Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2014, Special Issue (2014), Article ID 573075, 9 pages.

Weak Convergence Theorems for Bregman Relatively Nonexpansive Mappings in Banach Spaces

Chin-Tzong Pang, Eskandar Naraghirad, and Ching-Feng Wen

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Abstract

We study Mann type iterative algorithms for finding fixed points of Bregman relatively nonexpansive mappings in Banach spaces. By exhibiting an example, we first show that the class of Bregman relatively nonexpansive mappings embraces properly the class of Bregman strongly nonexpansive mappings which was investigated by Martín-Márques et al. (2013). We then prove weak convergence theorems for the sequences produced by the methods. Some application of our results to the problem of finding a zero of a maximal monotone operator in a Banach space is presented. Our results improve and generalize many known results in the current literature.

Article information

Source
J. Appl. Math., Volume 2014, Special Issue (2014), Article ID 573075, 9 pages.

Dates
First available in Project Euclid: 1 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1412177570

Digital Object Identifier
doi:10.1155/2014/573075

Mathematical Reviews number (MathSciNet)
MR3208630

Citation

Pang, Chin-Tzong; Naraghirad, Eskandar; Wen, Ching-Feng. Weak Convergence Theorems for Bregman Relatively Nonexpansive Mappings in Banach Spaces. J. Appl. Math. 2014, Special Issue (2014), Article ID 573075, 9 pages. doi:10.1155/2014/573075. https://projecteuclid.org/euclid.jam/1412177570


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References

  • W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953.
  • S. Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 274–276, 1979.
  • G. Chen and M. Teboulle, “Convergence analysis of a proximal-like minimization algorithm using Bregman functions,” SIAM Journal on Optimization, vol. 3, no. 3, pp. 538–543, 1993.
  • L. M. Brègman, “The relation method of finding the common point of convex sets and its application to the solution of problems in convex programming,” USSR Computational Mathematics and Mathematical Physics, vol. 7, pp. 200–217, 1967.
  • Y. Censor and A. Lent, “An iterative row-action method for interval convex programming,” Journal of Optimization Theory and Applications, vol. 34, no. 3, pp. 321–353, 1981.
  • Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178, pp. 15–50, Marcel Dekker, New York, NY, USA, 1996.
  • S. Reich, “A weak convergence theorem for the alternating method with Bregman distances,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178, pp. 313–318, Marcel Dekker, New York, NY, USA, 1996.
  • E. Naraghirad and J.-C. Yao, “Bregman weak relatively nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2013, article 141, 2013.
  • Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967.
  • R. T. Rockafellar, “Monotone operators and the proximal point algorithm,” SIAM Journal on Control and Optimization, vol. 14, no. 5, pp. 877–898, 1976.
  • R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the American Mathematical Society, vol. 149, pp. 75–88, 1970.
  • H. H. Bauschke, J. M. Borwein, and P. L. Combettes, “Bregman monotone optimization algorithms,” SIAM Journal on Control and Optimization, vol. 42, no. 2, pp. 596–636, 2003.
  • W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, Japan, 2000.
  • J. M. Borwein, S. Reich, and S. Sabach, “A characterization of Bregman firmly nonexpansive operators using a new monotonicity concept,” Journal of Nonlinear and Convex Analysis, vol. 12, no. 1, pp. 161–184, 2011.
  • V. Martín-Márquez, S. Reich, and S. Sabach, “Iterative methods for approximating fixed points of Bregman nonexpansive operators,” Discrete and Continuous Dynamical Systems: Series S, vol. 6, no. 4, pp. 1043–1063, 2013.
  • D. Butnariu and A. N. Iusem, Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization, vol. 40, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.
  • F. Kohsaka and W. Takahashi, “Proximal point algorithms with Bregman functions in Banach spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 3, pp. 505–523, 2005.
  • C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing, River Edge, NJ, USA, 2002.
  • R. T. Rockafellar, “Characterization of the subdifferentials of convex functions,” Pacific Journal of Mathematics, vol. 17, pp. 497–510, 1966.
  • R. T. Rockafellar, “On the maximal monotonicity of subdifferential mappings,” Pacific Journal of Mathematics, vol. 33, pp. 209–216, 1970.
  • D. Butnariu and E. Resmerita, “Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces,” Abstract and Applied Analysis, vol. 2006, Article ID 84919, 39 pages, 2006.
  • E. Naraghirad, W. Takahashi, and J.-C. Yao, “Generalized retraction and fixed point theorems using Bregman functions in Banach spaces,” Journal of Nonlinear and Convex Analysis, vol. 13, no. 1, pp. 141–156, 2012.
  • S. Reich and S. Sabach, “Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces,” in Fixed-Point Algorithms for Inverse Problems in Science and Engineering, vol. 49, pp. 299–314, Springer, New York, NY, USA, 2010. \endinput