×

Asymptotic properties of nonexpansive iterations in reflexive spaces. (English) Zbl 0951.47055

The authors study asymptotic behavior of a nonexpansive (resp., firmly nonexpansive) sequence \(x_n\) in a reflexive Banach space. They show that the set of weak limit points of the sequence \(x_n/n\) (resp., \(x_{n+1}-x_n\)) lies on a convex subset of the sphere centered at the origin of radius \(d=\lim_{n\to\infty}\|x_n/n\|\). This fact yields previous results of B. Djafari Rouhani [Proc. Am. Math. Soc. 117, No. 4, 951-956 (1993; Zbl 0784.47048), Proc. Am. Math. Soc. 123, No. 3, 771-777 (1995; Zbl 0827.47042)] and J. S. Jung and J. S. Park [Proc. Am. Math. Soc. 124, No. 2, 475-480 (1996; Zbl 0846.47039)]. Some potential applications (for instance to study of asymptotic behavior of nonexpansive mappings) are also discussed.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
46B10 Duality and reflexivity in normed linear and Banach spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baillon, J. B.; Bruck, R. E.; Reich, S., On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces, Houston J. Math., 4, 1-9 (1978) · Zbl 0396.47033
[2] Rouhani, B. Djafari, Asymptotic behavior of unbounded nonexpansive sequences in Banach spaces, Proc. Amer. Math. Soc., 117, 951-956 (1993) · Zbl 0784.47048
[3] Rouhani, B. Djafari, Asymptotic behavior of firmly nonexpansive sequences, Proc. Amer. Math. Soc., 123, 771-777 (1995) · Zbl 0827.47042
[4] Goebel, K.; Kirk, W. A., Topics in Metric Fixed Point Theory (1990), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0708.47031
[5] Goebel, K.; Reich, S., Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics, 83 (1984), Dekker: Dekker New York/Basel · Zbl 0537.46001
[6] James, R. C., Characterizations of reflexivity, Studia Math., 23, 205-216 (1964) · Zbl 0113.09303
[7] Jung, J. S.; Park, J. S., Asymptotic behavior of nonexpansive sequences and mean points, Proc. Amer. Math. Soc., 124, 475-480 (1996) · Zbl 0846.47039
[8] Kohlberg, E.; Neyman, A., Asymptotic behavior of nonexpansive mappings in normed linear spaces, Israel J. Math., 38, 269-275 (1981) · Zbl 0476.47045
[9] Pazy, A., Asymptotic behavior of contractions in Hilbert space, Israel J. Math., 9, 235-240 (1971) · Zbl 0225.54032
[10] Plant, A.; Reich, S., The asymptotics of nonexpansive iterations, J. Funct. Anal., 54, 308-319 (1983) · Zbl 0542.47045
[11] Reich, S., Asymptotic behavior of contractions in Banach spaces, J. Math. Anal. Appl., 44, 57-70 (1973) · Zbl 0275.47034
[12] Reich, S.; Shafrir, I., The asymptotic behavior of firmly nonexpansive mappings, Proc. Amer. Math. Soc., 101, 246-250 (1987) · Zbl 0649.47043
[13] Takahashi, W., The asymptotic behavior of nonlinear semigroups and invariant means, J. Math. Anal. Appl., 109, 130-139 (1985) · Zbl 0593.47057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.