Multiplicative Functional Equations

The multiplicative functional equation $$f(xy)=f(x)f(y)$$ may be identified with the exponential functional equation if the domain of functions involved is a semigroup. However, if the domain space is a field or an algebra, then the former is obviously different from the latter. It is well-known that the general solution $$f\ :\ \mathbb{R}\rightarrow \mathbb{R}$$ of the multiplicative functional equation $$f(xy)=f(x)f(y)\ {\rm is}\ f(x)=0, f(x)=1, f(x)=e^{A({\rm ln}|x|)}|{\rm sign{(x)}}|,\ {\rm and} f(x)=e^{A({\rm ln}|x|)}{\rm sign(x)} \ {\rm {for\ all\ }}x \varepsilon \mathbb{R}, \ {\rm where}\ A\ :\ \mathbb{R}\rightarrow \mathbb{R}$$ is an additive function and sign $$\mathbb{R}\rightarrow \{-1, 0,1\}$$ is the sign function. If we impose the continuity on solution functions $$f:\mathbb{R}\rightarrow \mathbb{R}$$ of the multiplicative equation, then $$f(x)=0,f(x)=1,f(x)=|x|^\alpha,\ {\rm and}\ f(x)= |x|^\alpha \ {\rm sign}(x)\,{\rm {for\ all}}\ x\,\varepsilon \mathbb{R}$$ , where α is a positive real constant. The first section deals with the superstability of the multiplicative Cauchy equation and a functional equation connected with the Reynolds operator. In Section 10.2, the results on i-multiplicative functionals on complex Banach algebras will be discussed in connection with the AMNM algebras which will be described in Section 10.3. Another multiplicative functional equation $$f(x^y)=f(x)^y$$ for real-valued functions defined on R will be discussed in Section 10.4. This functional equation is superstable in the sense of Ger. In the last section, we will prove that a new multiplicative functional equation $$f(x+y)=f(x)f(y)f(1/x+1/y)$$ is stable in the sense of Ger.