Weak uniform normal structure and iterative fixed points of nonexpansive mappings

• Autorzy: T. Domínguez Benavides , G. López Acedo , Hong-Kun Xu
• Języki publikacji: EN

Abstrakty

EN

This paper is concerned with weak uniform normal structure and iterative fixed points of nonexpansive mappings. Precisely, in Section 1, we show that the geometrical coefficient β(X) for a Banach space X recently introduced by Jimenez-Melado [8] is exactly the weakly convergent sequence coefficient WCS(X) introduced by Bynum [1] in 1980. We then show in Section 2 that all kinds of James' quasi-reflexive spaces have weak uniform normal structure. Finally, in Section 3, we show that in a space X with weak uniform normal structure, every nonexpansive self-mapping defined on a weakly sequentially compact convex subset of X admits an iterative fixed point.

Słowa kluczowe

EN

nonexpansive mapping; iterative fixed point; geometrical coefficients of Banach spaces; James' quasi-reflexive space; weak uniform normal structure

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 1995
• Tom: 68
• Numer: 1
• Strony: 17-23

Daty

wydano

1995

otrzymano

1994-01-03

Twórcy

autor

T. Domínguez Benavides

• Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Apdo. 1160, 41080 Sevilla, Spain

autor

G. López Acedo

• Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Apdo. 1160, 41080 Sevilla, Spain

autor

Hong-Kun Xu

• Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Apdo. 1160, 41080 Sevilla, Spain

Bibliografia

• [1] W. L. Bynum, Normal structure coefficients for Banach spaces, Pacific J. Math. 86 (1980), 427-436.
• [2] T. Domínguez Benavides, Weak uniform normal structure in direct-sum spaces, Studia Math. 103 (1992), 283-290.
• [3] T. Domínguez Benavides, Some properties of the set and ball measures of noncompactness and applications, J. London Math. Soc. 34 (1986), 120-128.
• [4] T. Domínguez Benavides and G. López Acedo, Lower bounds for normal structure coefficients, Proc. Roy. Soc. Edinburgh 121A (1992), 245-252.
• [5] M. Edelstein and R. C. O'Brien, Nonexpansive mappings, asymptotic regularity, and successive approximations, J. London Math. Soc. 17 (1978), 547-554.
• [6] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, 1990.
• [7] S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59 (1976), 65-71.
• [8] A. Jimenez-Melado, Stability of weak normal structure in James quasi reflexive space, Bull. Austral. Math. Soc. 46 (1992), 367-372.
• [9] M. A. Khamsi, James quasi reflexive space has the fixed point property, ibid. 39 (1989), 25-30.
• [10] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006.
• [11] E. Maluta, Uniformly normal structure and related coefficients for Banach spaces, Pacific J. Math. 111 (1984), 357-369.
• [12] P. M. Soardi, Schauder bases and fixed points of nonexpansive mappings, Pacific J. Math. 101 (1982), 193-198.