Bulletin of the
Korean Mathematical Society BKMS

We find the distributional solutions of the Wilson's functional equations \begin{align} u\circ T +u\circ T^\sigma -2u\otimes v&=0,\nonumber\\ u\circ T +u\circ T^\sigma -2v\otimes u&=0,\nonumber \end{align} where $u, v\in \mathcal D'(\mathbb R^n)$, the space of Schwartz distributions, $T(x, y)=x+y,\,\, T^\sigma (x, y)=x+\sigma y,\,\, x, y\in \mathbb R^n$, $\sigma$ an involution, and $\circ,\, \otimes$ are pullback and tensor product of distributions, respectively. As a consequence, we solve the Erd\"os' problem for the Wilson's functional equations in the class of locally integrable functions. We also consider the Ulam-Hyers stability of the classical Wilson's functional equations \begin{align} f(x+y) + f(x+\sigma y ) &= 2 f(x) g(y),\nonumber \\ f(x+y) + f(x+\sigma y ) &= 2 g(x) f(y)\nonumber \end{align} in the class of Lebesgue measurable functions.

Keywords: d'Alembert's functional equation, distributions, exponential function, Gelfand generalized function, involution, Wilson's functional equation

MSC numbers: 39B52, 46F15