ArticleOriginal scientific text
Title
A modulus for property (β) of Rolewicz
Authors 1, 1, 1
Affiliations
- Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Aptdo. 1160, Sevilla 41080, Spain
Abstract
We define a modulus for the property (β) of Rolewicz and study some useful properties in fixed point theory for nonexpansive mappings. Moreover, we calculate this modulus in spaces for the main measures of noncompactness.
Keywords
property (β), measures of noncompactness, normal structure, -spaces, uniform convexity
Bibliography
- [AKPRS] R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina and B. N. Sadovskiĭ, Measures of Noncompactness and Condensing Operators, Birkhäuser, 1992.
- [ADF] J. M. Ayerbe, T. Domínguez Benavides and S. Francisco Cutillas, Some noncompact convexity moduli for the property (β) of Rolewicz, Comm. Appl. Nonlinear Anal. 1 (1994), 87-98.
- [B1] J. Banaś, On modulus of noncompact convexity and its properties, Canad. Math. Bull. 30 (1987), 186-192.
- [B2] J. Banaś, Compactness conditions in the geometric theory of Banach spaces, Nonlinear Anal. 16 (1991), 669-682.
- [B] B. Beauzamy, Introduction to Banach Spaces and Their Geometry, North-Holland, 1986.
- [By] W. L. Bynum, A class of spaces lacking normal structure, Compositio Math. 25 (1972), 233-236.
- [DL] T. Domínguez Benavides and G. López Acedo, Lower bounds for normal structure coefficients, Proc. Roy. Soc. Edinburgh Sect. A 121 (1992), 245-252.
- [GK] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, 1990.
- [GS] K. Goebel and T. Sękowski, The modulus of noncompact convexity, Ann. Univ. Mariae Curie-Skłodowska Sect. A 38 (1984), 41-48 .
- [GGM] I. C. Gohberg, L. S. Goldenstein and A. S. Markus, Investigation of some properties of bounded linear operators in connection with their q-norms, Uchen. Zap. Kishinev. Un-ta 29 (1975), 29-36 (in Russian).
- [H] R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math. 4 (1980), 743-749.
- [K] K. Kuratowski, Sur les espaces complets, Fund. Math. 15 (1930), 301-309.
- [Ku] D. N. Kutzarova, k-(β) and k-nearly uniform convex Banach spaces, J. Math. Anal. Appl. 162 (1991), 322-338.
- [KMP] D. N. Kutzarova, E. Maluta and S. Prus, Property (β) implies normal structure of the dual space, Rend. Circ. Mat. Palermo 41 (1992), 353-368.
- [KP] D. N. Kutzarova and P. L. Papini, On a characterization of property (β) and LUR, Boll. Un. Mat. Ital. A (7) 6 (1992), 209-214.
- [M] R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Wiley Interscience, New York, 1976.
- [O] Z. Opial, Lecture Notes on Nonexpansive and Monotone Mappings in Banach Spaces, Center for Dynamical Systems, Brown University, 1967.
- [R1] S. Rolewicz, On drop property, Studia Math. 85 (1987), 27-35.
- [R2] S. Rolewicz, On Δ-uniform convexity and drop property, ibid. 87 (1987), 181-191.
- [S] B. N. Sadovskiĭ, On a fixed point principle, Funktsional. Anal. i Prilozhen. 4 (2) (1967), 74-76 (in Russian).
- [WW] J. H. Wells and L. R. Williams, Embeddings and Extensions in Analysis, Springer, Berlin, 1975.