Advertisement

On the split feasibility problem and fixed point problem of quasi-ϕ-nonexpansive mapping in Banach spaces

  • Zhaoli Ma
  • Lin Wang
  • Shih-sen Chang
Original Paper

Abstract

The purpose of this paper is to propose an algorithm to solve the split feasibility and fixed point problem of quasi-ϕ-nonexpansive mappings in Banach spaces. Without the assumption of semi-compactness on the mappings, it is proved that the sequence generated by the proposed iterative algorithm converges strongly to a common solution of the split feasibility and fixed point problems. As applications, the main results presented in this paper are used to study the convexly constrained linear inverse problem and split null point problem. Finally, a numerical example is given to support our results. The results presented in the paper are new and improve and extend some recent corresponding results.

Keywords

Split feasibility problem Quasi-ϕ-nonexpansive mapping Convergence Banach space 

Mathematics Subject Classification (2010)

47H09 47J25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Funding information

This work was supported by the National Natural Science Foundation of China (Grant No. 11361070).

References

  1. 1.
    Censor, Y., Bortfeld, T., Martin, B., Trofimov, T.: A unified approach for inversion problem in intensity-modolated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)CrossRefGoogle Scholar
  2. 2.
    Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications. Inverse Probl. 21, 2071–2084 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Censor, Y., Motova, A., Segal, A.: Pertured projections and subgradient projections for the multiple-sets split feasibility problems. J. Math. Anal Appl. 327, 1244–1256 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Xu, H.K.: A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 22, 2021–2034 (2006)CrossRefzbMATHGoogle Scholar
  5. 5.
    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
  6. 6.
    Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projection in a product space. Numer. Algor. 8, 221–239 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Byrne, C.: Iterative oblique projection onto convex subsets and the split feasibility problems. Inverse Probl. 18, 441–453 (2002)CrossRefzbMATHGoogle Scholar
  8. 8.
    Moudafi, A.: A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonlinear Anal. 74, 4083–4087 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Moudafi, A.: The split common fixed point problem for demi-contractive mappings. Inverse Probl. 26, 055007 (2010). (6pp)CrossRefzbMATHGoogle Scholar
  10. 10.
    Yao, Y., Postolache, M., Liou, Y.C.: Strong convergence of a self-adaptive method for the split feasibility problem. Fixed Point Theory Appl. Art No.201, 2013 (2013)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kraikaew, R., Saejung, S.: On split common fixed point problems. J. Math. Anal. Appl. 415, 513–524 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Qu, B., Xiu, N.: A note on the CQ algorithm for the split feasibility problem. Inverse Probl. 21, 1655–1665 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    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 No. 183 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Zhang, X.F., Wang, L., Ma, Z.L., Qin, L.J.: The strong convergence theorems for split common fixed point problem of asymptotically nonexpansive mappings in Hilbert spaces. J. Inequal. Appl. 2015, 1 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Takahashi, W., Yao, J.C.: Strong convergence theorems by hybrid method for the split common null point problem in Banach spaces. Fixed Point Theory Appl. 2015, 87 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Tang, J.F., Chang, S.S., Wang, L., Wang, X.R.: On the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings in Banach spaces. J. Inequal. Appl. 2015, 305 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Tian, X.J., Wang, L., Ma, Z.L.: On the split equality common fixed point problem for quasi-nonexpansive multi-valued mappings in Banach spaces. J. Nonlinear Sci. Appl. 9, 5536–5543 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ma, Z.L., Wang, L.: On the split equality common fixed point problem for asymptotically nonexpansive semigroups in Banach spaces. J. Nonlinear Sci. Appl. 9, 4003–4015 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Chen, J.Z., Hu, H.Y., Ceng, L.C.: Strong convergence of hybrid Bregman projection algorithm for split feasibility and fixed point problems in Banach spaces. J. Nonlinear Sci. Appl. 10, 192–204 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)zbMATHGoogle Scholar
  21. 21.
    Alber, Y.a.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 Monotonic Type, pp. 15–50. Marcel Dekker, New York (1996)Google Scholar
  22. 22.
    Alber, Ya.I., Reich, S.: An iterative method for solving a class of nonlinear operator equations in Banach spaces. Panamer. Math. J. 4(2), 39–54 (1994)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic Publishers, Dordrecht (1990)CrossRefzbMATHGoogle Scholar
  24. 24.
    Kamimura, S., Takahashi, W.: Strong convergence of a proximal-type algorithm in Banach space. SIAM. J. O ptim. 13, 938–945 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Reich, S.: A weak convergence theorem for the alternating method with Bregman distance. In: Kartsatos, A. G. (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotonic Type, pp. 313–318. Marcel Dekker, New York (1996)Google Scholar
  26. 26.
    Matsushita, S., Takahashi, W.: A strong convergence theorem for relatively nonexpansive mappings in Banach spaces. J. Approx. Theory 134, 257–266 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Chang, S.S., Joseph Lee, H.W., Chan, C.K., Yang, L.: Approximation theorems for total quasi-ϕ-asymptotically nonexpansive mappings with applications. Appl. Math. comput 218, 2921–2931 (2011)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Xu, H.K.: Inequalities in Banach spaces with applications. Nonlinear Anal. Theory Methods Appl. 16, 1127–1138 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Eicke, B.: Iteration methods for convexly constrained ill-posed problems in Hilbert space. Numer. Funct. Anal. Optim. 13, 413–429 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic Publishers Group, Dordrecht (1996)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of General EducationThe College of Arts and Sciences Yunnan Normal UniversityKunmingPeople’s Republic of China
  2. 2.College of Statistics and MathematicsYunnan University of Finance and EconomicsKunmingPeople’s Republic of China
  3. 3.Center for General EducationChina Medical UniversityTaichungTaiwan

Personalised recommendations