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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.
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참고문헌 (32)
참고문헌 신청
J. Aczel / 2006 / Lectures on Functional Equations and Their Applications / Dover Publications Inc.
J. Aczel / 1989 / Symmetric second differences in product form on groups, Topics in mathematical analysis / World Sci. Publ. :1~22
T. Angheluta / 1960 / Asupra unei ecuatii functionale cu trei functii necunoscute / Lucr. Sti. Inst. Politehn. Astr 5:23~30
A. Bahyrycz / 2013 / On solutions of the d’Alembert equation on a restricted domain / Aequationes Math 85(1-2):169~183
N. G. De Brujin / 1966 / On almost additive functions / Colloq. Math 15:59~63
J. Aczel / 1989 / Symmetric second differences in product form on groups, Topics in mathematical analysis / World Sci. Publ. :1~22
T. Angheluta / 1960 / Asupra unei ecuatii functionale cu trei functii necunoscute / Lucr. Sti. Inst. Politehn. Astr 5:23~30
A. Bahyrycz / 2013 / On solutions of the d’Alembert equation on a restricted domain / Aequationes Math 85(1-2):169~183
N. G. De Brujin / 1966 / On almost additive functions / Colloq. Math 15:59~63
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