fixed-point theorems

Fixed-point theorems are mathematical principles stating that under certain conditions, a function will have at least one point where the input equals the output. Imagine stretching or moving a shape without tearing it; a fixed point is like a spot on the shape that remains unchanged. These theorems are fundamental in fields like economics, computer science, and physics, helping prove the existence of solutions to equations and systems. Essentially, they assure us that specific processes or functions will have a stable point or equilibrium, where things stay constant.