The parameter λ in linear integral equations

[sarrah1]

Hi

the parameter $\lambda$ in linear integral equations say Fredholm that appears in front of the integral containing the kernel i.e.

$y(x)=f(x)+ \lambda \int_{a}^{b} \,k(x,t) y(t) dt $

can $\lambda$ be adjusted for the convergence of the method like in using Picard's successive approximations like imposing the condition $|\lambda|< 1/||K||$, where $K$ is the operator for $k(x,t)$ or is it fixed and thus can be united with the kernel, in which case one seeks the condition on the kernel itself, either satisfying the criteria of convergence i.e. whether it applies or not

thanks
I assume this time it's a very simple question
Sarrah

 

[Euge]

Hi sarrah,

I assume this time it's a very simple question
Sarrah

Your posts usually have problems that are ill-posed. If you present a PDE or an integral equation, it is necessary to indicate the (Banach) space under consideration and all other data that goes with it.

Here is an example of how you can post a well posed problem:

Consider the nonlinear Schrödinger integral equation

$$u(x,t) = e^{it\Delta}u_0 + i\lambda \int_0^t e^{i(t-t')\Delta}\lvert u\rvert^{p-1}(x,t') u(x,t')\, dt'$$

For $(x,t)\in \Bbb R^{n+1}$, $\lambda \neq 0$ and $p > 1$. If $u_0\in L^2(\Bbb R^n)$, for what values of $p$ and $q$ will the above equation have a unique solution in $C_t[-T,T]L^2_x(\Bbb R^n) \cap L_t^qL_x^{p+1}$ for some time $T > 0$? Explicitly, what would be an appropriate bound on $\lambda$ to ensure such a solution?

Now to answer your question -- you can do Picard iteration formally, but to speak of convergence, it is necessary to indicate the normed spaces which $y$ and $k$ belong to.

 

[sarrah1]

Hi sarrah,
Your posts usually have problems that are ill-posed. If you present a PDE or an integral equation, it is necessary to indicate the (Banach) space under consideration and all other data that goes with it.

Here is an example of how you can post a well posed problem:Now to answer your question -- you can do Picard iteration formally, but to speak of convergence, it is necessary to indicate the normed spaces which $y$ and $k$ belong to.

thank you very much Euge
Sarrah