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http://dx.doi.org/10.4134/CKMS.c170487

SOLUTIONS AND STABILITY OF TRIGONOMETRIC FUNCTIONAL EQUATIONS ON AN AMENABLE GROUP WITH AN INVOLUTIVE AUTOMORPHISM  

Ajebbar, Omar (Ibn Zohr University Faculty of Sciences)
Elqorachi, Elhoucien (Ibn Zohr University Faculty of Sciences)
Publication Information
Communications of the Korean Mathematical Society / v.34, no.1, 2019 , pp. 55-82 More about this Journal
Abstract
Given ${\sigma}:G{\rightarrow}G$ an involutive automorphism of a semigroup G, we study the solutions and stability of the following functional equations $$f(x{\sigma}(y))=f(x)g(y)+g(x)f(y),\;x,y{\in}G,\\f(x{\sigma}(y))=f(x)f(y)-g(x)g(y),\;x,y{\in}G$$ and $$f(x{\sigma}(y))=f(x)g(y)-g(x)f(y),\;x,y{\in}G$$, from the theory of trigonometric functional equations. (1) We determine the solutions when G is a semigroup generated by its squares. (2) We obtain the stability results for these equations, when G is an amenable group.
Keywords
Hyers-Ulam stability; semigroup; group; cosine equation; sine equation; involutive automorphism; multiplicative function; additive function;
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