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ON THE REAL-VALUED GENERAL SOLUTIONS OF THE D’ALEMBERT EQUATION WITH INVOLUTION
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 95 / Issue 2 / April 2017
- Published online by Cambridge University Press:
- 23 November 2016, pp. 260-268
- Print publication:
- April 2017
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- Article
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- You have access
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We find all real-valued general solutions
$f:S\rightarrow \mathbb{R}$ of the d’Alembert functional equation with involution for all
$$\begin{eqnarray}\displaystyle f(x+y)+f(x+\unicode[STIX]{x1D70E}y)=2f(x)f(y) & & \displaystyle \nonumber\end{eqnarray}$$
$x,y\in S$, where 
$S$ is a commutative semigroup and 
$\unicode[STIX]{x1D70E}~:~S\rightarrow S$ is an involution. Also, we find the Lebesgue measurable solutions 
$f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ of the above functional equation, where 
$\unicode[STIX]{x1D70E}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ is a Lebesgue measurable involution. As a direct consequence, we obtain the Lebesgue measurable solutions 
$f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ of the classical d’Alembert functional equation for all
$$\begin{eqnarray}\displaystyle f(x+y)+f(x-y)=2f(x)f(y) & & \displaystyle \nonumber\end{eqnarray}$$
$x,y\in \mathbb{R}^{n}$. We also exhibit the locally bounded solutions 
$f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ of the above equations.
