Stability and solution of two functional equations in unital algebras
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In this paper, we consider two functional equations: \begin{align}\label{1} &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} &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 \eqref{1} and \eqref{12} in unital Banach algebras.
Abstract
In this paper, we consider two functional equations: \begin{align}\label{1} &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} &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 \eqref{1} and \eqref{12} in unital Banach algebras.
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