Volterra Integral Equation as a Generalisation of Picard Theorem

[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.

Homework Statement

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.

Homework Equations

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.

 

[mighty2000]

Thank you for your post and for your interest in the Picard Theorem and Volterra integral equations. I am a physicist with a background in mathematical methods, and I would be happy to assist you in understanding this topic.

First of all, your approach to proving the convergence of the sequence \{y_n\}_{n\geq 0} is correct. However, the issue you are facing is due to a small mistake in your calculations. Let me explain further.

When you calculate the difference between successive approximations, you are integrating with respect to x', which is the variable of integration. However, in the second term of your integration, you are using x as the variable of integration, which is incorrect. The correct way to write the second term is:

\left|\int_0^xK(x,x')y_0(x') - K(x,x')y_0(x')dx'\right|

Now, we can see that the two terms inside the integral cancel out, leaving us with:

\left|\int_0^x0dx'\right| = 0

This means that the difference between successive approximations is 0, which shows that the sequence is converging to a solution of the Volterra integral equation.

I hope this helps to clarify your doubts. If you have any further questions, please don't hesitate to ask.