Stability of Bi-Additive Mappings and Bi-Jensen Mappings
Symmetry is repetitive self-similarity. We proved the stability problem by replicating the well-known Cauchy equation and the well-known Jensen equation into two variables. In this paper, we proved the Hyers-Ulam stability of the bi-additive functional equation <inline-formula><math xmlns=&...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-06-01
|
| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/13/7/1180 |
| _version_ | 1797528279070015488 |
|---|---|
| author | Jae-Hyeong Bae Won-Gil Park |
| author_facet | Jae-Hyeong Bae Won-Gil Park |
| author_sort | Jae-Hyeong Bae |
| collection | DOAJ |
| description | Symmetry is repetitive self-similarity. We proved the stability problem by replicating the well-known Cauchy equation and the well-known Jensen equation into two variables. In this paper, we proved the Hyers-Ulam stability of the bi-additive functional equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>+</mo><mi>w</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo>)</mo><mo>+</mo><mi>f</mi><mo>(</mo><mi>y</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></semantics></math></inline-formula> and the bi-Jensen functional equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>4</mn><mi>f</mi><mfenced separators="" open="(" close=")"><mfrac><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mspace width="0.166667em"></mspace><mfrac><mrow><mi>z</mi><mo>+</mo><mi>w</mi></mrow><mn>2</mn></mfrac></mfenced><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>z</mi><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>w</mi><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>y</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>z</mi><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>y</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>w</mi><mo>)</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> |
| first_indexed | 2024-03-10T09:55:58Z |
| format | Article |
| id | doaj.art-a0a7a7cd15204374a854840b430e1558 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T09:55:58Z |
| publishDate | 2021-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-a0a7a7cd15204374a854840b430e15582023-11-22T02:21:20ZengMDPI AGSymmetry2073-89942021-06-01137118010.3390/sym13071180Stability of Bi-Additive Mappings and Bi-Jensen MappingsJae-Hyeong Bae0Won-Gil Park1Humanitas College, Kyung Hee University, Yongin 17104, KoreaDepartment of Mathematics Education, College of Education, Mokwon University, Daejeon 35349, KoreaSymmetry is repetitive self-similarity. We proved the stability problem by replicating the well-known Cauchy equation and the well-known Jensen equation into two variables. In this paper, we proved the Hyers-Ulam stability of the bi-additive functional equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>+</mo><mi>w</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo>)</mo><mo>+</mo><mi>f</mi><mo>(</mo><mi>y</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></semantics></math></inline-formula> and the bi-Jensen functional equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>4</mn><mi>f</mi><mfenced separators="" open="(" close=")"><mfrac><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mspace width="0.166667em"></mspace><mfrac><mrow><mi>z</mi><mo>+</mo><mi>w</mi></mrow><mn>2</mn></mfrac></mfenced><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>z</mi><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>w</mi><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>y</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>z</mi><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>y</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>w</mi><mo>)</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula>https://www.mdpi.com/2073-8994/13/7/1180stabilitybi-additive mappingbi-Jensen mapping |
| spellingShingle | Jae-Hyeong Bae Won-Gil Park Stability of Bi-Additive Mappings and Bi-Jensen Mappings Symmetry stability bi-additive mapping bi-Jensen mapping |
| title | Stability of Bi-Additive Mappings and Bi-Jensen Mappings |
| title_full | Stability of Bi-Additive Mappings and Bi-Jensen Mappings |
| title_fullStr | Stability of Bi-Additive Mappings and Bi-Jensen Mappings |
| title_full_unstemmed | Stability of Bi-Additive Mappings and Bi-Jensen Mappings |
| title_short | Stability of Bi-Additive Mappings and Bi-Jensen Mappings |
| title_sort | stability of bi additive mappings and bi jensen mappings |
| topic | stability bi-additive mapping bi-Jensen mapping |
| url | https://www.mdpi.com/2073-8994/13/7/1180 |
| work_keys_str_mv | AT jaehyeongbae stabilityofbiadditivemappingsandbijensenmappings AT wongilpark stabilityofbiadditivemappingsandbijensenmappings |