# Volterra Integral Equation as a Generalisation of Picard Theorem

1. Oct 10, 2012

### Geremy

Hi Physics Forums, this is my first post here thanks in advance for any help hopefully I'll be able to return the favor.

1. The problem statement, all variables and given/known data

As a generalisation of the Picard Theorem: An integral equation of the form:

$y(x) = f(x) + \int_0^xK(x,x')y(x')dx'$ $(0 \leq x \leq b)$

where $f(x)$ and $K(x,x')$ are continuous is called a Volterra integral equation. Let $||f||$ and $||K||$ be upper bounds for
$0 \leq x \leq b$ and for $|K|$ on $0 \leq x' \leq x \leq b$, respectively.
Prove that the sequence $\{y_n\}_{n\geq 0}$ of functions deﬁned by $y_0(x) = f(x)$ and

$y_{n+1} = f(x) + \int_0^xK(x,x')y_{n}(x')dx'$ $n\geq0$
converges to a solution y(x) of the equation.

2. Relevant equations

3. The attempt at a solution
I'm trying to show that the difference between successive approximations $y_{n+1}-y_{n}$ tends to $0$ as $n\rightarrow \infty$ But the problem is that when I try to formulate an induciton hypothesis to prove this, it appears to diverge...
That is:

$\left|y_1 - y_0\right| = \left|\int_0^xK(x,x')y_0(x')dx'\right|$
$\leq \left|\int_0^xBdx'\right|$
$= \left|Bx\right|$ where B = ||f||.||K||
then:
$\left|y_2 - y_1\right| =\left|\int_0^xK(x,x')y_1(x') - K(x,x')y_0(x')dx'\right|$
$\leq \left|\int_0^xBx - Bdx'\right|$
$= \left|B(x^2 - x)\right|$

But going this way, the difference appears to be getting bigger between successive approximations.
I think my problems are stemming from integrating with respect to x' instead of x,
can someone please show me where I'm going wrong?

Thanks a lot for any help at all.