Open Access

Fixed point variational solutions for uniformly continuous pseudocontractions in Banach spaces

Fixed Point Theory and Applications20062006:69758

DOI: 10.1155/FPTA/2006/69758

Received: 27 June 2005

Accepted: 28 November 2005

Published: 1 March 2006


Let be a reflexive Banach space with a uniformly Gâteaux differentiable norm, let be a nonempty closed convex subset of , and let be a uniformly continuous pseudocontraction. If is any contraction map on and if every nonempty closed convex and bounded subset of has the fixed point property for nonexpansive self-mappings, then it is shown, under appropriate conditions on the sequences of real numbers , , that the iteration process , , , strongly converges to the fixed point of , which is the unique solution of some variational inequality, provided that is bounded.


Authors’ Affiliations

Department of Mathematics, Statistics, & Computer Science, University of Port Harcourt


  1. Deimling K: Zeros of accretive operators. Manuscripta Mathematica 1974,13(4):365–374. 10.1007/BF01171148MathSciNetView ArticleMATHGoogle Scholar
  2. Halpern B: Fixed points of nonexpanding maps. Bulletin of the American Mathematical Society 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0MathSciNetView ArticleMATHGoogle Scholar
  3. Kato T: Nonlinear semigroups and evolution equations. Journal of the Mathematical Society of Japan 1967, 19: 508–520. 10.2969/jmsj/01940508MathSciNetView ArticleMATHGoogle Scholar
  4. Lim T-C: On characterizations of Meir-Keeler contractive maps. Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 2001,46(1):113–120.View ArticleMathSciNetMATHGoogle Scholar
  5. Martin RH Jr.: Differential equations on closed subsets of a Banach space. Transactions of the American Mathematical Society 1973, 179: 399–414.MathSciNetView ArticleMATHGoogle Scholar
  6. Morales CH: On the fixed-point theory for local -pseudocontractions. Proceedings of the American Mathematical Society 1981,81(1):71–74.MathSciNetMATHGoogle Scholar
  7. Morales CH, Jung JS: Convergence of paths for pseudo-contractive mappings in Banach spaces. Proceedings of the American Mathematical Society 2000,128(11):3411–3419. 10.1090/S0002-9939-00-05573-8MathSciNetView ArticleMATHGoogle Scholar
  8. Moudafi A: Viscosity approximation methods for fixed-points problems. Journal of Mathematical Analysis and Applications 2000,241(1):46–55. 10.1006/jmaa.1999.6615MathSciNetView ArticleMATHGoogle Scholar
  9. Schu J: Approximating fixed points of Lipschitzian pseudocontractive mappings. Houston Journal of Mathematics 1993,19(1):107–115.MathSciNetMATHGoogle Scholar
  10. Udomene A: Path convergence, approximation of fixed points and variational solutions of pseudocontractions in Banach spaces. submitted to Nonlinear Analysis, TMA
  11. Xu H-K: Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society. Second Series 2002,66(1):240–256. 10.1112/S0024610702003332MathSciNetView ArticleMATHGoogle Scholar
  12. Xu H-K: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004,298(1):279–291. 10.1016/j.jmaa.2004.04.059MathSciNetView ArticleMATHGoogle Scholar


© Aniefiok Udomene. 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.