Abstract

We introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive nonself-mapping and the set of solutions of the variational inequality for an inverse-strongly-monotone mapping in a Hilbert space. Then we show that the sequence converges strongly to a common element of two sets. Using this result, we consider the problem of finding a common element of the set of zeros of a maximal montone mapping and the set of zeros of an inverse-strongly-montone mapping and the problem of finding a common element of the closed convex set and the set of zeros of the gradient of a continuously Frechet differentiable convex functional.

Author information

Contact details are reproduced from the original publication and may be historical.

Hideaki Iiduka

Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology,
Oh-Okayama, Meguro-ku, Tokyo 152-8522, Japan

Hideaki.Iiduka@is.titech.ac.jp

Wataru Takahashi

Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology,
Oh-Okayama, Meguro-ku, Tokyo 152-8522, Japan

Wataru@is.titech.ac.jp

Keywords

  • Metric projection
  • inverse-strongly-monotone mapping
  • nonexpansive nonself-mapping
  • variational inequality
  • strong convergence

References

  1. J. B. Baillon, G. Haddad: Quelques propriétés des opérateurs angle-bornés et n-cycliquement monotones, Israel J. Math. 26 (1977) 137–150.https://doi.org/10.1007/BF03007664Google Scholar
  2. F. E. Browder: Nonlinear monotone operators and convex sets in Banach spaces, Bull. Amer. Math. Soc. 71 (1965) 780–785.https://doi.org/10.1090/S0002-9904-1965-11391-XGoogle Scholar
  3. F. E. Browder: Nonlinear Operators and Nonlinear Equations of Evolutions in Banach Spaces, Proc. Sympos. Pure Math. 18-2, Amer. Math. Soc. Providence, R.I. (1976).https://doi.org/10.1090/pspum/018.2/9972Google Scholar
  4. F. E. Browder: The fixed point theory of multi-valued mappings in topological vector spaces, Math. Ann. 177 (1968) 283–301.https://doi.org/10.1007/BF01350721Google Scholar
  5. F. E. Browder, W. V. Petryshyn: Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967) 197–228.https://doi.org/10.1016/0022-247X(67)90085-6Google Scholar
  6. R. E. Bruck: Asymptotic convergence of nonlinear contraction semigroups in Hilbert space, J. Funct. Anal. 18 (1975) 15–26.https://doi.org/10.1016/0022-1236(75)90027-0Google Scholar
  7. R. E. Bruck: On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space, J. Math. Anal. Appl. 61 (1977) 159–164.https://doi.org/10.1016/0022-247X(77)90152-4Google Scholar
  8. A. Faik, A. Syam: Differential inclusions governed by a nonconvex sweeping process, J. Nonlinear and Convex Analysis 2 (2001) 381–392.Google Scholar
  9. B. R. Halpern: Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967) 957–961.https://doi.org/10.1090/S0002-9904-1967-11864-0Google Scholar
  10. H. Iiduka, W. Takahashi: Strong convergence theorems for nonexpansive mappings and inverse-strongly-monotone mappings, Nonlinear Anal., to appear.Google Scholar
  11. H. Iiduka, W. Takahashi, M. Toyoda: Approximation of solutions of variational inequalities for monotone mappings, Panamer. Math. J., to appear.Google Scholar
  12. J. L. Lions, G. Stampacchia: Variational inequalities, Comm. Pure Appl. Math. 20 (1967) 493–517.https://doi.org/10.1002/cpa.3160200302Google Scholar
  13. F. Liu, M. Z. Nashed: Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-Valued Anal. 6 (1998) 313–344.https://doi.org/10.1023/A:1008643727926Google Scholar
  14. S. Matsushita, D. Kuroiwa: Approximation of fixed points of nonexpansive nonself-mappings, Sci. Math. Jpn. 57 (2003) 171–176.Google Scholar
  15. K. Nakajo, W. Takahashi: Strong and weak convergence theorems by an improved splitting method, Comm. Appl. Nonlinear Anal. 9 (2002) 99–107.Google Scholar
  16. Z. Opial: Weak convergence of the sequence of successive approximation for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967) 591–597.https://doi.org/10.1090/S0002-9904-1967-11761-0Google Scholar
  17. R. T. Rockafellar: On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970) 75–88.https://doi.org/10.1090/S0002-9947-1970-0282272-5Google Scholar
  18. R. T. Rockafellar: Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976) 877–898.https://doi.org/10.1137/0314056Google Scholar
  19. W. Takahashi: Nonlinear variational inequalities and fixed point theorems, J. Math. Soc. Japan 28 (1976) 168–181.https://doi.org/10.2969/jmsj/02810168Google Scholar
  20. W. Takahashi: Nonlinear complementarity problem and systems of convex inequalities, J. Optim. Theory Appl. 24 (1978) 493–508.https://doi.org/10.1007/BF00932892Google Scholar
  21. W. Takahashi: Convex Analysis and Approximation of Fixed Points, Yokohama Publishers, Yokohama (2000) (Japanese).Google Scholar
  22. W. Takahashi: Nonlinear Functional Analysis, Yokohama Publishers, Yokohama (2000).Google Scholar
  23. W. Takahashi, M. Toyoda: Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118 (2003) 417–428.https://doi.org/10.1023/A:1025407607560Google Scholar

Suggested citation

H. Iiduka, W. Takahashi. “Strong Convergence Theorems for Nonexpansive Nonself-Mappings and Inverse-Strongly-Monotone Mappings.” Journal of Convex Analysis 11 (2004), No. 1, 69–79. https://doi.org/10.68381/jca11005

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Published by Heldermann Verlag, 2004. Rights now held by Banach Press LLC.