Fixed points and best approximation in Menger convex metric spaces

Ismat Beg; Mujahid Abbas

Archivum Mathematicum (2005)

  • Volume: 041, Issue: 4, page 389-397
  • ISSN: 0044-8753

Abstract

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We obtain necessary conditions for the existence of fixed point and approximate fixed point of nonexpansive and quasi nonexpansive maps defined on a compact convex subset of a uniformly convex complete metric space. We obtain results on best approximation as a fixed point in a strictly convex metric space.

How to cite

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Beg, Ismat, and Abbas, Mujahid. "Fixed points and best approximation in Menger convex metric spaces." Archivum Mathematicum 041.4 (2005): 389-397. <http://eudml.org/doc/249509>.

@article{Beg2005,
abstract = {We obtain necessary conditions for the existence of fixed point and approximate fixed point of nonexpansive and quasi nonexpansive maps defined on a compact convex subset of a uniformly convex complete metric space. We obtain results on best approximation as a fixed point in a strictly convex metric space.},
author = {Beg, Ismat, Abbas, Mujahid},
journal = {Archivum Mathematicum},
keywords = {fixed point; convex metric space; uniformly convex metric space; strictly convex metric space; best approximation; nonexpansive map; uniformly convex metric space; best approximation; nonexpansive map},
language = {eng},
number = {4},
pages = {389-397},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Fixed points and best approximation in Menger convex metric spaces},
url = {http://eudml.org/doc/249509},
volume = {041},
year = {2005},
}

TY - JOUR
AU - Beg, Ismat
AU - Abbas, Mujahid
TI - Fixed points and best approximation in Menger convex metric spaces
JO - Archivum Mathematicum
PY - 2005
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 041
IS - 4
SP - 389
EP - 397
AB - We obtain necessary conditions for the existence of fixed point and approximate fixed point of nonexpansive and quasi nonexpansive maps defined on a compact convex subset of a uniformly convex complete metric space. We obtain results on best approximation as a fixed point in a strictly convex metric space.
LA - eng
KW - fixed point; convex metric space; uniformly convex metric space; strictly convex metric space; best approximation; nonexpansive map; uniformly convex metric space; best approximation; nonexpansive map
UR - http://eudml.org/doc/249509
ER -

References

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  1. Aksoy A. G., Khamsi M. A., Nonstandard methods in fixed point theory, Springer, New York, Berlin, 1990. (1990) Zbl0713.47050MR1066202
  2. Aronszajn N., Panitchpakdi P., Extension of uniformly continuous transformations and hyper convex metric spaces, Pacific J. Math. 6 (1956), 405–439. (1956) MR0084762
  3. Ayerbe Toledano J. M., Dominguez Benavides T., Lopez Acedo G., Measures of noncompactness in metric fixed point theory, Birkhauser, Basel, 1997. (1997) Zbl0885.47021MR1483889
  4. Beg I., Azam A., Fixed points of asymptotically regular multivalued mappings, J. Austral. Math. Soc. Ser. A 53(3) (1992), 313–326. (1992) Zbl0765.54036MR1187851
  5. Beg I., Azam A., Common fixed points for commuting and compatible maps, Discuss. Math. Differential Incl. 16 (1996), 121–135. (1996) Zbl0912.47033MR1646650
  6. Berard A., Characterization of metric spaces by the use of their mid sets intervals, Fund. Math. 73 (1971), 1–7. (1971) MR0295300
  7. Blumenthal L. M., Distance Geometry, Clarendon Press, Oxford, 1953. (1953) Zbl0050.38502MR0054981
  8. Browder F. E., Nonexpansive nonlinear operators in Banach spaces, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041–1044. (1965) MR0187120
  9. Dotson W. G., On fixed points of nonexpansive mappings in non convex sets, Proc. Amer. Math. Soc. 38 (1973), 155–156. (1973) MR0313894
  10. Goeble K., Kirk W. A., Topics in metric fixed point theory, Cambridge Stud. Adv. Math. 28, Cambridge University Press, London, 1990. (1990) MR1074005
  11. Goeble K., Reich S., Uniform convexity, hyperolic geometry, and nonexpansive mappings, Marcel Dekker, Inc. New York and Basel (1984). (1984) MR0744194
  12. Gohde D., Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1995), 251–258. (1995) MR0190718
  13. Habiniak L., Fixed point theorem and invarient approximation, J. Approx. Theory 56 (1984), 241–244. (1984) 
  14. Hadzic O., Almost fixed point and best approximation theorems in H-Spaces, Bull. Austral. Math. Soc. 53 (1996), 447–454. (1996) MR1388593
  15. Khalil R., Extreme points of the unit ball of Banach spaces, Math. Rep. Toyama Univ. 4 (1981), 41–45. (1981) Zbl0473.46012MR0627961
  16. Khalil R., Best approximation in metric spaces, Proc. Amer. Math. Soc. 103 (1988), 579–586. (1988) Zbl0652.51019MR0943087
  17. Kirk W. A., A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006. (1965) Zbl0141.32402MR0189009
  18. Menger K., Untersuchungen über allegemeine Metrik, Math. Ann. 100 (1928), 75–63. (1928) MR1512479
  19. Prus B., Smarzewski R. S., Strongly unique best approximation and centers in uniformly convex spaces, J. Math. Anal. Appl. 121 (1978), 85–92. (1978) MR0869515
  20. Veeramani P., On some fixed point theorems on uniformly convex Banach spaces, J. Math. Anal. Appl. 167 (1992), 160–166. (1992) Zbl0780.47047MR1165265

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