Resolution of an integral equation with the Thue-Morse sequence

It is a classical fact that the exponential function is solution of the integral equation $\int_0^X f(x)dx + f(0) =f(X)$. If we slightly modify this equation to $\int_0^X f(x)dx+f(0)=f(\alpha X)$ with $\alpha\in ]0,1[$, it seems that no classical techniques apply to yields solutions. In this article, we consider the parameter $\alpha=1/2$. We will show the existence of a solution wich takes the values of the Thue-Morse sequence on the odd integers.