Let's say I want to solve a PDE that can be written as $(A - K)u=0$ where $A$ is an operator with a continuous inverse and $K$ is compact. Then I'd have a solution if I managed to find a fixed point for $A^{-1}K$, that is a $u$ such that $A^{-1}Ku=u$.
If I knew that, for some $R>0$, calling $B_R$ the ball of radius $R$ centered in $0$, $A^{-1}K (B_R) \subset B_R$ I could conclude via Brower fixed point theorem. An obvious sufficient condition for this to happen would be, for instance, $||A^{-1}K||<1$, or even $||A^{-1}||< a$ and $||K||< b$ with $ab<1$, but I was wondering if there were ways to generalise this result for a larger class of operators, if there was some trick that allowed you to use Brower or Schauder fixed point theorems in more situations.