Abstract
In this paper, we give an investigation of the asymptotics for the iterations associated with (c)-mappings acting on unbounded closed convex subsets of Banach spaces. In particular, the almost fixed point property (in short; AFPP) and conditions ensuring an ergodic type theorem for this class of mappings are established.
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References
Alspach, D.: A fixed point free nonexpansive map. Proc. Am. Math. Soc. 82, 423–424 (1981)
Ariza-Ruiz, D., Leustean, L., Lopez-Acedo, G.: Firmly nonexpansive mappings in classes of geodesic spaces. Trans. Am. Math. Soc. 336(8), 4299–4322 (2014)
Bae, J.S.: Fixed point theorems of generalized nonexpansive maps. J. Korean Math. Soc. 21(2), 233–248 (1984)
Baillon, J.B., Bruck, R.E., Reich, S.: On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces. Houst. J. Math. 4(1), 1–9 (1978)
Benavides, T.D.: The failure of the fixed point property for unbounded sets in $c_0$. Proc. Am. Math. Soc. 140(2), 645–650 (2012)
Bogin, J.: A generalization of a fixed point theorem of Goebel, Kirk and Shimi. Can. Math. Bull. 19(1), 7–12 (1976)
Browder, F.E.: Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. USA 54(4), 1041–1044 (1965)
Dowling, P.N., Lennard, C.J.: Every nonreflexive subspace of $L_1([0, 1])$ fails the fixed point property. Proc. Am. Math. Soc. 125(2), 443–446 (1997)
Dowling, P.N., Lennard, C.J., Turett, B.: Weak compactness is equivalent to the fixed point property in $c_0$. Proc. Am. Math. Soc. 132(6), 1659–1666 (2004)
Fekete, M.: Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Math. Z. 17, 228–249 (1923)
Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)
Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, New York (1984)
Göhde, D.: Zum Prinzip der kontraktiven Abbildung. Math. Nachr. 30, 251–258 (1965)
Hanebaly, E.: The fixed point property in Banach spaces via strict convexity and the Kadec–Klee property. J. Fixed Point. Theory. Appl. 20, 30 (2020)
Karlovitz, L.A.: On nonexpansive mappings. Proc. Am. Math. Soc. 55(2), 321–325 (1976)
Khamsi, M.A., Kirk, W.A.: An Introduction to Metric Spaces and Fixed Point Theory. A Wiley-Interscience Series of Texts, Monographs and Tracts, Pure and Applied Mathematics. Wiley, New York (2001)
Kirk, W.A.: A fixed point theorem for mappings which do not increase distances. Am. Math. Mon. 72, 1004–1006 (1965)
Kirk, W.A., Shahzad, N.: Orbital fixed point conditions in geodesic spaces. Fixed Point Theory 21(1), 221–238 (2020)
Lin, P.K.: Unconditional bases and fixed points of nonexpansive mappings. Pac. J. Math. 116(1), 69–76 (1985)
Maurey, B.: Points fixes des contractions sur un convexe fermé de $L^{1}$. Séminaire d’analyse fonctionnelle, École polytechnique, Palaiseau (1980–1981)
Matoušková, E., Reich, S.: Reflexivity and approximate fixed points. Stud. Math. 159(3), 403–415 (2003)
Pazy, A.: Asymptotic behavior of contractions in Hilbert spaces. Isr. J. Math. 9, 235–240 (1971)
Plant, A.T., Reich, S.: The asymptotics of nonexpansive iterations. J. Funct. Anal. 54, 308–319 (1983)
Ray, W.O.: The fixed point property and unbounded sets in Hilbert spaces. Trans. Am. Math. Soc. 258(2), 531–537 (1980)
Reich, S.: The almost fixed point property for nonexpansive mappings. Proc. Am. Math. Soc. 88(1), 44–46 (1983)
Reich, S., Shafrir, I.: The asymptotic behavior of firmly nonexpansive mappings. Proc. Am. Math. Soc. 101(2), 246–250 (1987)
Reich, S.: Asymptotic behavior of Contractions in Banach spaces. J. Math. Anal. Appl. 44, 57–70 (1973)
Reich, S.: The fixed point property for nonexpansive mappings. I. Amer. Math. Monthly. 83, 266–268 (1976)
Reich, S.: The fixed point property for nonexpansive mappings. II. Am. Math. Mon. 87, 292–294 (1980)
Reich, S.: Some remarks concerning contraction mappings. Can. Math. Bull. 14(2), 121–124 (1971)
Reich, S.: Fixed points of contractive functions. Boll. Un. Mat. Ital. 5, 26–42 (1972)
Reich, S., Shoikhet, D.: Nonlinear Semigroups, Fixed Points, and Geometry of Domains in Banach Spaces. Imperial College Press, London (2005)
Shafrir, I.: The approximate fixed point property in Banach and hyperbolic spaces. Isr. J. Math. 71(2), 211–223 (1990)
Smarzewski, R.: On firmly nonexpansive mappings. Proc. Am. Math. Soc. 113(3), 723–725 (1991)
Smyth, M.A.: The fixed point problem for generalised nonexpansive maps. Bull. Aust. Math. Soc. 55, 45–61 (1997)
Suzuki, T.: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. J. Math. Anal. Appl. 340, 1088–1095 (2008)
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The authors are very grateful to the anonymous referees for their valuable remarks and suggestions which helped us to improve the quality of this manuscript.
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This work is supported by the research team RPC (Controllability and Perturbation Results) in the laboratory of Informatics and Mathematics (LIM) at the university of Souk-Ahras (Algeria).
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Dehici, A., Redjel, N. On the asymptotics of (c)-mapping iterations . J. Fixed Point Theory Appl. 22, 99 (2020). https://doi.org/10.1007/s11784-020-00833-1
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DOI: https://doi.org/10.1007/s11784-020-00833-1