Fixed-point theorems are mathematical principles that guarantee the existence of points that remain unchanged under certain functions. In simpler terms, if you apply a specific function to a point, the output will be the same as the input. These theorems are essential in various fields, including mathematics, economics, and computer science. One of the most famous fixed-point theorems is Brouwer's Fixed-Point Theorem, which states that any continuous function mapping a compact convex set to itself has at least one fixed point. This concept helps in solving equations and optimizing problems where solutions need to be found within defined boundaries.