DOI QR코드

DOI QR Code

FIXED POINTS OF BSC-SEQUENCES

  • Received : 2016.12.01
  • Accepted : 2017.04.04
  • Published : 2017.10.31

Abstract

We call a sequence $(T_n)_n$ of bounded operators on a Banach space X, BSC-Sequence if it is a Cauchy sequence in the strong operator topology and is uniformly bounded below. We determine conditions under which such sequences has a fixed point.

Keywords

References

  1. R. Agarwal, M. Meehan, and D. O'Regan, Fixed point theory Appl., Cambridge Tracts in Mathematics, Cambridge Univ. Press, Cambridge, 2004.
  2. S. Banach, Sur les operations dans les ensembles abstracts ET leur applications aux equations integrals, Fund. Math., 3 (1922), 133-181. https://doi.org/10.4064/fm-3-1-133-181
  3. L. Barbet and K. Nachi, Sequences of contractions and convergence of fixed points, Ninth International Conference Zaragoza-Pau on Applied Mathematics and Statistics, 51-58, Monogr. Semin. Mat. Garca Galdeano, 33, Prensas Univ. Zaragoza, Zaragoza, 2006.
  4. F. Bonsall, Lectures on some fixed point theorem, Tata Inst. Fundam. Res. Stud. Math, Bombay, 1962.
  5. L. Brouwer, Uber Abbildungen von Mannigfaltigkeiten, Math. Ann. 70 (1912), 97-115.
  6. L. J. Ciric, Quasi-contraction in Banach spaces, Publ. Inst. Math. 21(35) (1977), 41-48.
  7. G. Costakis and A. Manoussos, J-class weighted shifts on the space of bounded sequences of complex numbers, Integral Equations Operator Theory 62 (2008), no. 2, 149-158. https://doi.org/10.1007/s00020-008-1621-6
  8. G. Costakis and A. Manoussos, J-class operators and hypercyclicity, J. Operator Theory 67 (2012), no. 1, 101- 119.
  9. M. Erturk and V. Karakaya, A note on n-tuplet fixed point theorems for contractive type mappings in partially ordered metric spaces, J. Inequal. Appl. 2013 (2013), DOI:10.1186/1029-242X-2013-196.
  10. N. S. Feldman, Perturbations of hypercyclic vectors, J. Math. Anal. Appl. 273 (2002), no. 1, 67-74. https://doi.org/10.1016/S0022-247X(02)00207-X
  11. G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), no. 2, 229-269. https://doi.org/10.1016/0022-1236(91)90078-J
  12. K. Grosse and G. Erdmann, Universal families and hypercyclic vectors, Bull. Amer. Math. Soc. 36 (1999), no. 3, 345-381. https://doi.org/10.1090/S0273-0979-99-00788-0
  13. S. Kakutani, A generalization of Tychonoff's fixed point theorem, Duke Math. J. 8 (1968), 457-459.
  14. R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968), 71-78.
  15. C. Kitai, Invariant closed sets for linear operators, Dissertation, Univ. of Toronto, 1982.
  16. A. Manoussos, Coarse topological transitivity on open cones and coarsely J-class and D-class operators, J. Math. Anal. Appl. 413 (2014), no. 2, 715-726. https://doi.org/10.1016/j.jmaa.2013.12.038
  17. S. Mishral, S. L. Singh, and S. Stofile, Stability of common fixed points in uniform spaces, Fixed Point Theory Appl. 2011 (2011), DOI:10.1186/1687-1812-2011-37.
  18. S. B. Nadler, Sequence of contractions and fixed point, Pacific J. Math. 27 (1968), no. 3, 579-586. https://doi.org/10.2140/pjm.1968.27.579
  19. A. B. Nasseri, J-class operators on certain Banach spaces, Dissertation, TU Dortmund, 2013.
  20. H. Poincare, Surless courbes define barles equations differentiate less, J. de Math. 2 (1886), 54-65.
  21. B. E. Rhoades, A fixed point theorem for generalized Metric space, Internat. J. Math. Sci. 19 (1996), no. 3, 457-460. https://doi.org/10.1155/S0161171296000658
  22. B. Sadovskii, On a fixed point principle, Funktsional. Anal. i Prilozen. 1 (1967), no. 2, 74-76.
  23. J. H. Shapiro, Notes on the dynamics of linear operators, www.joelshapiro. org/Pubvit/Downloads/LinDynamics/LynDynamics.html (2016).
  24. S. Sessa and B. Fisher, On common fixed points of weakly commuting mappings and set-valued mappings, internat. J. Math. & Math. Sci. 9 (1986), no. 2, 323-329. https://doi.org/10.1155/S0161171286000406
  25. G. Soleimani Rad, S. Shukla, and H. Rahimi, Some relations between n-tuple fixed point and fixed point results, RACSAM (2015); Doi:10.1007/s13398-014-0196-0.
  26. A. Soliman, Results on n-tupled fixed points in metric spaces with uniform normal structure, Fixed point Theory Appl. 2014 (2014), 168, 16 pp.; DOI:10.1186/1687-1812- 2014-168.