Any periodic function, for example sine or cosine or tangent, or any finite or infinite sum of multiple periodic functions which also yields a periodic function, can be expressed in the format which you are asking for. (Assuming that you are considering the sine solution you listed as a non-trivial solution)
For example, for $k=1$, you can transform the domain of the function $y=sin(x)$ to $y=sin(x/{(2\pi)})$, or for arbitrary $k$, $$y=sin(\frac{2\pi x}{k})$$
For any arbitrary function $g(x)$ which is periodic with period length $a$, dividing the domain by the $k$ desired and multiplying by period length will yield such a function.
$$ y = g(\frac{a x}{k}) $$
If you are looking for a non-periodic trivial solution, then it's a different story and answer, differential delay equations, as pointed out by Denis Serre above.