Question 25
A function $f : (0, \infty) \to (0, \infty)$ is called *contractive* if, for every numbers $x, y \in (0, \infty)$, we have $\lim_{n \to \infty} (f^n(x) - f^n(y)) = 0$, where $f^n = f \circ f \circ \dots \circ f$. Prove
a) If $f : (0, \infty) \to (0, \infty)$ is contractive, continuous and has a fixed point (there is $x_0 \in (0, \infty)$ such that $f(x_0) = x_0$), then $f(x) \gt x$, for $x \in (0, x_0)$, and $f(x) \lt x$, for all $x \in (x_0, \infty)$.
b) The function $f : (0, \infty) \to (0, \infty)$ defined by $f(x) = x + 1/x$ is contractive but has no fixed points.
a) If $f : (0, \infty) \to (0, \infty)$ is contractive, continuous and has a fixed point (there is $x_0 \in (0, \infty)$ such that $f(x_0) = x_0$), then $f(x) \gt x$, for $x \in (0, x_0)$, and $f(x) \lt x$, for all $x \in (x_0, \infty)$.
b) The function $f : (0, \infty) \to (0, \infty)$ defined by $f(x) = x + 1/x$ is contractive but has no fixed points.