 (1992), 369–396 \ref\key 26
  • J. Pejsachowicz, The homotopy theory of weighted mappings , Boll. Unione Mat. Ital. B (5), 14 (1977), 702–721 \ref\key 27 ––––, A Lefschetz fixed point theorem for multivalued weighted mappings , Boll. Unione Mat. Ital. A (5), 14 (1977), 391–397 \ref\key 28 ––––, Relation between the homotopy and the homology theory of weighted mappings , Boll. Unione Mat. Ital. B (5), 15 (1978), 285–302 \ref\key 29
  • J. Pejsachowicz and R. Skiba, Fixed point theory of multivalued weighted maps , Handbook of Topological Fixed Point Theory, Springer, Dordrecht (2005), 217–263 \ref\key 30
  • S. Plaskacz, On the solution sets for differential inclusions , Boll. Unione Mat. Ital. A (7), 6 (1992), 387–394 \ref\key 31 ––––, Periodic solutions of differential inclusions on compact subsets of $\Bbb R^n$ , J. Math. Anal. Appl., 148 (1990), 202–212 \ref\key 32
  • H.W. Siegberg and G. Skordev, Fixed point index and chain approximations , Pacific J. Math., 102 (1982), 455–486 \ref\key 33
  • R. Skiba, On the Lefschetz fixed point theorem for multivalued weighted mappings , Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 40 (2001), 201–214 \ref\key 34 ––––, Topological essentiality for multivalued weighted mappings , Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 41 (2002), 131–145 \ref\key 35 ––––, Fixed points of multivalued weighted maps , Lecture Notes in Nonlinear Analysis, 9 , Juliusz Schauder Center for Nonlinear Studies, Nicolaus Copernicus University, Toruń (2007), 1–148 \ref\key 36 ––––, Graph-approximation of multivalued weighted maps , Topol. Methods Nonlinear Anal., 29 (2007), 119–161 \ref\key 37
  • E.G. Sklyarenko and G. Skordev, Intger-valued fixed point index for acyclic maps on \romANR's, C.R. Acad Bulgare Sci., 57 (2004), 5–8 \ref\key 38
  • H.L. Smith and H.R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, American Mathematical Society, 118 (2011) \ref\key 39
  • E. Spanier, Algebraic Topology, Springer–Verlag, MacGraw–Hill, New York (1966)