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In this paper, we solve the following bi-quadratic s-functional inequality 1 \(\displaystyle \begin{array}{@{}rcl@{}} {} && \|f(x+y,z+w)+f(x+y,z-w)+f(x-y,z+w)+f(x-y,z-w)\\ && \quad - 4[f(x,z)+f(y,z)+f(x,w)+f (y,w)]\| \\ && \le \|s (f(x+y,z)+f(x-y,z)+f(x+y,w)+f(x-y,w)\\&& \quad -f(x,z+w)-f(x,z-w)-f(y,z+w)-f(y,z-w))\| , \end{array} \) where s is a fixed nonzero real or complex number with \(|s |< 2\) . Moreover, we prove the Hyers-Ulam stability of bi-quadratic derivations and bi-quadratic homomorphisms in Banach algebras, associated with the bi-quadratic s-functional inequality ( 1 ).

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Bi-quadratic Derivations and Bi-quadratic Homomorphisms in Banach Algebras

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  • 发表日期：

    2026年

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  • Jung Rye Lee,
  • Choonkil Park,
  • Michael Th. Rassias

摘要

In this paper, we solve the following bi-quadratic s-functional inequality 1 \(\displaystyle \begin{array}{@{}rcl@{}} {} && \|f(x+y,z+w)+f(x+y,z-w)+f(x-y,z+w)+f(x-y,z-w)\\ && \quad - 4[f(x,z)+f(y,z)+f(x,w)+f (y,w)]\| \\ && \le \|s (f(x+y,z)+f(x-y,z)+f(x+y,w)+f(x-y,w)\\&& \quad -f(x,z+w)-f(x,z-w)-f(y,z+w)-f(y,z-w))\| , \end{array} \) where s is a fixed nonzero real or complex number with \(|s |< 2\) . Moreover, we prove the Hyers-Ulam stability of bi-quadratic derivations and bi-quadratic homomorphisms in Banach algebras, associated with the bi-quadratic s-functional inequality ( 1 ).