International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 272137, 11 pages
Research Article

Normed Ordered and 𝐸 -Metric Spaces

1Department of Mathematical Sciences, UAEU, Al-Ain 17551, UAE
2Department of Mathematics, Hashemite University, Zarqa 13115, Jordan

Received 31 December 2011; Revised 23 February 2012; Accepted 26 February 2012

Academic Editor: Naseer Shahzad

Copyright © 2012 Ahmed Al-Rawashdeh 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.


In 2007, Haung and Zhang introduced the notion of cone metric spaces. In this paper, we define an ordered space 𝐸 , and we discuss some properties and examples. Also, normed ordered space is introduced. We recall properties of , and we discuss their extension to 𝐸 . We introduce the notion of 𝐸 -metric spaces and characterize cone metric space. Afterwards, we get generalizations of notions of convergence and Cauchy theory. In particular, we get a fixed point theorem of a contractive mapping in 𝐸 -metric spaces. Finally, by extending the notion of a contractive sequence in a real-valued metric space, we show that in 𝐸 -metric spaces, a contractive sequence is Cauchy.