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COMMON FIXED POINT AND INVARIANT APPROXIMATION IN MENGER CONVEX METRIC SPACES

  • Hussain, Nawab (DEPARTMENT OF MATHEMATICS KING ABDUL AZIZ UNIVERSITY) ;
  • Abbas, Mujahid (DEPARTMENT OF MATHEMATICS INDIANA UNIVERSITY, CENTRE FOR ADVANCED STUDIES IN MATHEMATICS AND DEPARTMENT OF MATHEMATICS LAHORE UNIVERSITY OF MANAGEMENT SCIENCES) ;
  • Kim, Jong-Kyu (DEPARTMENT OF MATHEMATICS, EDUCATION KYUNGNAM UNIVERSITY)
  • Published : 2008.11.30

Abstract

Necessary conditions for the existence of common fixed points for noncommuting mappings satisfying generalized contractive conditions in a Menger convex metric space are obtained. As an application, related results on best approximation are derived. Our results generalize various well known results.

References

  1. M. A. Al-Thagafi, Common fixed points and best approximation, J. Approx. Theory 85 (1996), no. 3, 318-323 https://doi.org/10.1006/jath.1996.0045
  2. M. A. Al-Thagafi and N. Shahzad, Noncommuting selfmaps and invariant approximations, Nonlinear Anal. 64 (2006), no. 12, 2778-2786 https://doi.org/10.1016/j.na.2005.09.015
  3. N. Aronszajn and P. Panitchpakdi, Extension of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405-439 https://doi.org/10.2140/pjm.1956.6.405
  4. I. Beg and M. Abbas, Fixed points and best approximation in Menger convex metric spaces, Arch. Math. (Brno) 41 (2005), no. 4, 389-397
  5. I. Beg and M. Abbas, Inequalities and fixed points in Menger convex metric spaces, Fixed point theory and applications. Vol. 7, 5-15, Nova Sci. Publ., New York, 2007
  6. I. Beg, D. R. Sahu, and S. D. Diwan, Approximation of fixed points of uniformly Rsubweakly commuting mappings, J. Math. Anal. Appl. 324 (2006), no. 2, 1105-1114 https://doi.org/10.1016/j.jmaa.2006.01.024
  7. A. Berard, Characterizations of metric spaces by the use of their midsets: Intervals, Fund. Math. 73 (1971/72), no. 1, 1-7 https://doi.org/10.4064/fm-73-1-1-7
  8. L. M. Blumenthal, Distance Geometry, Clarendon Press, Oxford, 1953
  9. R. Chugh and S. Kumar, Common fixed points for weakly compatible maps, Proc. Indian Acad. Sci. Math. Sci. 111 (2001), no. 2, 241-247 https://doi.org/10.1007/BF02829594
  10. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, 1990
  11. M. D. Guay, K. L. Singh, and J. H. M. Whitfield, Fixed point theorems for nonexpansive mappings in convex metric spaces, Nonlinear analysis and applications (St. Johns, Nfld., 1981), pp. 179-189, Lecture Notes in Pure and Appl. Math., 80, Dekker, New York, 1982
  12. N. Hussain, On strong convergence to common fixed points of uniformly $C_{q}$-commuting maps, submitted
  13. N. Hussain and G. Jungck, Common fixed point and invariant approximation results for noncommuting generalized (f, g)-nonexpansive maps, J. Math. Anal. Appl. 321 (2006), no. 2, 851-861 https://doi.org/10.1016/j.jmaa.2005.08.045
  14. N. Hussain, D. O'Regan, and R. P. Agarwal, Common fixed point and invariant approximation results on non-starshaped domains, Georgian Math. J. 12 (2005), no. 4, 659-669
  15. N. Hussain and B. E. Rhoades, $C_{q}$-commuting maps and invariant approximations, Fixed Point Theory Appl. 2006, Art. ID 24543, 9 pp https://doi.org/10.1155/FPTA/2006/24543
  16. R. Khalil, Extreme points of the unit ball of Banach spaces, Math. Rep. Toyama Univ. 4 (1981), 41-45
  17. R. Khalil, Best approximation in metric spaces, Proc. Amer. Math. Soc. 103 (1988), no. 2, 579-586
  18. G. Meinardus, Invarianz bei linearen Approximationen, Arch. Rational Mech. Anal. 14 (1963), 301-303 https://doi.org/10.1007/BF00250708
  19. K. Menger, Untersuchungen uber allgemeine Metrik, Math. Ann. 100 (1928), no. 1, 75-163 https://doi.org/10.1007/BF01448840
  20. D. O'Regan and N. Hussain, Generalized I-contractions and point wise R-subweakly commuting maps, Acta Math. Sinica., (2007) (in press) https://doi.org/10.1007/s10114-007-0935-7
  21. N. Shahzad, Invariant approximations and R-subweakly commuting maps, J. Math. Anal. Appl. 257 (2001), no. 1, 39-45 https://doi.org/10.1006/jmaa.2000.7274
  22. S. P. Singh, An application of a fixed-point theorem to approximation theory, J. Approx. Theory 25 (1979), no. 1, 89-90 https://doi.org/10.1016/0021-9045(79)90036-4

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  2. Fixed Point Approximation of Generalized Nonexpansive Mappings in Hyperbolic Spaces vol.2015, 2015, https://doi.org/10.1155/2015/368204
  3. Common fixed points for some generalized contraction pairs in convex metric spaces vol.2014, pp.1, 2014, https://doi.org/10.1186/1687-1812-2014-98