SEEMOUS 2020 Problem 4

Consider $0<a<T$ , $D=\mathbb{R}\backslash \{ kT+a\mid k\in \mathbb{Z}\}$ , and let $f:D\to \mathbb{R}$ a $T-$ periodic and differentiable function which satisfies $f' > 1$ on $(0, a)$ and $$f(0)=0,\lim_{\substack{x\to ax<a}}f(x)=+\infty \text{ and }\lim_{\substack{x\to a x<a}}\frac{f'(x)}{f^2(x)}=1.$$

• Prove that for every $n\in \mathbb{N}^*$ , the equation $f(x)=x$ has a unique solution in the interval $(nT, nT+a)$ , denoted $x_n$ .[/*]
• Let $y_n=nT+a-x_n$ and $z_n=\int_0^{y_n}f(x)\text{d}x$ . Prove that $\lim_{n\to \infty}{y_n}=0$ and study the convergence of the series $\sum_{n=1}^{\infty}{y_n}$ and $\sum_{n=1}^{n}{z_n}$ .