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Split common fixed point problems for demicontractive operators

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Abstract

In this paper, first, we introduce a new iterative algorithm involving demicontractive mappings in Hilbert spaces and, second, we prove some strong convergence theorems of the proposed method with the Armijo-line search to show the existence of a solution of the split common fixed point problem. Finally, we give some numerical examples to illustrate our main results.

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References

  1. Agarwal, R.P., Chen, J., Cho, Y.J.: Strong convergence theorems for equilibrium problems and weak Bregman relatively nonexpansive mappings in Banach spaces. J. Inequal. Appl. 2013:119 (2013)

  2. Anh, P.K., Hieu, D.V.: Parallel hybrid methods for variational inequalities, equilibrium problems and common fixed point problems. Vietnam J. Math. 44, 351–374 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bnouhachem, A.: Strong convergence algorithm for split equilibrium problems and hierarchical fixed point problems. The Sci. World J. Art. ID 390956, pp. 12 (2014)

  5. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. The Math. Student 63, 123–145 (1994)

    MathSciNet  MATH  Google Scholar 

  6. Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Prob. 18, 441–453 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Prob. 18, 103–120 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cegielski, A.: General method for solving the split common fixed point problem. J. Optim Theory Appl. 165, 385–404 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ceng, L.C., Cubiotti, P., Yao, J.C.: An implicit iterative scheme for monotone variational inequalities and fixed point problems. Nonlinear Anal. 69, 2445–2457 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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 

  11. Censor, Y., Elfving, T.: A multi-projection algorithms using Bregman projection in a product space. Numer. Algorithms 8, 221–239 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  12. Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59, 301–323 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Censor, Y., Segal, A.: The split common fixed point problem for directed operators. J. Convex Anal. 16(2), 587–600 (2009)

    MathSciNet  MATH  Google Scholar 

  14. Chang, S.S., Lee, H.W.J., Chan, C.K.: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. 70, 3307–3319 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chang, S.S., Leee, H.W.J., Chan, C.K., Wang, L., Qin, L.J.: Split feasibility problem for quasi- nonexpansive multi-valued mappings and total asymptotically strict pseudo-contractive mapping. Appl. Math. Comput. 219, 10416–10424 (2013)

    MathSciNet  MATH  Google Scholar 

  16. Cholamjiak, P., Suantai, S.: Iterative methods for solving equilibrium problems, variational inequalities and fixed points of nonexpansive semigroups. J. Glob. Optim. 57, 1277–1297 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Combettes, P.L. In: Hawkes, P. (ed.) : The convex feasibility problem in image recovery, in Advances in Imaging and Electron Physics, vol. 95, pp 155–270. Academic Press, New York (1996)

  18. Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming using proximal like algorithms. Math. Program. 78, 29–41 (1997)

    Article  MathSciNet  Google Scholar 

  19. Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)

    MathSciNet  MATH  Google Scholar 

  20. Cui, H., Wang, F.: Iterative methods for the split common fixed point problem in a Hilbert space. Fixed Point Theory Appl. 2014, 78 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Eslamian, M.: General algorithms for split common fixed point problem of demicontractive mappings. Optim. 65(2), 443–465 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Fang, Y.P., Huang, N.J.: Variational-like inequalities with generalized monotone mappings in Banach spaces. J. Optim. Theory Appl. 118 (2), 327–338 (2003). https://doi.org/10.1023/A:1025499305742

    Article  MathSciNet  MATH  Google Scholar 

  23. He, Z.H.: The split equilibrium problem and its convergence algorithms. J. Inequal. Appl. 2012, 162 (2012). https://doi.org/10.1186/1029-242X-2012-162

    Article  MathSciNet  MATH  Google Scholar 

  24. Hieu, D.V.: Parallel extragradient-proximal methods for split equilibrium problems. Math. Model. Anal. 21, 478–501 (2016)

    Article  MathSciNet  Google Scholar 

  25. Hieu, D.V., Muu, L.D., Anh, P.K.: Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings. Numer. Algorithms 73(1), 197–217 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kang, S.M., Cho, S.Y., Liu, Z.: Convergence of iterative sequences for generalized equilibrium problems involving inverse-strongly monotone mappings. J. Inequal. Appl. 2010, Art 827082 (2010)

    MathSciNet  MATH  Google Scholar 

  27. Katchang, P., Kumam, P.: A new iterative algorithm of solution for equilibrium problems, variational inequalities and fixed point problems in a Hilbert space. J. Appl. Math. Comput. 32, 19–38 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kazmi, K.R., Rizvi, S.H.: Iterative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem. J. Egypt Math. Soc. 21, 44–51 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kitkuan, D., Kumam, P., Padcharoen, A., Kumam, W., Thounthong, P.: Algorithms for zeros of two accretive operators for solving convex minimization problems and its application to image restoration problems. J. Comput. Appl. Math., In press. https://doi.org/10.1016/j.cam.2018.04.057

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Moudafi, A.: The split common fixed-point problem for demicontractive mappings. Inverse Prob. 26, 587–600 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Moudafi, A.: A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonlinear Anal. 74, 4083–4087 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Moudafi, A.: Split monotone variational inclusions. J. Optim. Theory Appl. 150, 275–283 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Plubtieng, S., Punpaeng, R.: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 336, 455–469 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  35. Qin, X., Cho, Y.J., Kang, S.M.: Convergence analysis on hybrid projection algorithms for equilibrium problems and variational inequality problems. Model Math. Anal. 14, 335–351 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Qin, X., Cho, Y.J., Kang, S.M.: Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal. 72, 99–112 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Qin, X., Shang, M., Su, Y.: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. Nonlinear Anal. 69, 3897–3909 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  38. 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 

  39. Shehu, Y., Cholamjiak, P.: Another look at the split common fixed point problem for demicontractive operators. RACSAM 110, 201–218 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  40. Stark, H.: Image recovery: theory and application. Academic Press, Orlando (1987)

    MATH  Google Scholar 

  41. Suantai, S., Cholamjiak, P.: Algorithms for solving generalized equilibrium problems and fixed points of nonexpansive semigroups in Hilbert spaces. Optim. 63, 799–815 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  42. Suantai, S., Pholasa, N., Cholamjiak, P.: The modified inertial relaxed CQ algorithm for solving the split feasibility problems. J. Indust Manag. Optim. 14(4), 1595–1615 (2018)

    MathSciNet  MATH  Google Scholar 

  43. Tada, A., Takahashi, W.: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133, 359–370 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  44. Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331, 506–515 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  45. Takahashi, S., Takahashi, W.: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 69, 1025–1033 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  46. Tang, Y.C., Peng, J.G., Liu, L.W.: A cyclic algorithm for the split common fixed point problem of demicontractive mappings in Hilbert spaces. Math. Model. Anal. 17, 457–466 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  47. Vinh, N.T., Cholamjiak, P., Suantai, S.: A new CQ algorithm for solving split feasibility problems in Hilbert spaces. Bull. Malays. Math. Sci. Soc. https://doi.org/10.1007/s40840-018-0614-0 (2018)

  48. Wang, F.: A new iterative method for the split common fixed point problem in Hilbert spaces. Optim. https://doi.org/10.1080/02331934.2016.1274991 (2017)

  49. Wang, F., Cui, H.: Convergence of a cyclic algorithm for the split common fixed point problem without continuity assumption. Math. Model. Anal. 18, 537–542 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  50. Wang, S.H., Gong, X.Y., Abdou, A., Cho, Y.J.: Iterative algorithm for a family of split equilibrium problems and fixed point problems in Hilbert spaces with applications. Fixed Point Theory Appl. 2016, 4 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  51. Wang, S.H., Gong, X.Y., Kang, S.M.: Strong convergence theorem on split equilibrium and fixed point problems in hilbert spaces. Bull. Malays. Math. Sci. Soc. 41(3), 1309–1326 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  52. Wang, S.H., Marino, G., Liou, Y.C.: Strong convergence theorems for variational inequality, equilibrium and fixed point problems with applications. J. Glob. Optim. 54, 155–171 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  53. Wang, S.H., Marino, G., Wang, F.H.: Strong convergence theorems for a generalized equilibrium problem with a relaxed monotone mapping and a countable family of nonexpansive mappings in a Hilbert space. Fixed Point Theory Appl. 2010, 230304 (2010)

    MathSciNet  MATH  Google Scholar 

  54. Wang, Y.M., Xiao, Y.B., Wang, X., Cho, Y.J.: Equivalence of well-posedness between systems of hemivariational inequalities and inclusion problems. J. Nonlinear Sci. Appl. 9, 1178–1192 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  55. Wen, D.J., Chen, Y.A.: General iterative methods for generalized equilibrium problems and fixed point problems of k-strict pseudo-contractions. Fixed Point Theory Appl. 2012, 125 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  56. Xiao, Y.B., Huang, N.J.: Generalized quasi-variational-like hemivariational inequalities. Nonlinear Anal. 69, 637–646 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  57. Xiao, Y.B., Huang, N.J., Cho, Y.J.: A class of generalized evolution variational inequalities in Banach space. Appl. Math. Lett. 25, 914–920 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  58. Xu, H.K.: Iterative algorithm for nonlinear operators. J. Lond. Math. Soc. 66, 1–17 (2002)

    Article  MathSciNet  Google Scholar 

  59. Xu, Y.H., Tang, Y.X., Guan, J.Y., Su, Y.: Cloud hybrid methods for solving split equilibrium and fixed point problems for a family of countable quasi-Lipschitz mappings and applications. J. Nonlinear Sci. Appl. 10, 752–770 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  60. Yao, Y., Agarwal, R.P., Postolache, M., Liu, Y.C.: Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem. Fixed Point Theory Appl. 2014, Art. 183 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  61. Yao, Y., Cho, Y.J.: A strong convergence of a modified Krasnoselskii-Mann method for nonexpansive mappings in Hilbert spaces. Math. Model. Anal. 15, 265–274 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  62. Yao, Y., Cho, Y.J., Liou, Y.C.: Hierarchical convergence of an implicit double net algorithm for nonexpansive semigroups and variational inequalities. Fixed Point Theory Appl. 2011, Art. 101 (2011)

    Article  MATH  Google Scholar 

  63. Yao, Y., Leng, L., Postolache, M., Zheng, X.: A unified framework for the two-sets split common fixed point problem in Hilbert spaces. J. Nonlinear Sci. Appl. 9, 6113–6125 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  64. Yao, Y., Leng, L., Postolache, M., Zheng, X.: Mann-type iteration method for solving the split common fixed point problem. J. Nonlinear Convex Anal. 18, 875–882 (2017)

    MathSciNet  MATH  Google Scholar 

  65. Yao, Y., Liou, Y.C., Postolache, M.: Self-adaptive algorithms for the split problem of the demicontractive operators. Optim. https://doi.org/10.1080/02331934.2017.1390747 (in press)

  66. Zhao, J., He, S.: Strong convergence of the viscosity approximation process for the split common fixed point problem of quasi-nonexpansive mappings. J. Appl. Math. 2012, Art. ID 438023 12 pp. (2012)

    MathSciNet  MATH  Google Scholar 

  67. Zho, J.: Solving split equality fixed-point problem of quasi-nonexpansive mappings without prior knowledge of operators norms. Optim. 64(12), 2619–2630 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The author would like to thank the Editors and anonymous referees for their valuable comments and suggestions which helped us very much in improving the original version of this paper. This project was supported by the Theoretical and Computational Science (TaCS) Center under C omputationaL and A pplied S cience for S mart I nnovation research C luster (CLASSIC), Faculty of Science, KMUTT.

The first author thanks for the support of Petchra Pra Jom Klao Doctoral Scholarship for Ph.d. student of King Mongkut’s University of Technology Thonburi (KMUTT). This work was completed while the first author visit Prof. Yoel Je Cho at Gyeongsang National University (GNU).

Funding

The authors received financial support provided by King Mongkut’s University of Technology Thonburi through the “KMUTT 55th Anniversary Commemorative Fund.”

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Authors and Affiliations

  1. KMUTT-Fixed Point Theory and Applications Research Group (KMUTT-FPTA), Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok, 10140, Thailand

    Anantachai Padcharoen & Poom Kumam

  2. KMUTTFixed Point Research Laboratory, Department of Mathematics, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok, 10140, Thailand

    Anantachai Padcharoen, Poom Kumam & Yeol Je Cho

  3. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, 40402, Taiwan

    Poom Kumam

  4. Department of Mathematics Education, Gyeongsang National University, Jinju, 52828, Korea

    Yeol Je Cho

  5. School of Mathematical Science, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, People’s Republic of China

    Yeol Je Cho

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  1. Anantachai Padcharoen
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  2. Poom Kumam
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  3. Yeol Je Cho
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Correspondence to Poom Kumam.

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Padcharoen, A., Kumam, P. & Cho, Y.J. Split common fixed point problems for demicontractive operators. Numer Algor 82, 297–320 (2019). https://doi.org/10.1007/s11075-018-0605-0

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