Let be a set and let be a function from to . A fixed point of is an element such that . A fixed point theorem is a theorem that asserts that every function that satisfies some given property must have a fixed point. If you have an equation and want to prove that it has a solution, and if it is hard to find that solution explicitly, then consider trying to rewrite the equation in the form and applying a fixed point theorem. This method can be applied not just to numerical equations but also to equations involving vectors or functions. In particular, fixed point theorems are often used to prove the existence of solutions to differential equations.
A knowledge of some of the main fixed point theorems (though these are also discussed in the article, and links are given to Wikipedia articles about them).
Let be an matrix with non-negative entries. A result with many applications is that must have an eigenvector with non-negative coefficients.
To prove this, let be the subset of that consists of all vectors such that each is non-negative and . If there exists such that , then we are done. Otherwise, we know that for every , has non-negative entries, not all of them zero. Let us write for the sum of the coefficients of . Then the map is a continuous map from to .
Now geometrically is a simplex of dimension , and therefore it is homeomorphic to a ball of dimension . The Brouwer fixed point theorem implies that has a fixed point, so there must be some such that . This is an eigenvector with eigenvalue , so the result is proved.
Let be a Banach space, and let and be linear operators on . If is invertible and , then the Contraction Mapping Theorem can be used to show that is also invertible. Informally, this says that if closely approximates an invertible operator, then is invertible.
To show that is invertible, we will show that for each in , there is a unique such that . Let be an element of . If for some in , then we have . Multiplying by , we have . To simplify notation, write and , so that .
Now we'll define a function on by . If we can show that is a contraction mapping, then the fixed point of will be , as gives us .
Let and be elements of . Then . By assumption, , so is a contraction mapping, which gives us the desired fixed point .
The proof of the contraction mapping theorem proceeds by iterating the contraction. In this case, since the contraction is a linear map, one can do this explicitly and see what one obtains. The first few iterates of the function above are , , , and we see that, because , this sequence is converging to the point .
If is the identity and , then this is particularly easy to see directly: the inverse of is , and one can justify it by noting that , which converges to . One can deduce the general case from this too, since . Therefore, the contraction mapping theorem is not essential in this example.