Consider the following Volterra integral equation
$$ g(t) = \int_0^t K_n(t,s)w_n(s) ds $$
where $g(t)$ and $K_n(t,s)$ are known(continuous) and $K_n(t,s)\geq K_{n+1}(t,s)$ for all $t,s$. Moreover, $K_n(t,s)$ converges to $K(t,s)$.
The conditions of $K_n(t,s)$ are as follows :
- $K_n(t,s)\neq 0 $ for each n and for all t,s.
- $\frac{\partial K_n(t,s)}{\partial t}$ is continuous for each n.
- $K(t,s)\neq 0 $ for all t,s.
Conditions 1, 2 are sufficient to gurantee the existence of solution $w_n(s)$.
Then, can we say that the solutions $w_n(s)$ also converges to some function $w(s)$?
If so, how can i prove it?