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 :

  1. $K_n(t,s)\neq 0 $ for each n and for all t,s.
  2. $\frac{\partial K_n(t,s)}{\partial t}$ is continuous for each n.
  3. $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?


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