Fixed Point Theory and Applications 
Volume 2004 (2004), Issue 4, Pages 317-336
doi:10.1155/S1687182004406068

On some Banach space constants arising in nonlinear fixed point and eigenvalue theory

Jürgen Appell,1 Nina A. Erzakova,2 Sergio Falcon Santana,3 and Martin Väth1

 1Mathematisches Institut, Universität Würzburg,, Am Hubland, Würzburg 97074, Germany
2Department of Mathematics, Moscow State Institute of Electronic Techniques, Zelenograd, K-498, Moscow 124498, Russia
3Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira, Las Palmas de Gran Canaria 35017, Spain
Received 8 June 2004

Abstract

As is well known, in any infinite-dimensional Banach space one may find fixed point free self-maps of the unit ball, retractions of the unit ball onto its boundary, contractions of the unit sphere, and nonzero maps without positive eigenvalues and normalized eigenvectors. In this paper, we give upper and lower estimates, or even explicit formulas, for the minimal Lipschitz constant and measure of noncompactness of such maps.