Afrika Matematika

, Volume 28, Issue 1–2, pp 295–309 | Cite as

Iterative solution of split variational inclusion problem in a real Banach spaces

Article
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Abstract

In this paper, we study split variational inclusion problem in real Banach spaces with a view to analyze an iterative method for obtaining a solution of the split variational inclusion problem in Banach spaces. We propose an Halpern type algorithm and with our algorithm, we state and prove a strong convergence theorem for the approximation of solution of split variational inclusion problem in the frame work of p-uniformly convex Banach spaces which are also uniformly smooth. Our results extend and complement many known related results in the literature.

Keywords

Strong convergence Split variational inclusion problem P-uniformly convex Uniformly smooth Maximal monotone operators Resolvent of maximal monotone 

Mathematics Subject Classification

49J53 65K10 49M37 90C25 
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Notes

Acknowledgments

The first author acknowledge with thanks, the bursary and financial support from the Department of Science and Technology and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DST-NRF COE-MaSS) Doctoral Bursary. Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the CoE-MaSS. Also, the authors are grateful to the anonymous referees whose suggestions and comments helped to improve the final version of this paper.

References

  1. 1.
    Alber, Y.I.: Metric and generalized projection operator in Banach spaces: properties and applications. In: Kartsatos, K. (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Vol 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15–50. Dekker, New York (1996)Google Scholar
  2. 2.
    Bauschke, H.H., Borwein, J.M., Combettes, P.L.: Bregman monotone optimization algorithms. SIAM J. Control Optim. 42, 596–636 (2003)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility theorem. Inverse Probl. 18, 441–453 (2002)CrossRefMATHGoogle Scholar
  4. 4.
    Byrne, C.: A unified treatment for some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Byrne, C., Censor, Y., Gibali, A., Reich, S.: Weak and strong convergence of algorithms for the split common null point problem. J. Nonlinear Convex Anal. 13, 759–775 (2012)MathSciNetMATHGoogle Scholar
  6. 6.
    Byrne, C., Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59, 301–323 (2012)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    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)CrossRefGoogle Scholar
  8. 8.
    Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projection in product space. Numer. Algorithms 8, 221–239 (1994)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer, Dordrecht (1990)CrossRefMATHGoogle Scholar
  10. 10.
    Combettes, P.L.: The convex feasibility problem in image recovery. Adv. Imaging Electron Phys. 95, 155–453 (1996)CrossRefGoogle Scholar
  11. 11.
    Khan, A.R., Abbas, M., Shehu, Y.: A general convergence theorem for multiple set split feasibility problem in Hilbert space. Carpathian J. Math. 31(3), 349–357 (2015)MathSciNetGoogle Scholar
  12. 12.
    Kazmi, K.R., Rizvi, S.H.: An iterative method for split variational inclusion problem and fixed point problem for nonexpansive mapping. Optim. Lett. doi: 10.1007/s11590-013-0629-2
  13. 13.
    Lin, L.-J., Chen, Y.-D., Chuang, C.-S.: Solutions for a variational inclusion problem with applications to multiple sets split feasibility problems. Fixed Point Theory Appl. 2013, 333 (2013). 10.1186/1687-1812-2013-333MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lin, L.-J.: Systems of variational inclusion problems and differential inclusion problems with applications. J. Glob. Optim. 44(4), 579–591 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Maing\(\acute{\rm {e}}\), P.E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set Valued Anal. 16, 899–912 (2008)Google Scholar
  16. 16.
    Mart\(\acute{\rm {i}}\)n-M\(\acute{\rm {a}}\)rquez, V., Reich, S., Sabach, S.: Right Bregman nonexpansive operators in Banach spaces. Nonlinear Anal. 75, 5448–5465 (2012)Google Scholar
  17. 17.
    Masad, E., Reich, S.: A note on the multiple-set split convex feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 8(3), 367–371 (2007)MathSciNetMATHGoogle Scholar
  18. 18.
    Moudafi, A.: Split monotone variational inclusions. J. Optim. Theory Appl 150, 275–283 (2011)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Moudafi, A.: The split common fixed point problem for demicontractive mappings. Inverse Probl. 26, 055007 (2010)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Phelps, R.P.: Convex Functions, Monotone Operators, and Differentiability, 2nd edn. Springer, Berlin (1993)MATHGoogle Scholar
  21. 21.
    Qu, B., Xiu, N.: A note on the CQ algorithm for the split feasibility problem. Inverse Probl. 21(5), 1655–1665 (2005)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Reich, S., Sabach, S.: Two strong convergence theorems for a proximal method in reflexive Banach spaces. Numer. Funct. Anal. Optim. 31, 22–44 (2010)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Sch\(\overset{..}{\rm {o}}\)pfer, F.: Iterative regularisation method for the solution of the split feasibility problem in Banach spaces. Ph.D thesis, Saabr\(\overset{..}{\rm {u}}\)cken (2007)Google Scholar
  25. 25.
    Sch\(\overset{..}{\rm {o}}\)pfer, 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)Google Scholar
  26. 26.
    Shehu, Y., Ogbuisi, F.U.: An iterative method for solving split monotone variational inclusion and fixed point problems. RACSAM (2015). doi: 10.1007/s13398-015-0245-3
  27. 27.
    Wen, D.-J., Chen, Y.-A.: Iterative methods for split variational inclusion and fixed point problem of nonexpansive semigroup in Hilbert spaces. J. Inequal. Appl. (2015). doi: 10.1186/s13660-014-0528-9
  28. 28.
    Xu, H.-K.: Iterative algorithms for nonlinear operators. J Lond. Math. Soc. 66(2), 240–256 (2002)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Xu, H.-K.: A variable Krasnoselskii–Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 22(6), 2021–2034 (2006)CrossRefMATHGoogle Scholar
  30. 30.
    Yang, Q.: The relaxed CQ algorithm solving the split feasibility problem. Inverse Probl. 20(4), 1261–1266 (2004)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Yang, Q., Zhao, J.: Generalized KM theorems and their applications. Inverse Probl. 22(3), 833–844 (2006)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Yao, Y., Jigang, W., Liou, Y.-C.: Regularized methods for the split feasibility problem. Abstr. Appl. Anal. 2012, 140679 (2012). doi: 10.1155/2012/140679

Copyright information

© African Mathematical Union and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of Mathematics, Statistics and Computer ScienceUniversity of KwaZulu-NatalDurbanSouth Africa
  2. 2.DST-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS)JohannesburgSouth Africa

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