tailieunhanh - Báo cáo hóa học: " Research Article The Locally Uniform Nonsquare in Generalized ` Cesaro Sequence Spaces"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article The Locally Uniform Nonsquare in Generalized ` Cesaro Sequence Spaces | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 162037 10 pages doi 2008 162037 Research Article The Locally Uniform Nonsquare in Generalized Cesaro Sequence Spaces Narin Petrot Department of Mathematics Faculty of Science Naresuan University Phitsanulok 65000 Thailand Correspondence should be addressed to Narin Petrot narinp@ Received 20 August 2008 Accepted 10 November 2008 Recommended by Martin J. Bohner We show that the generalized Cesaro sequence spaces possess the locally uniform nonsquare and have the fixed point property but they are not uniformly nonsquare. This result is related to the result of the paper by J. Falset et al. 2006 by giving the examples and the motivation to find the geometric properties that are weaker than uniformly nonsquare but still possess the fixed point property in any Banach spaces. Copyright 2008 Narin Petrot. 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. 1. Introduction In the whole paper N and R stand for the sets of natural numbers and of real numbers respectively. The space of all real sequence x x i TO1 is denoted by c . For a real normed space X ll H we denote by S X the unit sphere of X. We now give some definitions and basic concepts which will be used in this paper. A Banach space X - which is a subspace of 0 is said to be a Kothe sequence space if i for any x e 0 and y e X such that x i y i for all i e N we have x e X and llxll byb ii there is x e X with x i 0 for all i e N. An element x from a Kothe sequence space X is called order continuous if for any sequence xn in X the positive cone of X such that xn x for all n e N and xn 0coordinatewise we have xn 0. It is easy to see that x is order continuous if and only if II 0 0 . 0 x n 1 x n 2 . II 0 as n TO. 2 Journal of Inequalities and .