Tell me more ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
up vote 1 down vote favorite
1

Suppose that $f:\mathbb{R}\rightarrow (0,\infty)$ and $g:\mathbb{R}\rightarrow \mathbb{R}$ are two functions and satisfies the following relation \begin{equation} f(xy)=f(y)^{g(x)} \end{equation} for all $x,y\in \mathbb{R}$. Can we determine the all solutions of this funtional equation under some suitable conditions.

share|improve this question
Note that its easy to show that the function $g$ is a multiplicative map i.e; $g(xy)=g(x)g(y)$.. – Alisad Nov 11 '12 at 1:30
You can also show that there is some constant $C$ with $\frac{g(x)}{\text{log}f(x)}=C$ whenever $\text{log}f(x)\ne 0$, so $g(x)=C\text{log}f(x)$. This gives us the relation $\text{log}f(xy)=C\text{log}f(x)\text{log}f(y)$, so either $f(x)$ is constant or $f(0)=e$ and $f(1)=e^{1/C}$. – rayradjr Nov 11 '12 at 2:08
Thanks, by your idea the problem solve, since the general solution of the multiplicative functional equation studied by some athurs. – Alisad Nov 11 '12 at 2:50
@rayradjr, the usual thing is to ask you to post your comment as an answer, with whatever detail you may wish. That way Ali can accept your answer. – Will Jagy Nov 11 '12 at 3:00

1 Answer

up vote 0 down vote

By idea of @rayradjr, we can solve of my question, please see

Soon-Mo Jung, Hyers–Ulam–Rassias Stability of Functional Equations in Nonlinear Analysis, Springer Optimization and Its Applications 48, 2011, ISSN 1931-6828, page 227

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.