Exercise 1, § 1

Let g be a regulated function, and > 0 on ]0, +∞[.

a) Let u, v be two increasing functions on ]0, +∞[ such that u(x + 1) − u(x) = v(x + 1) − v(x) = g(x) for all x > 0. Show that if w = u − v one has

(Remark that, for a ≤ x ≤ y ≤ a + 1 one has u(y) − u(x) ≤ g(a).) In particular, if $\inf_{x > 0} g(x) = 0$ there exists at most one increasing solution of the equation $u(x + 1) - u(x) = g(x)$ taking a given value at a given point.

b) Assume that g is decreasing on ]0, +∞[. Show that the series

is absolutely and uniformly convergent on every compact interval contained in ]0, +∞[: if $\lambda = \lim_{x \to +\infty} g(x)$ then the function $u(x) = \varphi(x) + \lambda x$ is an increasing solution of the equation $u(x + 1) - u(x) = g(x)$. Show that for every increasing solution v of this equation one has $v(y) - v(x) \geq \varphi(y) - \varphi(x)$ for $0 < x < y$. What is the upper (resp. lower) envelope of the set of increasing solutions of the equation $u(x + 1) - u(x) = g(x)$ taking a given value at a given point? Show that this set reduces to a single element if and only if $\lambda = 0$.

c) Show that if g(x) is increasing and > 0 on [0, +∞[ there are infinitely many increasing solutions of the equation $u(x + 1) - u(x) = g(x)$ which take a given value at a given point.

d) Let $\psi(x)$ be the function defined on ]0, +∞[ by the conditions: $\psi(x) = 0$ for $0 \leq x < 1$, $\psi(x) = 1$ for $1 \leq x < 2$, $\psi(x) = n$ for $n - 1 + \frac{1}{n-1} \leq x < n + \frac{1}{n}$ ($n \geq 2$); let $g(x) = \psi(x + 1) - \psi(x)$. Show that $\psi$ is the unique increasing solution of the equation

such that $u(1) = 1$.