If $h(x)$ is such that it satisfies $$ h(x) = \lambda \int_a^b K(x,y) h(y)\, dy $$ Then for $\phi(x)$ a solution of $$ \phi(x) = f(x) + \lambda \int_a^b K(x,y)\phi(y)\, dy $$ it is true that $\phi + h$ is also a solution. But I am confused as to why this does not violate the uniqueness of solutions to these integral equations?
Asked
Active
Viewed 91 times
0
-
Why should uniqueness of solutions exist? – Carlos Eugenio Thompson Pinzón Oct 31 '13 at 18:27
-
Do you know anything about the kernel $K$? If, for example, $\lambda K$ is always non-negative, then you can apply Gronwall's inequality to imply that $h$ is zero, I believe. – Tom Oct 31 '13 at 18:28