Fixed Points in Function Spaces

If you look at the page on Wikipedia called “Fixed Point Theorems in Infinite Dimensional Spaces,” it talks about theorems about the existence and uniqueness of solutions to differential equations. This confused me for awhile, but now I think I understand what the connection is. First, let’s talk about what a fixed point is.

We say that x is a fixed point of a function f, if f(x) = x. That is, when you apply the function to that value, the value remains unchanged.

Let’s look at a few examples.

Example 1: \begin{pmatrix}7 && -4\\6 && -3\end{pmatrix}\begin{pmatrix} 2 \\ 3\end{pmatrix} = \begin{pmatrix}2 \\ 3\end{pmatrix}. When we multiply this matrix and vector, the vector remains unchanged. This also tells us that \begin{pmatrix}2 \\ 3\end{pmatrix} is an eigenvector of that matrix, and its corresponding eigenvalue is \lambda=1.

Example 2: Consider the function f(x) = x^2-2x. When x=3, f(x) = 3, since 3^2-2(3)=3. Since the input x=3 is unchanged after applying the function f(x), the point x=3 is called a fixed point of f.

Those two examples were for transformations from \mathbb{R}^2 to \mathbb{R}^2 and from \mathbb{R} to \mathbb{R} respectively. What about transformations between different sets of functions? Are there operations that don’t impact certain types of functions? Let’s look at a few examples. (In these examples the points are functions and the operations are all things that can be done to these functions.)

Example 3: Consider the operator \frac{d}{dx} that takes in a function in C^{\infty} and outputs the function’s derivative, which is also in C^{\infty}. A fixed point of this operator will be a function that doesn’t change after you take it’s derivative. The fixed point f(x) will have to satisfy \frac{d}{dx}f(x)=f(x). The only function that satisfies this property is f(x)=e^x. This means that the function f(x)=e^x is a fixed point of the operator \frac{d}{dx}. The existence and uniqueness of the fixed point of this function space is implies the existence and uniqueness of the solution to this differential equation.

Even differential equations that don’t look like their solutions are fixed points can be written to make it more clear. Here’s an example.

Example 4: Consider the differential equation f''(x) + 2f(x) = 0. We can add f(x) to both sides to get f''(x) + 3f(x) = f(x). Then we can rewrite this as (\frac{d^2}{dx^2}+3) f(x) = f(x), making it clearer how solving this differential equation is the same as finding a fixed point of an operator.

Now, all we have to do is figure out why function spaces are infinite dimensional. It’s not too hard to check that a function space like {C^\infty} is a vector space. The sum of two smooth functions is smooth, if you scale a smooth function, it’s smooth, … Now, if we take for granted the fact that for any vector space, there exists a basis for it, it’s not too hard to see why the basis for the set of all smooth functions is infinite. If we just think about the set of all polynomials (which are a subset of all smooth functions), that set has an infinite basis (1, x, x^2, x^3, ... ). Since a subset of the set of all continuous functions has infinite dimension, the set of all continuous functions must also have an infinite dimension.

Since solutions to differential equations can be thought of as inputs to an equation that are unchanged under some operator, it makes sense why a lot of physical systems can be described using differential equations. The operators in these cases are like the things happening to a particular object (things happening to money, things happening to the amount of fluid in a particular place), and the solution to the differential equation is a state of the object that is stable with respect to what is happening to it.

I hope this helped show the connection between fixed points and solutions to differential equations.


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