Abstract
In 1971, Pazy [Israel J. Math. 9 (1971), 235–240] presented a beautiful trichotomy result concerning the asymptotic behaviour of the iterates of a nonexpansive mapping. In this note, we analyze the fixedpoint- free case in more detail. Our results and examples give credence to the conjecture that the iterates always converge cosmically. The relationship to recent work by Lins [Proc. Amer. Math. Soc. 137 (2009), 2387–2392] is also discussed.
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Bauschke H. H.: The composition of finitely many projections onto closed convex sets in Hilbert space is asymptotically regular. Proc. Amer. Math. Soc. 131, 141–146 (2003)
Bauschke H. H., Borwein J. M.: Dykstra’s alternating projection algorithm for two sets. J. Approx. Theory 79, 418–443 (1994)
Bauschke H. H., Combettes P. L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)
H. H. Bauschke, V. Martín-Márquez, S. M. Moffat and X.Wang, Compositions and convex combinations of asymptotically regular firmly nonexpansive mappings are also asymptotically regular. Fixed Point Theory Appl. 2012 (2012), doi:10.1186/1687-1812-2012-53, 11 pages.
Beardon A. F.: Iteration of contraction and analytic maps. J. Lond. Math. Soc. (2) 41, 141–150 (1990)
Beardon A. F.: The dynamics of contractions. Ergodic Theory Dynam. Systems 17, 1257–1266 (1997)
Gaubert S., Vigeral G.: A maximin characterisation of the escape rate of non-expansive mappings in metrically convex spaces. Math. Proc. Cambridge Philos. Soc. 152, 341–363 (2012)
GeoGebra, http://www.geogebra.org.
Goebel K., Reich S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)
Karlsson A.: Non-expanding maps and Busemann functions. Ergodic Theory Dynam. Systems 21, 1447–1457 (2001)
Karlsson A.: Linear rate of escape and convergence in direction. In: Random Walks and Geometry, pp. 459–471. de Gruyter, Berlin (2004)
Lins B.: Asymptotic behavior of nonexpansive mappings in finite dimensional normed spaces. Proc. Amer. Math. Soc. 137, 2387–2392 (2009)
Mordukhovich B. S., Nam N. M.: An Easy Path to Convex Analysis and Applications. Morgan & Claypool Publishers, Williston, VT (2014)
Pazy A.: Asymptotic behavior of contractions in Hilbert space. Israel J. Math. 9, 235–240 (1971)
Plant A. T., Reich S.: The asymptotics of nonexpansive iterations. J. Funct. Anal. 54, 308–319 (1983)
S. Reich, Asymptotic behavior of contractions in Banach spaces. J. Math. Anal. Appl. 44 (1973), 57–50.
Reich S.: On the asymptotic behavior of nonlinear semigroups and the range of accretive operators. I. J. Math. Anal. Appl. 79, 113–126 (1981)
Reich S.: On the asymptotic behavior of nonlinear semigroups and the range of accretive operators. II. J. Math. Anal. Appl. 87, 134–146 (1982)
Rockafellar R. T.: Convex Analysis. Princeton University Press, Princeton (1970)
Rockafellar R. T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)
R. T. Rockafellar and R. J.-B. Wets, Variational Analysis. Corrected 3rd printing, Springer, New York, 2009.
von Neumann J.: On rings of operators. Reduction theory. Ann. of Math. (2) 50, 401–485 (1949)
Zarantonello E. H.: Projections on convex sets in Hilbert space and spectral theory. In: Contributions to Nonlinear Functional Analysis, pp. 237–424. Academic Press, New York (1971)
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Bauschke, H.H., Douglas, G.R. & Moursi, W.M. On a result of Pazy concerning the asymptotic behaviour of nonexpansive mappings. J. Fixed Point Theory Appl. 18, 297–307 (2016). https://doi.org/10.1007/s11784-015-0278-4
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DOI: https://doi.org/10.1007/s11784-015-0278-4