Problem 1

Find all functions

f

defined on the set of positive real numbers which take positive real values and satisfy the two conditions: (i)

f(xf(y)) = yf(x)

for all positive real numbers

x, y

; (ii)

f(x) \to 0

x \to \infty

Step 5 of 6: No fixed point other than

1

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a>1 \text{ fixed} \ \Rightarrow\ f\!\left(a^{2^k}\right)=a^{2^k}\xrightarrow[k\to\infty]{} \infty

Detailed analysis

Suppose some $a\ne1$ is a fixed point; replacing $a$ by $1/a$ if needed (Step 4), we may assume $a>1$ . By Step 4 with $b=a$ , $a^2$ is fixed; repeating, $a^{2^k}$ is fixed for every $k\ge0$ . As $k\to\infty$ , $a^{2^k}\to\infty$ , so $f(a^{2^k})=a^{2^k}\to\infty$ , contradicting condition (ii) that $f(x)\to0$ as $x\to\infty$ . Hence $1$ is the only fixed point of $f$ .