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About the linear functional equations: $f(x + a) = bf(x)$ and $f(ax) = bf(x)$, Marek Kuczma e Polyanin A.D. they got the respective solutions (http://eqworld.ipmnet.ru):

$f(x) = g(x)b^{x/a}$, where $g(x) = g(x + a)$ is an arbitrary periodic function with period $a$.

And

$f(x) = g(\log x)x^{\log b/\log a}$, where $g(x) = g(x +\log a)$ is an arbitrary periodic function with period log(a).

By the induction method I got the particular solutions: $f(x) = Cb^{x/a}$ and $f(x) = Cx^{\log b/\log a}$, where $C$ is an arbitrary constant. But I did not understand how they arrived at the generic solutions with the arbitrary periodic function "$g(x)$"?

Would anyone have a demonstration of how they arrived at the generic solutions including the arbitrary periodic functions?

I searched all over the net and found no proof and no book accessible. Thank You.

JaberMac
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1 Answers1

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I'll explain with a very simple example.

From the equation

$$f(x+1)=f(x)$$ you will conclude $$f(x)=c.$$

But as $x$ is a continuous variable it can take fractional values. Given that e.g. $f(0)$ and $f(0.3)$ are unrelated by the equation, you might very well have $f(0)=f(1)=f(2)=\cdots=4$ while $f(0.3)=f(1.3)=f(2.3)=\cdots=-5$.

In fact, $c$ is not a constant but a function of the fractional part of $x$, or if you prefer, $f$ is an arbitrary periodic function of period $1$.

For example,

$$f(x)=e^{\sin(2\pi x)}$$ or

$$f(x)=(x-\lfloor x\rfloor)^2$$ are solutions.

  • Yes, I understood how the periodic function in question works in isolation. But what I did not understand is how from the equation f(x + a) = bf(x) the solution was obtained with the associated arbitrary periodic function g(x)? Note that the arbitrary periodic function multiplies a particular solution. How to demonstrate this? – JaberMac May 17 '19 at 16:36
  • @JaberMac: apply my method to your equation. –  May 17 '19 at 16:51
  • Sorry, I still do not understand how I can apply the method in the original equation. There are constants a and b that I can not linearize. I got something like this: f(x + na) = (b^n)f(x). Replacing t:= x + na, we have f(t) = [b^(t-x)/a] f(t-na) – JaberMac May 18 '19 at 00:09
  • But in equation f(t) = [b^(t-x)/a] f(t-na) I can not see or extract the periodic function. Help. – JaberMac May 18 '19 at 00:20