On the Fixed Circle Problem on Metric Spaces and Related Results
The fixed-circle issue is a geometric technique that is connected to the study of geometric characteristics of certain points, and that are fixed by the self-mapping of either the metric space or of the generalized space. The fixed-disc problem is a natural resultant that arises as a direct outcome...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-04-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/12/4/401 |
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| author | Nabil Mlaiki Nihal Özgür Nihal Taş Dania Santina |
| author_facet | Nabil Mlaiki Nihal Özgür Nihal Taş Dania Santina |
| author_sort | Nabil Mlaiki |
| collection | DOAJ |
| description | The fixed-circle issue is a geometric technique that is connected to the study of geometric characteristics of certain points, and that are fixed by the self-mapping of either the metric space or of the generalized space. The fixed-disc problem is a natural resultant that arises as a direct outcome of this problem. In this study, our goal is to examine new classes of self-mappings that meet a new particular sort of contraction in a metric space. The common geometrical characteristic of the set of fixed points of any element of these classes is that a circle or even a disc, that is either termed the fixed circle or even the fixed disc of the appropriate self-map, is included within that set. In order to accomplish this, we establish two new classifications of contraction mapping: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>c</mi></msub></semantics></math></inline-formula>-contractive mapping and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>c</mi></msub></semantics></math></inline-formula>-expanding mapping. In the investigation of neural networks, activation functions with either fixed circles (or even fixed discs) are observed frequently. This demonstrates how successful our results with the fixed-circle (respectively, the fixed-disc) model were. |
| first_indexed | 2024-03-11T05:15:00Z |
| format | Article |
| id | doaj.art-6ad432fa564c40c6834dad6b4ed0edf2 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-11T05:15:00Z |
| publishDate | 2023-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-6ad432fa564c40c6834dad6b4ed0edf22023-11-17T18:19:50ZengMDPI AGAxioms2075-16802023-04-0112440110.3390/axioms12040401On the Fixed Circle Problem on Metric Spaces and Related ResultsNabil Mlaiki0Nihal Özgür1Nihal Taş2Dania Santina3Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi ArabiaDepartment of Mathematics, Izmir Democracy University, Izmir 35140, TurkeyDepartment of Mathematics, Balıkesir University, Balıkesir 10145, TurkeyDepartment of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi ArabiaThe fixed-circle issue is a geometric technique that is connected to the study of geometric characteristics of certain points, and that are fixed by the self-mapping of either the metric space or of the generalized space. The fixed-disc problem is a natural resultant that arises as a direct outcome of this problem. In this study, our goal is to examine new classes of self-mappings that meet a new particular sort of contraction in a metric space. The common geometrical characteristic of the set of fixed points of any element of these classes is that a circle or even a disc, that is either termed the fixed circle or even the fixed disc of the appropriate self-map, is included within that set. In order to accomplish this, we establish two new classifications of contraction mapping: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>c</mi></msub></semantics></math></inline-formula>-contractive mapping and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>c</mi></msub></semantics></math></inline-formula>-expanding mapping. In the investigation of neural networks, activation functions with either fixed circles (or even fixed discs) are observed frequently. This demonstrates how successful our results with the fixed-circle (respectively, the fixed-disc) model were.https://www.mdpi.com/2075-1680/12/4/401fixed pointfixed circlefixed disc |
| spellingShingle | Nabil Mlaiki Nihal Özgür Nihal Taş Dania Santina On the Fixed Circle Problem on Metric Spaces and Related Results Axioms fixed point fixed circle fixed disc |
| title | On the Fixed Circle Problem on Metric Spaces and Related Results |
| title_full | On the Fixed Circle Problem on Metric Spaces and Related Results |
| title_fullStr | On the Fixed Circle Problem on Metric Spaces and Related Results |
| title_full_unstemmed | On the Fixed Circle Problem on Metric Spaces and Related Results |
| title_short | On the Fixed Circle Problem on Metric Spaces and Related Results |
| title_sort | on the fixed circle problem on metric spaces and related results |
| topic | fixed point fixed circle fixed disc |
| url | https://www.mdpi.com/2075-1680/12/4/401 |
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