Abstract
The purpose of this paper is to study split feasibility problems and fixed point problems concerning left Bregman strongly relatively nonexpansive mappings in p-uniformly convex and uniformly smooth Banach spaces. We suggest an iterative scheme for the problem and prove strong convergence theorem of the sequences generated by our scheme under some appropriate conditions in real p-uniformly convex and uniformly smooth Banach spaces. Finally, we give numerical examples of our result to study its efficiency and implementation. Our result complements many recent and important results in this direction.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alber, Y.I.: Metric and generalized projection operator in Banach spaces: properties and applications, in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type vol 178 of Lecture Notes in Pure and Applied Mathematics, pp 15-50. USA, Dekker, New York, NY (1996)
Bauschke, H.H., Combettes, P.L.: Convex analysis and monotone operator theory in hilbert spaces. Springer, New York (2011)
Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Problems 18(2), 441–453 (2002)
Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems 20(1), 103–120 (2004)
Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numerical Algorithms 8(2-4), 221–239 (1994)
Censor, Y., Lent, A.: An iterative row-action method for interval convex programming. J. Optim. Theory Appl. 34, 321?353 (1981)
Censor, Y., Reich, S.: Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization. Optimization 37, 323–339 (1996)
Cioranescu, I.: Geometry of banach spaces, duality mappings and nonlinear problems. Kluwer Academic Dordrecht (1990)
Dunford, N., Schwartz, J.T.: Linear operators I. Wiley? Interscience, New York (1958)
Goebel, K., Kirk, W.A.: Topics in metric fixed point theory. In: Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, Cambridge, (1990)
Lindenstrauss, J., Tzafriri, L.: Classical banach spaces II. Springer, Berlin (1979)
Maingé, P.E.: The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces. Comput. Math. Appl. 59(1), 74–79 (2010)
Martín-Márquez, V., Reich, S., Sabach, S.: Bregman strongly nonexpansive operators in reflexive Banach spaces. J. Math. Anal. Appl. 400, 597–614 (2013)
Martín-Márquez, V., Reich, S., Sabach, S.: Right Bregman nonexpansive operators in banach spaces. Nonlinear Anal. 75, 5448–5465 (2012)
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)
Nakajo, K., Takahashi, W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372–379 (2003)
Qu, B., Xiu, N.: A note on the CQ algorithm for the split feasibility problem. Inverse Problems 21(5), 1655–1665 (2005)
Reich, S.: A weak convergence theorem for the alternating method with Bregman distances. In: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Marcel Dekker, New York, pp. 313-318 (1996)
Reich, S.: Book Review: Geometry of Banach spaces, duality mappings and nonlinear problems. Bull. Amer. Math. Soc. 26, 367–370 (1992)
Reich, S.: Extension problems for accretive sets in Banach spaces. J. Functional Anal. 26, 378–395 (1977)
Schöpfer, F.: Iterative regularization method for the solution of the split feasibility problem in Banach spaces. PhD thesis, Saarbrücken (2007)
Schöpfer, F., Schuster, T., Louis, A.K.: An iterative regularization method for the solution of the split feasibility problem in Banach spaces. Inverse Problems 24, 055008 (2008)
Shehu, Y.: Strong convergence theorem for Multiple Sets Split Feasibility Problems in Banach Spaces, Under review: Numerical Functional Analysis and Optimization
Shehu, Y.: A cyclic iterative method for solving Multiple Sets Split Feasibility Problems in Banach Spaces, Under review: Quaestiones Mathematicae
Shehu, Y., Ogbuisi, F. U., Iyiola, O. S.: Convergence Analysis of an iterative algorithm for fixed point problems and split feasibility problems in certain Banach spaces, In press: Optimization. doi:10.1080/02331934.2015.1039533
Takahashi, W.: Nonlinear Functional Analysis-Fixed Point Theory and Applications, Yokohama Publishers Inc., Yokohama. (in Japanese) (2000)
Takahashi, W.: Nonlinear functional analysis. Yokohama Publishers, Yokohama (2000)
Wang, F.: A new algorithm for solving the multiple-sets split feasibility problem in Banach spaces. Numer. Funct. Anal. Optim. 35, 99–110 (2014)
Xu, H.K.: Inequalities in Banach spaces with applications. Nonlinear Anal. 16(2), 1127–1138 (1991)
Xu, H.-K.: A variable Krasnoselskii-Mann algorithm and the multiple-set split feasibility problem. Inverse Problems 22(6), 2021–2034 (2006)
Yang, Q.: The relaxed CQ algorithm solving the split feasibility problem. Inverse Problems 20(4), 1261–1266 (2004)
Yang, Q., Zhao, J.: Generalized KM theorems and their applications. Inverse Problems 22(3), 833–844 (2006)
Yao, Y., Jigang, W., Liou, Y.-C.: Regularized methods for the split feasibility problem. Abstr. Appl Anal. 2012(ID), 140679 (2012). 13
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shehu, Y., Iyiola, O.S. & Enyi, C.D. An iterative algorithm for solving split feasibility problems and fixed point problems in Banach spaces. Numer Algor 72, 835–864 (2016). https://doi.org/10.1007/s11075-015-0069-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-015-0069-4
Keywords
- Strong convergence
- Split feasibility problem
- Uniformly convex
- Uniformly smooth
- Fixed point problem
- Left Bregman strongly nonexpansive mappings