Distributional solutions of Wilson's functional equations with involution and their Erd\"os' problem
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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.
Abstract
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.
참고문헌 (32)
- J. Aczel, "Lectures on Functional Equations and Their Applications" (2006), Dover Publications Inc.
- J. Aczel, "Symmetric second differences in product form on groups, Topics in mathematical analysis", pp.1~22 (1989), World Sci. Publ.
- T. Angheluta, "Asupra unei ecuatii functionale cu trei functii necunoscute", Lucr. Sti. Inst. Politehn. Astr, 5, pp.23~30 (1960)
- A. Bahyrycz, "On solutions of the d’Alembert equation on a restricted domain", Aequationes Math, 85(1-2), pp.169~183 (2013)
- N. G. De Brujin, "On almost additive functions", Colloq. Math, 15, pp.59~63 (1966)
- J. Chung, "A distributional version of functional equations and their stabilities", Nonlinear Anal, 62(6), pp.1037~1051 (2005)
- J. Chung, "Stability of exponential equations in Schwarz distributions", Nonlinear Anal, 69(10), pp.3503~3511 (2008)
- J. Chung, "Solution of several functional equations on nonunital semigroups using Wilson’s functional equations with involution", Abstr. Appl. Anal, 2014, pp.9~ (2014)
- J. Chung, "Stability of Wilson’s functional equations with involutions", Aequationes Math, 89(3), pp.749~763 (2015)
- J. K. Chung, "On a quadratic-trigonometric functional equation and some applications", Trans. Amer. Math. Soc, 347(4), pp.1131~1161 (1995)
- I. Corovei, "The cosine functional equation for nilpotent groups", Aequationes Math, 15(1), pp.99~106 (1977)
- I. Corovei, "The functional equation f(xy) + f(xy−1) = 2f(x)g(y) for nilpotent groups", Mathematica (Cluj), 22(45)(1), pp.33~41 (1980)
- I. Corovei, "The d’Alembert functional equation on metabelian groups", Aequationes Math, 57(2-3), pp.201~205 (1999)
- I. Corovei, "Wilson’s functional equation on metabelian groups", Mathematica (Cluj), 44(67)(2), pp.137~146 (2002)
- P. De Place Friis, "D’Alembert’s and Wilson’s equations on Lie groups", Aequationes Math, 67(1-2), pp.12~25 (2004)
- J. d’Alembert, "Addition au Memoire sur la courbe que forme une corde tendue mise en vibration", pp.355~360, Hist. Acad
- J. d’Alembert, "Memoire sur les principes de mecanique", pp.278~286, Hist. Acad. Sci
- P. Erdos, "Problem P310", Colloq. Math, 7, pp.311~ (1960)
- I. Fenyo, "Uber eine Losungsmethode gewisser Funktionalgleichungen", Acta Math. Acad. Sci. Hungar, 7, pp.383~396 (1956)
- I. M. Gelfand, "Generalized Functions II" (1968), Academic Press
- I. M. Gelfand, "Generalized Functions IV" (1968), Academic, Press
- L. Hormander, "The analysis of linear partial differential operator I" (1983), Springer–Verlag
- W. B. Jurkat, "On Cauchy’s functional equation", Proc. Amer. Math. Soc, 16, pp.683~686 (1965)
- S. Kaczmarz, "Sur lequation fonctionnelle f(x)+f(x+y) = Φ(y)f(x+y", 2), 1924, pp.6, 122~129
- P. K. Sahoo, "Introduction to Functional Equations" (2011), CRC Press
- L. Schwartz, "Theorie des Distributions" (1966), Hermann
- P. Sinopoulos, "Functional equations on semigroups", Aequationes Math, 59(3), pp.255~261 (2000)
- H. Stetkær, "Functional equations on abelian groups with involution", Aequationes Math, 54(1-2), pp.144~172 (1997)
- G. van der Lyn, "Sur l’equation fonctionnelle f(x + y) + f(x − y) = 2f(x)Φ(y)", Mathematica (Cluj), 16, pp.91~96 (1940)
- D. V. Widder, "The Heat Equation" (1975), Academic Press
- W. H. Wilson, "On certain related functional equations", Bull. Amer. Math. Soc, 26(7), pp.300~312 (1920)
- W. H. Wilson, "Two general functional equations", Bull. Amer. Math. Soc, 31(7), pp.330~333 (1925)