Minimum functional equation and some Pexider-type functional equation on any group
We discuss the solution to the minimum functional equation $ \begin{align} \min \{\, \eta(xy^{-1}), \eta(xy) \, \} = \eta(x)\eta(y), \qquad x, y \in G, \end{align} $ for a real-valued function $ \eta: G \to \mathbb{R} $ defined on arbitrary group $ G $. In addition, we examine the Pexider-...
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| Format: | Article |
| Language: | English |
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AIMS Press
2021-08-01
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| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2021656?viewType=HTML |
| _version_ | 1818902519935401984 |
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| author | Muhammad Sarfraz Yongjin Li |
| author_facet | Muhammad Sarfraz Yongjin Li |
| author_sort | Muhammad Sarfraz |
| collection | DOAJ |
| description | We discuss the solution to the minimum functional equation
$ \begin{align} \min \{\, \eta(xy^{-1}), \eta(xy) \, \} = \eta(x)\eta(y), \qquad x, y \in G, \end{align} $
for a real-valued function $ \eta: G \to \mathbb{R} $ defined on arbitrary group $ G $. In addition, we examine the Pexider-type functional equation
$ \begin{align} \max \{\, \eta(xy^{-1}), \eta(xy) \, \} = \chi(x)\eta(y)+\psi(x), \qquad x, y \in G, \end{align} $
where $ \eta $, $ \chi $ and $ \psi $ are real mappings acting on arbitrary group $ G $. We also investigate this Pexiderized functional equation that generalizes two functional equations
$ \begin{align} \max \{\, \eta(xy^{-1}), \eta(xy) \, \} = \eta(x)\eta(y), \qquad x, y \in G, \end{align} $
and
$ \begin{align} \min \{\, \eta(xy^{-1}), \eta(xy) \, \} = \eta(x)\eta(y), \qquad x, y \in G, \end{align} $
with the restriction that the function $ \eta $ satisfies the Kannappan condition. |
| first_indexed | 2024-12-19T20:36:57Z |
| format | Article |
| id | doaj.art-bef667a9b63b412dbc4615b2e4590d32 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-12-19T20:36:57Z |
| publishDate | 2021-08-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-bef667a9b63b412dbc4615b2e4590d322022-12-21T20:06:30ZengAIMS PressAIMS Mathematics2473-69882021-08-01610113051131710.3934/math.2021656Minimum functional equation and some Pexider-type functional equation on any groupMuhammad Sarfraz0Yongjin Li1School of Mathematics, Sun Yat-sen University, Guangzhou 510275, ChinaSchool of Mathematics, Sun Yat-sen University, Guangzhou 510275, ChinaWe discuss the solution to the minimum functional equation $ \begin{align} \min \{\, \eta(xy^{-1}), \eta(xy) \, \} = \eta(x)\eta(y), \qquad x, y \in G, \end{align} $ for a real-valued function $ \eta: G \to \mathbb{R} $ defined on arbitrary group $ G $. In addition, we examine the Pexider-type functional equation $ \begin{align} \max \{\, \eta(xy^{-1}), \eta(xy) \, \} = \chi(x)\eta(y)+\psi(x), \qquad x, y \in G, \end{align} $ where $ \eta $, $ \chi $ and $ \psi $ are real mappings acting on arbitrary group $ G $. We also investigate this Pexiderized functional equation that generalizes two functional equations $ \begin{align} \max \{\, \eta(xy^{-1}), \eta(xy) \, \} = \eta(x)\eta(y), \qquad x, y \in G, \end{align} $ and $ \begin{align} \min \{\, \eta(xy^{-1}), \eta(xy) \, \} = \eta(x)\eta(y), \qquad x, y \in G, \end{align} $ with the restriction that the function $ \eta $ satisfies the Kannappan condition.https://www.aimspress.com/article/doi/10.3934/math.2021656?viewType=HTMLminimum functional equationpexider functional equationkannappan conditionstrictly positive solution |
| spellingShingle | Muhammad Sarfraz Yongjin Li Minimum functional equation and some Pexider-type functional equation on any group AIMS Mathematics minimum functional equation pexider functional equation kannappan condition strictly positive solution |
| title | Minimum functional equation and some Pexider-type functional equation on any group |
| title_full | Minimum functional equation and some Pexider-type functional equation on any group |
| title_fullStr | Minimum functional equation and some Pexider-type functional equation on any group |
| title_full_unstemmed | Minimum functional equation and some Pexider-type functional equation on any group |
| title_short | Minimum functional equation and some Pexider-type functional equation on any group |
| title_sort | minimum functional equation and some pexider type functional equation on any group |
| topic | minimum functional equation pexider functional equation kannappan condition strictly positive solution |
| url | https://www.aimspress.com/article/doi/10.3934/math.2021656?viewType=HTML |
| work_keys_str_mv | AT muhammadsarfraz minimumfunctionalequationandsomepexidertypefunctionalequationonanygroup AT yongjinli minimumfunctionalequationandsomepexidertypefunctionalequationonanygroup |