Abstract
We study the recently introduced generalized split common null point problem in Hilbert spaces. In order to solve this problem, we propose two new parallel algorithms and establish strong convergence theorems for both of them. Our schemes combine the hybrid and shrinking projection methods with the proximal point algorithm.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agarwal, R.P., O’Regan, D., Sahu, D.R.: Fixed Point Theory for Lipschitzian-type Mappings with Applications. Springer, New York (2009)
Butnariu, D., Resmerita, E.: Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces. Abstr. Appl. Anal. 2006, 1–39 (2006)
Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18, 441–453 (2002)
Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 18, 103–120 (2004)
Byrne, C., Censor, Y., Gibali, A., Reich, S.: The split common null point problem. J. Nonlinear Convex Anal. 13, 759–775 (2012)
Censor, Y., Elfving, T.: A multi projection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221–239 (1994)
Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its application. Inverse Probl. 21, 2071–2084 (2005)
Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59, 301–323 (2012)
Censor, Y., Segal, A.: The split common fixed point problem for directed operators. J. Convex Anal. 16, 587–600 (2009)
Cui, H., Su, M.: On sufficient conditions ensuring the norm convergence of an iterative sequence to zeros of accretive operators. Appl. Math. Comput. 258, 67–71 (2015)
Dadashi, V.: Shrinking projection algorithms for the split common null point problem. Bull. Aust. Math. Soc. 99, 299–306 (2017)
Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, vol. 28. Cambridge University Press, Cambridge (1990)
Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)
Landweber, L.: An iterative formula for Fredholm integral equations of the first kind. Am. J. Math. 73, 615–624 (1951)
Masad, E., Reich, S.: A note on the multiple-set split convex feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 8, 367–371 (2007)
Moudafi, A.: The split common fixed point problem for demicontractive mappings. Inverse Probl. 26, 055007 (2010)
Reich, S., Tuyen, T.M.: Iterative methods for solving the generalized split common null point problem in Hilbert spaces. Optimization 69, 1013–1038 (2019)
Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33, 209–216 (1970)
Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)
Takahashi, S., Takahashi, W.: The split common null point problem and the shrinking projection method in Banach spaces. Optimization 65, 281–287 (2016)
Takahashi, S., Takahashi, W., Toyoda, M.: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl. 147, 27–41 (2010)
Takahashi, W.: The split feasibility problem and the shrinking projection method in Banach spaces. J. Nonlinear Convex Anal. 16, 1449–1459 (2015)
Takahashi, W.: The split common null point problem in Banach spaces. Arch. Math. 104, 357–365 (2015)
Tuyen, T.M.: A strong convergence theorem for the split common null point problem in Banach spaces. Appl. Math. Optim. 79, 207–227 (2019)
Tuyen, T.M., Ha, N.S., Thuy, N.T.T.: A shrinking projection method for solving the split common null point problem in Banach spaces. Numer. Algorithms 81, 813–832 (2019)
Tuyen, T.M., Ha, N.S.: A strong convergence theorem for solving the split feasibility and fixed point problems in Banach spaces. J. Fixed Point Theory Appl. 20, 140 (2018)
Wang, F.: A new algorithm for solving the multiple-sets split feasibility problem in Banach spaces. Numer. Funct. Anal. Optim. 35, 99–110 (2014)
Wang, F.: A new iterative method for the split common fixed point problem in Hilbert spaces. Optimization 66, 407–415 (2017)
Wang, F., Xu, H.-K.: Cyclic algorithms for split feasibility problems in Hilbert spaces. Nonlinear Anal. 74, 4105–4111 (2011)
Xu, H.-K.: A variable Krasnosel’skii–Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 22, 2021–2034 (2006)
Xu, H.-K.: Iterative methods for the split feasibility problem in infinite dimensional Hilbert spaces. Inverse Probl. 26, 105018 (2010)
Yang, Q.: The relaxed CQ algorithm for solving the split feasibility problem. Inverse Probl. 20, 1261–1266 (2004)
Acknowledgements
Simeon Reich was partially supported by the Israel Science Foundation (Grant 820/17), by the Fund for the Promotion of Research at the Technion and by the Technion General Research Fund. The second author was supported by the Science and Technology Fund of the Vietnam Ministry of Education and Training (B2019-TNA-14). Both authors are grateful to the referees for their detailed reports, useful comments and helpful suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Reich, S., Tuyen, T.M. Parallel iterative methods for solving the generalized split common null point problem in Hilbert spaces. RACSAM 114, 180 (2020). https://doi.org/10.1007/s13398-020-00901-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13398-020-00901-8