Fixed Point Theory is a branch of mathematics that studies the conditions under which a function will have points that remain unchanged when the function is applied. These points are called "fixed points." For example, if you have a function f(x) and a point x₀ such that f(x₀) = x₀, then x₀ is a fixed point of f. This theory has applications in various fields, including economics, computer science, and physics. One of the key results in Fixed Point Theory is the Brouwer Fixed Point Theorem, which states that any continuous function mapping a compact convex set to itself has at least one fixed point. This theorem is fundamental in understanding how systems behave and can be used to prove the existence of solutions in differential equations and optimization problems.