Abstract
LetT be a nonexpansive self-mapping of a closed convex subsetC of a real Hilbert space. In this paper we deal with the structure of the weak ω-limit set of iterates {T nx}, establish conditions under which it is invariant underT, and show that {T nx} converges weakly iffT has a fixed-point andT nx-Tn+1x→0 weakly.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. B. Baillon,Un exemple concernant le comportement asymptotique de la solution du problème du/dt + ∂ ϕ(u)∋ 0, preprint. 3
J. B. Baillon,Un théorème de type ergodique pour les contractions non linéaires dans un espace de Hilbert, C. R. Acad. Sci. Paris A-B280 (1975), A1511-A1514.
J. B. Baillon,Quelques propriétés de convergence asymptotique pour les contractions impaires, to appear in C. R. Acad. Sci. Paris.
H. Brézis and F. E. Browder,Nonlinear ergodic theorems, Bull. Amer. Math. Soc.82 (1976), 959–961.
F. E. Browder and W. V. Petryshyn,The solution by iteration of nonlinear functional equations in Banach spaces, Bull. Amer. Math. Soc.72 (1966), 571–575.
R. E. Bruck,Asymptotic convergence of nonlinear contraction semigroups in Hilbert space, J. Functional Analysis18 (1975), 15–26.
R. E. Bruck,On the strong convergence of an averaging iteration for the solution of operator equations involving monotone operators in Hilbert space, to appear in J. Math. Anal. Appl.
L. W. Cohen,On the mean ergodic theorem, Ann. of Math.41 (1940), 505–509.
C. M. Dafermos and M. Slemrod,Asymptotic behavior of nonlinear contraction semigroups, J. Functional Analysis13 (1973), 97–106.
M. Edelstein,On non-expansive mappings of Banach spaces, Proc. Camb. Phil. Soc.60 (1964), 439–447.
M. Edelstein,The construction of an asymptotic center with a fixed-point property, Bull. Amer. Math. Soc.78 (1972), 206–208.
M. Edelstein,Fixed point theorems in uniformly convex Banach spaces, Proc. Amer. Math. Soc.44 (1974), 369–374.
M. Edelstein and A. C. Thompson,Contractions, isometries and some properties of inner-product spaces, Indag. Math.29 (1967), 326–331.
H. Fong and L. Sucheston,On a mixing property of operators in L p spaces, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete28 (1974), 165–171.
G. G. Lorentz,A contrubition to the theory of divergent series, Acta Math.80 (1948), 167–190.
Z. Opial,Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc.73 (1967), 591–597.
A. Pazy,On the asymptotic behavior of iterates of nonexpansive mappings in Hilbert space, Israel J. Math.26 (1977), 197–204.
S. Reich,Nonlinear evolution equations and nonlinear ergodic theorems, preprint.
C.-L. Yen,On the rest points of a nonlinear nonexpansive semigroup, Pacific J. Math.45 (1973), 699–706.
A. Zygmund,Trigonometric Series, Vol. I, Cambridge University Press, 1959.
Author information
Authors and Affiliations
Additional information
Dedicated to the memory of my father
Supported by NSF Grant MCS 76-08217.
Rights and permissions
About this article
Cite this article
Bruck, R.E. On the almost-convergence of iterates of a nonexpansive mapping in Hilbert space and the structure of the weak ω-limit set. Israel J. Math. 29, 1–16 (1978). https://doi.org/10.1007/BF02760397
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02760397