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Browder's Convergence for One-Parameter Nonexpansive Semigroups

  • Shigeki Akiyama (a1) and Tomonari Suzuki (a2)
Abstract

We give the sufficient and necessary conditions of Browder's convergence theorem for one-parameter nonexpansive semigroups which was proved by Suzuki. We also discuss the perfect kernels of topological spaces.

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References
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[1] Belluce, L. P. and Kirk, W. A., Nonexpansive mappings and fixed-points in Banach spaces. Illinois J. Math 11(1967), 474479.
[2] Browder, F. E., Nonexpansive nonlinear operators in a Banach space. Proc. Nat. Acad. Sci. U.S.A. 54(1965), 10411044. doi:10.1073/pnas.54.4.1041
[3] Browder, F. E., Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces. Arch. Rational Mech. Anal. 24(1967), 8290.
[4] Bruck, R. E., A common fixed point theorem for a commuting family of nonexpansive mappings. Pacific J. Math. 53(1974), 5971.
[5] DeMarr, R., Common fixed points for commuting contraction mappings. Pacific J. Math. 13(1963), 11391141.
[6] Goebel, K. and Kirk, W. A., Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics 28, Cambridge University Press Cambridge, 1990.
[7] Kirk, W. A. and Sims, B., (eds.) Handbook of Metric Fixed Point Theory. Kluwer Academic Publishers, dordrecht, 2001.
[8] Kuratowski, K., Topology. I. Academic Press, New York, 1966.
[9] Lim, T. C., A fixed point theorem for families on nonexpansive mappings. Pacific J. Math. 53(1974), 487493.
[10] Suzuki, T., On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces. Proc. Amer. Math. Soc. 131(2003), no. 7, 21332136. doi:10.1090/S0002-9939-02-06844-2
[11] Suzuki, T., Common fixed points of one-parameter nonexpansive semigroups. Bull. London Math. Soc. 38(2006), no. 6, 10091018. doi:10.1112/S0024609306018893
[12] Suzuki, T., Browder's type convergence theorems for one-parameter semigroups of nonexpansive mappings in Banach spaces. Israel J. Math. 157(2007), 239257. doi:10.1007/s11856-006-0010-6
[13] Suzuki, T., Some comments about recent results on one-parameter nonexpansive semigroups. Bull. Kyushu Inst. Technol. Pure Appl. Math. 54(2007), 1326.
[14] Suzuki, T., Browder convergence and Mosco convergence for families of nonexpansive mappings. Cubo 10(2008), no. 4, 101108.
[15] Takahashi, W., Nonlinear Functional Analysis. Fixed Point Theory and its Applications. Yokohama Publishers, Yokohama, 2000.
[16] Willard, S., General Topology. Dover, Mineola, NY, 2004.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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