Taiwanese Journal of Mathematics

THE SET OF COMMON FIXED POINTS OF A ONE-PARAMETER CONTINUOUS SEMIGROUP OF NONEXPANSIVE MAPPINGS IS $F(\frac{1}{2} T(1) + \frac{1}{2} T(\sqrt{2}))$ IN STRICTLY CONVEX BANACH SPACES

Tomonari Suzuki

Full-text: Open access

Abstract

In this paper, we prove the following. Let $E$ be a strictly convex Banach space. Let $\{ T(t) : t \geq 0 \}$ be a one-parameter strongly continuous semigroup of nonexpansive mappings on a subset $C$ of $E$. Then \[ \bigcap_{t \geq 0} F(T(t)) = F\left( \frac{1}{2} T(1) + \frac{1}{2} T(\sqrt{2}) \right) \] holds, where $F(T(t))$ is the set of fixed points of $T(t)$ for each $t \geq 0$.

Article information

Source
Taiwanese J. Math., Volume 10, Number 2 (2006), 381-397.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500403831

Digital Object Identifier
doi:10.11650/twjm/1500403831

Mathematical Reviews number (MathSciNet)
MR2208273

Zentralblatt MATH identifier
1101.47044

Subjects
Primary: 47H20: Semigroups of nonlinear operators [See also 37L05, 47J35, 54H15, 58D07] 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30]

Keywords
nonexpansive semigroup common fixed point strict convexity irrational number

Citation

Suzuki, Tomonari. THE SET OF COMMON FIXED POINTS OF A ONE-PARAMETER CONTINUOUS SEMIGROUP OF NONEXPANSIVE MAPPINGS IS $F(\frac{1}{2} T(1) + \frac{1}{2} T(\sqrt{2}))$ IN STRICTLY CONVEX BANACH SPACES. Taiwanese J. Math. 10 (2006), no. 2, 381--397. doi:10.11650/twjm/1500403831. https://projecteuclid.org/euclid.twjm/1500403831


Export citation

References

  • [1.] S. Atsushiba and W. Takahashi, Strong convergence theorems for one-parameter nonexpansive semigroups with compact domains, in Fixed Point Theory and Applications, Volume 3, (Y. J. Cho, J. K. Kim and S. M. Kang Eds.), pp. 15-31, Nova Science Publishers, New York, 2002.
  • [2.] J. B. Baillon, Un théorème de type ergodique pour les contractions non linéaires dans un espace de Hilbert, C. R. Acad. Sci. Paris, Sér. A-B, 280 (1975), 1511-1514.
  • [3.] J. B. Baillon, Quelques properiétés de convergence asymptotique pour les semigroupes de contractions impaires, C. R. Acad. Sci. Paris, 283 (1976), A75-A78.
  • [4.] F. E. Browder, Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sci. USA, 53 (1965), 1272-1276.
  • [5.] F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. USA, 54 (1965), 1041-1044.
  • [6.] F. E. Browder, Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces, Arch. Ration. Mech. Anal., 24 (1967), 82-90.
  • [7.] R. E. Bruck, Properties of fixed-point sets of nonexpansive mappings in Banach spaces, Trans. Amer. Math. Soc., 179 (1973), 251-262.
  • [8.] R. E. Bruck, A common fixed point theorem for a commuting family of nonexpansive mappings, Pacific J. Math., 53 (1974), 59-71.
  • [9.] M. Edelstein and R. C. O'Brien, Nonexpansive mappings, asymptotic regularity and successive approximations, J. London Math. Soc., 17 (1978), 547-554.
  • [10.] D. Göhde, Zum Prinzip def kontraktiven Abbildung, Math. Nachr., 30 (1965), 251-258.
  • [11.] B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73 (1967), 957-961.
  • [12.] N. Hirano, Nonlinear ergodic theorems and weak convergence theorems, J. Math. Soc. Japan, 34 (1982), 35-46.
  • [13.] S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc., 59 (1976), 65-71.
  • [14.] S. Ishikawa, Common fixed points and iteration of commuting nonexpansive mappings, Pacific J. Math., 80 (1979), 493-501.
  • [15.] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72 (1965), 1004-1006.
  • [16.] M. A. Krasnosel'skiĭ, Two remarks on the method of successive approximations (in Russian), Uspehi Mat. Nauk 10 (1955), 123–127.
  • [17.] I. Miyadera and K. Kobayasi, On the asymptotic behaviour of almost-orbits of nonlinear contraction semigroups in Banach spaces, Nonlinear Anal., 6 (1982), 349-365.
  • [18.] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67 (1979), 274–276.
  • [19.] N. Shioji and W. Takahashi, Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces, Nonlinear Anal., 34 (1998), 87-99.
  • [20.] T. Suzuki, Strong convergence theorem to common fixed points of two nonexpansive mappings in general Banach spaces, J. Nonlinear Convex Anal., 3 (2002), 381-391.
  • [21.] T. Suzuki, On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces, Proc. Amer. Math. Soc., 131 (2003), 2133-2136.
  • [22.] T. Suzuki, Convergence theorems to common fixed points for infinite families of nonexpansive mappings in strictly convex Banach spaces, Nihonkai Math. J., 14 (2003), 43-54.
  • [23.] T. Suzuki, Common fixed points of two nonexpansive mappings in Banach spaces, Bull. Austral. Math. Soc., 69 (2004), 1-18.
  • [24.] T. Suzuki, Some remarks on the set of common fixed points of one-parameter semigroups of nonexpansive mappings in Banach spaces with the Opial property, Nonlinear Anal., 58 (2004), 441-458.
  • [25.] T. Suzuki, Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces, Fixed Point Theory Appl., 2005 (2005), 103-123.
  • [26.] T. Suzuki, Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl., 305 (2005), 227-239.
  • [27.] T. Suzuki, The set of common fixed points of a one-parameter continuous semigroup of mappings is $F \big( T(1) \big) \cap F \big( T(\sqrt2) \big)$, Proc. Amer. Math. Soc., 134 (2006), 673-681.
  • [28.] T. Suzuki, The set of common fixed points of an $n$-parameter continuous semigroup of mappings, Nonlinear Anal., 63 (2005), 1180-1190.
  • [29.] T. Suzuki, Common fixed points of commutative semigroups of nonexpansive mappings, arXiv Math. FA, 0404428 v1 23, April 2004.
  • [30.] T. Suzuki, Browder's type convergence theorem for one-parameter semigroups of nonexpansive mappings in Hilbert spaces, submitted.
  • [31.] T. Suzuki, Browder's type convergence theorems for one-parameter semigroups of nonexpansive mappings in Banach spaces, Israel J. Math., to appear.
  • [32.] T. Suzuki, Mosco convergence of the sets of fixed points for one-parameter nonexpansive semigroups, submitted.
  • [33.] T. Suzuki, Browder's type strong convergence theorems for infinite families of nonexpansive mappings in Banach spaces, submitted.
  • [34.] T. Suzuki and W. Takahashi, Strong convergence of Mann's type sequences for one-parameter nonexpansive semigroups in general Banach spaces, J. Nonlinear Convex Anal., 5 (2004), 209-216.
  • [35.] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000.
  • [36.] R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. $($Basel$)$, 58 (1992), 486-491.