Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-01-24T01:21:06.892Z Has data issue: false hasContentIssue false
  • English
  • Français

A strong convergence theorem for contraction semigroups in Banach spaces

Published online by Cambridge University Press:  17 April 2009

Hong-Kun Xu
Hong-Kun Xu
Affiliation:
School of Mathematical Sciences University of KwaZulu-Natal, Westville Campus, Private Bag X54001, Durban 4000, South Africa e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We establish a Banach space version of a theorem of Suzuki [8]. More precisely we prove that if X is a uniformly convex Banach space with a weakly continuous duality map (for example, lp for 1 < p < ∞), if C is a closed convex subset of X, and if F = {T (t): t ≥ 0} is a contraction semigroup on C such that Fix(F) ≠ ∅, then under certain appropriate assumptions made on the sequences {αn} and {tn} of the parameters, we show that the sequence {xn} implicitly defined by

for all n ≥ 1 converges strongly to a member of Fix(F).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 72 , Issue 3 , December 2005 , pp. 371 - 379
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Browder, F.E., Fixed point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1272–1276.Google ScholarPubMed
[2]Browder, F.E., ‘Convergence theorems for sequences of nonlinear operators in Banach spaces’, Math. Z. 100 (1967), 201–225.CrossRefGoogle Scholar
[3]Bruck, R.E., ‘A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces’, Israel J. Math. 32 (1979), 107–116.CrossRefGoogle Scholar
[4]Goebel, K. and Reich, S., Uniform convexity, hyperbolic geometry, and nonexpansive mappings (Marcel Dekker, New York, 1984).Google Scholar
[5]Lim, T.C. and Xu, H.K., ‘Fixed point theorems for asymptotically nonexpansive mappings’, Nonlinear Anal. 22 (1994), 1345–1355.CrossRefGoogle Scholar
[6]Reich, S., ‘Strong convergence theorems for resolvents of accretive operators in Banach spaces’, J. Math. Anal. Appl. 75 (1980), 287–292.CrossRefGoogle Scholar
[7]Shioji, N. and Takahashi, W., ‘Strong convergence theorems for asymptotically nonexpansive mappings in Hilbert spaces’, Nonlinear Anal. 34 (1998), 87–99.CrossRefGoogle Scholar
[8]Suzuki, T., ‘On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces’, Proc. Amer. Math. Soc. 131 (2002), 2133–2136.CrossRefGoogle Scholar
[9]Xu, H.K., ‘Approximations to fixed points of contraction semigroups in Hilbert spaces’, Numer. Funct. Anal. Optim. 19 (1998), 157–163.CrossRefGoogle Scholar
[10]Xu, H.K., ‘Iterative algorithms for nonlinear operators’, J. London Math. Soc. 66 (2002), 240–256.CrossRefGoogle Scholar