Bihomomorphisms and biderivations in Lie Banach algebras

Tae Hun Kim; Ha Nuel Ju; Hong Nyeong Kim; Seong Yoon Jo; Choonkil Park; Tae Hun Kim; Ha Nuel Ju; Hong Nyeong Kim; Seong Yoon Jo; Choonkil Park

doi:10.3934/math.2020145

AIMS Mathematics

2020, Volume 5, Issue 3: 2196-2210. doi: 10.3934/math.2020145

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Research article

Bihomomorphisms and biderivations in Lie Banach algebras

1.
Mathematics Branch, Seoul Science High School, Seoul 03066, Korea
2.
Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

Received: 04 October 2019 Accepted: 25 February 2020 Published: 27 February 2020
MSC : 39B52, 47B47, 39B62, 17B40

In this paper, we solve the following bi-additive $s$-functional inequality $ \begin{array}{*{20}{c}}{\left\| {f(x - y, y + z) + f\left( {y + z, z - x} \right) + f\left( {z + x, x - z} \right) - f\left( {x - y, x + y} \right)} \right.}\\\;\;\;\;\;\;\;\;\;\;\;{ \le \left\| {s\left( {f\left( {y - z, z + x} \right) + f\left( {z + x, x - y} \right) + f\left( {x + y, y - x} \right) - f\left( {y - z, y + z} \right)} \right)} \right\|, }\end{array}~~~~~~\left( {0.1} \right) $ where $s$ is a fixed nonzero complex number satisfying $|s| \lt 1$. Furthermore, we prove the Hyers-Ulam stability of bihomomorphisms and biderivations in Lie Banach algebras associated with the bi-additive $s$-functional inequality (0.1).
- Hyers-Ulam stability,
- bi-additive s-functional inequality,
- Lie Banach algebra,
- bihomomorphism,
- biderivation
Citation: Tae Hun Kim, Ha Nuel Ju, Hong Nyeong Kim, Seong Yoon Jo, Choonkil Park. Bihomomorphisms and biderivations in Lie Banach algebras[J]. AIMS Mathematics, 2020, 5(3): 2196-2210. doi: 10.3934/math.2020145

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Abstract

In this paper, we solve the following bi-additive $s$-functional inequality $ \begin{array}{*{20}{c}}{\left\| {f(x - y, y + z) + f\left( {y + z, z - x} \right) + f\left( {z + x, x - z} \right) - f\left( {x - y, x + y} \right)} \right.}\\\;\;\;\;\;\;\;\;\;\;\;{ \le \left\| {s\left( {f\left( {y - z, z + x} \right) + f\left( {z + x, x - y} \right) + f\left( {x + y, y - x} \right) - f\left( {y - z, y + z} \right)} \right)} \right\|, }\end{array}~~~~~~\left( {0.1} \right) $ where $s$ is a fixed nonzero complex number satisfying $|s| \lt 1$. Furthermore, we prove the Hyers-Ulam stability of bihomomorphisms and biderivations in Lie Banach algebras associated with the bi-additive $s$-functional inequality (0.1).

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