Ekonomi dan Keuangan Indonesia Vol 39, No. 3, 1991

Proof of the Brouwer Fixed-Point Theorem: A Simple-Calculus Approach

Irsan Azhary Saleh

ABSTRAK

Kendatipun ieorema titik-tetap Brouwer merupakan landasan penting untuk memahami model-model keseimbangan umum yang relatif telah dikenal luas, namun konsep dan pembuktian dari teorema yang bersangkutan ini kurang mendapat perhatian atau hampir tidak pernah dianalisa sebeiumnya di Indone• sia. Tulisan ini bertujuan untuk membuktikan teorema titik tetap Brouwer dengan menggunakan pendekatan kalkulus, suatu metode pembuktian yang relatif sangat sederhana jika dibandingkan dengan metode-metode lain yang secara konvenstonal telah sering digunakan, yakni antara lain misalnya: teori homologi, bentuk-bentuk diferensial, argumen kombinatorial, serta topologi geometris.

217

1 Saleh

1. INTRODUCTION

The Brouwer fixed-point theorem is one of the most important results in modern mathematics. It is easy to state but hard to prove. The statement is easy enough to be understood by anyone, even by someone who cannot add or substract. This is in essence what the Brouwer theorem says in everyday terms: Think of a person is sitting down with a cup of coffee. Gently and continuously he or she swtrls the coffee about in the cup. He or she then puts the cup down, and lets the motion subside. When the coffee is still, Brouwer says there is ai least one point in the coffee that has returned to the exact spot in the cup where it was when the person first sat down. Regardless of its simplicity in this everyday terms, however, the proof of the Brouwer theorem has been usually difficult that it has 'been taught only in graduate courses on . The Brouwer theorem has many applications in various mathe• matical disciplines. In mathematical analysis, as an example, it is an essential tool in the theory of ordinary differential equations. In its extension by Schauder and Leray, the fixed-point theorem is used to establish previous• ly unattainable results in the theory of nonlinear partial differential equations and integral equations, Browder (1967), Brezis and Browder (1975). In mathe• matical economics, as another example, in its primitive application the Brouwer theorem is used to prove the existence of an equilibrium in a pure exchange economy. Although this type of an economy is very simplistic and unrealistic, the existence proof contains the essential features of arguments for existence in more general equilibrium models. One of the famous interesting applications of the Brouwer theorem outside of topology is a proof of the minimax theorem of game theory given by von Neumann in 1928 (see Nikaido, 1970, Chapter 7).

It should be noted that in terms of the last two examples the Brouwer fixed-point theorem, as it was originally stated in I9I0 and commonly proved, is a pure existence theorem. The theorem, however, was later put in a more useful form. Extensions in this direction were initiated by von Neumann (1938) in connection with general economic equilibrium model, which is often considered as the first sophisticated application of the Brouwer fixed-point theorem to an economic problem. Somewhat later, Kakutani (I94I) generalized the Brouwer theorem to correspondences, which not only allows numerous applications in economic theory and its related fields but also has stimulated further extensions by authors, such as: Eilenberg and Montgomery (1946) working on multivalued transformation which relies on algebraic topological notions; Nash (1950) proving the existence of equilibria for games where the players' preferences are representable by continuous quasi-concave utilities and the strategy sets are simplexes; Debreu (1952) proving the existence of equilibrium for abstract economics; Arrow and Debreu (1954) proving the existence of Walrasian equilibrium of an economy; Moore (1968) replacing the term semi-continuity with hemi-continuity in

218 Proof of the Brouwer Fixed-Point Theorem: A Simple-Calculus Approach

referring to correspondences, which helps to avoid confusion with semi-conti• nuity of real-valued functions; and, even further, less than two decades ago. Scarf (1973) who has developed a remarkable algorithm that can easily be implemented on computers to give a method of approximating a fixed-point for a continuous mapping of a price vector into itself. This fixed-point algo• rithm, for practical purposes and policy analysis, has contributed significantly in solving complex and computable general equilibrium models. Comparable to or in point of Scarf algorithm are, among others, works due to Kuhn (1969), MacKinnon (in Karamardian, 1977), and Todd (1980). |

In Indonesia it seems to be evident that, for the last few years, there has been a growing interest in the methods of general equilibrium analysis, in particular with regard to the topics of computable general equilibrium model. Such an interest is relevant to the current tendency of globalization at almost any level of the economy, because via general equilibrium approach one can explore how to solve complex models which simulate many of the simul• taneous interdependencies which exist in an economy. Of course, there are many situations in which the theoretically and computationally simpler methods of partial equilibrium analysis are appropriate. However, there are certain problems where the income and other-price effects operate to shift the supply and demand curves used in partial equilibrium analysis and where these feedback effects are the central feature of interest. Examples of this are the formation of a tariff customs union or a change in the tax on one major production input such as capital. In this type of problem, a general equi• librium model is appropriate because we are interested in what effects the policy change will have on many different markets and because what happens in other markets crucially affects any one market of interest. Furthermore, it is a characteristic of such policy changes that they are not marginal so that linear approximation (as in economies with leontief technologies) is a poor way to specify relevant demand and supply functions, Cornwall (1984).

We are now already aware that Brouwer fixed point theorem (and understanding its proof) is a basic mathematical tool used in showing the existence of solution concepts in game theory and economics, especially in regard to general equilibrium analysis. Unfortunately, the exposition on this, at least in Indonesia, seems to be quite limited and rather abandoned. Therefore, as the title reflects, the purpose of this paper is to give a relative nonrigorous note for understanding the proof of the Brouwer fixed-point theorem. While there are many excellent readings available on the topic we are discussing about, most of them are inaccessible to a typical well-trained economist. Generally speaking, the reason, why proofs of the Brouwer fixed- point theorem have been difficult because they have used combinatorial argu• ments, differential forms, theory, or geometric topology. That is why, in this paper, we are taking a somewhat significantly different direction; all we need to now is just calculus, plus a supporting tool of analysis called contraction-mapping principle which is going to be discussed in Section II. In Section III, we can then proceed with the formal proof of the Brouwer

219 Saleh theorem. John Milnor (1978) deserves the full credit for his initiating work on using calculus for the proof of the theorem. The bulk of Section III is, in essence, a recast of Milnor's article which is only four pages long! So, what the author is trying to do is to rewrite out Milnor's original article in great detail and, hopefully, in much less rigorous fashion.

II. CONTRACTIONS MAPPING PRINCIPLE

Fixed-point theorem is a one that refers to an equation

x = /(x). (1)

Usually, the theorem gives conditions for the existence of a solution. The / may be thought of as a mapping. Then a solution x is a point that the mapping leaves fixed. There are many varieties of fixed point theorems. Some gives conditions for uniqueness or multiplicity of solutions. Some fixed-point theorems are constructive; most are not. The Brouwer theorem is not. It just tells us a solution exists; it is up to us to find it.

If it is true that equation (1) is a fixed-point equation, one can then infer that every equation is a fixed point equation. The reason is simple. Suppose this equation is given: ^(x) = 0. Then we can write the fixed-point equation

x = x + g(x) (2) or, if we prefer,

X = x-83s(x).

In general we can write the equation ^(x) = 0 cis the fixed-point equation

x = x + (j>[g{x)] / (3) provided that

The contraction mapping principle is an exception among fixed-point theorems: it is constructive. We can use the method of proof to construct a solution. The method is called the method of successive approximations.

In its elementary form, the contraction mapping theorem says this: Let M be a closed set of real numbers, and suppose the function / maps M into itself. Finally, suppose / is a contraction mapping:

\fia)-fm<&\a-b\, (4)

220 Proof of the. Brouwer Fixed-Point Theorem: A Simple-Calculus Approach

where 0 <6 <1. Conclusion: The mapping / has a unique fixed point in Af. Uniqueness is easy to prove. If there were two fixed points, a and 6, then the contraction property implies

|a-6| = |/(a)-/(6)|<%-6|. (5)

Then a - 6 = 0 because 0 < ^ < 1. To prove the existence of a fixed point, we will use an iterative scheme. Start with any XQ in M, and compute the successive approximations

^n + i = /K) (n = 0,l,2,...). (6)

We will prove that the sequence converges.

That will suffice. If x^-x x, then the limit x must lie in M because all the x„ lie in AI and M is closed {M contains all its limit points). Next, (4) implies

\f{xj-f{x)\

To prove that x„ converges, we substract from equation (6) the same equation with n replaced by n — 1:

(8)

The contraction property implies

K + l-^n\<^K-^n-l\ (n = l,2,...).

Then |x„ ^ ^ - x„| < e2|x„ _ j - x„ _ 2I, etc. At last, we get

K + i-^„l<^"K-*ol (n=l,2,...). (9)

For all q> p this implies

q-l q-l J2 (*„ + l-*n) n = p

<|xi-Xo|(eP + eP + i-t-... + ^9-i) (10)

< Ixj-Xo|5IP(l-0)-^0 as p-Hoo.

221 Saleh

Since — Xp -» 0 for 9 > p -» oo, the sequence converges, and so we have constructed a fixed point. m. A CALCULUS PROOF OF THE BROUWER FIXED POINT THEOREM

Brouwer's theorem says this: Let f(x) be any that maps the ball |x| < 1 into itself. Then there is a fixed-point, x, which is mapped into itself, x = f(x).

We will use the term unit ball for the n-dimensional solid comprising all points x whose coordinates satisfy

xl + - + xl

(In other words, [xj^ < 1, where x-x= |x|^ = Y^xJ). We will call this ball B". The surface of B" is the sphere S"'^ comprising all points x whose coordinates satisfy

xl + ... + xl=l.

Thus, in 3-dimensional space is the 2-dimensional sphere that bounds the unit ball B^. . If n = 1, the unit ball is the line segment — 1 < x < 1. If /(x) maps B^ into itself, we get Figure 1. Each point x goes to a point /(x). Draw an arrow from x to /(x); the arrow has length zero if x is fixed point.

• >- • >- •<——• -1 /(-I) /

Figure 1

Suppose there were no fixed point. Then the arrow at — 1 would point to the right; the arrow at + 1 would point to the left. Let x move continuously from — 1 to +1, and watch the arrow from x to /(x). Somewhere between — 1 and + 1 the arrow has to flip from right to the left. This is impossible if the arrow is continuous and nowhere zero.

In analytical terms, the arrow .is represented by the difference /(x) — X. This function is positive at x = — 1; it is negative at x = +1. Since

222 Proof of the Brouwer Fixed-Point Theorem: A Simple-Calculus Approach

f(x) — X is continuous, it must equal zero at some intermediate point c. Then /(c) = 0, and c is the required fixed point. If n = 2, we get Figure 2. Each point x goes to another point f(x) in the ball B^. Draw the arrow from each jjoint x to its f(x). On the boundary all the arrows point inward. Brouwer's theorem says some arrow in the ball has length zero.

Notice that we need the hypothesis of continuity. Otherwise, we could let f(x) be, say, a rotation through 90°; that would leave the center fixed, so we could agree to map the center x = 0 into some other point f(0) ^ 0. Then f(x) would be a discontinuous function with no fixed point.

Figure 2

Already for n = 2 the Brouwer theorem is somewhat difficult. For n > 2, all proofs have been difficult, or they required advanced specialized preparation. But now we have the astonishing proof by John Milnor (1978), which depends on this fact: the function (1-t-t^)"^^ is not a polynomial if n is an odd integer. Why should that have anything to do with the Brouwer's fixed-point theorem? To answer this, we begin with a puzzle: in n dimensions, construct a continuous of unit to the sphere 5"'^. For n = 2 the solution is easy. Look at Figure 3. At each point u on we construct the v(u) = ( — U2,u^). Then

|y| = 1 and v • u = 0.

For n = 3 we have 3-space, as in the real world. Regard the earth as the unit ball. Can we put a continuous direction field on the surface? One will bet we cannot. Think about it; we are bound to have trouble at the north pole or somewhere else.

223 Saleh

Figure 3

How about n = 4? Now we cannot visualize the problem, but we can easily solve it. If we use the trick we used for n = 2, first on «j,«2 then on U3, U4, we get the unit tangents

v(u) = (-U2,"i; -«4'"3)- .

Clearly this trick works for all even n. We get this continuous field of tangents:

•v(u) = (-U2,"i; «„-i)- (11)

But if n is odd, the hairy-ball theorem says there is no solution:

Theorem 1. If n ts odd, then there is no continuous field of non-zero tangents to the unit sphere 5""^.

PROOF. For |u| - 1, let v(u) be a field of unit tangents:

U-V(u)=:0, |v(u)| = l for H=:l.

For the moment, assume that v(u) is continuously differentiable. Let A be the spherical shell (or annulus)

A:\

v(ru) = rv(u) (\

This definition makes v(x) continuously differentiable in A, with

224 Proof of the Brouwer Fixed-Point Theorem: A Simple-Calculus Approach

X • v(x) = 0 and lv(x)| = |x| = r.

Now let < be a small real constant, and define the mapping

XH^x-(-tv(x) (xGA). (14)

This maps A into some set A^. (Can we guess what A^ is? Look at Figure 4). We have to show that the mapping (14) is one-to-one. Since v(x) is continuously differentiable, there is some constant A > 0 for which

|v(x)-v(y)|

X +

|x-y| = |<| |v(y)-v(x)|

This implies x = y if |<| < 1/A. Thus, for small \t\ the mapping is one-to-one.

X + fv(x)

Figure 4

Now we will show that the image A^ is just another annulus. If |x| = r, then the image has length

|x+tv(x)| =r(l+t2)l/2, (16)

since x and v(x) are orthogonal vectors of length r. This identity shows that the sphere or radius j- maps into the sphere of radius r(l + t^)^^^. Now we need to show that the first sphere maps onto the second,

225 Saleh

which means that every point in the second sphere is the image of some point in the first sphere. In other words, we need to prove that the following equation must have a solution:

x+tv(x)=x, (17)

where

i < |x| < I and |x| = (1 + <2)l/2 |^|_

• Here x is given, and we must find x.

First suppose |x| = 1. Write the equation in the form

x = x-tv(x). (18)

We will use the contraction-mapping principle discusssed in the previous section. The function on the right maps the shell A into itself if |t| < |, for then

|tv(x)|<||v(x)|=ir

and so

i^|xi-i<|x-Mx)|<|£|+i = i. "

If also |<| < 1/A, then

IMx)-My)l

and so the function x —tv(x) is a contracting mapping on A. This mapping has a fixed point, which solves the equation (18) if |x| = 1.

If |x| 7^ 1, define the unit vector x^ = x/|x|. Then we can solve this equation for x^:

xl-P.

If we multiply this equation by |x|, we get the required solution x = |x|x^.

Now we have completed a proof that, for small t, the function

f(x) = x-htv(x) (19)

maps to shell A one-to-one onto the shell

.4,: i(l + t2)l/2<^<3p^^2)l/2

226 Proof of the Brouwer Fixed-Point Theorem: A Simple-Calculus Approach

What is the volume |.4j|? In n dimensions, since A^ is geometrically similar to A, we have

\A,\ = il + t^)"/^\A\, (20) where the constant \A\ is the volume of n-dimensional shell | < r < |. For instance, if n = 3,

1-41 =H(|f

But suppose we use calculus. The function f(x) maps A one-to-one onto Aj. Therefore,

det (21) if the Jacobian determinant is positive. Now the definition (19) gives

(22) where 6- — dxjdx^ = 1 or 0. Thus the jacobian matri.x (22) tends to 1 as t -* 0, and so the determinant tends to 1.

As a function of the parameter t, each component (22) is a linear function. Therefore, the Jacobian determinant is some polynomial

det

If we integrate over x, we get another polynomial:

(23) where 6^1. is the integral of aj.(x) over the annulus A.

EXAMPLE. If n = 2 let

/ \ 1 \ f(x) = +1

227 Saleh

The Jacobian determinant equals

1 -t = l-b<2. t 1

Integrating over x for T < X < |, we compute

|A,|=|A|(l + <2)

where |A| = 7r[(|)2-(1)2] = 2Tr. In general formula (23) says |Af| is polynomial in t. But formula (20) says jAjl is a constant times (1 + t2]n/2 These conclusions are inconsistent if n is odd. Thus, we have proved that for odd n the sphere 5"'^ has no field of unit tangents v(x) if v(x) is continuously differentiable. Finally, we have to remove the assumption of differentiability. Let v(x) be any continuous field of non-zero tangents to 5""^. Using v(x), we will construct a differentiable tangent field w(x). First extend the definition of v(x) to the whole space by the formula

v(ru) = rv(u) (0 < r < oo. |u| = 1).

Now consider v(x) in the closed n-dimensional cube

C: - l

Using the Weierstrass approximation theorem in the cube C, we can approxi• mate each component v-{x^,..., x„) by some polynomial p,(Xj,..., x^), and we can make this approximation as good as we like in the whole cube C. Since the cube C includes the unit sphere, we can make p so close to v that

p-(p-u)u760 for |u| = l. (24)

(Note that this expressions tends to v as p-»v. Remember, v 76 0 for |u{ = 1). If p = p(u), the vectors (24) constitute an infinitely differentiable non-zero tangent field w(u) on the unit sphere. If we require unit tangents, just form w/|w|. Now our proof for differentiable tangents gives the full result: there can be continuous field of non-zero tangents to 5""^ if n is odd. That proves the hairy-ball theorem. Hopefully, the details did not make us lose track of the idea, which was this: The function x -|- tv(x) maps the annulus .4 onto the annulus

A, = (l+<2)i/2^,

228 Proof of the Brouwer Fixed-Point Theorem: A Simple-Ctilculus Approach

because x • v = 0. The volume of Aj in n dimensions is

|AJ = (H-<2)"/2|^|.

But calculus says |AJ is a polynomial in t. Therefore n is even.

Now we can prove the Brouwer fixed-point theorem.

This is the idea: Suppose the Brouwer theorem is false in n dimen• sions. Then we will construct a field of non-zero vectorns in the Ball B" that are normal to sphere S"'^. We will then regard the ball B" as the equatorial disk inside the higher-dimensional B""*"^. By stereographic projections from the north and the south poles, we construct a continuous field of tangents to the sphere 5". But we know this is impossible if n is even. But Brouwer's theorem for n dimensions implies Brouwer's theorem for n — 1 dimensions, as we we will see. This proves Brouwer's theorem for all n, even or odd. ^

Theorem 2. Let f(x) map the unit ball B" continuously into itself. Then there is a fixed point x = f(x).

PROOF. Suppose f(x) has no fixed point. Let |x| < 1; let y = ffx), and form all the vectors z = x — y. Then z ^ 0, and on the unit sphere the vectors z point outward:

x-z = X-(x-y) = 1-x-y > 0 if |x| = 1. (25)

Why? Because

0< |x-y|2 = |x|2-Hy|2_2x-y< -2x-y.

We will now construct a field of vectors w that are continuous and non-zero in B", with w(x) = x if |x| = 1. Define

w = x-Ay (y = l(x)) (26) where A is the scalar

A = i^^i^. (27) 1-x-y

The denominator is non-zero, by (25), and so w(x) is a continuous function of X in B".

On the sphere |x| we have A = 0, and so w = x. This says w(x) is the outward unit normal at the surface point x. j

It remains to show w /: 0 inside the ball. If w = 0, multiply the equa• tion (26) by the denominator of the fraction A. This gives the equation

229 Saleh

0 = (l-x-y)x-(l-x-x)y. (28)

But w = 0 says x = Ay, which gives

(x • y)x = A2(y - y)y - (x • x)y.

Now (28) becomes 0 = — x + y, which we have ruled out. Therefore w y^ 0 in the ball B". Now regard B" as the equatorial disk inside the ball B""*"^. Using the vectors w in the disk B", we will construct a field of tangents to the sphere 5", which is the boundary of B""*"^.

First we will work on the southern hemisphere. Look at Figure 5. We are looking at a side view of the ball B""*"^. Our eyes are at the level of the equator, so the equatorial disk B" looks to us like a line segment. From the north pole, N, we project each point x in the disk onto a point u in the southern hemisphere. We are going to construct a tangent v at the point u.

N

Figure 5

All this n-dimensional visualizing can be written as formulas in what follows:

If X has the coordinates x^,...,x^ in n dimensions, then

x=(xi,...,x„,0) - (29)

in n + I dimensions. The north pole is

N = (0,...,0,I). (30)

The point x lies on the line segment between N and the u in the southern hemisphere. Therefore, for some 6 = ^(x) between 0 and I, we have

230 Proof of the Brouwer Fixed-Point Theorem: A Simple-CEdcnlus Approach

x = {l + 9)fi + 9u. (31)

The coordinates satisfy these equations:

x- = 9u- (i = !,...,«) (32)

Given the Z;, we can solve for the tt,- and 9. First, we write

u, = xJ9 (i =!,...,«); u„^^ = {9-\)l9. (33)

Since |u| = 1 on the sphere 5", we require

t = 1

Multiplying by 9'^, we find 9'^ = |x|2 -|- - 1)2, and so

-|(|xP + l). (34)

Note that 9 In equation (31), this says x is closer to u than to N. But that is clear from figure 5. Now we are going to construct a tangent v at the point u. yVe have projected a point x in the equatorial disk B" onto the point u in the southern hcdf of 5". We will now use the vector w(x) defined in (26), satisfying

wGB", W76O, w = x if |x| = I. (35)

If |x| < I, we can construct a small line segment

x(<) = x-btw(x) (0

If we project this segment stereographically from N, we get a small arc

u(t) = u(x +

If we regard t as time, the point x(t) has velocity w(x) in the disk B". The projection u(t) then has the velocity

v = ^u(t) for t = 0. (38) The vector v is tangent to S" at the point u. That follows by differen tiating the identity u{t) • u(<) = I.

231 Saleh

To show V ^ 0, we write the projection identity (31) as a function of time:

x(t) = il-9{t))fi + eit)u(t) (0

If we take derivatives at t = 0, we get

w= -9'li-\-9'ii + 9v.

Then

v = ri[w + &'(N-u)]. (40)

If 9' - 0, then v / 0 because w 76 0; if 0' 76 0, then v 76 0 because

^„ + i=^"'[0 + m-«„ + i)]9^0, (41) since ^ j = 0 and u„ ^ j < 0. We have shown v 7^ 0 and v • u = 0 if u,, ^ j < 0. Now let u approach the equator u,^^j=0. According to the projection identity (31), this happens if

9-*l, |x|-»l, x-u-»0.

Then formula (30) gives, in the limit,

v=[w + ^'(N-x)] (42) as u approaches a point x on the equator. But then w = x, by (35); and

0'=^i(l+x.x) = w.x=l, by (34) and (35). Therefore,

v = [x+N-x] = N if «„ + i = 0. (43)

This completes the definition of v as a non-zero field of tangents to the closed southern hemisphere in S". At the equator the tangents v are unit vectors pointing straight up.

As a function of u, the vector v is continuous. Why? All the functions 9,9', w, and u are continuous functions of x, as we have seen. Then (40) says v is a continuous function of x, since 9 >^. But (32) gives x as a continuous function of u. And so v is a continuous function of u.

What we have done so far? First, we assumed the Brouwer theorem

232 Proof of the Brouwer Fixed-Point Theorem: A Simple-Ceilculus Approach false for ball B". We then regarded B" as the equatorial disk in B" + V By stereographic projection from the north pole, we constructed a continuous field of non-zero tangents to the lower half of 5". For u on the equator the tangent v(u) equals a unit vector pointing straight up (v = N).

We used the north pole and projected down. Suppose we had used the south pole and projected up. What we have got? By symmetry, we would have got a continuous field of non-zero tangents to the upper half of 5"; call these tangents v"*'(u). By symmetry, for u on the equator the tangent V*" (u) equals a unit vector pointing straight down (v = S). Look at Figure 6 and compare it to Figure 5. As we can see, vmakes a perfect mismatch with v on the equator:

v + (u)= -v(u) for «„^i = 0(|u| = l). (44)

That is just what we do not want. What we want is a continuous field of tangents on the whole sphere 5": the limits from above and below have to match on the equator.

northern half of S"i

Figure 6

But we can get that. We have defined v(u) in the lower hemisphere. We can define v(u) in the ttpper hemisphere as follows:

v(u)= -v + (u) for u„ + i>0.

This turns the mismatch (44) into a match, and now we have a continuous field of non-zero tangents on the whole sphere S". But the hairy-ball theorem says that is impossible if n = 2, 4, 6, Therefore, the Brouwer fi.xed-point theorem must be true for the ball B" if n = 2,4,6, • I

What about, n = 1, 3, 5, ? Then the hairy-ball theorem is no help. On the contrary, if n = 1, 3, 5, , then 5" does have a continuous field of

233 Saleh tangents. In what follows we could see that the Brouwer's theorem is true for all n, either even or odd. Let n = 1, 3, 5, Let f(x) map B" continuously into itself; we want a fixed-point x = f(x). For y in B" ^ define the function

g(y) = (/i(x),...,/„(x),o) where we set y- = x- for j = \,...,n. The function g(y) first vertically projects y onto the point x in the equatorial disk j/„ q. j =0, then applies the mapping f from the disk into itself. The function g(y) maps B" ^ continuously into itself. By what we have proved, g(y) must have a fixed point if n-|- 1 = 2, 4, 6, The fixed point satisfies y = g(y), which says

2'j = ^j = W (j=L...,n); 2/„ + i=0.

Thus the Brouwer's theorem for B"^ implies the Brouwer's theorem for B", and so the proof is done.

2.34 Proof of the._Brouwer Fixed-Point Theorem: A Simple-Calculus Approach

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