Caristi fixed-point theorem

The Caristi Fixed-Point Theorem is a mathematical theorem that says that under certain conditions, you can always find a "fixed point" in a system. A fixed point is like an imaginary spot in a system that never changes. For example, imagine a trampoline - if you keep throwing a ball on the trampoline, the ball will keep bouncing back up, but never end up in the same exact spot again. The Caristi Fixed-Point Theorem says that if certain conditions are true, then you can find a single spot on the trampoline that the ball will always come back to, no matter how many times you throw it.