Operator Form for Volterra Linear Integral Equations

[sarrah1]

Hi
I know Fredholm linear integral equations i.e.
$y(x)=f(x)+\lambda\int_{a}^{b} \,k(x,s)y(s)ds$
can be written compactly in the operator form as $y=f+\lambda Ky$
this facilitates many proofs. Usually there is a condition on $\lambda$ for the convergence of say successive approximations method using operators confortably.
To carry the same proof on Volterra equation although I know how to do it yet very tedious since it involves integrals of functions.
my question is there an operator like form for Volterra equation.
$y(x)=f(x)+\lambda\int_{a}^{x} \,k(x,s)y(s)ds$
N.B. In the proof for Fredholm $\lambda$ must be restricted which is not the case for Volterra like equations, the iterates we usually obtain in Volterra is
$|y(n)-y|<\lambda^n M^n x^n/n!$ so convergence guaranteed irrespective of $\lambda$.
How can I reach such conclusion using operators.
very grateful
thanks
Sarrah

 

[jvicens]

Hello Sarrah,

Thank you for your forum post. I am a scientist who specializes in integral equations, and I am happy to help you with your question.

To answer your question, yes, there is an operator form for the Volterra integral equation. It is given by:

$y=f+\lambda Kx$

where $Kx=\int_{a}^{x} \,k(x,s)y(s)ds$.

This form is similar to the operator form for Fredholm equations, but the only difference is that the upper limit of the integral is now $x$ instead of $b$. This form can also be used to prove convergence of successive approximations, as you mentioned in your post.

To see how this form can be used to prove convergence, let's consider the equation $y=f+\lambda Kx$ and its successive approximations $y_n=f+\lambda K^nx$. We can rewrite this as:

$y_n=f+\lambda K^{n-1}Kx$

Using the properties of operators, we can write $K^{n-1}Kx$ as $K^n x$. Therefore, we have:

$y_n=f+\lambda K^nx$

This is the same form as the successive approximations method for Fredholm equations, and we can use the same proof to show that the sequence converges to the solution of the Volterra equation.

I hope this helps answer your question. Let me know if you have any further questions or need clarification. Good luck with your research!