×

Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. (English) Zbl 1134.47052

Summary: We prove a strong convergence theorem by the hybrid method for a family of nonexpansive mappings which generalizes the theorems of K.Nakajo and W.Takahashi [J. Math.Anal.Appl.279, No.2, 372–379 (2003; Zbl 1035.47048)]. Furthermore, we obtain another strong convergence theorem for the family of nonexpansive mappings by a hybrid method which is different from the one of Nakajo and Takahashi. Using this theorem, we get some new results for a single nonexpansive mapping or a family of nonexpansive mappings in a Hilbert space.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H20 Semigroups of nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Citations:

Zbl 1035.47048
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baillon, J. B.; Brézis, H., Une remarque sun le compertement asymptotique des semigroupes nonlinéaires, Houston J. Math., 2, 5-7 (1976) · Zbl 0318.47039
[2] Brézis, H., Opérateurs maximaux monotones, Mathematics Studies, vol. 5 (1973), North-Holland: North-Holland Amsterdam · Zbl 0252.47055
[3] Halpern, B., Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73, 957-961 (1967) · Zbl 0177.19101
[4] Kamimura, S.; Takahashi, W., Approximating solutions of maximal monotone operators in Hilbert spaces, J. Approx. Theory, 106, 226-240 (2000) · Zbl 0992.47022
[5] Mann, W. R., Mean value methods in iteration, Proc. Amer. Math. Soc., 4, 506-510 (1953) · Zbl 0050.11603
[6] Nakajo, K.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279, 372-379 (2003) · Zbl 1035.47048
[7] Nakajo, K.; Shimoji, K.; Takahashi, W., Strong convergence to common fixed points of families of nonexpansive mappings in Banach spaces, J. Nonlinear Convex Anal., 8, 11-34 (2007) · Zbl 1125.49024
[8] Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73, 591-597 (1967) · Zbl 0179.19902
[9] Rockafellar, R. T., Characterization of the subdifferentials of convex functions, Pacific J. Math., 17, 497-510 (1966) · Zbl 0145.15901
[10] Rockafellar, R. T., On the maximal monotonicity of subdifferential mappings, Pacific J. Math., 33, 209-216 (1970) · Zbl 0199.47101
[11] Shimizu, N.; Takahashi, W., Strong convergence to common fixed points of families of nonexpansive mappings, J. Math. Anal. Appl., 211, 71-83 (1997) · Zbl 0883.47075
[12] Takahashi, W., Nonlinear Functional Analysis (2000), Yokohama Publishers: Yokohama Publishers Yokohama
[13] Takahashi, W., Convex Analysis and Approximation of Fixed Points (2000), Yokohama Publishers: Yokohama Publishers Yokohama, (in Japanese)
[14] Takahashi, W., Introduction to Nonlinear and Convex Analysis (2005), Yokohama Publishers: Yokohama Publishers Yokohama, (in Japanese)
[16] Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. Math., 58, 486-491 (1992) · Zbl 0797.47036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.