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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 574614, 19 pages
http://dx.doi.org/10.1155/2011/574614
Research Article

The Fixed Point Property in π‘πŸŽ with an Equivalent Norm

1Matemáticas Básicas, Centro de Investigación en Matemáticas (CIMAT), Apartado Postal 402, 36000 Guanajuato, GTO, Mexico
2Departamento de Matemáticas Aplicadas, Universidad del Papaloapan (UNPA), 68400 Loma Bonita, OAX, Mexico

Received 7 June 2011; Accepted 27 August 2011

Academic Editor: Elena Litsyn

Copyright © 2011 Berta Gamboa de Buen and Fernando Núñez-Medina. 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

We study the fixed point property (FPP) in the Banach space 𝑐0 with the equivalent norm ‖⋅‖𝐷. The space 𝑐0 with this norm has the weak fixed point property. We prove that every infinite-dimensional subspace of (𝑐0,‖⋅‖𝐷) contains a complemented asymptotically isometric copy of 𝑐0, and thus does not have the FPP, but there exist nonempty closed convex and bounded subsets of (𝑐0,‖⋅‖𝐷) which are not πœ”-compact and do not contain asymptotically isometric 𝑐0β€”summing basis sequences. Then we define a family of sequences which are asymptotically isometric to different bases equivalent to the summing basis in the space (𝑐0,‖⋅‖𝐷), and we give some of its properties. We also prove that the dual space of (𝑐0,‖⋅‖𝐷) over the reals is the Bynum space 𝑙1∞ and that every infinite-dimensional subspace of 𝑙1∞ does not have the fixed point property.