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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 616989, 7 pages
http://dx.doi.org/10.1155/2014/616989
Research Article

Monotonicity and the Dominated Farthest Points Problem in Banach Lattice

Faculty of Mathematics, Yazd University, Yazd, Iran

Received 11 October 2013; Revised 18 February 2014; Accepted 20 February 2014; Published 27 March 2014

Academic Editor: Adrian Petrusel

Copyright © 2014 H. R. Khademzadeh and H. Mazaheri. 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 introduce the dominated farthest points problem in Banach lattices. We prove that for two equivalent norms such that X becomes an STM and LLUM space the dominated farthest points problem has the same solution. We give some conditions such that under these conditions the Fréchet differentiability of the farthest point map is equivalent to the continuity of metric antiprojection in the dominated farthest points problem. Also we prove that these conditions are equivalent to strong solvability of the dominated farthest points problem. We prove these results in STM, reflexive STM, and UM spaces. Moreover, we give some applications of the stated results in Musielak-Orlicz spaces and over nonatomic measure spaces in terms of the function . We will prove that the Fréchet differentiability of the farthest point map and the conditions and in reflexive Musielak-Orlicz function spaces are equivalent.