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A variant of Wilson's functional equation on semigroups
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Let $S$ be a semigroup. We determine the complex-valued solutions of the following functional equation \[f(xy)+\mu (y)f(\sigma (y)x) = 2f(x)g(y),\ x,y\in S,\] where $\sigma:S\rightarrow S$ is an automorphism, and $\mu :S\rightarrow \mathbb{C}$ is a multiplicative function such that $\mu (x\sigma (x))=1$ for all $x\in S$.
Abstract
Let $S$ be a semigroup. We determine the complex-valued solutions of the following functional equation \[f(xy)+\mu (y)f(\sigma (y)x) = 2f(x)g(y),\ x,y\in S,\] where $\sigma:S\rightarrow S$ is an automorphism, and $\mu :S\rightarrow \mathbb{C}$ is a multiplicative function such that $\mu (x\sigma (x))=1$ for all $x\in S$.
참고문헌 (9)
- J. Acz´el, "Encyclopedia of Mathematics and its Applications, 31" (1989), Cambridge Univ. Press
- Omar Ajebbar, "Variants of Wilson's functional equation on semigroups", 대한수학회논문집, 35(3), pp.711~722 (2020), 대한수학회[DOI]
- B. Ebanks, "The sine addition and subtraction formulas on semigroups", Acta Math. Hungar, 164(2), pp.533~555 (2021)[DOI]
- B. Ebanks, "Around the sine addition law and d’Alembert’s equation on semigroups", Results Math, 77(1), pp.14~ (2022)[DOI]
- E. Elqorachi, "Solutions and stability of a variant of Wilson’s functional equation", Proyecciones, 37(2), pp.317~344 (2018)[DOI]
- B. Fadli, "A variant of Wilson’s functional equation", Publ. Math. Debrecen, 87(3-4), pp.415~427 (2015)[DOI]
- K. Sabour, "Wilson’s functional equation with an endomorphism", Math-Recherche et Application, 15, pp.32~39 (2016)
- H. Stetkær, "Functional equations on groups" (2013), World Sci. Publ
- H. Stetkær, "A variant of d’Alembert’s functional equation", Aequationes Math, 89(3), pp.657~662 (2015)[DOI]
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