I am new to non-linear integral equations and want to know how to solve non-linear Fredholm integral equations of the following form generally:
$u(x)=c+\lambda\int_{0}^{1}u^n(y)dy$ where $c,\lambda\in \mathbb{R}$ and $n\in \mathbb{N}$ with $n\geq 1$.
I know that typically these are solved via a substitution method using the Resolvent Kernel function. The representation formula for the solution is given as follows: $u(x)=1+\lambda\int_{0}^{1}R(x,t;\lambda)u^n(y)dy$, and the resolvent Kernel is given by:
$R(x,t;\lambda)=\sum_{m=1}^{\infty}\lambda^{m-1}K_m(x,t)$
Are the $K_m$ kernels always $1$, and how do we go from here. The non-linear term of $u^n(y)$ is throwing me off.
EXTRA
As an aside/bonus, I am assuming the solution $u$ will be an $L^2$ function, but I also want to know what happens if we consider Sobolev spaces like $H^2$ and other function spaces, if the solution methodology changes.