Is the following condition necessary for the integral equation $$u(x) = f(x)+\lambda\int K(x,t)u(t)dt$$ to have a continuous solution: $f(x) \neq 0$, is real and continuous in the interval $[a,b]$?
When $f(x) = 0$, the integral equation will become homogeneous. So I think this should be a necessary condition. Am I right? Please suggest me