Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2012, Special Issue (2012), Article ID 902437, 12 pages.

Approximation of Common Fixed Points of a Sequence of Nearly Nonexpansive Mappings and Solutions of Variational Inequality Problems

D. R. Sahu, Shin Min Kang, and Vidya Sagar

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Abstract

We introduce an explicit iterative scheme for computing a common fixed point of a sequence of nearly nonexpansive mappings defined on a closed convex subset of a real Hilbert space which is also a solution of a variational inequality problem. We prove a strong convergence theorem for a sequence generated by the considered iterative scheme under suitable conditions. Our strong convergence theorem extends and improves several corresponding results in the context of nearly nonexpansive mappings.

Article information

Source
J. Appl. Math., Volume 2012, Special Issue (2012), Article ID 902437, 12 pages.

Dates
First available in Project Euclid: 3 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.jam/1357180326

Digital Object Identifier
doi:10.1155/2012/902437

Mathematical Reviews number (MathSciNet)
MR2948120

Zentralblatt MATH identifier
1251.65084

Citation

Sahu, D. R.; Kang, Shin Min; Sagar, Vidya. Approximation of Common Fixed Points of a Sequence of Nearly Nonexpansive Mappings and Solutions of Variational Inequality Problems. J. Appl. Math. 2012, Special Issue (2012), Article ID 902437, 12 pages. doi:10.1155/2012/902437. https://projecteuclid.org/euclid.jam/1357180326


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References

  • R. P. Agarwal, D. O'Regan, and D. R. Sahu, Fixed Point Theory for Lipschitzian-Type Mappings with Applications, Topological Fixed Point Theory and Its Applications, Springer, New York, NY, USA, 2009.
  • D. R. Sahu, “Fixed points of demicontinuous nearly Lipschitzian mappings in Banach spaces,” Commentationes Mathematicae Universitatis Carolinae, vol. 46, no. 4, pp. 653–666, 2005.
  • A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.
  • H. K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.
  • G. Marino and H. K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006.
  • Y. Liu, “A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert spaces,” Nonlinear Analysis, Theory, Methods and Applications, vol. 71, no. 10, pp. 4852–4861, 2009.
  • X. Qin, M. Shang, and S. M. Kang, “Strong convergence theorems of modified Mann iterative process for strict pseudo-contractions in Hilbert spaces,” Nonlinear Analysis, Theory, Methods and Applications, vol. 70, no. 3, pp. 1257–1264, 2009.
  • S. Wang, “Convergence and weaker control conditions for hybrid iterative algorithms,” Fixed Point Theory and Applications, vol. 2011, no. 1, article 3, 14 pages, 2011.
  • S. Wang, “Two general algorithms for computing fixed points of nonexpansive mappings in Banach spaces,” Journal of Applied Mathematics, vol. 2012, Article ID 658905, 11 pages, 2012.
  • S. Wang and C. Hu, “Two new iterative methods for a countable family of nonexpansive mappings in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 852030, 12 pages, 2010.
  • M. Tian, “A general iterative algorithm for nonexpansive mappings in Hilbert spaces,” Nonlinear Analysis, Theory, Methods and Applications, vol. 73, no. 3, pp. 689–694, 2010.
  • I. Yamada, “The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), D. Butnariu, Y. Censor, and S. Reich, Eds., vol. 8, pp. 473–504, Studies in Computational Mathematics, Amsterdam, The Netherlands, 2001.
  • L. C. Ceng, Q. H. Ansari, and J. C. Yao, “Some iterative methods for finding fixed points and for solving constrained convex minimization problems,” Nonlinear Analysis, Theory, Methods and Applications, vol. 74, no. 16, pp. 5286–5302, 2011.
  • K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,” Nonlinear Analysis, Theory, Methods and Applications, vol. 67, no. 8, pp. 2350–2360, 2007.
  • N. C. Wong, D. R. Sahu, and J. C. Yao, “A generalized hybrid steepest-descent method for variational inequalities in Banach spaces,” Fixed Point Theory and Applications, vol. 2011, Article ID 754702, 28 pages, 2011.
  • K. Goebel and W. A. Kirk, Topics on Metric Fixed Point Theory, Cambridge University Press, Cambridge, UK, 1990.
  • H. K. Xu and T. H. Kim, “Convergence of hybrid steepest-descent methods for variational inequalities,” Journal of Optimization Theory and Applications, vol. 119, no. 1, pp. 185–201, 2003.
  • N. C. Wong, D. R. Sahu, and J. C. Yao, “Solving variational inequalities involving nonexpansive type mappings,” Nonlinear Analysis, Theory, Methods and Applications, vol. 69, no. 12, pp. 4732–4753, 2008.