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A New Self-Adaptive Method for the Multiple-Sets Split Common Null Point Problem in Banach Spaces

Abstract

In this paper, we study the multiple-sets split common null point problem (MSCNPP) in Banach spaces. We introduce a new self-adaptive algorithm for solving this problem. Under suitable conditions, we prove a strong convergence theorem of the proposed algorithm. An application of the main theorem to the multiple-sets split feasibility problem in Banach spaces is also presented. Finally, we provide the numerical experiments which show the efficiency and implementation of the proposed method.

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References

  1. Alber, Y., Ryazantseva, I.: Nonlinear Ill-Posed Problems of Monotone Type. Springer, Dordrecht (2006)

    MATH  Google Scholar 

  2. Alofi, A.S., Alsulami, S.M., Takahashi, W.: Strongly convergent iterative method for the split common null point problem in Banach spaces. J. Nonlinear Convex Anal. 17, 311–324 (2016)

    MathSciNet  MATH  Google Scholar 

  3. Alber, Y.I.: Metric and generalized projection operators in banach spaces: Properties and applications. In: Kartsatos, A.G. (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp 15–50. Marcel Dekker, New York (1996)

  4. Aoyama, K., Kohsaka, F., Takahashi, W.: Three generalizations of firmly nonexpansive mappings: Their relations and continuity properties. J. Nonlinear Convex Anal. 10, 131–147 (2009)

    MathSciNet  MATH  Google Scholar 

  5. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. Springer, New York (2017)

    MATH  Book  Google Scholar 

  6. Bello Cruz, J.Y., Shehu, Y.: An iterative method for split inclusion problems without prior knowledge of operator norms. J. Fixed Point Theory Appl. 19, 2017–2036 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  7. Bregman, L.M.: The relaxation method for finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7, 200–217 (1967)

    MathSciNet  Article  Google Scholar 

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

    MATH  Article  Google Scholar 

  9. Byrne, C., Censor, Y., Gibali, A., Reich, S.: The split common null point problem. J. Nonlinear Convex Anal. 13, 759–775 (2012)

    MathSciNet  MATH  Google Scholar 

  10. Cegielski, A.: Application of quasi-nonexpansive operators to an iterative method for variational inequality. SIAM J. Optim. 25, 2165–2181 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  11. Cegielski, A., Reich, S., Zalas, R.: Regular sequences of quasi-nonexpansive operators and their applications. SIAM J. Optim. 28, 1508–1532 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  12. Cegielski, A., Reich, S., Zalas, R.: Weak, strong and linear convergence of the CQ-method via the regularity of Landweber operators. Optimization 69, 605–636 (2020)

    MathSciNet  MATH  Article  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

  14. Censor, Y., Lent, A.: An iterative row-action method for interval convex programming. J. Optim. Theory Appl. 34, 321–353 (1981)

    MathSciNet  MATH  Article  Google Scholar 

  15. 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 

  16. 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)

    MathSciNet  MATH  Article  Google Scholar 

  17. Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: The Split Feasibility Model Leading to a Unified Approach for Inversion Problems in Intensity-Modulated Radiation Therapy. Technical Report. Department of Mathematics, University of Haifa, Haifa (2005)

  18. Cholamjiak, P., Suantai, S., Sunthrayuth, P.: An iterative method with residual vectors for solving the fixed point and the split inclusion problems in Banach spaces. Comput. Appl. Math. 38, 12 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  19. Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic, Dordrecht (1990)

    MATH  Book  Google Scholar 

  20. Combettes, P.L.: The convex feasibility problem in image recovery. Adv. Imaging Electron. Phys. 95, 155–270 (1996)

    Article  Google Scholar 

  21. Dadashi, V.: Shrinking projection algorithms for the split common null point problem. Bull. Aust. Math. Soc. 96, 299–306 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  22. Eslamian, M.: Split common fixed point and common null point problem. Math. Methods Appl. Sci. 40, 7410–7424 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  23. Eslamian, M., Eskandani, G.Z., Raeisi, M.: Split common null point and common fixed point problems between Banach spaces and Hilbert spaces. Mediter. J. Math. 14, 119 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  24. Hendrickx, J.M., Olshevsky, A.: Matrix p-norms are NP-hard to approximate if \(p\neq 1, 2,\infty \). SIAM J. Matrix Anal. Appl. 31, 2802–2812 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  25. Kuo, L.-W., Sahu, D.R.: Bregman distance and strong convergence of proximal-type algorithms. Abstr. Appl. Anal. 2013, 590519 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  26. López, G., Martín-Márquez, V., Wang, F., Xu, H.-K.: Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Probl. 28, 085004 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  27. Maingé, P.-E.: Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 325, 469–479 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  28. Maingé, P.-E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 16, 899–912 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  29. Raeisi, M., Zamani Eskandani, G., Eslamian, M.: A general algorithm for multiple-sets split feasibility problem involving resolvents and Bregman mappings. Optimization 67, 309–327 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  30. Reich, S.: A weak convergence theorem for the alternating method with Bregman distances. In: Kartsatos, A.G. (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp 313–318. Marcel Dekker, New York (1996)

  31. Reich, S., Sabach, S.: Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces. Nonlinear Anal. TMA 73, 122–135 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  32. Reich, S., Sabach, S.: A projection method for solving nonlinear problems in reflexive Banach spaces. J. Fixed Point Theory Appl. 9, 101–116 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  33. Reich, S., Tuyen, T.M.: A new algorithm for solving the split common null point problem in Hilbert spaces. Numer Algor. 83, 789–805 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  34. Reich, S., Tuyen, T.M.: Two projection methods for solving the multiple-set split common null point problem in Hilbert spaces. Optimization 69, 1913–1934 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  35. Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33, 209–216 (1970)

    MathSciNet  MATH  Article  Google Scholar 

  36. Sahu, D.R., O’Regan, D., Agarwal, R.P.: Fixed Point Theory for Lipschitzian-type Mappings with Applications. Topological Fixed Point Theory and Its Applications, vol. 6. Springer, New York (2009)

    MATH  Book  Google Scholar 

  37. Schöpfer, F., Schuster, T., Louis, A.K.: An iterative regularization method for the solution of the split feasibility problem in Banach spaces. Inverse Probl. 24, 055008 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  38. Shehu, Y.: Strong convergence theorem for multiple sets split feasibility problems in Banach spaces. Numer. Funct. Anal. Optim. 37, 1021–1036 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  39. Stark, H.: Image Recovery: Theory and Application. Academic Press, Orlando, FL (1987)

    MATH  Google Scholar 

  40. Suantai, S., Shehu, Y., Cholamjiak, P.: Nonlinear iterative methods for solving the split common null point problem in Banach spaces. Optim. Methods Softw. 34, 853–874 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  41. Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)

    MATH  Google Scholar 

  42. Takahashi, W.: The split common null point problem for generalized resolvents in two Banach spaces. Numer. Algor. 75, 1065–1078 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  43. Tang, Y.: New inertial algorithm for solving split common null point problem in Banach spaces. J. Inequal. Appl. 2019, 17 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  44. Tang, Y.: Convergence analysis of a new iterative algorithm for solving split variational inclusion problems. J. Ind. Manag. Optim. 16, 945–964 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  45. Tang, Y., Gibali, A.: New self-adaptive step size algorithms for solving split variational inclusion problems and its applications. Numer. Algor. 83, 305–331 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  46. Tang, Y., Sunthrayuth, P.: An iterative algorithm with inertial technique for solving the split common null point problem in Banach spaces. Asian-Eur. J. Math (2021)

  47. Tuyen, T.M.: A strong convergence theorem for the split common null point problem in Banach spaces. Appl. Math. Optim. 79, 207–227 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  48. Tuyen, T.M., Sunthrayuth, P., Trang, N.M.: An inertial self-adaptive algorithm for the genealized split common null point problem in Hilbert spaces. Rend. Circ. Mat. Palermo, II. Ser (2021)

  49. Tuyen, T.M., Thuy, N.T.T., Trang, N.M.: A strong convergence theorem for a parallel iterative method for solving the split common null point problem in Hilbert spaces. J. Optim. Theory Appl. 183, 271–291 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  50. Xu, H.-K.: Inequalities in Banach spaces with applications. Nonlinear Anal. TMA 16, 1127–1138 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  51. Yang, Q.: On variable-step relaxed projection algorithm for variational inequalities. J. Math. Anal. Appl. 302, 166–179 (2005)

    MathSciNet  MATH  Article  Google Scholar 

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Acknowledgements

T.M. Tuyen was supported by the Science and Technology Fund of the Vietnam Ministry of Education and Training (B2022-TNA-23). P. Cholamjiak was supported by University of Phayao and Thailand Science Research and Innovation grant no. FF65-UoE001. P. Sunthrayuth was supported by Rajamangala University of Technology Thanyaburi (RMUTT). The authors would like to thank the anonymous referees for their valuable comments and suggestions which helped to improve the quality of the manuscript.

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Correspondence to Pongsakorn Sunthrayuth.

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Tuyen, T.M., Cholamjiak, P. & Sunthrayuth, P. A New Self-Adaptive Method for the Multiple-Sets Split Common Null Point Problem in Banach Spaces. Vietnam J. Math. (2022). https://doi.org/10.1007/s10013-022-00574-3

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  • DOI: https://doi.org/10.1007/s10013-022-00574-3

Keywords

  • Banach space
  • Strong convergence
  • Maximal monotone
  • Split common null point problem

Mathematics Subject Classification (2010)

  • 47H09
  • 47H10
  • 47J25
  • 47J05