Introduction

  • Simeon Reich
  • Alexander J. Zaslavski
Part of the Developments in Mathematics book series (DEVM, volume 34)

Abstract

Let X be a complete metric space. According to Baire’s theorem, the intersection of every countable collection of open dense subsets of X is dense in X. This rather simple, yet powerful result has found many applications. In particular, given a property which elements of X may have, it is of interest to determine whether this property is generic, that is, whether the set of elements which do enjoy this property contains a countable intersection of open dense sets. Such an approach, when a certain property is investigated for the whole space X and not just for a single point in X, has already been successfully applied in many areas of Analysis. In this chapter we discuss several recent results in metric fixed point theory which exhibit these generic phenomena.

Keywords

Porosity 

References

  1. 13.
    Ayerbe Toledano, J. M., Dominguez Benavides, T., & López Acedo, G. (1997). Measures of noncompactness in metric fixed point theory. Basel: Birkhäuser. CrossRefMATHGoogle Scholar
  2. 43.
    Cohen, J. E. (1979). Bulletin of the American Mathematical Society, 1, 275–295. CrossRefMATHGoogle Scholar
  3. 45.
    Covitz, H., & Nadler, S. B. Jr. (1970). Israel Journal of Mathematics, 8, 5–11. MathSciNetCrossRefMATHGoogle Scholar
  4. 49.
    de Blasi, F. S., & Myjak, J. (1976). Comptes Rendus de L’Académie Des Sciences. Paris, 283, 185–187. MATHGoogle Scholar
  5. 51.
    de Blasi, F. S., & Myjak, J. (1998). Journal of Approximation Theory, 94, 54–72. MathSciNetCrossRefMATHGoogle Scholar
  6. 66.
    Goebel, K., & Kirk, W. A. (1983). In Contemporary mathematics: Vol. 21. Topological methods in nonlinear functional analysis (pp. 115–123). CrossRefGoogle Scholar
  7. 67.
    Goebel, K., & Kirk, W. A. (1990). Topics in metric fixed point theory. Cambridge: Cambridge University Press. CrossRefMATHGoogle Scholar
  8. 68.
    Goebel, K., & Reich, S. (1984). Uniform convexity, hyperbolic geometry, and nonexpansive mappings. New York: Dekker. MATHGoogle Scholar
  9. 81.
    Kirk, W. A. (1982). Numerical Functional Analysis and Optimization, 4, 371–381. MathSciNetCrossRefMATHGoogle Scholar
  10. 94.
    Lim, T.-C. (1974). Bulletin of the American Mathematical Society, 80, 1123–1126. MathSciNetCrossRefMATHGoogle Scholar
  11. 102.
    Nadler, S. B. Jr. (1969). Pacific Journal of Mathematics, 30, 475–488. MathSciNetCrossRefMATHGoogle Scholar
  12. 107.
    Nussbaum, R. D. (1990). SIAM Journal on Mathematical Analysis, 21, 436–460. MathSciNetCrossRefMATHGoogle Scholar
  13. 114.
    Rakotch, E. (1962). Proceedings of the American Mathematical Society, 13, 459–465. MathSciNetCrossRefMATHGoogle Scholar
  14. 116.
    Reich, S. (1972). Bollettino dell’Unione Matematica Italiana, 5, 26–42. MATHGoogle Scholar
  15. 119.
    Reich, S. (1978). Journal of Mathematical Analysis and Applications, 62, 104–113. MathSciNetCrossRefMATHGoogle Scholar
  16. 123.
    Reich, S. (2005). In Proceedings of CMS’05, computer methods and systems (pp. 9–15). Krakow. Google Scholar
  17. 124.
    Reich, S., & Shafrir, I. (1990). Nonlinear Analysis, 15, 537–558. MathSciNetCrossRefMATHGoogle Scholar
  18. 129.
    Reich, S., & Zaslavski, A. J. (1999). Nonlinear Analysis, 36, 1049–1065. MathSciNetCrossRefMATHGoogle Scholar
  19. 131.
    Reich, S., & Zaslavski, A. J. (2000). Comptes Rendus Mathématiques de L’Académie Des Sciences. La Société Royale du Canada, 22, 118–124. MathSciNetMATHGoogle Scholar
  20. 132.
    Reich, S., & Zaslavski, A. J. (2000). Mathematical and Computer Modelling, 32, 1423–1431. MathSciNetCrossRefMATHGoogle Scholar
  21. 142.
    Reich, S., & Zaslavski, A. J. (2001). In Handbook of metric fixed point theory (pp. 557–575). Dordrecht: Kluwer Academic. CrossRefGoogle Scholar
  22. 144.
    Reich, S., & Zaslavski, A. J. (2002). In Set valued mappings with applications in nonlinear analysis (pp. 411–420). London: Taylor & Francis. Google Scholar
  23. 145.
    Reich, S., & Zaslavski, A. J. (2002). Set-Valued Analysis, 10, 287–296. MathSciNetCrossRefMATHGoogle Scholar
  24. 180.
    Zaslavski, A. J. (2001). Calculus of Variations and Partial Differential Equations, 13, 265–293. MathSciNetCrossRefMATHGoogle Scholar
  25. 182.
    Zaslavski, A. J. (2010). Optimization on metric and normed spaces. New York: Springer. CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Simeon Reich
    • 1
  • Alexander J. Zaslavski
    • 1
  1. 1.Department of MathematicsTechnion-Israel Institute of TechnologyHaifaIsrael

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