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Abstract and Applied Analysis
Volume 2015, Article ID 165053, 7 pages
Research Article

The Best Approximation Theorems and Fixed Point Theorems for Discontinuous Increasing Mappings in Banach Spaces

1 School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China
2 College of Information Sciences and Engineering, Shandong Agricultural University, Tai’an, Shandong 271018, China
3 Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia

Received 20 June 2014; Revised 30 September 2014; Accepted 30 September 2014

Academic Editor: Poom Kumam

Copyright © 2015 Dezhou Kong 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.


We prove that Fan’s theorem is true for discontinuous increasing mappings in a real partially ordered reflexive, strictly convex, and smooth Banach space . The main tools of analysis are the variational characterizations of the generalized projection operator and order-theoretic fixed point theory. Moreover, we get some properties of the generalized projection operator in Banach spaces. As applications of our best approximation theorems, the fixed point theorems for non-self-maps are established and proved under some conditions. Our results are generalizations and improvements of the recent results obtained by many authors.