Abstract and Applied Analysis

General Split Feasibility Problems in Hilbert Spaces

Mohammad Eslamian and Abdul Latif

Full-text: Open access

Abstract

Introducing a general split feasibility problem in the setting of infinite-dimensional Hilbert spaces, we prove that the sequence generated by the purposed new algorithm converges strongly to a solution of the general split feasibility problem. Our results extend and improve some recent known results.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2012), Article ID 805104, 6 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393442754

Digital Object Identifier
doi:10.1155/2013/805104

Mathematical Reviews number (MathSciNet)
MR3034910

Zentralblatt MATH identifier
1266.65091

Citation

Eslamian, Mohammad; Latif, Abdul. General Split Feasibility Problems in Hilbert Spaces. Abstr. Appl. Anal. 2013, Special Issue (2012), Article ID 805104, 6 pages. doi:10.1155/2013/805104. https://projecteuclid.org/euclid.aaa/1393442754


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References

  • P. L. Combettes, “The convex feasibility problem in image recovery,” in Advances in Imaging and Electron Physics, P. Hawkes, Ed., vol. 95, pp. 155–270, Academic Press, New York, NY, USA, 1996.
  • 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.
  • L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 4, pp. 2116–2125, 2012.
  • L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “An extragradient method for solving split feasibility and fixed point problems,” Computers & Mathematics with Applications, vol. 64, no. 4, pp. 633–642, 2012.
  • Y. Censor, T. Elfving, N. Kopf, and T. Bortfeld, “The multiple-sets split feasibility problem and its applications for inverse problems,” Inverse Problems, vol. 21, no. 6, pp. 2071–2084, 2005.
  • F. Wang and H.-K. Xu, “Cyclic algorithms for split feasibility problems in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 12, pp. 4105–4111, 2011.
  • H.-K. Xu, “Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces,” Inverse Problems, vol. 26, no. 10, Article ID 105018, p. 17, 2010.
  • Y. Censor, A. Motova, and A. Segal, “Perturbed projections and subgradient projections for the multiple-sets split feasibility problem,” Journal of Mathematical Analysis and Applications, vol. 327, no. 2, pp. 1244–1256, 2007.
  • 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.
  • H. H. Bauschke and J. M. Borwein, “On projection algorithms for solving convex feasibility problems,” SIAM Review, vol. 38, no. 3, pp. 367–426, 1996.
  • Y. Alber and D. Butnariu, “Convergence of Bregman projection methods for solving consistent convex feasibility problems in reflexive Banach spaces,” Journal of Optimization Theory and Applications, vol. 92, no. 1, pp. 33–61, 1997.
  • 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.
  • 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.
  • 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.
  • Y. Dang and Y. Gao, “The strong convergence of a KM-CQ-like algorithm for a split feasibility problem,” Inverse Problems, vol. 27, article 015007, p. 9, 2011.
  • F. Wang and H.-K. Xu, “Approximating curve and strong convergence of the $CQ$ algorithm for the split feasibility problem,” Journal of Inequalities and Applications, vol. 2010, Article ID 102085, 13 pages, 2010.
  • L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Mann type iterative methods for finding a common solution of split feasibility and fixed point problems,” Positivity, vol. 16, no. 3, pp. 471–495, 2012.
  • A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.
  • S.-S. Chang, J. K. Kim, and X. R. Wang, “Modified block iterative algorithm for solving convex feasibility problems in Banach spaces,” Journal of Inequalities and Applications, vol. 2010, Article ID 869684, 14 pages, 2010.
  • H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002.
  • 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.
  • K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28, Cambridge University Press, Cambridge, UK, 1990.