There is an integral equation
$$ y(x)=\cos(\rho x) + \int_a^xq(t)y(t-a)\frac{\sin(\rho(x-t))}{\rho}dt $$
The solution of it could be represented in the form
$$ y = \sum_{k=0}^\infty y_k $$
and there is a recursive expression
$$ y_0(x) = \cos(\rho x) $$ $$ y_{k+1}(x)=\int_a^xq(t)y_k(t-a)\frac{\sin(\rho(x-t))}{\rho}dt $$
How is it possible to prove that the series $\sum_{k=0}^\infty y_k$ has only a finite number of non-zero elements?