SpringerLink
Log in

Basic Concepts

  • Chapter
  • First Online:
Topology and Approximate Fixed Points

Part of the book series: Developments in Mathematics ((DEVM,volume 71))

  • 796 Accesses

Abstract

This chapter collects well known concepts and results that will play a major role in constructing approximate fixed point theory in the remaining chapters. We note that we will reference the appropriate source papers after Sect. 1.2.8 (before this subsection well known results are presented so that the book is self contained). A brief introduction on fixed point theory is given at the end of this chapter.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (Canada)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. D.E. Alspach, A fixed point free nonexpansive map**. Proc. Amer. Math. Soc. 82, 423–424 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  2. A.V. Archangelskii, Topological Spaces of Functions. Moscow State Univ., 1989 (in Russian) (Kluwer Acad. Publ., Dordrecht, 1992). (in English)

    Google Scholar 

  3. S. Banach, Sur les opérations dans les ensembles abstraits et leurs applications. Fund. Math. 3, 133–181 (1922)

    Article  MathSciNet  MATH  Google Scholar 

  4. C.S. Barroso, The approximate fixed point property in Hausdorff topological vector spaces and applications. Discrete Contin. Dyn. Syst. 25, 469–479 (2009)

    Article  MathSciNet  Google Scholar 

  5. C.S. Barroso, O.F.K. Kalenda, P.-K. Lin, On the approximate fixed point property in Hausdorff topological vector spaces and applications. Math. Z. 271(3–4), 1271–1285 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. J.M. Bayod, J. Martinez-Maurica, Ultrametrically injective spaces. Proc. Amer. Math. Soc. 101(3), 571–576 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  7. T.D. Benavides, A renorming of some nonseparable Banach spaces with the fixed point property. J. Math. Anal. Appl. 350, 525–530 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Bourgain, D.H. Fremlin, M. Talagrand, Pointwise compact sets of Baire measurable functions. Amer. J. Math. 100, 845–886 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  9. R.D. Bourgin, Geometric Aspects of Convex Sets with the Radon-Nikodým Property. Lecture Notes in Math., vol. 993 (Springer-Verlag, 1983)

    Google Scholar 

  10. L.E.J. Brouwer, Über Abbildungen von Mannigfaligkeiten. Math. Ann. 71, 97–115 (1910)

    Google Scholar 

  11. L.E.J. Brouwer, An intuitionist correction of the fixed point theorem on the sphere. Proc. R. Soc. Lond. A Math. Phys. Sci. 213, 1–2 (1952)

    MathSciNet  MATH  Google Scholar 

  12. F.E. Browder, Fixed point theorems for nonlinear semicontractive map**s in Banach spaces. Arch. Rat. Mech. Anal. 21, 259–269 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Caristi, Fixed point theorems for map**s satisfying the inwardness condition. Trans. Amer. Math. Soc. 30 (part 3), 1527–1534 (1997)

    Google Scholar 

  14. G. Darbo, Punti uniti in trasformazioni a codimino non compatto. Rend. Sem. Mat. Univ. Padova. 24, 84–92 (1955)

    MathSciNet  MATH  Google Scholar 

  15. G. Díaz, Monte subspaces in the countable projective limits of Lp(μ)-spaces. Canad. Math. Mat. Bull. 32, 169–176 (1989)

    Article  Google Scholar 

  16. M. Fabian, Gâteaux Differentiability of Convex Functions and Topology: Weak Asplund Spaces (Wiley-Intersscience, New York, 1997)

    MATH  Google Scholar 

  17. M. Fitting, Metric methods, three examples and a theorem. J. Logic Program. 21, 113–127 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Furi, A. Vignoli, Fixed points for densifying map**s. Rendi. Accademie. Naz. Lincei. 47, 465–467 (1969)

    MathSciNet  MATH  Google Scholar 

  19. D. Göhde, Zum prinzip der kontraktiven abbildung. Math. Nachr. 30, 251–258 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  20. P. Habala, P. Hájek, V. Zizler, Introduction to Banach Spaces I, II (Matfyzpress, Prague, 1996)

    MATH  Google Scholar 

  21. U. Heckmanns, Aspects of Ultrametric Spaces, volume 109 of Queen’s Papers in Pure and Applied Mathematics (Queen’s University, Kingston, 1998)

    MATH  Google Scholar 

  22. P. Hitzler, A.K. Seda, Generalized ultrametrics, domains and an application to computational logic. Irish Math. Soc. Bull. 41, 31–43 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. P. Hitzler, A.K. Seda, The fixed point theorems of Priesse-Crampe and Ribenboim in logic programming. Fields Inst. Commun. 32, 219–235 (2002)

    MATH  Google Scholar 

  24. R.C. James, A separable somewhat reflexive Banach space with non separable dual. Bull. Amer. Math. Soc. (N.S.) 80, 738–743 (1974)

    Google Scholar 

  25. S. Kakutani, Two Fixed-point theorems concerning bicompact convex sets. Proc. Imp. Acad. 14(7), 242–245 (1938)

    MathSciNet  MATH  Google Scholar 

  26. O.F.K. Kalenda, Spaces not containing l1 have weak approximate fixed point property. J. Math. Anal. Appl. 373, 134–137 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. M.A. Khamsi, V. Kreinovich, D. Misane, A new method of proving the existence of answer sets for disjunctive logic programs, in Proceeding of the Workshop on Logic Programming with Incomplete Information, Vancouver, B. C., October (1993), pp. 227–244

    Google Scholar 

  28. W.A. Kirk, A fixed point theorem for map**s which do not increase distance. Am. Math. Mon. 72, 1004–1006 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  29. W.A. Kirk, N. Shahzad, Fixed Point Theory in Distance Spaces (Springer International Publishing, Cham, 2014)

    Book  MATH  Google Scholar 

  30. M. Krötzsch, Generalized ultrametric spaces in quantitative domain theory. Theoret. Comput. Sci. 368(1–2), 30–49 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. C. Kuratowski, Sur les espaces complets. Fund. Math. 15, 301–309 (1930)

    Article  MATH  Google Scholar 

  32. J. Lindenstrauss, C. Stegall, Examples of separable spaces which do not contain l1 and whose duals are non-separable. Stud. Math. 54, 81–105 (1975)

    Article  MATH  Google Scholar 

  33. R.D. Nussbaum, The fixed point index and fixed point theorems for k-set contractions. Doctoral dissertation, University of Chicago, 1969

    Book  MATH  Google Scholar 

  34. W.V. Petryshyn, Structures of the fixed point sets of the k-set contractions. Arch. Rat. Mech. Anal. 40, 312–328 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  35. S. Priess-Crampe, Der Banachsche Fixpunktsatz für ultrametrishe Räume. Results Math. 18, 178–186 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  36. S. Priess-Crampe, Remarks on some theorems of functional analysis, in Ultrametric Functional Analysis (Clermont-Ferrand, 2004), Contemp. Math., vol. 384 (American Mathematical Society, Providence, RI, 2005), pp. 235–246

    Google Scholar 

  37. S. Priess-Crampe, P. Ribenboim, Fixed point, combs and generalized power series. Abh. Math. Semin. Univ. Hambg. 63, 227–244 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  38. S. Priess-Crampe, P. Ribenboim, Generalized ultrametric spaces. I. Abh. Math. Semin. Univ. Hambg. 66, 55–73 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  39. S. Priess-Crampe, P. Ribenboim, Generalized ultrametric spaces. II. Abh. Math. Semin. Univ. Hambg. 67, 19–31 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  40. S. Priess-Crampe, P. Ribenboim, The common fixed point theorem for ultrametric spaces. Geom. Dedicate 72, 105–110 (1998)

    Article  MATH  Google Scholar 

  41. S. Priess-Crampe, P. Ribenboim, Fixed point and attractor theorems for ultrametric spaces. Forum Math. 12, 53–64 (2000)

    MathSciNet  MATH  Google Scholar 

  42. S. Priess-Crampe, P. Ribenboim, Ultrametric Dynamics. Illinois J. Math. 55, 287–303 (2011)

    Google Scholar 

  43. S. Priess-Crampe, P. Ribenboim, The approximation to a fixed point. J. Fixed Point Theory Appl. 14, 41–53 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  44. P. Ribenboim, The new theory of ultrametric spaces. Periodica Math. Hung. 32, 103–111 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  45. B.N. Sadovskiĭ, On a fixed point principle. Funktsional. Anal. i Prilozhen 1(2), 74–76 (1967). (in Russian)

    Google Scholar 

  46. J. Schauder, Der fixpunktsatz funktional räumen. Stud. Math. 2, 361–370 (1930)

    Article  Google Scholar 

  47. E. Schören, On immediate extensions of ultra metric spaces. Results Math. 29, 287–303 (1996)

    Google Scholar 

  48. A.K. Seda, Quasi-metrics and the semantics of logic programs. Fundam. Inform. 29, 97–117 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  49. A. Tychonoff, Ein fixpunktsatz. Math. Ann. 111, 767–676 (1935)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Sciences, Mathematics University of Sfax, Sfax, Tunisia

    Afif Ben Amar

  2. School of Mathematical and Statistical Sciences, National University of Ireland, Galway, Ireland

    Donal O’Regan

Authors
  1. Afif Ben Amar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Donal O’Regan
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Ben Amar, A., O’Regan, D. (2022). Basic Concepts. In: Topology and Approximate Fixed Points. Developments in Mathematics, vol 71. Springer, Cham. https://doi.org/10.1007/978-3-030-92204-7_1

Download citation

Publish with us

Policies and ethics

Search

Navigation