1

22

3

4

5

6

7

81

92

103

11

11.1

11.2

11.3

11.4

11.5

11.6

11.7

11.8

11.9

18 C16530128

243

59 3

1

3

X0, X1, , Xn n1(n 1)

pi( 0), i0, 1, , n p0p1+pn

ipii0, 1, , n

0, 1, , n p0, p1, , pn

p0p1pn1 1

xi

p0 x0p1 x1pn xn 0

p0p1pn

0 x0 1 x1 n xn 0

0 n1

2

1

11.10

11.11

244

3 X, Y, Z, px , py , pz

1 px , py , pz 3

(1, 0, 0), (0, 1, 0), (0, 0, 1)

2 1

1 1

2

2

(0,0,1)

(0,1,0)(1,0,0)

px py

pz

1 3

245

59 3

2

2

2

1 p0, p1, , pn n 1 (1,

0, 0, , 0), (0, 1, , 0), , (0, 0, , 1)

4

22

1Sperner2

6

6

6 6 36

6

K K K

1 123 1

2 K

123

K 123

1 2

1

246

2

3

12 K 1

2

12

1 1

1 2 12 1

n 12

1

1 2

2 1

2 1

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2

K 12 K

1

3 2

3 2

3

3 2

247

59 3

2 12

2

3 1

23

K

12

1 123

12

1

1

1 1

1

1 1

1

1

2

2

2

2

2

22

2

3

3

33

3

3

3

3

3

248

K 12

1 1

12

3

1 3

3

n n

n K

3 (1, 0, 0), (0, 1, 0), (0, 0, 1) 2

n1 (1, 0, 0, , 0), (0, 1, , 0), , (0, 0, , 1)

n 0 1

2 3

1 42

2

2

3

K0n

1 n1 0 n

249

59 3

2 K v n1 n1

v v

3 K v n2

v

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2K

K n 0

n

n0

0 0 1

n1

n1

01n1 K n1 n1

01n1 n1

1 2

K n

0 n1 K n1

2 0 n1

2

3 n

0 n1 n1 n

1 n 1 n 2

250

0 n n

n1

K n1 0 n1

1 0 n

n

0 n1

n1 K n1 0 n1

2 0 n

n 1

1 1 1

0 n n

1

2 0 n

n

0 n

n

1

0 n1

0 n n

0 n n

2

251

59 3

0 n

n

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n 2

0

n n

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n

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1

1

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2

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2

22

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3

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33

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3

3

0 252

0 n

1

4

2 2

0 x1 y

2 z 3

1Brouwer f

m (x0, x1, , xn)

xi 0, i0, 1, , n x0 x1 xn 1 (x0, x1, , xn)

f (x0, x1, , x n) x0x1xn1

1x0 x0 (x0, x1, , xn) 0

2x0 x0 x1 x1 (x0, x1, , xn) 1

3x0 x0x1 x1 x2 x2 (x0, x1, , xn) 2

4 x i xi i (i0, 1, , n)

i (i0, 1, , n)

x i xi x0 x1 xn 1 x0x1 x n 1

i (i0, 1, , n) x i xi

1 (1, 0, 0, , 0) (x0, x1, , x n) x0 1

x0 1 (1, 0, 0, , 0)

253

59 3

0

2 (0, 1, 0, , 0) (x0, x1, , x n) x1 1

x0 x0 x11 (0, 1, 0, , 0)

1

3 (0, 0, 1, , 0) (x0, x1, , x n) x2 1

x0 x0x1 x1 x2 1 (0, 0, 1, , 0)

2

4

xn 0 n1 n1

n1 n3 (x0, x1, , 0)

(x0, x1, , x n) xn 0 xn xn

n (x0, x1, , 0) 0 n1

(x0, x1, , 0, xn) n1 (0, x1, x2, , xn)

0 xn1 xn0

n2 n3

(x0, x1, , 0, 0) (x0, x1, , xn1, xn)

xn1 xn0 x n xn x n1 xn1 n

n1 0 n2 1

0 n

0 n n

1 n

(x 0m, 0 , x 1

m, 0 , , x nm, 0), (x 0

m, 1 , x 1m, 1 , , x n

m, 1), , (x 0m, n , x 1

m, n , , x nm, n)

m 0, 1, , n 1

m n 0 n

m

1 x0x2x50 025

254

m 1

2m

1 1

1 m

n1 (x0, x1, , xn)

f i0, 1, , n (x 0m, i, x 1

m, i, ,

xnm, i) (x 0

m, i, x 1m, i, , x n

m, i) (x0, x1, , xn) (x0, x1, , xn)

(x 0m, 0, x 1

m, 0, , xnm, 0) 0 x 0

m, 0 x 0m, 0 f

m (x 0m, i, x 1

m, i, , xnm, i) (x0, x1, , xn)

(x 0m, i, x 1

m, i, , x nm, i) (x0, x1, , xn)

x0 x0

(x0m, i, x1

m, i, , xnm, i) (x0, x1, , xn) x 0

m, 0 x0m, 0

(x0x0)

x1 x1, , xn xn x0x1xn x0x1xn 1

(x0, x1, , xn)(x0, x1, , xn) (x0, x1, , xn)

5

1

2n1 X0, X1, , Xn

(p0, p1, , pn) fi (p0, p1, , pn), i0,

1, , n

2 1

255

59 3

p0 f0 p1 f1 pn fn 0

fi Xi

(i 0, 1, , n) fi (p0, p1, , pn)0

(p0, p1, , p

n )

(p0, p1, , pn) n1 v(v0, v1, , vn)

vi pi fifi 0

vi pifi 0

(v0, v1, , vn)

( 0, 1, , n)

i (p0, p1, , pn)v0v1vn1

v1

i 0i 0, 1, , n

0 1 n1

( 0, 1, , n) fi

( 0(p0, p

1, , p

n ), 1

(p0, p1, , p

n ), , n(p

0, p

1, , p

n ))(p

0, p

1, , p

n ) (p

0, p

1, , p

n )

i vi pi v

1 i vi (p0, p

1, , p

n )p

i

1 1 pi 0 vi (p0,

p1, , pn )pi fi (p0, p1, , pn )0

i pi 0 1

p0 f0p1 f1p

n fn 0

1 v0 p0, v1 p

1, ,vn p

n fi (p

0, p

1, ,

256

pn ) 0

i fi (p0, p

1, , p

n ) 0 p

0 f0p

1 f1p

n fn

0 p1 fi (p0, p

1, , p

n ) 0

pi 0 fi (p0, p1, , pn )0

v

2

3 X

Y A ((xA, yA)), B ((xB, yB)) B

A AB

A B B A

AB

3

3AB

257

59 3

B A B A

AB

AB A

1

p ((px, py)) p

pp p AA

A A A

B

p p A B

B p 4

A p B

p

p B 5 A

A p

1 B A

AB

A B B A p p

A A A

4 1

5Bp

258

A

A C C p p

A p

A

X Y (x, y) A (xA, yA)B (xB,

yB) xA xB Y A B xA

xB B A xA xB Y

Y

(1, 1) (1, 0)

(1, 1) (1, 0)

(1, 0) (1, 1) X 1

(1, 1) (1x, 2) x

(1, 1) x 0

(1 x, 2) (1, 2) (1, 1)

6

6

6Walrass Existence Theorem and Brouwers Fixed Point Theorem, 8 1962, pp. 59-62.

259

59 3

3

(0,1, ,n)

p (p0, p1, , pn)

zi (p)i(p)pi (p)i0, 1, ,n 2

(p)

p i2

pii(p) i1n

i0n(p)

zi

(2) pi 0 n

i0

n

pi zi i0

n

pii(p)(p)p i

2i0

n

pii(p)i0

n

p i2

pii(p) i1n

i0n p i

2i0

n

pii(p)

i0

n

pii(p)i0

n

0

zi

7 p (p0, p1, , p

n )

i (p) (p) pi (zi(p

)0)

pi 0 i (p)(p) pi i (p

)

pi 0 i (p) 0

i i (p)(p) pi 0 n

7 1

in pipi

i ( 0, 1, , n) 0 i i0

260

i(p)

i1

n

(p)i0

n

pi

i (p) i1

n i1n pi 1 (p)1

i (p) pi

p

pi i 1 n1

pn1i0

n1

pi n(p)1i0

n1

i(p)zn(p) 1i0

n1

i(p) (p) 1i0

n1

pi (3)

pii(p) i0

n1 pi) i1n1(1 i(p)) i0

n1(1p i

2 i0n1 pi)

2 i0n1(1

(p) (4)

pn zn

pn zn 1i0

n1

pi 1i0

n1

i(p) (p) 1i0

n1

pi2 (5)

(2) pi 0 n1

i0

n1

pi zi pii(p)i0

n1

(p) p i2

i0

n1

(6)

(5)(6)

i0

n

pi zi pii(p)

i0

n1

1i0

n1

pi 1i0

n1

i(p) (p)p i

2i0

n1

1i0

n1

pi2

(4) i0n pi zi 0 (3)

7

n1m1 pi (i 0,

261

59 3

1, , n), qj (j 0, 1, , m) 2

l

l p0 y0lp1 y1

lpn ynlq0 x0

lq1x1lqm xm

l

yil xj

l l i j

k l skl (0 skl 1)

k

p0 x0

kp1 xk1pn xn

kq0 y0kq1 y1

kqm ymk sk

lll

xik i yj

k j l skll

p0 x0kp1 x1

kpn xnkq0 y0

kq1 y1kqm ym

k

skl

l(p0 y0

lp1 y1l pn yn

lq0 x0lq1x1

lqm xml)

0 262

kskl1 1

p0k x0kp1k x1

kpnk xnk q0k y0

kq1k y1kqm k ym

k

l( p0 y0

lp1 y1lpn yn

lq0 x0lq1x1

lqmxml )

i k

xik l yi

l XiYi

j k yjk t xjl

YiXi

p0(X0Y0)p1(X1Y1)pn(XnYn)

q0(X0Y0)q1(X1Y1)pm(XmYm)0

n1 nm2 pj , j

n1 , n2, , nm2 nm2

1

8 1

3

263

59 3

(1) ab a b a, b

(2)

(3)

(1) (2) n

7

X A X A

1

A

X

1 X

X

X

X

X

1

1

2 3

4

n n

2D

264

u xD x u

D D

l xD x l

D D

ul 1

3D

s D D u s u

D sup D

D t D

D l t l D inf D

D D

D D D

D D D

D

DE

(1) DEDEDE

(2)x Dy E x y

D E

1

2

265

59 3

2

S 2

AS BS

S A B

A B A

S S s s

21

( s) s S

A A

B

S

2

AS BS

S A B

A B A

S S s s

21

(s ) s S

A A

B

266

4ab a b

a, b

a, b 1

a, b

S a,

a, bs supS s W

(sW) b b S s b W

(ss) W 0

0

s S s S

S a, U V1, V2, ,

Vr

t a, b t s

a, ta, (s, s) V1V2 VrW

a, t tS sS

sb s S

s b (s , s) W s t s t S

t s21 b t s

s S bSa, b U

267

59 3

(1, 1) (1n1

, 1n1

) (n2, 3, )

(1, 1) (1, 1)

(1n1

, 1n1

)

(1n1

, 1n1

) (1, 1) (1, 1)

5

2 XY (XY ) XY 1

3

2 2 1

n( 3) n 1

3f X Y A X

Y V f 1(V) X f

2

f 1(V) f V

f 1(V) xX|f(x)V

V f 1(V)

f V X f 1(V)

V Y Y V

V

Y V

YY

2 2

3

268

3

2

f(A) Y f(A) f

A

f(A) Y A V

f 1(V)

f 1(V) A A

V V1, V2, , Vk

A f 1(V1) f 1(V2) f 1(Vk)

f(A) V1V2Vk f(A)

4X, Y K Y U XYX

Y 8VxXxK U V X

xV U yK (x, y) Dy

Ey , DyEy U XY DyEy K

K Ey1Ey2Eyk y y1, y2, , yk

Nx Dy1Dy2Dy k Nx X

NxKi1k

(NxEyi)i1k

(Dyi Eyi)U

Nx V NxKU x Nx x

V xV Nx Nx V V Nx

V X

5

2 XY

8 3

269

59 3

XY Z XY

Z XY

x X xY Y XY

yY (x, y)

3XY

U1, U2, , Ur xY U1U2Ur

X x xY U1U2Ur

Vx x Vx 4 Vx

VxY U1U2Ur

x X Vx Vxx X X x

Vx XXVx1 Vx2 Vxr

x1, x2, , xr Vx X

Vx1 Vx2 Vxr X XY j1, 2, , r Vxj Y

XY

6

X x1, x2, x3, Fn

xn, xn1, x n2,

xn, xn1, xn2, xn, xn1, xn2,

xn, xn1, xn2,

270

F1, F2, F3,

Vn X \ Fn Fn Vn X

V1, V2, V3,

V1 V2 V1V2 F1 F2

X \ (F1F2) Vn F1, F2, F3,

F1, F2, F3,

Vn X

Vn V1, V2, V3, X X

Vn V1 V2 V3

n XVn Fn n

F1, F2, F3, n1 Fn

Fn p p Fn

0 d(xj, p) xj( j n) p xn, xn1,

x n2, xn, xn1, x n2, p

9

n 2 x(x1, x2, , xn)y(y1, y2, , yn)

d(x, y) (x1y1)2+(x2y2)2(xnyn)2

x1, x2, 1 d (xn1, p)1 xn1 1

n1 1p F1 xnn n1 d(xn2, p) 21

xn2(n2 n1) p Fn11

xn3 , xn4 , xnj 1( j1) xnn nj d (xnj 1 , p)

j11

xn1 , xn2 , j p

9 26

271

59 3

9 2

5 X Y yf (x) x0 X

2

(1)x x0 f(x) f(x0)

(2) 2 xy d(x, y)

0 d(x0, x ) d( f(x0), f(x ))

0

yf(x) x0 (1)

d(x0, x ) d( f(x0)f(x )) 0

0

d(x0, x) d( f(x0)f(x))

x x

n n1 x 1 x1, x2, , xn

n

d(x0, xn) n

1 d( f(x0)f(xn))

n d(x0, xn)

d( f(x0), f(xn)) f(x) (1) x0

0

yf(x) x0 (2) X x1,

x2, , xn n x0 1

Y f(x1), f(x2), , f(xn) f(x0)

0 N

n N d( f(x0)f(xn))

n f(x) (2)

0

0 272

d(x0, xn) d( f(x0)f(xn))

x1, x2, , xn x0

N

n N d(x0, xn)

N n N d( f(x0), f(xn))

n n

d(x0, xn) d( f (x0)f (xn))

(2) f(x)

(1)

X Y f(x) X X

X x

BX(x, )

5 (2)

0 x BX(x0, ) f(x )BY( f(x0), )

0

6f X Y Y V

f 1(V) X f X

5

f V Y x f 1(V)

f 1(V) BX(x, ) f1(V)

f(x) V V

BY( f(x), )Vf xBX(x, )

f(x)BY( f(x), ) x BX(x, )

x f(x)V x f 1(V)

f Y V f 1(V) X

Y V f 1(V) X

273

59 3

x X BY( f(x), )

Y f 1(BY ( f(x), )) X

x BX(x, ) f1(BY ( f(x), ))

f (BX(x, )) BY( f(x), )

x BX(x, ) f(x) BY( f(x), )

f

7f X Y Y G

f 1(G) X f X

Y G X \ f 1(G) f 1(Y \G) f 1(G)

X G Y f

Y \G f 1(Y \G)

G f 1(G)

G f 1(G) Y \G

f 1(Y \G) f

8X, Y, Z fX Y, gY Z X YY

Z X Z f g g

fX Z

V Z g g1(V )

f f 1(g1(V )) f 1(g1(V ))

g V Z Y X

gf V X (gf )1(V )

gf

274

10 3

9(1)X, Y n m

XY U X

Y

U (v, w)

(x, y) XYd(x, v)1, d(y, w)2U (7)

1 0, 2 0

(2) X, Y (1) XY U

XY

U (v, w) v V, w W VW U

X V Y W

(1) XY U U (v, w)

(x, y) XYd((x, y) (v, w))U (8)

XY

x(x1, x2, , xn) v(v1, v2, , vn) y(y1, y2, , ym) w(w1, w2, , wm)

d(x, v) (x1v1)2+(x2v2)2(xnvn)2

d(y, w) (y1w1)2+(y2w2)2(ymwm)2

(x, y)(x1, x2, , xn, y1, y2, , ym), (v, w)(v1, v2, , vn, w1, w2, , wm)d((x, y)(v, w))

(x1v1)2+(x2v2)2(xnvn)2(y1w1)2+(y2w2)2(ymwm)2

U (v, w)(8)

1 2 11

2222 d(x, v) 1, d(y, w)2

d((x, y) (v, w)) (x, y) U (7)

1, 2

275

59 3

(x, y) XYd(x, v) 1, d(y, w)2U

1, 2 d((x, y) (v, w)) d(x, v) 1

d(y, w)2 (x, y) U (8)

(2)U XY (1)

1 0, 2 0

(x, y) XYd(x, v) 1, d(y, w)2U

V x Xd(x, v) 1, W y Yd(y, w)2

V X W Y vV, wW, VW U

U XY U (v, w) X, Y V,

W vV, wW VW U (v, w)

xd(x, v)1V

yd(y, w)2W

1 0, 2 0

(x, y) XYd(x, v) 1, d(y, w)2VW U

(1) U XY

11

11.1

0 1 1

1 1

276

2

1 2

2 3

3

2

3

4 5 , 4

5 n

n 1

2 2 2 x (x1, x2)y (y1, y2)

2

xy (x1y1)2+(x2y2)

2 (xiyi)2

i1

2

23

2 x(x1, x2, x3)y(y1, y2, y3) 2

xy (x1y1)2+(x2y2)

2+(x3y3)2 (xiyi)

2i1

3

2

n 2 x (x1, x2, , xn) y (y1, y2, , yn)

xy (xiyi)2

i1

n

n

(1) x, y n xy 0

277

59 3

(2) x, y n xyyx

(3) x, y, z n xy yz xz

(4)xy 0 xyx y

(1), (2), (4) (3) 2

1 3

2

(aibi)2

i1

n

i1

n

ai22

i1

n

ai bii1

n

bi22

ai bii1

n

ai2

i1

n

bi2

i1

n

(aibi)2

i1

n

i1

n

ai22

i1

n

ai bii1

n

bi2

ai22

i1

n

ai2

i1

n

bi2

i1

n

bi2

i1

n

ai2

i1

n

bi2

i1

n2

ai xiyi, bi yizi

(xizi)2

i1

n

(xiyi)2

i1

n

(yizi)2

i1

n

n 3 x x1, x2, , xn, y y1,

y2, , yn, z z1, z2, , zn

xz xy + yz

4 X V

V v V x Xxv V

278

0

X

1 (0, 1)

1

0 v 0

212

10 x xv 2

1 (0, 1)

[0, 1]

1

0 1

11.2

5 X

x, y

d(x, y) d

(X, d)

(1) x, y X d(x, y) 02

(2) x, y X d(x, y) d(y, x)

(3) x, y, z X d(x, z) d(x, y)d(y, z)

(4) x y d(x, y) 0 2

2

279

59 3

n

1 x

1 x 1 x 0

6(X, d) X x r 0

BX (x, r) X x r

BX (x, r) xXd(x , x) r

x r X

3

7(X, d) X V

V v V BX (v, ), V 0

4

10X d x0 X

r 0 r BX (x0, r) X

x BX (x0, r) 0 BX (x, ) BX (x0, r)

r d(x, x0) d(x, x0) rx BX (x0, r)

0 xBX (x, )

d(x , x0) d(x , x)d(x, x0)d(x, x0)r

xBX(x0, r) BX(x, ) BX(x0, r) BX(x, )

BX(x0, r) BX(x0, r)

1

280

[0, 1]

y f(x), yg(x)(y , x [0, 1]) 2

d(f, g)

[0, 1]max f(x)g(x)

[0, 1] f(x) g(x)

[0, 1]

f(x)g(x) f(x) g(x) f(x) g(x)

d(f, g) 0

[0, 1] f(x)g(x) d(f, g)0

f(x) g(x) d(f, g)0 3 y f(x), yg(x), yh(x)

d(f, h)[0, 1]

max f(x)h(x) [0, 1]

max f(x)g(x)g(x) h(x)

[0, 1]

max f(x)g(x) [0, 1]

max g(x)h(x) d(f, g)d(g, h)

[0, 1]

11.3

8 X

X

(1) X

(2)

(3)

X X (X,

) X

281

59 3

11X X

(1) X

X X x

X

(2) X A U A

U x U x

A V BX (x, ) V

V U BX (x, ) U U

(3) V1, V2, , Vk X

V V1V2 Vk x j x Vj

V1, V2, , Vk j

BX (x, j) Vj 1, 2, , k 1, 2, , k

k j 0

x Vj j

BX (x, ) BX (x, j) Vj BX (x, ) V V

(3)

1n (n1

, 1n1

)

n 1, 2, , n ( n1

, 1n1

)

[0, 1]

0 282

11.4

9X X

F X \ FF X

X X X

X X

1 F[0, 1]

\ F( , 0)(1, ) ,

v x

xv \ F 0

F

1 [1, 1]1 1

(1, 1)1 1 (1, 1]1 1

X [2, 1] (1, 1]

1 (3) 1

[2, 1] [2, 1]

X

X

12

(1) X

(2)

283

59 3

(3)

(2)

(3)

(2) 1

n [ n1

, 1n1

] n3, 4, ,

0 1 n [ n1

, 1n1

]

n

(0, 1)

11.5

10 X

x y X x Uy V UVU

V U V

13

XdxyX

21

d(x, y) x, y BX (x, ) BX (y, )

x, y

BX (x, )BX (y, )

d(x, z), d(y, z) zX z BX (x, ) BX (y, )

d(x, y) d(x, z) + d(y, z)

221

d(x, y) BX (x, )BX (y, )

X

284

11.6

X A X A V

VA A A A

VA

V

A A

2 V1, V2 V1A, V2A V1 A

V2 A

2 (V1 A)(V2 A) A V1

V2 (V1V2)A

VA V

A

A

2 V1, V2 V1A, V2A (V1A) (V2A)

A V1 V2 (V1V2)A

14X X A

A

A X A V1V2

Vr V1,V2, ,Vr

A B X V VA

A A V1V2 Vr

A V1A, V2A, , VrA

15A X A

285

59 3

A X X \ A

X A

X \ A X X

A

14

A

16X K X

x X \ K x V, KW, VW X

V, W

X y K x Vx, y, y Wx, y,

Vx, y Wx, y Vx, y, Wx, y K

K Wx, y1 Wx, y2 Wx, yr y1,y2,

, yrK y K Wx, y K

K K

VVx, y1 Vx, y2 Vx, yr, WWx, y1 Wx, y2 Wx, yr

V, W x V, KW VW Vx, y1

Vx, y2 Vx, yr Vx, yi(i1, 2, , r) Wx, y1 , Wx, y2, ,

Wx, yr WWx, y1 Wx, y2 Wx, yr

17

K X

16xX \ KxVx, VxKVx

X \ K x Vx

Vx K K Vx

X \ K X \ K K

286

11X d X A

x, y A d(x, y) K K

A diamA

diamA A x, y d(x, y)

7K n K

K n

17

K m Bm n

m Bm xn x m x x

x

Bmm n K K

Bm1 Bm1 Bmk m1, m2, , mk Bm

M m1, m2, , mk K BM K

K K

C L

C (x1, x2, , xn)nL xj L.j1, 2, , n

4 [L, L]

C [L, L] n 5 C

K C

15K

287

59 3

11.7

12 X x1, x2, X p

p U N

j NN xj U

p p U

p

1

18 X 1

(xjj ) 1, 2, 3,

(xj) (xj) p q 2 X

p U, q V , UV U, V p, q

j N1 j N2 xj U, xj V

N1, N2 xj UVUV

1

1 1, 2, 1, 2,

1, 2, 1 2 1, 2, 3, 4, 5,

19X F X (xjj ) F (xj

j ) X p p F

p F F X \ F

F X \ F (xjj ) p

j N xj X \ F N

xj F p F

288

20f X 1

f

X

3 f(X) m

(m, m) m 1

f(X)

I1, I2, , IkM (M,

M) I1, I2, , Ik f(X)(M, M)

f

21f X

xX f(u) f(x) f(v)Xu, v

minf f(x)xX , Msup f(x)xX minf f(x)xX

xX m f(x) Msup f(x)xX

xX M f(x) mM

f(x)

X x f(x) xX

f(x) M xX g(x)Mf(x)1

g(x)Mf(x)1

y f(x) zMy1

M f(x)

M f(x) M M f(x) M M

20 f(v)M v

X x X f(x) m x X

h(x) f(x)m1

289

59 3

m f(x) m f(x) m

m f(x) m m f(u)m u

X m M f(u) f(x) f(v)

11.8

13X x X N x X

x U U N U N x

22X X V V

X

V V

V

V v Uv, Uv V Uv

V Uv V v

v Uv Uv Uv V Uv

V

14

(1) X A A

XXA

(2) X A A X

X A A0

A

(1)A A

(2) F A F F

290

A

(1) A A0 A

(2) U A U A0 A0 U

23X A A x1, x2, x3, X

p p A

F A j xj F

p F 19

p

p A

1 (0, 1)1, 21

, 31

, 0

0 (0, 1) [0, 1]

11.9

15X d

X x1, x2, x3,

j N, k N j, k d(xj, xk)

N

X X

X

(0, 1) 21

, 31

,

n(n1) 1 n

n1

n11

59 3

24X X

X

x1, x2, x3, X X p

xn1, xn2, xn3 p

x1, x2, x3, 0 m Nn N

m, n d(xm, xn) 21N2

1

p N

nj N nj d(xnj, p) 21

n N

d(xn, p) d(xn, xnj)d(xnj, p)212

1

x1, x2, x3, p

X

6

160 X

X X

X X Bi (i1, 2, , k)

X XB1B2 Bk

X A A(B1 A) (B2 A) (Bk

A) Bi A A

25X X

0 292

X

X 3 X

X 3

X m k, n k, mn d(xm, xn)

k1 x1, x2, , xk1 xm

BX (xm, ) 23X

mk BX (xm, ) X

BX (xm, ) xk m k

d(xm, xk) k

x1, x2,

X X

26X A A

diamAdiam

A diam A diam x, y

d(x, x), d(y, y) A x, y

x d(x, x)

A x BX (x, ) A

x

d(x, y) d(x, x)d(x, y)d(y, y)diam A2

0 d(x, y) diamA2

d(x, y) diam A d(x, y)diam A0

21d(x, y)diam A d(x, y) diam A2

diam diam A diam A diam diam A diam

293

59 3

27

X X

X

X x1, x2, x3, X

X

0 X 26 X

X

X X

0

X

F1 F2 F3 Fn 2n

1

X F1 ,

F2 , F3,

X 21 F1 X

F1 0

F1

2214

1 F2 n

n xn Fn mn xm, xn

xm, xn Fn diamFn2n1 d(xm, xn)2n

1

x1, x2, x3, X p

294

Fn m n xm Fn n

pFn

X

V pV BX (p, ) X p

BX (p, )V V

n 2n1

p Fn

diamFn FnBX (p, ) V

Fn x1, x2, x3,

X

8X d 3

(1)X

(2)X

(3)X

6 242527 (1) (2) (2) (3) (3) (1)

(1), (2), (3)

11.10

28(X, d) d

X X

1

X 1 X

x x 2x B(x, 2x) 1

295

59 3

x x x

B(x, x) X x B(x, x) X

X

B(x1, 1)B(x2, 2) B(xr, r)X, i xii1, 2, , r

x1, x2, , xr 1, 2, , r

A X u

u 1 r i B(xi, i) A

v

d(v, xi) d(v, u)d(u, xi)i 2i

AB(xi, 2i) B(xi, 2i) 1

A

X X

1

X, Y dX, dY fX Y X

Y f

0 0 dX (x, x) X x, x

dY (f(x), f(x)) x, x

9X, Y X X

Y

dX, dY X, Y fX Y X

Y yY

VyxXdY (f(x), y) 21

f(x) y 21 x y 2

1

296

Y BY (y, 21) Vyf

1(BY (y, 21)) BY (y, 2

1)

f y Vy X

xX xVf (x)

Vy VyyY X

28 X

Vy 0 x, x dX (x, x)

x, xx x 2 dX (x, x)

xVy xVy y Y

dY (f(x), y) 21, dY (f(x), y) 2

1

dY ( f(x), f(x)) dY(f(x), y)dY(f(x), y)

fX Y

11.11

17 X X f(x) X

x x idX A

X A A X

iA X 1 Y

X Y

18XY X

Y h

(1) h Y X h1

(2)h h1

f(x) x 1

297

59 3

Y y Y x

y Y x 1

f(x) x 1 X Y 1 1

h Y

h f(x) x

h

Y 1 x

X Y X Y

X Y h h1

X h X, Y h1

X h1hidX

h1 h Y hh1idY

19xyz

R

(1)xRx

(2)xRy yRx

(3)xRy yRz xRz

29

X X X

X X f X Y

f 1 Y X X Y

Y X

X Y Y Z fX Y gY

Z f, g f 1, g1

298

gf (gf)1f1g1

gf X Z X Z

2

2 2

1t (t1) 2

t

r(0)

( r)

r

r

31 X Y(1, 1) X

Y

f(x)1 xx

, xX

X Y f(0)0, x

f(x) 1, x f(x) 1

y1 xx

x x 0 y 0 x1yy

x 0 y 0

299

59 3

x1yy

x1 y

y

f

f 1(y)1 y y

f1(0)0, y 1 f1(y) , y 1

f1(y)

300

The Doshisha University Economic Review Vol.59 No.3

Abstract

Yasuhito TANAKA, Introduction to General Equilibrium Theory with a Proof of

Brouwer's Fixed Point Theorem

I present an introductory explanation of the proof of the existence of

general equilibrum mainly in an exchange economy. Present elementary proofs

of Sperner's lemma and Brouwer's fixed point theorem are also indicated. The

appendices provide expositions of related mathematical concepts.

301

1223456781921031111.111.211.311.411.511.611.711.811.911.1011.11

Abstract