Abstract
In this paper, we introduce and study an iterative method to approximate a common solution of split variational inclusion problem and fixed point problem for a nonexpansive mapping in real Hilbert spaces. Further, we prove that the sequences generated by the proposed iterative method converge strongly to a common solution of split variational inclusion problem and fixed point problem for a nonexpansive mapping which is the unique solution of the variational inequality problem. The results presented in this paper are the supplement, extension and generalization of the previously known results in this area.
Similar content being viewed by others
References
Moudafi, A.: Split monotone variational inclusions. J. Optim. Theory Appl. 150, 275–283 (2011)
Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59, 301–323 (2012)
Moudafi, A.: The split common fixed point problem for demicontractive mappings. Inverse Probl. 26 055007 (6pp) (2010)
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)
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)
Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in product space. Numer. Algorithms 8, 221–239 (1994)
Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse probl. 18, 441–453 (2002)
Combettes, P.L.: The convex feasibility problem in image recovery. Adv. Imaging Electron Phys. 95, 155–453 (1996)
Crombez, G.: A hierarchical presentation of operators with fixed points on Hilbert spaces. Numer. Funct. Anal. Optim. 27, 259–277 (2006)
Crombez, G.: A geometrical look at iterative methods for operators with fixed points. Numer. Funct. Anal. Optim. 26, 157–175 (2005)
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)
Byrne, C.: A unified treatment for some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)
Lopez, G., Martin-Marquez, V., Xu, H.K.: Iterative algorithms for the multi-sets 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 (2010)
Goebel, K., Kirk, W.A.: Topics on Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)
Xu, H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004)
Acknowledgments
The authors are very thankful to the referees for their constructive, detailed and helpful comments and suggestions toward the improvement of the manuscript.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kazmi, K.R., Rizvi, S.H. An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping. Optim Lett 8, 1113–1124 (2014). https://doi.org/10.1007/s11590-013-0629-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-013-0629-2