Open Access

Moduli and Characteristics of Monotonicity in Some Banach Lattices

  • Paweł Foralewski1,
  • Henryk Hudzik1Email author,
  • Radosław Kaczmarek1 and
  • Miroslav Krbec2
Fixed Point Theory and Applications20102010:852346

DOI: 10.1155/2010/852346

Received: 7 December 2009

Accepted: 10 February 2010

Published: 31 May 2010

Abstract

First the characteristic of monotonicity of any Banach lattice is expressed in terms of the left limit of the modulus of monotonicity of at the point . It is also shown that for Köthe spaces the classical characteristic of monotonicity is the same as the characteristic of monotonicity corresponding to another modulus of monotonicity . The characteristic of monotonicity of Orlicz function spaces and Orlicz sequence spaces equipped with the Luxemburg norm are calculated. In the first case the characteristic is expressed in terms of the generating Orlicz function only, but in the sequence case the formula is not so direct. Three examples show why in the sequence case so direct formula is rather impossible. Some other auxiliary and complemented results are also presented. By the results of Betiuk-Pilarska and Prus (2008) which establish that Banach lattices with and weak orthogonality property have the weak fixed point property, our results are related to the fixed point theory (Kirk and Sims (2001)).

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Authors’ Affiliations

(1)
Faculty of Mathematics and Computer Science, Adam Mickiewicz University
(2)
Institute of Mathematics, Academy of Sciences of the Czech Republic

Copyright

© Paweł Foralewski et al. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.