On Global Univalence Theorems

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

977

T. Parthasarathy

On

Global Univalence Theorems

Springer-Verlag Berlin Heidelberg New York 1983

Author

T. Parthasarathy Indian Statistical Institute, Delhi Centre ?, S.J.S. Sansanwal Marg., New Delhi 110016, India

AMS Subject Classifications (1980): 26-02, 26 B10, 90-02, 90A14, 90 C 30 ISBN 3-540-11988-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-38?-11988-4 Springer-Verlag NewYork Heidelberg Berlin

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1983 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210

PREFACE This volume of lecture notes contains results on global univalent mappings. Some of the material of this volume had been given as seminar talks at the Department of Mathematics,

~hiversity of Kansas, Lawrence during 1978-79 and at

the Indian Statistical Institute, Delhi Centre during 1979-80. Even though the classical local inverse function theorem is well-known, C~le-Nikaido's global univalent results obtained in (1965) are not known to many mathematicians that I have sampled.

Recently some significant contributions have

been made in this area notably by ~arcia-Zangwill Soarf-Hirsch-Chilnisky

(1980).

(1979), Mas-Colell

(1979) and

Global univalent results are as important as local

univalent results and as such I thoughtit is worthwhile to make these results well-known to the mathematical community at large.

Also I believe that there are

very many interesting open problems which are worth solving in this branch of Mathematics.

I have also included a number of applications from different

disciplines like Differential Equations, Mathematical EconomiNcs, Mathematical Programming, Statistics etc.

Some of the results have appeared only in Journals

and we are bringing them to-gether in one ~lace. These notes contain some new results.

For example Proposition 2, Theorem 4

in Chapter II, Theorem 4, Theorem 5 in Chapter III, Theorem 2" in Chapter V, Theorem 8 in Chapter VI, Theorem 2 in Chapter VII, Theorem 9 in Chapter VIII are new results. It is next to impossible to cover all the known results on global univalent mappings for lack of space and time.

For example a notable omission could be the

role played by univalent mappings whose domain is complex numbers.

We have also

not done enough justice to the oroblem when a PL-function will be a homeomorphism in view of the growing importance of such functions.

We have certainly given

references where an interested reader can ~et more information. I am grateful to Professors : Andreu Mas-Colell, Ruben Schr~mm, Albrecht Dold and an anonymous referee for their several constructive suggestions on various parts

IV

of this material.

I am also grateful to Professor David Cale for the example given

at the end of Chapter II and Professor L. Salvadori for some useful discussion that I had with him regardin~ Chapter VII. Moreover I wish to thank the Indian Statistical Institute, Delhi Centre for ~roviding the facilities and the atmosphere necessary and conducive for such work. Finally I express my sincere thanks to Mr. V.P. Sharma for his excellent and painstakin~ work in t~ming several revisions of the manuscript, Mr. Mehar Lal who tyoed a preliminary version of this manuscriot and Mr. A.N. Sharma who heloed me in filling many s~rmbols.

1 DECKW~ER 1982

T. PART F~AK&T RY INDIAN STATISTICAL INSTITL~E DELIYI CKNTRE

CONTENTS PREFACE INTRODUCTION CFAPTER

CFAPTER

I

II

CFAPTER III

CFAPTER

C FAPTER

CFAPTER

IV

V

VI

CFAPTER VII

:

:

:

:

PRELIMINARIES AND STATE...WENTOF THE PROBLEM

1

Classical inverse function theorem

1

Invariance of domain theorem

3

Statement of the oroblem

3

P-N3[TRICES AND N-N~TRICES

I0

Is AB a P-matrix or N-matrix when A and B are P-matrices or N-matrices

12

F[NDAMENTAL GLOBAL LNIVALENCE RESLGTS OF CALE-NIKAIDO-INADA

:

17

Fundamental global univalence theorem

20

Univalent results in R ~

22

GLOBAL }DMEOMORFI{ISN~$ BETWEEN FINITE DIMENSIONAL SPACES

28

Plastock's theorem

29

More-Rheinboldt's theorem and its consequences

34

Light open mappings and homeomorphisms

36

: SCARF'S CONJECT~E AND ITS VALIDITY

:

6

Characterization of P-matrices

41

Carcia-Zangwill's result on univalent mappings

41

Mas-Colell's univalent result

44

GLOBAL INIVALENT RESULTS IN R 2 AND R 3

49

Univalent mappings in R 2

49

Univalent results in R ~

57

ON THE GLOBAL STABILITY OF AN AtTONOMOU3 SYSTEM ON THE PLANE

59

Proof of theorem 2

6O

Vidossich's contribution to Olech's oroblem on stability

65

VI

CFAPTER

VIII

CHAPTER

CFAPTER

IX

X

: ~TIVALENCE FOR MJIPPINCS WITH LEOh~IEF TYPE JACOBIANS

68

thivalence for dominant diagonal mappings

68

Interrelation between P-prooerty and M-property

71

Lhivalence for comoosition of two functions

74

: ASSORTED APPLICATIONS OF LNIVALENCE MAPPING RESb~TS

77

An aoolication in Mathematical Economics

77

On the distribution of a function of several random variables

79

On the existence and uniqueness of solutions in non-linear complimentarity theory

81

An aDolication of Hadamard's inverse function theorem to algebra

85

On the infinite divisibility of multivariate gamma distributions

86

: FLRT~R

~NERALIZATIONS AND REMARKS

9O

A generalization of the local inverse function theorem

9O

Monotone functions and univalent functions

92

On PL functions

93

On a ~lobal univalent result when the Jacobian vanishes

96

Injectivity of quasi-isometric mappings

99

REFERENCES

i01

INDEX

105

INTRODLDTION Let ~ be a subset of R n and let F be a differentiable function from ~ to R n. We are looking for nice conditions that will ensure the equation F(x) = y to have at most one solution for all y s R n.

In other words we want the equation F(x) = y

to have a unique solution for every y in the range of F. Classical inverse function theorem says that if the Jacobian of the mao does not vanish then if F(x) = y has a solution x*, then x* is an isolated solution, that is, there is a neighbourhood of x* which contains no other solution.

In the

global univalence problem, we demand x* to be the only solution throughout ~. It is a fascinating fact, why the ~lobal univalence problem had not been posed or at any rate solved before ~adamard in 1906, which of course is a very late stage in the development of Analysis.

It is funny and actually baffling, how much

misunderstanding associated with the global univalence problem survived right into the middle of the twentieth century.

A brief history of this may not be out

of place here. Paul Samuelson in (1949) gave as sufficient condition for uniqueness, that the Jaeebian should not vanish and it was pointed out by A. was faulty.

Turing that this statement

However Paul Samuelson's economic intuition was correct and in his

case all the elements of the Jacobian were essentially one-signed and this condition combined with the non-vanishing determinant, turns out to be sufficient to guarantee uniqueness in the large. Paul Samuelson (1953) then stated that non-vanishing of the leading minors will suffice for global univalence in ~eneral.

But Nikaido produced a counter

example to this assertion and he went on to show that global univalence prevails in any convex region provided the Jacobian matrix is a quasi-positive definite matrix. Later,

C~le proved that it is sufficient for uniqueness in any rectangnlar region

provided the Jacobian matrix is a P-matrix, that is, every principal minor is positive.

In fact this culminated in the well-known article of ~le-Nikaido

(1965)

which is the main source of inspiration for the present writer. I should mention two other articles.

The article of Banach-Mazur

(1934) gives

probably the first proof of a relevant result formulated with the demands of rigour still valid to-day.

The more recent article by Palais (1959) covers a much wider

area than the article of Banach and Mazur. There are several approaches one can consider to the global univalent problem. For examole the approach could be via linear inequalities, monotone functions or PL functions. inequalities.

Throughout we have followed more or less the approach through linear

VIMI

In most of the theorems the conditions for global univalence are very stringent and therefore often not satisfied in applications. conditions of the theorem in practice.

Another problem is to verify the

In general it is hard to obtain necessary

and sufficient conditions for global univalence results. further research in this area.

There is lot of room for

C~le-Nikaido's global univalent theorem is valid

even if the partial derivatives are not continuous whereas Mas-Colell's results as well as C~rcia-Zangwill's results demand the partial derivatives to be continuous. One of the major open problems in this area is the following:

Can continuity of

the derivatives in Mas-Colell's results be dispensed with (altogether or at least in part) or alternatively - are there counter example~ following:

Amother problem is the

Is the fundamental global univalent result due to C~le-Nikaido valid in

any compact convex region? As alrea~v pointed out in some of the applications complete univalence is not warranted but in which some weaker univalence enunciations can nevertheless be made. In this connection I would like to cite at least two important papers one by Chua and Lam and the other by Schramm. Because of the lack of a text on the global univalence and since the results are available only in articles scattered in various journals or in texts devoted to other subjects

(for example Economics), I felt the need for writing this notes

on global univalent mappings.

In the next ten chapters with the exception of the

first two chapters, various results on global univalent mappings as well as their applications are discussed.

Also many examples are given and several open problems

are mentioned which I believe will interest research workers. Prerequisites needed for reading this monograph are real analysis and matrix theory.

Fere are a few suggestions.

[i].

W.Rudin (1976), Principles of Nathematical Analysis, Third Edition (International Student Edition) McGraw-Hill, Koyakusha Ltd.

[2].

F.R. Gantmacher (1959), The Theory of Matrices Vols. I and II, Chelsea Publishing Company, New York.

[3].

C.R. Rap (1974), Linear Statistical Inference and its Applications, Second Edition, Wiley Eastern Private Limited, New Delhi (Especially Chapter I dealing with 'Algebra of vectors and matrices').

[4].

G.S. Rogers (1980), Matrix derivatives, Marcel Dekker, New York and Basel (Actually only chapters 13 and 14 have the Jacobian and its orooerties as their central topic while 11 and 12 refer to the general theory).

[5].

W.Fleming (1977), Functions of several variables, Second Edition, SpringerVerlag, Heidelberg-New York. Some knowledge of algebraic topology will be useful (especially degree theory)

and we have mentioned a few references in Chapter IV.

CHAPTER

I

PRELIMINARIES AND STATEMENT OF THE PROBLEM

Abstract :

In this chapter we will collect some well-known results like classical

inverse function theorem, domain invariance theorem etc for ready reference (without proof).

We will then give the statement of the problem considered in this monograph

cite a few results and make some remarks. Classical inverse function theorem : C Rn

to

Let F be a transformation from an open set

R n. We will say that F is locally univalent, if for every x s ~ there

exists a neighbourhood U x of x such that F I U

(=F restricted to Ux ) is one-one.

Inverse function theorem gives a set of sufficient conditions for F to be locally univalent.

We come across such problems in various situations.

for a given y, there exists an

x°

such that F(x o) = y.

For example, suppose

We may like to know whether

there are points x other than Xo, contained in a small neighbourhood around x ° satisfying F(x) = y. is unique locally.

Classical inverse function theorem asserts that the solution In order to state the inverse function theorem we need the

following. Definition : A transformation F is differentiable at t

if there exists a linear

O

transformation L (depending on t o) such t h a t lim h÷0

1 ~-~

[F(to+h)-F(to)-L(h)]

Here Ilhll stands for the usual vector norm.

=

0

The linear transformation L is called

the differential of F at to and is often denoted by DF(to).

Write F = (fl,f2,...,fn)

where each fi is a real-valued function from ~. We denote their partial derivatives ~f. 4 1 as fJ. = i ~x. " J Remark i : A transformation F is differentiable at t

if and only if each of its O

components f. is differentiable at t i

Remark 2 :

for i = 1,2,...,n. O

If F is differentiable at to, then the matrix of the linear transformation

L is simply the Jacobian matrix J of partial derivatives fJ(to).

Definition : C (q).

Call F a transformation of class q, q ~ 0

That is, for every

if each

fi

is of class

fi(i = 1,2,...,n) all the partial derivatives upto order

q exist and are continuous over its domain.

We are now ready to state the (local) inverse function theorem. Local inverse function theorem : set

~ ~'R n

into

an open set A o ~

R n.

Let F be a map of class C (q), q > i from an open

If the Jacobian at t o s ~ does not vanish, then there exists

~ containing t o such that :

(i)

FIA °

is one-one, that is, F restricted to A ° is univalent.

(ii)

F(A e) is an open set.

(iii) The inverse G of F[A ° is of class C (q). (iv)

JG(X) = (JF(t)) -I where F(t) = x, t s A o.

matrix evaluated at x.

Remark i :

Here JG(X) denotes the Jacobian

Proof of this may be found in Fleming

In one dimension the situation is simpler.

[17].

If F is a real-valued

function with domain an open interval ~, then F -I (=inverse map of F) exists if F is strictly monotone.

Also F will be strictly monotone if F'(t) ~ 0 for all t E ~,

and in fact G'(x) = F ' ~I JF(t) takes the place

where x = F(t). of F'(t).

In higher dimensions the Jacobian

The situation here is more complicated.

For

example, the non-vanishing of the Jacobian does not guarantee that F has a (global) inverse as in the univariate case.

However, if JF(to) does not vanish at to, we

can find a small neighbourhood A ° containing t o such that F restricted to A ° will have an inverse.

In other words we can only assert local inverse.

This is precisely

part of the statement of inverse function theorem. If one is interested in just the local univalence we have the following theorem (proof may be found in

[44]).

Local univalent theorem : connected subset of R n. (i)

Let F:~ C R n ÷ R n be a mapping where ~ is an open

We have the following:

If F is differentiable at a point t o s ~ and JF(to) # 0, then there is a

neighbourhood U of t o such that F(y) = F(to) , y s U ~ (ii)

y = to •

If F is continuously differentiable in a neighbourhood of an interior point

t o of ~

and JF(to) # 0, then there is a neighbourhood U of t o where F is univalent,

that is, F(y) = F(z), y,z s U ~

y = z.

We are now ready to state the following: Theorem on invarianoe of interior points :

Let F:O ÷ R n be a differentiable map

with non-vanishing Jacobian, where ~ is an open region in R n .

Then the image set

F(~) is also an open region. For a proof see Nikaido

[44].

This result is true not only for differentiable

mappings with nonvanishing Jacobians but also for homeomorphic mappings from a

region of R n into R n.

That is the content of the following classical theorem due

to Brouwer. Invariance

of domain theorem:If ~ is open in R n and F:~ ÷ R n is one-one and

continuous,

then F(~) is open and F is a homeomorohism.

Definition

:

For a proof see

A mapping F:~ ÷ R n is called a local homeomorphism

a neighbourhood

of t is mapped homeomorphically

if for each t s ~,

by F onto a neighbourhood

It is clear that if F:~ ÷ R n is a continuously non-vanishing

[30].

differentiable

Jacobian it follows from local inverse-function

univalent theorem that F is a local homeomorphism.

of F(t).

function with

theorem or local

We will introduce one more

definition. Definition

:

Let F:~ ÷ R n be a continuous mapping where ~ is an open region in R n

with the property that each y s F(~) has a neighbourhood of F-I(v)

is mapped homeomorphically

onto V by F.

map and (~,F) is called a covering space for F(~).

In this case, the cardinal number

n of the set F-l(y) is the same for all y s F(~). F is called a finite covering,

Remarks

:

It is well-known

V such that each component

Then F is called a covering

If n is a finite integer,

or more specifically,

then

an n-covering.

that every covering map F:~ ÷ R n is a homeomorphism

is connected and that every homeomorphic

if

onto function F:~ ÷ R n is a covering map

and every covering map is a local homeomorphism.

However the converse is not true.

A local homeomorphism need not be a covering map and a covering map need not be a homeomorphic function.

onto function.

A 1-covering map is necessarily

The following result is well-known

A theorem on covering space :

Let X and Y be connected,

spaces (for example X = Y = Rn). F is a homeomorohism [ Here F:X ÷ Y

Furthermore

a homeomorphic

onto

[48]. locally Dathwise connected

suppose Y is simply connected.

Then

of X onto Y if and only if (X,F) is a covering space of Y.

is a map from X to Y].

We need this result especially in chapter IV where sufficient conditions are given in order that a map F from R n to R n will be a hemeomorphism results on degree theory, we freely use from chapter VI in references

for degree theory are

Statement of the problem

:

[ 48].

onto R n.

For

Other good

[13,59,63].

Let F : ~ r - R n ÷ R n be a differentiable

F to be globally one-one throughout ~.

What conditions

map.

We want

should we impose on the

map F and the region ~ so that F is globally one-one ? Remark i : univariate

Non-vanishing case.

of the Jacobians

alone will not suffice except in the

See the example of Gale and Nikaido given in chapter III.

4

Remark 2 :

Even in R I non-vanishing

for global univalence.

of the derivative

is not a necessary condition

For example f(x) = x 3 is globally univalent

whereas its derivative vanishes at x = 0.

throughout R I

In general it appears difficult or

hopeless to derive necessary conditions whenever global univalence prevails. We will cite now a few typical results to give the reader some idea about this monograph.

Fundamental

global univalence theorem

be a differentiable

:

(Gale-Nikaido-lnada)

: Let F : O C R

mapping where ~ is a rectangular region in R n.

n ÷ Rn

Then F is

globally univalent in ~ if either one of the following conditions holds good. (a)

J(x) (= Jacobian of F at x) is a P-matrix for every x E O.

(b)

J(x) is an N-matrix and the partial derivatives

A global univalent theorem in R 3

[ Parthasarathy ]

are continuous

:

for all x s ~.

Let F be a differentiable

map

from a rectangular region O C R 3 to R 3 with its Jacobian J having the following two properties

for every x E ~:

(a)

diagonal entries are negative and off-diagonal

(b)

Every principal minor of order 2 x 2 is negative.

Then F is univalent

Plastock's

theorem

entries are positive.

in ~.

:

Let F:R n ÷ R n

Suppose J does not vanish at any x s R n. f

o then F is a homeomorphism

inf

be a continuously

differentiable

map.

If (I/lIJ(x)-llI)dt

=

11x11=t

of R n onto R n.

In fact F is a diffeomorphism.

(Here llx[l stands for the usual Euclidean distant norm and IIAII = suP11Aull A an

for

n x n matrix and u an n vector with norm one). In order to state

Definition

:

disconnected

MoAuley's theorem we need the following definition.

Call a continuous mapping F:O ÷ R n light if F-I(F(x)) for each x E ~.

is totally

[ Here we will assume ~ to be a unit ball].

open if for each U open in ~, F(U) is open relative to F(O).

Call F

Denote by S F the

set of points x E ~ such that F is net locally one-one at x.

McAuley's

Theorem

:

Suppose that F is a light open mapping of a unit ball ~ in

R n onto another unit ball B in R n such that (I) F -I F(8~) = 8~ (3) FIS F is one-one

8B

(4) for each component C of B-S F there is a nonempty V in

C open relative to B such that FIF-I(v) Scarf's conjecture

(2) F(8O) =

:

Let F:O C R n ÷ R n

is one-one.

Then F is a homeomorphism.

be continuously

differentiable

on a compact

rectangle ~ with det J(x) ~ C for every x s ~

for every x s ~.

(= boundary of ~).

Further suppose J(x) is a P-matrix

Then F is one-one

throughout ~.

This conjecture was proved by three different set of researchers Garcia-Zangwill, Mas-Colell and Scarf et al.

This result is an significant generalization of

Gale-Nikaido's theorem.

Schramm's theorem :

Let ~ be an x-simple domain in the (x,y)-plane, ~ its boundary.

Let F = (f,g) : ~ ÷ R 2 be a differentiable map, ~ the minimum and B the maximum of f on Z.

Suppose the Jacobian of F is an NVL matrix for each z s ~ and for each

u s (~,~),~uppose at most two points z s ~ satisfy f(z) = u. to

~ \ (A(~)UA(~))

Remark i :

is univalent where A(u) = (z:z s ~ a n d

Then F restricted f(z) = u~.

Results obtained so far on global umivalence are not complete and we

have mentioned several interesting open problems throughout the monograph.

For

example it is not known whether Gale-Nikaido's result holds good in any compact convex regions.

In chapter VIII and IX we have given various applications of

univalent results in other areas like differential equations, Economics, Mathematical programming, Algebra etc.

Remark 2 : All the theorems cited above with the exception of McAuley's theorem depend on the choice of a fixed coordinate system. conditions on the Jacobian matrix.

This is so because we place

Though one may argue that this may not be the

most natural approach to the problem under consideration, the present writer feels that this method yields useful results in many problems that arise in practice. See Chapter VII and Chapter IX in this connection.

Also in some special cases the

matrix conditions turn out to be necessary as well - see for example theorem I and theorem 6 in chapter VIII.

Also, the present writer feels that it is not difficult

to check these matrix conditions in a given problem.

CHAPTER

II

P-MATRICES AND N-MATRICES

Abstract

:

In this chapter we will give a geometric

We will give some properties

of N-matrices.

characterization

global univalence results due to Gale, Nikaido and Inada. interrelation between P-matrices

of P-matrices.

These facts we need later to prove We will also see the

and positive quasi-definite

matrices.

examine the question whether P-property holds good under multiplication P-matrices

Finally we (sum) of two

- this kind of result is useful in determining when the composition

F o G (sum, F+G) of two univalent Let A be an

functions

is univalent.

n × n matrix with entries real numbers.

matrices with complex entries. if the associated quadratic

We will not consider

If A is a symmetric matrix then A is positive definite

form x'Ax ~ 0, for any x different from O.

denotes the transpose of the vector x.

Here prime

It is well known that a symmetric matrix A

is positive definite if and only if every orincipal minor of A is positive. we drop the symmetric assumption results?

In other words,

for every x ~ 0.

from A.

In such situations

Suppose

can we prove similar

suppose A has the following property,

namely x'Ax ~ 0

They can we assert that every principal minor of A is positive?

Another interesting question is to characterize matrices whose principal minors are positive.

Next we will answer these questions.

Characterization not necessarily

of P-matrices

:

We will start with a few definitions.

Let A be a

symmetric real n × n matrix.

Definition

:

Call A a P-matrix if every principal minor of A is positive.

Definition

:

Call A a positive quasi-definite

Definition

:

Call A an N-matrix if every principal minor of A is negative.

N-matrices

are divided into two categories:

(i)

matrix if x'Ax ~ 0 for every x # 0. Further

An N-matrix is said to be of the first category if A has at least one positive

element. (ii)

An N-matrix is said to be of the second category if every element of A is

non-positive. Definition

:

Call A a Leontief-type matrix if the off-diagonal

non-positive. We will make a few quick remarks.

entries are

Remark i :

Every positive quasi-definite matrix is necessarily a P-matrix (we will

give a proof of this fact after characterizing the class of P-matrices). converse is not necessarily true as the following example shows.

But the

12 Let A = [ 0 1 ]"

Then (Au,u) = Ul+2UlU2+U2 2 2 (where u = (Ul,U2)) and (Au,u) = 0 whenever u I = - u 2. Thus A is a P-matrix but not positive quasi-definite. positive quasi-definite matrix if and only if ( - ~' In this example ( - - ) Remark 2 :

=

I [i

Also observe that A is a ) is a positive definite matrix.

i i ] is a singular matrix.

The following example shows that every positive quasi-definite matrix 2 2 Let A = [ 3 2 ] Clearly (Au,u) = Ul+5UlU2+8u2 > 0

need not be positive definite. for any u # O.

Hence A is positive quasi-definite but not a positive definite matrix

as A is not symmetric. Remark 3 :

First category N-matrices share some properties in common with P-matrices

as we shall see below. category N-matrices.

However there is a nice characterization for symmetric second In order to do that we need the following definition.

matrix A, merely positive definite if (i) x'Ax< 0

Call a

there exists some vector x such that

and (ii) whenever x'Ax < 0, this will imply Ax < 0 or Ax > 0-in other words

Ax is onesigned.

The result then is the following.

the second kind then A is merely positive definite. (simple) negative eigenvalue.

If A is a symmetric N-matrix of Furthermore A has exactly one

Proofs of these results may be found in Rao

[62 ].

We are now ready to prove some results on P-matrices. Theorem I :

Let A be a P-matrix or anN-matrix of the first category.

Then the

system of linear inequalities Ax
O has only the trivial solution x = O. Game theoretic interoretation of theorem i :

Theorem I says that the minimax value

of the matrix game A (as well as the minimax value of every principal submatrix C of A) is positive, brovided A is a P-matrix or an N-matrix of the first kind. can he seen as follows. or equal to zero. columns). A'y ~ 0

This

Suppose yon Neumannvalue of the matrix game is less than

(We will assume minimizer chooses rows and maximizer chooses

We have a probability vector y for the minimizer such that y'A ~ 0 (prime denotes transpose).

If A is a P-matrix or an N-matrix so is A'.

Thus we have got a nontrivial non-negative vector y satisfying A'y ~ 0 dicts theorem i and consequently value of A must be positive. value of A' is positive.

See

or

which contra-

It is also clear that

[49, 54 ] for details regarding game theory and

for results relating game theory and M-matrices.

[61 ]

We follow the proof as given in

Nikaido

[44].

Proof of Theorem I :

First we will prove when A is a P-matrix.

induction principle.

For n = I, clearly Theorem i, is true.

We will use

So assume theorem 1

for n = k, prove that it holds good for n = k+l (Here n refers to the order of the square matrix A).

Let x > 0.

allX I . . . .

°

That is,

+ a12x 2 + .-. + al,k+ I Xk+ I .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

%+l,lXl+~k+l,2X2+

.

.

.

.

.

.

.

.

.

.

.

.

.

j 0

.

+ %+l,k+l~k+l < 0

Since aii > 0, we can increase if necessary x i such that one of the inequality becomes an equality.

We will assume without loss of generality

allX I + a12x2 + ... + al,k+ 1 Xk+ 1 a21x I + a22x2 + ... + a2,k+ 1 Xk+ 1 . . . .

o

.

.

.

.

.

.

.

.

.

.

.

.

.

.

ak+l,lXl+ak+l,2X2+ Using the first equality, resulting

inequalities

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

=

0


0 and hence x I = 0.

Now assume A is an N-matrix of first category.

a < 0, d < 0, and ad-bc < 0. A is of first category, A-IAx = x < 0.

This means b

b > 0 and c > 0.

But x > 0 by hypothesis,

This

Clearly order of an N-matrix Suppose A =

a [c

bd ] where

and c should be of the same sign. Consequently A -I > 0.

therefore x = 0.

Since

If Ax < 0 then

This proves the theorem

As before assume the result for n = k where k > 2 and prove it holds

good for n = k+l.

As A has at least one positive element, we can imitate the proof

verbatim given for P-matrices

till we get the matrix C = (a~j)i,j = 2,3,...,k+i.

Observe that det A = all det C. that det C > 0.

Since all < 0, det A = all det C < O, it follows

In fact one can check that C is a P-matrix.

the first part of the proof x i = 0 ¥ i ~ 2. equality.

C = (a~j)

the proof of theorem i when A is a P-matrix.

of first category should be at least 2 x 2 .

when n = 2.

Plainly the matrix

Hence by induction hypothesis,

ting this in the first equality, terminates

The

one can eliminate x I from the other inequalities.

can be written as

This terminates

Hence it follows from

Since all # 0,x I = 0 from the first

the proof of theorem i for N-matrices

of first kind.

Remark I :

Geometrically,

theorem I says the following:

Any non-trivial non-negative

vector cannot be mapped to a vector in the negative orthant when A is a P-matrix or an N-matrix of the first kind. Remark 2 :

Theorem I is valid for any matrix A which has non-negative

is A -I ~ 0.

Characterization

class of matrices.

inverse - that

results are available in the literature for such

In particular

if A is a Leontief type matrix then A -I ~ 0 if and

only if there exists some x > 0 such that Ax ~ 0. Remark 3 :

A result on linear inequalities

For any given matrix D not necessarily alternatives

holds.

asserts the following

Either the inequalities x'D < 0

the inequality Dy ~ 0

has a non-negative

trivial solution. poverse,

This statement

is the following,

has only a

is equivalent to the fact that A has a left

is due to Charnes et. al.

inequalities

solution or

conclusion of theorem i can be

For any matrix A, suppose the system Ax ~ O, x ~ 0

that is there exist non-negative

observation

has a semipositive

solution.

In view of this result on linear inequalities, viewed as follows:

E See 18, pp. 49].

a square matrix exactly one of the following

matrices N,M such that NA = I+M.

[ 8].

This

Another feature of the result on linear

which says that von Neumann value of a P-matrix game

is positive. Theorem 2 :

Suppose A is a P-matrix or an N-matrix of first category.

Then there

exists a positive vector Yo ~ 0 such that Ay ° ~ 0. Proof

:

From theorem i, the matrix D = (A,-I) has no semi-positive

x'D < 0 and consequently for some y ~ 0.

solution x with

from the above remark it follows that Dy = (A,-l)y > 0

Here y = (yl,Y2,...,yn,

Yn+l' Y n + 2 ' ' ' " Y 2 n )"

Then clearly Ay ° ~ z ~ 0 where z = (Yn+l' Y n + 2 ' ' ' " Y 2 n )"

Define Yo = (YI'Y2"'Yn)

This terminates

the proof

of theorem 2. Corollary I :

Let S n = {x : X ~

n 2 ~ x i = i}. i=l

O,

or an N-matrix of first category.

Let A be a P-matrix or

Then there exists an a ~ 0 such that

for every x s S .

Max (Ax)i>a_ l~i~n

n

Proof :

From theorem I, fer every x s S n it follows that Ax has at least one

coordinate strictly positive. Min x ~ S

Since

Max (Ax) i is continuous l~i~n

Max (Ax) = Max (Ax°) i for some x ° E S l~i~n i l 0 for i = 1,2,...,n. First j#i we will prove that det A ~ 0.

Then we will show det A is positive.

Suppose

n

j=l

a.. x. = 0 for i = 1,2, . lJ J

Define

,n .where x . .

rxil 0

=

max

l~l -

l 0.

di

dk

n = - ~ akj x.. j=l J j ~k

akk x k

det

a~

~ 0,

A # 0.

diagonal Note

that

clearly

l%T

Now

then det

0 + + ~.

say .

In case ~ = 0, then x will be zero vector which will contradict the

fact that x is a non-zero vector.

Since

(Xl,X2, .. .,xn) is a non-zero vector.

~ 0 %.

it will

(A+pI)

If det impossible.

shown has

that

this

Hence

det

a positive

is a continuous

A ~ 0

We have

n Then lakkXk I < I ~ akj xjl < ~ ~klakjldj < 0 akk dk. --j=l -- j j ~k

That is 0 ~ 0 which

be

A + pl also

Suppose 0 > 0.

will det

dominant

function imply

A > 0.

is impossible.

A > 0.

det

in

a positive

diagonal

for

0 and

that

a positive

dominant

every

det

(A + pe I) = 0 for

If A has

This shows that

If A has

p > 0.

(A +

some

dominant

pl) ÷ + ~ as

0 O > 0 which diagonal,

principal submatrix C of A also has the same property and consequently P-matrix.

This terminates the proof of the proposition.

enough to identify several classes of P-matrices. symmetric N-matrices of the second kind.

is every

A is a

[Thus we are fortunate

We are able to characterize only

The situation is far from complete in the

case of non-symmetric N-matrices ].However one can prove the following elementary

12 result for N-matrices of order 3 × 3 of the first kind Preposition 2 :

[53].

Let A be an N-matrix of the first kind with order 3 × 3.

Then

A will contain exactly four positive elements and five negative elements. Proof :

Since A is an N-matrix of the first kind and since A is of order 3 × 3, it

is clear that the number of positive elements is either 2 or 4 or 6. 2 positive elements.

Suppose A has

Then A will be of the form : +

A

Clearly det A > 0

=

which is impossible.

the off-diagonal entries are positive.

Similar contradiction will be reached if Thus it follows that A contains exactly four

positive elements and five negative elements.

This terminates the proof of

proposition 2. Remark i :

If A is a positive definite or a quasi-positive definite matrix and if

Q is a non-singular matrix then it follows that Q'AQ, is a positive definite or a quasi-positive definite matrix.

Such a result is not valid in general for P-matrices

(though it is true if Q is a diagonal matrix with all entries +I or -I. This fact is used during the course of our proof of Theorem 3). Let A = [ 1 0 -42 ] and Q =

[04

Ii ] "

Then Q'AQ =

[ 146 -12_i] "

Clearly A is a P-matrix while Q'AQ is not a

P-matrix. Remark 2 :

If A is an N-matrix of the first kind and if Q is a diagonal matrix

with all entries +I or -I then Q'AQ is an N-matrix but it may be of second kind. Open problem :

Let A be a positive definite matrix.

Can one find a diagonal matrix

D with diagonal entries strictly positive such that DA-I is a (weak) N-matrix? [We call a matrix B, a weak N-matrix if every principal minor of B is non-positive]. If the answer to this question is in the affirmative, this will settle an old conjecture of Paul Levy regarding the infinite divisibility of multivariate gamma distribution - See Paranjape

[51].

A final remark on N-matrices of the first kind:

If A is an n x n, N-matrix of

the first kind, then every principal minor of order less than or equal to n-i of A -I is positive.

(Note that det A -I < O since det A < 0).

Is AB a P-matrix or N-matrix when A and B are P-matrices or N-matrices ?

In this

section we are interested in examining whether the product of two P-matrices (or N-matrices) will be a P-matrix (or N-matrix).

In this connection we have the

13

following

[ 53].

Theorem 4 :

(i)

There exist two P-matrices (N-matrices) A and B such that AB is

not a P-matrix (N-matrix). (ii)

If A and B are 2 x 2 Leontief type P-matrices (N-matrices of the same kind)

then AB is a P-matrix. (iii)

If A and B are 3 x 3 Leontief type P-matrices then AB is a P-matrix but AB

need not be of the Leontief type. (iv)

If A and B are n x n Leontief type B-matrices and AB is also of Leontief type

then AB is a P-matrix.

However this result is not valid if we do not assume that AB

is of Leontief type when n > 4 [ see (iii) above]. Proof :

(i) A =

-~ ], B =

[0i

[2i

0] 1

Then AB = [-~

-~]

is not a P-matrix.

If we assume all the entries to be positive then clearly AB is a P-matrix if A and B are 2 × 2 P-matrices.

However such a result is not true for 3 x 3 P-matrices.

Let

111

I 2

,

4

=

12

i

2

i

i

1

2

Then

AB

=

13 i 7

~7

9

ii

21

29

Clearly A and B are positive-definite matrices but AB is not a P-matrix as the principal minor

4] is negative. 9

[

(ii) Proof of this is elementary and we omit it. necessary.

However a word of caution is

If A,B are 2 x 2 matrices with A, an N-matrix of the first kind, B and

N-matrix of the second kind then AB need not be

neither a P-matrix nor an N-matrix

as the following example demonstrates.

[-i i

AB = (iii)

[-2 - ]

Let

A =

2] B = -i '

[-3 -I

-7] -2

then

is neither a P-matrix nor an N-matrix.

We will prove that AB is a P-matrix provided A and B are 3 x 3 Leontief

type P-matrices.

A

=

Let -al 2

-al 3 ~

[~ i i

a22

-a23|

L-a31

-a32

a33 j

Here a.. and b.. are all non-negative. ij

Ij

'

B

=

bll

-bl 2

-bl 3

-b21

b22

-b23

L-b31

-b32

b33

It is clear that det(AB) = det A. det B > 0

and further the diagonal entries are positive in AB. every principal minor of order 2 is positive.

We need only to check that

Let us take the leading principal

14

minor C of order 2 from AB.

C

: la c

where

Then

+

a13

b31

b

+

a13

b32]

+

a23

b31

d

+

a23

b32j

la

all a12 L -a21

c Clearly Then

ad - bc > 0.

a22

bll

-b21

-b21

b22

Also note b : -all b12 -a12 b22 < 0

and c = -a21bll-a22b21 < O.

det C = (ad-bc) - a13 b32 c - a23 b32 b + a23 b32 a + a13 b31 d

>

O.

Similarly one can show the other two principal minors of order 2 in AB are positive. Thus we have shown that AB is a P-matrix. (iv)

We will show that AB is a P-matrix if A,B are Leontief t~/pe P-matrices and if

AB is of Leontief type.

Since A and B are Leontief type P-matrices it follows that

A -I and B -I are non-negative.

Consequently

(AB) -I satisfies Stolper-Samuelson

(AB) -I is non-negative.

In other words

condition and therefore AB is a P-matrix.

We

will now give a counter example to show that this result is false if we drop the assumption AB is of Leontief type when n > 4.

A

=

Let,

110ii iI li l°il i3i 0

~

0

0

0 0

_0

Then,

AB

:

-

and B

=

2

0

I

-4

i

0

0

0

7/8

2/8

0

0

-4

0

-I

0

0

-

One can check that A and B are Leontief type P-matrices but AB is not of Leontief type. 2,

Also note that AB is not a P-matrix as one of the principal minors of order

i [ 7/8

3 2/8 ]

is not positive.

This terminates the proof of theorem 4.

As already pointed out elsewhere, the motivation for proving theorem 4 is simply

15

that it helps us to determine when F o G will be univalent given that F and G are differentiable and univalent in R n.

Gale-Nikaido's fundamental result asserts

global univalence of a differentiable mapping F when the Jacobian of the map F is a P-matrix.

Note that the Jacobian associated with F o G is equal to the product

of the Jacobian matrices associated with F and G. in another chapter.

We will prove the following result

If F and G are differentiable maps from R 3 (or R 2) to R 3 (or R 2)

and if their Jacobians are P-matrices everywhere then F o G is a P-function and consequently one-one throughout R 3 (or R2).

From theorem 4 we may be able to

conclude such a result is valid in R n for any n provided we assume that the Jacobian associated with F o G is of Leontief type. If A and B are 2 × 2 Leontief type P-matrices then A+B is of Leontief type but need not be a P-matrix as the following simple example demonstrates• •

i ] is not a P-matrix.

Let A =

[~

-~]

Such results will be

useful to determine when F+G is one-one given that F and G are one-one, differentiable functions.

Observe the following result which is easy to prove.

If A and B

are

n x n Leontief type P-matrices and if there exists a non-negative vector x ~ O such that (A+B)x > 0

then A+B

is a Leontief type P-matrix.

[ Here (A+B)x > 0 means,

every component of the vector (A+B)x is positive]• It may not be out of place to mention other interesting equivalent properties of P-matrices. Theorem 5 :

We will state it in the form of a theorem without proof

Let A be an n × n

real matrix.

[ 56].

Then each of the following conditions

is equivalent to the statement : A is a P-matrix (i)

Every real eigenvalue of each principal submatrix of A is positive.

(ii)

A+D is nonsingular for each nonnegative diagonal matrix D

(iii) For each x ~ 0 there exists a nonnegative diagonal matrix D such that x,ADx > 0 (iv)

A does not reverse the sign of any nontrivial vector.

(v)

For each signature matrix S there exists an x > 0

Remark

such that SASx > 0.

I : We have already shown that (iv) is equivalent to the statement that A

is a P-matrix.

We can use this fact to prove the equivalence of other statements.

When A is of Leontief type, proof can be given using ideas from game theory-interested readers should refer to Raghavan available for N-matrices. problems: (I)

[ 61].

Such a beautiful characterization is not

We will close this chapter by mentioning two open

Suppose A is a P-matrix.

Does it mean A is positive stable?

[ Call

a matrix A positive stable if the real part of each eigenvalue of A is positive]. Clearly this is so if either A is a Leontief type P-matrix or if A is a 2 × 2 P-matrix.

(2) Suppose A is a positive definite matrix.

Does there exist a diagonal

matrix D with entries strictly positive such that DA-I is a (weak) N-matrix (that

16

is principal minors of (DA-I) is non-positive).

Again this result is true if A is

a 2 × 2 positive definite matrix. David Gale has communicated to the author the following example which answers the first question raised above in the negative.

A

Then A has eigenvalues

Let

lit!f0

=

l+s ,

E-½ + i ~ / 2

with negative real parts if s < ½

and

of course A is a P-matrix. We also wish to add that if A is a Dositive definite or quasi-positive definite matrix then A is positive stable. definite case.

We will indicate the proof in the quasi-positive

In fact as already remarked elsewhere

if and only if (A+A')/2 is a P-matrix. eigenvalues of (A+A')/2.

Let ~m

and

Let ~ be any eigenvalue of A. Xm-< Real part of ~ --< ~M

Since

A

~ > 0, it follows that A is positive stable. m

is quasi-positive definite

XM be the minimal and maximal Then it is well-known that

CHAPTER

III

FUNDAMENTAL GLOBAL UNIVALENCE RESULTS OF GALE-NIKAIDO-INADA

Abstract:

In this chapter we will derive fundamental global univalence results of

Gale-Nikaido-lnada for differentiable maps in rectangular regions whose Jacobian matrices are either P-matrices or N-matrices.

Using mean-value theorem of differen-

tial calculus we will obtain global univalence results for differentiable maps whose Jacobian matrices are quasi-positive definite matrices, in convex regions. result is due to Gale and Nikaido.

This

Finally we present two new results on univalent

mappings in R 3 . Let F be a differentiable mad from ~ R

n to R n.

We are interested in finding

suitable conditions such that F is univalent throughout ~.

Non-vanishing of the

Jacobian matrices will not suffice as we shall see below. approaches to the problem under consideration.

There are at least two

One approach places topological

assumptions on the map and the other places further conditions on the Jacobian matrices.

We will study the former in the next chapter and the latter in the

present chapter.

Global univalence theorems :

In order to state and prove results on univalent

mappings we will adopt the following notation and terminology. Gale and Nikaido's original paper. = {x:x ~ R n, a i ~ x i ~ bi).

We closely follow

Let ~ be a rectangular region in R n, where

Here ai,b i are real numbers where we may allow some

or all of them to assume -~ or +~.

Let F:~ ÷ R n be a mapping defined

by

F(x) = (fl(x), f2(x),...,fn(X)) where each fi(x) is a real-valued function defined on ~.

Then F is said to be differentiable in ~ just in case every fi has a total

differential ~fij(x)dxj in D - in other words for x, as~, fi(x)=fi(a)+ 2f + o(llx-all)(i=l,2

.....

n),

where o ( t l x - a l l ) / l l x - a l l

÷

0

as

x ÷ a.

The Jacobian matrix J of the mapping F is given by Jex) =

(a)(x.-a.)

j ij

a

a

llfij(x)ll.Differen-

tiability of F clearly implies continuity of F and it also implies partial differentiability of f!si and in fact fij(x) = ~fi(x)/~xj. the boundary points.

However, one has to be careful at

Yet the Jacobian can be defined by means of the coefficients of

the total differential at those points.

Now we are ready to prove a non-linear

counterpart of theorem I of chapter II.

Theorem i :

Let F be a differentiable mapping from ~ to R n.

matrix J(x) of the mapping F is a P-matrix for every x s ~.

Suppose the Jacobian Then for any a and x in

~, the inequalities F(x) < F(a) and x > a have only one solution namely x = a.

18

Proof :

We prove this theorem by induction on n, the common dimension of both the

argument and image vectors x,F(x). X = {x:x s ~ Clearly a ~ X.

and

Plainly theorem is true for n = I. F(x) i F ( a ) ,

x h a} .

We are through if we show that X contains only a.

we first prove that a is an isolated ooint of X. F(x)-F(a)

lim X

+

II

_ J(a)

llx_al I

we have

x-a

~

II = 0.

there is a positive number 6 > 0 by corollary i, chapter II

such that for any x > a, some coordinate of J(a) in any nei~hbourhood

7 x-a 1~

of a, some component of F(x)-F(a)

This shows that a is an isolated point of X.

Clearly b > a.

With that in mind,

As F is differentiable,

a

Since J(a) is a P-matrix,

in ~.

Let

Define y r - x

as follows

~ > 0.

Consequently,

must be oositive for x > a

Suppose b c X with b ~ a.

.

Y = (x : a < x < b, F(x) < F(a)) Plainly Y is compact and since a is an isolated ooint, Y - (a) is also compact. be a minimal element of Y - {a) in the sense that no other element y E Y -

Let

{a)

fulfills y ~ x (Note that such an x can be obtained by minimizing the sum of the components

of y when y ranges over Y - {a)).

As ~ s Y - {a), only two possibilities

can occur (i) x > a, (ii) x > a and x has some comoonent equal to the corresponding component of a.

Case (i) :

x > a.

Theorem 2 of cha~ter II ensures the existence of a vector u

satisfying u < 0 and J(x) u < 0 (for J(x) is a P-matrix). As u < 0, ~

a

and ~ is a rectangular

Moreover by differentiability

F(x(t))-F(x) tliull

we olease by letting t aoproach 0. t,F(x(t))

_ j(~) ~ _ ~

Case (ii)

:

Since J(x) u < 0 it follows for small positive

the minimality of x.

respectively.

Note that x(t) < x

This rules out the oossibility

Now we will apply induction principle.

generality that

small positive t,x(t)s~.

can be made as small as

< F(~) < F(a) and consequently x(t) s Y - (a}.

this contradicts

Define x(t) = x + tu.

region for sufficiently

We will assume without loss of

Xl = al where Xl and a I are the first coordinates We now define a new differentiable = ((x2,x3,...,Xn)

and

of case (i).

of x and a

mapping G:~ ÷ R n-I

where

: (al,x 2 ..... Xn) s ~}

and gi(x2,x3,...,Xn) Jacobian matrix of G is plainly a principal P-matrix

.

Further

= fi(al,x2,x3,...,Xn) , i = 2,3 .... ,n submatrix of J(x) and hence it is a

19

gi(x2,x 3 .-.,Xn ) ~ x. i

gi(a2,a3,..-,a n)

> a. --

for

i = 2,3,...,n.

1

Therefore we must have xi : ai for i = 2,3,...,n by induction hypothesis. contradicts x # a.

Hence case (ii) is also impossible.

This

This terminates the proof

of theorem i. For N-matrix we have the following [44].

Theorem 2 :

Let F be a continuously differentiable mapping from 9 ÷ R n (n > 2).

Suppose J(x) is an N-matrix of the first category for each x s 9.

Then F(x) < F(a)

and x > a imply x : a.

Proof :

Proof of this theorem is almost the same as that of theorem i except that

one has to exercise caution for two reasons:

(i) induction will start when n = 2

(ii) Althouth every principal submatrix of an N-matrix is also an N-matrix they need not be of the same category.

In other words we not only have to show that the

Jacobian of the new mapping G to be an N-matrix but also to be of the first category. (i)

We will now Drove theorem 2 when n = 2.

all the partial derivatives are continuous. fll J(x)

As F is continuously differentiable

Consider the Jacobian matrix J(x) : fl

=

21

f22~

As J(x) is of first category, fll < 0, f22 < 0 and f21 > 0, f12 > 0.

As all the

partial derivatives are continuous, they will be of definite sign throughout x s ~. Define G = (f2,fl). G(x) < G(a). (ii)

Jacobian matrix of G is a P-matrix.

Clearly F(x) ~ F(a) iff

Using theorem I, we can complete the argument.

We will now show that the Jacobian of the map G which appears under

case (ii) of theorem I, is an N-matrix of the first kind in order to complete the induction argument of the present theorem. theorem i.

The rest of the proof is same as that of

Define J(x2,x3,...,x n) = principal minor of J(x) got by omitting the

first column and the first row where x = (al,x2,...,xn). Suppose every element of J is non-positive.

Clearly J is an N-matrix.

If flj(Xl,X2,...,xn) > 0 for j=2,3,...,n

then flj(X) > 0 for every x s ~ - this is due to the fact that all partial derivatives should be of definite signs (as already pointed out in (i)). But since Xl = al'xi ~ ai, i = 2,...,n with strict inequality for atleast one i ~ 2, we have fl(x) > fl(a) contradicting F(x) ~ F(a).

Therefore flj (Xl'X2'''"Xn) ~ 0 for some Jo"

Define

a vector v which has zero everywhere except at the Joth coordlnate where it is i. This vector would be a nontrivial solution to J(x)v < 0 which contradicts theorem i

20

of chanter II.

Hence Jmust be s~ N-matrix of the first kind.

This completes the

discussion for theorem 2.

Remark :

Theorem I (as well as 2) can be proved under somewhat weaker restrictions

on the Jacobian matrix.

For example theorem i holds good if for every x, J(x) as

well as its principal submatrices and their transpose matrices considered as matrix games have positive minimax values.

We have already pointed out that every P-matrix

has this proDerty but the converse is not true.

For examole the matrix [~

~] is

not a P-matrix but every game that can be constructed out of this in the manner described as above has oositive value.

In order to Drove the main result due to

Gale Nikaido and Inada we need the following definition.

Definition :

A mapping F:O + R n is a P-function if for any x,y s ~, x W y, there

is an index k = k(x,y) E {l,2,...,n} such that (xk-Yk)(fk(x)-fk(y)) > 0. a P-function is a I-i function

Trivially

[42].

Fundamental Global Univalence Theorem (Gale-Nikaido-lnada)

:

differentiable mapping where ~ is a rectangular region in R n.

Let F:~ ÷ R n be a Then F is globally

univalent on ~ if either one of the following corLditions holds good. (a)

J(x) is a P-matrix

V x c

(b)

J(x) is an N-matrix and the partial derivatives are continuous for all x s ~. In fact, F is a P-function under condition (a).

Proof :

(under condition a) : We assume that J(x) is a P-matrix ¥ x s ~.

show that F is a P-function, using induction principle.

We will

For n = i, result is obvious.

We will assume the result to be true for n-I and show that it is true for n. x,y s ~ be such that x ~ y.

Suppose (xi-Yi)(fi(x)-fi(y)) ~ 0.

that x i ~ Yi for all i = 1,2,...,n.

Let

We will also assume

For if x i = Yi for some i, we can construct

a mapping G similar to that as in theorem I, and then use induction hypothesis to exhibit an index k such that (xk-Yk)(fk(x)-fk(y)) > 0.

Let D be a diagonal matrix

whose i-th diagonal entry is +i or -i according as x i > Yi or x i < Yi"

Let H be

a mapping from D(~) ~ R n defined as follows: H(z) = DoFo D(z) for every z ~ D(~). Plainly D(~) is a rectangular region and the Jaeobian matrix of the mapping H is again a P-matrix.

Let x* = D(x) and y* = D(y).

Since (xi-Yi)(fi(x)-fi(y)) ~ 0

and x i ~ Yi for all i, it follows that H(x*) ~ H(y*) where x* ~ y*, but this contradicts theorem I.

This terminates the proof of the first part.

Proof (under condition b) :

We will now prove univalence when J(x) is an N-matrix.

Suppose F(x) = F(y) and x ~ y.

Let D be a diagonal matrix with i-th diagonal entry

+ i or -i according as x i ~ Yi or x i < Yi"

As in part (a), let H be a mapping from

21

D(~) ÷ R n defined by H = D o F o D.

It is clear that the Jacobian JH of H will be

an N-matrix, the partial derivatives will be continuous and they will have definite signs.

Also throughout, JH will be of the same category.

then H(x*) = H(y*) and x* = D(x) ~ y *

If JH is of first category,

= D(y) imply (from theorem 2) that x* = y* or

x = y which contradicts ~ u r assumption x ~ y. member of JH will be strictly negative.

If JH is of second category then every

If x* ~ y*, then H(x*) ~ H(y*), with the

inequality strict in some coordinate, contradicting the fact that H(x*) = H(y*). Hence as before x* = y* or x = y contradicting x ~ y.

This finishes the proof of

part (b) and the theorem.

Remark I : an N-matrix.

Continuity of the partial derivatives is crucially used when J(x) is It is not clear whether the continuity of the partial derivatives can

be omitted from the theorem.

Remark 2 :

We have assumed ~ to be a rectangular region in theorem 1,2 and the

fundamental theorem.

How far can one relax this assumption?

assume O to be a convex region?

Will it suffice if we

The following theorem gives a partial answer, when

the Jacobian is everywhere a positive quasidefinite matrix - in this case proof is also extremely simple which uses mean value theorem of differential calculus of a single variable.

Theorem 3 :

Let ~ C R

n be a convex set and F:~ ÷ R n be a differentiable map, with

its Jacobian everywhere positive (or negative) quasidefinite in ~.

Then F is

univalent.

Proof :

We will prove only for positive quasi-definite case as the proof for negative

quasi-definite case is similar. that F(a) ~ F(b). assumption.

Suppose a,b s ~ with a ~ b.

Let x(t) = t a +

By convexity, x(t) s D for t E [O,i].

f.(b))(l > t > 0). i

We will directly show

(l-t)b = b+ t(a-b) = b+ th where h = a-b ~ 0 by Let @(t) = ~i hi(fi (x(t)) -

Differentiating with respect to t, we have @'(t)= Z ~h.h.f..(x(t)) j i 1 J lJ

which is positive everywhere by positive quasidefiniteness of the Jacobian. ~(0) = O, ~(i) ¢ O.

Since

That is 2hi(fi(a)-fi(b)) # 0 or f.(a)1 ~ fi (b) for some i.

Consequently F(a) ~ F(b).

This proves F is univalent.

As an application we have the following corollary.

Corollary (Noshira, 1934) :

Let g(z) be an analytic (complex) function of a complex

variable z in a convex region ~ of the complex plane. has positive real part in ~.

Proof :

Then g(z) is univalent in

Suppose its derivative g'(z) ~.

Let g(z) = u(x,y) + i v(x,y) where i2 = - I and z = x+ iy.

Then g'(z) =

22 ~u

~v

~x+ i ~ .

Note~1~ > 0.

Now, consider the mapping F : ~ ÷ R 2 defined by

F(x,y) = (u(x,y), v(x,y)).

Its Jacobian is given by

J=

?v

?v

However from the Cauchy-Riemann equations we have ~u ~x

~v ~u ~y ' ~y

~v ~x

Hence

JJ2

xu°I

is a P-matrix for

~U

~--> 0. 4x

This implies that J is positive quasidefinite and from theorem 3 one can conclude that F is univalent.

This terminates the proof of the corollary.

The following example shows that theorem 3 may not hold good in a non-convex region.

Let g(z) = z + ~ (z # 0). Z

Let ~ be the common exterior portion of two

circles of radius @ having their centres at ½ and - ½

respectively.

One can

verify that the Real part of g'(z) is positive in ~, as

Re(g'(z))

= ([Z-½12-¼)(Iz~212-¼)

+ (Ira(z)) 2

lzr 4 However g ( i )

= g(-i)

simply connected. subregion

= 0 and i, In fact

of ~ containing

-i

~ ~.

T h e o r e m 3 may f a i l

i n t h e a b o v e e x a m p l e we c a n t a k e i and -i where theorem 3 will

even if

the region

is

any simply connected

fail.

In the next chapter we will prove univalence results which place analytical conditions on the map F.

Also we will prove an important result of More and

P~heinboldt which uses results from degree theory.

Univalent results in R 3 : We will present two univalent results for differentiable maps in R 3.

One result is valid for rectangular regions and the other for any

convex region in R 3 [53]. We will start with a simple example. g(x,y,z) = x 2 y 2 and h(x,y,z) = y2+ z 2.

2 2 Let F = (f,g,h) where f(x,y,z) = x + z , Let ~ be any convex region in the strictly

positive orthant which includes points of the form (4, ½, z). J =

Ix°:I 2x

2y

m 0

2y

2z~

and

J+ 2J'

-

In this example,

[i x 2y y

2

23 Observe that J is a P-matrix but J+ J'

is not a P-matrix since the 2 x 2 leading

minor is (equal to 4xy-x 2) is negative at x = 4,y=½.Thus the Jacobian matrix is not positive-quasidefiniteandit is not difficult to check that the map F is one-one in any convex region in the strictly positive orthant.

In the example note that f

does not depend on y, g does not depend on z and h does not depend on x. Theorem 4 : Let F = (f,g,h) be a differentiable map from a convex region D c R 3 to R 3. z.

Suppose f does not depend on x,g does not depend on y and h does not depend on Further suppose the partial derivatives fy, fz' gx' gz' hx and hy are positive

throughout D.

Then F is univalent over ~.

Proof : Jacobian matrix J of F can be written as :

I O J =

Suppose F(a) = F(b) with a W b.

fy

gx

0

hx

hy

Let ~ =

(al,a2,a3) and b = (bl,b2,b3).

discuss essentially two cases (i) a i ~ b i and

a3 ~ b 3.

f~i1

for

We need to

i = 1,2,3 and (ii) a I ~ bl, a 2 ~ b 2

(Other cases can be reduced to one of these two cases).

Case (i) : a i ~ b i

for i = 1,2,3.

independent of the first variable.

Note that f(al,a2,a 3) = f(bl,a2,a 3) since f is We will assume without loss of generality a 2 ~ b 2.

Then, since fy ~ 0 and fz ~ 0, f(al,a2,a 3) = f(bl,a2,a 3) ~ f(bl,b 2,a 3)_>f(bl,b2,b3). In other words f(al,a2,a3) ~ f(bl,b2,b3) which contradicts our assumption that

F(a)

=

F(b).

Case (ii) : a I ~ bl, a 2 ~ b 2 a2 ~ b2, a 3 ~ b 3. matrix D =

i _

and

a 3 ~ b 3.

First we will prove when a I ~ bl,

As in the proof of the fundamental theorem define a 3 x 3 diagonal and the map H:D(~) ÷ R 3 when H(u) = D o F o D(u).

It is

0

not difficult to check that the first row of the Jacobian matrix associated with H will have the following property.

First entry will be identically zero while the

second and third entries will be negative.

Let D(a) = a* and D(b) = b*.

Then

H(a*) = H(b*) with a* ~ b* and as in case (i) we can argue to show that hl(a*) ~hi(b*) (Here h I is the first component function of H) which leads to a contradiction. In case a I ~ bl, a 2 > b2, a 3 = b 3 then clearly f(al,a2,a 3) = f(bl,a2,a 3) > f(bl,b2,a 3) = f(bl,b2,b 3) which again contradicts our assumption that F(a) = F(b). This terminates the proof of theorem 4.

24

Remark I :

One interesting feature of this result as well as the next is that we

have not explicitly assumed that the Jacobian is non-vanishing.

Automatically this

is satisfied because of our other conditions.

Remark 2 :

We do not know whether a similar result is true in R n for n > 3.

At the

present we do not have a counter example in R 4. We are ready to state our next theorem [53].

Theorem 5 :

Let F be a differentiable map from a rectangular region 2 C R

3 to R 3

with its Jacobian J having the following two properties for every x E 2 :

(i)

diagonal entries are negative and off-diagonal entries are positive.

(ii)

Every principal minor of order 2 x 2 is negative.

Then F is univalent in the

rectangular region. Before proving this result we would like to make the following observations. First note that the Jacobian is not an N-matrix.

It can be seen as follows.

Signs

of the entries of J can be written out explicitly because of condition (i).

Y ~ ÷ In other words -J is of Leontief type matrix.

Also one can easily check by expanding

through first row that determinant J > O - this is a consequence of condition ( i ) and (ii).

Second, we need to prove theorem i in this situation.

crucially depends on theorem i of chapter II.

Lemma i : (i) (ii)

Proof of theorem I

We first prove the following.

A be a 3 x 3 matrix with the following two properties.

diagonal entries are negative and off-diagonal entries are positive. Principal minors of order 2 x 2 are negative. Ax < 0

Proof :

and

x > 0

Suppose A x ~

A

=

0,

x ~ 0

and

all

a12

a13

a21

a22

a23

~31

a32

a33

I

which is clearly impossible

So we will assume x. > 0 l

Then the system of inequalities

has only the trivial solution

since

x # 0.

Let

x = 0.

x = (Xl,X2,X3). a11

i

I

If x 3 = 0 then L a21

I

ll

a12|

21

a22~

for i = 1,2,3.

!

We have,

is an N-matrix of the first kind.

25

allXl

+

a 12x2

+ a 13x3

a21x I +

a22x2

+

< 0 --

a23x 3
2.

Then clearly 3D is a connected set.

differentiable with positive Jacobian throughout ~.

We will also assume ~ C

Rn

Also suppose F is continuously In view of Kestelmants result,

in order F to be one-one it is enough if we check that F is one-one on the boundary 80.

This naturally raises the following important question:

Is F univalent if we

simply assume that the Jacobian is a P-matrix for only those x which belong to the

27

boundary of D ? Scarf has conjectured that this question will have an affirmative answer.

Indeed this conjecture has been verified recently by three sets of

researchers: (i) C.Garcia and W.Zangwill (ii) G.Chichilnisky, M,Hirsch and H.Scarf and (iii) A, Mas-Colell.

This will form the subject matter of a subsequent chapter.

Garcia-Zangwill's proof of this conjecture depends on norm-coerciveness theorem while the proof of Mas-Colell depends on the Poincare index theorem of differential topology.

It is not clear how one can use Kestelman's result directly to give an

alternative proof of Scarf's conjecture. We pose another related question: "Is F univalent when the Jacobian is an N-matrix for every x s ~ ? answer in R n for n ~ 3.

".

This is certainly true in R 2 but we do not know the

CHAPTER

IV

GLOBAL HOMEONORPHISMS BETWEEN FINITE DIMENSIONAL SPACES

Abstract :

It is well-known from covering space theory that global homeomorphism

problem can be reduced to finding conditions for a local homeomorphism to satisfy the line lifting property.

We will show that this property is equivalent to a

limiting condition (which in many cases easy to verify) which we call by L.

We will

use this condition L to derive several results on global homeomorphisms due to Roy Plastock.

We will prove an approximation theorem due to Nere and Rheinboldt and

this result will then be used to prove Gale-Nikaido's theorem under weaker assumptions. In the last section we will prove a result

due to

NcAuley for light open mappings.

We will end this chapter with an old conjecture of Whyburn.

Line lifting property equivalent to condition (L) : We will start with a few definitions.

Let F be a map from R n to R n,

Definition : A continuous mad F is said to be proper if F-I(K) is compact whenever K is compact.

Definition :

Let ~ R

n be open and connected.

if for each line L(t) = (l-t)y I + x

Then F:q ÷ R n lifts lines in F(~)

ty 2 (0 < t < i) in F(~) and for every point

s F-l(yl ) there is a path P (t) such that P (0) = x

Remark : PJt)

and F(P (t)) = L(t).

If F is a local homeomorphism and F lifts lines in F(~), then the path

in the above definition is unique for each ~. Let ~ be open and connected in R n.

Let F:~ ÷ R n be continuous.

We now

introduce the condition (L).

Condition (L) :

Whenever P(t), 0 < t < b, is a path satisfying F(P(t)) = L(t) for

0 ~ t < b (where L(t) = (l-t)Yl+ ty 2 is any line in Rn), then there is a sequence t i ÷ b as i ~ ~ such that

limit P(t i) exists and is in ~. i÷~

We need the following well-known results for the proof of Plastook's theorem

[55]. Lemma :

Let X and Y be connected, locally pathwise connected spaces where X and

YC

Furthermore let Y be simply connected.

R n.

Then F is a homeomorphism of X

onto Y if and only if F is a covering map of X onto Y.

2g

For definitions regarding simply connected regions see [33]~

Theorem (Hermann) :

Let O C R n be open and connected, F:~ + R n.

In order that

F is a covering map of ~ onto F(O) it is necessary and sufficient that (i) F is a local homeomorphism and (ii) F lifts lines in F(D). For a proof of this theorem see Hermann [28] or Plastoek (p. 170-171,

[55]).

We now have the following result due to Plastock.

Plastook's Theorem :

Let F : ~ C R

n ÷ Rn

be a local homeomorphism.

Then condition

(L) is both necessary and sufficient for F to be a homeomorphism of D onto R n.

Proof :

In view of Hermann's theorem, to prove the sufficiency, it is enough if

we show that F lifts lines in F(D).

Let L(t) be any line in F(D) with L(O) = y and

let x E F-l(y). We can find an s > 0 and a oath P(t)(=F-I(L(t))), F(P(t)) = L(t) for 0 < t < E.

Let c ( < l )

0 < t < s with P(O) = x and

be the largest number for which P(t) can

be extented to a continuous oath for 0 < t < c and satisfyin~ F(P(t)) = L(t), 0 ~ t < c.

Let z =

limit P(t i) and observe that this limit exist for the map F t. ÷ e 1

satisfies condition (L).

By continuity, F(z) = L(c).

on which F is a homeomorphism.

Let U be a nei~hbourhood of z

There exists N o such that P(t i) c U for i ~ N o,

Also there exists a g > 0 and a oath Q(t) defined for c-g 0 for all x s ~ and K ~ l ( x ) . Garcia and Zangwill isK prove univalence when F satisfies the S property. This S property permits certain principal minors to be negatiw~ on the boundary.

We will now give an example of a

one-one function F whose Jacob~Lan is not everywhere a P-matrix. function defined as follows: f2(x,y) = x+y.

Let F:R 2 ÷ R 2 be a

F = (fl,f2), where fl(x,y) = x 3 / 3 + x ( y 2 - ½ ) - y and

Then

I

x2+ y2 ½

J (x,y) =

L

2xy_l I Plainly J(x,y)is not a P-matrix.

1

Also det ~x,y)=(x-y)2+ ½ > 0 V (x,y) E R 2.

1 Nowever J(x,y)is a P-matrix if x 2 y 2 > ½.

Consequently F satisfies the conditions of theorem 2 and hence F is one-one throughout R 2 .

Mas-Colell's univalent result :

Theorem 3 :

We will now prove the following theorem of Mas-Colell.

Let F:~ ÷ R n be a continuously differentiable function where ~ is a

compact convex polyhedron of full dimension.

For every nonempty subspace

let ~L: Rn + L denote the perpendicular projection map.

If for every x s

L~R

n

~ and

subspace L C R n spanned by a face of K (that is, the translation to the origin

of

the minimal affine space containing K)which includes x, the map H L DF(x) : L ÷ L has a positive determinant

(that is the linear map H L- DF(x) preserves orientation

where DF(x) stands for the derivative map of F at x) then F is one-one on ~ and consequently a homeomorphism. Several remarks will be in order now. convexDo~hedral set. theorem on univalence

Here ~ is assumed to he simply a compact

As such this theorem includes Gale-Nikaido's fundamental mappings.

Conditions imposed on this theorem are coordinate

free, in the sense that their formulation does not rely on a previous choosing of coordinates.

Proof of this theorem will depend on the following known results from

45

degree theory.

Theorem (a) :

Let F,G:C C

of an open bounded set C.

R n ÷ R n be two continuous maps whereC is the closure Suppose there exists a homotopy H:SC x [0,I] such that

N(x,O) = F(x), H(x,l) = G(x) for all x s 8C.

If y s R n such that H(x,t) ~ y for all

x E 8C, t ~ [0,i], then deg(F,C,y) = deg (O,C,y). For a proof see Schwartz 1964, p. 93 Non-linear functional analysis, Lecture notes Courant Inst. of Math. Sei., New York.

Theorem (b) :

Let F:D ~ R n + R n

be continuously differentiable on the open set D

and C an open bounded set such that C ~

D.

Given a coordinates system define

A = {x s C, Jacobian at x is singular}. If y ~ F(SC U A ) is empty and deg(F,C,y)

=

then either F={xsclF(x)=y}

0 or F consists of finitely many points x I , . . . , x m and

m

deg(F,C,y) =

~ sign det J(xJ). (For a proof see p. 159, [48]). j=l

As in theorem 1, it suffices to prove that F is one-one restricted to n ° = interior of ~; otherwise we can always enlarge the domain of F which will contain ~ in its interior and similar to ~ and satisfying all the other conditions of theorem 3.

For every x s R n let s(x) c 9 be the foot of x, that is, s(x) is the

unique element of minimizing

l lx-sll

for s s ~.

Plainly s(x) = x for x ~ ~.

We now

extend F:~ ÷ R n to the whole of R n by letting a function F:R n ÷ R n be defined as F(x) = F(s(x)) + x-s(x).

For any y s F(~) define Fy(X) = F(x)-y.

We will now state

and prove two lemmas that are needed for a proof of theorem 3.

{x:llxll =

Lemma i :

Let S r =

r) and B r = {x:Ilxll < r} be the sphere and ball of

radius r.

Then for any y s F(Q), and r sufficiently large, Fy restricted to S r has

degree one, that is, it is homotopic to the identity in S

r

with respect to R N

R n ' ~ {0}.

Proof :

Clearly the homotopy bridge H(x,t) is given by H(x,t) = t Fy(X)+ (l-t)l(x)

where t E [O,i], l(x) = x.

We have to only check that H(x,t) ~ 0 for any t s [O,i]

and x s Sr, if r is sufficiently large.

We will verify this by simply showing that

X.Fy(X) > 0 for any y E F(~) and x s S r when r is sufficiently large. choose any r >

max l lF(z)-z-Yll = s. zs~,ysF (~)

In fact

Then

x.~ (x) = l lxl 12 - x. (s(x)+ y-F(s(x))) Y

> llxll 2 -

ITxll

This finishes the proof of lemma i.

IIs(x)+ y-F(s(x)ll ~

r 2-rs

> O.

46

Lemma 2 :

Let K~be a polyhedron and F satisfy the hypothesis of theorem 3.

A = (x e Rn:F

is not continuously differentiable at x).

Let

Then if x ~ A, ID#(x) I

(= determinant of the linear map DF(x)) is positive.

Proof :

Let x ~ A.

Then x-s(x) is perpendicular to a single face of ~, which, of

course, includes s(x).Let L be the subspace spanned by this face and L ± orthogonal to L.

the subspace

For small v s L,s(x+ v) = s(x)+ v and so F(x+ v) = F(s(x)+v)+x+v-s(x);

hence DF(x)v = DF(s(x))v. Consequently DF (x)v ~ v.

For v s L ~

, s(x+v) = s(x) and so, F(x+v)=F(s(x))+x+v-s(x).

Choose an orthogonal coordinate system whose k first

coordinates generate L, J(x), the matrix of DF with respect to this coordinate system, takes the form

where JkF(S(X)) are the first k columns of J(x) (= Jacobian matrix for F).

Therefore

IDF(x) I = IJ(x) I = IJkk F(s(x)) I where Jkk F(s(x)) are the first k rows of JkF(S(X)). However Jkk~(S(X)) is the matrix of HL.DF(s(x)):L ÷ L and by hypothesis

IJkkF(S(X))I>0

and this terminates the proof of the lemma. We are now set to prove theorem 3.

Proof of theorem 3 "

Let A be the set (as defined in lemma 2) of those points x s R n

at which F is not continuously differentiable.

This set contains no open set.

This

is a consequence of the fact that s(x) is continuously differentiable except at those x which are contained in a finite number of hyperplanes.

Since F is lipschitzian,

F(A) contains no open set. Choose r > 0, sufficiently large so that lemma i holds good. Fvl(O) ~

A = @.

Also suppose

In other words we are assuming Fvl(0) lies entirely in the region

of differentiability.

From theorem (a) and theorem (b) it follows that

deg(Fy,Br,0)

~

deg(l, Br, 0)

=

Z sign det J F (x) = I. Y

Here the summation is extended over those x which belong to B r with Fy(X) = 0. Note that F-I(0) • B r ~ @ for any y s F(~). Also note that we are writing Y [As such we will assume for this proof a coordinate system is given to

det JFy(X). us].

Invoking lemma 2, we infer F-I(0) g~ B is a singleton set. y r As remarked earlier we will only show that F restricted to interior ~ = ~o is

one-one.

Suppose there are Xl,X 2 s ~o with x I ~ x 2 and F(x I) = F(x2).

disjoint open sets U I ~ Xl, U 2 ~ x 2

with F(U I) g~ F(U 2) ~ @ open.

tains no open set there is a y s F(U I) ~ F(U 2) such that y ~ #(A).

One can find

Since F(A) conConsequently

47

FyI(O)~A

= ¢.

Note that F-I(Y) C Fyl(0) t'~ B r and the latter set is not a

singleton set as F-l(y) has at least two elements.

This contradiction establishes

that F restricted to ~o must be one-one and this terminates the proof of theorem 3. Mas-Colell's proof of theorem 3 makes use of Poincare-Y~pf index theorem [see pages 35-41 John Milnor's Topology from the differentiable view point (1965), the U~iversity Press Virginia].

Instead we use theorem (a) and theorem (b).

We

now have the following theorem due to Nas-Colell.

Theorem 4 :

Let ~ C R n be a compact, convex set of full dimension with smooth

boundary $~ (= C I boundary) and F : ~ ÷ R n For each x s ~ ,

let T

x

a continuously differentiable function.

denote the tangent plane at x.

If the Jacobian J(x) has a

positive determinant at each x ~ ~ and if for each x E $~, J(x) is positive quasi-definite on Tx (that is is positive for every v s Tx with v # 0), then F is one one (and consequently a homeomorphism).

Proof :

We will first show that if x E ~

has a positive determinant. 3.

and L C Tx is a subspace then H L . D F ( x ) : L ÷ L

In other words this theorem is a consequence of theorem

Assume that we are given an orthogonal coordinate system such that the first

k coordinates generate L and the n-th is perpendicular to Tx and let J(x) denote the Jacobian matrix.

Then Jn_l,n_l(X), the matrix obtained by omitting the n-th

row and n-th column (by hypothesis) is positive quasidefinite. matrix is a P-matrix [see examples of P-matrines in chapter II]. Jkk(X) the matrix of HL.DF(x) determinant.

: L ÷ L

Fowever, any such This applies to

and yields the fact that Jkk(X) has a positive

Now one can complete the proof by approximatin~ ~ by a polyhedran ~'

and the above fact implies that the hypotheses of theorem 3 are satisfied for ~'. Hence F is i-I on ~.

Remark I :

This completes the proof of theorem 4.

Will theorem 4 be true if one assumes that J(x) is weakly positive

quasi-definite on T ? It is easy to see that the map F will be one-one on ~o = x interior of ~, by noting that the maps G = F(x) + s l(x) are one-one over ~ where s > 0 and l(x) = x. boundary.

Also the map G will be one-one throughout ~ including the s It is not clear whether F is one-one throughout ~. In other words this

raises the following question:

Let ~ be a closed convex region with nonempty

interior and let F:~ + R n be a one-one map throughout ~o = (interior of ~). other conditions should be imposed on ~ ~.

What

and F so that F will be one-one throughout

[see Kestelman's result given in chapter III].

In fact what we want is an

approximation theorem similar to theorem 4 in chapter IV which will hold good for closed region ~ (with nonempty interior).

In particular we would like to know

whether theorem 4 holds good when J(x) is weakly positive quasidefinite on Tx. Remark 2 : Nas-Colell's conditions on J(x) at the boundary of ~ neither imply nor

48

are implied by the fact J(x) is a P-matrix. One can give a slight generalization of theorem 2.

For that we need the

following

Definition :

Let A be an

n x n

matrix.

Call A a weak P-matrix if det A > 0 and

every other principal minor is non-negative. Theorem 2' [53] : rectangle in R n.

Let F:R n ÷ R n be continuously differentiable.

suppose J(x) is a weak P-matrix for all x s R n \ ~ . Proof :

Let ~ be a bounded

Suppose that the determinant J(x) > 0 for every x E R n and further Then F is one-one.

We will assume without loss of generality that D is a compact rectangle.

Let G (x) = F(x) + sl(x) when l(x) = x. every s > 0.

From theorem 2, each GE(x) is one-one for

Note that GE(x) ÷ F(x) as E + 0.

Hence we can conclude from More-

Rheinboldt's result that F(x) is one-one in R n. This terminates the proof of theorem 2' In fact theorem 2 can be further generalized as follows.

We allow determinant

J(x) to vanish on a set S of isolated points in R n. Theorem 2"[53]

:

Let n # 2 and let F:R n ÷ R n be continuously differentiable.

be a bounded rectangle in Rn.

Suppose that the determinant J(x) is positive for

all x except on a set S of isolated points where it vanishes.

Further suppose that

every principal minor of J(x) is non-negative for all x s R n \ ~ . Remark : the proof.

Let

Then F is one-one.

We will not attempt to prow~ this result but indicate the main steps in Using (Lemma 2, pp. 244-245 in [20]) we can construct a function G having

the following property (i) det G > 0 for every x except on a set of isolated points and (ii) G will be norm-coercive.

Now invoking a result of Chua and Lam (theorem

2.1, pp. 602-208 in [9]), we can conclude that G is one-one and consequently F is one-one.

Since Chua-Lam's theorem is valid only for n ~ 2, we have to impose this

condition in our theorem 2".

For n = 2, Chua-Lam's theorem is not true but we do

not know whether theorem 2" is true or false when n = 2.

It is worthwhile to look

at the following example which serves as a counter example in R 2 to theorem 2.1 in [9]. Let F=(f,g) where f(x,y) = x2-y 2 and g(x,y) = 2xy. Then the Jacobian is given by J = [ ~

-2Y2x] and det J = 4 ( x 2 y 2) which vanishes only when x = y = 0.

However F is not univalent since F(I,I) = F(-I,-I). But the diagonal entries of J do change sign and as such this will not be a counter example to theorem 2". We close this chapter by mentioning once again the following open problem: Can continuity of the derivatives in Mas-Colell's result be dispensed with or alternatively - are there counter examples ?

CFAPTER

VI

GLOBAL ~IVALENT RESULTS IN R 2 AND R 3

Abstract :

In this chapter we prove global univalent results obtained by Gale

Nikaido and Schramm when F is a map in R 2 .

Some extensions are indicated in R 3 .

Howeve~ the following two interesting open problems posed by Gale-Nikaido remain unanswered : (i) Suppose F is a differentiable map from a rectangular region ~ C R 3 to R 3.

Suppose the Jacobian is non-vanishing and every entry in the Jacobian is

non-negative.

Is F one-one?

gular region ~ C R

3 to R 3.

(ii) Suppose F is a differentiable map from a rectanSuppose every principal minor of the Jacobian is

non-vanishing for every x s ~.

Is F one-one ?

Gale and Nikaido have shown that

both (i) and (ii) together imply that F is one-one in any rectangular region in R 3. We have shown that (i) together with the assumption that the diagonal entries are identically zero will imply that F is one-one in any open convex region in R3-this result supplements the result obtained by Gale and Nikaido.

We can weaken our

assumptions in rectangular regions in R 3 using Garcia-Zangwill's result given in the previous chapter.

Univalent mappings in R 2 :

There are four results which we oresent in this section

which are due to Gale, Nikaido and Schramm.

The results are rather fragmentary but

are presented, for they suggest possible generalizations - in certain directions. In the first two results we relax conditions on the Jacobian matrix while in the third ~ is Assumed to be a region bounded by an x-curve. Let F:~ ÷ R 2 be a mapping given by F(x,y) = (f(x,y), g(x,y)), is a region in R 2.

(x,y) s ~ where

The first result concerns itself with one-signed principal

minors [19].

Theorem i : (a)

Let ~ be an arbitrary rectangular region either closed or not closed.

Suppose F has continuous partial derivatives and suppose none of the principal minors of the Jacobian vanishes. (b)

Then F is univalent.

Let ~ be an open rectangular region.

continuous.

Suppose the Jacobian does not vanish and no diagonal entries of the

Jacobian matrix change signs.

Proof (a) :

Then F is univalent.

Since F has continuous partial derivatives, every principal minor will

keep the same sign throughout ~. negative. f,

Let the partial derivatives of F be

That is, they will be everywhere positive or

We will assume without loss of generality that the diagonal entries

gy of the Jacobian matrix to be positive.

In case the Jacobian is also positive

50

then from Gale-Nikaido's fundamental theorem, F is univalent. is negative.

This means fx gy < fy gx"

be of the same sign.

Suppose the Jacobian

Since fx,gy > 0 it follows fy and gx should

In case fy and gx > 0 consider the map F(x,y) = (g(x,y),f(x,y)).

Then the Jacobian of F is a P-matrix and F is univalent in other words F is univalent. In case fy < 0, gx < 0 consider the map C(x,y) = (-f(x,y), -g(x,y)). Jaeobian of G is an N-matrix of the first kind. hence) F is univalent.

Then the

In this case also (G is univalent and

Similar proofs can be given for (b) also.

This terminates

the proof of theorem i. In the next theorem we will assume one-signedness of the entries in some row of the Jacobian matrix to assert univalence.

Theorem 2 (a) :

Let ~ be an arbitrary rectangular region.

partial derivatives and the Jacobian never vanishes.

Suppose F has continuous

If there are real numbers a

and b such that afx + bg x and afy + bgy do not change sign and one of them does not vanish, then F is univalent. (b)

~

Let ~ be an open rectangular region and let F be differentiable.

Jaeobian is everywhere either positive or negative.

Suppose the

If there are real numbers a

and b not both of them zero such that af x + bgx and afy + bgy do not change sign in ~, then F is univalent.

Proof :

(a)

We will assume a ~ 0 for by hypothesis one of the functions afx + bgx,

afy+ bgy does not vanish.

Observe that the mapping F(x,y) = (a f(x,y)+ bg(x,y),

g(x,y)) is univalent if and only if the original mapping F is univalent.

Jacobian

of F is given by

=

I afx + bgx

afy + bgy1

Jr

!

L

gx

gy

Then det J~ = a det JF" Since afx + bg x and afy + bgy do not change sign and one of them does not vanish and since the partial derivatives are continuous, we will assume afx+bg x ~ 0 and afy + bgy > 0.

In view of this analysis we may finally assume that the original

mapping F has the following properties f ~ 0 and f ~ 0. x y-Now suppose that F is not univalent, so that F(p,q) = F(r,s) = (a,B) for some distinct points (p,q), (r,s) s ~. will imply

Clearly q ~ s; for if q = s then p ~ r and this

fx = 0 for some point contradicting our assumption fx ~ 0 "

fixed y satisfying q ~ y ~ s [We assume q ~ s]. = f(r,s) ~ f(r,y). f(x,y) = ~.

Choose any

Since fy -~- 0, a = f(p,q) ~ f(p,y),

By the continuity of f we have an x between p and r such that

Since fx ~ 0, for each y in[q,s] there will exist a unique x = ¢(y) such

51

that f(¢(y),y) .= ~.

Since f(x,y) has continuous derivatives, ¢ is also continuously

differentiable, its derivative is given by ¢'(y) = -fy(¢(y),y)/fx(¢(y),y). G(y) = g(¢(y),y).

Define

Then G is continuously differentiable and its derivative is

given by, G'(y) = IJl/fx evaluated for x = ¢(y) where IJl denotes the determinant of the Jacobian.

Since IJI/fx # 0

, G is strictly monotonic in [q,s].

contradicts the fact that G(q) = G(s) = B.

This

Therefore F must be univalent.

This

terminates the proof of part (a) of theorem 2.

Proof (b) :

This can be reduced to case (a) as follows.

In any case we can assume

f > 0 and f > 0 throughout ~ (which is assumed to be an open rectangle). x -y -Suppose F(p,q) = F(r,s) for two distinct points (p,q), (r,s) belonging to ~.

Since

is an open set we can find real numbers ~i,Bi (i = 1,2) and an open set U containing (p,q) and (r,s) such that U = {(x,y) : ~ i ( x < ~i

and

~2 < y < B2}

Since a I < p, r < BI and ~2 < q,s < B2 for a suitably chosen small positive number s we can find an open subset ~

of U containing (p,q),

~i < x < BI, ~2 - S~l 0 and fy > O. h

> 0 and h > 0 for V(x,y) s A. Clearly the mapping H satisfies all the conditions x y -stipulated in part (a) of theorem 2 and hence H is univalent in A. But A contains

two distinct points

(p,q-sp) and (r,s-sr) which are mapped to the same point under H

and hence we arrive at a contradiction.

Therefore F is one one in ~ and this

terminates the proof of theorem 2.

Remark :

It is not clear how to formulate theorem 2 in higher dimensions

(when n > 2).

To state the next theorem we need the following [68].

Definition :

An x-simple domain in the (x,y)-plane is the interior domain determ-

ined by an x-simple curve, which is a Jordan curve cut in two points at most by every line parallel to the x-axis, except possibly by lines passing through points on the curve which have extremal values of y. y-simple domain are defined.

Analogously y-simple curves and

52

4 Observe that an x-simple curve can be put in the form U ~. such that for i i suitable mappings si:[c,d] ÷ R with c < d and s I < s 2 in (c,d) we have ~l={(sl(y),y): c < y < d},~ 2 = ~x,c):sl(c) < x < s 2 ( c ) ) ~ 3 = ~4 = {(x'd):Sl(d) < x < s2(d)}.

(s2(y),y) : c < y < d},

The following definition helps us to identify one

of the two subarcs seoarated by two points on a Jordan curve.

Definition : ~\{P'Q}

Let P ~ Q be two points on a Jordan curve ~ and ~i,~2 the subarcs with

= ~I ~

~2"

When ~ is a subset of X i for i = i or i = 2 define (P,Q,X)=~i;

when U is a proper subarc of ~ which contains

Definition :

A square matrix over the reals is called an NVL matrix when its leading

minors do not vanish.

Theorem 3 :

We are now ready to state and orove the following [68].

Let ~ be an x-simple domain in the (x,y)-olane,~ its boundary.

F = (f,g) : ~ ÷ R 2 ~.

Xi for i = I or i = 2 define (D'P'Q)=Xi"

Let

be a differentiable map,~ the minimum and B the maximum of f on

Suppose the Jacobian of F is an NVL matrix for each z s ~ and for each u s (~,~),

suppose at most two points z s Z satisfy f(z) = u. (a)

the sign of

(b)

defining A(u) = (z:z E ~ and f(z) = u}, both A(~) and A(B) are subarcs of ~ and

(c)

F restricted to

Remarks :

f x

Then

in ~ is constant

~\(A(a) UA(B))

is univalent.

When conditions on univalence are arranged in the partial order of

stringency of either the local conditions or the boundary conditions the above result will occupy an intermediate position - when we compare it with the GaleNikaido's fundamental theorem and Kestelman's result.

Clearly every P-function is

an NVL function and NVL functions have invertible derivatives.

Contrary to a

conjecture of Samuelson (see Gale-Nikaido's example) NVL functions are not unconditionally injective. Also note that theorem 3 asserts the univalence of F in the whole of ~, which is not necessarily a rectangular region and which need not be convex.

We will now

prove theorem 3.

Proof (a) :

Let c,d,Sl,S2,~l,~2,~ 3 and ~4 have the same meaning as defined above.

By the Darboux property sign fx(.,y) is a function X ef y only, for c < y < d.

We

will now show that the set {y:y c (c,d) and X(y) = i} is either empty or open in (c,d).

Let Yo be an element with X(y o) = i.

This means fx(.,yo ) > 0 where yoS(C,d).

Since s 2 > Sl, we have f(s2(Yo),y o) > f(sl(Yo),Yo).

As f is continuous for ly-yol

small enough f(s2(y),y) > f(sl(y),y) and hence fx(.,y) > 0.

This proves

(a) of

53

theorem 3-

Without loss of generality,

assume f

to be positive in ~.

Consequently

X

f(.,c) and f(.,d) are non-decreasing. We will now prove

(b). From (a) we have A(~) ~

meets Z3 at the end points at most.

Z, A(~) meets ZI and A(~)

Anologous assertions hold good for A(B).

To

show that A(~) is an arc, note first that f restricted to Z has exactly two solutions to f(z) = u when ~ < u < ~ since f-u changes sign on Z.

[In other words if f(zl)= ~,

f(z 2) = ~ where Zl,Z 2 are on ~ and since one can travel in two different directions from z I to z2, f(z) = u has at least two solutions on ~. f(z) = u can have at most two solutions]. z2sA(~) the arc ~ = (£i ~

However by hypothesis,

Now we shall orove that whenever ZlSA(~) ,

~2 ~ ~4,Zl,Z2) belongs to A(~).

Let 0 < s < ~ - ~ and let

~s,~ a be the subarcs of Z on which f < ~ + s, f ~ ~ + c, respectively. zI s ~ ' z2 s X similarly.

and ~ C

This proves

We will now prove

~ •

Since s is arbitrary f(~) = ~ .

A(~) is treated

(b) of theorem 3. (c). Since the given region is bounded by a Jordan curve and

since fX is assumed to be positive to prove the following:

Therefore

(without loss of generality)

in 2, it is sufficient

(i) For ~ < u < ~ we have A(u)={(h(u,y),y):y.(u) 0.

Th~s shows that a(u,.) is

This terminates the proof of theorem 3.

Theorem 3 does not ensure the univalence of F on ~

simple example shows.

Let f(x,y) = x(l-y) and g(x,y) = y and

The mapping F = (f,g) transforms

~

the hypothesis of theorem 3. f(x,y) = x2y and g(x,y) = xy 2.

Let

~ = [1,2] x [0,i].

into the triangle with vertices

(0,i) the upper side of ~ shrinking into (0,I). the rectangle fx = (l-y) = IJI > 0.

as the following

(I,0), (2,0),

Also note that in the interior of

We will now give an example which will satisfy ~ = [0,i] x [0,i] with ~ interior of

~.

Let

Clearly the map F = (f,g) transforms two adjoining

sides of [0,i] x [0,I] into (0,0) and satisfies the conditions of theorem 3.

The

54

following corollary is helpful in drawing conclusions about the behaviour of F on ~.

Corollary

:

Suppose F satisfies the set of conditions of theorem 3 from which the

following condition is deleted: satisfy f(z) = u".

"For each u s (~,B) let two points z s £

Suppose F restricted to ~ is not one-one.

at most

Then for all u in

some subinterval of (~,B) the equation f(restricted to £) = u has more than two solutions. Proof of the corollary follows from theorem 3.

Remark :

To illuminate the corollary,

and Nikaido.

Let F(x,y) = (e2X-y23,

let us again look at the example due to C~le 4e2Xy-y 3) over

~ = [O,i] x [-2,2].

Here

£i = {(O,y):-2 < y < 2}, £3 ={(l,y):-2 < y < 2~£ 2 = {(x,c):O < x < l,o = - 2} and £4 = {(x,d):O < x < i, d=2} .

It turns out both

F(£ I)

Jordan curves symmetrical with respect to the x-axis.

and F ( £ 2 ~ £

3 ~£4

) are

Also F(£ I) lies in the

interior determined by F ( £ 2 ~ £

3 ~£4

the curves.

in a certain interval the equation fl£ = u has more

Thus for each

u

) except for the origin which belongs to both

than two solutions.

Theorem 4 :

In addition to conditions of theorem 3, suppose that for both Yo = c

and Yo = d either sl(Y o) = s2(Y o) or f(.,yo ) has a nonvanishing derivative in (Sl(Yo) , s2(Yo)) and F is NVL on £%£2\£4 .

Then F is one-one on ~ (= interior plus

the boundary).

Proof :

In view of theorem 3(c) it is enough to prove that F is one one on both

A(~) and A(B).

We will prove theorem 4 when Sl(C) = s2(c), Sl(d) = s2(d). A similar

proof can be given in the other case. A(~)C

£I or A(~) = {(sl(y),y)

From theorem 3(b) we can conclude that

: y.(~) < y < y*(~)} for some y.(~) and y*(~) s R.

Assume y.(~) < y*(~) and define 7(~,y) = g(sl(y),y).

In order to complete the proof

of the theorem it is enough if we show that 7y(~,.) # 0 in (y.(~),y*(~)). Yo s (y.(~), y*(~)) and let f and g have partial derivatives

Let

(P,Q) and (M,N) at

z° = (sl(Yo),Yo). From the local implicit function theorem it follows that s!(Yo)l exists and it is given by s~(y o) = - ~ • Also 7y(~,y o) = (PN~Q}____~)# O as F is NVL at z . o

This terminates the proof of theorem 4.

(Note that F'(z o) is unique since P = fx,M = gx' Q = - Ps~ and N = ((Pyy+QM) /P). We will now present a result due to Nikaido

[45].

Let F = (f(x,y),g(x,y))

be a

mapping from R 2 to R 2 where f and g have continuous partial derivatives throughout R 2"

55 Theorem 5 :

Suppose there are 4 positive numbers ml,m 2 and MI,M 2 such that

m I ! Ifx[ !

M1

and m2 ~ throughout R 2 .

fxgy-fygx I ~ M 2

Then the system of equations f(x,y)

:

aI

g(x,y)

=

a2

has exactly one solution in R 2 for any given constants al,a 2.

In other words F

is one-one and onto R 2.

Proof :

Observe that the existence and uniqueness of a solution of a single

equation f(x) = a in the single unk1~own x are clearly true if m _< If'(x) l < M(- ~ < x < ~)

for some positive m and M.

If this condition is satisfied, one can

see from the mean value theorem in calculus that either lim x÷+oo

lim x÷+ao

f(x) = + ~ or

f(x) = ~ ~ holds true, depending on the sign of the derivative f' (x).

H~nce,

f(x) can be equated to a at some value of x because of the continuity of f(x). Moreover the solution is unique from the montonicity of f(x). Suppose the conditions of theorem are met.

This means given a I and y,f(x,y)

= a I has a unique solution from the first paragraph of the proof. exists a function ¢(y) = x, such that f(¢(y),y) = a I.

That is there

Also one can check by the

local implicit function theorem that @ has continuous derivative @' with respect to y which will satisfy, fx(¢(y),y)¢'(y) + fv(¢(y),y) = 0.

Let G(y) = g(¢(y),y).

Then

u

G'(y) = gx(@(y),y)@'(y)+

gy(@(y),y).

f Hence it follows that G'(y) = - gx f-~x+ gy

m2 M2 ~ii 0) .

Then the solution (x,y) = (0,0) is globally asymptotically stable.

(Note that we

have slightly altered the notation; instead of writing fl,f2 we have written (f,g) and instead of writing (Xl,X2) we have written (x,y)). Proof of theorem 2 uses results from Olech [47] More and Rheinboldt [42] and Garcia-Zangwill [20]. Then we will give several examples to illustrate the sharpness of the results.

In particular we will give an example where theorem 2 is applicable

but not theorem i.

Proof of Theorem 2 : We will give a proof of theorem 2.

In view of Theorem 3,

(PP. 395 in [47]) it is enough if we show that the mapping F is globally one-one under each of the assumptions (a), (b) and (c).

We will write fx,fy instead of

~f ~x

Sf etc. Suppose (a) holds. That is we are given that the set D=[(x,y):fxgy 0

for some point in R 2 ~ D .

If (a) holds good,

since fx + gy < 0 by hypothesis it follows that fx < 0 and gy < 0 throughout R 2 ~ D . Invoking theorem 3 (PP. 246 in [20]) we may conclude that -F = (-f, -g) is globally univalent or that F is globally univalent in R 2 . If (~ holds good, then we define F (x,y) for every s > 0 F ( x , y ) = F ( x , y ) - (Ex,Ey).

as follows.

61

Observe that the JE = Jacobian of F

=

I fx-s

fY I

gx s

2

- S(fx+gy) - gxfy ~ 0

for every s ~ 0

trace of Js = fx + gy - 2s ~ 0.

Now det JE =

fxgY +

gy-~ since fx + gy ~ 0

and det J ~ 0.

Also

Furthermore product of the diagonal entries of Js

is given by

(fx-~)(gy-E) = fxgy - ~(f+gy) + 2 which is strictly positive throughout R2\ D.

In other words F s (x,y) satisfies all

the conditions of theorem 3 of Garcia-Zangwill [20] and consequently Fs(x,y) is globally univalent in R 2 for every s ~ 0.

That is -Fs(x,y) = -F(x,y) + (sx,sy) is

globally univalent in R 2 for every s ~ 0.

Also Jacobian of -F is non-vanishing and

R 2 is an open set.

Thus we may infer from theorem 5.9 of More and Rheinboldt [42]

that -F(x,y) or F(x,y) is globally univalent in R 2.

This terminates the proof under

condition (a). Suppose (b) holds good.

If the set E is bounded, since the Jacobian of F is

positive it follows that D ~ E or the set D is bounded and we are in case (a). Suppose (c) holds good.

Since fxgy ~ 0

we have fy gx ~ 0 throughout R 2 ~

G.

throughout R 2 ~

G and since det J ~ 0,

As F is of C (I) class and R 2 ~ G is a connected

set, it follows that f and gx will keep the same sign throughout R 2 ~ G. Suppose Y 2 fy ~ 0 and gx ~ 0 throughout R ~G. Let H = (g,-f). Then the Jacobian JH of H is a P-matrix - (that is every principal minor of JH is p~s~tive) throughout R2~G. Hence from Garcia-Zangwill's result it follows that N is globally univalent or F is one-one throughout R 2. throughout R 2 ~ G .

A similar proof can be given when fy ~ 0 and gx ~ 0

Thus we see that each one of the conditions (a), (b) and (c)

imply that the map F is univalent in R 2 and consequently (x,y) = (0,0) is ~lobally asymptotically stable.

This terminates the proof of theorem 2.

We will now deduce theorem I from theorem 2.

In order to do that it is enough

to verify that condition [0] implies one of the conditions (a), (b) and (c) of theorem 2.

If fx gy ~ 0

throughout R2~ condition (c) is trivially satisfied as G 2 is an empty set. If f g 0 throughout R , condition (a) is met as D is an empty x y 2 ~ R2 set. If f g ~ 0 throughout R it will imply f g 0 is and D will be an empty yx xy set. Suppose f g ~ 0 throughout R 2. We w~ll consider two cases (i) fy ~ 0 and

y x(2) fy ~ 0 and gx ~ 0. gx ~ 0 in R 2 and Then condition (a) is met for the map H. Then again condition (a) is met for L.

Under case (i) define a new map H = (-g,f). Under case (2) define a new map L=(g,-f). We may conclude from theorem 2, (x,y)=(0,0)

is globally asymptotically stable. One has the following corollary which can be deduced from theorem 2.

62

Corollary i :

Let the autonomous system satisfy (i) and (ii) and F(0) = (0,0).

Let F = (f,g) be a C (I) function.

Supoose fxgy ~ 0 throughout R 2.

Then the

solution (x,y) = (0,0) is globally asymptotically stable.

Remark :

Clearly corollary i includes Markus and Yamabe's result as well as

Hartman's result.

Also Olech's condition given in theorem i need not be satisfied.

We will utilize this corollary to construct an example where theorem i may not be applicable.

Examples:

In this section we will give several examples to illustrate the limita-

tions or the sharpness of the results known for the global asymptotic stability of the solutions for the autonomous system in the plane.

In the first two examples

Olech's condition given in theorem i will not be satisfied but conditions given in theorem 2 will be satisfied.

Fourth example will illustrate that condition (ii)

trace of J < 0 cannot be weakened to trace of J ~ 0, to get the asymptotic stability results.

If condition (i) is replaced by (i') det J < 0 throughout R 2 and retain

condition (ii) then one can have more than one critical points - we will demonstrate this in the third example.

The next two examples will show that it is possible to

have det J = 0 at some points and yet (0,0) to be a globally asymptotic solution. In other words condition

(i) or the map F to be one-one need not be necessary.

In

these two examples we will use the following result due to Nartman and 01ech (Theorem 2.1, pp. 155 in [27]).

Theorem 3 :

Let F be a continuously differentiable function from R 2 to R 2.

F(0) = 0 and F(x) # 0 whenever x # 0. solution of (S). F evaluated at x.

Suppose

Suppose x = 0 is a locally asymptotic stable

Let ~l(X) and ~2(x) be the characteristic roots of the Jacobian of Suppose~l(X) +

~2(x) ~ 0 for all x s R 2 and further suppose

Ixl If(x)] > constant > O whenever Ixl > constant > 0.

Then x = 0 is globally

asymptotically stable. Our last example will illustrate that theorem 3 may fail if ~l(X) + X 2 (x) admits positive values.

x2 Example i :

Let fl(Xl,X2) = - x I + log(l+x~) + x 2 and f 2 ( x l , x 2 ) = - x

- e

+ i.

Here 2x I

-I+

(

2 )

1

i+ x I J

= x2

2 - Bx I

Clearly det J > 0 and trace J < 0.

Note that

-e

~fl ~x I

~f2 2~x = 0

if

xI = i

and

63

~fl 8x2

~f2 8Xl

= 0 if x I = 0.

8fl 8xI

Also

8f2 8x2 ~ 0

throughout

R 2.

Here all the

conditions of corollary i are satisfied (but 01ech's condition given in theorem i is not satisfied).

Example 2 :

Consequently x = 0 is globally asymptotically stable.

9

Let

fl(Xl'X2 ) = - [ ~3-+ Xl(X~L - ½)-x2]

f2(Xl,X2)

=_

[xfx2].

Here the Jacobian J is given by 2

2

-(Xl+X 2 - ½1

-2XlX2+l]-l

J= -I Clearly det J > 0, trace J < 0. bounded set. applicable.

In this example the set D given in theorem 2, is a

2 In fact D = ~f~,Xl,X 2, : x I + x 2 2 _< ½}.

Here also theorem i is not

We can conclude from theorem 2 that x = 0 is globally asymptotically

stable.

Example 3 : This example is a slight modification of Gale-Nikaido given in (pp 82, [19]). Let fl(Xl,X2)

=

Xl 2 - 2 e + 3 x2 - i

f2(xl,x2)

=

x2

eXl

- x~ .

xI

Xl

Xl 2 e -3x 2

x2e Clearly det J < 0 and trace J < O.

However F = (fl,f2) is not globally univalent

since F(O,I) = F(0,-I) = (0,0). This example shows that global univalent problem is false if condition (i) is replaced by the condition det J < 0 throughout R 2. already remarked global univalent problem still remains open. of

the same example will serve

A modification

as a counter example if one attempts to prove

global univalent problem in R 3 under (i) and (ii).

Example 4 :

If we weaken condition (ii) to trace J ~ 0

then global asymptotic

As

84

stability may not hold good as the following example shows. -Xl+2X 2 and f2(xl,x2) = - Xl+X 2.

Set fl(Xl,X2) =

Here

Clearly det J > 0 and trace J < 0 throughout R 2.

However here x = 0 is not even

locally asymptotically stable though it is stable (see theorem 9.1, pp. 411 in [4]).

Example 5 :

The following example will show that condition

(i) namely det J > 0

is not a necessary condition for x = 0 to be globally asymptotically stable. fl(Xl,X2)

=

- Xl(l+ x~)

f2(xl,x2)

=

- x2(l+ x~).

Jfix

Let

and

12xlx21

[-2XlX2 2 22= Here det J = (I+ x 2)(I+ x 2) - 4XlX 2

0 if x I ± i and x 2 = ± I.

Consequently condition

(i) is violated.

It is not hard to check that x = 0 is locally asymptotically stable. = 2 (see pp. 440 in [4]). Also one can easily check that ~I + ~2 - (2 + x ~ + x 2) < 0.

Hence all the conditions of theorem 3 are satisfied and thus x = 0 is globally asymptotically

stable.

Alternatively one can use theorem 9.5 in pP. 439, [4] to

demonstrate that x = 0 is globally asymptotically stable.

Example 6 :

The following example shows that x = 0 may be globally asymptotically

stable but that F need not be globally one-one in R 2. x3 fl(Xl,X2) = - ~ +

Let

2XlX ~

f2(xl,x2) = - x~

J

x I + 2x 2

4XlX 2

0

2 -3x 2

=

1

65

det J = 3x2(2x 2 -

x I) ~ 0 if x 2 # 0 and 2x~ ~

x~

and trace J ~ 0.

F = (fl,f2) is not one-one since F(1,½) = F(-I,½) = (0,-½). IIF(x) ll ~ constant ~ 0 whenever

llxll ~ constant ~ 0.

Also

One can check that

All the conditions of

theorem 3 are satisfied and thus x = 0 is globally asymptotically stable.

[One

can check that x = 0 is locally asymptotically stable by Liapounou's second method for details see pp. 440 in (4)].

Example 7 :

The following example demonstrates that theorem 3 is probably the best

available result in R 2 for global asymptotic solution. fl(Xl,X2)

=

f2(xl,x2)

= _ x2 •

Let s ~ 0

and ~ ~ 0.

Let

sx I - ox~

Then the Jacobian

Ji

iiJ

In this example x = 0 is locally asymptotically stable but not globally asymptotically stable.

Also note that ~l(X) + ~2(x) = - i + s -2 ~Xl, may take both positive and

negative values.

In other words one of the conditions of theorem 3 is not satisfied.

In fact other conditions are met.

This shows that in general we may not be able

to relax the condition "~I + X2 ~ 0"

if we want globally asymptotic solution.

Vidossich's contribution to 01ech's problem on stability :

Vidossich's gives

another set of sufficient conditions under which F is one-one.

In fact we have the

following [71].

Theorem 4 :

Let F:R n ÷ R n

be of class C (I) and all eigenvalues of the Jacobian J(x)

of F have negative real parts, then F is one-one if any of the following conditions is met: (a)

there exist two positive constants ~ and B such that trace J(x) ~ - ~ IIJ(x) II ~ B

(b)

for every x E R n

and

or alternatively,

for each y s R n, there exists s ~ 0 such that for the topological degree Y for s ~ Sy, B(O,s) being the open ball with

we have deg(F,B(O,s),y) = I centre 0 and radius

Remarks :

s.

Theorem 4 is true for any n.

When (a) is satisfied one can show that F

66 is proper which will in turn imply that F is one-one - this can be seen from Hadamard's theorem or from Caccioppoli [6]. Also condition (b) is satisfied if there exists ~

o

> 0

and

0 ~ k < i such that

Ilx-r(x)II ~ kll~ll, (llxllm%). This fact follows from the homotopy invariance of the topological degree by considering l-k(l-F),

(0 < ~ < i).

Proof of theorem 4 : We will prove under condition (b). Suppose F is not one-one. This means for some

Yos

Rn' F-I(yo) will contain at least two points say

u,v.

Since F is locally one-one and R n is separable, F-l(y o) contains at most countably many points.

There exists s > 0 such that

> maX{ay o, 1 1 u l 1 , 1 1 v l l ~ and no points of F-l(y o) has norm equal to s, since otherwise F-l(yo) would contain a subset with the same cardinality of a non-trivial interval of R contrary to the countability of F-l(yo).

Since the closure of B(O,c) is compact and F-l(yo) fhB(O,E)

is discrete it follows that F-l(yo) lh B(O,c) must be finite : let x I .... ,xm points.

Clearly Ilxil I < a

for i = 1,2 .... ,m.

be its

Hence we can find for each i an

open set U i of x i such that the closed sets Ui are pairwise disjoint and U i ~ B(O,s). Let C-(x)

Then, F(x) = Yo

-- x +

F (x)-y °

if and only if G(x) = x.

.

Observe that G(x) has no fixed point in

m

B(O,s)~M

U i.

By the additivity of the topological degree, we have m

deg(l-G, B(O,~),0)=

~ dega-a, Ui,0) i=l

where the right hand side is well defined since no point of F-l(yo ) belongs to ~U i. We will now show that deg(l-G, Ui,0 ) = i, i = 1,2, ...,m. JG(Xi) = Jacobian of G. (~-l)x= JF(Xi)X. -i < 0

Let ~ be an eigenvalue of

Then there exists x ~ 0 such that (l+JF(Xi))x = kx

or

It follows that (~-i) is an eigenvalue of JF(Xi) and by hypothesis

or X < i.

It implies that JG has no positive eigenvalue greater than one.

Therefore a well-known theorem of Leray-Schauder deg(l-G, Ui,0) = i for i = 1,2 .... ,m.

(See pp 162-163 [48]) implies that

It is known that

deg(g,B(O,s),y) = deg(g-y, B(0,s),0) for any continuous function g and any point y provided that the equation g(x) = y has no solution x with

Iixll

--

~.

Therefore we have :

67

deg(F,B(O,s),y o)

=

deg(F-Yo, B(O,s),0)

=

deg(l-G, B(O,E),O) m

--

X degII-G, ~i,0) i=l m .

But m _> 2, since u,v s F-l(y o) g~ B(O,s), hence we have a contradiction to our assumption

(b).

This terminates the proof of theorem 4.

We now have the following theorem on stability.

Theorem 5 :

The solution x = 0 of the autonomous equation x' = F(x) in the plane

is globally asymptotically

stable if F is of class C (I), F(0) = 0, the eigenva!ues

of J(x) all have negative real parts and F satisfies

Proof :

(a) or (b) of theorem 4.

Follows from theorem 4 and Olech's theorem.

Remark i :

Example 6 shows that the solution x = 0 may be globally asymptotic stable,

without any of the conditions imposed by theorem 4 being satisfied.

Remark 2 :

Global asymptotic stability in R n has been studied by Hartman [25] and

Hartman and Olech [27].

Hartman considers the case when (J+ J')/2 is negative

definite while Nartman and Olech places conditions on the eigenvalues of the Jacobian matrix.

Interested readers should refer to their works for further details.

We close this chapter by inviting the readers to prove the global univalent problem or give a counter example.

CHAPTER

VIII

UNIVALENCE FOR MAPPINGS WITH LEONTIEF TYPE JACOBIANS

Abstract

:

In this chapter we prove several results on univalence

Leontief type Jacobians. Gale-Nikaido,s

for mappings with

The first result is in some sense a sort of converse to

theorem on univalence.

Here we prove a result due to Gale-Nikaido

and this says that if F and F -I are differentiable

and if F -I is monotonic

increasing

then the Jacobian of F is a P-matrix provided the Jacobian matrix of F is of Leontief type.

The second result due to Nikaido

says that there exists a unique solution to

F(x) = 0 provided its domain is non-negative

orthant and the Jacobian matrix is of

Leontief type satisfying certain uniform diagonal dominance property. present related results on M-functions Rheinboldt.

Then we

and inverse isotone maps due to More and

Finally we give some results on the univalence of the composition of

maps F and G when their Jacobians are of Leontief type.

In particular we show that

F o G is a P-function when F and G are maps from R 3 to R 3 with their Jacobians Leontief type P-matrices

throughout.

We give an example to show that F o G need

not be a P-function in R 4.

Univalence for dominant diagonal mappings section.

:

Two results will be presented

The first result shows that if both F and F -I are differentiable

F -I is monotonic

and if

increasing then the Jacobian matrix of F has to be a P-matrix,

provided the Jacobian matrix of F is of Leontief type. and Nikaido.

in this

This result is due to Gale

The second result due to Nikaido says that there exists a unique

solution to F = 0 provided the domain of F is ~ = non-negative

orthant and the

Jacobian matrix is of Leontief type satisfying certain uniform diagonal dominance property.

We will say that a matrix A is of Leontief type if the off-diagonal

entries of A are nonpositive. initiated by well-known

Such matrices play a prominent role in studies

economist Leontief.

that the following two conditions (i)

A is a P-matrix

(ii)

There exists a vector x > 0

For Leontief type matrices we have seen

are equivalent.

such that

Ax > 0.

In chapter II we have defined the notion of dominant diagonal.

That is, a

matrix A is said to possess a dominant diagonal if there is a set of positive numbers n d.(i = 1,2 ..... n) such that a..d. > ~ la..Id, 1 ii i ~--7 i$ condition is equivalent

for all

i = 1,2 ..... n.

This

to the condition that there exists a set of positive numbers

69 c. (j = 1,2,...,n) such that J n

aj j cj

>

~ laijlc i i=l

for all

j = 1,2 ..... n.

i#j Equivalence follows from the fact that any one of the above conditions implies that A is a P matrix. If A, whose elements are functions defined on a set ~ has dominant diagonal throughout the set ~, the weights d. in general will be a function of w s ~. di(w) ~ d i for all w s ~

then we say A has a uniformly dominant diagonal.

If

We say A

has a uniformly dominant diagonal in the strong sense if there are positive numbers d i and ci, i = 1,2,...,n such that d A ~

c where d = (dl,d2,...,dn) , A is an n x n

matrix and c = (Cl,C 2 ..... Cn). [Entries of A are functions of w].

We are now ready

to state the following theorems.

Theorem i :

Let F : ~ ÷ R n be a differentiable mapping where D is an open region

in R n and Jacobian matrix is of Leontief type.

Suppose F -I is differentiable and

monotonic increasing (that is F(a) < F(b) implies a < b).

Then the Jacobian matrix

of F is a P-matrix.

Theorem 2 :

Let F:R+n ÷ R n be a differentiable map whose Jacobian matrix is of (Here R+n = non-negative orthant). Suppose Jacobian matrix of F has a uniformly dominant diagonal in the strong sense. Further suppose fi(x) ~ 0

Leontief type.

whenever the i-th component of x is zero. F(x)).

(Here fi(x) is the i-th component of

Then F(x) = 0 has a unique solution.

In other words F(x) = a has a unique

solution for every a s R+n . Proof of Theorem i :

Since ~ is open in R n, by the invariance of domain, F(~) is

also an open set in R n.

Let b be any point of ~ such that b* = F(b).

any strictly positive vector.

Let u* be

Since F(~) is open, there exists s > 0 such that

x*(~) = b * + X u * s F(~) for all ~ with l~I < s.

Let F-l(x*(~)) = x(~).

Then since

F-I is differentiable and monotonic increasing, x(~) is differentiable a n d ~ x ( ~ ) for

IXI

< s.

0

Differentiating x*(X) = F(x(~)) at ~ = 0, we have J(b)x'(0) = u* > 0.

Since J(b) is of Leontief type, and since x'(0) > 0 and u* > 0 it follows that J(b) has to be a P-matrix. at b).

(Here as usual J (b) stands for the Jacobian matrix evaluated

This terminates the proof of theorem i.

Proof of theorem 2 : chapter VI.

Proof of this result is similar to the proof of theorem 5 in

Proof is based on induction on n, which will establish both the existence

and uniqueness of the solution of the system of equations F(x) = 0.

If n = i,

we are looking for a solution of a single equation of a scalar variable fl(x) = 0.

70

We are given two positive constants d I and c I such that d I fll(X) > c I for all x ~ 0.

This implies the first derivative fll(X) of fl(x) is positive and in

fact fll(X) > (Cl/dl). and fl(x) ÷ + ~ fl(x) = 0

Hence it follows that fl(x) is strictly monotonic increasing

as x ÷ ~.

Since from hypothesis fl(0) _< 0, it follows that

has a solution as fl is continuous and the solution is unique as fl is

strictly monotonic. Assume the result to be true for n-l.

Let us write F(x) = 0 explicitly as

follows: fi(xl,x2,...,Xn_l,Xn) = 0

for

i = 1,2,...,n-I

fn(Xl,X2,...,Xn_l,Xn) = 0 Fix xn and consider the system of equations in the (n-l) variables Xl,X2,...,Xn_l. The system fi(xl,...,Xn_l,Xn) = 0 for i = 1,2,...,n-i satisfies all the hypothesis of the theorem including the strong uniform dominant diagonal property. end, let Jn-i denote principal that of the original system.

To this

submatrix of order n-i in the upper left corner of Since (dl,...,dn)J ~ (Cl,...,c n) for some d ~ 0,e ~ 0

and since the off-diagonal entries of J are non-positive it follows that (dl,...,dn_l)Jn_l ~ (Cl,..-,Cn_l). Hence by induction hypothesis the system of equations fi(xl,...,Xn_l,Xn) = 0 for i = 1,2,...,n-l, for every fixed non-negative Xn, has a tmique solution, Xl,...,Xn_ I which will be functions of xn.

Let xj = gj( Xn), j = 1,2,...,n-I which of course will

satisfy fi(gl(Xn) , g2(Xn),...,gn_l(Xn),Xn) = 0,i = 1,2,...,n-I for all x n_> 0.

Let g(Xn) = fn(gl(Xn),g2(Xn) ,...,gn_l(xn),xn).

Now it is easy to

conclude that F(Xl,...,xn) = 0 has a unique solution if and only if g(xn) = 0 is uniquely solvable in the non-negative unknown x n. to one-variable case.

Thus we have reduced the problem ~ cn - these

We will now show that g(0) ~ 0 and d n ~

two facts will complete the proof,

n

Obviously g(0) = fn(gl(0),...,gn_l(0),0) ~ 0. dJ > c

and

Observe

J is of Leontief type.

Hence it follows that j-i is non-negative.

Thus

(dl,d2,...,dn)J ~ (Cl,...,Cn) or (dl,...,d n) ~(Cl,...,Cn)J-i n

Now dn > 111"=cifmn

where (fin) is the n-th column of j-l.

fin~ 0 or

and

Since

e i ~ 0, dn ~ en fnn = On

det Jn-i det J

71 d

det J > c n det J n - i n

Note that dg dxn

_

det J det Jn-i

Hence g(Xn) = 0 will have a unique solution and consequently F (x) = 0 will have a unique solution.

This terminates the proof of theorem 2.

Interrelation between P-property and M-property :

We will now introduce the

notion of M-function similar to the notion of P-function introduced in chapter III and prove that if F is a differentiable map over the rectangle and if the Jacobian is a P-matrix of the Leontief type then F must be an M-function.

Definition :

Consider a mapping F : ~ C ~

x ~ y

Rn ÷ Rn

whenever

with the following properties

(a)

F(x) ~ F(y)

x,y s

(b)

the functions T~:(t s Rllx+ te j E ~) + R I defined by Tij(t) = fi(x+teJ) are monotonic decreasing as functions of t where i # j and e

i

= (I,0,0,...,0), e 2

=

(0,I,O,...~0) etc. [Here f. as usual denotes 1

the ith component of F]. Then F is said to be an M-function.

Property (a) is usually called inverse isotone

and (b) is called off-diagonally antitone.

This notion is a nonlinear generalization

of Leontief type P-matrices introduced and studied by More and Rheinboldt.

We are

ready to prove the following theorem.

Theorem 3 :

Let F:~ C R n ÷ R n

be a differentiable map on the rectangle ~.

the Jacobian matrix of F is a P-matrix of the Leontief type for every x s ~.

Suppose Then

F is an N-function, in the sense described above.

Remark :

It is known from Gale-Nikaido's result that F is a P-function.

that F is off-diagonally antitone.

Also note

Theorem 3 is an immediate consequence of the

following result.

Theorem 4 :

Any off-diagonally antitone P-function F:~ ~ R n ÷ R n on a rectangle

is an M-function.

Proof :

We need only to verify property (a) of the above definition of an M-function.

That is, we have to establish that F is inverse isotene. some x,y s ~.

Let A = (i:x i > yi }.

empty and let A = {l,2,...,m}.

Suppose F(x) < F(y) for

If A is empty we are through.

Define

Suppose A is not

72

gi(tl,t2 .... ,tm) = fi(tl,t2 ..... tm,Ym+l,...,Y n) where i = 1,2,...,m.

Note that

gi(Yl,y2, ---,Fro) = fi(y) ~_ fi(x) ~_ gi(xl,x2 ..... Xm). Here the last inequality follows from the fact that F is off-diagonally antitone and the first inequality by hypothesis.

Hence we can conclude that

(Yi-Xi)(gi(Yl ..... ym)-gi(x I ..... Xm)) ~_ 0, Set x' = (Xl,X2,...,Xm, Ym~l,...,yn ).

Clearly x' ~ y

i = 1,2,...,m.

and

(Yi-x~)(fi(y) - f±(x,)) ~ 0 ~i = 1,2 ..... m and (Yi-X.~)(fi(y) - fi(x')) = 0)i = m + l ..... n

but this contradicts the fact that F is a P-ftunction. of theorem 4.

Remark :

This terminates the proof

Now theorem 3 follows from theorem 4.

We need the fact that ~ is a rectangle to conclude that x' s ~.

It is not

clear whether this result is valid if ~ is a convex region and not necessarily a rectangular region.

We have lot of freedom when

~ is a rectangular region.

One

can restate theorem I as follows, for convex maps in terms of property (a).

Theorem 5 : set ~.

Let F : ~ C R

n ÷ Rn

be convex and differentiable on the open convex

Then F is inverse isotone if and only if the Jacobian matrix J(x) is inver-

tible for each x s ~

and J(x) -I ~ 0

for each x E ~ (that is every entry in the

inverse matrix is non-negative).

Proof :

Suppose F is inverse isotone.

x s ~ and suppose J(x)h ~ 0.

That is F(x) ~ F(y)

implies

x ~ y.

Let

Since ~ is open, there is a 0 ~ 0, such that x + Oh s ~.

m

It fellows from the convexity of F F(x+eh)-F(x) ~ 0J(x)h ~ 0 The last inequality follows because 0 ~ 0

and

is inverse isotone, x+ eh ~ x

In other words J(x)h ~ 0 implies h > 0.

or

h ~ 0.

J(x)h _> 0

Consequently it follows that J(x) is non-singular.

by assumption.

Now we will prove J(x) -I _> 0.

In order to do that, let y_> 0 then we have J(x)(J(x) -I y) ~_ 0 implies J(x) -I y > 0.

Since F

which in turn

Hence J(x) -I _> 0.

Conversely suppose J(x) -I ~ 0

for every x s ~.

Let F(x) _> F(y) for some x,ys~

From the convexity of F it follows that J(x)(x-y) >_ F(x)-F(y) >_ 0.

That is

73

J(x)(x-y)

> 0

or J

(x) -1 J (x) (x-y) = x-y ~ 0.

This terminates the proof of

theorem 5.

Remark

:

If in addition we assume J(x) is of Leontief type in theorem 5 then F

is inverse isotone if and only if J(x) is a P-matrix for each x s ~. isotone concept is equivalent provided F -I exists.

Inverse

to fact that the mapping F -I is monotonic

increasing

If we drop the assumption of convexity of F as well as

convexity of ~ we have the following theorem.

Theorem 6 :

Let F : ~ C

R n ÷ R n be inverse isotone and differentiable

set ~.Suppose the Jacobian J(x) of F is of Leontief type.

on the open

Then J(x) -I is a

P-matrix for any x E ~ at which J(x) is non-singular.

Proof

:

Suppose x s D with J(x) non-singular.

In order to show J(x) is a P-matrix

it is enough if we show J(x) -I ~ 0 since J(x) is of Leontief type. we will prove the following J(x)h> 0 be seen as follows.

This can

First we will consider the case J(x)h~ O. limit t÷0

Hence F(x+ th)-F(x)

As in theorem 5

for some h s R n will imply h ~ 0.

[F (x+ th)-F (x) ]

~ 0 for sufficiently

isotonicity that h > 0.

=

J(x)h > 0 .

+

small t > 0 and consequently by the inverse

Now consider J(x)h > 0 for some h E R n.

i h k = h + ~ J(x) -I e~ k = 1,2 ....

Define

Here e denotes the vector with all entires one.

Clearly J(x)h k > 0 which in turn implies from the first part h k > O, for all k and hence h ~ 0 since h k ÷ h as k ÷ ~ proof of theorem 6.

This shows J(x) -I > 0.

Observe that theorem 6 is essentially

This terminates

the

the same as theorem i.

For open rectangular regions one can prove the following theorem.

Theorem 7 : set ~.

Let F:~ r- R n ÷ R n

be a differentiable

map on the open rectangular

Suppose J(x) is of Leontief type and suppose J(x) is non-singular

for each

x s ~ .

Then F is inverse isotone if and only if J(x) is a P-matrix for every x s ~.

Proof

If F is inverse isotone then from theorem 6, it follows that J(x) is a

:

P-matrix for every x c ~. fundamental

Now suppose J(x) is a P-matrix.

theorem F is univalent

by induction argument on n.

in ~.

From Gale-Nikaido's

We will prove that F is inverse isotone

When n = i, clearly F is inverse isotone.

In general,

if F(a) ~ F(b) where a = (al,a2,...,a n) and b = (bl,b2,...,bn),

then for some k we

have a k ~

We may assume without

bk

this is a consequence of theorem i, chapter III.

less of generality k = I that is a I ~ b I. Jaeobian matrix are non-positive,

Since the off-diagonal

we have for i ~ i,

entries of the

74

fi(bl,a2 .... ,an ) ~ fi(al,a2,--.,an ) ~ fi(bl,b2,...,bn )Here the second inequality follows from F(a) ~ F(b). G:~n_l c

We now define a new map

Rn-I ÷ R n-I by the rule G(x2,x3~...,Xn) = (f2(bl,X2 .... ,Xn),...,fn(bl,X2,

...,Xn)) where ~n-I is the image of ~ under the projection (Xl,X2,...,xn) ÷ (x2,...~Xn).

Now G(a2,...~an) ~ G(b2,...,bn) and the Jacobian matrix of G is again

a P-matrix of the Leontief type.

Fence by induction hypothesis a. ~ b. for i ~ 2. i

We already have a I ~ b I and hence a i ~ b i for every i.

--

i

This proves F is inverse

isotone and this terminates the proof of Theorem 7.

Univalence for composition of two functions :

In this section we will prove in R 3,

that if F and G are differentiable functions whose Jacobian matrices are Leontief type P-matrices then F o G is a P-function and hence univalent in R 3. can prove the following result in Rn.

However one

If F and G are differentiable functions and if

Jacobian matrices of F and G are Leontief type P-matrices and further if the Jacobian of F o G is also of Leontief type then F o G is a P-function and hence univalent in Rn.

The results we prove in this section are similar to theorem 3.4

in More and Rheinboldt [42]. We are ready to prove the following [53].

Theorem 8 :

Let F and G be C (I) differentiable maps from R n ÷ Rn.

Jacobians of F and G are Leontief type P-matrices. the composition map F o G is of Leontief type.

Suppose

Further suppose Jacobian of

Then F o G is a P-function and hence

univalent in R n.

Proof :

Let H = F o G.

Observe that the Jacobian JH of H can be computed by means

of the relation JH = JF JG where the elements of JF as well as JG will be evaluated at the appropriate points. P-matrix throughout R n.

Theorem 9 :

From theorem 4, chapter II we can conclude that JH is a

Invoking Gale°Nikaido's theorem, we have the desired result.

Let F and G be C (I) differentiable maps from R 3 to R 3.

Jacobians of F and G are Leontief type P-matrices.

Suppose

Then F o G and G o F are

P-functions and hence univalent in R 3.

Proof : We will prove F o G is a P-function.

Let H = F o G.

From theorem 4,

chapter II, we can conclude that the Jacobian of H is a P-matrix and consequently F o G is a P-function.

Similar proof may be given for the univalence of G o F.

From theorem 2, chapter V we can deduce the following:

75

Theorem I0 :

Let F and G be C (I) differentiable maps from R n to R n.

compact rectangle in R n. Rn\

~ respectively.

Let D be a

Further suppose Jacobians of F and G are P-matrices in

Then F o G is univalent in H n.

We gave an example in chapter II (see theorem 4) to show that the product of two Leontief type P-matrices in R 4 need not be either a P-matrix or a Leontief type matrix.

That example will correspond to the following functions in R4: F(x,y,z,w)

:

i (x-z, [ y-w, z, w)

O(x,y,z,w)

(x-y, -x+ 2y, -4y+ z, -x+w).

(Fo G) (x,y,z,w)

(x+ 3y-z,7 x +

Then o

~-

w, -4y+ z, -x+ W ) .

In this example Jacobian of F and G are Leontief type P-matrices. of F o G is a non-singular

However Jacobian

(constant) matrix which is not a P-matrix.

easy to see in this example that F o G is one-one throughout R 4. following question:

Also it is

This raises the

Suppose F and G are differentiable C (I) function in R 4 whose

Jacobian matrices are Leontief type P-matrices.

Is F o G a P-function in R47

The

answer is "no" and the example given above is a counter example and it can be seen as follows.

Jacobian matrix of F o G is a constant matrix A given by

A

=

1

3

? B-

-8-

2

-i

o

o

-1

0

-4

1

o

-i

0

0

i

Since A is not a P-matrix, then exists a non-trivial vector u ° = (x°, yO, zo, wO), such that (Au°)~ u~ ~ 0

for i = 1,2,3,4 [Here prime denotes the transpose vector

and u~ stands for the ith component of the vector u°]. I

Let 0 be the zero vector.

Note that (F o G)(u °) = Au °'

Clearly u ° ~ 0 and (F o C) (0) = O.

Hence it follows

that F o G is not a P-function. We will now deviate a little bit and start with two examples.

The first is an

example of a differentiable P-function whose Jacobian is not a P-matrix. F:R 2 ÷ R 2 where F = (f,g) with f(x,y) = x3-y and g(x,y) = x+y 3. check that F is a P-function throughout R 2.

However Jacobian

One can easily J

of

3x 2 J

-i

= I

is not a P-matrix at x = y = O.

3y 2

I

=

Consider

i + 9x2y 2 > 0

F

given by

76

The second example is the following. This says that if the Jacobian is a weak P-matrix in a rectangular region, F need not be a P-function. 0 Here J = [-I

Consider F = (f,g) where f(x,y) = y, g(x,y) = y-x.

1 i ] is a weak P-matrix but F is not a P-function.

Gale-Nikaido's

fundamental

theorem that this result is true if J is a P-matrix.

We will now introduce the concept of P

Definition

:

A mapping F : ~ c

there is an index k = k(x,y) For Po-funetions

Theorem ii :

Let F : ~ c

o

I :

function which is weaker than P-function

R n ÷ R n is a P -function if for any x,y ~ ~, x # y, o such that (xk-Yk)(fk(x)-fk(y)) ~ 0 and x k # Yk"

one can prove the following theorem

Rn ÷ Rn

be a differentiable

for every x s ~ where ~ is an open rectangle.

Remark

We know from

[42].

P -function with det J(x) > 0

Then F-l°is again a P -function. o

We will not attempt to give a proof but we will indicate the proof.

Let D be any diagonal matrix with entires non-negative.

Then one can check that

F(x) + Dx is a Po-function which will in turn imply J(x) + D is a weak P-matrix. other words F(x) + Dx is univalent

for every such D.

In

This in turn implies that

F -I is a P -function. o Remark 2 :

A differentiable

Jacobian.

Po-function need not possess a weak P-matrix as its

It will only be a P -matrix, o non-negative.

that is, every principal minor is

We will close this chapter with a conjecture of Nore-Rheinboldt

[42]:

For

any continuous,

injective P -function F:~ c R n ÷ R n on an open rectangle ~,F -I o is also a Po-function. This result certainly holds good for the linear case and

for F-differentiable

P -functions. In order to settle this conjecture it is enough o if one can show that F(x) + Dx is one-one for every diagonal matrix D with non-

negative entries.

CHAPTER

IX

ASSORTED APPLICATIONS OF I~IVALENCE MAPPING RESULTS

Abstract :

In this chapter we will give various applications of the univalence

results proved in the earlier chapters. results are quite handy and useful.

There ar~ several areas where univalence

The first application will deal with a problem

in Mathematical Economics where we will give a set of sufficient conditions due to Nikaido and Mas-Colell which will ensure factor price equalization.

The second

application deals with the distribution of a function of several independent random variables.

As a third application we will consider a ~roblem in nonlinear complim-

entarity theory due to Kojima and Megiddo. theorem to Algebra.

Next we give an application of ~damard's

In the fifth we consider the problem of deciding whether a

certain multivariate gamma distribution is infinitely divisible. weak N-matrices play an important role.

In this situation

There arel various other applications

(for

example to nonlinear net-work theory) but we will not attempt to exhaust all of them for lack of time and space.

[We have already seen a nice application of univalent

results in stability theory in chapter VII].

kn application in Mathematical Economics : As a first application we will consider a well known nroblem in mathematical economics first suggested by Samuelson which in a sense prompted ~le-Nikaido to ~rove their fundamental result on univalence mappings.

In this problem one seeks conditions implying that countries facing

the same prices for goods in foreign trade will have the same factor prices.

To be

precise we will give a set of sufficient conditions which will not only ensure factor price equalization but also a set of equilibrium factor prices under any given set of good prices.

Put differently, the condition ensures the complete invertibility

of the determination of good prices by factor prices, giving rise to the inverse unique determination of factor prices by arbitrary good prices. We will now explain the terminology that will be used.

Let wj > 0, Pi > 0

denote the price of the jth factor and the ith good respectively where j,i=l,2,...,n. Let ci(wl,w2,...,w n) be n cost functions which hay@ continuous non-negative partial derivatives cij = 3oi/3w j

and c i 's are positively homogeneous of degree one, that is

ci(Xw I .... ,Xwn) = ~ci(Wl,...,w n) for all X > 0 and wj > 0.

Let ~ij = cij wj/ci -

this quantity is the relative share of the jth factor in the ith good sector, because

n ~ j=l

c.. wj = e i. iJ

[The last equality follows from the fact that c.'s are 1

positively homogeneous of degree one].

By 'equalization of factor prices' we mean

78

the uniqueness of solution of the system of equations ci(wl,w2,...,w n) = Pi (i = 1,2,...,n). Nikaido

We are ready to state the complete invertibility theorem due to

[45].

Complete invertibility theorem :

Suppose ci(wl,w2,...,w n) are defined for all

w. > 0, take positive values everywhere, have continuous partial derivatives c.. 3 ~J which are non-negative everywhere and c.'s are positively homogeneous of degree one. i Further suppose the relative shares matrix A = (aij) where ~ij = cij wj/ci has the upper left-hand corner principal minors whose absolute values are bounded from below by some number

absolute value of det

all .

.--

C~kl

"'"

alk 1 h @,(k = 1,2,...,n)

.

&kk

Then for any given set of positive good prices Pi > 0, there exists a unique set of factor prices wj > 0 satisfying ci(wl,w2,...,w n) = Pi for i = 1,2,...,n. X.

Proof :

Set f. = log c. and w. = e J i 1 j functions defined on R n

In other words we are defining n new xI

fi(xl,x2,...,Xn)

= log ci(e

x2 , e

Xn) ,...,e

.

It is clear that the system of equations ci(wl,...,w n) = Pi has a unique solution for Pi > 0 if and only if the system of equations fi(xl,x2,...,Xn) a i = log Pi has a unique solution in R n.

the conditions imposed on theorem 6 in chapter VI. 3f i ~x. 3

= a i where

We will now verify that fi's satisfy Note that

x. xI x2 x xI xn - e J cij(e ,e ,...,e n)/ci(e ,...,e ) X.

so that

f.. = ~.. if evaluated for w. = e 3 in R n. Hence ~ given in the theorem m3 13 3 will serve as a lower bound. Also observe by homOgeneity, 2 a.. = i for j ~J e. = Z c.. w. and further by hypothesis ~.. > 0. i 13 3 iJ -the principal minors of (fij) are bounded above.

Fence 0 < a.. < I and consequently -- 13 -Therefore the system fi = ai has

a unique solution for any given Pi > 0 from theorem 6 chapter VI.

This terminates

the proof of the complete invertibility theorem.

Remark :

Mas-Colell has substantially generalized the complete invertibility

theorem.

He shows that the restrictions on the principal minors are irrelevant;

that matters is that that the determinant of laijln~m be uniformly bounded away from zero.

For details see Mas-Colell

[36].

all

79

We will now look at another example from input-output model recently considered by Chander.

Chander's model is similar to the model considered earlier

by I.W. Sandberg [1973, Econometrica pp. 1167-1182].

We assume that the economy is

divided into n industrial sectors each of which produces a single kind of good that is traded, consumed and invested in the economy.

The interrelations in such an

economy may be described by the system of equations

xi

-

n ~ a..(x ) = c. j=I ij j l

for

i = 1,2,...,n

where x i denotes the quantity of good i produced in the ith sector, aij(x j) represents the total amount of good i used as input for producing x. units of good j. Hence, J for each i the total amount of good i available for final consumption, export and investment is x i -

n ~ j i

demand vector.

a..(xj). ij

The vector (Cl,C2,...,c n) is called the final

Here the problem is to find the vector x given the demand vector c.

Also one would be interested in computing x. (i)

We will make two assumptions:

for each i and j, aij(.) is continuously differentiable on [O,~), aij(O) = 0

and a!.(a) > 0 ij --

for all ~ > 0

where prime denotes the derivative.

(ii) There

n exists Pi > 0, i = 1,2,...,n and v > 0 such that Pi~ ~ Pi a!.(~) + v for all i=l iJ ~ [0,~) and j=l,2 ..... n. We now have the following:

Theorem :

(for input-output model)

:

Under assumptions (i) and (ii) the mapping

A : R n ÷ R n where A(x) = ( nZ aij(xj), i = 1,2,...,n) j=l

is a contraction mapping.

Furthermore for every c s R~, there exists a unique x s Rn+ such that x-A(x) = c and for any x(0) s R+, n the sequence (x(t)) 0 defined by the Jacobi iterates x(t+l)

=

A(x(t)) + c, t > 0.

converges to x. For a proof of this result one can refer to Chander [7].

Alternatively one can

use Nikaido's result on uniform diagonal property in the strong sense.

Here we are

making stronger assumption and this enables us to compute the vector x to any degree of accuracy that we want.

On the distribution of a function of several random variables : statistics we come across problems of the following type.

Usually in

If XI,X2,...,Xn are

mutually independent random variables then we would like to know the distribution of a function u(XI,X2,...,X n) of the random variables XI,X2,...,Xn or we may even be interested in the joint distribution of F(XI,X2,...,Xn) = (uI(XI,X2,--.,Xn), u2(XI,X 2 .... ,Xn)...Un(Xl,...,Xn)).

To fix ideas and show how univalence results are

useful in such a situation we will work out a simple example.

Let XI,X2,X 3 be

80 mutually independent random variables each having a gamma distribution with B = i. Then the joint distribution of XI,X2,X 3 is given by 3 : i=iH ~

~(Xl'X2'X3)

=

I

~i-i -x. xi e l, 0 _< x.1 -< ~

0

and ~.I > 0

otherwise.

Let Ul(Xl,X2,X 3)

=

Xl/(Xl+X2+X 3)

u2(xl,x2,x 3)

=

xJ(xl+x2+x 3)

u3(xl,x2,x 3)

=

xI+ x2 + x3

and

Problem is to find the joint distribution of (UI,U2,U 3) (we are using capital letters to denote random variables).

In order to do that, first we have to check

whether the map F = (Ul,U2,U3) is one-one in the effective domain of (Xl,X2,X3). In this example we have to verify whether F is one-one in the positive orthant of R 3.

We now write down the Jacobian matrix

J

=

u3-x I 2 u3

xI 2 u3

xI 2 u3

x2 2 u3

u 3-x 2 2 u3

x2 2 u3

i

i

i

One can easily check that the Jaeobian matrix is a P matrix for every x in the positive orthant of R 3.

In fact in this example one can easily write down the

inverse map. In fact F-l(ul,u2,u 3) = (UlU3, u2u3, u3(l-Ul-U2)) where (Ul,U2,U 3) is a point in the range of F. Now one can write down the joint distribution of UI, U2, U3 by means of the following formula. Y(Ul,U2,U 3) whenever u I > O, u 2 > 0, Ul+U 2 < i

and

I %(UlU 3, u2u 3, u3 (l_Ul_U2)) = ~-~ 0 < u3 < ~

and Y = 0 otherwise.

Explicit

expression for ~(Ul,U2,U 3) is given by a2-1 u2

al+~2+~3-I ~ - I u3 uI

~3-i (l-Ul-U 2)

-u e

3/(r(al)r(~2)r(a3))

81

In particular joint distribution h of (UI,U2) is given by

h(Ul,U 2) =

r (~i+~2+~3) ~i-I ~2-i ~3-i u2 r(~l)F(o2)r(~3) Ul (l-Ul-U 2)

when 0 < Ul,U 2 and u I + u 2 < i

an@

h = 0 otherwise.

Random variables UI,U2 that

have a joint distribution of this form are said to have a Dirichlet distribution with parameters ~i' ~2' ~3"

One can easily check that the marginal of U I or U2

is a beta distribution.

On the existence and uniqueness of solutions in Nonlinear Complimentarity theory: Let F : R n ÷ R n be a continuous mapping where R+ is the non-negative orthant of R + n n" We are interested in finding a non-negative vector z s R+n such that F(z) E R+n and z.F(z) = 0. F(z).

~re

z.F(z) stands for the inner product between the two vectors z and

We call this problem the complimentarity problem associated with F.

The

CP (= Complimentarity Problem) associated with F is said to be globally uniquely solvable if for any vector q s R n the solution.

CP associated with F(.) + q has a unique

We will follow the approach of Megiddo and Kojima to give a solution

to the problem under consideration. G is a map from R n to R n.

Define G(x) = F(x+) + x- where ii i

x +i

Let G be an extension of F defined as follows.

if

xi > 0

if

xi < 0

=

x and

Let F:R~ ÷ R n be a continuous mapping.

x

i

if

xi < 0

if

xi > 0

=

0

Then the CP associated with F is globally

uniquely solvable if and only if for every q c R n, there is a unique x = x(q) s R+n such that F(x) + q s Rn+ and fi(x) + qi = 0 for those i where x.1 > 0. This then is equivalent to the existence of a unique z = z(q) s R n such that F(z+) + q = - z- or or G(z) = - q. Theorem i :

Now we have the following.

Let F:R~ ÷ R n be a coni~inuous madding.

Then the CP associated with F

is globally uniauely solvable if and only if extension G of F is a homeomorphism of R n onto itself.

Remark :

Suppose F:R n ÷ R n is a co~atinuous mapping such that the CP associated with

F is globally uniquely solvable.

T~en the solution of the CP associated with F(.)+ q

is a continuous function of the vector q for every q E R n.

This is a consequence

of the fact that, if z = G-l(q ) then x = z+ is a solution for the CP associated with F(.) + q.

Now we shall prove the following theorem giving sufficient conditions

on the map F such that the CP associated with F+ q will have at most one solution for every q s R n.

82

Theorem 2 : (a)

If F:R~ + R n is a continuously differentiable function such that

the Jaeobian associated with F is a P-matrix for every x e R+n then the CP associated with F+ q has at most one solution for every q ~ R n. (b)

If F:R~ ÷ R n

is a differentiable mapping such that all the principal miners

of the Jacobian matrix of F are bounded between 6 and 6 -1 for some 0 < 6 < I, then the CP associated with F is globally uniquely solvable.

Proof (a) :

From the fundamental theorem of Gale and Nikaido it follows that F

is a P-function in R+n.

That is for every x # y we have max(xi-Y i) (fi(x)-fi(y)) > O.

Consequently there can be at most one solution to the complimentarity problem, x~ This can be seen as follows.

O, F(x) _> 0

and

x.F(x)

= 0

Suppose x # y with

x _> O, F(x) _~ 0

and

x.F(x)

=

0

y_~ o, F(y) 3 o

and

y.F(y)

:

0 .

Consider for any i, (xi-Yi) (fi (x)-fi (y)) =

xifi(x) + Yifi(y) - xifi(y) - Yifi(x)

= - xif i(y) - Yifi(x) < 0 This contradicts the fact that F is a P-function. also a P-function for any q s R n. most one solution. Proof (b) :

Since F is a P-function F+ q is

Consequently CP associated with F+ q also has at

This terminates the proof of (a) of theorem 2.

We will now prove that the extension map G of F is a homeomorphism.

Then from theorem i it will follow that F is globally uniquely solvable. G(z) = F(z+) + z- for any z ~ R n.

Recall

Observe that the mappings x ÷ x+ and x + x- are

differentiable in points x such that x. ~ 0 (i = 1,2,...,n), it follows that G is i differentiable in the interior of every orthant of R n. However we can approximate G by differentiable functions G ( x ) functions.

Specially, for any ~ > 0 I t(~,~) =

0 (~+ ~)2/4~

and then apply Gale-Nikaido's theorem to these and any real number ~ let if

~ 0. Since G

is univalent in R n, u = v and consequently S = T.

Hence it follows that

x = y since G is univalent in the interior of every orthant. Assume, by induction, that G(x) = G(y) and II(x) l + Suppose x ~ y with G(x) = G(y) and ll(x) I + suppose II(x) I ~ II(y) l.

ll(y) I ~ k

ll(y) I = k+ i.

imply

We shall show first G is locally univalent.

a neighbourhood of x such that

y ~ N x and l(u) C

x = y.

Without loss of generality, Let N x denote

l(x) for every u s N x.

We claim

that G is univalent in N . Let u and v be two distinct points of N . We distinguish x x First l(u) = l(x). In this case there exists an orthant QS such that

two cases.

x,u,v s QS and since G is univalent in every orthant, G(u) # G(v). l(u)

~

l(x).

In this case ll(u) l +

hypothesis implies G(u) # G(v).

II(v) l ~ 211(x) I ~ k + l ,

so that the induction

Fence G is univalent on N .

It follows from the invariance theorem of d o m a i n t h a t contains G(y).

Second,

G(Nx) is an open set which

Since G is continuous at y, there exists w ~ N

such that l(w) =

and G(w) s G(Nx).

Thus, we can find a u s N x such that G(w) = G(u).

[l(w) l +

k+ i, it follows from the induction hypothesis that w = u which is

ll(u) l
0

and y~ = 0 if Yi ~ 0; yi = 0 if Yi ~ 0 and yi = - Yi if Yi ~ 0.

Similarly

we write

4 dxZ Then, +

-

+

Yi = Yi-Yi

'

4 i-x

xi =

-

+

yi,y i ~ 0

,

,x

> 0

-

and

yiy i = 0

for

i = 1,2,...,n

and

x.x.

for

i = 1,2,...,n

+i-

--

= 0 -I-

-

-

for each i = 1,2,...,n,

i = 1,2,...,n.

Hence y+.x + = y-.x- = 0 [Here dot refers to the inner product

that xiY i = 0 = xiY i for

Since y = Ax, we have y+Ax - + = y--Ax- = qo

the vectors].

x # 0, x+ # x-.

it follows

4-

Since xiY i < 0

between

Hence one can conclude =

y+

and

x+.y+

=

o

Ax- + qo

=

Y-

and

x-.y-

=

0

qo

Ax+

qo has a unique

has two solutions

which contradicts

solution.

(say).

Since

from,

~++qo

Ax+

.

1

our assumption

Fence A is a P-matrix.

that CP associated

This terminates

with

the proof

of the corollary. In theorem 2(b) we have shown that the CP-associated provided

all the principal

minors of the Jacobian

and 6 -1 for some 0 < ~ < I. may fail without

The following

this condition.

example

with F is globally

solvable

matrix of F are bounded between demonstrates

that this result

Let F = (f,g,h) be a map from R +3 to R 3 where

f(xl,x2,x 3)

=

xI x2 -x 2 x (e - e - e + l)e 3

g(xl,x2,x 3)

=

x2 xI -x I x + l)e 3 (e - e - e

h (Xl,X2,X 3 )

=

x3 .

In this case Jacobian matrix turns out to be xI e

x e 3 -x I

xI

J

=

(-e

x2 (-e

+e

0 Clearly all the principal

x~ )e 5

-x + e x3

e

x e

0

x 2)e 3

x x I -x 2 x2 e 3(e -e -e + i) x x 2 x I -x I e 3(e -e -e + I)

I

minors of J are all bounded below by one but some principal

85 Xl minors

(for example e

x3 e ) are not bounded above.

theorem 2(b) are not met.

Thus the conditions

imposed on

We will now show that the CP associated with F is not

globally uniquely solvable.

Let q = (e,e,l) and G = F(Xl,X2,X3)- q.

associated with G is not solvable:

Then the CP

Suppose the CP associated with G has a solution.

This means that that there exists a z e R+3 such that G(z) s R+3 with z.G(z) = 0. Let z = (Zl,Z2,Z3).

Since G(z) s __R~' it follows that z3-1 _> 0

or

z 3 _> i.

z.G(z) = 0, and z 3 >_ I third coordinate of G(z) = O or z3-1 = 0 or z 3 = i.

Since Now

we have, zI (e

z2

-z 2

- e

- e

+ l)e > e

and z2 (i÷ e since O(z) s R+3.

zI

-z I

- e

- e

)e >_e

From these inequalities we have, zI e

z2 -e

-z 2 -e

>

0

>

0

and

z2 e

zI -e

-z I -e

-z I Adding these two inequalities

we get, -(e

thus we arrive at a contradiction has a solution.

-z 2 + e

) > 0 which is impossible and

to the supposition that the CP associated with G

Thus in this example F is not globally uniquely solvable.

Note that the CP associated with F+ q for any q ~ R 3 is feasible, exists z _> 0 such that F(z) + q > 0. (oo,~,oo).

An application of Hadamard's

This is met here since

inverse function theorem to Algebra

result of Hadamard we will demonstrate with the structure of a commutative

[22] :

x(~y)

=

~xy

(ii)

x(y+ z) = xy + xz

(iii)

xy = 0

(iv)

xy : yx.

~

By using a

More precisely we

consider the possibility of defining an operation of "multiplication"

(i)

(x,y) ÷ xy

:

where ~ is a scalar.

x = 0

or

y = 0

We now have the following theorem.

Theorem 3 : satisfies

For

n > 3

(i) - (iv).

=

that Euclidean n-space cannot be endowed

division algebra when n > 3.

of R n which obeys the following axioms

that is, there

limit F(l,l,a) a + oo

there is no operation of multiplication

on R n which

86

Remark

:

It is clear that when n = i or 2, we can define a multiplication

satisfying

(i) through

(iv) and also associative

possible to do so in R n for n > 3.

law.

Theorem 3 says that it is not

We will use a special case of a theorem due to

Fadamard to prove theorem 3 - this ~roof is due to Gordon

Hadamard's theorem n £ 3.

:

operation

Let F:R n - {0} ÷ R n - {0}

Then F is a diffeomorphism

be a

of R n - {0} onto

proper and the Jacobian of F never vanishes.

[21, 22].

C (I) differentiable

map with

R n - {0} if and only if F is

[For a proof of Hadamard's

theorem

see [21]]. The following two examples when n = i or n = 2.

show that Fadamard's theorem may fail to hold good

When n = I, f(x) = x 2 is a proper map with derivative non-zero

for x ~ 0 and clearly it is not one-one. 2 2 f(xl,x 2) = x I - x 2 and g(xl,x2) = 2XlX 2. and its Jacobian is non-vanishing

When n = 2, define F(Xl,X2) = (f,g) where One can easily check that F is proper

in R~-{0}.

Also F is not one-one since F (i,i) =

F(-1,-l). Remark

:

Proof of Hadamard's

connected when

theorem depends on the fact that Rn-{0}

is simply

n > 3.

Proof of theorem 3 :

(Proof due to Gordon)

We will show that axioms

(iv) imply that the map x ÷ x 2 is a homeomorohism when n > 3 absurd since axiom

(i) requires that

(-x) 2 = x2).

Let F be the map from R n - {0} to R n - (0} where F(x) = x 2. well-defined map because of axiom check that F is continuous

(i) through

(which is obviously

(iii).

and proper.

This is a

Using the relevant axioms one can easily Using axiom

(iv) we can compute dFx(V)

(-- the differential F and x operating on v) and it is seen that dFx(V)

=

lim h÷0

{ ~i (F(x+hv)-F(x))}

= xv+

vx = 2xv.

Hence axiom (iii) implies that dF is non-singular for all x s Rn-(0} so that x F(x) = x 2 is a homeomorphism via Fadamard's theorem which of course leads to a contradiction.

Hence there cannot exist a multiplication

operation in R n for n >_ 3

satisfying the four axioms.

This terminates

the proof of theorem 3.

On the infinite divisibility

of multivariate

gamma distributions

:

in this section

we will consider the problem of deciding whether a certain multivariate distribution Paranjape.

is infinitely divisible.

We will give a sufficient

gamma

condition due to

In this situation weak N-matrices play an important role.

Let X = (XI,X2,...,x ~) be a multivariate

normal random vector with zero mean

vector and positive definite variance covariance matrix E.

The characteristic

87

2 2 2 xI x2 x function of ( 2 ' 2 .... ' P2 ) is given by hp (t) = h(tl,t2,...,tp) = II-T21-½ where T is a diagonal matrix with diagonal elements itl,...,itp, i = -~.

Call hp(t)

infinitely divisible if (hp(t)) a is a characteristic function for every a > 0.

Paul

Levy conjectured that h (t) is not infinitely divisible. However Vere-Jones proved P h2(t) is infinitely divisible, and Moran and Vere-Jones established hp(t) is infinitely divisible if ~ = (i-0)I + 0Epp with 0 > 0, I = identity matrix and E matrix with every entry equal to one. Also Griffiths obtained a necessary and PP sufficient conditions for h3(t) to be infinitely divisible. We will present a result due to Paranjape which in some sense unifies the known results. posed by Levy is still open.

The problem

We will now prove the following:

Infinite divisibility theorem : If for a set of positive constants Cl,C2,...,Cp, the matrix (diag (Cl,C2,...,Cp)Z -I - I) is a weak N-matrix (that is all the principal minors are non-positive) then hp(t) is infinitely divisible.

Remark : If f(t) is a characteristic function then for 0 < ~ < i, (l-~)/(l-~(f(t)) is infinitely divisible.

If Cl,C2,...,Cp are positive constants, then h2(t)p can be

written as hp2(t)

=

P P H gj(tj)Iz-ll H c./det (I+A) j=l i=l !

where gj(tj) = (l-icjtj)-I and A = (diag(cl,c2,...,Cp)Z-l-l)diag(gl(t l),g2(t2),...,

gp(tp)). Proof :

(of infinite divisibility theorem).

Observe that det (I+ A) = I + ~ tr.A i=l m

where tr.A denotes the sum of all principal minors of order i, I < i < p of the matrix A.

Write P = diag(cl,c2,...,Cp)~ -I - I.

A typical rth order principal minor

of A is equal to r j=l

~. (tk) P(kl,k 2 ..... k r) J J

where P(kl,k2,...,k r) is the (kl,k2,...,kr)-th principal minor of P. I = ~ r=l

~ (-P(kl,k2,...,kr)). (kl,...,k r)

of the theorem, it follows that

I > 0.

Set

Since each P(kl,k2,...,kr) ~ 0 by hypothesis

Therefore

88 p

r

det(I+ A) = i-~ [

~

(-P(k I . . . . . kr)/k ) fI gk.(tk.)]

r=l (kl,...,kr) Since -P(kl,...,kr)/k ~ 0

and

j=l

Z(-P(kl,k2,...,kr))/k

~

j

= I, it follows that the

expression in the right hand side which appears within the brackets in the expansion of det (I+ A), is a multivariate characteristic function. hE(t )=det (E-1)

Define

P

H c./det(l+ A). i=l z P

Since hp(O) = 1, h~(O) = 1

and consequently det(E -I)

ci/(1-~) = i.

Since the

i=l numerator is positive, I-X > 0

or X < i. Thus h*(t) has the representation of the P form (l-~)/(l-~f(t)) where f(t) is a characteristic function and therefore h*(t) P is infinitely divisible. Consequently h~(t) is infinitely divisible or hp(t) is infinitely divisible. divisibility of

This terminates the proof of the theorem on infinite

hp(t). i

Remark : The following result due to Rao gives an equivalent condition for a matrix A to be a weakly N-matrix.

A nonpositive symmetric matrix A(# 0) is merely

positive subdefinite if and only if it is a weakly N-matrix. Rao [62] x,x'Ax < 0

For a proof see

[Call a real symmetric matrix A positive subdefinite if for any vector implies Ax < 0

or

Ax > 0.

A positive submatrix which is not positive

semidefinite is called a merely positive subdefinite matrix].

Examples :

We will prove

h2(t) is always infinitely divisible by actually checking

the condition given in the theorem.

If ~-I = I~

one can easily check, diag (ci,c2)~-i-I = Lb0

1

bla I

i I cI = ~ , c 2 =

choose

Clearly

principal

/c minors are non-positive.

Hence h2(t) is infinitely divisible.

we will look at Griffith's result in three dimensions where

Z = (l-p)l + PE33 =

p

I

p

p

p

i

As another example

89

1-p 2 (1-p)2(1+ 2p)

T.- 1 =

p2 -p (l-p) 2 (1+ 2p)

p2 -p (1-p)2(l+ 2p)

1-p 2

p2 -p

p2 -p (l-p) 2 (1+ 2p)

(l-p) 2 (Z+2p)

(l-p) 2 (1+ 2p)

p2 -p

1-p 2

(l-p) 2 (l+ 2p)

(z-p) 2 (i+ 2p)

p2 -p (1-p)2 (1+ 2p)

Choose cI = c2 = c 3 = (l-p)2 (I+ 2p)/(l-p2). weak N-matrix provided p > 0. is of the form (l-p)l + PE33 h3(t) is infinitely divisible.

Therefore

Then (diag(cl,c2,e3)~-l-l) will be a

h3(t)

where p ~ O.

will be infinitely divisible provided

In general it is not known whether

CPAPTER

X

FIRTI~ER 6ENERALIZATIONS AND REMARKS

Abstract :

In this chapter we will first discuss a generalization of local inverse

function theorem due to Clarke and Pedamard's theorem due to Pourciau when F is a h

~

Lipschitzian function but not necessarily a C ~I) function. the notion of monotone functions.

We will then discuss

Next we will say something about PL functions.

Finally we will discuss global univalent results when the Jacobian is allowed to vanish.

Main aim of this chaoter is to indicate oossible generalizations in global

univalent results.

A generalization of the local inverse function theorem :

Here we will describe a

nice generalization of the local inverse function theorem due to Clarke. F : R n ÷ R n satisfy a Lipschitz condition in a neighbourhood of a point

Let Yo

in Rn"

That is for some constant d, for all x and y near Yo' we have

where II'II denotes as usual the Euclidean norm. the partial derivatives exist.

Let J denote the Jacobian whenever

We will metrize the vector space M of n x n matrices

with the norm,

11AII

= max laij I

A

=

where

Definition :

(aij), I < i < n

and

I < j < n.

The generalized Jaeobian of F at Yo denoted by ~F (yo), is the convex

hull of all matrices A of the form,

A

=

lim n÷~

J (yn)

where Yn converges to Yo and F is differentiable at

Remark I :

Yn

for each n.

It follows from Rademacher's theorem that F is almost everywhere

differentiable near Yo"

Furthermore J(y) is bounded near Yo since F satisfies a

Lipschitz condition in a neighbourhood of Yo"

Remark 2 :

The generalized Jacobian 8F(y o) is a nonempty compact convex subset,

in the vector space M of matrices.

91

Remark 3 :

If F is C (I), 8F(y o) reduces to J(yo ) .

Definition :

The generalized Jacobian 8F (yo) is said to be of maximal rank if every

A in ~F(Yo ) is of maximal rank. We are ready to state the theorem due to Clarke.

Ceneralized local inverse function theorem :

If ~F(y o) is of maximal rank, then

there exist onen sets U and V of Yo and F(Yo) respectively, and a Lipschitzian function C:V ÷ R n (i)

G(F(u))

=

~

for every

u s U

(ii)

F(G(v))

=

v

for every

v c V

For a proof of this result see Clarke [12].

In view of remark 3, it is clear that

when F is C (I), Q is necessarily C (I). Also observe that it is not sufficient to assume that J is of maximal rank whenever it exists, as the function Ixl, (n=l) demonstrates.

A simple example to which the above theorem applies, n = 2, is the

followinz: F(x,y) = (IxI+y, 2x+ly]) near 3F(O,O)

:

s

[ [2

(0,0).

Here one can check t h a t ,

1

t ] : - 1 < s < i, -i < t < I]

Immediately one can raise the following question in view of this result due to Clarke : Is it possible to formulate a ~lobal univalent result when F satisfies Lipschitz condition but F not necessarily a C (I) function ?. In fact Pourciau recently has shown that Hadamard's theorem holds good for locally Lioschitzian maps.

In order

to state the theorem due to Pourciau [58] we need to slightly extend the concept of generalized Jacobian that was given earlier. Let F:R n ÷ R n.

Call F locally LiDschitzian nrovided each point x has a

neighbourhood U where some positive number M satisfiesIIF(Zl)-F(z2)ll < for all Zl,Z 2 E U. measure.

MIIZl-Z211

As already remarked F' (x) exists a.e. with respect to Lebesgue

Moreover almost every x is a Lebes~ule point of the derived mapping F'.

By definition such x satisfy

l~mit s + 0

1 ~(B~(x)) /

IIF'(z)-F'(x)fld~(z) = 0

BE(x)

~mre £(Be(x)) stands for the Lebesgne measure of Be(x) centre x).

(= Ball of radius e with

Let L(F') stand for the set of all these Lebesgue points and let p E R n.

Then the generalized derivative 8F(p) of F at o is the non-empty, comoact convex subset

~ A6(F,p) of L(Rn,Rn), where A6(F, p) denotes the collection 6>O Co----n{F'(x) : x e B6(p) ~

L(F')} .

This extra condition allows us to ignore null sets in forming the generalized

92

derivative.

With this notion of generalized derivative, Pourciau has proved the

local inverse function theorem stated in chaoter I for Lipschitz functions.

For

details see [57].We are now ready to state the following:

Lipschitzian F~damard Theorem : into R n and let M > O.

Suppose F is a locally Lipschitzian map from R n

If ~F(D) is invertible and IIA-III < M for each p ~ R n

and each A in 8F(p), then F is a homeomorDhism from R n onto R n. For a proof of this result see [58].

Remark I :

It appears that some of the results proved in chapter IV, in particular

Plastock's results might be suitably formulated for locally Lipschitzian maps.

Remark 2 :

F~damard's theorem is generalized to Banach spaces by Caccioppoli [6]

and Plastock has proved some of his results for Banach soaces.

As such can we

assert that Lipschitzian Fadamard's theorem holds good for Banach soaces?

Remark 3 :

Is it possible to formulate Mas-Colell's or Oarcia-Zangwill's global

univalent result for locally Lioschitzian maos ?

Monotone functions and univalent mappings :

In the introduction we have mentioned

as one of the approaches for tacklingglobal univalent problem, is via monotone functions.

In this section we will present an example of that aoproach.

Definition : x,y E R n.

A mapping F:Rn-~ R n

is monotone if (F (x)-F (y))'(x-y) > O

Call F strictly monotone if (F(x)-F(y))'(x-y) > 0

for every

whenever x # y.

Call

F uniformly monotone if there is a ~ > 0 so that (F(x)-F(y))'(x-y) > 6 (x-y)(x-y) for all

x,y E R n.

[Here as usual prime denotes transpose]. This concept is a natural non-linear generalization of positive definiteness. The following proposition is easy to prove.

Proposition i :

Let ~ be an open convex set in R n.

Let F:O ÷ R n

be a C (I) map.

Then (i)

F is monotone if and only if J is

positive semi-definite for all

(ii)

If J is quasi-positive definite for all x E ~, then F is strictly monotone on ~.

(iii) F is uniformly monotone on ~Q if and only if there is a ~ > 0 so that h'Jh > ~h'h for all x ~ ~ and h ~ R n.

x s ~.

93

Proof :

We will prove (i) and (iii) simultaneously.

monotone.

Then for any x E ~

and

h'J(x) h =

_>

Suppose F is uniformly

h ~ R n, we have h'lim t+O lim t+O

[F(x+ th)-F(x)]

t -2 ~IIthll 2 = ~h'h

If F is monotone then $ = 0 and J(x) is positive semi-definite. h'J(x)h> ~h'h.

Conversely suppose

Then from the mean-value theorem we have, (x-y)'F(x)-F(y))

=

i S (x-y)'J(y+ t(x-y))(x-y)dt 0 ~(x-y)'(x-y)

so that F is monotone or uniformly monotone depending on whether ~ = 0 or

6 ~ O.

Finally~ if J(x) is quasi-positive definite for all x s ~, and x ~ y, then the above integrand is positive for all t E [C,l] and hence F is strictly monotone.

This

terminates the proof of Proposition i. Proof of this proposition is taken from Ortega and Rheinboldt [48].

Remark :

Observe that (ii) of Proposition I is given in Chapter III.

However the

conditions do not suffice for the existence of solutions as the examole F (x) = ex in one-dimension shows.

We may guarantee existence, however, by strengthening the

monotonicity assumption.

Theorem i :

If F:R n ÷ R n

is a C (I) map and uniformly monotone on R n then F is

one-one and onto R n and consequently F is a homeomorohism. From the given hypothesis one can easily verify thatlIJ(x)-iII ~ x s R n.

[Here the norm of the matrix is taken as the Z2 norm].

theorem will imply that F is one-one and onto.

i ~

for all

Now F~damard's

For details of the proof see

142 or 48].

Remark i :

If F is continuous and montone on the open, convex set ~, then for any

b ~ R n, the solution set S = ~x ~ ~ : F(x) = b) is convex if it is not empty.

Remark 2 :

Suppose that F,G : R n ÷ R n

F is uniformly monotone. R n onto R n.

are both C (I) maps and montone and that

Then the map H defined by : H = F+ G is a homeomorphism of

In particular if A is a quasi-positive definite matrix, H = A + G is a

homeomorphism.

(See problem E

On PL-funetions

:

5.4-5 in [48]).

Because of the growing importance (as well as simplicity) of

94

piecewiselinear Kojima-Saigal

functions, in this section we will prove a result due to

[32].

In order to do that we need some preliminaries.

Let S be a closed convex polyhedral subset of R n, and let ~ be a class of closed

convex polyhedral subsets of S which partition S.

We will assume E contains

only finitely many members.

Definition : (a)

Call (S,Z) a subdivided polyhedron of dimension n if :

elements of Z are n-dimensional convex and polyhedral and are called pieces.

For

(b)

any two

members of Z are either disjoint or meet on a common face.

(c)

the union of the pieces in Z is S.

our result S = R n.

Let (Rn,Z) be a subdivided polyhedron, and let F

:

Rn ÷ R n

be piecewise linear and continuously differentiable on this subdivision, that is, PC I with affine on each piece of Z.

Since Z contains a finite number of pieces,

outside some compact region, points of R n lie in some unbounded piece in Z.

Let

these unbounded pieces be numbered al,~2,...,Uk and let F(x) l~" = Aix - a i for i some n x n matrices Ai, and n vectors a i.

Theorem 2 :

Then we have the following:

Suppose that the Jacobian matrix of each piece of linearity of F has

a positive determinant.

Also, let there exist

a matrix B such that (l-t)B+ tA i

is non-singular for each t E [0,i] and i = 1,2,...,k.

Then F is a homeomorphism

of R n onto R n.

Remark :

Let z s Rn,

containing z. for each

such that det JF (z) is positive

Then, there exists s > 0

6 s (0,~).

Proof of theorem 2 :

(negative) for every

such that deg (F,B~(z), F(z)) ~ i (~ - i)

For a ~roof of this remark see [32].

Let y s R n

be arbitrary.

Now consider the homotopy

H(x,t) : (l-t)Bx + t(F(x)-y), t s [0,i]. We will first prove that H-I(0) has no unbounded component.

Assume the contrary

this means that for some ai we can find a sequence (xm,tm) s ~--i(0), m such that x m s ~i and llxmll ÷ ~. t -~ t*, t* E [0,i] and x* # 0.

1,2,...

Also on some subsequence xm/llxmll ÷ x*, ~ence we have

m

(l-t*)Bx* + t* A.x* = 0 ]

95

which is a contradiction to our hypothesis that (l-t)B + tA. is non-singular for 1

each t s [0,i].

Since H-I(0) is bounded for each y, and det B > 0, from the

homotopy invariance theorem the degree of F(x)-y is + i for all y. remark it follows that F is one-one and onto. Now we will give two examples. are satisfied.

From the above

This terminates the proof of theorem 2.

In the first example conditions of theorem 2

In the second example one of the conditions of theorem 2 will be

violated but the map will be a homeomorphism.

Example I :

Consider F:R 2 + R 2

where

=f(x-y,y)

if

x~ 0

if

x < 0

F (x,y) ~(2x-y,y) In this example A I = [0

i ]' A2 = [0

i ] and let B = [

check that (l-t)B + tA. is n o n - s i n ~ l a r is a homeomo~hism

].

One can easily

for each t c [0,i] s~_d i = 1,2.

~nce

F

of mR2 onto R 2.

We need the following lemma for examole 2.

Lemma :

-i Let A I = [-2

0 1 ]'

-i A2 = [ 2

0 i ]'

i A3 = [0

there does not exist any matrix B such that tB + t s [O,i]

Proof :

-2 -i ]

and

i A 4 = [0

2 -i ]"

Then

(l-t)A i is non-singular for every

and i = 1,2,3,4.

One can possibly give a direct proof but our proof depends on theorem 2.

Consider the following diagram, which represents a piecewise linear function F in R 2.

~+ For example

F(x,y)

=

(-x,-2x+y)

when

y>x>0

=

(x+ 2y, -y)

when

0~-y~x.

96

Clearly the given AI, A2, A3, A 4

are the Jacobians of F in different regions.

Also note that F(½, I) = F(½,-I) = (-½, 0) and consequently the given homeomorphism.

F

is not a

Hence from theorem 2 we can conclude that there cannot exist a

matrix B with tB + (l-t)A i nonsingular for every t s [0,I] and i = 1,2,3,4.

This

terminates the proof of the lemma. We will use this lemma in our next example to show that the condition given in theorem 2 is not a necessary condition for an F to be a homeomorphism.

Example 2 :

See the diagram below.

In this example R 2 is divided into 6 regions

and in each region we have given the Jacobian of the function F.

Clearly F is a

homeomorphism but this F does not satisfy the condition imposed in theorem 2.

In

other words this example demonstrates that the condition "tB+ (l~t)A i is non-singular for every t s [0,i] and i = 1,2,...,k for some matrix B" is not a necessary condition for homeomorphism.

I [0

-2 _l ]

-i 0 [21 ] Because of the growing i~oortance of PL-functions it would be nice to give univalence conditions that are both necessary and sufficient.

Recently Schramm in

fact has given in [70] necessary and sufficient conditions for PL-functions to be a homeomorphism.

Among other results he proves that when the determinants

associated with a PL-funetion F:Rn ÷ R n have the same sign, F is an open mapping. Also he is able to give bounds for the number of solutions of certain PL-equations. For details see [69].

¢~le-Nikaido's results have also been extended by Kojima-

Saigal for piecewise continuously differentiable, pcl-fumctions [32].

On a global univalent result when the Jacobian vanishes :

In this section we will

give a global tunivalent result due to Chua and Lam [9] when the Jacobian is allowed

97

to vanish on a set of isolated points. thereby extending

Gale-Nikaido's

Kojima-Saigal

fundamental

have shown a similar result

theorem.

For the proof of Chua and

Lam we need the following results on local homeomorphism where the Jacobian may be allowed to vanish. (i)

Let F be C I map from an open set U e - R n + R n and n > 3.

t s U\{t

}

where t

O

t .

is an isolated point.

Then F is a local homeomorphism

(ii)

[ii].

Let F be a C (n) map from R n to R n where n ~ 3.

zeros of the Jacobian.

If Rn_ I is zero-dimensional

This result is due to Church

Remarks

:

Let Rn_ I denote the set of then F is a local homeomorphism.

[i0].

It is interesting to note that both the results are valid when n > 3.

For n = 2, the analytic function f(z) = z 2 is an obvious counter example. of

at

O

This result is due to Church and Hemmingsen

O

Suppose J(t) # 0,

(i) and (ii) depend on the fact that the set of branch points

points at which F fails to be a local homeomorphism)

Proofs

(= collection of

forms a oerfect set when n ~ 3.

We are ready to state the following

Theorem 3 :

(Chua and Lam)

and J(x) is singular)

Let F:R n ÷ R n

cient for F to be a homeomorphism

Proof

(i)

det J(x) > 0

(2)

F is norm-coercive.

:

be a C I mao where n # 2.

and I c = (x:x ~ I).

are suffi-

of R n onto Rn:

for all x s I c and I is at most a set of isolated points.

First we will prove theorem 3 when n = i.

Suppose F:R I + R I

This means we can find x I # x 2 with F(x I) = F(x2). exists an interval

Let l = { x : x s R n

Then the following conditions

Condition

is not one-one

(i) implies that there

(c,d) with x I ~ c < d ~ x 2 such that F'(x) > 0 on (c,d).

Since

F is C I, we have x2

F(x 2)-F(x l)

d

= / F'(x)dx~ x1

which is a contradiction. Since F is norm coercive,

c

Fence F is one-one and consequently F is a homeomorphism. F is onto R

and this terminates

We will now consider the case when n > 3. an open set N u about u in R n with N u ¢~ I = (u}.

quoted above that F is a local homeomorphism

the proof when n = i.

Let u be a point in I.

There exists

Since det J(x) ~ 0 on N u except

at the isolated point u where the Jacobian vanishes,

non-singular

/ F'(x)dx>O

it follows from the results

on N u for each u s I.

Since J(x) is

for every x s I c, it follows that F is a local homeomorphism

for all n ~ 3.

Hence we can conclude that F is a homeomorohism

norm-coercive and a local homeomorphism).

on R n

of R n (as F is

98

Proof of the theorem will be complete if we show that F is onto R n • Suppose F(R n) is a proper subset of R n.

Observe F(R n) is open as R n is open.

be a boundary point of F (Rn) and let ~ •

.

n

Since F is a finlte covering mapping on R , F number of components.

N~ are open and connected. onto M b.

-I

(Mb) has a finite and a non-zero

Let N b be a component of

Let N~ = F(Rn)g] F-I(Mb).

Let b s R n

an open connected neighbourhood of F (b).

F-I(Mb ) that contains the point b.

Since F is continuous F -I is open.

Fence, both N b and

Also observe that F maps both N b and N~ topologically

Clearly Nbf~ F(Rn)# ~

as the point b belongs to both N b and F(Rn).

follows that N b C ~ N ~ ~ ~, for otherwise there will be at least one point

xI

It in

N b (] F(R n) and a point x 2 in N~ such that F(x I) = F(x 2) s M b and F restricted to F(R n) will not be one-one, which is a contradiction. we have N b = N~. R n.

As both N b and N~ are connected,

Hence it follows that F(R n) cannot be an open proper subset of

That is F(Rn), is closed in R n.

and it is non-empty.

We have F(R n) which is both open and closed

Therefore we can conclude that F(R n) = R n.

This completes

the proof of theorem 3.

Remark i :

We have used theorem 3 in chaoter V to prove theorem 2".

We have given

an example there which will serve as a counter example to theorem 3 in R 2.

For

extensions of Gale-Nikaido's univalent results and other results along these lines we refer the readers to [32], [9].

Another result which is closely related to

theorem 3 is the following theorem due to Cronin and

Theorem 4 :

McAuley [14].

Let A and B be the interiors of two unit balls in R n and F a continuously

differentiable light and open map from A to B. of A and B respectively].

Suppose F(A\A)

[Here A and B stand for the closure

= B\B

and F I(A\A) is a homeomorphism.

Let J(x) stand for the Jacobian evaluated at x s A as usual, and let =

{x s A

:

J(x) = O}

=

(x s A

:

J(x) > O)

J

=

{x s A

:

J(x) < O}

J1

=

~+~

J O

J

+

~ g~ A

If dim Jl < (n-l), then F is a homeomorphism.

Proof :

See pp. 406-408

Remark i :

in Cronin and

McAuley [14].

This result enables us to study subsets of the set of points Jo where the

Jacobian is zero instead of dealing directly with the branching or singular set which is a subset of J . O

99

Remark 2 :

The following example shows that there are many homeomorphie onto

mappings with their Jacobian vanishing on an (n-l) dimensional set.

The following

simple example is a case in point. Let F:R 3 ÷ R 3

This function

F

be defined by Yl

=

fl (x)

=

x~

Y2

=

f2 (x)

=

x~

Y3

=

f3(x) = x I + x 2 + x 3.

has a global inverse on R3; namely :

:

:

Hence, F : R 3 + R 3

is a homeomorphic onto mapping.

2 2 But det J(x) = 9x I x 2 vanishes

on two 2-dimensional hyperplanes : one defined by x I = 0 and the other defined by x 2 = O.

This shows that the condition "dim Jl < n-l" imposed in theorem 4 is not

necessary.

The set J

Remark 3 :

contains the singular set or branch set - see

Ma~uley [37].

O

But it x s J

the map F can be locally one-to-one at x as the example of the map O

Yl

=

x~

Y2

=

x~

in a neighbourhood of (0,O) shows.

Remark 4 :

It may be possible to strengthen theorem 3 as well as theorem 4 with the

help of theorem 6 (as well as theorem 7) of chapter IV but we have not made any attempt to do so.

Injectivity of quasi-isometric mappings : Let ~ be an open subset of X.

Let X,Y stand for real Banach spaces.

Following John, a mapping F:~ ÷ Y is said to be

(m,M)-isometric if it is a local homeomorohism one-one)

for which M > D+F(x)

respectively,

(that is continuous, open and locally

and 0 < m < D-F(x) where D+F(x) and D-F(x) are

the upper and lower limits l lF(y)-F(x)II/IIy-xll

as

y + x.

Less

precisely, F is called quasi-isometric if it is (m,M) - isometric for some m, M. We have then the following theorem due to

Gevirtz.

100 Theorem 5 : Let ~ C - X

Let X and Y be Banach spaces [In particular we can take X = Y = R n]"

be an open ball and let F:~ ÷ Y be an (m,M) isometric mapping.

Then

F is one-one if any one of the following conditions is satisfied.

(a)

M/m < ~o x

=

[x+

where

U o is the unique real root of the equation

(25x2-8x)½]/2(3x2-x).

(b)

X is a Hilbertspace

and M/m < ~

.

(c)

X and Y are Hilbertspaces and M/m < (i + ~)½.

For a proof of this result see J.Cevirtz [Proc. Amer. Math. Soc. 85 (1982), 345-349]. For related results see also F.John [On quasi-isometric mappings I, II Communications to Pure and Appl. Math. 1968, 1969

21,22 pp. 77-110, 265-278].

We will close this chapter by mentioning an

old conjecture due to Jacobi.

[This conjecture was brought to my attention by Professor Garcia]. be a polynomial map.

That is

n variables Zl,Z2,...,z n.

F = (fl,f2,...,fn)

Let F : ~ n + (~n

where each fi is a polynomial in

If F has a polynomial inverse

G = (gl,g2,...,gn),

9f. 1 then the determinant o~ the Jacobian matrix ( ~7. ) is a non-zero constant. J follows from the chain rule. Zi

= gi(fl,f2,...,fn), 6ij

-

This

Since F o G is the identity, we have

so ~ gi(fl'f2'''''fn ) - L

~zj

~gi t=l-~Zt

~ft (fl'

"" "'fn )" 9zj

This shows that the product ~f (~gi "~z. (fl' "'" 'fn ) )" (~zl.) J J is the identity matrix.

Thus the Jacobian determinant of F is a non-vanishing

polynomial, hence a constant.

The Jacobian conjecture :

Now we are ready to state

Let F: ~ n ÷ ~ n

determinant is a non-zero constant.

be a polynomial map such that the Jacobian

Then F has a polynomial inverse.

For more

information on this conjecture see the following articles: D.Wright (1981), On the Jacobian conjecture, ILL.Jour. Math. 25, 423-439. H.Bass, E.H. Connell and D. Wright (1982), The Jacobian conjecture - reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. 7, 287-330.

REFERENCES [I]

S.Banach and S.Mazur (1934), tber mehradeutige stetige abbildnngen, Studia Math 5, 174-178.

[2]

M.S.Berger (1977) Non-linearity and functional analysis, Academic Press, Yew York.

[3]

L.Bers (1956) Topology, Courant Institute of Mathematical Sciences, New York University, New York.

[4]

W.E. Boyce and R.C. DiPrima (1977) Elementary differential equations, Third Edition, John Wiley and Sons, New York.

[5]

F.E.Browder (1954) Covering spaces, fiber spaces and local homeomorphisms, Duke Math Jour 21, 329-336.

[6]

R.Caccioppeli (1932) Sugli elementi uniti delle transformazioni funzionali:un teerema di esistenza e di unicita ed alucune sue applicazioni, Rend. Sem. Math. Padova 3, 1-15.

[7]

P.Chander, The non-linear input-output model, To appear in Jour. Econ. Theory,

[8]

A.Charnes, W.Raike and J.Stuz (1975) Poverses and the economic global unicity theorems of Gale and Nikaido, Zeitschrift for Ooerations Research 19, 115-121.

[9]

L.O.Chua and Y.F.Lam (1972) Global hemeomorphism of vector valued functions, Jeur. Math. Analy and its Appl. 39, 600-624.

[I0]

P.T.Church (1963) Differentiable ooen maps on manifolds, Trans. Amer. Math. Sec. 109, 87-100.

[Ii]

P.T.Church and E.Femmingsen (1960) Light open maps on n-manifolds, Duke. Math. Jour. 27/ 527-536.

[12]

F.F. Clarke (1976) On the inverse function theorem, Pac. Jour. Math. 61, 97-102.

[13]

J.Cronin (1964) Fixed ~oints and topological degree in non-linear analysis, Providence Amer. Math. Soc.

[14]

J.Cronin and L.F. McAuley (1966) Whyburn's conjecture for some differentiable maps, Prec. Natl. Acad. Sci.,U.S.A., 56, 405-412.

[15]

Deimling (1974), Niehtlineare Gleichungen und Abbildungslehre, Springer-Verlag (Hochschul text).

[16]

N.Dunferd and J.T.Schwartz (1963) Linear Operators II, Spectral theory, Self-adjoint operators in Filbert space, Interscience, New York.

[17]

W.Fleming (1977) Functions of several variables, Second Edition, Springer Verlag, Feidelberg - New York.

[18]

D.Gale (1960) The theory of linear ecomic models, Mcgraw F~II Book Company, New York.

102

[19]

D.C~Ie and H.Nikaido (1965) The Jacobian matrix and global univalence of mappings, Math. Ann. 159, 81-93.

[20]

C.B.Garcia and W.I.Zangwill (1979) On univalence and P-matrices, Lin. Alg. and its Appl. 24, 239-250.

[21]

W.B. Cordon (1972) On the diffeomorphisms of euclidean space, Amer. Math. Monthly 79, 755-759.

[22]

W.B. Cordon (1977) An application of Hadamard's inverse function theorem to Algebra, Amer. Math. Monthly 84, 28-29.

[23]

J.Fadamard (1906) Sur les transformations ponctuelles, Bull. Soc. Math. France 34, 71-84.

[24]

J.Fadamard (1910) Sur quelques aoDlications de d'indice de Kronecker in "Introduction a la Theorie des Fonctions d'une Variable" by J.Tannery, pp. 437-477. Ferman, Paris.

[25]

P.F~rtman (1961) On stability in the large for systems of ordinary differential equations, Can. J. Math. 13, 480-492.

[26]

P.Nartman (1964) Ordinary differential equations, John Wiley, New York.

[27]

P.F~rtman and C.01ech (1962) On global asymptotic stability of solutions of differential equations, Trans. Amer. Math. See. 104, 154-178.

[28]

R.Hermann (1968) Differential geometry and the calculus of variations, Math. in. Sci and Engineering Vol. 49, Academic Press, New York.

[29]

R.V.Hogg and A.T.Craig (1970) Introduction to Mathematical Statistics, The Macmillan Company, Collier - Macmillan Limited, London pp 139-141.

[30]

W.Hurewioz and H.Wallman (1948) Dimension Theory, Princeton University Press.

[31]

H.Kestelman (1971) Mapping with non-vanishing Jacobian, Amer. Math. Monthly 78, 662-663.

[32]

M.Kojima and R.Saigal (1979) A study of PC (1) homeomorphisms on subdivided polyhedrons, SIAM J. Math. Anal. i0, 1299-1312.

[33]

C.Kosniewski (1980) A first course in algebraic topology, Cambridge University Press.

[34]

L.Markus and H.Yamabe (1960) Global stability criteria for differential systems, Osaka Math. J. 12, 305-317.

[35]

A.Mas-Colell (1979) F~meomornhisms of compact convex sets and the Jacobian matrix, SIAM J. Math. Anal. i0, 1105-1109.

[36]

A.Mas-Colell (1979) Two ProPositions on the global univalence of systems of cost function, C~neral Equilibrium, Growth and Trade, Edited by J.Green and J.Scheinlsman, Academic Press, New York.

[37]

L.F.MoAuley (1965) Concerning a conjecture of Whyburn on light open mappings, Bull. Amer. Math. Soc. 71, 671-674.

[38]

L.F.McAuley (1966) Conditions under which light open mappings are homeomorphisms, Duke Math. Jour. 33, 445-452.

103

[39]

L.W.McKenzie (1960) Matrices with dominant diagonals and economic theorie, Mathematical Methods in Social Sciences, Stanford University Press.

[40]

N.Megiddo and M.Kojima (1977) On the existence and uniqueness of solutions in non-linear complimentarity theory, Math. Programming 12, 110-130.

[41]

J.Milnor (1965) Topology from the differentiable view point, The University Press of Virginia, Charlottesville, Virginia.

[42]

J.More and W.C.Rheinboldt (1973) On P- and S- functions and related classes of n-dimensional non-linear mappings, Linear Alg. and its Appl. 6, 45-68.

[43]

K.G.Murty (1971) On a characterization of P-matrices SlAM J on Appl. Math. 20, 378-383.

[44]

F.Nikaide (1968) Convex structures and Economic Theory, Academic Press, New York and London.

[45]

H.Nikaido (1972) Relative shares and factor price equalization, Jour. of Intl. Economics 2, 257-264.

[46]

H.Nikaido (1975) Economic adjustments under non-comoetetive pricing, adaptive Economic Models, Academic Press, New York.

[47]

C.Olech (1963) On the global stability of an autonomous system in the plane, Contr. diff. equations I, 389-400.

[48]

J.N.Ortega and W.C.Rheinboldt (1970) Iterative solution of non-linear equations in several variables, Academic Press, New york.

"[49]

G.Owen (1968) Game Theory, W.B.Saunders Company, Philadelphia.

[50]

R.S.Palais (1959) Natural operations on differential forms, Trans. Amer. Math. Soc. 92, 125-141.

[51]

S.R.Paranjape (1978) Simpler oroofs for the infinite divisibility of multivariate gamma distributions, Sankhya Series A 40, 393-398.

[52]

T.Parthasarathy (1982) On the global stability of an autonomous system on the plane, Technical report No. 8203, Indian Statistical Institute, New Delhi [submitted for publication].

[53]

T.Parthasarathy, On univalent mappings (under preparation).

[54]

T.Parthasarathy and T.E.S.Raghavan (1971) Some topics in two-person g a m e s , American Elsevier Company, New York.

[55]

R.Plastock (1974) Homeomorphisms between Banach spaces, Trans. Amer. Math. Soe. 200, 169-183.

[56]

R.J.Plemmons (1977) M-~atrix characterizations l-Nonsingular M-matrices, Lin. Alg. And its Appl. 18, 175-188.

[57]

B.H.Pourciau (1977) Analysis and Optimization of Lipschitz Continuous Mappings, Jour. Opt. Theo~j and its Appl. 22, 311-351.

[58]

B.H.Poureiau (1982) N~damard's theorem for locally Lipschitzian maps, Jour. Math. Analy Appl. 85, 279-285.

[59]

Rado-Reichelderfer (1955) Continuous transformations in analysis, SpringerVerlag.

104

[60]

M.Radulesou and S.Radulescu (1980) Global inversion theorems and applications to differential equations, Non-linear Analysis, Theory, Methods and Applications 4, 951-965.

[61]

T.E.S. Raghavan (1978) Completely mixed games and M-Matrices, Lin.Alg. And its Appl. 21, 35-45.

[62]

P.S.S.N.V.P. Rao (1977) A characterization of merely positive subdefinite matrices and related results, Sankhya Series A, 39, 387-395.

[63]

Rothe (1976) Expository introduction to some aspects of degree theory, published in "Nonlinear function analysis and differential equations" (edited by Cesari, Kannan, Schuur), Marcel Dekker, Lectures in pure and applied mathematics No. 19.

[64]

P.A.Samuelson (1953) Prices of factors and goods in general equilibrium, Rev. Econ. Stud. 21, 1-20.

[65]

Collected Scientific papers of Paul A.Samuelson, Vol. II edited by J.E. Stiglitz (1965), The M.I.T. Press, Cambridge, Massachusetts pp. 902-908.

[66]

H.Samuelson,R.M.Thrall and O.Wesler (1958) A partition theorem of euclidean n-space, Proc. Amer. Math. Soc. 9, 805-807.

[67]

A.Sard (1942) The measure of critical values of differentiable maps, Bull. Amer. Math. Soc. 48, 883-890.

[68]

R.Schramm (1978) Boundary conditions for the univalenoe of NVL-transformations, Math. Ann. 236, 191-198.

[69]

R.Sohramm (1979) Bounds for the number of solutions of certain piecewise linear equations, Proceedings of a conference on Geometry and Differential Geometry, Haiffa University, Springer Verlag, Heidelberg, New York.

[70]

R.Schramm (1980) On piecewise linear functions and piecewise linear equations, Math. O.R. 5, 510-522.

[71]

G.Vodossich (1980) Two remarks on the stability of ordinary differential equations, Non-linear Analysis, Theory, Methods and Appl. 4, 967-974.

[72]

G.T.Whyburn (1942) Analytic Topology, Amer. Math. Soc.

[73]

G.T.Whyburn (1958) Topological Analysis, Princeton University Press.

[74]

G.T.~yburn (1951) An open mapping approach to ~rwitz's theorem, Trans. Amer. Math. Soc. 71, 113-119.

[75]

G.T.Whyburn (1961) Open mappings on 2-dimensional manifolds, Jour. Math. and Mech. 10, 181-198.

I~DEX Browder

29

Local univalent theorem

Caceioppoli

92

Local inverse function theorem

Ch~nder

79

Invariance of domain theorem

Chua-Lam

48,96

Inverse isotone

2 2,91 3 71

Clarke

90

Jacobian

Condition (L)

28

John

99

Complete with resoect to arc length

30

Kojima-Saigal

94

Complimentarity problem

81

Leontief type Jaeobians

68

Kestelman

1,17

Covering soace

3

Critical points

59

Leontief type matrix

Cronin

35

Light mao

37

Line liftin~ property

28 92

Differentiable

I

Diffeomorphism

30

LipschitzianFadamard theorem

Diriehlet distribution

81

Local homeomorohism

Fund~mental global univalence theorem

4,20

Cele-Nikaido-lnada

4,26

Garcia C~rcia-Zangwill Ceneralized Jacobian

6,68

3

Local k-extension

33

Local univalence

2

Mas-Colell

41,78

P-matrices

6

N-matrices

6

N-matrices first kind

6

N-matrices second kind

6

i00 41 90,91

Gevirtz

99

Globally asymptotically stable

59

Globally uniquely solvable

81

Cordon

86

PL functions Positive quasidefinite matrix Positive dominant diagonal matrix Hadamard

26,27

93 6,21 ll

30,86 Merely positive definite matrix

~artman

59

Inada

17

Infinite divisibility theorem

87

7

NVL-Matrix

52

Uniformly dominant diagonal matrix

69

106

Megiddo-Kojima McAuley

83 36,98

Samuelson

77

Samuelson-Thrall-Wesler

83

M-function

71

Scarf

4

Monotone function

92

Schramm

4

Strictly monotone function

92

(Positive) Stable

Uniformly monotone function

92

Stolper-Samuelson

Markus-Yamabe

59

Subdivided polyhedron of dimension n

More-Rheinboldt

15 (condition)

Transformation of class q

34,74

}{ultiplieation operation

85

Nikaido

54

I0 94 i

Univalent result when Jacobian vanishes

96

Univalent (results in R 2 and R 3)

49

Vidossich

65

Non-linear complimentarity theory 81 Norm-coercive mapping

41

Norm-coerciveness theorem

41

Noshira

21

Off-diagonally antitone

yon NeumarLn value

7

Weak P-matrix

48

71

Weakly positive quasi-definite matrix

36

Olech

59

x-simple domain

51

Open map

37

x-simple curve

51

Ortega-Rheinboldt

45

P-function

20

Paranjape

87

Parthasarathy

4

Plastock

4,29

Pourciau

92

Proper mapping

28

Quasi-isometric mappings

99

Radulescu

33

Raghavan

15

Rao

88

Rectangular region

17

Reverse the sign of a vector

i0

Rothe

35