Solution $y(x)$ for $\sin(x) = \int_0^{2\pi} \max(y(t), y(x+t)) dt$

Question

I've been banging my head against a wall for a few weeks to find a feasible solution for $y(x)$.

$$\sin(x) = \int_0^{2\pi} \max(y(t), y(x+t)) dt$$

I don't think there is an unique solution, but I will accept any solution. I've tried various numerical minimization methods, convolution, guess the solution, etc, but nothing seems to work.

Does anybody have any ideas?

Answer 1

If $x=0$ then \begin{equation*} 0= \sin(0)=\int_{0}^{2\pi}\max(y(t),y(t))\, \mathrm{d}t = \int_{0}^{2\pi}y(t)\, \mathrm{d}t . \end{equation*} If $x = \frac{3\pi}{2}$ then \begin{equation*} -1 = \sin\left(\frac{3\pi}{2}\right) = \int_{0}^{2\pi}\max\left(y(t),y\left(\frac{3\pi}{2}+t\right)\right)\, \mathrm{d}t \ge \int_{0}^{2\pi}y(t)\, \mathrm{d}t = 0. \end{equation*} Consequently, it does not exist any solution $y(x)$.