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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 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.
  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 solution $w_n(s)$ also converges to some function $w(s)$?

If so, how can i prove it?

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  • $\begingroup$ Please specify the exact conditions you are putting on your kernels in order to guarantee existence of unique solutions, otherwise I could just take $K_n$ converging monotonely to zero. $\endgroup$ – Yemon Choi Dec 24 '14 at 12:34
  • $\begingroup$ I have revised my question. Please see again. $\endgroup$ – user155214 Dec 27 '14 at 11:30

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