Iteration changes discontinuity into smoothness (Ⅱ): oscillating case
It has been shown that a self-mapping with exactly one removable or jumping discontinuity may have a $ C^1 $ smooth iterate of the second-order. However, some examples show that a self-mapping with exactly one oscillating discontinuity may also have a $ C^1 $ smooth iterate of the second-order, indi...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-02-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2023441?viewType=HTML |
| _version_ | 1811161978032357376 |
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| author | Tianqi Luo Xiaohua Liu |
| author_facet | Tianqi Luo Xiaohua Liu |
| author_sort | Tianqi Luo |
| collection | DOAJ |
| description | It has been shown that a self-mapping with exactly one removable or jumping discontinuity may have a $ C^1 $ smooth iterate of the second-order. However, some examples show that a self-mapping with exactly one oscillating discontinuity may also have a $ C^1 $ smooth iterate of the second-order, indicating that iteration can turn a self-mapping with exactly one oscillating discontinuity into a $ C^1 $ smooth one. In this paper, we study piecewise $ C^1 $ self-mappings on the open interval $ (0, 1) $ having only one oscillating discontinuity. We give necessary and sufficient conditions for those self-mappings whose second-order iterates are $ C^1 $ smooth. |
| first_indexed | 2024-04-10T06:22:37Z |
| format | Article |
| id | doaj.art-4656a55bd38442ba869fe26acab8c78a |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-10T06:22:37Z |
| publishDate | 2023-02-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-4656a55bd38442ba869fe26acab8c78a2023-03-02T01:10:20ZengAIMS PressAIMS Mathematics2473-69882023-02-01848793881010.3934/math.2023441Iteration changes discontinuity into smoothness (Ⅱ): oscillating caseTianqi Luo0Xiaohua Liu 1School of Mathematics and Physics, Leshan Normal University, Leshan 614000, ChinaSchool of Mathematics and Physics, Leshan Normal University, Leshan 614000, ChinaIt has been shown that a self-mapping with exactly one removable or jumping discontinuity may have a $ C^1 $ smooth iterate of the second-order. However, some examples show that a self-mapping with exactly one oscillating discontinuity may also have a $ C^1 $ smooth iterate of the second-order, indicating that iteration can turn a self-mapping with exactly one oscillating discontinuity into a $ C^1 $ smooth one. In this paper, we study piecewise $ C^1 $ self-mappings on the open interval $ (0, 1) $ having only one oscillating discontinuity. We give necessary and sufficient conditions for those self-mappings whose second-order iterates are $ C^1 $ smooth.https://www.aimspress.com/article/doi/10.3934/math.2023441?viewType=HTMLiterationoscillating discontinuityc<sup>1</sup> smoothpiecewise c<sup>1</sup> smooth |
| spellingShingle | Tianqi Luo Xiaohua Liu Iteration changes discontinuity into smoothness (Ⅱ): oscillating case AIMS Mathematics iteration oscillating discontinuity c<sup>1</sup> smooth piecewise c<sup>1</sup> smooth |
| title | Iteration changes discontinuity into smoothness (Ⅱ): oscillating case |
| title_full | Iteration changes discontinuity into smoothness (Ⅱ): oscillating case |
| title_fullStr | Iteration changes discontinuity into smoothness (Ⅱ): oscillating case |
| title_full_unstemmed | Iteration changes discontinuity into smoothness (Ⅱ): oscillating case |
| title_short | Iteration changes discontinuity into smoothness (Ⅱ): oscillating case |
| title_sort | iteration changes discontinuity into smoothness ii oscillating case |
| topic | iteration oscillating discontinuity c<sup>1</sup> smooth piecewise c<sup>1</sup> smooth |
| url | https://www.aimspress.com/article/doi/10.3934/math.2023441?viewType=HTML |
| work_keys_str_mv | AT tianqiluo iterationchangesdiscontinuityintosmoothnessiioscillatingcase AT xiaohualiu iterationchangesdiscontinuityintosmoothnessiioscillatingcase |