Advertisement

Relatively Nonexpansive Operators with Respect to Bregman Distances

  • Simeon Reich
  • Alexander J. Zaslavski
Part of the Developments in Mathematics book series (DEVM, volume 34)

Abstract

The following problem often occurs in functional analysis and optimization theory, as well as in other fields of pure and applied mathematics: given a nonempty, closed and convex subset K of a Banach space X and an operator T:KK, do the sequences iteratively generated in K by the rule x k+1=Tx k converge to a fixed point of T no matter how the initial point x 0K is chosen? It is well known that this indeed happens, in some sense, for “standard” classes of operators (e.g., certain nonexpansive operators and operators of contractive type). In this chapter we show that the question asked above has an affirmative answer even if the operator T is not contractive in any standard sense, but is relatively nonexpansive with respect to a Bregman distance induced by a convex function f. More precisely, we prove that in appropriate complete metric spaces of operators which are relatively nonexpansive with respect to Bregman distances there exists a subset which is a countable intersection of open and everywhere dense sets such that for any operator belonging to this subset, all its orbits converge strongly.

References

  1. 24.
    Browder, F. E. (1976). Proceedings of symposia in pure mathematics: Vol. 18. Nonlinear operators and nonlinear equations of evolution in Banach spaces (Part 2). Providence: Am. Math. Soc. CrossRefzbMATHGoogle Scholar
  2. 27.
    Butnariu, D., & Iusem, A. N. (1997). In Contemporary mathematics: Vol. 204. Recent developments in optimization theory and nonlinear analysis (pp. 61–91). CrossRefGoogle Scholar
  3. 28.
    Butnariu, D., & Iusem, A. N. (2000). Totally convex functions for fixed points computation and infinite dimensional optimization. Dordrecht: Kluwer Academic. CrossRefzbMATHGoogle Scholar
  4. 29.
    Butnariu, D., Censor, Y., & Reich, S. (1997). Computational Optimization and Applications, 8, 21–39. MathSciNetCrossRefzbMATHGoogle Scholar
  5. 30.
    Butnariu, D., Reich, S., & Zaslavski, A. J. (1999). Numerical Functional Analysis and Optimization, 20, 629–650. MathSciNetCrossRefzbMATHGoogle Scholar
  6. 31.
    Butnariu, D., Iusem, A. N., & Burachik, R. S. (2000). Computational Optimization and Applications, 15, 269–307. MathSciNetCrossRefzbMATHGoogle Scholar
  7. 32.
    Butnariu, D., Iusem, A. N., & Resmerita, E. (2000). Journal of Convex Analysis, 7, 319–334. MathSciNetzbMATHGoogle Scholar
  8. 33.
    Butnariu, D., Reich, S., & Zaslavski, A. J. (2001). Journal of Applied Analysis, 7, 151–174. MathSciNetCrossRefzbMATHGoogle Scholar
  9. 37.
    Censor, Y., & Lent, A. (1981). Journal of Optimization Theory and Applications, 34, 321–353. MathSciNetCrossRefzbMATHGoogle Scholar
  10. 38.
    Censor, Y., & Reich, S. (1996). Optimization, 37, 323–339. MathSciNetCrossRefzbMATHGoogle Scholar
  11. 39.
    Censor, Y., & Zenios, S. A. (1997). Parallel optimization. New York: Oxford University Press. zbMATHGoogle Scholar
  12. 68.
    Goebel, K., & Reich, S. (1984). Uniform convexity, hyperbolic geometry, and nonexpansive mappings. New York: Dekker. zbMATHGoogle Scholar
  13. 122.
    Reich, S. (1996). In Theory and applications of nonlinear operators of accretive and monotone type (pp. 313–318). New York: Dekker. Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Simeon Reich
    • 1
  • Alexander J. Zaslavski
    • 1
  1. 1.Department of MathematicsTechnion-Israel Institute of TechnologyHaifaIsrael

Personalised recommendations