The generalized Day norm. Part II. Applications

Monika Budzyńska, Aleksandra Grzesik, Mariola Kot

Abstract


In this paper we prove that for each \(1< p, \tilde{p} < \infty\), the Banach space \((l^{\tilde{p}}, \left\|\cdot\right\|_{\tilde{p}})\) can be equivalently renormed in such a way that  the Banach space \((l^{\tilde{p}},\left\|\cdot\right\|_{L,\alpha,\beta,p,\tilde{p}})\) is LUR and has a diametrically complete set with empty interior. This result extends the Maluta theorem about existence of such a set in \(l^2\) with the Day norm. We also show that the Banach space \((l^{\tilde{p}},\left\|\cdot\right\|_{L,\alpha,\beta,p,\tilde{p}})\) has the weak fixed point property for nonexpansive mappings.

Keywords


Diametrically complete set; Day norm, fixed point; Kadec-Klee property; LUR space; nonexpansive mapping; non-strict Opial property; 1-unconditional Schauder bases

Full Text:

PDF

References


Ayerbe Toledano, J. M., Domınguez Benavides, T., López Acedo, G., Measures of Noncompactness in Metric Fixed Point Theory, Birkhauser, 1997.

Baillon, J.-B., Schoneberg, R., Asymptotic normal structure and fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 81 (1981), 257-264.

Budzyńska, M., Grzesik, A., Kot, M., The generalized Day norm. Part I. Properties, Ann. Univ. Mariae Curie-Skłodowska Sect. A 71 (2) (2017), 33-49.

Goebel, K., Kirk, W. A., Topics in Metric Fixed Point Theory, Cambridge University Press, 1990.

Holmes, R. B., Geometric Functional Analysis and Its Applications, Springer, 1975.

Kadec, M. I., On the connection between weak and strong convergence, Dopovidi Akad. Nauk Ukrain. RSR 9 (1959), 949-952.

Kirk, W. A., A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006.

Klee, V., Mappings into normed linear spaces, Fund. Math. 49 (1960/1961), 25-34.

Lin, P.-K., Unconditional bases and fixed points of nonexpansive mappings, Pacific J. Math. 116 (1985), 69-76.

Lindenstrauss, J., Tzafriri, L., Classical Banach Spaces I and II, Springer, 1977.

Maluta, E., A diametrically complete set with empty interior in a reflexive LUR space, J. Nonlinear Conv. Anal. 18 (2017), 105-111.

Maluta, E., Papini, P. L., Diametrically complete sets and normal structure, J. Math. Anal. Appl. 424 (2015), 1335-1347.

Mariadoss, S. A., Soardi, P. M., A remark on asymptotic normal structure in Banach spaces, Rend. Sem. Mat. Univ. Politec. Torino 44 (1986), 393-395.

Moreno, J. P., Papini, P. L., Phelps, R. R., Diametrically maximal and constant width sets in Banach spaces, Canad. J. Math. 58 (2006), 820-842.

Singer, I., Bases in Banach Spaces I, Springer, 1970.

Smith, M. A., Some examples concerning rotundity in Banach spaces, Math. Ann. 233 (1978), 155-161.

Smith, M. A., Turett, B., A reflexive LUR Banach space that lacks normal structure, Canad. Math. Bull. 28 (1985), 492-494.




DOI: http://dx.doi.org/10.17951/a.2017.71.2.51
Date of publication: 2017-12-18 20:31:33
Date of submission: 2017-12-16 23:00:58


Statistics


Total abstract view - 852
Downloads (from 2020-06-17) - PDF - 493

Indicators



Refbacks

  • There are currently no refbacks.


Copyright (c) 2017 Monika Budzyńska, Aleksandra Grzesik, Mariola Kot