Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2012, Special Issue (2012), Article ID 438023, 12 pages.

Strong Convergence of the Viscosity Approximation Process for the Split Common Fixed-Point Problem of Quasi-Nonexpansive Mappings

Jing Zhao and Songnian He

Full-text: Open access

Abstract

Very recently, Moudafi (2011) introduced an algorithm with weak convergence for the split common fixed-point problem. In this paper, we will continue to consider the split common fixed-point problem. We discuss the strong convergence of the viscosity approximation method for solving the split common fixed-point problem for the class of quasi-nonexpansive mappings in Hilbert spaces. Our results improve and extend the corresponding results announced by many others.

Article information

Source
J. Appl. Math., Volume 2012, Special Issue (2012), Article ID 438023, 12 pages.

Dates
First available in Project Euclid: 3 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.jam/1357180296

Digital Object Identifier
doi:10.1155/2012/438023

Mathematical Reviews number (MathSciNet)
MR2904520

Zentralblatt MATH identifier
1319.47065

Citation

Zhao, Jing; He, Songnian. Strong Convergence of the Viscosity Approximation Process for the Split Common Fixed-Point Problem of Quasi-Nonexpansive Mappings. J. Appl. Math. 2012, Special Issue (2012), Article ID 438023, 12 pages. doi:10.1155/2012/438023. https://projecteuclid.org/euclid.jam/1357180296


Export citation

References

  • Y. Censor and T. Elfving, “A multiprojection algorithm using Bregman projections in a product space,” Numerical Algorithms, vol. 8, no. 2-4, pp. 221–239, 1994.
  • C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,” Inverse Problems, vol. 18, no. 2, pp. 441–453, 2002.
  • C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems, vol. 20, no. 1, pp. 103–120, 2004.
  • P.-E. Maingé, “The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces,” Computers & Mathematics with Applications, vol. 59, no. 1, pp. 74–79, 2010.
  • B. Qu and N. Xiu, “A note on the CQ algorithm for the split feasibility problem,” Inverse Problems, vol. 21, no. 5, pp. 1655–1665, 2005.
  • H.-K. Xu, “A variable Krasnosel'skii-Mann algorithm and the multiple-set split feasibility problem,” Inverse Problems, vol. 22, no. 6, pp. 2021–2034, 2006.
  • Q. Yang, “The relaxed CQ algorithm solving the split feasibility problem,” Inverse Problems, vol. 20, no. 4, pp. 1261–1266, 2004.
  • Q. Yang and J. Zhao, “Generalized KM theorems and their applications,” Inverse Problems, vol. 22, no. 3, pp. 833–844, 2006.
  • Y. Yao, W. Jigang, and Y.-C. Liou, “Regularized methods for the split feasibility problem,” Abstract and Applied Analysis, vol. 2012, Article ID 140679, 13 pages, 2012.
  • H.-K. Xu, “Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces,” Inverse Problems, vol. 26, no. 10, Article ID 105018, 2010.
  • Y. Yao, Y.-C. Liou, and S. M. Kang, “Two-step projection methods for a system of variational inequality problems in Banach spaces,” Journal of Global Optimization. In press.
  • Y. Yao, R. Chen, and Y.-C. Liou, “A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1506–1515, 2012.
  • Y. Yao, Y.-J. Cho, and Y.-C. Liou, “Hierarchical convergence of an implicit doublenet algorithm for nonexpansive semigroups and variational inequalities,” Fixed Point Theory and Applications, vol. 2011, article 101, 2011.
  • J. Zhao and Q. Yang, “Several solution methods for the split feasibility problem,” Inverse Problems, vol. 21, no. 5, pp. 1791–1799, 2005.
  • Y. Censor and A. Segal, “The split common fixed point problem for directed operators,” Journal of Convex Analysis, vol. 16, no. 2, pp. 587–600, 2009.
  • S. Măruşter and C. Popirlan, “On the Mann-type iteration and the convex feasibility problem,” Journal of Computational and Applied Mathematics, vol. 212, no. 2, pp. 390–396, 2008.
  • A. Moudafi, “A note on the split common fixed-point problem for quasi-nonexpansive operators,” Nonlinear Analysis, vol. 74, no. 12, pp. 4083–4087, 2011.
  • P.-E. Maingé, “Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization,” Set-Valued Analysis, vol. 16, no. 7-8, pp. 899–912, 2008.