Can a Neumann Series Be Proposed for the Resolvent Kernel in Iterative Methods?

[eljose]

let be the integral equation:

$$f(x)=\lambda \int_{0}^{1}dyK(x,y)f(y)$$

where the Kernel is known and "lambada" is an small eigenvalue... the question is if i want to solve i propose the iterative scheme:

$$f_{n+1} (x)=\lambda \int_{0}^{1}dyK(x,y)f_n(y)$$

My question is if in this case i can propose a "Neumann series" for the resolvent Kernel R(x,y) in the form...

$$R= \sum_{n=0}^{\infty} \lambda ^ n K^{n}$$

for K^n the n-th iterated kernel since the equation is HOmogeneus (all the functions involved except the Kernel are unknown) or if we have enough with the iteration procedure to solve the equation..thanks.

 

[HallsofIvy]

"lambda" is an eigenvalue. "Lambada" is a dance!

 

[pezze]

"lambda" is an eigenvalue. "Lambada" is a dance!

 

[lokofer]

- Ooooh...¡what a wonderful and marvelous joke¡¡..I ask myself daily why you are here wasting your time when you could be a "millionaire" Hollywood comediant like Eddie Murphy or appear on "Saturday Night LIfe"...

 

[HallsofIvy]

I keep asking myself that!