Strong Proximal Continuity and Convergence
In several situations the notion of uniform continuity can be strengthened to strong uniform continuity to produce interesting properties, especially in constrained problems. The same happens in the setting of proximity spaces. While a parallel theory for uniform and strong uniform convergence was r...
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2013/412796 |
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| author | Agata Caserta Roberto Lucchetti Som Naimpally |
| author_facet | Agata Caserta Roberto Lucchetti Som Naimpally |
| author_sort | Agata Caserta |
| collection | DOAJ |
| description | In several situations the notion of uniform continuity can be strengthened to strong uniform continuity to produce interesting properties, especially in constrained problems. The same happens
in the setting of proximity spaces. While a parallel theory for uniform and strong uniform convergence was recently developed, and a notion of proximal convergence is present in the literature, the notion of strong proximal convergence was never considered. In this paper, we propose several possible convergence notions, and we provide complete comparisons among these concepts and the notion of strong uniform convergence in uniform spaces. It is also shown that in particularly meaningful classes of functions these notions are equivalent and can be considered as natural definitions of strong proximal convergence. Finally we consider a function acting between two proximity spaces and we connect its continuity/strong continuity to convergence in the respective hyperspaces of a natural functor associated to the function itself. |
| format | Article |
| id | doaj-art-82ed94a8eb0746bea832c3749bb6e5b0 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-82ed94a8eb0746bea832c3749bb6e5b02025-02-03T01:22:08ZengWileyAbstract and Applied Analysis1085-33751687-04092013-01-01201310.1155/2013/412796412796Strong Proximal Continuity and ConvergenceAgata Caserta0Roberto Lucchetti1Som Naimpally2Department of Mathematics, Seconda Università degli Studi di Napoli, 81100 Caserta, ItalyDepartment of Mathematics, Politecnico di Milano, 20133 Milano, Italy96 Dewson Street, Toronto, ON, M3J 1P3, CanadaIn several situations the notion of uniform continuity can be strengthened to strong uniform continuity to produce interesting properties, especially in constrained problems. The same happens in the setting of proximity spaces. While a parallel theory for uniform and strong uniform convergence was recently developed, and a notion of proximal convergence is present in the literature, the notion of strong proximal convergence was never considered. In this paper, we propose several possible convergence notions, and we provide complete comparisons among these concepts and the notion of strong uniform convergence in uniform spaces. It is also shown that in particularly meaningful classes of functions these notions are equivalent and can be considered as natural definitions of strong proximal convergence. Finally we consider a function acting between two proximity spaces and we connect its continuity/strong continuity to convergence in the respective hyperspaces of a natural functor associated to the function itself.http://dx.doi.org/10.1155/2013/412796 |
| spellingShingle | Agata Caserta Roberto Lucchetti Som Naimpally Strong Proximal Continuity and Convergence Abstract and Applied Analysis |
| title | Strong Proximal Continuity and Convergence |
| title_full | Strong Proximal Continuity and Convergence |
| title_fullStr | Strong Proximal Continuity and Convergence |
| title_full_unstemmed | Strong Proximal Continuity and Convergence |
| title_short | Strong Proximal Continuity and Convergence |
| title_sort | strong proximal continuity and convergence |
| url | http://dx.doi.org/10.1155/2013/412796 |
| work_keys_str_mv | AT agatacaserta strongproximalcontinuityandconvergence AT robertolucchetti strongproximalcontinuityandconvergence AT somnaimpally strongproximalcontinuityandconvergence |