Fixed Point s

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Mathematical World • Volum e 2

Fixed Point s Yu. A. Shashki n

Translated from the Russian by Viktor Minachi n

http://dx.doi.org/10.1090/mawrld/002

K). A . IIIAIIIKH H

HEnO^BH5KHBIE TOHKH

«HAYKA», MOCKBA, 198 9

Translated from th e Russian by Viktor Minachin

1991 Mathematics Subject Classification. Primar y 01-01, 54-01 , 54H25; Secondary 54C15.

Library of Congress Cataloging-in-Publication Dat a

Shashkin, IU . A. (IUrii Alekseevich )

[Nepodvizhnye tochki. English ]

Fixed points/Yu . A . Shashkin; translated fro m th e Russian b y Viktor Minachin .

p. cm.—(Mathematical world , ISSN 1055-9426 ; v. 2)

Includes bibliographical references .

ISBN 0-8218-9000-X (alk . paper )

1. Fixed poin t theory . I . Title. II . Series. QA329.9.S5313 199 1 91-2899 4 515'.7248—dc20 CI P

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Table o f Content s

Preface vi i

Chapter 1 Continuou s Mapping s o f a Closed Interva l an d

a Square 1

Chapter 2 Firs t Combinatoria l Lemm a 5

Chapter 3 Secon d Combinatoria l Lemma , o r Walk s

through th e Rooms in a House 7

Chapter 4 Sperner' s Lemma 9

Chapter 5 Continuou s Mappings , Homeomorphisms , an d the Fixed Poin t Propert y 1 5

Chapter 6 Compactnes s 2 1 Chapter 7 Proo f of Brouwer's Theorem fo r a Closed Inter -

val, th e Intermediat e Valu e Theorem, an d Ap -plications 2 5

Chapter 8 Proo f o f Brouwer' s Theorem fo r a Square. .. 3 3

Chapter 9 Th e Iteratio n Metho d 3 9

Chapter 10 Retractio n 4 3

Chapter 11 Continuou s Mapping s o f a Circle , Homotopy ,

and Degre e of a Mapping 4 7

Chapter 12 Secon d Definitio n o f the Degre e of a Mapping 5 3

Chapter 13 Continuou s Mapping s o f a Sphere 5 5

Chapter 14 Theore m o n Equalit y o f Degree s 6 1

Solutions and Answers 6 5

References 7 7

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Preface

Applying mathematics means , in many cases , solving equations . I f that is the case , then th e important thin g to know is whether a particular equatio n has a solution o r not . Th e presence o f solution s i s guaranteed b y so-called existence theorems. Le t / b e a function o f the real variable x , continuou s in th e close d interva l [a , b] an d assumin g value s o f differen t sign s a t it s endpoints. The n the equation

(0.1) / (* ) = <>

has at least one solution insid e the interval. Existence theorems are often expresse d in the form o f "fixed point " prin -

ciples. Fo r example, le t us view equation (0.1 ) in the following way . Writ e (0.1) in the form Xf(x) + x = x , where A is a positive parameter. Denotin g A/(JC) + x b y F(x) w e get the equation

(0.2) F{x) = x.

Choose th e value o f k i n suc h a way that al l the values o f F li e inside the interval [a , b] . Equatio n (0.2 ) can now be looked upo n as follows. Th e function F map s th e point (th e real number ) x fro m th e interva l [ a, b ] into th e point F(x) = y o f th e same interval , which , i n general , doe s not coincide with x . In other words, the mapping F take s the point x int o the point y . However , i f a point x 0 i s a solution o f (0.2) , then i t stay s where it was; that is , it is a fixed point . Th e same poin t i s evidently a solution of equation (0.1 ) as well.

Therefore, i n geometrica l terms , a theore m ensurin g th e existenc e o f a solution for equation (0.2) is formulated a s the following fixed point principle: if F i s a continuous functio n mappin g a closed interva l int o itself , the n the function ha s at leas t one fixed point. Becaus e F i s an arbitrary continuou s function, thi s property is actually that of the closed interval itself; it does not depend o n the choice o f a particular mappin g an d i s called th e fixed point property.

The problem o f solving the system of two equations

f{x,y) = a, g{x,y) = b

for unknow n x an d y ca n be reduced to the problem of whether a mapping of a square or, say, a disk int o itsel f ha s a fixed point.

vii

vm PREFACE

Fixed poin t theorem s hav e numerou s application s i n mathematics . Mos t of th e theorems ensurin g th e existence o f solution s fo r differential , integral , operator, o r othe r equation s ca n b e reduce d t o fixed point theorems . The y are also used in new areas of mathematical applications, e.g., in mathematical economics, game theory, etc .

The subject of this book is essentially one single problem: whethe r a closed interval, square , disc, or sphere has the fixed point property .

The theory o f fixed points belongs to topology (se e [3] and [5]) , a par t o f mathematics created a t the end o f the nineteenth century , and makes exten-sive use o f suc h topologica l notion s a s continuity , compactness , homotopy , and the degree of a mapping.

Another ai m o f th e boo k i s t o sho w ho w combinatoria l consideration s related t o decompositio n (triangulation ) o f figures into distinc t part s calle d faces (simplexes ) adjoinin g eac h othe r i n a regula r fashio n ar e use d i n thi s theory.

Three names should be mentioned here . The first i s the famou s Frenc h mathematicia n H . Poincar e (1854-1912) ,

the founder o f the fixed point approach , who had deep insight into its futur e importance fo r problem s o f mathematica l analysi s an d celestia l mechanics , and too k a n activ e rol e i n it s development . Poincar e wa s the first t o appl y the combinatoria l approac h t o topology , usin g triangulations o f geometrica l figures into simplexes .

The second is the Dutch mathematician L.E.Y . Brouwer (1881-1966) . H e introduced th e topologica l notion s use d i n thi s book , amon g the m thos e o f homotopy an d th e degre e o f a mapping . H e als o prove d th e fixed poin t theorems fo r a square, a sphere, and thei r ^-dimensiona l counterparts .

The third is the German mathematician E. Sperner (1906-1980), who back in 192 8 proved the combinatorial geometri c lemma on the decomposition o f a triangl e (a s well as of an y «-dimensiona l simple x i n general) , which play s an importan t rol e in the theory of fixed points.

The autho r would lik e to expres s hi s deep gratitude t o V . G. Boltyansky , whose suggestions led to substantial improvement s o f the first version o f th e manuscript. H e i s als o indebte d t o E . G. Pytkeye v fo r usefu l discussion s during the preparation o f thi s book .

References

1. M. B. Balk and V. G. Boltyansky, Geometry of masses, ("Kvant" series, vol. 67) , "Nauka", Moscow , 1987 . (Russian )

2. V . G . Boltyansky , Method of iterations, Kvan t 3 (1983) , 16-21 , 37 . (Russian)

3. V . G . Boltyansk y an d V . A. Yefremovich, Topology through pictures, ("Kvant" series, vol. 21), "Nauka", Moscow, 1982 ; German transl., An-schanliche Kombinatorische Topologie, VEB Deutschen Verla g Wiss. , Berlin, and F . Vieweg & Sohn, Braunschweig, 1986 .

4. N . Ya. Vilenkin , Method of successive appproximations, (Popula r lec -tures in mathematics, vol. 35), "Nauka", Moscow, 1968 ; English transl. of 1s t ed. , Successive approximation, Pergamo n Press , Oxford , an d Macmillan, New York, 1964 .

5. N. E . Steenrod an d W . G. Chinn , First concepts of topology, Random House, New York and Toronto , 1966 .

6. Hug o Steinhaus , Kalejdoskop matematyczny, 2n d ed. , Pahstwow e Zaklady Wydawnict w Szkolnych , Warsaw , 1954 ; Russia n transl. , "Nauka", Moscow , 1981.

7. , Problems and arguments, "Mir" , Moscow , 1974 . (Russian )

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