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Implicit and explicit algorithms for solving the split feasibility problem

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Abstract

Very recently, Dang and Gao (Inverse Probl 27:015007, 2011) introduced a KM-CQ algorithm with strong convergence for the split feasibility problem. In this paper, we will continue to consider the split feasibility problem. We present two algorithms. First, we introduce an implicit algorithm. Consequently, by discretizing the continuous implicit algorithm, we obtain an explicit algorithm. Under some weaker conditions, we show the strong convergence of presented algorithms to some solution of the split feasibility problem which solves some special variational inequality. As special cases, we obtain two algorithms which converge strongly to the minimum norm solution of the split feasibility problem. Results obtained in this paper include the corresponding results of Dang and Gao (2011) and extend a recent result of Wang and Xu (J Inequalities Appl 2010, doi:10.1155/2010/102085).

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References

  1. Censor Y., Elfving T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221–239 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Byrne C.: Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Probl. 18, 441–453 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Censor Y., Bortfeld T., Martin B., Trofimov A.: A unified approach for inversion problems in intensity modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)

    Article  Google Scholar 

  4. Censor Y., Elfving T., Kopf N., Bortfeld T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 21, 2071–2084 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Censor Y., Motova A., Segal A.: Perturbed projections and subgradient projections for the multiple-sets split feasibility problem. J. Math. Anal. Appl. 327, 1244–1256 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Xu H.K.: A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 22, 2021–2034 (2006)

    Article  MATH  Google Scholar 

  7. Wang, F., Xu, H.K.: Cyclic algorithms for split feasibility problems in Hilbert spaces. Nonlinear Anal. (2011). doi:10.1016/j.na.2011.03.044

  8. Lopez G., Martin V., Xu H.K.: Iterative algorithms for the multiple-sets split feasibility problem. In: Censor, Y., Jiang, M., Wang, G. (eds) Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems, pp. 243–279. Medical Physics Publishing, Madison (2009)

    Google Scholar 

  9. Xu H.K.: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 26, 105018 (2010)

    Article  Google Scholar 

  10. Qu B., Xiu N.: A note on the CQ algorithm for the split feasibility problem. Inverse Probl. 21, 1655–1665 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Yang Q.: The relaxed CQ algorithm solving the split feasibility problem. Inverse Probl. 20, 1261–1266 (2004)

    Article  MATH  Google Scholar 

  12. Byrne C.: An unified treatment of some iterative algorithm algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Zhao J., Yang Q.: Several solutionmethods for the split feasibility problem. Inverse Probl. 21, 1791–1799 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Martinez-Yanes C., Xu H.K.: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal. 64, 2400–2411 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Nakajo K., Takahashi W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372–379 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yao Y., Liou Y.C., Marino G.: A hybrid algorithm for pseudo-contractive mappings. Nonlinear Anal. 71, 4997–5002 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Marino G., Xu H.K.: Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces. J. Math. Anal. Appl. 329, 336–349 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kim T.H., Xu H.K.: Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal. 64, 1140–1152 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang, F., Xu, H.K.: Approximating curve and strong convergence of the CQ algorithm for the split feasibility problem. J. Inequalities Appl. (2010). doi:10.1155/2010/102085

  20. Dang Y., Gao Y.: The strong convergence of a KM-CQ-like algorithm for a split feasibility problem. Inverse Probl. 27, 015007 (2011)

    Article  MathSciNet  Google Scholar 

  21. Chinchuluun, A., Pardalos, P.M., Enkhbat, R., Tseveendorj, I. (eds.): Optimization and Optimal Control Theory and Applications Series: Springer Optimization and its Applications, vol. 39. Springer (2010)

  22. Floudas C.A., Pardalos P.M.: Encyclopedia of Optimization, 2nd edn. Springer, New York (2009)

    Book  MATH  Google Scholar 

  23. Geobel, K., Kirk, W.A.: Topics in metric fixed point theory. In: Cambridge Studies in Advanced Mathematics, vol. 28. Cambridge University Press (1990)

  24. Lu X., Xu H.K., Yin X.: Hybrid methods for a class of monotone variational inequalities. Nonlinear Anal. 71, 1032–1041 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Suzuki T.: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl. 2005, 103–123 (2005)

    Article  MATH  Google Scholar 

  26. Xu H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Science, Tianjin Polytechnic University, Tianjin, 300387, China

    Xin Yu

  2. Department of Mathematics, King Abdul Aziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    Naseer Shahzad

  3. Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, China

    Yonghong Yao

Authors
  1. Xin Yu
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  2. Naseer Shahzad
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  3. Yonghong Yao
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Correspondence to Naseer Shahzad.

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Yu, X., Shahzad, N. & Yao, Y. Implicit and explicit algorithms for solving the split feasibility problem. Optim Lett 6, 1447–1462 (2012). https://doi.org/10.1007/s11590-011-0340-0

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  • DOI: https://doi.org/10.1007/s11590-011-0340-0

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