Stability of Maximum Functional Equation and Some Properties of Groups
In this research paper, we deal with the problem of determining the function <inline-formula><math display="inline"><semantics><mrow><mi>χ</mi><mo>:</mo><mi>G</mi><mspace width="3.33333pt"></mspace><mo>...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-11-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/12/12/1949 |
| _version_ | 1797546780583264256 |
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| author | Muhammad Sarfraz Qi Liu Yongjin Li |
| author_facet | Muhammad Sarfraz Qi Liu Yongjin Li |
| author_sort | Muhammad Sarfraz |
| collection | DOAJ |
| description | In this research paper, we deal with the problem of determining the function <inline-formula><math display="inline"><semantics><mrow><mi>χ</mi><mo>:</mo><mi>G</mi><mspace width="3.33333pt"></mspace><mo>→</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, which is the solution to the maximum functional equation (MFE) <inline-formula><math display="inline"><semantics><mrow><mo movablelimits="true" form="prefix">max</mo><mrow><mo>{</mo><mspace width="0.166667em"></mspace><mi>χ</mi><mrow><mo>(</mo><mi>x</mi><mi>y</mi><mo>)</mo></mrow><mo>,</mo><mi>χ</mi><mrow><mo>(</mo><mi>x</mi><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mo>}</mo></mrow><mo>=</mo><mi>χ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>χ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> when the domain is a discretely normed abelian group or any arbitrary group <i>G</i>. We also analyse the stability of the maximum functional equation <inline-formula><math display="inline"><semantics><mrow><mo movablelimits="true" form="prefix">max</mo><mrow><mo>{</mo><mspace width="0.166667em"></mspace><mi>χ</mi><mrow><mo>(</mo><mi>x</mi><mi>y</mi><mo>)</mo></mrow><mo>,</mo><mi>χ</mi><mrow><mo>(</mo><mi>x</mi><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mo>}</mo></mrow><mo>=</mo><mi>χ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mi>χ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and its solutions for the function <inline-formula><math display="inline"><semantics><mrow><mi>χ</mi><mo>:</mo><mi>G</mi><mspace width="3.33333pt"></mspace><mo>→</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, where <i>G</i> be any group and also investigate the connection of the stability with commutators and free abelian group <i>K</i> that can be embedded into a group <i>G</i>. |
| first_indexed | 2024-03-10T14:34:03Z |
| format | Article |
| id | doaj.art-7b85df7a2560477f8da0f554ea8f4908 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T14:34:03Z |
| publishDate | 2020-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-7b85df7a2560477f8da0f554ea8f49082023-11-20T22:21:47ZengMDPI AGSymmetry2073-89942020-11-011212194910.3390/sym12121949Stability of Maximum Functional Equation and Some Properties of GroupsMuhammad Sarfraz0Qi Liu1Yongjin Li2School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, ChinaSchool of Mathematics, Sun Yat-Sen University, Guangzhou 510275, ChinaSchool of Mathematics, Sun Yat-Sen University, Guangzhou 510275, ChinaIn this research paper, we deal with the problem of determining the function <inline-formula><math display="inline"><semantics><mrow><mi>χ</mi><mo>:</mo><mi>G</mi><mspace width="3.33333pt"></mspace><mo>→</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, which is the solution to the maximum functional equation (MFE) <inline-formula><math display="inline"><semantics><mrow><mo movablelimits="true" form="prefix">max</mo><mrow><mo>{</mo><mspace width="0.166667em"></mspace><mi>χ</mi><mrow><mo>(</mo><mi>x</mi><mi>y</mi><mo>)</mo></mrow><mo>,</mo><mi>χ</mi><mrow><mo>(</mo><mi>x</mi><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mo>}</mo></mrow><mo>=</mo><mi>χ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>χ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> when the domain is a discretely normed abelian group or any arbitrary group <i>G</i>. We also analyse the stability of the maximum functional equation <inline-formula><math display="inline"><semantics><mrow><mo movablelimits="true" form="prefix">max</mo><mrow><mo>{</mo><mspace width="0.166667em"></mspace><mi>χ</mi><mrow><mo>(</mo><mi>x</mi><mi>y</mi><mo>)</mo></mrow><mo>,</mo><mi>χ</mi><mrow><mo>(</mo><mi>x</mi><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mo>}</mo></mrow><mo>=</mo><mi>χ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mi>χ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and its solutions for the function <inline-formula><math display="inline"><semantics><mrow><mi>χ</mi><mo>:</mo><mi>G</mi><mspace width="3.33333pt"></mspace><mo>→</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, where <i>G</i> be any group and also investigate the connection of the stability with commutators and free abelian group <i>K</i> that can be embedded into a group <i>G</i>.https://www.mdpi.com/2073-8994/12/12/1949maximum functional equationsdiscretely normed abelian groupstability of functional equation |
| spellingShingle | Muhammad Sarfraz Qi Liu Yongjin Li Stability of Maximum Functional Equation and Some Properties of Groups Symmetry maximum functional equations discretely normed abelian group stability of functional equation |
| title | Stability of Maximum Functional Equation and Some Properties of Groups |
| title_full | Stability of Maximum Functional Equation and Some Properties of Groups |
| title_fullStr | Stability of Maximum Functional Equation and Some Properties of Groups |
| title_full_unstemmed | Stability of Maximum Functional Equation and Some Properties of Groups |
| title_short | Stability of Maximum Functional Equation and Some Properties of Groups |
| title_sort | stability of maximum functional equation and some properties of groups |
| topic | maximum functional equations discretely normed abelian group stability of functional equation |
| url | https://www.mdpi.com/2073-8994/12/12/1949 |
| work_keys_str_mv | AT muhammadsarfraz stabilityofmaximumfunctionalequationandsomepropertiesofgroups AT qiliu stabilityofmaximumfunctionalequationandsomepropertiesofgroups AT yongjinli stabilityofmaximumfunctionalequationandsomepropertiesofgroups |