Generalized Iterated Function Systems on <i>b</i>-Metric Spaces
An iterated function system consists of a complete metric space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>d</mi><m...
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MDPI AG
2023-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/13/2826 |
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| author | Izabella Abraham Radu Miculescu |
| author_facet | Izabella Abraham Radu Miculescu |
| author_sort | Izabella Abraham |
| collection | DOAJ |
| description | An iterated function system consists of a complete metric space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>)</mo></mrow></semantics></math></inline-formula> and a finite family of contractions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>f</mi><mi>n</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow></semantics></math></inline-formula>. A generalized iterated function system comprises a finite family of contractions defined on the Cartesian product <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>X</mi><mi>m</mi></msup></semantics></math></inline-formula> with values in <i>X</i>. In this paper, we want to investigate generalized iterated function systems in the more general setting of <i>b</i>-metric spaces. We prove that such a system admits a unique attractor and, under some further restrictions on the <i>b</i>-metric, it depends continuously on parameters. We also provide two examples of generalized iterated function systems defined on a particular <i>b</i>-metric space and find the corresponding attractors. |
| first_indexed | 2024-03-11T01:35:25Z |
| format | Article |
| id | doaj.art-a0ba7ed65fcc4f27af78f7fd27008fd3 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T01:35:25Z |
| publishDate | 2023-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-a0ba7ed65fcc4f27af78f7fd27008fd32023-11-18T17:01:54ZengMDPI AGMathematics2227-73902023-06-011113282610.3390/math11132826Generalized Iterated Function Systems on <i>b</i>-Metric SpacesIzabella Abraham0Radu Miculescu1Faculty of Mathematics and Computer Science, Transilvania University, Iuliu Maniu Street 50, 500091 Braşov, RomaniaFaculty of Mathematics and Computer Science, Transilvania University, Iuliu Maniu Street 50, 500091 Braşov, RomaniaAn iterated function system consists of a complete metric space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>)</mo></mrow></semantics></math></inline-formula> and a finite family of contractions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>f</mi><mi>n</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow></semantics></math></inline-formula>. A generalized iterated function system comprises a finite family of contractions defined on the Cartesian product <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>X</mi><mi>m</mi></msup></semantics></math></inline-formula> with values in <i>X</i>. In this paper, we want to investigate generalized iterated function systems in the more general setting of <i>b</i>-metric spaces. We prove that such a system admits a unique attractor and, under some further restrictions on the <i>b</i>-metric, it depends continuously on parameters. We also provide two examples of generalized iterated function systems defined on a particular <i>b</i>-metric space and find the corresponding attractors.https://www.mdpi.com/2227-7390/11/13/2826fractalsiterated function systemsfixed pointsattractor<i>b</i>-metric spaces |
| spellingShingle | Izabella Abraham Radu Miculescu Generalized Iterated Function Systems on <i>b</i>-Metric Spaces Mathematics fractals iterated function systems fixed points attractor <i>b</i>-metric spaces |
| title | Generalized Iterated Function Systems on <i>b</i>-Metric Spaces |
| title_full | Generalized Iterated Function Systems on <i>b</i>-Metric Spaces |
| title_fullStr | Generalized Iterated Function Systems on <i>b</i>-Metric Spaces |
| title_full_unstemmed | Generalized Iterated Function Systems on <i>b</i>-Metric Spaces |
| title_short | Generalized Iterated Function Systems on <i>b</i>-Metric Spaces |
| title_sort | generalized iterated function systems on i b i metric spaces |
| topic | fractals iterated function systems fixed points attractor <i>b</i>-metric spaces |
| url | https://www.mdpi.com/2227-7390/11/13/2826 |
| work_keys_str_mv | AT izabellaabraham generalizediteratedfunctionsystemsonibimetricspaces AT radumiculescu generalizediteratedfunctionsystemsonibimetricspaces |