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Set-Valued Analysis

, Volume 8, Issue 4, pp 361–374 | Cite as

Weak and Strong Convergence of Solutions to Accretive Operator Inclusions and Applications

  • Shoji Kamimura
  • Wataru Takahashi
Article

Abstract

Our purpose in this paper is to approximate solutions of accretive operators in Banach spaces. Motivated by Halpern's iteration and Mann's iteration, we prove weak and strong convergence theorems for resolvents of accretive operators. Using these results, we consider the convex minimization problem of finding a minimizer of a proper lower semicontinuous convex function and the variational problem of finding a solution of a variational inequality.

accretive operator resolvent iteration strong convergence weak convergence 

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References

  1. 1.
    Brézis, H. and Lions, P. L.: Produits infinis de resolvants, Israel J. Math. 29 (1978), 329–345.Google Scholar
  2. 2.
    Browder, F. E.: Semicontractive and semiaccretie nonlinear mappings in Banach spaces, Bull. Amer. Math. Soc. 74 (1968), 660–665.Google Scholar
  3. 3.
    Bruck, R. E.: A strongly convergent iterative solution of 0 ε U(x) for a maximal monotone operator U in Hilbert space, J. Math. Anal. Appl. 48 (1974), 114–126.Google Scholar
  4. 4.
    Bruck, R. E. and Passty, G. B.: Almost convergence of the infinite product of resolvents in Banach spaces, Nonlinear Anal. 3 (1979), 279–282.Google Scholar
  5. 5.
    Halpern, B.: Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967), 957–961.Google Scholar
  6. 6.
    Jung, J. S. and Takahashi, W.: Dual convergence theorems for the infinite products of resolvents in Banach spaces, Kodai Math. J. 14 (1991), 358–364.Google Scholar
  7. 7.
    Khang, D. B.: On a class of accretive operators, Analysis 10 (1990), 1–16.Google Scholar
  8. 8.
    Mann, W. R.: Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506–510.Google Scholar
  9. 9.
    Nevanlinna, O. and Reich, S.: Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces, Israel J. Math. 32 (1979), 44–58.Google Scholar
  10. 10.
    Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591–597.Google Scholar
  11. 11.
    Passty, G. B.: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space, J. Math. Anal. Appl. 72 (1979), 383–390.Google Scholar
  12. 12.
    Pazy, A.: Remarks on nonlinear ergodic theory in Hilbert space, Nonlinear Anal. 6 (1979), 863–871.Google Scholar
  13. 13.
    Reich, S.: On infinite products of resolvents, Atti Acad. Naz. Lincei 63 (1977), 338–340.Google Scholar
  14. 14.
    Reich, S.: An iterative procedure for constructing zeros of accretive sets in Banach spaces, Nonlinear Anal. 2 (1978), 85–92.Google Scholar
  15. 15.
    Reich, S.: Constructing zeros of accretive operators, Appl. Anal. 8 (1979), 349–352.Google Scholar
  16. 16.
    Reich, S.: Constructing zeros of accretive operators II, Appl. Anal. 9 (1979), 159–163.Google Scholar
  17. 17.
    Reich, S.: Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67 (1979), 274–276.Google Scholar
  18. 18.
    Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980), 287–292.Google Scholar
  19. 19.
    Rockafellar, R. T.: Characterization of the subdifferentials of convex functions, Pacific J. Math. 17 (1966), 497–510.Google Scholar
  20. 20.
    Rockafellar, R. T.: On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970), 75–88.Google Scholar
  21. 21.
    Rockafellar, R. T.: Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976), 877–898.Google Scholar
  22. 22.
    Shioji, N. and Takahashi, W.: Strong convergence theorems of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (1997), 3641–3645.Google Scholar
  23. 23.
    Takahashi, W.: Nonlinear Functional Analysis, Kindai-Kagakusha, Tokyo, 1988 (Japanese).Google Scholar
  24. 24.
    Takahashi, W. and Kim, G. E.: Approximating fixed points of nonexpansive mappings in Banach spaces, Math. Japon. 48 (1998), 1–9.Google Scholar
  25. 25.
    Takahashi, W. and Ueda, Y.: On Reich's strong convergence theorems for resolvents of accretive operators, J. Math. Anal. Appl. 104 (1984), 546–553.Google Scholar
  26. 26.
    Tan, K. K. and Xu, H. K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993), 301–308.Google Scholar
  27. 27.
    Wittmann, R.: Approximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992), 486–491.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Shoji Kamimura
    • 1
  • Wataru Takahashi
    • 2
  1. 1.Department of Mathematical and Computing SciencesTokyo Institute of TechnologyTokyoJapan
  2. 2.Department of Mathematical and Computing SciencesTokyo Institute of TechnologyTokyoJapan

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