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ON THE REAL-VALUED GENERAL SOLUTIONS OF THE D’ALEMBERT EQUATION WITH INVOLUTION

  • JAEYOUNG CHUNG (a1), CHANG-KWON CHOI (a2) and SOON-YEONG CHUNG (a3)

Abstract

We find all real-valued general solutions $f:S\rightarrow \mathbb{R}$ of the d’Alembert functional equation with involution

$$\begin{eqnarray}\displaystyle f(x+y)+f(x+\unicode[STIX]{x1D70E}y)=2f(x)f(y) & & \displaystyle \nonumber\end{eqnarray}$$
for all $x,y\in S$ , 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
$$\begin{eqnarray}\displaystyle f(x+y)+f(x-y)=2f(x)f(y) & & \displaystyle \nonumber\end{eqnarray}$$
for all $x,y\in \mathbb{R}^{n}$ . We also exhibit the locally bounded solutions $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ of the above equations.

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The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (no. 2015R1D1A3A01019573). The third author was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (no. NRF-2015R1D1A1A01059561).

Footnotes

References

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[1] Aczél, J. and Dhombres, J., Functional Equations in Several Variables (Cambridge University Press, New York–Sydney, 1989).
[2] Baker, J. A., ‘The stability of the cosine equation’, Proc. Amer. Math. Soc. 80 (1980), 411416.
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[4] Chung, J., ‘Distributional method for the d’Alembert equation’, Arch. Math. 85 (2005), 156160.
[5] Chung, J., ‘Distributional solutions of Wilson’s functional equations with involution and their Erdös’ problem’, Bull. Korean Math. Soc. 53 (2016), 11571169.
[6] d’Alembert, J., ‘Addition au mémoire sur la courbe que forme une corde tendue mise en vibration’, Hist. Acad. Berlin 6 (1750), 355360.
[7] Jung, S.-M., Hyers–Ulam–Rassias Stability of Functional Equations in Nonlinear Analysis (Springer, New York, 2011).
[8] Sahoo, P. K. and Kannappan, Pl., Introduction to Functional Equations (CRC Press, Boca Raton, FL, 2011).
[9] Sinopoulos, P., ‘Functional equations on semigroups’, Aequationes Math. 59 (2000), 255261.
[10] Stetkær, H., Functional Equations on Groups (World Scientific, Singapore, 2013).
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ON THE REAL-VALUED GENERAL SOLUTIONS OF THE D’ALEMBERT EQUATION WITH INVOLUTION

  • JAEYOUNG CHUNG (a1), CHANG-KWON CHOI (a2) and SOON-YEONG CHUNG (a3)

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