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Proper 1-ball contractive retractions in Banach spaces of measurable functions

Published online by Cambridge University Press:  17 April 2009

D. Caponetti ,
A. Trombetta  and
G. Trombetta
D. Caponetti
Affiliation:
Department of Mathematics, University of Palermo, Via Archirafi, 34, 90123 Palermo, Italy e-mail: d.caponetti@math.unipa.it
A. Trombetta
Affiliation:
Department of Mathematics, University of Calabria, 87036 Arcavacata di Rende (CS), Italy e-mail: aletromb@unical.it
G. Trombetta
Affiliation:
Department of Mathematics, University of Calabria, 87036 Arcavacata di Rende (CS), Italy e-mail: trombetta@unical.it
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In this paper we consider the Wośko problem of evaluating, in an infinite-dimensional Banach space X, the infimum of all k ≤ 1 for which there exists a k-ball contractive retraction of the unit ball onto its boundary. We prove that in some classical Banach spaces the best possible value 1 is attained. Moreover we give estimates of the lower H-measure of noncompactness of the retractions we construct.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 72 , Issue 2 , October 2005 , pp. 299 - 315
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Appell, J., Erkazova, N.A., Santana, S. Falcon and Väth, M., ‘On some Banach space constants arising in nonlinear fixed point and eigenvalue theory’, Fixed Point Theory Appl. 4 (2004), 317–336.Google Scholar
[2]Appell, J. and Zabrejko, P.P., Nonlinear superposition operators (Cambridge University Press, Cambridge, 1990).Google Scholar
[3]Bennett, C. and Sharpley, R., Interpolation of operators, Pure and Applied Maths 129 (Boston Academic Press, Boston MA, 1988).Google Scholar
[4]Benyamini, Y. and Sternfeld, Y., ‘Spheres in infinite-dimensional normed spaces are Lipschitz contractible’, Proc. Amer. Math. Soc. 88 (1983), 439–445.Google Scholar
[5]Capone, C. and Fiorenza, A., ‘On small Lebesgue spaces’, J. Funct. Spaces Appl. 3 (2005), 73–89.Google Scholar
[6]Caponetti, D. and Trombetta, G., ‘On proper k-ball contractive retractions in the Banach space BC([0, ∞))’, Nonlinear Func. Anal. Appl. (to appear).Google Scholar
[7]Fiorenza, A., ‘Duality and reflexivity in grand Lebesgue spaces’, Collect. Math. 51 (2000), 131–148.Google Scholar
[8]Goebel, K., ‘On the minimal displacement of points under Lipschitzian mappings’, Pacific J. Math. 45 (1973), 151–163.Google Scholar
[9]Goebel, K., ‘On the problem of retracting balls onto their boundary’, Abstr. Appl. Anal. 2 (2003), 101–110.Google Scholar
[10]Goebel, K. and Kirk, W.A., Topics in metric fixed point theory (Cambridge, 1990).Google Scholar
[11]Iwaniec, T. and Sbordone, C., ‘On the integrability of the Jacobian under minimal hypotheses’, Arch. Rational Mech. Anal. 119 (1992), 129–143.Google Scholar
[12]Lewicki, G. and Trombetta, G., ‘Almost contractive retractions in Orlicz spaces’, Bull. Austral. Math. Soc. 68 (2003), 353–369.Google Scholar
[13]Nowak, B., ‘On the Lipschitzian retraction of the unit ball in infinite-dimensional Banach spaces onto its boundary’, Bull. Acad. Polon. Sci. 27 (1979), 861–864.Google Scholar
[14]Rao, M.M. and Ren, Z.D., Theory of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics 146 (Marcel Dekker, Inc., New York, 1991).Google Scholar
[15]Rudin, W., Real and complex analysis (McGraw-Hill Book Co., New York, 1987).Google Scholar
[16]Trombetta, G., ‘k-set contractive retractions in spaces of continuous functions’, Sci. Math. Jpn. 59 (2004), 121–128.Google Scholar
[17]Trombetta, A. and Trombetta, G., ‘On the existence of (γp) k-set contractive retractions in Lp[0, 1] spaces, 1 ≤ p < ∞’, Sci. Math. Jpn. 56 (2002), 327–335.Google Scholar
[18]Väth, M., Ideal spaces, Lect. Notes in Math. 1664 (Springer-Verlag, Berlin, 1997).Google Scholar
[19]Väth, M., ‘On the minimal displacement problem of γ-Lipschitz maps and γ-Lipschitz retractions onto the sphere’, Z. Anal. Anwendungen 21 (2002), 901–914.Google Scholar
[20]Wośko, J., ‘An example related to the retraction problem’, Ann. Univ. Mariae Curie-Sktodowska 45 (1991), 127–130.Google Scholar