Open Access

Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition

Fixed Point Theory and Applications20062006:74503

DOI: 10.1155/FPTA/2006/74503

Received: 2 January 2006

Accepted: 22 February 2006

Published: 1 June 2006

Abstract

We prove the existence of coincidence point and common fixed point for mappings satisfying generalized weak contractive condition. As an application, related results on invariant approximation are derived. Our results generalize various known results in the literature.

[123456789101112131415161718]

Authors’ Affiliations

(1)
Department of Mathematics and Centre for Advanced Studies in Mathematics, Lahore University of Management Sciences

References

  1. Ahmed MA: Common fixed point theorems for weakly compatible mappings. The Rocky Mountain Journal of Mathematics 2003,33(4):1189–1203. 10.1216/rmjm/1181075457MathSciNetView ArticleMATHGoogle Scholar
  2. Alber YaI, Guerre-Delabriere S: Principle of weakly contractive maps in Hilbert spaces. In New Results in Operator Theory and Its Applications, Oper. Theory Adv. Appl.. Volume 98. Edited by: Gohberg I, Lyubich Yu. Birkhäuser, Basel; 1997:7–22.View ArticleGoogle Scholar
  3. Beg I, Abbas M: Fixed points and best approximation in Manger convex metric spaces. Archivum Mathematicum 2005, 41: 389–397.MathSciNetMATHGoogle Scholar
  4. Beg I, Abbas M: Fixed-point theorems for weakly inward multivalued maps on a convex metric space. Demonstratio Mathematica 2006,39(1):149–160.MathSciNetMATHGoogle Scholar
  5. Beg I, Azam A: Common fixed points for commuting and compatible maps. Discussiones Mathematicae, Differential Inclusions 1996,16(2):121–135.MathSciNetMATHGoogle Scholar
  6. Brosowski B: Fixpunktsätze in der Approximationstheorie. Mathematica—Revue d'Analyse Numérique et de Théorie de l'Approximation 1969, 11 (34): 195–220.MathSciNetMATHGoogle Scholar
  7. Hussain N, Khan AR: Common fixed-point results in best approximation theory. Applied Mathematics Letters 2003,16(4):575–580. 10.1016/S0893-9659(03)00039-9MathSciNetView ArticleMATHGoogle Scholar
  8. Jungck G: Common fixed points for commuting and compatible maps on compacta. Proceedings of the American Mathematical Society 1988,103(3):977–983. 10.1090/S0002-9939-1988-0947693-2MathSciNetView ArticleMATHGoogle Scholar
  9. Jungck G, Rhoades BE: Fixed points for set valued functions without continuity. Indian Journal of Pure and Applied Mathematics 1998,29(3):227–238.MathSciNetMATHGoogle Scholar
  10. Kamran T: Coincidence and fixed points for hybrid strict contractions. Journal of Mathematical Analysis and Applications 2004,299(1):235–241. 10.1016/j.jmaa.2004.06.047MathSciNetView ArticleMATHGoogle Scholar
  11. Kaneko H, Sessa S: Fixed point theorems for compatible multi-valued and single-valued mappings. International Journal of Mathematics and Mathematical Sciences 1989,12(2):257–262. 10.1155/S0161171289000293MathSciNetView ArticleMATHGoogle Scholar
  12. Meinardus G: Invarianz bei linearen Approximationen. Archive for Rational Mechanics and Analysis 1963, 14: 301–303.MathSciNetMATHGoogle Scholar
  13. Pant RP: Common fixed points of noncommuting mappings. Journal of Mathematical Analysis and Applications 1994,188(2):436–440. 10.1006/jmaa.1994.1437MathSciNetView ArticleMATHGoogle Scholar
  14. Rhoades BE: Some theorems on weakly contractive maps. Nonlinear Analysis 2001,47(4):2683–2693. 10.1016/S0362-546X(01)00388-1MathSciNetView ArticleMATHGoogle Scholar
  15. Sessa S: On a weak commutativity condition of mappings in fixed point considerations. Institut Mathématique. Publications. Nouvelle Série 1982, 32(46): 149–153.MathSciNetMATHGoogle Scholar
  16. Shahzad N: Invariant approximations and R-subweakly commuting maps. Journal of Mathematical Analysis and Applications 2001,257(1):39–45. 10.1006/jmaa.2000.7274MathSciNetView ArticleMATHGoogle Scholar
  17. Shahzad N: Generalized I- nonexpansive maps and best approximations in Banach spaces. Demonstratio Mathematica 2004,37(3):597–600.MathSciNetMATHGoogle Scholar
  18. Singh SP: An application of a fixed-point theorem to approximation theory. Journal of Approximation Theory 1979,25(1):89–90. 10.1016/0021-9045(79)90036-4MathSciNetView ArticleMATHGoogle Scholar

Copyright

© I. Beg and M. Abbas. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.