Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 852346, 22 pages
doi:10.1155/2010/852346
Research Article

Moduli and Characteristics of Monotonicity in Some Banach Lattices

Paweł Foralewski,1 Henryk Hudzik,1 Radosław Kaczmarek,1 and Miroslav Krbec2

1Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland
2Institute of Mathematics, Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Prague 1, Czech Republic

Received 7 December 2009; Accepted 10 February 2010

Academic Editor: Mohamed Amine Khamsi

Copyright © 2010 Paweł Foralewski et al. 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.

Abstract

First the characteristic of monotonicity of any Banach lattice X is expressed in terms of the left limit of the modulus of monotonicity of X at the point 1. 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 δ^m,E. 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 X with ε0,m(X)<1 and weak orthogonality property have the weak fixed point property, our results are related to the fixed point theory (Kirk and Sims (2001)).