Normed Ordered and -Metric Spaces

In 2007, Haung and Zhang introduced the notion of cone metric spaces. In this paper, we define an ordered space , and we discuss some properties and examples. Also, normed ordered space is introduced. We recall properties of ？, and we discuss their extension to . We introduce the notion of -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 -metric spaces. Finally, by extending the notion of a contractive sequence in a real-valued metric space, we show that in -metric spaces, a contractive sequence is Cauchy.