Normed Ordered and 𝐸-Metric Spaces
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 ℝ, and we discuss their extension to 𝐸. We introduce the notion of 𝐸-metri...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2012/272137 |
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| author | Ahmed Al-Rawashdeh Wasfi Shatanawi Muna Khandaqji |
| author_facet | Ahmed Al-Rawashdeh Wasfi Shatanawi Muna Khandaqji |
| author_sort | Ahmed Al-Rawashdeh |
| collection | DOAJ |
| description | 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 ℝ, 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. |
| format | Article |
| id | doaj-art-3386ee8e2518454d9feed7769894ae97 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-3386ee8e2518454d9feed7769894ae972025-02-03T01:13:00ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252012-01-01201210.1155/2012/272137272137Normed Ordered and 𝐸-Metric SpacesAhmed Al-Rawashdeh0Wasfi Shatanawi1Muna Khandaqji2Department of Mathematical Sciences, UAEU, Al-Ain 17551, UAEDepartment of Mathematics, Hashemite University, Zarqa 13115, JordanDepartment of Mathematics, Hashemite University, Zarqa 13115, JordanIn 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 ℝ, 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.http://dx.doi.org/10.1155/2012/272137 |
| spellingShingle | Ahmed Al-Rawashdeh Wasfi Shatanawi Muna Khandaqji Normed Ordered and 𝐸-Metric Spaces International Journal of Mathematics and Mathematical Sciences |
| title | Normed Ordered and 𝐸-Metric Spaces |
| title_full | Normed Ordered and 𝐸-Metric Spaces |
| title_fullStr | Normed Ordered and 𝐸-Metric Spaces |
| title_full_unstemmed | Normed Ordered and 𝐸-Metric Spaces |
| title_short | Normed Ordered and 𝐸-Metric Spaces |
| title_sort | normed ordered and 𝐸 metric spaces |
| url | http://dx.doi.org/10.1155/2012/272137 |
| work_keys_str_mv | AT ahmedalrawashdeh normedorderedandemetricspaces AT wasfishatanawi normedorderedandemetricspaces AT munakhandaqji normedorderedandemetricspaces |