๐๐-Ward Continuity
A function ๐ is continuous if and only if ๐ preserves convergent sequences; that is, (๐(๐ผ๐)) is a convergent sequence whenever (๐ผ๐) is convergent. The concept of ๐๐-ward continuity is defined in the sense that a function ๐ is ๐๐-ward continuous if it preserves ๐๐-quasi-Cauchy sequences; that is, (๐(...
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Wiley
2012-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/680456 |
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| author | Huseyin Cakalli |
| author_facet | Huseyin Cakalli |
| author_sort | Huseyin Cakalli |
| collection | DOAJ |
| description | A function ๐ is continuous if and only if ๐ preserves convergent sequences; that is, (๐(๐ผ๐)) is a convergent sequence whenever (๐ผ๐) is convergent. The concept of ๐๐-ward continuity is defined in the sense that a function ๐ is ๐๐-ward continuous if it preserves ๐๐-quasi-Cauchy sequences; that is, (๐(๐ผ๐)) is an ๐๐-quasi-Cauchy sequence whenever (๐ผ๐) is ๐๐-quasi-Cauchy. A sequence (๐ผ๐) of points in ๐, the set of real numbers, is ๐๐-quasi-Cauchy if lim๐โโ(1/โ๐)โ๐โ๐ผ๐|ฮ๐ผ๐|=0, where ฮ๐ผ๐=๐ผ๐+1โ๐ผ๐, ๐ผ๐=(๐๐โ1,๐๐], and ๐=(๐๐) is a lacunary sequence, that is, an increasing sequence of positive integers such that ๐0=0 and โ๐โถ๐๐โ๐๐โ1โโ. A new type compactness, namely, ๐๐-ward compactness, is also, defined and some new results related to this kind of compactness are obtained. |
| format | Article |
| id | doaj-art-9db8b6f2cb534268911f80efd44b42bd |
| institution | OA Journals |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-9db8b6f2cb534268911f80efd44b42bd2025-08-20T02:20:29ZengWileyAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/680456680456๐๐-Ward ContinuityHuseyin Cakalli0Department of Mathematics, Maltepe University, Marmara Education Village, 34857 Istanbul, TurkeyA function ๐ is continuous if and only if ๐ preserves convergent sequences; that is, (๐(๐ผ๐)) is a convergent sequence whenever (๐ผ๐) is convergent. The concept of ๐๐-ward continuity is defined in the sense that a function ๐ is ๐๐-ward continuous if it preserves ๐๐-quasi-Cauchy sequences; that is, (๐(๐ผ๐)) is an ๐๐-quasi-Cauchy sequence whenever (๐ผ๐) is ๐๐-quasi-Cauchy. A sequence (๐ผ๐) of points in ๐, the set of real numbers, is ๐๐-quasi-Cauchy if lim๐โโ(1/โ๐)โ๐โ๐ผ๐|ฮ๐ผ๐|=0, where ฮ๐ผ๐=๐ผ๐+1โ๐ผ๐, ๐ผ๐=(๐๐โ1,๐๐], and ๐=(๐๐) is a lacunary sequence, that is, an increasing sequence of positive integers such that ๐0=0 and โ๐โถ๐๐โ๐๐โ1โโ. A new type compactness, namely, ๐๐-ward compactness, is also, defined and some new results related to this kind of compactness are obtained.http://dx.doi.org/10.1155/2012/680456 |
| spellingShingle | Huseyin Cakalli ๐๐-Ward Continuity Abstract and Applied Analysis |
| title | ๐๐-Ward Continuity |
| title_full | ๐๐-Ward Continuity |
| title_fullStr | ๐๐-Ward Continuity |
| title_full_unstemmed | ๐๐-Ward Continuity |
| title_short | ๐๐-Ward Continuity |
| title_sort | ๐๐ ward continuity |
| url | http://dx.doi.org/10.1155/2012/680456 |
| work_keys_str_mv | AT huseyincakalli nฮธwardcontinuity |