Stability and solution of two functional equations in unital algebras
Main Article Content
Abstract
In this paper, we consider two functional equations:
\begin{align}\label{1}
(1) \qquad&h(\mathcal{F}(x,y,z)+2x+y+z)+h(xy+z)+yh(x)+yh(z)\nonumber\\
&=h (\mathcal{F}(x,y,z)+2x+y)+h(xy)+yh(x+z)+2h(z),
\end{align} \begin{align}\label{12}
(2) \qquad&h(\mathcal{F}(x,y,z)-y+z+2e)+2h(x+y)+h(xy+z)+yh(x)+yh(z)\nonumber\\
&=h(\mathcal{F}(x,y,z)-y+2e)+2h(x+y+z)+h(xy)+yh(x+z),
\end{align} without any regularity assumption for all $x,y,z$ in a unital algebra $A$, where $\mathcal{F}:A^3\rightarrow A$ is defined by \begin{align*}
\mathcal{F}(x,y,z):=h(x+y+z)-h(x+y)-h(z)
\end{align*} for all $x,y,z\in A$. Also, we find general solutions of these equations in unital algebras. Finally, we prove the Hyers-Ulam stability of (1) and (2) in unital Banach algebras.
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