Abstract
In this paper, we prove Browder’s type convergence theorems for one-parameter strongly continuous semigroups of nonexpansive mappings in Banach spaces.
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J. B. Baillon, Quelques aspects de la théorie des points fixes dans les espaces de Banach. I, II. (in French), Séminaire d’Analyse Fonctionnelle (1978–1979), Exp. No. 7–8, 45 pp., École Polytech., Palaiseau, 1979.
S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fundamenta Mathematicae 3 (1922), 133–181.
L. P. Belluce and W. A. Kirk, Nonexpansive mappings and fixed-points in Banach spaces, Illinois Journal of Mathematics 11 (1967), 474–479.
M. S. Brodskii and D. P. Mil’man, On the center of a convex set (in Russian), Doklady Akademii Nauk SSSR 59 (1948), 837–840.
F. E. Browder, Fixed-point theorems for noncompact mappings in Hilbert space, Proceedings of the National Academy of Sciences of the United States of America 53 (1965), 1272–1276.
F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proceedings of the National Academy of Sciences of the United States of America 54 (1965), 1041–1044.
F. E. Browder, Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces, Archive for Rational Mechanics and Analysis 24 (1967), 82–90.
R. E. Bruck, Nonexpansive retracts of Banach spaces, Bulletin of the American Mathematical Society 76 (1970), 384–386.
R. E. Bruck, A common fixed point theorem for a commuting family of nonexpansive mappings, Pacific Journal of Mathematics 53 (1974), 59–71.
D. Göhde, Zum Prinzip def kontraktiven Abbildung, Mathematische Nachrichten 30 (1965), 251–258.
J.-P. Gossez and E. Lami Dozo, Some geometric properties related to the fixed point theory for nonexpansive mappings, Pacific Journal of Mathematics 40 (1972), 565–573.
J. L. Kelley, General Topology, Van Nostrand Reinhold Company, New York, 1955.
W. A. Kirk, A fixed point theorem for mappings which do not increase distances, The American Mathematical Monthly 72 (1965), 1004–1006.
T. C. Lim, A fixed point theorem for families on nonexpansive mappings, Pacific Journal of Mathematics 53 (1974), 487–493.
Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bulletin of the American Mathematical Society 73 (1967), 591–597.
S. Reich, Asymptotic behavior of contractions in Banach spaces, Journal of Mathematical Analysis and Applications 44 (1973), 57–70.
S. Reich, Product formulas, nonlinear semigroups, and accretive operators, Journal of Functional Analysis 36 (1980), 147–168.
S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, Journal of Mathematical Analysis and Applications 75 (1980), 287–292.
S. Reich, On asymptotic behavior of nonlinear semigroups and the range of accretive operators, Journal of Mathematical Analysis and Applications 79 (1981), 113–126.
S. Reich, Convergence, resolvent consistency, and the fixed point property for nonexpansive mappings, Contemporary Mathematics 18 (1983), 167–174.
N. Shioji and W. Takahashi, Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces, Nonlinear Analysis 34 (1998), 87–99.
T. Suzuki, On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces, Proceedings of the American Mathematical Society 131 (2003), 2133–2136.
T. Suzuki, The set of common fixed points of a one-parameter continuous semigroup of mappings is \( F(T(1)) \cap F(T(\sqrt 2 )) \), Proceedings of the American Mathematical Society 134 (2006), 673–681.
T. Suzuki, Common fixed points of one-parameter nonexpansive semigroups in strictly convex Banach spaces, Abstract and Applied Analysis 2006 (2006), Article ID 58684, 1–10.
T. Suzuki, Browder’s type convergence theorem for one-parameter semigroups of nonexpansive mappings in Hilbert spaces, Proceedings of the Fourth International Conference on Nonlinear Analysis and Convex Analysis (W. Takahashi and T. Tanaka Eds.), Yokohama Publishers, to appear.
W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000.
W. Takahashi and Y. Ueda, On Reich’s strong convergence theorems for resolvents of accretive operators, Journal of Mathematical Analysis and Applications 104 (1984), 546–553.
B. Turett, A dual view of a theorem of Baillon, in Nonlinear Analysis and Applications (St. Johns, Nfld., 1981), Lecture Notes in Pure and Applied Mathematics, Vol. 80, Dekker, New York, 1982, pp. 279–286.
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The author is supported in part by Grants-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology.
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Suzuki, T. Browder’s type convergence theorems for one-parameter semigroups of nonexpansive mappings in Banach spaces. Isr. J. Math. 157, 239–257 (2007). https://doi.org/10.1007/s11856-006-0010-6
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DOI: https://doi.org/10.1007/s11856-006-0010-6