This is what the Internet says:

Brouwer's fixed point theorem is named after the Dutch mathematician Ruiz Brouwer (English: L. E. J. Brouwer). Brouwer's fixed point theorem shows that there exists a point x0 for a continuous function f that satisfies certain conditions in topological space, so that f(x0) = x0. The simplest form of Brouwer's fixed point theorem is the function f that projects from the disk D to itself. The more general theorem applies to all functions projected from the convex compact subset of Euclidean space to itself.

In the sequence, a 1 = 1, a2 = 2, a (n+2) =-a (n+1)+2an (the brackets after a represent subscripts) find a general term.

At that time, I remembered a method of this body: A (n+2)+A (n+ 1)-2AN = 0 after the original shape is deformed.

Let x 2+x -2 = 0 to get X=-2 or 1, so {a (n+ 1)-an} is a sequence with a public ratio of -2; {a (n+ 1)+2an} is a sequence with the public ratio of 1.

And then solve it at the same time

The above methods should be said to be characteristic root method and fixed point method.

Characteristic root:

For the recurrence formula of multiple continuous terms (excluding constant terms), it can be simplified to the (n- 1) order equation of X. 。

Namely: A0 * an+a1* an+1+A2 * an+2+... AK * an+k can be written as:

a0+a 1x+a2x^2+...akx^(k- 1)=0

Then find the roots (both real roots and imaginary roots are acceptable). Different terms are written as c * x (n- 1), and the same term is written as an algebraic expression about n. How many roots are there? The degree of n is the number of roots minus 1. For example, find x 1 = 2, x2 = 3, x3 = 3 and x4 = 6.

Fixed point:

For example, given a 1= 1, a(n+ 1)= 1+2/an (n is greater than or equal to 1), find an.

a(n+ 1)=(an+2)/an(*)

Let an = x and a (n+1) = x.

x=(x+2)/x

x^2-x-2=0

x 1=2，x2=- 1

{(an-2)/(an+ 1)} is a geometric series.

Order (an-2)/(an+ 1)=bn

b(n+ 1)/bn =[(a(n+ 1)-2)/(a(n+ 1)+ 1)]/[(an-2)/(an+ 1)]

(replace a with an (n+1)

=- 1/2

b(n+ 1)=(- 1/2)bn

b 1=- 1/2

bn=(- 1/2)^n=(an-2)/(an+ 1)

an=[2+(- 1/2)^n]/[ 1-(- 1/2)^n],n>； = 1

Note: Fractional recursion in the form of a(n+ 1)=(Aan+B)/(Can+D), where a and c are not 0, can be solved by fixed point method. Let a(n+ 1)=an=x and substitute it into the quadratic equation about x.

(1) If two x 1 are not equal to x2, then one {(an-x 1)/(an-x2)} is a geometric series, and the common ratio is obtained by two quotients.

(2) If two x 1 equals x2, {1/(an-x 1)} is arithmetic progression, the tolerance is calculated by the difference between the two terms.

If there is no solution, we must find another way.

Moreover, the fixed point is generally only used when the score is up and down once, and it will not work if there is a second possibility.

When you open the map and find your position, you may not know it, but you have verified an important theorem in mathematics-the "compressed image theorem" in the "fixed point principle". If your map is still very irregular and seriously deformed, then you have done something that mathematicians think is very difficult-finding a fixed point in a complicated situation.

Solving equations is undoubtedly a very important problem in mathematics, such as algebraic equations, functional equations and differential equations. These equations can be rewritten as? (x)=x form, which is the fixed point principle. Mathematicians have proved many existence theorems of fixed points, but it is still very difficult to find fixed points specifically, except in special cases.

The fixed point principle has many forms and involves many branches of mathematics. Please refer to 1 2 for the introduction of popular science, so I won't discuss it in detail here.

The fixed point principle has very intuitive geometric significance. This paper tries to give you an intuitive understanding of the abstract concept of fixed point principle through several examples.

Examples of fixed points:

Example 1

Suppose you have an accurate ideal meter scale A, reduce A to B (not required to be evenly proportional), and then put it on A at will. At this time, the scales of A and B are likely to be different at the same position. For example, 10cm of B may be on 15cm of A, but the fixed point principle tells us that there must be a point on B, and A and B have the same scale, which is the so-called "fixed point". Describe this process in mathematical language: if a line segment undergoes continuous transformation F, but every point is still on this line segment, that is, F(A) is contained in A, then there must be a point C that remains unchanged, that is, F (c) = C

If scaled down, we can easily prove this proposition 3, p 136 by geometric method. Generally speaking, we can intuitively prove that:

Let the parameter of A be T, and the compression transformation F: A→B(A includes B), and let F be differentiable and v=dF/dt. Imagine two ants, A and B, climbing to the end of A and B respectively. A moves at a uniform speed of 1, and B moves at a variable speed of V, so A and B are on the same scale of A and B at the same time. Intuitively, T, A and B must have an intersection moment, and the intersection point is the "fixed point".

(But if you find this in detail, it will become very difficult with the complexity of F)

Example 2

Let's look at the two-dimensional scene again: reduce the map A, such as the map of China, and record it as B. Put B on the original map A at will, and every point on the map B will have a new position on A. Maybe B's Beijing is in A's Shanghai position and Nanjing is in Chengdu position. But the fixed point principle tells us that there must be a point on B that has not moved, that is, this point represents the same position on two maps A and B.

If scaled down, we can prove 3, p 138 by plane geometry (not easy). Generally speaking, it is difficult for us to give a simple and intuitive proof of this example (if we have studied the interval set theorem, we can use this theorem to prove it, and the idea of proof can refer to a Weibo dialogue listed later), but we can give a very intuitive explanation:

Imagine that you have an accurate and ideal GPS, but the screen is severely deformed, so a map of China with reduced deformation is displayed on the screen. If we regard China as a big map A and the map on the GPS screen as a miniature map B, then the point on the screen that shows your current position is the so-called "fixed point".

In fact, you use a map to find your position, which is the process of finding a fixed point (nearby). If your map is very irregular, then you are doing something difficult mathematically, find a fixed point (nearby).

Example 3

Let's look at a three-dimensional example: we compress an ideal cake A into B from all directions (not necessarily uniform in all directions) and move it arbitrarily inside A, then the fixed point principle tells us that there must be a point in the cake that has no displacement, that is, a fixed point.

Similar to Example 2, intuitively, we can understand it as follows:

Think of China and the sky above 1000 meters as a big cake. Suppose you have a three-dimensional accurate and ideal GPS in the future, suppose you are suspended in the air (by plane, hot air balloon? ), you can think of this three-dimensional GPS as a compressed cake, and your current position displayed by this GPS is this fixed point.

After reading the above three examples, we can find that they are only intuitive descriptions of the same problem in one, two and three dimensions. In this process, the image is compressed. Therefore, this phenomenon is called "contraction image theorem" in mathematics and is one of many "fixed point principles". The "contraction image theorem" has a more general expression 1, 2.

There is an interesting conversation about compressing pictures in Weibo:

In fact, this dialogue describes the idea of proving the theorem of compressed images. The theorem of compressed image can be proved similarly, and the compression function can be expressed as f, that is, f: a → a, B=B( 1)=F(A), B(n)=F(B(n- 1)). If f is similar to the above three examples, then B(n).

Let's look at another very intuitive example and an interesting but incredible inference.

Example 4

Mathematicians are always curious and always try to discuss a wider range of problems. According to this mindset, it is natural to ask, what will happen on the sphere? The problem of spherical surface is much more complicated because of its different spatial structure. We have the following conclusions:

Brouwer theorem: If F is a continuous mapping from a (two-dimensional) sphere to the sphere itself, there must be a point C, so F(c)=c or–c ... That is to say, F has either a fixed point or a point mapped to its diameter point (connected to another point on the sphere through the diameter of C).

The proof of this proposition is very complicated and requires algebraic topology theory, even if it is difficult to understand intuitively (at least I can't describe it intuitively like the previous example). However, using this proposition, mathematicians (not meteorologists) can get a very interesting and intuitive conclusion: at any moment, there is always no wind on the earth (the horizontal wind speed is zero).

This assertion seems to have nothing to do with Brouwer theorem, but it is not difficult to prove it. We can prove this assertion by reducing to absurdity:

For convenience, we assume that the earth is a unit sphere (radius is 1), the points on the sphere can be regarded as unit vectors, and the mapping from sphere to sphere can be regarded as the mapping between unit vectors. Suppose that at a certain moment, the wind speed V(x) (horizontal vector) at any point X on the earth is not zero, then the vector V(x) is perpendicular to the vector ox from the center O to the point X, and V(x)/ |V(x)|, where |V(x)| is the length of the vector V(x) and can be regarded as one on the (unit) sphere. So we can construct a mapping f: ox → v (x)/| v (x) | from the (unit) sphere to itself. It is not difficult to see that this mapping maps the vector ox to a vector perpendicular to it, so it is neither itself nor its counterpart-ox, which is contrary to Brouwer theorem. So the assumption is not true, so there must be no wind at any time on the earth.

More generally, the above Brouwer theorem and our assertion are still valid for all even-dimensional spheres.

However, the above proposition does not hold true for odd-dimensional spheres, and Brouwer theorem has a more complicated form.

For odd-dimensional spheres, we can use linear algebra to construct a counterexample:

Using an even 2n-order orthogonal matrix O without real characteristic roots, it naturally acts on 2n-dimensional Euclidean space, and it transforms the odd 2n- 1 dimensional unit sphere into itself. It is easy to prove that there is no such point, it is fixed (the characteristic root is the characteristic vector of 1) or it becomes a radial point (the characteristic root is the characteristic vector of-1) under the action of O.

Note: Brouwer theorem is incorrectly expressed in some popular science books and online (such as 1, 3): if F is a continuous mapping from a sphere to the sphere itself, then there must be a point that makes F(x)=x, that is, there is a fixed point.

Note: As a general practice, in order to avoid confusion, periods are not used in China mathematics textbooks or test papers. Instead, use the English period "." As a popular science article, this article does not follow this convention.

It's a mess, because I'm also ignorant. ....