Abstract and Applied Analysis

Iterative Schemes for Fixed Point Computation of Nonexpansive Mappings

Rudong Chen

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Abstract

Fixed point (especially, the minimum norm fixed point) computation is an interesting topic due to its practical applications in natural science. The purpose of the paper is devoted to finding the common fixed points of an infinite family of nonexpansive mappings. We introduce an iterative algorithm and prove that suggested scheme converges strongly to the common fixed points of an infinite family of nonexpansive mappings under some mild conditions. As a special case, we can find the minimum norm common fixed point of an infinite family of nonexpansive mappings.

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 469270, 12 pages.

Dates
First available in Project Euclid: 1 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1364845133

Digital Object Identifier
doi:10.1155/2012/469270

Mathematical Reviews number (MathSciNet)
MR2910734

Zentralblatt MATH identifier
1237.65054

Citation

Chen, Rudong. Iterative Schemes for Fixed Point Computation of Nonexpansive Mappings. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 469270, 12 pages. doi:10.1155/2012/469270. https://projecteuclid.org/euclid.aaa/1364845133


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References

  • A. Sabharwal and L. C. Potter, “Convexly constrained linear inverse problems: iterative least-squares and regularization,” IEEE Transactions on Signal Processing, vol. 46, no. 9, pp. 2345–2352, 1998.
  • Y.-L. Cui and X. Liu, “Notes on Browder's and Halpern's methods for nonexpansive mappings,” Fixed Point Theory, vol. 10, no. 1, pp. 89–98, 2009.
  • Y. Yao and H.-K. Xu, “Iterative methods for finding minimum-norm fixed points of nonexpansive mappings with applications,” Optimization, vol. 60, no. 6, pp. 645–658, 2011.
  • Y. Yao, R. Chen, and H.-K. Xu, “Schemes for finding minimum-norm solutions of variational inequalities,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 7-8, pp. 3447–3456, 2010.
  • H. H. Bauschke, “The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 202, no. 1, pp. 150–159, 1996.
  • F. E. Browder, “Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces,” Archive for Rational Mechanics and Analysis, vol. 24, pp. 82–90, 1967.
  • F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967.
  • H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.
  • S. A. Hirstoaga, “Iterative selection methods for common fixed point problems,” Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 1020–1035, 2006.
  • P.-E. Mainge, “Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 469–479, 2007.
  • Y. Yao, R. Chen, and J.-C. Yao, “Strong convergence and certain control conditions for modified Mann iteration,” Nonlinear Analysis. Theory, Methods & Applications, vol. 68, no. 6, pp. 1687–1693, 2008.
  • M. Kikkawa and W. Takahashi, “Approximating fixed points of infinite nonexpansive mappings by the hybrid method,” Journal of Optimization Theory and Applications, vol. 117, no. 1, pp. 93–101, 2003.
  • H. Zegeye and N. Shahzad, “Viscosity approximation methods for a common fixed point of finite family of nonexpansive mappings,” Applied Mathematics and Computation, vol. 191, no. 1, pp. 155–163, 2007.
  • Y. Yao, Y. C. Liou, and G. Marino, “Strong convergence of two iterative algorithms for nonexpansive mappings in Hilbert spaces,” Fixed Point Theory and Applications, Article ID 279058, 7 pages, 2009.
  • Y. Yao, Y. J. Cho, and Y.-C. Liou, “Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems,” European Journal of Operational Research, vol. 212, no. 2, pp. 242–250, 2011.
  • S.-S. Chang, “Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1402–1416, 2006.
  • M. A. Noor, “Some developments in general variational inequalities,” Applied Mathematics and Computation, vol. 152, no. 1, pp. 199–277, 2004.
  • M. A. Noor, K. I. Noor, and E. Al-Said, “Iterative methods for solving nonconvex equilibrium variational inequalities,” Applied Mathematics & Information Sciences, vol. 6, no. 1, pp. 65–69, 2012.
  • M. A. Noor, “Extended general variational inequalities,” Applied Mathematics Letters, vol. 22, no. 2, pp. 182–186, 2009.
  • M. A. Noor, “Some aspects of extended general variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID 303569, 16 pages, 2012.
  • P. K. F. Kuhfittig, “Common fixed points of nonexpansive mappings by iteration,” Pacific Journal of Mathematics, vol. 97, no. 1, pp. 137–139, 1981.
  • K. Shimoji and W. Takahashi, “Strong convergence to common fixed points of infinite nonexpansive mappings and applications,” Taiwanese Journal of Mathematics, vol. 5, no. 2, pp. 387–404, 2001.
  • Y. Yao, Y.-C. Liou, and J.-C. Yao, “Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings,” Fixed Point Theory and Applications, Article ID 64363, 12 pages, 2007.
  • K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990.
  • T. Suzuki, “Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces,” Fixed Point Theory and Applications, no. 1, pp. 103–123, 2005.
  • H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002.