Define the operator by
where is a continuous complex valued function on the unit square. Also assume that
How to show that given , the integral equation
has exactly one continuous solution ?
Note: I only know introductory measure theory and functional analysis, so I am suppose to do this problem without heavy integral function theory.
One way to get started is to show that K is compact and its operator norm because we are given that Then we know that is invertible and its inverse is given by the Neumann series. At this point, I'm not sure how to proceed.