Ulam Type Stability of ?-Quadratic Mappings in Fuzzy Modular ∗-Algebras
In this paper, we find the solution of the following quadratic functional equation <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><msub><mo>∑</mo><mrow><mn>1</mn><mo>≤</mo><mi>i<...
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MDPI AG
2020-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/9/1630 |
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| author | Hark-Mahn Kim Hwan-Yong Shin |
| author_facet | Hark-Mahn Kim Hwan-Yong Shin |
| author_sort | Hark-Mahn Kim |
| collection | DOAJ |
| description | In this paper, we find the solution of the following quadratic functional equation <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><msub><mo>∑</mo><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo><</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></msub><mi>Q</mi><mfenced separators="" open="(" close=")"><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><msub><mi>x</mi><mi>j</mi></msub></mfenced><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mi>Q</mi><mfenced separators="" open="(" close=")"><msub><mo>∑</mo><mrow><mi>j</mi><mo>≠</mo><mi>i</mi></mrow></msub><msub><mi>x</mi><mi>j</mi></msub><mo>−</mo><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><msub><mi>x</mi><mi>i</mi></msub></mfenced></mrow></semantics></math></inline-formula>, which is derived from the gravity of the <i>n</i> distinct vectors <inline-formula><math display="inline"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>x</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> in an inner product space, and prove that the stability results of the <inline-formula><math display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula>-quadratic mappings in <inline-formula><math display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula>-complete convex fuzzy modular ∗-algebras without using lower semicontinuity and <inline-formula><math display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>-homogeneous property. |
| first_indexed | 2024-03-10T16:10:59Z |
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| id | doaj.art-c2f9eec7e32a430e9cb9adbb8738774f |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T16:10:59Z |
| publishDate | 2020-09-01 |
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| spelling | doaj.art-c2f9eec7e32a430e9cb9adbb8738774f2023-11-20T14:28:00ZengMDPI AGMathematics2227-73902020-09-0189163010.3390/math8091630Ulam Type Stability of ?-Quadratic Mappings in Fuzzy Modular ∗-AlgebrasHark-Mahn Kim0Hwan-Yong Shin1Department of Mathematics, Chungnam National University, 99 Daehangno, Yuseong-gu, Daejeon 34134, KoreaDepartment of Mathematics, Chungnam National University, 99 Daehangno, Yuseong-gu, Daejeon 34134, KoreaIn this paper, we find the solution of the following quadratic functional equation <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><msub><mo>∑</mo><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo><</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></msub><mi>Q</mi><mfenced separators="" open="(" close=")"><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><msub><mi>x</mi><mi>j</mi></msub></mfenced><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mi>Q</mi><mfenced separators="" open="(" close=")"><msub><mo>∑</mo><mrow><mi>j</mi><mo>≠</mo><mi>i</mi></mrow></msub><msub><mi>x</mi><mi>j</mi></msub><mo>−</mo><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><msub><mi>x</mi><mi>i</mi></msub></mfenced></mrow></semantics></math></inline-formula>, which is derived from the gravity of the <i>n</i> distinct vectors <inline-formula><math display="inline"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>x</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> in an inner product space, and prove that the stability results of the <inline-formula><math display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula>-quadratic mappings in <inline-formula><math display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula>-complete convex fuzzy modular ∗-algebras without using lower semicontinuity and <inline-formula><math display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>-homogeneous property.https://www.mdpi.com/2227-7390/8/9/1630fuzzy modular ∗-algebrasmodular ∗-algebras?-quadratic derivationΔ<sub>2</sub>-condition<i>β</i>-homogeneous property |
| spellingShingle | Hark-Mahn Kim Hwan-Yong Shin Ulam Type Stability of ?-Quadratic Mappings in Fuzzy Modular ∗-Algebras Mathematics fuzzy modular ∗-algebras modular ∗-algebras ?-quadratic derivation Δ<sub>2</sub>-condition <i>β</i>-homogeneous property |
| title | Ulam Type Stability of ?-Quadratic Mappings in Fuzzy Modular ∗-Algebras |
| title_full | Ulam Type Stability of ?-Quadratic Mappings in Fuzzy Modular ∗-Algebras |
| title_fullStr | Ulam Type Stability of ?-Quadratic Mappings in Fuzzy Modular ∗-Algebras |
| title_full_unstemmed | Ulam Type Stability of ?-Quadratic Mappings in Fuzzy Modular ∗-Algebras |
| title_short | Ulam Type Stability of ?-Quadratic Mappings in Fuzzy Modular ∗-Algebras |
| title_sort | ulam type stability of quadratic mappings in fuzzy modular ∗ algebras |
| topic | fuzzy modular ∗-algebras modular ∗-algebras ?-quadratic derivation Δ<sub>2</sub>-condition <i>β</i>-homogeneous property |
| url | https://www.mdpi.com/2227-7390/8/9/1630 |
| work_keys_str_mv | AT harkmahnkim ulamtypestabilityofquadraticmappingsinfuzzymodularalgebras AT hwanyongshin ulamtypestabilityofquadraticmappingsinfuzzymodularalgebras |