Functional inequalities and pairs of hom-derivations and homomorphisms
Main Article Content
Abstract
In this paper, we introduce and solve the following additive-additive $(s,t)$-functional inequality
\begin{eqnarray}\label{0.1} && \left\|2 g\left(\frac{x+y}{2}\right)-g(x)-g(y) \right\| +\left\| 2h\left(\frac{x+y}{2}\right)+ 2h \left(\frac{x-y}{2}\right)- 2h (x) \right\| \\ && \le \left\|s\left( g\left(x+y\right) -g(x) -g(y) \right)\right\|+ \left\|t \left( h(x+y) + h(x-y) -2 h(x) \right) \right\| , \nonumber
\end{eqnarray}
where $s$ and $t$ are fixed nonzero complex numbers with $|s|+|t| <1$.
We define a pair of hom-derivation and homomorphism in complex Banach algebras, and using the direct method and the fixed point method, we prove the Hyers-Ulam stability of pairs of hom-derivations and homomorphisms in complex Banach algebras, associated to the additive-additive $(s,t)$-functional inequality (\ref{0.1}) and the following functional inequality
\begin{eqnarray}\label{0.2} \| g(xy)-g(x) h(y) - h(x) g(y) \| +\| h(xy) - h(x) h(y) \| \le \varphi(x,y).
\end{eqnarray}
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