Strong convergence theorems by hybrid methods for maximal monotone operators and relatively nonexpansive mappings in Banach spaces

Journal of Convex Analysis

Volumn 16, Issue 3-4, 2009, Pages 791-806

  Inoue, Go ^a   Takahashi, Wataru ^a   Zembayashi, Kei ^a  

^a Tokyo Institute of Technology ^*  (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 73349110388     PISSN: 09446532     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (29)

1. 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, A. G. Katrosatos (ed.), Marcel Dekker, New York (1996) 15-50.
2. H. Brézis, P. L. Lions: Produits infinis de résolvantes, Israel J. Math. 29 (1978) 329-345.
3. O. Güler: On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optimization 29 (1991) 403-419.
4. B. Halpern: Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967) 957-961.
5. S. Kamimura, W. Takahashi: Approximating solutions of maximal monotone operators in Hilbert spaces, J. Approximation Theory 106 (2000) 226-240.
6. S. Kamimura, W. Takahashi: Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim. 13 (2002) 938-945.
7. S. Kamimura, F. Kohsaka, W. Takahashi: Weak and strong convergence theorems for maximal monotone operators in a Banach space, Set-Valued Anal. 12 (2004) 417-429.
8. F. Kohsaka, W. Takahashi: Strong convergence of an iterative sequence for maximal monotone operators in a Banach space, Abstr. Appl. Anal. 2004 (2004) 239-249.
9. F. Kohsaka, W. Takahashi: Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces, SIAM J. Optim. 19 (2008) 824-835.
10. P. L. Lions: Une méhode itérative de résolution d'une inéquation variationnelle, Israel J. Math. 31 (1978) 204-208.
11. W. R. Mann: Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953) 506-510.
12. B. Martinet: Régularisation d'inéquations variationnelles par approximations successives, Rev. Franç. Inform. Rech. Opér. 4 (1970) 154-159.
13. S. Matsushita, W. Takahashi: Weak and strong convergence theorems for relatively non-expansive mappings in Banach spaces, Fixed Point Theory Appl. 2004 (2004) 37-47.
14. S. Matsushita, W. Takahashi: A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approximation Theory 134 (2005) 257-266.
15. K. Nakajo, W. Takahashi: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl 279 (2003) 372-379.
16. G. B. Passty: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space, J. Math. Anal. Appl. 72 (1979) 383-390.
17. S. Reich: A weak convergence theorem for the alternating method with Bregman distance, in: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G. Katrosatos (ed.), Marcel Dekker, New York (1996) 313-318.
18. S. Reich: Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67 (1979) 274-276.
19. R. T. Rockafellar: On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970) 75-88.
20. R. T. Rockafellar: Monotone operators and the proximal point algorithm, SIAM J. Control Optimization 14 (1976) 877-898.
21. N. Shioji, W. Takahashi: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (1997) 3641-3645.
22. M. V. Solodov, B. F. Svaiter: Forcing strong convergence of proximal point iterations in a Hilbert space, Math. Program., Ser. A 87 (2000) 189-202.
23. W. Takahashi: Nonlinear Functional Analysis, Yokohama Publishers, Yokohama (2000).
24. W. Takahashi: Convex Analysis and Approximation of Fixed Points, Yokohama Publishers, Yokohama (2000), (in Japanese).
25. W. Takahashi, Y. Takeuchi, R. Kubota: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 341 (2008) 276-286.
26. W. Takahashi, K. Zembayashi: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal., Theory Methods Appl. 70 (2009) 45-57.
27. R. Wittmann: Approximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992) 486-461.
28. C. Zǎlinescu: On uniformly convex functions, J. Math. Anal. Appl. 95 (1983) 344-374.
29. C. Zǎlinescu: Convex Analysis in General Vector Spaces, World Scientific, Singapore (2002).