×

Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. (English) Zbl 1140.47041

The author introduces the following condition. A mapping \(T\) on a subset \(C\) of a Banach space \(E\) satisfies condition (C) if \(\frac12\| x-Tx\| \leq\| x-y\|\) implies that \(\| Tx-Ty\|\leq\| x-y\|\) for all \(x,y\in C\). This notion lies strictly between nonexpansiveness and quasi-nonexpansiveness. He also proves some fixed point theorems and convergence theorems for such mappings.

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
47J25 Iterative procedures involving nonlinear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baillon, J. B., Quelques aspects de la théorie des points fixes dans les espaces de Banach. I, II, (Séminaire d’Analyse Fonctionnelle (1978-1979), vols. 7-8 (1979), École Polytech.: École Polytech. Palaiseau), 45 pp. (in French) · Zbl 0414.47040
[2] Browder, F. E., Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Natl. Acad. Sci. USA, 53, 1272-1276 (1965) · Zbl 0125.35801
[3] Browder, F. E., Nonexpansive nonlinear operators in a Banach space, Proc. Natl. Acad. Sci. USA, 54, 1041-1044 (1965) · Zbl 0128.35801
[4] Connell, E. H., Properties of fixed point spaces, Proc. Amer. Math. Soc., 10, 974-979 (1959) · Zbl 0163.17705
[5] Diaz, J. B.; Metcalf, F. T., On the structure of the set of subsequential limit points of successive approximations, Bull. Amer. Math. Soc., 73, 516-519 (1967) · Zbl 0161.20103
[6] van Dulst, D., Equivalent norms and the fixed point property for nonexpansive mappings, J. London Math. Soc., 25, 139-144 (1982) · Zbl 0453.46017
[7] Edelstein, M.; O’Brien, R. C., Nonexpansive mappings, asymptotic regularity and successive approximations, J. London Math. Soc., 17, 547-554 (1978) · Zbl 0421.47031
[8] Goebel, K.; Kirk, W. A., Iteration processes for nonexpansive mappings, Contemp. Math., 21, 115-123 (1983) · Zbl 0525.47040
[9] Goebel, K.; Kirk, W. A., Topics in Metric Fixed Point Theory, Cambridge Stud. Adv. Math., vol. 28 (1990), Cambridge Univ. Press · Zbl 0708.47031
[10] Göhde, D., Zum Prinzip def kontraktiven Abbildung, Math. Nachr., 30, 251-258 (1965) · Zbl 0127.08005
[11] Gossez, J.-P.; Lami Dozo, E., Some geometric properties related to the fixed point theory for nonexpansive mappings, Pacific J. Math., 40, 565-573 (1972) · Zbl 0223.47025
[12] Ishikawa, S., Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc., 59, 65-71 (1976) · Zbl 0352.47024
[13] Kirk, W. A., A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72, 1004-1006 (1965) · Zbl 0141.32402
[14] Kirk, W. A., Caristi’s fixed point theorem and metric convexity, Colloq. Math., 36, 81-86 (1976) · Zbl 0353.53041
[15] Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73, 591-597 (1967) · Zbl 0179.19902
[16] Prus, S., Geometrical background of metric fixed point theory, (Kirk, W. A.; Sims, B., Handbook of Metric Fixed Point Theory (2001), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 93-132 · Zbl 1018.46010
[17] Subrahmanyam, P. V., Completeness and fixed-points, Monatsh. Math., 80, 325-330 (1975) · Zbl 0312.54048
[18] Suzuki, T., Krasnoselskii and Mann’s type sequences and Ishikawa’s strong convergence theorem, (Takahashi, W.; Tanaka, T., Proceedings of the Third International Conference on Nonlinear Analysis and Convex Analysis (2004), Yokohama Publishers), 527-539 · Zbl 1088.47512
[19] Suzuki, T., Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces, Fixed Point Theory Appl., 2005, 103-123 (2005) · Zbl 1123.47308
[20] Suzuki, T., Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl., 305, 227-239 (2005) · Zbl 1068.47085
[21] T. Suzuki, A generalized Banach contraction principle which characterizes metric completeness, Proc. Amer. Math. Soc., in press; T. Suzuki, A generalized Banach contraction principle which characterizes metric completeness, Proc. Amer. Math. Soc., in press · Zbl 1145.54026
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.