Consider the first-order equation given by $y' = f(x, y)$ in $I := [x_0, b]$ subject to the initial condition $y(x_0) = y_0$, assuming $f$ is continuous everywhere it is defined and there exists $M > 0$ such that
$$|f(x, y_1) - f(x, y_2)| \leq M|y_1 - y_2|^{a} \quad \text{for some } 0 < a < 1$$
and for each fixed $x \in [x_0, b]$. Comment on the uniqueness of the solutions for such a problem.
I know some things about holder continuity i.e. if the exponent $a > 1$, the function has to be constant and if $a = 1$, the function is Lipschitz continuous which has unique solution. If $a > 0$, then the function is uniformly continuous. I am not sure how to solve this problem. We can say that a solution exists locally by Picard theorem, but what about uniqueness?