On the existence of fixed points for typical nonexpansive mappings on spaces with positive curvature. (English) Zbl 1491.47041

Summary: We show that the typical nonexpansive mapping on a small enough subset of a \(\mathrm{CAT} (\kappa)\)-space is a contraction in the sense of E. Rakotch [Proc. Am. Math. Soc. 13, 459–465 (1962; Zbl 0105.35202)]. By typical we mean that the set of nonexpansive mapppings without this property is a \(\sigma\)-porous set and therefore also of the first Baire category. Moreover, we exhibit metric spaces where strict contractions are not dense in the space of nonexpansive mappings. In some of these cases we show that all continuous self-mappings have a fixed point nevertheless.


47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
54E52 Baire category, Baire spaces
54E40 Special maps on metric spaces


Zbl 0105.35202
Full Text: DOI arXiv


[1] C. Bargetz and M. Dymond, \( \sigma \)-porosity of the set of strict contractions in a space of non-expansive mappings, Israel J. Math. 214 (2016), 235-244. · Zbl 1354.54016 · doi:10.1007/s11856-016-1372-z
[2] C. Bargetz, M. Dymond and S. Reich, Porosity results for sets of strict contractions on geodesic metric spaces, Topol. Methods Nonlinear Anal. 50 (2017), 89-124. · Zbl 1474.54062
[3] Y. Benyamini and Y. Sternfeld, Spheres in infinite-dimensional normed spaces are Lipschitz contractible, Proc. Amer. Math. Soc. 88 (1983), 439-445. · Zbl 0518.46010 · doi:10.1090/S0002-9939-1983-0699410-7
[4] M.R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften, vol. 319, Springer-Verlag, Berlin, 1999. · Zbl 0988.53001
[5] F.E. Browder, Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sci. USA 53 (1965), 1272-1276. · Zbl 0125.35801
[6] D. Burago, Y. Burago and S. Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. · Zbl 0981.51016
[7] F.S De Blasi and J. Myjak, Sur la convergence des approximations successives pour les contractions non linéaires dans un espace de Banach, C.R. Acad. Sci. Paris Sér. A-B 283 (1976), Aiii, A185-A187. · Zbl 0332.47028
[8] F.S. De Blasi and J. Myjak, Sur la porosité de l’ensemble des contractions sans point fixe, C.R. Acad. Sci. Paris Sér. I Math. 308 (1989), 51-54. · Zbl 0657.47053
[9] A. Denjoy, Leçons sur le Calcul des Coefficients d’une Série Trigonométrique. \romTome II. Métrique et Topologie d’Ensembles Parfaits et de Fonctions, Gauthier-Villars, Paris, 1941. · Zbl 0063.01081
[10] J. Dugundji, An extension of Tietze’s theorem, Pacific J. Math. 1 (1951), 353-367. · Zbl 0043.38105 · doi:10.2140/pjm.1951.1.353
[11] R. Espínola and A. Fernández-León, \( \text{CAT}(k)\)-spaces, weak convergence and fixed points, J. Math. Anal. Appl. 353 (2009), 410-427. · Zbl 1182.47043
[12] T. Ezawa, Convergence to a common fixed point of a finite family of nonexpansive mappings on the unit sphere of a Hilbert space, (2020), preprint (arXiv:2002.04305).
[13] K. Goebel and W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, Cambridge, 1990. · Zbl 0708.47031
[14] K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics, vol. 83, Marcel Dekker, Inc., New York, 1984. · Zbl 0537.46001
[15] M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75-263. · Zbl 0634.20015
[16] J.S. He, D.H. Fang, G. López and C. Li, Mann’s algorithm for nonexpansive mappings in \(\text{CAT}(\kappa)\) spaces, Nonlinear Anal. 75 (2012), 445-452. · Zbl 1319.47053
[17] \plsc B. Pi¹tek, The fixed point property and unbounded sets in spaces of negative curvature, Israel J. Math. 209 (2015), 323-334. · Zbl 1327.54049
[18] E. Rakotch, A note on contractive mappings, Proc. Amer. Math. Soc. 13 (1962), 459-465. · Zbl 0105.35202 · doi:10.1090/S0002-9939-1962-0148046-1
[19] S. Reich and I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal. 15 (1990), 537-558. · Zbl 0728.47043 · doi:10.1016/0362-546X(90)90058-O
[20] S. Reich and A.J. Zaslavski, The set of noncontractive mappings is \(\sigma \)-porous in the space of all nonexpansive mappings, C.R. Acad. Sci. Paris Sér. I Math. 333 (2001), 539-544. · Zbl 1001.47036 · doi:10.1016/S0764-4442(01)02087-0
[21] S. Reich and A.J. Zaslavski, Two porosity theorems for nonexpansive mappings in hyperbolic spaces, J. Math. Anal. Appl. 433 (2016), 1220-1229. · Zbl 1326.54048 · doi:10.1016/j.jmaa.2015.08.043
[22] J.H. Wells and L.R. Williams, Embeddings and Extensions in Analysis, Springer-Verlag, New York, Heidelberg, 1975; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84. · Zbl 0324.46034
[23] L. Zajíček, On \(\sigma \)-porous sets in abstract spaces, Abstr. Appl. Anal. (2005), 509-534. · Zbl 1098.28003
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.