A note on the relationship between the isotone assumption of the Abian-Brown fixed point theorem and Abian’s most basic fixed point theorem
In a recent paper Xie et al. (Fixed Point Theory Appl. 2013:192, 2013) gave several extensions and some applications of the Abian-Brown (AB) fixed point theorem. While the AB fixed point theorem and its extensions (as well as other related fixed point theorems) assume that the mapping is isotone, th...
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| Format: | Article |
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SpringerOpen
2014
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| Online Access: | https://eprints.nottingham.ac.uk/47162/ |
| _version_ | 1848797478330040320 |
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| author | Luckraz, Shravan |
| author_facet | Luckraz, Shravan |
| author_sort | Luckraz, Shravan |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | In a recent paper Xie et al. (Fixed Point Theory Appl. 2013:192, 2013) gave several extensions and some applications of the Abian-Brown (AB) fixed point theorem. While the AB fixed point theorem and its extensions (as well as other related fixed point theorems) assume that the mapping is isotone, this note shows that for single-valued finite maps this condition relates to the acyclicity of the map, which in turn relates to Abian’s (Nieuw Arch. Wiskd. XVI:184-185, 1968) most basic fixed point theorem for finite sets. |
| first_indexed | 2025-11-14T20:04:31Z |
| format | Article |
| id | nottingham-47162 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T20:04:31Z |
| publishDate | 2014 |
| publisher | SpringerOpen |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-471622020-05-04T16:47:30Z https://eprints.nottingham.ac.uk/47162/ A note on the relationship between the isotone assumption of the Abian-Brown fixed point theorem and Abian’s most basic fixed point theorem Luckraz, Shravan In a recent paper Xie et al. (Fixed Point Theory Appl. 2013:192, 2013) gave several extensions and some applications of the Abian-Brown (AB) fixed point theorem. While the AB fixed point theorem and its extensions (as well as other related fixed point theorems) assume that the mapping is isotone, this note shows that for single-valued finite maps this condition relates to the acyclicity of the map, which in turn relates to Abian’s (Nieuw Arch. Wiskd. XVI:184-185, 1968) most basic fixed point theorem for finite sets. SpringerOpen 2014-05-29 Article PeerReviewed Luckraz, Shravan (2014) A note on the relationship between the isotone assumption of the Abian-Brown fixed point theorem and Abian’s most basic fixed point theorem. Fixed Point Theory and Applications, 2014 (1). p. 129. ISSN 1687-1812 Isotone assumption; Fixed point theorem https://doi.org/10.1186/1687-1812-2014-129 doi:10.1186/1687-1812-2014-129 doi:10.1186/1687-1812-2014-129 |
| spellingShingle | Isotone assumption; Fixed point theorem Luckraz, Shravan A note on the relationship between the isotone assumption of the Abian-Brown fixed point theorem and Abian’s most basic fixed point theorem |
| title | A note on the relationship between the isotone assumption of the Abian-Brown fixed point theorem and Abian’s most basic fixed point theorem |
| title_full | A note on the relationship between the isotone assumption of the Abian-Brown fixed point theorem and Abian’s most basic fixed point theorem |
| title_fullStr | A note on the relationship between the isotone assumption of the Abian-Brown fixed point theorem and Abian’s most basic fixed point theorem |
| title_full_unstemmed | A note on the relationship between the isotone assumption of the Abian-Brown fixed point theorem and Abian’s most basic fixed point theorem |
| title_short | A note on the relationship between the isotone assumption of the Abian-Brown fixed point theorem and Abian’s most basic fixed point theorem |
| title_sort | note on the relationship between the isotone assumption of the abian-brown fixed point theorem and abian’s most basic fixed point theorem |
| topic | Isotone assumption; Fixed point theorem |
| url | https://eprints.nottingham.ac.uk/47162/ https://eprints.nottingham.ac.uk/47162/ https://eprints.nottingham.ac.uk/47162/ |