A Characterization of Strong Completeness in Fuzzy Metric Spaces
Here, we deal with the concept of fuzzy metric space <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi mathvariant="script">X</mi> <mo>,</mo> <mi mathvariant="script">M</mi>...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2020-05-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/8/6/861 |
| _version_ | 1797567005603135488 |
|---|---|
| author | Valentín Gregori Juan-José Miñana Bernardino Roig Almanzor Sapena |
| author_facet | Valentín Gregori Juan-José Miñana Bernardino Roig Almanzor Sapena |
| author_sort | Valentín Gregori |
| collection | DOAJ |
| description | Here, we deal with the concept of fuzzy metric space <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi mathvariant="script">X</mi> <mo>,</mo> <mi mathvariant="script">M</mi> <mo>,</mo> <mo>∗</mo> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, due to George and Veeramani. Based on the fuzzy diameter for a subset of <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">X</mi> </semantics> </math> </inline-formula>, we introduce the notion of strong fuzzy diameter zero for a family of subsets. Then, we characterize nested sequences of subsets having strong fuzzy diameter zero using their fuzzy diameter. Examples of sequences of subsets which do or do not have strong fuzzy diameter zero are provided. Our main result is the following characterization: a fuzzy metric space is strongly complete if and only if every nested sequence of close subsets which has strong fuzzy diameter zero has a singleton intersection. Moreover, the standard fuzzy metric is studied as a particular case. Finally, this work points out a route of research in fuzzy fixed point theory. |
| first_indexed | 2024-03-10T19:35:27Z |
| format | Article |
| id | doaj.art-301cb94f41614c76b4fbce66da550602 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T19:35:27Z |
| publishDate | 2020-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-301cb94f41614c76b4fbce66da5506022023-11-20T01:45:20ZengMDPI AGMathematics2227-73902020-05-018686110.3390/math8060861A Characterization of Strong Completeness in Fuzzy Metric SpacesValentín Gregori0Juan-José Miñana1Bernardino Roig2Almanzor Sapena3Instituto de Investigación para la Gestión Integrada de Zonas Costeras, Universitat Politècnica de València, C/ Paranimf, 1, 46730 Grao de Gandia, SpainDepartament de Ciències Matemàtiques i Informàtica, Universitat de les Illes Balears, Carretera de Valldemossa km. 7.5, 07122 Palma, SpainInstituto de Investigación para la Gestión Integrada de Zonas Costeras, Universitat Politècnica de València, C/ Paranimf, 1, 46730 Grao de Gandia, SpainInstituto de Investigación para la Gestión Integrada de Zonas Costeras, Universitat Politècnica de València, C/ Paranimf, 1, 46730 Grao de Gandia, SpainHere, we deal with the concept of fuzzy metric space <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi mathvariant="script">X</mi> <mo>,</mo> <mi mathvariant="script">M</mi> <mo>,</mo> <mo>∗</mo> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, due to George and Veeramani. Based on the fuzzy diameter for a subset of <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">X</mi> </semantics> </math> </inline-formula>, we introduce the notion of strong fuzzy diameter zero for a family of subsets. Then, we characterize nested sequences of subsets having strong fuzzy diameter zero using their fuzzy diameter. Examples of sequences of subsets which do or do not have strong fuzzy diameter zero are provided. Our main result is the following characterization: a fuzzy metric space is strongly complete if and only if every nested sequence of close subsets which has strong fuzzy diameter zero has a singleton intersection. Moreover, the standard fuzzy metric is studied as a particular case. Finally, this work points out a route of research in fuzzy fixed point theory.https://www.mdpi.com/2227-7390/8/6/861fuzzy metricCauchy sequence(strong) convergencecompletenessfuzzy diameter |
| spellingShingle | Valentín Gregori Juan-José Miñana Bernardino Roig Almanzor Sapena A Characterization of Strong Completeness in Fuzzy Metric Spaces Mathematics fuzzy metric Cauchy sequence (strong) convergence completeness fuzzy diameter |
| title | A Characterization of Strong Completeness in Fuzzy Metric Spaces |
| title_full | A Characterization of Strong Completeness in Fuzzy Metric Spaces |
| title_fullStr | A Characterization of Strong Completeness in Fuzzy Metric Spaces |
| title_full_unstemmed | A Characterization of Strong Completeness in Fuzzy Metric Spaces |
| title_short | A Characterization of Strong Completeness in Fuzzy Metric Spaces |
| title_sort | characterization of strong completeness in fuzzy metric spaces |
| topic | fuzzy metric Cauchy sequence (strong) convergence completeness fuzzy diameter |
| url | https://www.mdpi.com/2227-7390/8/6/861 |
| work_keys_str_mv | AT valentingregori acharacterizationofstrongcompletenessinfuzzymetricspaces AT juanjoseminana acharacterizationofstrongcompletenessinfuzzymetricspaces AT bernardinoroig acharacterizationofstrongcompletenessinfuzzymetricspaces AT almanzorsapena acharacterizationofstrongcompletenessinfuzzymetricspaces AT valentingregori characterizationofstrongcompletenessinfuzzymetricspaces AT juanjoseminana characterizationofstrongcompletenessinfuzzymetricspaces AT bernardinoroig characterizationofstrongcompletenessinfuzzymetricspaces AT almanzorsapena characterizationofstrongcompletenessinfuzzymetricspaces |