Exercise 3, § 1

a) Let $g$ be a function defined and admitting a continuous $k^{th}$ derivative on ]0, $+\infty[$, such that $g^{(k)}$ is decreasing on this interval and that $\lim_{x \to +\infty} g^{(k)}(x) = 0$. Show that there exists one and only one solution $u$ of the equation $u(x + 1) - u(x) = g(x)$ which admits an increasing $k^{th}$ derivative on ]0, $+\infty[$ and takes a given value at a given point (use exerc. 1 b)).

b) Let $\alpha$ be an arbitrary real number. Let $S_\alpha$ be the function defined on ]0, $+\infty[$ satisfying the relation

such that $S_\alpha(1) = 0$, and, further, that the derivative of $S_\alpha$ of order equal to the integer part of $(\alpha + 1)^+$ is increasing (it is a unique function by $a$) ). Show that and for every integer $p \geq 1$, where $C_\alpha$ is a constant, which, when $\alpha \neq -1$, is equal to (cf. VII, p.318, prop. 2 and VI, p. 291, exerc. 3).