Brouwer Degree and Applications - sorbonne-universite · Inﬂuenced by Brouwer’s construction...

Brouwer Degree and Applications

George Dinca and Jean Mawhin

January 17, 2009

Chapter 1

Introduction

Degree theory for continuous mappings from R to R can of course be traced toBolzano’s intermediate value theorem for continuous functions [32]. Implicit in someof Gauss’ proofs of the fundamental theorem of algebra [144, 145], the concept ofindex or winding number around a closed curve, a forerunner of the notion of degreefor continuous mappings from R2 to R2, was first explicitely defined and developedby Cauchy in a series of memoirs published between 1831 and 1837 [52, 53, 54, 55],first devoted to the special case of holomorphic functions from C to C and then, in1833 (published 1837), to C1 mappings from R2 to R2. See [185], p. 134-136 for aninteresting discussion and references. As shown in [272], Cauchy’s index was widelyused by Poincare in his early work on the qualitative theory of nonlinear di!erentialequations [319].

Cauchy’s index was generalized in 1869 by Kronecker [222] to C1 mappingsf = (f1, f2, . . . , fn) from Rn intoRn, on the compact (n! 1)-dimensional manifoldF!1(0) associated to a smooth mapping F : Rn " R and the regular value 0 ofF. He defined the ‘characteristic of (F, f)’, nowadays called Kronecker’s index of fon F!1(0), by the integral (written here in the modern terminology of di!erentialforms)

iK [f, F!1(0)] =1

µn!1

!

F!1(0)#f#!n

n"

j=1

(!1)j!1fj df1 $ . . . $ #dfj $ . . . dfn (1.1)

where µn!1 is the (n! 1)-dimensional measure of the unit sphere Sn!1 in Rn, and#dfj means that the factor dfj is missing. One of Kronecker’s motivation was someextensions to systems of equations of the results of Sturm, Liouville and Hermite onthe number of real roots of polynomial equations or systems [377, 378, 379, 380, 172].See [185] and [366] for more details, as well as [272] for Poincare’s use of Kronecker’sindex in his qualitative study of nonlinear di!erential equations.

Influenced by Brouwer’s construction (using simplicial methods) of a degree the-ory for continuous mappings between twooriented compact manifolds of the samefinite dimension [39], Hadamard[162] extended in 1910 Kronecker’s integral to con-tinuous mappings and more general (n ! 1)-dimensional compact manifolds. See

3

4 CHAPTER 1. INTRODUCTION

[133] for the role of Hadamard in the genesis of Brouwer’s ideas. After Brouwer’spioneering paper, degree theory was essentially developed using algebraic topologi-cal tools (see e.g. [6, 152]), with the exception of two papers of Nagumo [292, 293],who based his analytical construction of the (localized) Brouwer degree dB [f, D, z]of a continuous mapping f : D " Rn, with D open bounded and z %& f(!D), onapproximating f by smooth mappings g and z by regular values u of g, for whichone can define

dB[g, D, u] ="

x"g!1(u)

sign Jg(x),

with Jg the Jacobian of g. Nagumo’s approach, which also uses the structure ofcompact 1-dimensional manifolds, has been adopted in the monographs [10, 29, 30,94, 174, 221, 282, 368, 400].

Inspired by de Rham cohomology theory based on di!erential forms [87], Heinz[171] proposed in 1959 another analytical approach to Brouwer degree. He firstdefined dB[f, D, z] for f smooth, z %& f(!D), by the integral

!

Dc(#f(x)! z#)Jf(x) dx, (1.2)

where the continuous function c has support in ]0, min!D #f(·) ! z#[, and thenapproximated a continuous f through smooth mappings. Heinz’ approach, whichavoids the explicit use of di!erential forms, and does not discuss the connection ofintegral (1.2) with Kronecker’s one (1.1) when both are defined, has been adoptedin many monographs [1, 22, 89, 90, 130, 138, 198, 254, 315, 328, 333, 353].

A variant of it, due to Lax (see [297]), explicitely uses the language of di!erentialforms, and can be found in the monographs [23, 29, 30, 95, 130, 297, 370, 407].

In this book, we present Heinz’ approach in a somewhat simpler, shorter andbetter motivated way, by linking it to the Kronecker index (Proposition 2.6.1), andusing the language of di!erential forms, at the level of an advanced calculus treatise(for example [373]). The needed versions of Stokes formula are

!

Dd" =

!

!D", (1.3)

when the open bounded set D ' Rn has a regular oriented boundary !D and " isa di!erential (n ! 1)-form of class C1 (to relate Kronecker and Heinz definitions),and

!

Dd" = 0. (1.4)

when the C1 di!erential (n! 1)-form " has compact support in an arbitrary opensetD ' Rn (to justify Heinz definition and prove the homotopy invariance of de-gree). The homotopy invariance of the degree is proved using an unpublished resultof Tartar [382], quoted in the Introduction of [130]. When written in terms of di!er-ential forms, this result (our Lemma 3.2.2) immediately follows from an elementarycomputation of exterior calculus (Lemma 3.2.1). In the setting of Kronecker index,

5

a similar approach was used recently by Hatziafratis and Tsarpalias [170], at theexpense of much longer and tedious computations. An extension of Lemma 3.2.2from di!erential n-forms in Rn to di!erential k-forms in Rn is given in [275].

6 CHAPTER 1. INTRODUCTION

Chapter 2

Kronecker index

2.1 Definition

Let n ( 2 be an integer,

# := dy1 $ . . . $ dyn (2.1)

be the volume n-form in Rn, and let $ be the solid angle (n! 1)-form

$ :=n"

j=1

(!1)j!1yj dy1 $ . . . $ #dyj $ . . . $ dyn, (2.2)

where #dyj means that dyj is missing. One has

# = d [(1/n)$] . (2.3)

Let D ' Rn be a bounded open set with oriented smooth boundary !D and f &C1(D, Rn) be such that 0 %& f(!D). Then

µ := min!D#f# > 0. (2.4)

Definition 2.1.1 The Kronecker index iK [f, !D] is defined by

iK [f, !D] =1

µn!1

!

!Df# $#y#!n $

%(2.5)

=1

µn!1

!

!D

&

'n"

j=1

(!1)j!1 fj

#f#n df1 $ . . . $ #dfj $ . . . $ dfn

(

) ,

where dfk is the 1-di!erential form

dfk =n"

i=1

!ifk dxk (1 ) k ) n)

7

8 CHAPTER 2. KRONECKER INDEX

and µn!1 denotes the (n! 1)-dimensional measure of the unit sphere

Sn!1 = {y & Rn : #y# = 1}

for the Euclidian norm # · # in Rn.

Explicitely

µn!1 =

*+

,

2"n/2

(n2 !1)!

if n ( 2 is even

2n"(n!1)/2(n!12 )!

(n!1)! if n ( 3 is odd.

Remark 2.1.1 If n = 1 and D = ]a, b[ , the Kronecker index of a continuousfunction f : [a, b]" R such that f(a)f(b) %= 0 can be accordingly defined by

iK [f, ! ]a, b[ ] =1

2

-f(b)

|f(b)| !f(a)

|f(a)|

.=

1

2[sign f(b)! sign f(a)]. (2.6)

Hence,

iK [f, ! ]a, b[ ] = 0 if f(a)f(b) > 0

iK [f, ! ]a, b[ ] = 1 if f(a) < 0 < f(b)

iK [f, ! ]a, b[ ] = !1 if f(b) < 0 < f(a).

This definition is compatible with (2.1.1) if one uses the usual conventions of calling0-form on [a, b] any real continuous function g on [a, b], defining the (oriented)integral of g on the boundary of ]a, b[ by g(b) ! g(a), and taking the countingmeasure as 0-measure of !]a, b[ .

2.2 The planar case : winding number

For n = 2, formula (2.5) reduces to a line integral, already defined by Cauchy, andalso named the winding number or the Poincare index of f around !D

iK [f, !D] =1

2%

!

!D

f1 df2 ! f2 df1

f21 + f2

2

.

If !D has the parametric representation & : [0, 2%] " R2 (with &(0) = &(2%)),then

iK [f, !D] =1

2%

! 2"

0

f1[&(s)] ddsf2[&(s)]! f2[&(s)] d

dsf1[&(s)]

f21 [&(s)] + f2

2 [&(s)]ds. (2.7)

Example 2.2.1 If D = B(1) ' R2, and !D = S1 is parametrized by &(s) =(cos s, sin s) (s & [0, 2%]), then for f = I,

iK [I, S1] =1

2%

! 2"

0

cos2 s + sin2 s

cos2 s + sin2 sds = 1

and the vector field f(cos s, sin s) = (cos s, sin s) rotates once anticlockwise aroundthe origin when s varies from 0 to 2%.

2.2. THE PLANAR CASE : WINDING NUMBER 9

Example 2.2.2 If f(x1, x2) = (x1,!x2), then

iK [f, S1] =1

2%

! 2"

0

! cos2 s! sin2 s

cos2 s + sin2 sds = !1

and the vector field f(cos s, sin s) = (cos s,! sin s) rotates once clockwise aroundthe origin when s varies from 0 to 2%.

Example 2.2.3 If f(x1, x2) = (x21 ! x2

2, 2x1x2), then

f(cos s, sin s) = (cos 2s, sin 2s),

iK [f, S1] =1

2%

! 2"

0

2 cos2 2s + 2 sin2 2s

cos2 2s + sin2 2sds = 2

and the vector field f(cos s, sin s) = (cos 2s, sin 2s) rotates twice anticlockwise a-round the origin when s varies from 0 to 2%.

One can also give a complex formulation of Kronecker index. Set f = f1 + if2,df = df1 + idf2. Then

1

2%i

!

!D

df

f=

1

2%i

!

!D

df1 + idf2

f1 + if2=

1

2%i

!

!D

(f1 ! if2)(df1 + idf2)

#f#2

=1

2%i

!

!D

f1df1 + f2df2

#f#2 +1

2%

!

!D

f1df2 ! f2df2

#f#2= iK [f, !D], (2.8)

because

1

2%i

!

!D

f1df1 + f2df2

#f#2 =1

2%i

!

!D

d(#f#2)2#f#2 =

1

2%i

!

!Dd[log #f#] = 0.

Consequently

iK [f, !D] =1

2%i

! 2"

0

!1f [&(s)]&$1(s) + !2f [&(s)]&$

2(s)

f [&(s)]ds

=1

2%i

! 2"

0

!#f [&(s)]

f [&(s)]ds, (2.9)

where !#f denotes the tangential derivative of f along !D.

The complex formulation allows a simple proof, due to Ahlfors [3], of the factthat that iK [f, !D] is an integer.

Lemma 2.2.1 If !D has the parametric representation & : [0, 2%] " R2, theniK [f, !D] is an integer.


Proof. Let " := &1 + i&2, and let, for t & [0, 2%],

h(t) :=

! t

0

dds [f("(s))]

f("(s))ds

g(t) := e!h(t)f("(t)).

One has

g$(t) = e!h(t) d

dt[f("(t))]!

ddt [f("(t))]

f("(t))e!h(t)f("(t)) = 0,

for all t & [0, 2%], so that g(0) = g(2%), which easily implies eh(2") = 1, and

! 2"

0

dds [f("(s))]

f("(s))ds = h(2%) = 2%mi

for some m & Z.

2.3 The case where f : !D ' Rn " Sn!1

Let again n ( 2 be an integer and D ' Rn be a bounded open set with orientedsmooth boundary !D. Now let g & C2(D, Rn) be such that

#g(x)# = 1 for all x & !D. (2.10)

Let us show that, in this case, it is easy to express iK [g, !D] through an integralover D. We denote by Jg the Jacobian of g.

Lemma 2.3.1 If condition (2.10) holds, then

iK [g, !D] =n

µn!1

!

DJg(x) dx. (2.11)

Proof. Using formulas (2.5), (2.3) and Stokes formula, we get

iK [g, !D] =1

µn!1

!

!Dg#$ =

1

µn!1

!

Dd[g#$] =

1

µn!1

!

Dg#[d$]

=n

µn!1

!

Dg## =

n

µn!1

!

Ddg1 $ . . . $ dgn

=n

µn!1

!

DJg(x) dx.

In the special case where n = 2, we can express iK [g, !D] as follows.

2.3. THE CASE WHERE F : !D ' RN " SN!1 11

Lemma 2.3.2 If g & C1(D ' R2, R2) is such that #g(x)# = 1 for all x & !D, andif !D has the parametric representation & : [0, 2%]" R2, then

iK [g, !D] =1

2%

! 2"

0{g[&(s)] $ !#g[&(s)]} ds, (2.12)

where !#g is the tangential derivative of g along !D,

Proof.

iK [g, !D] =1

2%

! 2"

0

/g1[&(s)]

d

dsg2[&(s)]! g2[&(s)]

d

dsg1[&(s)]

0ds

=1

2%

! 2"

0{g1[&(s)]!# g2[&(s)]! g2[&(s)]!#g1[&(s)]} ds

=1

2%

! 2"

0{g[&(s)] $ !#g[&(s)]} ds.

Now, !#g * & only depends upon the values of g on !D. Therefore, if f &C2(!D, S1) and !D is parametrized by & : [0, 2%] " R2, then, by analogy withformula (2.12), we can introduce the following definition.

Definition 2.3.1 The Kronecker index iK [f, !D] of f with respect to !D is

iK [f, !D] :=1

2%

! 2"

0{f [&(s)] $ !#f [&(s)]} ds (2.13)

where !# denotes the tangential derivative of f with respect to !D.

Combining this definition with Lemma 2.3.1, we then have the following result.

Proposition 2.3.1 If g & C2(D, R2) is any C2-extension of f to D, then

iK [f, !D] =1

%

!

D[!1g(x) $ !2g(x)] dx. (2.14)

Proof. Using (2.11) and (2.12), we get

iK [f, !D] =1

2%

! 2"

0{f [&(s)] $ !#f [&(s)]} ds =

1

2%

! 2"

0{g[&(s)] $ !#g[&(s)]} ds

=1

2%

!

!D(g1dg2 ! g2dg1) = iK [g, !D] =

1

%

!

DJg(x) dx

=1

%

!

D[!1g(x) $ !2g(x)] dx.


2.4 A minimum problem from Ginzburg-Landau’s

theory

Ginzburg-Landau’s theory provides a model very suitable for the study of phasetransitions occuring in supraconductors and superfluidity [26]. We suppose throughthe whole section that D ' R2 be open, bounded, regular and simply connected,and that g & C2(!D, S1). We are looking for a function u = u1 + iu2 : D " C

which minimizes the Dirichlet or energy integral!

D#+u(x)#2 dx

(where

+u = !1u + i!2u,

#+u#2 = |!1u|2 + |!2u|2 = (!1u1)2 + (!1u2)

2 + (!2u1)2 + (!2u2)

2

under the restrictions

|u(x)| =$u2

1(x) + u22(x)

%1/2= 1 a.e. in D, u = f on !D.

If H1(D, C) denotes the Sobolev space of functions u & L2(D, C) such that !1u and!2u belong to L2(D, C), with the usual norm, let

H1f (D, S1) = {u & H1(D, C) : |u(x)| = 1 a.e. on D, u = f on !D}

where the boundary value is understood in the sense of trace. Thus, our minimumproblem can be expressed in the following way

minu"H1

f (D,S1)

!

D#+u(x)#2 dx. (2.15)

If a solution exists, the corresponding Euler-Lagrange equation is the coupled non-linear elliptic system

!#u = #+u#2u in D, u = f on !D.

However, in order that problem (2.15) makes sense, it is necessary first to prove thatH1

f (D, S1) %= ,. We will see that the Kronecker index iK [f, !D] plays a fundamentalrole in this respect. This topological invariant is also responsible of the apparitionof quantified vorticity e!ects comparable to those observed in supraconductors. Letus first give a useful generalization of Proposition 2.3.1.

Proposition 2.4.1 If f & C2(!D, S1) and u & H1(D, C) is such that u = f on!D, then

iK [f, !D] =1

%

!

D[!1u(x) $ !2u(x)] dx. (2.16)

2.4. A MINIMUM PROBLEM FROM GINZBURG-LANDAU’S THEORY 13

Proof. Let u0 & C2(D, C) be such that u0 = f on !D, and u & H1(D, C) besuch that u = f on !D. Clearly u ! u0 & H1

0 (D, C), and therefore there existsa sequence (un) in D(D, C) such that un " u ! u0 strongly in H1(D, C). Then1un := un + u0 & C2(D, C) is equal to f on !D. Using Proposition 2.3.1,

iK [f, !D] =1

%

!

D[!11un(x) $ !21un(x)] dx

for all n & N, and the result follows by letting n"-.

We need another result, essentially due to Eilenberg [99].

Proposition 2.4.2 If f : !D " S1 is smooth and such that iK [f, !D] = 0, thenthere exists a smooth function &0 : !D" R2 such that f = ei$0 on !D.

Proof. If iK [f, !D] = 0, thenWe are now ready to prove the Bethuel-Brezis-Helein theorem showing the

role of iK [f, !D] in the study of minimization problem (2.15).

Theorem 2.4.1 H1f (D, S1) %= , if and only if iK [f, !D] = 0.

Proof. Suppose that H1f (D, S1) %= ,, and let u & H1

f (D, S1), i.e. u & H1(D, C),

u21(x) + u2

2(x) = 1 for a.e. x & D, (2.17)

and u = f on !D in the sense of trace. It is well known that if v & H1(D, R) .L%(D, R) and G & C1(R, R), then G * u & H1(D, R) and !i(G * v) = (G$ * v)!iv(i = 1, 2). Clearly, ui & H1(D, R) . L%(D, R) (i = 1, 2) and hence, for G(t) = t2,u2

i & H1(D, R) (i = 1, 2) and

!j(u2i ) = 2ui!jui (i, j = 1, 2).

By di!erentiating the identity (2.17), we obtain

u1!1u1 + u2!1u2 = 0, u1!2u1 + u2!2u2 = 0,

i.e. 2!1u1 !1u2

!2u1 !2u2

32u1

u2

3= 0

with (u1, u2) %= 0. Therefore,

!1u $ !2u = Ju = 0

and Proposition 2.4.1 implies that

iK [f, !D] =1

%

!

D[!1u(x) $ !2u(x)] dx = 0.

Conversely, suppose that iK [f, !D] = 0. Using Proposition 2.4.2, we can writef = ei$0 for some smooth &0 : !D " R. Denote by 1& the harmonic extension of &0

to D, i.e. the unique solution of the linear Dirichlet problem

#1& = 0 in D, 1& = &0 on !D.

Then 1u = eie$ & H1(D, C), |1u(x)| = 1 for x & D, and 1u(x) = ei$0(x) = f(x) forx & !D. Consequently, 1u & H1

f (D, C).


2.5 An exact form

Writing iK [f, !D] as an integral over D when #f(x)# is not constant over !D ismore delicate, and requires some preparation.

Let 0 < '0 < µ0, a & Ck([0, +-[, R), (k ( 0) be such that

supp a ' ['0, µ0],

and consider the di!erential n-form (with n a positive integer)

#a := a(#y#) dy1 $ . . . $ dyn = a(#y#)#. (2.18)

The following lemma shows that #a is exact on Rn.

Lemma 2.5.1 The equality

#a = d$A (2.19)

with

$A := A(#y#)$, (2.20)

A & Ck+1([0, +-[, R) and supp A ' ['0, +-[, holds if and only if

rA$(r) + nA(r) = a(r) (r ( '0), (2.21)

i.e. if and only if A is given by

A(0) = 0, A(r) =1

rn

! r

0a(s)sn!1 ds (r > 0). (2.22)

Proof. Let A & Ck+1([0, +-[, R), with supp A ' ['0, +-[. $A & Ck+1(Rn, R) and$A = 0 on B['0]. For y %= 0, we have

d$A = d[A(#y#)] $ $ + A(#y#) d$ = A$(#y#)n"

k=1

yk

#y# dyk $ $ + nA(#y#)#

= [#y#A$(#y#) + nA(#y#)]#.

Consequently, #a = d$A if and only if, for all r & ['0, +-[, A satisfies (2.21). Forr > 0, this is equivalent to

d

dr[rnA(r)] = rn!1a(r),

i.e. to A given by formula (2.22). Clearly, supp A ' ['0, +-[.

Corollary 2.5.1 If we assume furthermore that! +%

0a(r)rn!1 dr = 0, (2.23)

then supp A ' ['0, µ0].

2.6. AN EQUIVALENT FORMULA 15

Proof. For r > µ0, we have, by formula (2.22),

A(r) =1

rn

! r

0a(s)sn!1 ds =

1

rn

! µ0

0a(s)sn!1 ds =

1

rn

! +%

0a(s)sn!1 ds = 0.

2.6 An equivalent formula

The Kronecker integral in the right-hand member of (2.5) is associated to the dif-ferential (n! 1)-form $A with A(r) = r!n, which is not defined at 0 and does notvanish near 0. However, in (2.5), one only has to consider the values of f(x) near!D, i.e. values of f(x) for which #f(x)# ( µ. Hence, let

0 < '0 < µ0 < µ,

and define B & C1([0, +-[, R) by

B(r) =

*+

,

0 if r & [0, '0]positive if r & ]'0, µ0[

r!n if r & [µ0, +-[.(2.24)

Notice that, for r > µ0, we have

rB$(r) + nB(r) = 0. (2.25)

Proposition 2.6.1 If B is given by (2.24), the function b & C([0, +-[, R) definedby

b(r) = rB$(r) + nB(r) (2.26)

is such that

supp b ' ['0, µ0],

! +%

0b(r)rn!1 dr = 1,

and, if f & C2(D, Rn),

iK [f, !D] =1

µn!1

!

Df##b =

1

µn!1

!

Db(#f(x)#)Jf (x) dx. (2.27)

Proof. For x & !D, we have #f(x)# ( µ > µ0, and hence B(#f(x)#) = #f(x)#!n.Using Lemma 2.5.1 and Stokes formula (1.3), we get

iK [f, !D] =1

µn!1

!

!Df# $#y#!n $

%=

1

µn!1

!

!Df#$B

=1

µn!1

!

Dd[f#$B] =

1

µn!1

!

Df#[d$B ] =

1

µn!1

!

Df##b.


It is clear from its definition and (2.25) that b(r) = 0 for r & [0, '0[/ ]µ0, +-[, andthat! +%

0b(r)rn!1 dr =

! +%

0[rnB$(r) + nrn!1B(r)] dr =

! +%

0

d

dr[rnB(r)] dr = 1.

Remark 2.6.1 In the construction above, the functions b and B can be taken assmooth as we want.

Chapter 3

Brouwer degree

3.1 Smooth mappings

Let now n ( 1 be an integer, D be open and bounded in Rn and f & C2(D, Rn) besuch that 0 %& f(!D). We again define µ > 0 by (2.4), and take 0 < '0 < µ0 < µ.The following definition is due to Heinz [171].

Definition 3.1.1 The Brouwer degree dB [f, D] is defined by

dB [f, D] =

!

Df##c =

!

Dc(#f(x)#)Jf (x) dx, (3.1)

where c & C([0, +-[, R) is such that supp c ' ['0, µ0] and!

Rn

c(#x#) dx = 1

2i.e.

! +%

0c(r)rn!1 dr =

1

µn!1

3. (3.2)

To justify this definition, it su$ces to notice that if 1c & C([0, +-[, R) satisfies thesame conditions as c, then a = c! 1c is such that

! +%

0a(r)rn!1 dr = 0.

Hence, by Corollary 2.5.1, supp A ' ['0, µ0], with A given by (2.22), so that,!

Df##c !

!

Df##ec =

!

Df##a =

!

Df#[d$A] =

!

Dd[f#$A] = 0,

using (1.4), and the fact that #f(x)# > µ0 in a neighbourhood of !D.

Remark 3.1.1 It follows immediately from Definition 3.1.1 that dB [f, ,] = 0.

Remark 3.1.2 Proposition 2.6.1 implies that, if !D is smooth enough, one has

iK [f, !D] = dB[f, D]. (3.3)

17

18 CHAPTER 3. BROUWER DEGREE

Proposition 3.1.1 If the open set D$ ' D is such that 0 %& f(D \ D$), then

dB[f, D] = dB[f, D$].

In particular, if 0 %& f(D), then dB[f, D] = 0.

Proof. By assumption, if µ$ := min!D" #f#, and µ$$ := minD\D" #f#, then 0 < µ$$ <µ and µ$$ < µ$. If we take 0 < '0 < µ0 < µ$$ in the function c defining dB[f, D], wehave c(#f(x)#) = 0 if x & D \ D$, and

dB [f, D] =

!

Dc(#f(x)#)Jf (x) dx =

!

D"

c(#f(x)#)Jf (x) dx = dB[f, D$],

because c is also good for defining DB[f, D$].

It is convenient to introduce a new notation.

Definition 3.1.2 If D ' Rn is open and bounded, f & C2(D, Rn) and z %& f(!D),the Brouwer degree dB [f, D, z] is defined by

dB[f, D, z] = dB [f(·)! z, D].

Of course, dB[f, D] = dB[f, D, 0].

Definition 3.1.3 z & Rn is a regular value for f if Jf (x) %= 0 when x & f!1(z).

So any z with f!1(z) empty is a regular value.

Proposition 3.1.2 If z %& f(!D) is a regular value for f, then

dB[f, D, z] ="

x"f!1(z)

sign Jf (x) = N+ !N!, (3.4)

where N+ (resp. N!) denotes the number of elements of f!1(z) with positive (resp.negative) Jacobian.

Proof. If f!1(z) = ,, the result is trivial. If z %& f(!D) is a regular value for f andf!1(z) %= ,, then f!1(z) ' D is compact, and is discrete by the inverse functiontheorem. Hence it is finite, namely f!1(z) = {x1, . . . , xm}. By the inverse functiontheorem again, there exists, for each 1 ) j ) m, an open neighbourhood Uj ' D ofxj such that f is a di!eomorphism on Uj, and Uj . Uk = , if j %= k. Thus the setsf(Uj) are open neighbourhoods of z, as well as N = .m

j=1f(Uj), so that N 0 B[z; r]

for some r > 0. As f(x) %= z for all x & K := D \ /mj=1Uj , Proposition 3.1.1 gives

dB [f, D, z] = dB[f,/mj=1Uj , z] =

!

&mj=1Uj

c(#f(x)! z#)Jf(x) dx

=m"

j=1

!

Uj

c(#f(x)! z#)Jf(x) dx =m"

j=1

dB[f, Uj , z],

3.1. SMOOTH MAPPINGS 19

where we have taken

µ0 < min{r, minD\&m

j=1Uj

#f(·)! z#}

in the definition of c. The change of variables formula in an integral gives

d[f, Uj , z] =

!

Uj

c(#f(x)! z#)Jf(x) dx

= sign Jf (xj)

!

Uj

c(#f(x)! z#)|Jf(x)| dx

= sign Jf (xj)

!

f(Uj)!zc(#y#) dy.

Now, for y %& f(U j)! z, we have #y! z# ( r ( µ0, and hence c(#y! z#) = 0. Thus!

f(Uj)!zc(#y ! z#) dy =

!

Rn

c(#y ! z#) dy = 1,

which gives (3.4).

Remark 3.1.3 Proposition 3.1.2 explains the name of “algebraic number of zeros”sometimes given to the degree.

Remark 3.1.4 For each regular value z %& f(!D), dB[f, D, z] is an integer.

Corollary 3.1.1 If A : Rn " Rn is a linear isomorphism, then

dB[A, D, z] =

/sign det A if z & A(D)0 if z %& A(D).

Now this last result can be expressed in terms of the eigenvalues of A. Recall thatif "1, . . . ,"r are the real eigenvalues of A, with respective multiplicities m1, . . . , mr

and if µ1, µ1, . . . , µs, µs are the non real eigenvalues of A, with respective multiplic-ities n1, . . . , ns, then r + 2s = n and

det (A! µI) =r4

j=1

("j ! µ)mj

s4

k=1

(µk ! µ)nk(µk ! µ)nk

and hence

det A =r4

j=1

"mj

j

s4

k=1

|µk|2nk .

Consequently, if z & A(D),

dB[A, D, z] = sign detA = (!1)Pp

j=1 mj (3.5)

where the sum is over the negative eigenvalues "1, . . . ,"p of A. Of course one canuse as well the sum over the negative characteristic values of A.


3.2 An exact form and homotopy invariance

For D ' Rn open and bounded, let w & C1(Rn, R), F & C2(D 1 [0, 1], Rn), and

dFj =n"

j=1

!jF (·,") dxj .

The following lemma shows that !%{F (·,")#[w(y)#]} is an exact form.

Lemma 3.2.1 For each " & [0, 1], we have

!% [(w * F ) dF1 $ . . . $ dFn]

= d

&

'(w * F )

5

6n"

j=1

(!1)j!1!%Fj dF1 $ . . . $7dFj $ . . . $ dFn

8

9

(

) . (3.6)

Proof. We first notice that

!%dFj = !%

:n"

k=1

!kFj dxk

;=

n"

k=1

!%!kFj dxk =n"

k=1

!k!%Fj dxk = d[!%Fj ].

Now

!% [(w * F ) dF1 $ . . . $ dFn]

= !%[(w * F )] dF1 $ . . . $ dFn + (w * F )!%[dF1 $ . . . $ dFn]

=n"

j=1

[(!jw * F )!%Fj ] dF1 $ . . . $ dFn

+ (w * F )

&

'n"

j=1

dF1 $ . . . $ !%dFj $ . . . $ dFn

(

)

=n"

j=1

(!1)j!1(!jw * F ) dFj $ !%Fj dF1 $ . . . $7dFj $ . . . $ dFn

+ (w * F )

*+

,

n"

j=1

(!1)j!1d[!%Fj ] $ dF1 $ . . . $7dFj $ . . . $ dFn

<=

>

=n"

j=1

(!1)j!1

:n"

k=1

(!kw * F ) dFk

;$ !%Fj dF1 $ . . . $7dFj $ . . . $ dFn

+ (w * F )

*+

,

n"

j=1

(!1)j!1d?!%Fj dF1 $ . . . $7dFj $ . . . $ dFn

@<=

>

= d[w * F ] $

&

'n"

j=1

(!1)j!1!%Fj dF1 $ . . . $7dFj $ . . . $ dFn

(

)

3.2. AN EXACT FORM AND HOMOTOPY INVARIANCE 21

+ (w * F ) d

&

'n"

j=1

(!1)j!1!%Fj dF1 $ . . . $7dFj $ . . . $ dFn

(

)

= d

&

'(w * F )

5

6n"

j=1

(!1)j!1!%Fj dF1 $ . . . $7dFj $ . . . $ dFn

8

9

(

) .

Define now

Iw[F (·,")] :=

!

D(w * F ) dF1 $ . . . $ dFn

=

!

DF (·,")#[w(y) dy1 $ . . . $ dyn] (3.7)

=

!

Dw[F (x,")]JF (·,%)(x,") dx.

The following result is due to Tartar [130, 382], who stated and proved it withoutusing di!erential forms.

Lemma 3.2.2 If w & C1(Rn, R) and F & C2(D 1 [0, 1], Rn) are such that

F (!D 1 [0, 1]) . supp w = ,, (3.8)

then Iw[F (·,")] is independent of " on [0, 1].

Proof. Using Lemma 3.2.1, Assumption (3.8) and Stokes formula (1.4), we get

d

d"Iw [F (·,")] =

!

D!%[(w * F ) dF1 $ . . . $ dFn]

=

!

Dd

&

'(w * F )

5

6n"

j=1

(!1)j!1!%Fj dF1 $ . . . $7dFj $ . . . $ dFn

8

9

(

) = 0.

Corollary 3.2.1 If F & C2(D 1 [0, 1], Rn) is such that 0 %& F (!D 1 [0, 1]), thendB[F (·,"), D] is independent of " over [0, 1].

Proof. By assumption,µ := inf

!D'[0,1]#F# > 0,

and, if 0 < '0 < µ0 < µ, the corresponding function c in Definition 3.1.1 (which canalways be chosen of class C1) is such that w(y) = c(#y#) has the property (3.8).Thus, by Lemma 3.2.2, dB[F (·,"), D] = Iw[F (·,")] is independent of " in [0, 1].


Corollary 3.2.2 If f, g & C2(D, Rn) are such that

( := max!D#f ! g# < µ := min

!D#f#, (3.9)

then dB [f, D] = dB[g, D].

Proof. Let us define F : D 1 [0, 1]" Rn by

F (x,") = (1! ")f(x) + "g(x).

Then, for all (x,") & !D 1 [0, 1], we have

#F (x,")# ( #f(x)# ! #g(x)! f(x)# ( µ! ( > 0.

The result follows from Corollary 3.2.1.

Corollary 3.2.3 For each f & C2(D, Rn) and z %& f(!D), dB [f, D, z] is an integer.

Proof. By Sard’s theorem, there exists a regular value v such that

#v ! z# < µ = min!D#f(·)! z#.

Hence,max

D#f(·)! z ! [f(·)! u]# = #z ! u# < µ,

and, using Corollary 3.2.2 and Proposition 3.1.2 we get

dB [f, D, z] = dB[f, D, u] & Z.

3.3 Continuous mappings

Let D ' Rn be open and bounded, f & C(D, Rn) and z %& f(!D), so that

µ := min!D#f(·)! z# > 0.

Let g and h be two mappings in C2(D, Rn) such that maxD #g ! f# < µ/4 andmaxD #h! f# < µ/4. They exist by Weierstrass’ approximation theorem. Then

min!D#g(·)! z# > µ/2, max

D#h! g# < µ/2,

and hence, by Corollary 3.2.2,

dB [g, D, z] = dB[h, D, z]. (3.10)

Definition 3.3.1 If D ' Rn is open and bounded, f & C(D, Rn) and z %& f(!D),the Brouwer degree dB [f, D, z] is defined by

dB[f, D, z] = dB[g, D, z], (3.11)

where g & C2(D, Rn) is such that maxD #g ! f# < µ/4.

The definition is justified by (3.10) showing that the right-side of (3.11) does notdepend upon the choice of the approximating mapping g.

3.4. BASIC PROPERTIES 23

3.4 Basic properties

We now easily extend the basic properties of Brouwer degree to the continuous case.First the excision property.

Theorem 3.4.1 If the open subset D$ ' D is such that z %& f(D \ D$), then

dB[f, D, z] = dB[f, D$, z].

Proof. Indeed, taking the approximating g in Definition 3.3.1 such that

maxD#g ! f# < min

D\D"

#f(·)! z#,

one has minD\D" #g(·)! z# > 0. By Proposition 3.1.1 applied to g, we get

dB[g, D, z] = dB[g, D$, z],

and hence the result, as g is clearly a good approximation for defining dB[f, D, z]and dB[f, D$, z].

The existence property follows directly by taking D$ = ,.

Corollary 3.4.1 If z %& f(D), then dB [f, D, z] = 0. Equivalently, if dB[f, D, z] %= 0,there exists at least one x & D such that f(x) = z.

We now extend the homotopy invariance property.

Theorem 3.4.2 If F & C(D 1 [0, 1], Rn) and z %& F (!D 1 [0, 1]), then

dB[F (·,"), D, z]

is independent of " on [0, 1].

Proof. Letµ := min

!D'[0,1]#F (·, ·)! z# > 0.

Then there exists G & C2(D 1 [0, 1]) such that maxD'[0,1] #G! F# < µ/4. Conse-quently, for any " & [0, 1], we have, by definition

dB[F (·,"), D, z] = dB [G(·,"), D, z],

and, by Proposition 3.2.1, the right-hand member is independentof " on [0, 1].

Remark 3.4.1 It is clear that, in Theorem 3.4.2, one can replace [0, 1] by anarbitrary compact interval [a, b].

A direct consequence is Rouche’s property.


Corollary 3.4.2 If f, g & C(D, Rn), z %& f(!D),

max!D#f ! g# < µ := min

!D#f(·)! z#,

thendB[f, D, z] = dB[g, D, z].

Proof. Define F (x,") = (1! ")f(x) + "g(x) and apply Theorem 3.4.2.

Corollary 3.4.3 If f, g & C(D, Rn), z %& f(!D), f = g on !D, then

dB[f, D, z] = dB[g, D, z].

Corollary 3.4.4 If f & C(D, Rn) z %& f(!D), then, for any y & Bz(min!D[f(·)]!z), dB[f, D, y] = dB [f, D, z].

Proof. Apply Corollary 3.4.2 to f(·)! z and f(·)! y at 0.

Corollary 3.4.5 If f & C(D, Rn) is such that 0 %& f(!D) and f(D) ' Y, where Yis a proper vector subspace of Rn, then dB[f, D, 0] = 0.

Proof. Let y %& Y be such that #y# < min!D #f#. Then, using Corollary 3.4.4 andthe existence theorem 3.4.1, we get dB [f, D, 0] = dB[f, D, y] = 0.

Corollary 3.4.6 Let f & C(D, Rn). Then dB[f, D, z] is constant on each connectedcomponent of Rn \ f(!D).

Proof. Each connected component of Rn \ f(!D) is open and connected and hencepath-connected. If # is such a connected component and if y & #, z & #, thereexists a continuous path ) : [0, 1] " # such that )(0) = y, )(1) = z. For each(x,") & !D 1 [0, 1], one has f(x) %= )("), and hence

dB[f, D, y] = dB [f, D, )(0)] = dB[f ! )(0), D, 0] = dB[f ! )(1), D, 0]

= dB [f ! z, D, 0] = dB [f, D, z].

Remark 3.4.2 The common value of dB[f, D, z] for z belonging to a connectedcomponent # of Rn \ f(!D) is often denoted by

dB[f, D,#].

Corollary 3.4.7 If f & C(D, Rn), z %& f(!%) and dB [f, D, z] %= 0, then f(%) is aneighborhood of z.

Proof. By Corollary 3.4.6, dB[f, D, z] is constant, and hence non zero, on the (open)connected component of Rn \ f(!D) containing z. Existence property 3.4.1 impliesthat equation f(x) = y has at least one solution for each y belonging to this con-nected component, which is therefore contained in f(D).

3.4. BASIC PROPERTIES 25

We finally prove the additivity property.

Theorem 3.4.3 Suppose that there exists a sequence (Dj)j of open and mutuallydisjoint subsets of D. If z %& f(D \/%j=1Dj), then dB[f, Dj , z] = 0 for all but finitelymany j, and

dB[f, D, z] ="

j

dB[f, Dj , z].

Proof. By assumption,

0 < µ0 := minD\&#

j=1Dj

#f(·)! z#

and

µ0 ) µ := min!D#f(·)! z#, µ0 ) µj := min

!Dj

#f(·)! z# (j = 1, 2, . . .).

Lett g & C2(D, R2) be such that maxD #f ! g# < µ0/4, so that

3µ0

4) 1µ := min

!D#g(·)! z#, 3µ0

4)Aµj := min

!Dj

#g(·)! z# (j = 1, 2, . . .),

and

dB[f, D, z] = dB[g, D, z], dB[f, Dj , z] = dB[g, Dj, z] (j = 1, 2, . . .). (3.12)

Let 0 < r < 1µ. By Sard’s theorem, there exists a regular value u & B[z; r] ' D off. By Corollary 3.4.2, this implies that

dB[g, D, z] = dB[g, D, u], dB[g, Dj , z] = dB [g, Dj, u] (j ( 1). (3.13)

Now, u being regular, g!1(u) is finite, and hence contained in a finite number ofthe Dj’s, namely Dj1 , . . . , Djm . By Theorem 3.4.1 and Definition 3.1.1,

dB [g, D, u] = dB [g,/mi=1Dji , u] =

m"

i=1

dB[g, Dji , u]. (3.14)

The result follows from (3.13), (3.14) and (3.12).

It is now easy to compute the Brouwer degree of a continuous function f :[a, b]" R2 when a < b and f(a)f(b) %= 0. Let g : R" R be defined by

g(x) = f(r) +f(b)! f(a)

b! a(x! a).

g is a$ne and equal to f on the boundary of ]a, b[, so that, by Corollary 3.4.3,

dB[f, ]a, b[ , 0] = dB[g, ]a, b[ , 0].

Consequently, if f(a)f(b) > 0, 0 %& g([a, b]) and dB[g, ]a, b[, 0] = 0, if f(a) < 0 < f(b)(resp. f(b) > 0 > f(a)), g has a unique zero in ]a, b[ at which g$ > 0 (resp. g$ < 0),and dB [g, ]a, b[, , 0] = 1 (resp. dB [g, ]a, b[ , 0] = !1). We deduce immediately thefollowing result.


Proposition 3.4.1 Let a < b and f : [a, b]" R continuous.

dB [f, ]a, b[ , 0] =

*+

,

0 if f(a)f(b) > 01 if f(a) < 0 < f(b)!1 if f(b) > 0 > f(a).

In other words,

dB [f, ]a, b[ , 0] =1

2

-f(b)

|f(b)| !f(a)

|f(a)|

.. (3.15)

Corollary 3.4.8 Let p(x) =Bn

k=0 akxk be a real polynomial with an %= 0. Then,for all su!ciently large R > 0,

dB [p, ]!R, R[ , 0] =

/0 if n is evenan|an| if n is odd.

3.5 The case where f : !D ' Rn " Sn!1

If D ' Rn is open and bounded, and if f : !D " Sn!1 is continuous, it follows fromCorollary 3.4.3 that if g : D " Rn and h : D " Rn are two continuous extensionsof f to D, dB [g, D] and dB[h, D] are defined and equal, as g and h coincide over!D. Hence the following definition is justified.

Definition 3.5.1 If D ' Rn is open and bounded, and if f : !D " Sn!1 iscontinuous, the Brouwer degree dB[f, !D] is defined by

dB [f, !D] = dB [g, D], (3.16)

where g : D " Rn is any continuous extension of f to D.

Of course dB[f, !D] inherits the properties of the Brouwer degree dB[g, D], exceptthat the existence property takes the following form.

Proposition 3.5.1 If D ' Rn is open and bounded, f & C(!D, Sn!1) and f isnot onto, then dB [f, !D] = 0.

Proof. Let y & Sn!1 \ f(!D). Using Tietze-Dugundji’s extension theorem there

exists a continuous extension 1f : Rn " B(1) of f. Consider the homotopy F :B(1)1 [0, 1]" Rn defined by

F (x,") = (1! ") 1f(x)! "y.

If there exists (x,") & !D 1 [0, 1] such that F (x,") = 0, then, because 1f = f on!D,

(1! ") = (1! ")#f(x)# = "#y# = ",

so that " = 12 and 1

2 [f(x) ! y] = 0, which is impossible. Hence, it follows from thehomotopy invariance property 3.4.2 and the definition of dB [f, !D] that

dB[f, !D] = dB[ 1f, D] = dB[F (·, 0), D] = dB[F (·, 1), D] = dB[!y, D] = 0.

3.5. THE CASE WHERE F : !D ' RN " SN!1 27

Proposition 3.5.1 can of course be expressed in the equivalent contraposed way.

Corollary 3.5.1 If D ' Rn is open and bounded, f & C(!D, Sn!1) and dB [f, !D]%= 0, f is onto.

The following formula for the degree of fixed point free mappings of the sphereinto itself was obtained in 1912 by Brouwer [39].

Proposition 3.5.2 If g & C(Sn!1, Sn!1) has no fixed point, dB [g, Sn!1] = (!1)n.

Proof. Let 1g : Rn " B(1) be a continuous extension of f given by Tietze-Dugundji’stheorem, and consider the homotopy F : B(1)1 [0, 1]" Rn defined by

F (x,") = (1! ")1g(x) ! "x.

If F (x,") = 0 for some (x,") & !B(1) 1 [0, 1] = Sn!1 1 [0, 1], then, as 1g = g on!B(1),

(1! ") = (1! ")#g(x)# = "#x# = ",

so that " = 12 and 1

2 [g(x)!x] = 0, which is impossible by assumption. Consequently,using the definition of dB[g, Sn!1], the homotopy invariance property 3.4.2, andCorollary 3.1.1, we get

dB[g, Sn!1] = dB[1g, B(1)] = dB[F (·, 0), B(1)] = dB[F (·, 1), B(1)]

= dB[!I, B(1)] = (!1)n.

The equivalent contraposed version of Proposition 3.5.2 is Brouwer’s fixedpoint theorem on spheres [39].

Corollary 3.5.2 Any mapping g & C(Sn!1, Sn!1) such that dB [g, Sn!1] %= (!1)n

has at least one fixed point.

This is in particular the case for any g & C(Sn!1, Sn!1) which is not onto. Onthe other hand, there exist mappings g & C(Sn!1, Sn!1) without fixed point, forexample, when n = 2, any non-trivial rotation, or, for any n, the antipodal mappingx 2" !x, and there exist also mappings g having fixed points and degree (!1)n, forexample the identity for n even.

As the same boundary dependence property of course holds for Kronecker index,we can similarly define the Kronecker index of a C1 mapping f : !D " Sn!1.

Definition 3.5.2 If D ' Rn is open and bounded, !D is smooth and if f &C1(!D, Sn!1), the Kronecker index iK [f, !D] is defined by

iK [f, !D] = iK [g, !D], (3.17)

where g : D " Rn is any C1 extension of f to D.

HencedB[f, !D] = iK [f, !D]

when f & C1(!D, Sn!1) and !D is su$ciently smooth.

Chapter 4

Uniqueness of degree

4.1 Introduction

The aim of this chapter is to show that there exists only one application d fromthe set of (D, f) with D ' Rn open and bounded, f : D " Rn continuous and 0 %&f(!D), into the set Z of the integers, which satisfies the following three properties :

(A 1) (Normalization) d[I, D] = 1 if 0 & D.

(A 2) (Additivity) d[f, D] = d[f, D1] + d[f, D2] if D1 and D2 are disjoint opensubsets of D such that 0 %& f(D \ (D1 /D2)).

(A 3) (Homotopy invariance) If F & C(D 1 [0, 1], Rn) and 0 %& F (!D 1 [0, 1]),then dB[F (·,"), D] is independent of " on [0, 1].

The existence of such a function, namely the Brouwer degree dB, has been proved inthe previous chapter. It remains to prove the uniqueness, and this was first shownindependently by Fuhrer [139] and Amann-Weiss [12], by reducing the problem toa very special case.

We first notice that, by taking D1 = D and D2 = , in Property (A 2), we obtain

d[f, ,] = 0,

and by taking then D2 = , in the same property, we get the(P 1) (Excision property) : If 0 %& f(D \ D1), then d[f, D] = d[f, D1].

4.2 Reduction to the linear case

Let f & C(D, Rn) and 0 %& f(!D). Then µ := min!D #f# > 0. The Weierstrassapproximation theorem implies that there exists g & C(D) . C%(D) such that

maxD#f ! g# < µ. (4.1)

29

30 CHAPTER 4. UNIQUENESS OF DEGREE

The function F & C(D 1 [0, 1], Rn) defined by

F (x,") = (1! ")f(x) + "g(x)

is such that, for each x & !D and " & [0, 1], one has

#F (x,")# ( #f(x)# ! "#f(x)! g(x)# ( #f(x)# ! #f(x)! g(x)# > 0.

Consequently, Property A 3 implies that d[f, D] = d[g, D] and hence d is uniquelydetermined by its values on C% functions.

Now, by Sard’s theorem, there exists a regular value z for g such that #z# < µ.Hence the function G & C(D 1 [0, 1], Rn) defined by

G(x,") = g(x)! "z

is such that, for each x & !D and " & [0, 1], one has

#G(x,")# ( #g(x)# ! "#z# ( #g(x)# ! #z# > 0.

Consequently, Property A 3 implies that d[g, D] = d[g(·)! z, D], and d is uniquelydetermined by its values on C% functions for which 0 is a regular value.

For such a function, it follows from the argument of the beginning of the proofof Proposition 3.1.2 that g!1(0) = {x1, . . . , xm} finite. If g!1(0) = ,, we haved[g, D] = d[g, ,] = 0. If g!1(0) %= ,, there exists mutually disjoint open balls Ui

centered at xi (1 ) i ) m), contained in D and, using Property A 2, we obtain

d[g, D] =m"

j=1

d[g, Uj ].

Thus d[g, D] is uniquely determined by the values of d[g, Uj ] where Uj is an openball in D centered at a non-degenerate zero xj of g and containing no other zero ofg.

For such an xj , we have

g(x) = g$xj(x! xj) + r(x) := A(x! xj) + r(x),

with limx(xj r(x)/#x! xj# = 0. As 0 is a regular value for g, det A %= 0, and thereexists c > 0 such that

#A(y)# ( c#y#

for all y & Rn. Furthermore, using excision property (P 1), we can reduce if necessarythe radius of Uj in such a way that

#r(x)# ) c

2#x! xj#

4.3. UNIQUENESS FOR LINEAR MAPPINGS 31

for all x & Uj . Hence, if H & C(D 1 [0, 1], Rn) is defined by

H(x,") = (1! ")g(x) + "A(x ! xj),

we have, for all (x,") & !Uj 1 [0, 1],

#H(x,")# ( #A(x! xj)# ! "#r(x)# ( c

2#x! xj# > 0,

which implies, using Property A 3, that

d[g, Uj] = d[A(·! xj), Uj]. (4.2)

Now, as xj is the only zero of A(· ! xj), Property A 2 implies that

d[A(·! xj), B(r)] = d[A(·! xj), Uj] (4.3)

for any r > 0 is such that Uj ' B(r). Furthermore if J & C(Rn 1 [0, 1], Rn) isdefined by

J(x,") = A(x! "xj),

we have J(x,") %= 0 for all (x,") & !B(r)1 [0, 1], because J(x,") = 0 if and only ifx = "xj , and "xj & B(r) for all " & [0, 1]. Consequently, by Property A 3, we have

d[A(·! xj), B(r)] = d[A, B(r)],

which, together with (4.2) and (4.3), shows that d[g, Uj ] is uniquely determined bythe values of d[A, B(r)] where A is a nonsingular linear mapping.

4.3 Uniqueness for linear mappings

We show in this section that properties A 1 to A 3 imply that if A : Rn " Rn is alinear nonsingular mapping and r > 0, then

d[A, B(r)] = sign det A. (4.4)

We first recall a classical result of linear algebra.

Lemma 4.3.1 Let A be a real n1 n matrix such that det A %= 0, let "1, . . . ,"p beits possible negative eigenvalues, with respective multiplicities m1, . . . , mp. Then Rn

can be written as the direct sum Rn = N 3M of two vector subspaces, such that

(a) N and M are invariant under A.

(b) A|N only has the eigenvalues "1, . . . ,"p and A|M has no negative eigenvalues.

(c) dim N =Bp

k=1 mk.


Under the above assumptions, we have

det (A! µI) = (!1)np4

k=1

(µ! "k)mk

s4

j=p+1

(µ! µj)qj ,

where the µj are the nonnegative eigenvalues of A with respective multiplicities qj .A being real, the complex eigenvalues occur in conjugate pairs, and hence

det A = (!1)mp4

k=1

|"k|mk

s4

j=p+1

µqj

j ,

where m =Bp

k=1 mk. Consequently

sign det A = (!1)m. (4.5)

Assume first that A has no negative eigenvalues. Then, for all " & [0, 1], onehas

det ["A + (1! ")I] %= 0,

and, using Properties A 1 and A 3 with F (x,") = "A + (1! ")I, we get

d[A, B(r)] = d[I, B(r)] = 1 = (!1)0,

so that (4.4) follows from (4.5).

Assume now that N %= {0} and suppose first that m = dim N is even. Denoterespectively by P and Q = I!P the linear projectors P : Rn " N, Q : Rn "M.Then A = AP + AQ, with AP (Rn) ' N and AQ(Rn) ' M by the invariance ofN and M under A. Furthermore, AP has only negative eigenvalues et AQ has nonegative eigenvalues. Let us define the homotopy F : Rn 1 [0, 1]" Rn by

F (x,") = "Ax + (1! ")(!Px + Qx).

If F (x, 0) = 0, then Px = 0, Qx = 0, and hence x = 0. If F (x,") = 0 for some" & ]0, 1], then by projecting on the subspaces N and M, and using their invarianceunder A, we get

APx =1! "

"Px, AQx = !1! "

"Qx,

which is only possible if Px = 0 and Qx = 0. Consequently, Property A 3 impliesthat

d[A, B(r)] = d[!P + Q, B(r)]. (4.6)

Now, since m is non zero and even, one can find a linear mapping B : N " N suchthat B2 = !I|N . The eigenvalues of B are imaginary, as easily checked. Considerthe homotopies G and H respectively defined by

G(x,") = "BPx! (1! ")Px + Qx, H(x,") = "BPx + (1! ")Px + Qx.


They trivially only vanish at 0 when " = 0. If " & ]0, 1] and G(x,") = 0, (resp.H(x,") = 0), then

BPx =1! "

"Px, Qx = 0, (resp. BPx = !1! "

"Px, Qx = 0),

and hence Px = 0, Qx = 0. Consequently, Property A 3 implies that

d[!P + Q, B(r)] = d[BP + Q, B(r)] = d[P + Q, B(r)] = 1 = (!1)m. (4.7)

It follows from (4.5), (4.6) and (4.7) that (4.4) holds.

It remains to consider the case where m = dim N = 2q + 1 for some integerq ( 0. Then we can write N = N1 3 N2, with dim N1 = 1 and dim N2 = 2r,and corresponding linear projectors P1 : N " N1, Q1 = I ! P1 : N " N2. Then,P = P1P + Q1P, and, following the line of case where m is even, we introduce thesequence of homotopies

F (x,") = "Ax + (1! ")(!Px + Qx),

1G(x,") = "(!P1Px + BQ1Px)! (1! ")Px + Qx,

1H(x,") = !P1Px + "BQ1Px + (1! ")Q1Px + Qx,

where B : N2 " N2 is linear and such that B2 = !I|N2 . One checks like above thatnone of them vanishes on !B(r) 1 [0, 1], and hence

d[A, B(r)] = d[!P + Q, B(r)] = d[!P1P + BQ1P + Q, B(r)]

= d[!P1P + Q1P + Q, B(r)] = d[!P2 + Q2, B(r)], (4.8)

if we setP2 = P1P, Q2 = P2P + Q,

so that P2 and Q2 = I ! P2 are the projectors associated to the decompositionRn = N1 3 (N2 3M). Of course, using Property A 2, we also have

d[!P2 + Q2, B(r)] = d[!P2 + Q2, B1(0, r) + B2(0, r)], (4.9)

where B1(0, r) = B(r) .N1 and B2(0, r) = B(r) . (N2 3M).

We reduce this situation to a one-dimensional one. Namely, if D ' N1 is openand bounded, and if g : D " N1 is continuous and such that 0 %& g(!D), let usdefine d1[g, D] by

d1[g, D] = d[g * P2 + Q2, D + B2(0, r)],

so that, in particular,

d[!P2 + Q2, B1(0, r) + B2(0, r)] = d1[!I, B1(0, r)]. (4.10)


Then, clearly, using Property A 1,

d1[I, D] = d[I, D + B2(0, r)] = 1 if 0 & D. (4.11)

Now, g(P2x) + Q2x = 0 is and only if Q2x = 0 and x = P2x is a zero of g. Hence,if D1 and D2 are disjoint open subsets of D and 0 %& g(D \ (D1 /D2)), we have

0 %& (g * P2 + Q2)([D + B2(0, r)] \ [(D1 + B2(0, r)) / (D2 + B2(0, r))]),

so that Property A 2 implies that

d1[g, D] = d[g * P2 + Q2, D + B2(0, r)]

= d[g * P2 + Q2, D1 + B2(0, r)] + d[g * P2 + Q2, D2 + B2(0, r)]

= d1[g, D1] + d2[g, D2]. (4.12)

Finally, if G : D 1 [0, 1]" N1 is continuous and such that 0 %& G(!D 1 [0, 1]), then0 %& (G * P2 + Q2)(!(D + B2(0, r)) 1 [0, 1]), and, by Property A 3, we get

d1[G(·,"), D] = d[G(P1(·),") + Q2, D + B2(0, r)] is independent of ". (4.13)

Finally we show that

d1[!I, D] = !1 = (!1)2q+1, (4.14)

for any bounded open neigbourhood D of 0 in N1, which, together with (4.8), (4.9)and (4.10), will conclude the proof. We have N1 = {*e : * & R} for some e & Rn

with #e# = 1. Let us consider the open bounded sets

D = {*e : * & ]! 2, 2[}, D1 = {*e : * & ]! 2, 0[}, D2 = {*e : * & ]0, 2[},

and let h : N1 " N1 and K : N1 1 [0, 1]" N1 be respectively defined by

h(*e) = (|*|! 1)e, H(*e,") = "(|*| ! 2)e + e.

Since h(0) = !e %= 0, and K(*e,") = e %= 0 for all (*,") & {±2}1 [0, 1], we have,by (4.11), (4.12) and (4.13),

0 = d1[e, D] = d1[K(·, 0), D] = d1[K(·, 1), D]

= d1[h, D] = d1[h, D1] + d1[h, D2]. (4.15)

Now, on D1, h(*e) = !(* + 1)e := g(*e), with g(*e) %= 0 on D \ D1, so that

d1[h, D1] = d1[g, D1] = d1[g, D].


Using the homotopy L : D 1 [0, 1]" N1 defined by

L(*e,") = !(* + ")e,

we see immediately that L(*e,") %= 0 for (*,") & {±2} 1 [0, 1], and hence, by(4.13), we get

d1[g, D] = d1[!I, D],

so that

d1[h, D1] = d1[!I, D]. (4.16)

In a similar way, on D2, h(*e) = (*! 1)e := f(*e), with f(*e) %= 0 on D \ D2, sothat

d1[h, D2] = d1[f, D2] = d1[f, D].

Using the homotopy M : D 1 [0, 1]" N1 defined by

M(*e,") = (*! ")e,

we see immediately that M(*e,") %= 0 for (*,") & {±2} 1 [0, 1], and hence, by(4.13), we get

d1[f, D] = d1[I, D] = 1,

so that

d1[h, D2] = 1. (4.17)

From (4.15), (4.16) and (4.17), we obtain

0 = d1[!I, D] + 1,

and henced1[!I, B(r) .N1] = !1.

Chapter 5

Brouwer index

5.1 Definition

One can localize the concept of Brouwer degree in the neighbourhood of an isolatedpoint of f!1(z).

Definition 5.1.1 Let f & C(D, Rn), z & Rn, and y be an isolated element ofD . f!1(z). The Brouwer index of f at y is defined by

iB[f, y] = dB [f, Br(y), z] = dB[f, Br(y); f(y)], (5.1)

where r > 0 is such that {y} = Br(y) . f!1(z).

Example 5.1.1 If L : Rn " Rn is linear and invertible, then, for each z & Rn,one has, with y = L!1(z),

iB[L, y] = sign det L. (5.2)

Using formula (3.5), one can write this result as

iB[L, 0] = sign det L = (!1)m (5.3)

where m is the sum of the multiplicities of the negative eigenvalues of L, or, equiv-alently the sum of the multiplicities of the negative characteristic values of L.

If follows easily from Theorem 3.4.1 that the right-hand member of formula (5.1)does not depend upon the choice of r. A direct consequence of the definition andTheorem 3.4.3 is the following

Proposition 5.1.1 Let f & C(D, Rn) and z %& f(!D) be such that f!1(z) is finite.Then

dB [f, D, z] ="

y"f!1(z)

iB[f, y]. (5.4)

The assumptions are in particular satisfied when f is of class C1 and z is a regularvalue.

37

38 CHAPTER 5. BROUWER INDEX

5.2 Computation

The index is easily computed when f is of class C1 in the neighbourhood of y.

Theorem 5.2.1 Let y & Rn, D an open neighbourhood of y and f & C(D, Rn) besuch that f is di"erentiable at y and Jf (y) %= 0. Then

iB[f, y] = sign Jf (y). (5.5)

Proof. As f $y is invertible, f $

y(u) %= 0 for all u & !B(1), and hence there exists µ > 0such that #f $

y(u)# ( µ for all u & !B(1). Consequently, if x %= 0,

#f $y(x)# =

CCCCf$y

2#x# x

#x#

3CCCC ( µ#x#.

On the other hand, there exists r0 > 0 such that, whenever #x! y# ) r0, one has

#f(x)! f(y)! f $y(x! y)# ) µ

2#y ! x#.

Hence, for all 0 < r ) r0, " & [0, 1] and all x & !Br(y), one has

#(1!")[f(x)! f(y)] +"f $y(x! y)# = #f $

y(x! y)+ (1!")[f(x)! f(y)! f $y(x! y)]#

( #f $y(x! y)# ! #f(x)! f(y)! f $

y(x! y)# ( µ

2#x# > 0.

It then follows from Theorem 3.4.2 that, for all 0 < r ) r0, one has

dB[f, Br(y), f(y)] = dB[f(·)! f(y), Br(y), 0] = dB[f $y(·! y), Br(y), 0] (5.6)

= dB[f $y, B(r), 0], (5.7)

and the result follows from the definition of the Brouwer index and formula (5.2).

Using Theorem 5.2.1 and Proposition 5.1.1, we obtain the following slight gen-eralization of Proposition 3.1.2.

Corollary 5.2.1 Let D ' Rn be open and bounded, f & C(D, Rn).C1(D, Rn) andz %& f(!D) be a regular value for f. Then

dB[f, D, z] ="

x"f!1(z)

sign Jf (x),

and the sum is finite.

An important class for which the index at 0 exists is the one of nondegeneratepositive homogeneous continuous mappings.

5.2. COMPUTATION 39

Definition 5.2.1 We say that h & C(Rn, Rm) is positive homogeneous of or-der * > 0 if, for each x & Rn and each " > 0, one has

h("x) = "&h(x). (5.8)

It follows from the definition that h(0) = 0. Any linear mapping is of coursepositive homogeneous of order one.

This concept can be generalized as follows. Let p ( 1 be an integer, nj ( 1 beintegers, *j > 0 be positive real numbers (j = 1, . . . , p), and n =

Bpj=1 nj .

Definition 5.2.2 We say that the continuous mapping

h : Rn " R

n = Rn1 1 . . .1 R

np , x 2" h(x) = (h1(x), . . . , hp(x)) (5.9)

is positive homogeneous of orders *1, . . . ,*p if, for each j = 1, . . . , p, themapping hj : Rn " Rnj is positive homogeneous of order *j, i.e. if for each x & Rn

and each " > 0, one has

h("x) = ["&1h1(x), . . . ,"&php(x)]. (5.10)

Definition 5.2.3 We say that the positive homogeneous mapping (5.9) is nonde-generate if h!1(0) = {0}.

Consequently, if (5.9) is nondegenerate, Corollary 3.4.1 implies that dB [h, B(r), 0]exists for all r > 0 and is independent of r. The common value is the Brouwer indexiB[h, 0].

Proposition 5.2.1 If (5.9) is nondegenerate, there exists µ > 0 such that, for allu & !B(1) ' Rn, one has

p"

j=1

#hj(u)# ( µ. (5.11)

Proof. By the nondegeneracy assumption, h(u) %= 0, and henceBp

j=1 #hj(u)# > 0

for all u & !B(1), and hence there exists µ > 0 such thatBp

j=1 #hj(u)# ( µwhenever u & !B(1).

Proposition 5.2.2 Let % be an open neighbourhood of 0, f & C(%, Rn11 . . .1Rnp)such that

limx(0

fj(x)! hj(x)

#x#&j= 0 (j = 1, . . . , p), (5.12)

for some positive homogeneous non-degenerate mapping h & C(Rn, Rn) of type(5.9). Then

iB[f, 0] = iB[h, 0]. (5.13)


Proof. By Proposition 5.2.1, there exists µ > 0 be such that h satisfies condition(5.11). Consequently, for each u & !B(1), there exists j(u) & {1, . . . , p} suchthat #hj(u)(u)# ( µ

p , and, by continuity, there exists +(u) > 0 such that, for all

x & !B(1) .Bu(+(u)), one has

#hj(u)(x)# ( µ

2p,

By compactness, there exists a finite family {S1, . . . , Sm} of closed sets Sk ' !B(1)and a finite family of points {u1, . . . , um} such that uk ' Sk ' Buk(+(uk)), !B(1) =/k

j=1Sk and hence such that

#hj(uk)(x)# ( µ

2pfor all x & Sk (1 ) k ) m). (5.14)

By assumption (5.12), there exists r0 & ]0, 1] such that, for all j = 1, . . . , p, and allx & B(r0), one has

#fj(x)! hj(x)# ) µ

4p#x#&j .

Let r & ]0, r0], x & !B(r), and k & {1, 2, . . . , m} such that x)x) & Sk. Then, for any

" & [0, 1] one has, letting jk = j(uk),

#(1! ")hjk (x) + "fjk(x)# = #x#&jk

CCCChjk

2x

#x#

3! "

hjk(x)! fjk(x)

#x#&jk

CCCC

( #x#&jk

-CCCChj

2x

#x#

3CCCC!CCCC

hjk(x) ! fjk(x)

#x#&jk

CCCC

.

( µ

4p#x#&j(k) > 0,

and hence (1 ! ")h(x) + "f(x) %= 0. If follows from Theorem 3.4.2 that, for all0 < r ) r0, one has

dB[f, B(r), 0] = dB[h, B(r), 0] = iB[h, 0],

and the result follows.

5.3 The Brouwer index at infinity

Let f & C(Rn, Rn) and z & Rn be such that there exists R0 > 0 for which

f(x) %= z whenever #x# ( R0. (5.15)

Hence, by Corollary 3.4.1, dB [f, B(R), z] is defined for each R ( R0 and does notdepend upon R. This justifies the following

5.3. THE BROUWER INDEX AT INFINITY 41

Definition 5.3.1 Let f & C(Rn, Rn) and z & Rn be such that condition (5.15)holds for some R0. Then the Brouwer index at infinity of f at z is defined by

iB[f,-, z] = dB [f, B(R), z]

for any R ( R0.

We use the notation iB[f,-] for iB[f,-, 0].

Proposition 5.3.1 Let f & C(Rn, Rn) be such that there exists a continuous non-degenerate homogeneous mapping h : Rn " Rn of type (5.9) for which

lim)x)(%

fj(x) ! hj(x)

#x#&j= 0 (j = 1, . . . , p). (5.16)

Then, for each z & Rn, one has

iB[f,-, z] = iB[h, 0]. (5.17)

Proof. By assumption and Proposition 5.2.1, there exists µ > 0 such that inequality(5.11) holds for all x & Rn. By condition (5.16), there exists R0 > 0 such that

#fj(x)! zj ! hj(x)# ) µ

4p#x#&j (j = 1, . . . , p)

whenever #x# ( R0. Proceeding like in the proof of Proposition 5.2.2, we can showthat if x & Rn is such that #x# ( R0 and x

)x) & Sk for some k & {1, . . . , m}, and if

" & [0, 1], we have

#(1! ")hjk(x) + "[fjk(x)! zjk ]# ( µ

4p#x#&j(k) > 0

and hence (1 ! ")h(x) + "[f(x) ! z] %= 0. So Theorem 3.4.2 implies that, for eachR ( R0,

dB[f, B(R), z] = dB[f(·)! z, B(R), 0] = dB [h, B(R), 0] = iB[h, 0],

and formula (5.17) follows.

Corollary 5.3.1 Let f & C(Rn, Rn) be such that there exists a linear invertiblemapping L : Rn " Rn for which

lim)x)(%

f(x)! L(x)

#x# = 0. (5.18)

Then, for each z & Rn, one has

iB[f,-, z] = sign det L. (5.19)

Chapter 6

Degree in finite-dimensionalvector spaces

6.1 Composition with linear isomorphisms

Let A, B : Rn " Rn be linear isomorphisms, D ' Rn be an open bounded set,g : D " Rn be continuous and z %& Ag(!D). Then the mapping f := A * g *B is continuous on B!1(D) = B!1(D) and z %& f(!B!1(D)) = f(B!1(!D)),which is equivalent to A!1z %& g(!D). Consequently, both dB [f, B!1(D), z] anddB[g, D, A!1z] are defined. The following Lemma relates those two Brouwer degrees.

Lemma 6.1.1 Under the above assumptions,

dB [A * g *B, B!1(D), z] = sign det (AB) · dB[g, D, A!1z]. (6.1)

Proof. From the definition of Brouwer degree for continuous mappings, Sard’s lem-ma and Weierstrass approximation theorem, we can assume without loss of gener-ality that g & C2(D, Rn) and that z is a regular value for A * g *B. Hence,

dB[A * g *B, B!1(D), z] ="

x"(A*g*B)!1(z)

sign JA*g*B(x)

="

x"(A*g*B)!1(z)

sign [det A · Jg(Bx) · det B]

= (sign det AB)"

y"g!1(A!1z)

sign Jg(y)

= (sign det AB) dB [g, D, A!1z].

43

44 CHAPTER 6. DEGREE IN FINITE-DIMENSIONAL VECTOR SPACES

6.2 Definitions and basic properties

Let X be an n-dimensional real topological vector space. It is well known thatif (*1, · · · ,*n) is a base in X, and (e1, · · · , en) the canonical base in Rn, then thelinear mapping

h : X " Rn, x =

n"

j=1

xj*j 2" h(x) =

n"

j=1

xjej

is a homeomorphism.Let now D ' X be open and bounded, f : D " X continuous and z & X

such that z %& f(!D). Then h * f * h!1 is a continuous mapping from the closureof the open bounded set h(D) ' Rn to Rn such that h(z) %& h * f * h!1(!h(D)).Consequently, dB[h * f * h!1, h(D), h(z)] is well defined. If now (,1, · · · ,,n) isanother base in X, and if

g : X " Rn, x =

n"

j=1

xj,j 2" h(x) =

n"

j=1

xjej ,

is the corresponding linear homeomorphism, then dB[g * f * g!1, g(D), g(z)] is welldefined. Now

g * f * g!1 = g * h!1 * h * f * h!1 * h * g!1,

so that, if we set m = h * g!1, then m : Rn " Rn is a linear homeomorphism and

g * f * g!1 = m!1 * (h * f * h!1) *m.

We can therefore apply Lemma 6.1.1 to obtain

dB[g * f * g!1, g(D), g(z)]

= dB[m!1 * (h * f * h!1) *m, g(D)), g(z)]

= dB[m!1 * (h * f * h!1) *m, m!1(m * g)(D)), g(z)]

= dB[m!1 * (h * f * h!1) *m, m!1h(D)), m!1(h(z))]

= dB[h * f * h!1, h(D), h(z)]. (6.2)

This independence of the Brouwer degree with respect to the choice of the basejustifies the following definition.

Definition 6.2.1 Let X be a n-dimensional topological vector space, D ' X openand bounded, f : D " X continuous and z & X such that z %& f(!D). The Brouwerdegree dB[f, D, z] is defined by the formula

dB [f, D, z] = dB[h * f * h!1, h(D), h(z)]

where

h : X " Rn, x =

n"

j=1

xj*j 2"n"

j=1

xjej

6.3. THE BROUWER INDEX 45

is the linear homeomorphism associated to a base (*1, · · · ,*n) of X and the canon-ical base (e1, · · · , en) of Rn.

It is easy to show, from this definition, that the degree in space X conserves theproperties of degree in Rn.

Suppose now that X and Z are two n-dimensional topological vector spaces,D ' X is open and bounded, f : D " Z is continuous and z %& f(!D). Choosingbases (*1, · · · ,*n) and (,1, · · · ,,n) in X and Z respectively, and denoting by h :X " Rn and g : Z " Rn linear homeomorphisms constructed like above, we seethat the Brouwer degree dB [g * f * h!1, h(D), g(z)] is well defined. If we change

bases, i.e. homemorphisms, then we have, with 1h : X " Rn and 1g : Z " Rn,

g!1 * g * f * h!1 * h = f = 1g!1 * 1g * f * 1h!1 * 1h,

and hence

1g * f * 1h!1 = m * g * f * h!1 * 1m,

where m := 1g * g!1 : Rn " Rn and 1m := h * 1h!1 : Rn " Rn are linear homeomor-phisms. Then, by Lemma 6.1.1, we obtain, like above that

dB[1g * f * 1h!1,1h(D),1h(z)] = sign (det m · det 1m)dB[g * f * h!1, h(D), h(z)],

and this relation can be interpreted as defining a Brouwer degree for f betweenoriented vectors spaces X and Z.

6.3 The Brouwer index

One can extend the Brouwer index to this more general situation. Let X, Z betwo n-dimensional topological vector spaces, which we suppose oriented if they aredi!erent. Let D ' X be an open bounded set.

Definition 6.3.1 Let f : D " Z be continuous, z & Z, and y be an isolated elementof D . f!1(z). The Brouwer index of f at y is defined by

iB[f, y] = dB [f, By(r), z] = dB[f, By(r), f(y)], (6.3)

where r > 0 is such that {y} = By(r) . f!1(z).

If follows easily from the excision property of degree that the right-hand memberof formula (6.3) does not depend upon the choice of r.

Example 6.3.1 If L : X " Z is linear and invertible, then, for each z & Z, onehas, with y = L!1(z),

iB[L, y] = sign det gLh!1, (6.4)

where h : X " Rn and g : Z " Rn are linear homeomorphisms of the type intro-duced above.


A direct consequence of Definition 6.3.1 and of Theorem 5.2.1 is the followingformula for computing the index at y when f is of class C1 in a neighborhood of yand its Jacobian is not zero.

Theorem 6.3.1 Let y & X, D ' X an open neighborhood of y and f & C(D, Z)be such that f is di"erentiable at y and f $(y) : X " Z is invertible. Then, ifh : X " Rn and g : Z " Rn are linear homeomorphisms of the type introducedabove,

iB[f, y] = sign Jgfh!1(h(y)). (6.5)

Proof. By definition and excision property 3.4.1,

iB[f, y] = dB [f, By(r), f(y)] = dB [gfh!1, h[By(r)], g(f(h(y)))]

= dB [gfh!1, Bh(y)(-), g(f(h(y)))]

for su$ciently small r > 0, and - > 0 such that Bh(y)(-) ' h(By(r)). Then, usingformulas (5.6) we get

dB[gfh!1, Bh(y)(-), g(f(h(y)))] = dB[(gfh!1)$h(y), Bh(y)(-), 0]

= sign Jgfh!1(h(y)).

A direct consequence of the definition and of the additivity of degree is thefollowing

Proposition 6.3.1 Let f : D ' X " Z be continuous, and z %& f(!D) be suchthat f!1(z) is finite. Then

dB[f, D, z] ="

y"f!1(z)

iB[f, y]. (6.6)

The assumptions are in particular satisfied when f is of class C1 and z is a regularvalue.

6.4 Reduction formulas

Suppose now that X is a n-dimensional topological vector space, Y ' X a vectorsubspace of dimension m < n, D ' X is open and bounded, c : D " Y is continuousand z & Y \ c(!D). Let f := I ! c. Notice that each solution of equation f(x) = zis such that x = c(x) + z & Y and hence a relation could be expected betweendB[f, D, z] and dB[f |Y , D . Y, z]. This is the conclusion of the first reductionformula due to Leray and Schauder.

Theorem 6.4.1 Let X be a n-dimensional topological vector space, Y ' X a vectorsubspace of dimension 1 ) m < n, D ' X be open and bounded, c : D " Ycontinuous and z & Y \ c(!D). If f := I ! c, then

dB [f, D, z] = dB[f |Y , D . Y, z]. (6.7)

6.4. REDUCTION FORMULAS 47

Proof. From the definitions of degrees and Weierstrass approximation theorem, wecan assume that X = Rn, Y = Rm = {x & Rn : xm+1 = · · · = xn = 0}, and c is ofclass C2. Now,

f $(x) = I ! c$(x) =

2Im ! (!jci(x))1+i,j+m ! (!jci(x))1+i+m;m+1+j+n

0 In!m

3

so that

Jf (x) = det [Im ! (!jci(x))1+i,j+m],

and hence, if x & Y,

Jf (x) = Jf |Y (x). (6.8)

Therefore, if z & Y is a regular value for f |Y , we have Jf |Y (x) %= 0, and hence zis also a regular value for f because if f(x) = z, then x & Y . D and, by (6.8),Jf (x) %= 0. Thus, by Sard lemma applied to f |Y and the above reasoning, we canassume that z & Y is a regular value for both f |Y and f. Consequently,

dB[f, D, z] ="

x"f!1(z)

sign Jf (x) ="

x"f |!1Y (z)

sign Jf |Y (x) = dB [f |Y , D . Y, z].

We can deduce from the previous result a second reduction formula, provedin a more general setting in [265].

Theorem 6.4.2 Let X and Z be n-dimensional topological vector spaces, L : X "Z a linear mapping with N(L) %= {0}, Y ' Z a vector subspace such that Z =Y 3R(L), D ' X an open bounded set, r : D " Y a continuous mapping such that0 %& (L + r)(!D). Then, for each isomorphism J : N(L) " Y, and each projectorP : X " X such that R(P ) = N(L), one has, if f := L + r,

dB[f, D, 0] = iB[L + JP, 0] · dB[J!1r|N(L), D .N(L), 0]. (6.9)

Proof. Let Q : Z " Z be the projector such that R(Q) = Y and N(Q) = R(L). Wefirst notice that if (L + JP )x = 0, then, by applying Q and I !Q to the equation,we obtain the equivalent system

Lx = 0, JPx = 0

which immediately implies that x = 0. Thus L + JP : X " Z, one-to-one, is ontoand iB[L+JP, 0] is well defined and has absolute value one. Furthermore, if z & Y,then, again by projection on Y and R(L), one gets

(L + JP )(x) = z 4 Lx = 0, JPx = z 4 x = Px = J!1z,


so that

(L + JP )!1z = J!1z.

Consequently,

f = L + r = L + JP + r ! JP = (L + JP )[I + (L + JP )!1(r ! JP )]

= (L + JP )(I ! P + J!1r).

Using Lemma 6.1.1, Theorem 6.4.1 and the definitions above, we get

dB[f, D, 0] = dB [g * f * h!1, h(D), 0]

= dB[g * (L + JP ) * h!1 * h * (I ! P + J!1r) * h!1, h(D), 0]

= sign det [g * (L + JP ) * h!1] · dB[h * (I ! P + J!1r) * h!1, h(D), 0]

= iB[L + JP, 0] · dB[I ! P + J!1r, D, 0]

= iB[L + JP, 0] · dB[(I ! P + J!1r)|N(L), D .N(L), 0]

= iB[L + JP, 0] · dB[J!1r|N(L), D .N(L), 0].

Remark 6.4.1 Formula (6.9) implies in particular that

|dB [L + r, D, 0]| =DDdB[J!1r|N(L), D .N(L), 0]

DD .

6.5 Computation of the index in degenerate case

The computation of the index of f at y when f $y is not invertible is more complicated.

We state and prove here a generalization, due to B. Laloux [237], of a result of V.B.Melamed [280].

Theorem 6.5.1 m and k being nonnegative integers, and D ' X being an openneighborhood of 0, assume that f & C(D, Z) can be written in the form

f(x) = Lx +k"

i=0

Hm+i(x) + R(x),

where L : X " Z is linear and not invertible, Hm+i is homogeneous of order m + i(i = 0, . . . , k), and R is such that

limx(0

R(x)

#x#m+k= 0. (6.10)

Assume furthermore that the following conditions hold.

(i) For i %= k, Hm+i(N(L)) ' R(L).

6.5. COMPUTATION OF THE INDEX IN DEGENERATE CASE 49

(ii) Hm+k(x) %& R(L) for any x & N(L) \ {0}.

(iii) There exists * > 0 such that, for all i %= k and all x, y & D, one has

#Hm+i(x)!Hm+i(y)# < *#x! y#max{#x#m+i!1, #y#m+i!1}.

(iv) m > k + 1.

Then iB[f, 0] is well defined and

iB[f, 0] = iB[L + JP, 0] · iB[J!1QHm+k|N(L), 0] (6.11)

where Q : Z " Z is any projector such that N(Q) = R(L), and J : N(L)" R(Q)any isomorphism.

Proof. Let us define the homotopy F : D 1 [0, 1]" Z by

F (x,") = Lx + (1! ")QHm+k(x) + "

Ek"

i=0

Hm+i(x) + R(x)

F.

Let - > 0 be such that B(-) ' D, " & [0, 1], and let x & !B(-) be a possible zero ofF (·,"). Then, by applying Q and (I !Q) to both members, we see x is a solutionof the system

(1! ")QHm+k(x) + "

Ek"

i=0

QHm+i(x) + QR(x)

F= 0

L(I ! P )x + "(I !Q)

Ek"

i=0

Hm+i(x) + R(x)

F= 0,

where P : X " X is a projector onto N(L), or equivalently

QHm+k(x) + "

Ek!1"

i=0

QHm+i(x) + QR(x)

F= 0

L(I ! P )x + "(I !Q)

Ek"

i=0

Hm+i(x) + R(x)

F= 0, (6.12)

Noticing that there exists , > 0 such that #L(I!P )x# ( ,#(I!P )x# for all x & X,we deduce from the second equation in (6.12) and from assumptions (6.10) and (iii),that there exist -1 > 0 and p > 0 such that

#(I ! P )x# ) "p-m (6.13)

whenever x is a solution such that #x# = - & ]0, -1]. Now, if -2 > 0 is such that2p-m!1

2 < 1, we also have, for any - & ]0, min{-1, -2}, and x such that #x# = -,

#Px# ( #x# ! #(I ! P )x# ( -! -

2=

-

2> p-m ( #(I ! P )x#, (6.14)


and

#Px# ) #x#+ #(I ! P )x# ) - +-

2=

3-

2. (6.15)

Now, using the fact that, from Assumption (ii), Hm+i(Px) = 0 (i = 1, . . . , k ! 1),we deduce from the first equation in (6.12) that

#x#1!m!kQHm+k(x)

= !"#x#1!m!kn

Ek!1"

i=0

[QHm+i(x) !QHm+i(Px)] + QR(x)

F

or, equivalently,

L(I ! P )x! #x#1!m!kQHm+k(x) (6.16)

= L(I ! P )x + "#x#1!m!kn

Ek!1"

i=0

[QHm+i(x)!QHm+i(Px)]

F.

Now, letting y = x/#x# & !B(1), we get

#L(I ! P )x! #x#1!m!kQHm+k(x)# = #x##L(I ! P )y !QHm+k(y)#

and

L(I ! P )y !QHm+k(y) = 0 4 L(I ! P )y = 0, QHm+k(y) = 0

4 y & N(L), QHm+k(y) = 0.

Consequently, from Assumption (ii), L(I ! P )y ! QHm+k(y) %= 0 for y & !B(1),and hence there exists . > 0 such that

#L(I ! P )y !QHm+k(y)# ( . for y & !B(1). (6.17)

Introducing (6.17) in (6.16), we obtain, using Assumption (iii),

-. ) #L(I ! P )x# + *k!1"

i=0

(3/2)m+i!1-i!k#(I ! P )x#+ -1!m!kq(-)#x#m+k

where q(-)" 0 if -" 0. Consequently, if 0 < - ) min /-1, -2, we have

. ) #L#p-m!1 + *pk!1"

i=0

(3/2)m+i!1-m+i!k!1 + q(-).

Now, there exists 0 < -3 ) min{-1, -2 such that

#L#p-m!1 + *pk!1"

i=0

(3/2)m+i!1-m+i!k!1 + q(-) < .

6.5. COMPUTATION OF THE INDEX IN DEGENERATE CASE 51

whenever - & ]0, -3] which implies that F (x,") %= 0 whenever #x# = - & ]0, -3] and" & [0, 1]. Hence, the homotopy invariance property implies that

iB[f, 0] = dB[f, B(-), 0] = dB[F (·, 1), B(-), 0] = dB[F (·, 0), B(-), 0]

= dB[L + QHm+k, B(-), 0].

Using the second reduction formula 6.4.2, we obtain

dB[L + QHm+k, B(-), 0] = iB[L + JP, 0] · dB[J!1QHm+k|N(L), B(-) .N(L), 0]

= iB[L + JP, 0] · iB[J!1QHm+k|N(L), 0],

where J : N(L)" R(Q) is any isomorphism.

The special case of Theorem 6.5.1 where k = 0 is of interest, and can be tracedto M.A. Krasnosel’skii [213]. For other results in this direction, see [403].

Corollary 6.5.1 m > 1 being an integer, and D ' X being an open neighborhoodof 0, assume that f & C(D, Z) can be written in the form

f(x) = Lx + Hm(x) + R(x),

where L : X " Z is linear and not invertible, Hm is homogeneous of order m andR is such that (6.10) holds. If Hm(x) %& R(L) for any x & N(L) \ {0}, iB[f, 0] iswell defined and

iB[f, 0] = iB[L + JP, 0] · iB[J!1QHm|N(L), 0] (6.18)

where Q : Z " Z is any projector such that N(Q) = R(L), and J : N(L)" R(Q)any isomorphism.

Example 6.5.1 Consider the mapping f : R4 " R4 defined, in complex notations(R4 5 C2) by

f1(z1, z2) = z21 + z2

1z2, f2(z1, z2) = z1 + z1z2 + z32 ,

so that, in the notations of Corollary 6.5.1, we have

f(z1, z2) = L(z1, z2) + H2(z1, z2) + R(z1, z2),

where

L =

20 00 1

3, H2(z1, z2) =

2z21

z1z2

3, R(z1, z2) =

2z21z2

z32

3.

Consequently, N(L) = C 1 {0}, R(L) = {0} 1 C, P (z1, z2) = (z1, 0) = Q(z1, z2),R(Q) = C 1 {0}. As QH2(z1, 0) = (z2

1 , 0) %& R(L) for z1 %= 0, all the conditions ofCorollary 6.5.1 are satisfied, and, taking J = I we get L + P = I and hence, usingexample 2.2.3,

iB[f, 0] = iB[z21 , 0] = iB[(x2

1 ! x22, 2x1x2), 0] = 2.

Chapter 7

Continuation theorems

7.1 Extended homotopy invariance property

If [a, b] is a compact interval, we consider the trace of the usual topology of Rn+1

on Rn 1 [a, b]. Let D ' Rn 1 [a, b] be a bounded set which is open for the relativetopology of Rn 1 [a, b], i.e. which is the trace on Rn 1 [a, b] of an open set of Rn+1.Its relative boundary is denoted by !D, and, for each " & [a, b], let

D% := {x & Rn : (x,") & D}, (!D)% := {x & R

n : (x,") & !D}.

Notice that !D% ' (!D)%, and the inclusion may be strict. Let F & C(D, Rn)satisfying the condition

z %& F (!D). (7.1)

Then, for each " & [a, b], z %& F (!D%), and dB [F (·,"),D%, z] is well defined. Weshow that it is independent of ".

Theorem 7.1.1 Let F & C(D, Rn) and z & Rn verify condition (7.1). ThendB[F (·,"),D%, z] is constant on [a, b].

Proof. For " & [a, b] fixed, let

N% = {x & D% : F (x,") = z} ' D%.

By assumption, N% . (!D)% = , so that N% . !D% = ,. As N% is compact in D,there exists an open neighbourhood O% of N% in D% with

N% ' O% ' O% ' D%

and some '0 > 0 such that

O% 1 ["! '0," + '0] ' D.

53

54 CHAPTER 7. CONTINUATION THEOREMS

Of course, for, say, " = a, one replaces [" ! '0," + '0] by [a, a + '0]. We showthat there exists '1 & ]0, '0] such that for each µ & ["! '1," + '1] . [a, b], and eachsolution x of equation F (x, µ) = z, one has

(x, µ) & O% 1 (["! '1," + '1] . [a, b]).

If it is not the case, we can find a sequence (µn) in [a, b] with |µn ! "| ) 1/n and asequence (xn) such that F (xn, µn) = z and

(xn, µn) %& O% 1 (["! (1/n)," + (1/n)] . [a, b]).

Going if necessary to a subsequence, we can assume that xn " x and µn " ",so that, by continuity, F (x,") = z, and, by the openess of O%, x %& O%. But thenx %& N%, a contradiction. For each µ & [" ! '1," + '1] . [a, b], dB[F (·, µ),O%, z] iswell defined, and, by Theorem 3.4.2, independent of µ. Furthermore, by Theorem3.4.1, one has, for all µ & ["! '1," + '1] . [a, b],

dB[F (·, µ),O%, z] = dB [F (·, µ),Dµ, z].

Hence dB[F (·, µ),Dµ, z] is locally constant on [a, b], and hence constant there.

Remark 7.1.1 Of course, one has a result similar to Theorem 7.1.1 for a continuousmapping F : D ' X " Y where X and Z are oriented topological vector spaces ofthe same finite dimension and D an open bounded set of X 1 [a, b].

7.2 Leray-Schauder continuation theorem

To complete the above result by some information on the structure of the solutionset of the homotopy, we need a result from the theory of metric spaces, often referredas Whyburn’s lemma.

Lemma 7.2.1 Let K be a compact metric space, and K1, K2 two closed disjoinedsubsets of K. Then

i) either there exists a component (maximal closed connected part) of K meetingK1 and K2;

i) or there exist two disjoint compacts #K1 and #K2 such that K = #K1 / #K2 and

Ki ' #Ki, i = 1, 2.

An easy and useful consequence of Whyburn lemma is the following Corollary.

Corollary 7.2.1 Let M be a metric space, K a compact subset of M . Assume thatK0 and K1 are two disjoint closed subsets of K such that no connected componentof K intersect both. Then there exists an open bounded set U such that

K0 ' U, U .K1 = ,, !U .K = ,.

7.2. LERAY-SCHAUDER CONTINUATION THEOREM 55

The following result is called the Leray-Schauder continuation theorem[251].

Theorem 7.2.1 Let F & C(D, Rn), z & Rn verify condition (7.1), and let

S := {(x,") & D : F (x,") = z}

and, for each " & [a, b] let

S% = {x & Rn : (x,") & S}.

If dB[F (·,"),D%, z] %= 0 for some " & [a, b], there exists a connected component C ofS which meets both Sa 1 {a} and Sb 1 {b}.

Proof. By Theorem 7.1.1, dB[F (·,"),D%, z] is constant, and hence nonzero for each" & [a, b]. As S is compact, for each " & [a, b], the set S% 1 {"} is a closed subset ofS. Hence, the sets

K1 := (Da 1 {a}) . S, K2 := (Db 1 {b}) . S

are disjoint closed non-empty subsets and S. If no component of S meets bothDa 1 {a} and Db 1 {b} then Lemma 7.2.1 implies the existence of compact sets #K1

and #K2 such that

K1 ' #K1, K2 ' #K2, #K1 . #K2 = ,, S = #K1 / #K2.

Let' := (1/2)dist (#K1, #K2), O := {(x,") & D : dist [(x,"), #K1] < '}.

O is bounded, open, such that O . K2 = , and S . !O = ,. The conditions ofTheorem 7.1.1 are satisfied for O, so that dB [F (·,"),O%, z] is constant for each" & [a, b]. But, by Corollary 3.4.1, we have

dB [F (·, a),Oa, z] = dB [F (·, a),Da, z] %= 0,

because Oa contains all the solutions of equation F (x, a) = z. On the other hand,Ob = ,, and hence dB[F (·, b),Ob, z] = 0, a contradiction.

The Leray-Schauder continuation theorem can be written in the form of analternative.

Theorem 7.2.2 Let D ' Rn 1 [a, b] an open bounded set and F : D " Rn becontinuous. Assume that Sa . (!D)a = , and that

dB [F (·, a),Da, 0] %= 0.

Then S contains a connected component C intersecting Sa1{a} and !D/(Sb1{b}).


Proof. By Theorem 7.2.1, the result is proved if S . !D = ,. If S1 := S . !D %= ,,then S1 is a closed subset of S disjoint from S0 = Sa 1 {a}. Hence, by Whyburn’slemma, either there exists a connected component C of S which connects S0 andS1, and the result is proved, or there exist two disjoint closed sets C0 and C1

containing respectively S0 and S1 and such that S = C0 / C1. We can then findan open neighborhood V0 of C0 contained in D and such that V0 . C1 = ,, and inparticular V0 . !D = ,. Thus all condition of Theorem 7.2.1 with D replaced by C0

are satisfied, and C0 contains a connected component C0 which meets Sa1 {a} andSb 1 {b}.

Remark 7.2.1 Of course, one has results similar to Theorems 7.2.1 and 7.2.2 fora continuous mapping F : D ' X " Y where X and Z are oriented topologicalvector spaces of the same finite dimension and D an open bounded set of X1 [a, b].

7.3 The case of an unbounded set

We can now formulate a variant of Theorem 7.2.2 for the case where D is unbounded(and in particular when D = Rn 1 [a, b]).

Theorem 7.3.1 Let D ' Rn 1 [a, b] be an open unbounded set and F : D " X becontinuous. Let

S = {(x,") & D : F (x,") = 0},and, for each " & [a, b] let

S% = {x & Rn : (x,") & S}.

Assume that Sa is bounded, that Sa . (!D)a = , and that

dB[F (·, a),Da, 0] %= 0.

Then S contains a connected component C intersecting Sa 1 {a} and which eitherintersects !D / (Sb 1 {b}) or is unbounded.

Proof. Let n0 be a positive integer such that Sa ' B(n0). For each integer n ( n0,Dn = D.(B(n)1 [a, b]) is an open bounded set for which the conditions of Theorem7.2.2 are satisfied and therefore there exists xn & Sa and a connected component Cn

of S containing xn and meeting !Dn/[(Sb.B(n))]1{b}). By the compactness of Sa,the sequence (xn) has an accumulation point x0 & Sa. Assume that the connectedcomponent C0 of S containing x0 does not meet !D / (Sb 1 {b}). We have to provethat C0 is unbounded. If it is bounded and A is an open bounded subset of D suchthat A 0 C0 / (Sa 1 {a}), then Theorem 7.2.2 applied to A instead of D impliesthat S . !A %= ,. Thus, {(x0, a)} and S . !A satisfy the conditions of Whyburn’stheorem and, by assumption, cannot be connected by a connected component ofS. Consequently, by Corollary 7.2.1, there is an open neighbourhood V0 of (x0, a)in A such that !V0 . S = ,. But there exists n1 ( n0 such that xn1 & V0 andB(n1) 1 [a, b] 0 A. Consequently, Cn1 meets both V0 and (Rn 1 [a, b]) \ V0, andhence Cn . !V0 %= ,, a contradiction with S . !V0 = ,.

7.3. THE CASE OF AN UNBOUNDED SET 57

We notice the following useful consequence of Theorem 7.3.1.

Corollary 7.3.1 Let F : Rn 1 [a, b] " Rn be continuous. Assume that Sa isbounded and that

dB[f(·, a),%, 0] %= 0,

for some open bounded set % 0 Sa. Then S contains a connected component Cintersecting Sa 1 {a} and which either intersects Sb 1 {b} or is unbounded.

The example of n = 1, [a, b] = [0, 1], F (x,") = " ! 12 sinx, shows that the

conclusion of Corollary 7.3.1 does not necessary hold if we drop the assumptionthat Sa is bounded.

Remark 7.3.1 Of course, one has results similar to Theorem 7.3.1 and Corollary7.3.1 for a continuous mapping F : D ' X " Y where X and Z are orientedtopological vector spaces of the same finite dimension and D an open unboundedset of X 1 [a, b].

Chapter 8

Fixed points and zeros

8.1 Semilinear equations

We deduce here some useful existence theorems based on Theorem 7.2.1. Let X, Zbe oriented topological vector spaces of dimension n, L : X " Z a linear mapping,D ' X 1 [0, 1] an open bounded set and N : D " Z a continuous mapping. Let

S := {(x,") & D : Lx + "N(x,") = 0}

Theorem 8.1.1 Assume that there exists a linear mapping A : X " Z such thatthe following conditions hold.

1. L + A is invertible;

2. Lx + (1! ")A + "N(x,") %= 0 for each x & (!D)% and each " & ]0, 1];

3. 0 & D0.

Then

dB[L + (1! ")A + "N(·,"),D%, 0] = dB [L + A,D%, 0]

= sign det (L + A) (" & [0, 1]),

and S contains a continuum C connecting (0, 0) to D1, along which " takes all valuesin [0, 1]. In particular the equation Lx+N(x, 1) = 0 has at least one solution in D1.

Proof. Because L + A is invertible and 0 & D0, we have

dB[L + A, D0, 0] = sign det (L + A) = ±1,

Lx + (1 ! ")Ax + "N(x,") %= 0 for each (x,") & !D, and the result follows fromTheorem 7.2.1 applied to F : D " Z defined by

F (x,") = Lx + (1! ")Ax + "N(x,").

59

60 CHAPTER 8. FIXED POINTS AND ZEROS

When L is invertible, the simplest choice for A in Theorem 8.1.1 is A = 0. As,in this case, #Lx# %= 0 for x %= 0, we have, for all x & X,

#Lx# ( l#x# with l := min)x)=1

#L(x)# > 0. (8.1)

Corollary 8.1.1 Assume that L : X " Z is linear and invertible, and that g :B(R) ' X " Z is continuous and such that

#g(x)# ) lR for x & B(R). (8.2)

Then equation

Lx + g(x) = 0

has at least one solution in B(R). Furthermore, if L + g has no zero on !B(R),then

|dB [L + g, B(R), 0]| = 1.

Proof. Let us take N(x,") = g(x) in Theorem 8.1.1. If there exists x & !B(R) suchthat Lx + g(x) = 0, the result is proved. If not, let (x,") & B(R) 1 [0, 1] be apossible solution of equation

Lx + "g(x) = 0.

We already know that x %& !B(R) for " = 1. For " & [0, 1[ we have

l#x# ) #Lx# = #"g(x)# <) lR,

and hence #x# < R. The result follows from Theorem 8.1.1 with D = B(R)1 [0, 1].

Corollary 8.1.2 Assume that L : X " Z is linear and invertible, and that g :X " Z is continuous and such that, for some positive ,

#g(x)# ) *#x#+ , for x & X, (8.3)

where

0 ) * < l. (8.4)

Then equation

Lx + g(x) = 0

has at least one solution in B(R). Furthermore, for - > 'l!& , we have

|dB[L + g, B(-), 0]| = 1.

8.2. BROUWER’S FIXED POINT THEOREM AND EXTENSIONS 61

Proof. If we take R = 'l!& and x & B(R), then, using (8.3) and (8.4),

#g(x)# ) *

l ! *, + , = l

,

l! *= lR,

and we can apply Corollary 8.1.1 for the existence conclusion. Now, for any possible(x,") & X 1 [0, 1] such that

Lx + "g(x) = 0,

we have

l#x# ) #Lx# = #"g(x)# ) *#x# + ,

and hence #x# ) 'l!& = R, so that, for - > R, the homotopy invariance theorem

3.4.2 implies that

dB [L + g, B(-), 0] = dB[L, B(-), 0] = ±1.

An immediate consequence of Corollary 8.1.1 is Brouwer’s fixed point the-orem [39].

Corollary 8.1.3 Any continuous g : B(R) ' X " B(R) ' X has at least onefixed point.

Proof. Apply Corollary 8.1.1 with L = !I, so that l = 1.

8.2 Brouwer’s fixed point theorem and extensions

We show now that the existence of a fixed point of a continuous mapping g holdsunder more general conditions upon g and more general sets than closed balls.

Definition 8.2.1 A Hausdor" topological space S has the fixed point propertyif any continuous mapping g : S " S has at least one fixed point.

Thus, Brouwer’s fixed point theorem insures that B(R) ' Rn has the fixed pointproperty. On the other hand, for an closed annulus A[r, R] := {x & Rn : r ) #x# )R} in Rn, a non-trivial rotation around 0 is continuous but has no fixed point inA[r, R], and hence A[r, R] has not the fixed point property.

Proposition 8.2.1 If the Hausdor" topological space A has the fixed point propertyany Hausdor" topological space B homeomorphic to A has the fixed point property.

Proof. Let g : B " B be continuous, and h : A " B be a homeomorphism. Thenk = h!1 * g * h maps continuously A into itself, and has at least one fixed point x#,namely x# = h!1 * g * h(x#). So h(x#) = g(h(x#)), i.e. h(x#) is a fixed point of g.


We therefore have the following extended version of Brouwer’s fixed pointtheorem.

Corollary 8.2.1 Any Hausdor" topological space homeomorphic to the unit closedball B(1) ' Rn has the fixed point property.

In particular any closed ball, closed n-intervall, closed n-simplex, or any compactconvex subset of Rn has the fixed point property.

Remark 8.2.1 The proof of theorem 8.1.1 only uses the fact that g maps !B(R)into B(lR). Hence, the following generalized version of Brouwer’s fixed point theo-rem, already noticed by G.D. Birkho! and Kellogg in 1922 [28], proved by Knaster-Kuratowski-Mazurkiewicz in 1929 [203], but generally known as Rothe’s fixedpoint theorem holds : Any continuous mapping g : B(R) ' Rn " Rn such thatg(!B(R)) ' B(R) has at least one fixed point in B(R).

The assumption of Brouwer’s fixed point theorem means that, for any x & B(R),the segment joining x to g(x) lies in B(R). The assumption of Rothe’s fixed pointtheorem means that the same is required only for any x & !B(R). We prove nowa special case of a result of Leray and Schauder [251]showing that the existence ofa fixed point still holds if, for any x & !B(R), the segment joining x to g(x) is notcontained in the normal half line {µx : µ > 1}. This is Leray-Schauder’s fixedpoint theorem in Rn.

Theorem 8.2.1 Any continuous mapping g : B(R) ' Rn " Rn such that g(x) %=µx for all x & !B(R) and all µ > 1 has at least one fixed point in B(R).

Proof. If g has a fixed point on !B(R) the result is proved. If not, let us introduceagain the homotopy H : B(R)1 [0, 1]" Rn defined by H(x,") = x!"g(x). ClearlyH(x, 0) = x %= 0 for x & !B(R), and, by assumption, H(x, 1) = x!g(x) %= 0 for eachx & !B(R). Now, if H(x,") = 0 for some (x,") & !B(R)1 ]0, 1[ , then "!1x = g(x),a contradiction with the assumption.

Remark 8.2.2 As shown essentially by H. Schaefer [345] in 1955, Leray-Schau-der’s fixed point theorem 8.2.1 follows from Brouwer’s fixed point theorem 8.1.3applied to the continuous mapping h : B(R)" B(R) defined by

h(x) =

/f(x) if #f(x)# ) R,

R)f(x))f(x) if #f(x)# > R .

If the fixed point x# of h is such that #f(x#)# > R, it contradicts Leray-Schauder’shypothesis with " = R/#f(x#)#. Hence, #f(x#)# ) R and, by construction of h, x#

is a fixed point of f.

8.3. NON-EXISTENCE THEOREMS 63

8.3 Non-existence theorems

Brouwer’s degree also provides the non-existence of continuous mappings havingspecific properties.

Definition 8.3.1 If B ' A ' H, where H is a Hausdor" topological space, B is aretract of A if there exists a continuous mapping r : A" B such that r|B = IB .

The following result, which can be traced to Bohl [31], was rediscovered inde-pendently by Borsuk [35] and is called Borsuk’s no-retraction theorem.

Theorem 8.3.1 There is no retraction r : B(R) ' Rn " !B(R).

Proof. If such a retraction r exists, then, by Corollary 3.4.3, we have

dB[r, B(R), 0] = dB[I, B(R), 0] = 1

and hence, r has at least one zero in B(R), a contradiction with #r(x)# = R foreach x & B(R).

Remark 8.3.1 Theorem 8.3.1 also follows from Brouwer’s fixed point theorem8.1.3. Indeed, if a retraction r : B(R)" !B(R), exists, then !r : B(R)" !B(R)has at least one fixed point x# by Brouwer’s fixed point theorem 8.1.3. By assump-tion R = # ! r(x#)# = #x##, so that x# & !B(R), and hence x# = r(x#). Thus,r(x#) = !r(x#), i.e. r(x#) = 0, a contradiction. On the other hand, Theorem 8.3.1also implies Theorem 8.1.3, because if g : B(R)" B(R) has no fixed point, and if,for each x & B(R), we denote by r(x) the intersection with !B(R) of the orientedhalf-line joining g(x) to x, then r : B(R) " !B(R) is continuous and equal toidentity on !B(R), i.e. is a retraction.

When n = 2m ( 2 is even, the continuous mapping v : R2m " R2m defined by

v(x) = (!x2, x1,!x4, x3, . . . ,!x2m, x2m!1)

is such that #v(x)# = 1 and 6v(x), x7 = 0 for any x & !B(1). In other words, v is acontinuous non-vanishing tangent vector field to !B(1).

The following non-existence theorem, which can be traced to to Poincare [319]and Brouwer [40]. is called the hedgehog or hairy ball theorem and states thatno such vector field can exist when n is odd.

Theorem 8.3.2 If n is odd, there exists no continuous mapping v : !B(R) ' Rn "!B(1) ' Rn such that

6v(x), x7 = 0

for all x & !B(R).


Proof. If such a mapping v exists, let 1v : B(R) " Rn be a continuous extension ofv to B(R), and let us define the homotopy H : B(R)1 [0, 1]" Rn by

H(x,") = (cos%")x ! (sin%")1v(x).

If (x,") & !B(R)1 ([0, 1] \ {1/2}), then

6H(x,"), x7 = (cos%")R2 %= 0,

and hence H(x,") %= 0. If x & !B(R) H(x, 1/2) = !v(x) %= 0. Hence, using thehomotopy invariance theorem 3.4.2 and Corollary 3.1.1, we get

1 = dB [I, B(R), 0] = dB[H(·, 0), B(R), 0] = dB[H(·, 1), B(R), 0]

= dB [!I, B(R), 0] = (!1)n = !1,

a contradiction.

8.4 Zeros of continuous mappings

Brouwer degree allows to prove a n-dimensional version of Bolzano’s intermedi-ate value theorem, first stated and proved, using Kronecker index, in 1883 by H.Poincare [320, 321], then forgotten, rediscovered by S. Cinquini [63] with an incor-rect proof in 1940, and proved to be equivalent to Brouwer’s fixed point theorem byC. Miranda [286] in 1941. It is usually refered as Poincare-Miranda’s theorem.

Theorem 8.4.1 Any continuous map f : P ' Rn " Rn of a n-dimensional closedinterval P =

Gni=1[ai, bi] with faces

P!i : {x & P : xi = ai}, P+

i : {x & P : xi = bi} (1 ) i ) n),

such that

fi(x) ) 0 (x & P!i ), fi(x) ( 0 (x & P+

i ) (1 ) i ) n), (8.5)

has at least one zero in P.

Proof. If f has a zero on !P =Hn

i=1(P!i / P+

i ), the result is proved. If not, let usdefine the homotopy H : P 1 [0, 1]" Rn by

Hi(x,") = (1! ")

2xi !

ai + bi

2

3+ "fi(x) (1 ) i ) n).

We already know that H(x, 1) = f(x) %= 0 on !P. Now, if " & [0, 1[ and x & !P ,then x & P±

i for some 1 ) i ) n, and, if, say, x & P!i , we have

Hi(x,") = (1! ")

2ai ! bi

2

3+ "fi(x) < 0,

8.4. ZEROS OF CONTINUOUS MAPPINGS 65

and hence H(x,") %= 0. Similarly Hi(x,") > 0 if x & P+i . Consequently, us-

ing the homotopy invariance theorem 3.4.2 and Corollary 3.1.1, we get, with a =(a1, . . . , an), b = (b1, . . . , bn),

dB [f, int P, 0] = dB [H(·, 1), int P, 0] = dB[H(·, 0), int P, 0]

= dB [I ! a + b

2, int P, 0] = 1

and the existence of a zero of f in int P follows from the existence theorem 3.4.1.

The following result, called Poincare-Bohl existence theorem following Ha-damard [162], was never stated in this way by Poincare or Bohl, but used byHadamard in the first published proof of Brouwer’s fixed point theorem.

Theorem 8.4.2 If f : B(R) ' Rn " Rn is continuous and such that 6f(x), x7 ( 0for all x & !B(R), then f has at least one zero in B(R).

Proof. If f has a zero on !B(R), the result is proved. If not, let us define thehomotopy H : B(R)1 [0, 1]" Rn by H(x,") = (1! ")x + "f(x). By assumption,H(x, 1) %= 0 for all x & !B(R). For (x,") & !B(R)1 [0, 1[ , we have

6H(x,"), x7 = (1! ")R2 + "6f(x), x7 ( (1! ")R2 > 0,

and hence H(x,") %= 0. Consequently, using the homotopy invariance theorem 3.4.2,we get

dB[f, B(R), 0] = dB[H(·, 1), B(R), 0] = dB [H(·, 0), B(R), 0]

= dB[I, B(R), 0] = 1,

and f has at least one zero in B(R).

A useful consequence of Theorem 8.4.2 is the following surjectivity theorem.

Corollary 8.4.1 If g : Rn " Rn be continuous and strongly coercive, i.e. such that

6g(x), x7#x# " +- as #x# " +-, (8.6)

then g is onto.

Proof. Let y & Rn. Using (8.6), we can find R > 0 such that

6g(x)! y, x7 ( 6g(x), x7 ! #y##x# ( 0

whenever #x# ( R. Hence, by Theorem 8.4.2 applied to f(x) = g(x) ! y, there isat least one x & B(R) such that g(x) = y.


8.5 A Brouwer-Poincare-Bohl’s theorem

When L : X " Z is not invertible, a suitable choice of A in Theorem 8.1.1 it to takeA = JP, where P is a projector onto N(L) and J a isomorphism between N(L)and a supplementary subspace to R(L).

Corollary 8.5.1 Let L : X " Z be a linear mapping, P : X " X, Q : Z " Zprojectors such that

R(P ) = N(L), N(Q) = R(L), (8.7)

J : N(L) " R(Q) an isomorphism, D ' X 1 [0, 1] an open bounded set, andN : D " Z a continuous mapping. Assume that the following conditions hold.

(A) Lx + (1! ")JPx + "N(x) %= 0 for each (x,") & !D.

(B) 0 & D0.

Then

SJP = {(x,") & D : Lx + (1! ")JPx + "N(x) = 0}

contains a compact connected component CJP along which " takes all values in [0, 1].In particular equation

Lx + Nx = 0 (8.8)

has at least one solution in D1.

Proof. We have

(L + JP )x = 0 4 Lx = 0, JPx = 0 4 x & N(L), Px = 0

4 x = 0,

and the result follows from Theorem 8.1.1 with A = JP .

We use Corollary 8.5.1 to obtain a unique statement containing and connectingBrouwer’s fixed point theorem 8.1.3 and Poincare-Bohl’s theorem 8.4.2.

Theorem 8.5.1 Let X be a normed vector space and Z a Hilbert space of the samefinite dimension, with inner product 6·, ·7, L : X " Z be linear, P : X " X, Q :Z " Z be projectors such that

N(L) = R(P ), R(L) = N(Q),

J : N(L)" R(Q) an isomorphism, and let l > 0 be such that

#L(I ! P )x#Z ( l#(I ! P )x#X for all x & X. (8.9)

Let - > 0, R > 0,

D(,R := {x & X : #Px#X < -, #(I ! P )x#X < R},

and N : D(,R " Z be continuous. Assume that the following conditions hold.

8.5. A BROUWER-POINCARE-BOHL’S THEOREM 67

(i) #(I !Q)N(x)#Z ) lR for all x & D(,R.

(ii) 6QN(x), JPx7 ( 0 whenever #Px#X = - and #(I ! P )x#X ) R.

Then L + N has at least one zero in D(,R.

Proof. If L + N has a zero such that #Px#X ) - and #(I ! P )x#X = R, or hasa zero such that #Px#X = - and #(I ! P )x#X ) R, then the theorem is proved.Then we can assume that

Lx + Nx = 0 8 #(I ! P )#X < R and #Px#X < -. (8.10)

We apply Corollary 8.5.1 with D = D(,R 1 [0, 1]. Let " & [0, 1] and x & D(,R be apossible zero of L + (1! ")JPx + "N. Then

L(I ! P )x + "(I !Q)N(x) = 0, (8.11)

and

(1! ")JPx + "QN(x) = 0. (8.12)

From (8.11), (8.9) and Assumption (i), we deduce that, for " & [0, 1[

l#(I ! P )x#X ) #L(I ! P )x#Z = #"N(x)#Z < lR,

and hence

#(I ! P )x#X < R. (8.13)

By (8.10), we can assume that (8.13) also holds for " = 1. From (8.12), we have

(1! ")#JPx#2Z + 6QN(x), JPx7 = 0,

and Assumption (ii) and (8.10) imply that we cannot have #Px#X = -, so that#Px#X < -. Consequently, for each " & [0, 1], each possible zero of L + (1 !")JPx + "N belongs to D(,R, and the result follows from Corollary 8.5.1.

Remark 8.5.1 If X = Z is a finite-dimensional Hilbert space, L = 0, then P =Q = I, D(,R = B(-), condition (8.9) and Assumption (i) are trivially satisfied, andAssumption (ii) becomes condition (16.2.2) if we choose J = I or J = !I. Hencewe recover Poincare-Bohl’s theorem 8.4.2.

Remark 8.5.2 If L is invertible, then P = Q = 0, D(,R = B(R), Assumption (ii)is trivially satisfied, and the remaining assumptions

#Lx# ( l#x# for all x & X.

#N(x)# ) lR for all x & B(R)

are those of Corollary 8.1.1. Brouwer’s fixed point theorem, which corresponds itselfto X = Z a finite-dimensional normed space, L = !I and l = 1.


8.6 Perturbed non-invertible linear mappings

The situation is more complicated when L is not invertible. Then

dim N(L) = codim R(L) > 0

and we denote by Q : Y " Y a linear projector such that N(Q) = R(L).

Theorem 8.6.1 Assume that the following conditions hold;

1. Lx + "N(x,") %= 0 for each x & (!D)% and each " & ]0, 1];

2. QN(x, 0) %= 0 for each x & (!D)0;

3. dB [QN(·, 0)|N(L),D0 .N(L), 0] %= 0;

Then

|dB[L + "N(·,"),D%, 0]| = |dB[QN(·, 0),D0 .N(L), 0]|, (" & [0, 1]),

and S contains a continuum C connecting D0 to D1, along which " takes all valuesin [0, 1]. In particular the equation Lx+N(x, 1) = 0 has at least one solution in D1.

Proof. Let us define F : D " Z by

F (x,") = Lx + (1! ")QN(x,") + "N(x,").

For " & ]0, 1], we have, by projections on the supplementary subspaces R(Q) andN(Q),

F (x,") = 0 4 QN(x,") = 0, Lx + "(I !Q)N(x,") = 0

4 Lx + "N(x,") = 0.

For " = 0, we have

F (x, 0) = 0 4 Lx + QN(x, 0) = 0 4 Lx = 0, QN(x, 0) = 0

4 x & N(L), QN(x, 0) = 0.

Consequently, Assumptions 1 and 2 imply that 0 %& F (!D). On the other hand, itfollows from Theorem 6.4.2 that

dB [F (·, 0),D0, 0]

= dB [L + QN(·, 0),D0, 0] = ±dB[QN(·, 0)|N(L),D0 .N(L), 0].

The result follows from Theorem 7.2.1 applied to F.

Chapter 9

Fixed points and variationalinequalities

9.1 Browder’s fixed point theorem

We show in this chapter that, although Brouwer’s fixed point theorem is a finite-dimensional tool, it provides results for mappings in locally convex topological vec-tor spaces. This is due to two classical tools from topology and functional analysis:

1) if E is a Hausdor! topological vector space, the topology induced on anyfinite-dimensional vector subspace of E by the topology of E coincides withthe usual Euclidian topology.

2) for any finite open covering of a compact set one can find a partition of unitysubordinated to this covering.

The combination of those ideas was used by Browder [45] in proving a fixed pointtheorem for multivalued mappings. If B is a set, recall that 2B denotes the set ofsubsets of B. A multivalued mapping f of A into B is a mapping of A into 2B. Westate and prove Browder’s fixed point theorem for multivalued mappings.

Theorem 9.1.1 Let K be a non-empty compact convex subset of a Hausdor" topo-logical vector space E and T : K " 2K a mapping such that

(i) for each x & K, T (x) is a nonempty convex subset of K;

(ii) for each y & K, T!1(y) = {x & K : y & T (x)} is open in K.

Then there exists x0 & K such that x0 & T (x0).

Proof. The family {T!1(y) : y & K} is a covering of K, because, for each x &K, x & T!1(T (x)), and hence an open covering by Assumption (ii). Since K is

69

70 CHAPTER 9. FIXED POINTS AND VARIATIONAL INEQUALITIES

compact, there is a finite family {y1, . . . , yn} ' K such that K =Hn

j=1 T!1(yj).Let {,1, . . . ,,n} be a partition of unity subordinate to this covering, namely, each,j : K " [0, 1] is continuous, vanishes outside of T!1(yj), and

Bnj=1 ,j(x) = 1 for

all x & K. Define the continuous mapping p : K " K by p(x) =Bn

j=1 ,j(x)yj .

Let K0 =IBn

j=1 "jyj : "j ( 0 (1 ) j ) n),Bn

j=1 "j = 1J

be the closed convex set

spanned by the yj. By construction, p is a continuous mapping from K0 into itself.From the version of Brouwer’s fixed point theorem given in Corollary 8.1.3, p hasa fixed point x0. To conclude the proof, it su$ces to show that, for each x & K,p(x) & T (x). Let x & K; since ,j(x) = 0 if x %& T!1(yj), and T (x) is convex, then

p(x) ="

1+j+n:x"T!1(yj)

,j(x)yj ="

1+j+n:yj"T (x)

,j(x)yj & T (x).

9.2 Variational inequalities

A consequence of Theorem 9.1.1 is a version of Hartman-Stampacchia’s theo-rem for variational inequalities [169] in a locally convex topological vector space,also proved independently by Karamardian [193] in the special case of a simplex inRn.

Definition 9.2.1 A locally convex topological vector space is a Hausdor" topo-logical vector space E such that its origin possesses a neighborhood base consistingof convex sets or, equivalently, a Hausdor" topological vector space in which anyneighborhood of any point x & E contains a convex neighborhood of x.

Theorem 9.2.1 Let K be a compact convex subset of a locally convex topologicalvector space E, and T : K " E# be continuous. Then there exists u0 & K such that

6T (u0), v ! u07 ( 0 (9.1)

for all v & K.

Proof. Suppose that the assertion of the theorem is false. Then, for each u & K,there exists v & K such that 6T (u), v ! u7 < 0. Define T : K " 2K by

T (u) = {v & K : 6T (u), v ! u7 < 0}.

For each u & K, T (u) is non-empty, convex and open, being the inverse image bythe continuous map v " 6T (u), v ! u7 of the open interval ]!-, 0[ . By Theorem9.1.1, there exists u0 & K such that u0 & T (u0), i.e. such that

0 = 6T (u0), u0 ! u07 < 0,

a contradiction.

9.2. VARIATIONAL INEQUALITIES 71

Remark 9.2.1 A condition of the type (9.1) is called a variational inequality.Such relation occurs for example in the study of the minimum on a closed set C ofa function & & C1(C, R) Namely, if there exists u0 & C such that &(u0) = infC &,then

6+&(u0), v ! u07 ( 0 (9.2)

for all v & C, where +& denotes the gradient of &. This follows from the fact that,given v & C, the real function " : [0, 1] " R defined by "(t) = &(u0 + t(v ! u0))reaches its minimum on [0,1] at t = 0, and hence

"$(0) = 6+&(u0), v ! u07 ( 0.

The converse is true if C is convex, because, in this case,

&(u) ( &(u0) + 6+&(u0), u! u07 ( &(u0)

for all u & C.

Another elementary occurence of variational inequalities is in the complemen-tary problem of nonlinear programming : given a continuous mapping F : R+

n "Rn, where R+

n = {x & Rn : xi ( 0 (1 ) i ) n)}, find u0 & R+n such that

F (u0) & R+n and 6F (u0), u07 = 0. The following characterization is due to Kara-

mardian [193].

Proposition 9.2.1 u0 is a solution of the complementary problem if and only if

u0 & R+n, 6F (u0), v ! u07 ( 0 for v & R+

n. (9.3)

Proof. If u0 is a solution of the complementary problem, then F (u0) & R+n and

hence 6F (u0), v7 ( 0 for all v & R+n, so

6F (u0), v ! u07 = 6F (u0), v7 ( 0

for all v & R+n. On the other hand, is u0 satisfies the variational inequality (9.3),

then, taking v = u0 + ei & Rn+, with ei = (0, . . . , 1, . . . , 0) the ith unit vector, we

obtain

0 ) 6F (u0), ei7 = Fi(u0) (1 ) i ) n),

i.e. F (u0) & R+n. In particular, 6F (u0), u07 ( 0, and, taking v = 0 in (9.3), we

obtain 6F (u0),!u07 ( 0, and u0 solves the complementary problem.

The Hilbert space version of Hartman-Stampacchia’s theorem is of interest.

Corollary 9.2.1 Let H be a real Hilbert space with inner product (·, ·), K ' Ha compact convex set and T : H " H a continuous mapping. Then there existsu0 & K such that

(T (u0), v ! u0) ( 0

for all v & K.


Proof. By Riesz representation theorem, for each u & H, there exists Ju & H# suchthat

6Ju, v7H,H$ = (u, v) (v & H),

and J : H " H# is linear, bijective and isometric. Now,

6JT (u), w7H,H$ = (T (u), w)

for all u & H and w & H, and Theorem 9.2.1 applied to JT : H " H# implies theexistence of u0 & H such that

0 ) 6JT (u0), v ! u07H,H$ = (T (u0), v ! u0)

for all v & K.

Remark 9.2.2 A classical result of the theory of Hilbert spaces states that if C isa non-empty closed convex subset of a real Hilbert space H with inner product (·, ·),then, for each f & H there exists a unique u0 := PCf which minimizes the distancev " #v ! f# for v & C. Furthermore, u0 = PCf is characterized by the property

u0 & C, (u0 ! f, v ! u0) ( 0 for all v & C. (9.4)

Corollary 9.2.1 extends this result from the special mapping u 2" u! f to a generalcontinous mapping T : H " H , but for compact convex sets only.

In turn, Corollary 9.2.1 implies an extension of Brouwer fixed point to somemappings in a possibly infinite-dimensional space, namely the Hilbert space versionof Schauder’s fixed point theorem [348].

Corollary 9.2.2 Let H be a real Hilbert space, K ' H a compact convex set, andg : K " K a continuous mapping. Then g has at least one fixed point.

Proof. If we apply Corollary 9.2.1 to T = I ! g, we obtain the existence of u0 & Ksuch that

((I ! g)(u0), v ! u0) ( 0 (9.5)

for all v & K. Taking v = g(u0), we obtain (I ! g)(u0) = 0.

Remark 9.2.3 In an infinite-dimensional Hilbert space, the statement of Corollary9.2.2 is false if we replace the compact convex set K by a closed ball. For example,for H = l2, the mapping g defined on B(1) ' l2 by

g(x) = (K

1! #x#2, x1, x2, . . .), (9.6)

9.3. HEMIVARIATIONAL INEQUALITIES 73

with #x# =LB%

j=1 x2j

M1/2, is continuous and maps B(1) into itself (indeed into

!B(1).) But it has no fixed point, because if x# = f(x#), then #x## = #f(x#)# = 1and

x#1 = 0, x#

2 = x#1, x

#3 = x#

2, . . .

i.e. x# = 0, a contradiction. This example was given by Tychonov [391] as ananswer to a question of Ulam.

Using the construction used in the finite-dimensional case to prove the equiv-alence between Brouwer’s fixed point theorem and the no-retraction theorem, thisresult implies that the mapping R defined on B(1) ' l2 by

r(x) =

N#x#2 ! 1

Og(x) + 2(1! 6x, g(x)7)x

#x#2 ! 26x, g(x)7+ 1,

with g given in (8.3.1), is a retraction of B(1) ' l2 onto !B(1) ' l2.

Remark 9.2.4 It may be of interest to notice that Corollary 9.2.1 can also bededuced from Schauder’s fixed point theorem in a Hilbert space and the characteri-zation of the projection on a closed convex set given in Remark 9.2.2. Indeed, givenT : K " H, let us define g : K " K by g(x) = PK(I ! T )(x), where PK is theprojection on K defined in Remark 9.2.2. By Corollary 9.2.2, there exists u0 suchthat u0 = PK(I ! T )(u0). Then, applying (9.4) to f = (I ! T )(u0), we get

(u0 ! (I ! T )(u0), v ! u0) ( 0

for all v & K, which is nothing but (9.5).

9.3 Hemivariational inequalities

The hemivariational inequalities have been introduced in mechanics by P.D.Panagiotopoulos [310]-[311]. Their derivation is based upon the generalized gra-dient of F.H. Clarke [69] and the corresponding mechanical notion of nonconvexsuperpotential. In what follows, we give, following [91] and [93], an existence re-sult for noncoercive hemivariational inequalities. The main tools are Hartman-Stampacchia’s theorem 9.2.1 for variational inequalities in finite-dimensional spacesand a limit process.

Let D ' Rn be open and bounded, V a real Hilbert space such that V 'L2(D) ' V #, the injection of V in L2(D) being compact. We denote the corre-sponding duality mapping by 6·, ·7V . Assume that

(i) a : V 1 V " R is a bilinear, continuous form such that a(u, u) ( 0 for allu & V.

(ii) , & L%(R) is such that


(a) ,(/ ± 0) exists for any / & R.

(b) there exists /0 such that

ess sup!%<)<!)0,(/) ) 0 ) ess inf)0<)<+%,(/). (9.7)

(iii) K ' V is a closed, convex subset such that 0 & K, and l & V #.

Under those assumptions, we consider

Problem (P). Prove the existence of some u & K which satisfies

a(u, v ! u) +

!

DDCj(u(x)) · (v(x) ! u(x)) dx ( 6l, v ! u7V for all v & K. (9.8)

In (9.8),

j(/) :=

! )

0,(t) dt (/ & R), (9.9)

and DCj(u(x)) · (v(x) ! u(x)) means Clarke’s derivative of j at u(x) in direction(v(x) ! u(x)).

Recall at this point that given a locally Lipschitzian function f : X " R, with Xa locally convex Hausdor! topological space, the directional Clarke’s derivativeis defined by

DCf(x) · h = limy(x

sup%(0+

f(y + "h)! f(y)

",

and Clarke’s gradient of f at x is defined by

!Cf(x) = {x# & X# : DCf(x) · h ( 6x#, h7X for all h & X}.

According to a result of K.C. Chang [57], Clarke’s generalized gradient of the locallyLipschitzian function (9.9) is given by

!Cj(/) = [,(/),,(/)], (9.10)

where

,(/) = limµ(0+

,µ(/), ,

µ(/) = ess inf|)1!)|+µ,(/1), (9.11)

,(/) = limµ(0+

,µ(/), ,µ(/) = ess sup|)1!)|+µ,(/1). (9.12)

In order to prove that Problem (P) has a solution, we first consider a regularizedfinite-dimensional problem. Denote by F the family of all finite-dimensional vector


subspaces of V. Given ' & ]0, 1[ and F & F , Problem (P*,F ) consists in proving theexistence of some u*,F & F .K such that

a(u*,F , v ! u*,F ) +

!

D,*(u

*,F (x))(v(x) ! u*,F (x)) dx

( 6l, v ! u*,F 7V for all v & F .K. (9.13)

In (9.13), ,* is defined as follows. If p & D(] ! 1, 1[), p ( 0,P

Rp(/) d/ = 1 is a

mollifier, ' & ]0, 1[ and p*(/) := '!1p('!1/) for all / & R, then,

,* := p* 0 ,,

where 0 denotes the convolution product.

Proposition 9.3.1 Assume that conditions (i) to (iii) above hold and that thereexists R > 0 such that

{u & K : a(u, u) +

!

Du(x),*(u(x)) dx ) 6l, u7V for some ' & ]0, 1[ }

' B(R). (9.14)

Then problem (P*,F ) has at least one solution. Moreover,

S :=Q

*" ]0,1[ ,F"F

{u*,F solves (P*,F )} ' B(R). (9.15)

Proof. The boundedness (9.15) of S is a consequence of the a priori estimate (9.14).Indeed, taking v = 0 in (9.13), one gets

a(u*,F , u*,F ) +

!

Du*,F (x),*(u

*,F (x)) dx ) 6l, u*,F 7V ,

and, according to (9.14), u*,F & B(R) for all ' & ]0, 1[ and all F & F . In order toprove the existence of some u*,F & F . K satisfying (9.13), let us first define thecontinuous operator T*,F : F " F # by

6T*,F (u), v7V = a(u, v) +

!

D,*(u(x))v(x) dx ! 6l, v7V . (9.16)

From Hartman-Stampacchia’s theorem 9.2.1, for any positive n & N, there existsu*,F

n & K . F .B(n) such that

6T*,F (u*,Fn ), v ! u*,F

n 7V ( 0 for all v & K . F .B(n). (9.17)

Taking v = 0 in (9.17), we obtain

6T*,F (u*,Fn ), u*,F

n 7V = a(u*,Fn , u*,F

n ) +

!

Du*,F

n (x),*(u*,Fn (x)) dx ! 6l, u*,F

n 7F

) 0,


and then, by (9.14), u*,Fn & K . F . B(R). Consequently, passing if necessary to

a subsequence, one may assume that u*,Fn 1 u*,F & K . F . B(R), and u*,F is a

solution of problem (P*,F ) (for more details, see [93]).

We now prove the main existence theorem of [91]-[93].

Theorem 9.3.1 If assumptions (i) to (iii) above hold as well as the a priori esti-mate condition (9.14), problem (9.8) has at least one solution.

Proof. For any ' & ]0, 1[ and any F0 & F, let us define

V *F0

:=Q

F0,F"F

{u*,F : u*,F is a solution to P*,F }.

Since V *F0' K . B(R), the same is true for its weak closure V *

F0

w. Now, for each

' & ]0, 1[ , the family {V *F0

w: F0 & F} has the finite intersection property, and

hence there exists u* &R

F0"F V *F0

w. Clearly, {u* : ' & ]0, 1[ } ' K . B(R), which

implies the existence of a sequence ('m) in ]0, 1[ converging to zero and such thatu*m 1 u0 & K . B(R), with u0 a solution of problem P. The principal steps inproving this results are the following ones :

(a) for any m & N, u*m satisfies the relation

a(u*m , v ! u*m) +

!

D,*m(u*m(x))(v(x) ! u*m(x)) dx ( 6f, v ! u*m7V

for all v & K.(b) by using the nonnegativity and the continuity of a, one has

lim supm(%

a(u*m , v ! u*m) ) a(u0, v ! u0).

(c) For all v & K, one has

lim supm(%

!

D,*m(u*m(x))(v(x) ! u*m) dx )

!

DDCj(u0(x)) · (v(x) ! u0(x)) dx.

If (a), (b), (c) are true, then the result follows by taking the upper limit in(a). While (a) and (b) are relatively easy to be checked, we proceed as follows toobtain (c). The properties of , imply the existence of two constants -1 > 0 and-2 = #,#L# such that

/,*(/) ( 0 whenever |/| > -1 and ' & ]0, 1[ . (9.18)

/,*(/) ( !-1-2 for all / & R and ' & ]0, 1[ . (9.19)

We refer to [93] for the details. From (9.18) and (9.19), we deduce that

,*m(x)(u*m(x))(v(x) ! u*m(x)) ) -1-2 + #,#L##v#


for a.e. x & D. Due to Fatou’s lemma, in order to prove (c), it is su$cient to provethat

!

Dlim supm(%

,*m(u*m(x))(v(x) ! u*m(x)) dx

)!

DDCj(u0(x)) · (v(x) ! u0(x)) dx. (9.20)

In order to prove (9.20) we shall show that, for any y & R,

lim supm(%

,*m(u*m(x))(y ! u*m(x)) dx ) DCj(u0(x)) · (y ! u0(x)) (9.21)

for a.e. x & D. Indeed, since the injection of V into L2(D) is compact and u*m 1 u0

in V, it follows that u*m " u0 in L2(D), and, going if necessary to a subsequence,u*m(x)" u0(x) a.e. in D. Thus, for the proof of (9.21), it is su$cient to show thatfor any x & D such that u*m(x) " u0(x), and to any convergent subsequence (yk)of (,*m(u*m)(x))(y ! u*m(x)), inequality

limk(%

yk ) DCj(u0(x)) · (y ! u0(x)) (9.22)

holds. First, we show that

lim infm(%

,*m(u*m(x)) and lim supm(%

,*m(u*m(x))

belong to !Cj(u0(x)). As a consequence, we shall have

lim infm(%

,*m(u*m(x))z ) DCj(u0(x))z

lim supm(%

,*m(u*m(x))z ) DCj(u0(x))z (9.23)

for all z & R. For the proof, let x & D with u*m(x) " u0(x) and µ > 0 be given.Then there exists mµ,x such that, for any m ( mµ,x one has

|u*m(x) ! u0(x)| ) µ

2, 0 ) 'm )

µ

2.

Consequently,

,*m(u*m(x)) ) ess sup|t!u!m (x)|+*m,(t) ) ess sup|t!u0(x)|+µ,(t) = ,µ(u0(x)).

Therefore,

lim supm(%

,*m(u*m(x)) ) ,µ(u0(x)),

and, passing to the limit with µ" 0+, we deduce

lim supm(%

,*m(u*m(x)) ) ,(u0(x)).


Analogously, one obtains

,(u0(x)) ) lim infm(%

,*m(u*m(x)).

Thus,

lim supm(%

,*m(u*m(x)) and

lim infm(%

,*m(u*m(x)) & [,(u0(x)),,(u0(x))] = !Cj(u0(x)),

and inequalities (9.23) hold. Now we are able to prove (9.22). If y = u0(x), theny ! u*m(x)" 0. Because of |,*(t) ) #,#L# for any t & R and any ' > 0, it followsthat yk " 0. Thus (9.22) is verified. If now y > u0(x), then

limk(%

yk =

2lim

k(%,*mk

(u*mk (x))

3(y ! u0(x)) ) lim sup

m(%,*m(u*m(x))(y ! u0(x))

) DCj(u0(x)) · (y ! u0(x)),

the last inequality being justified by (9.23). A similar procedure can be used ify < u0(x), using the lower limit this time. So (9.21) is proved, and then (9.20) isalso proved, implying the assertion (c). Taking into account (b) and (c), the resultfollows by taking the upper limit in (a).

Finally, we formulate su$cient conditions in order that the a priori estimateassumption (9.14) holds.

Theorem 9.3.2 Let K = span(K). Assume that the bilinear form a is continuous,nonnegative and semicoercive, i.e. that, in addition to assumptions (i) to (iii), onehas

(iv) K. ker (a) is finite dimensional and a is coercive over [K. ker (a)]-, (theorthogonal being considered in K), namely

a(u, u) ( .2#u#2 for all u & [K . ker (a)]- and some . > 0.

Then condition (9.14) is satisfied over K if one of the following conditions hold :

1. K . ker (a) = {0},

2. K . ker (a) %= {0} and

(v) ,(!-) := limt(!% ,(t) < ,(+-) := limt(+% ,(t)

(vi)P

D[,(+-))+(x) ! ,(!-))!(x)] dx > 6l, )7V for all ) & [K . ker (a)] \{0}.

Condition (vi) is a Landesman-Lazer-type condition first introduced by A.C.Lazer and E.M. Landesman in 1969 [241] for bounded nonlinear perturbations ofnon-coercive linear Dirichlet problems. The proof of Theorem 9.3.2, which can befound in [92], is inspired by the one of P.J. McKenna and J. Rauch [279], whoconsider a case where dim ker (a) = 1.

9.4. KY FAN FIXED POINT THEOREM AND CONSEQUENCES 79

9.4 Ky Fan fixed point theorem and consequences

We will now state and prove an extension, due to Ky Fan [111], of Brouwer’s fixedpoint theorem to some multivalued mappings of a compact convex set of a locallyconvex topological vector space. A few concepts and results about locally convextopological vector spaces will be required [330].

Definition 9.4.1 If E is a vector space, C ' E is said to be circled if "C ' Cwhenever |"| ) 1.

Lemma 9.4.1 If E is a locally convex topological vector space, there is a neighbor-hood base at 0 of E consisting of open, convex and circled sets.

Lemma 9.4.2 Let A and B be non-empty, disjoint and convex subsets of a reallocally convex topological vector space E, such that A is closed and B is compact.Then there exists a closed real hyperplane in E strictly separating A and B. In otherwords, there exists a linear and continuous functional f : E " R and * & R suchthat

f(x) < * < f(y) for each x & A and y & B.

Letting

H<& := {x & E : f(x) < *}, H>& := {x & E : f(x) > *},

it is clear that H<& and H>& are disjoint, open and convex, and Lemma 9.4.2 saysthat A ' H<& and B ' H>&.

Definition 9.4.2 If X and Y are Hausdor" topological spaces, and T : X " 2Y isa multivalued mapping from X into Y, we say that T is upper semi-continuous onX if, to each point x0 of X and an arbitrary neighborhood V to T (x0) in Y, thereexists a neighborhood U of x0 in X such that, for each x & U, T (x) ' V .

The following characterization of upper semi-continuity is of interest (see [45]).

Lemma 9.4.3 Let K and K1 be compact topological spaces and T : K " 2K1 amulti-valued mapping from K into K1 such that T (x) is closed for each x & K.Then T is upper semi-continuous on K if and only if its graph

G(T ) := {[x, y] & K 1K1 : x & K, y & T (x)}

is closed in K 1K1.

Proof. Suppose first that T is upper semi-continuous on K. We shall prove that(K 1 K1) \ G(T ) is open in K 1 K1. Let [u, w] & (K 1 K1) \ G(T ). Since w &K1 \ T (u), the regularity of K1 implies the existence of neigborhoods V1 of w andV2 of T (u) such that V1 . V2 = ,. Using the upper semi-continuity of T, thereexists a neighborhood U of u in K such that, for all x & U, T (x) ' V2. Hence theneighborhood U 1 V1 of [u, w] does not intersect G(T ), and G(T ) is closed.


Suppose conversely that G(T ) is closed in K 1 K1 and hence compact. Letu & K and V an open neighborhood of T (u). If T were not upper semi-continuousat u, then, for each neighborhood U of u in K we would have

G(T ) . [U 1 (K1 \ V )] %= ,.

But

{G(T ) . [U 1 (K1 \ V )] : U neighborhood of u}

is a family of compact sets having the finite intersection property, and hence

G(T ) .S

U

[U 1 (K1 \ V )] %= ,.

Any point [x, y] of this intersection is such that x = u and y & T (x) \ V, a contra-diction with T (u) ' V .

We can now state and prove Ky Fan’s fixed point theorem [111], using anargument due to F. Terkelsen [384].

Theorem 9.4.1 Let K be a nonempty compact convex subset of a locally convextopological vector space E and let T : K " 2K be an upper semi-continuous multi-valued function of K into K, such that T (u) is non empty, closed and convex foreach u & K. Then there exists u0 & K such that u0 & T (u0).

Proof. Denote by U the set of all open convex and circled neighborhoods of 0 andlet us introduce on the set N# 1 U = {(i, U) : i & N#, U & U} the order structuredefined as follows :

(i, U) ) (j, V ) 4 i ) j and U 0 V.

Endowed with this order structure, ((N#,U),)) is directed. Indeed, if (i, U), (j, V ) &N# 1 U , then (k, U . V ) with k ( max(i, j) belongs to N# 1 U too, and

(i, U) ) (k, U . V ), (j, V ) ) (k, U . V ).

The net

N# 1 U " U , (i, U) 2" 1

iU := Ui

is decreasing, in the sense that

(i, U) ) (j, V )8 Ui =1

iU 0 1

jV = Vj .

Let (i, U) be arbitrarily chosen in N#1U . Since {x+Ui : x & K} is an open coveringof K, there exists a finite set {x(i,U),j & K : j & J(i, U)} such that

K 'Q

j"J(i,U)

(x(i,U),j + Ui).

9.4. KY FAN FIXED POINT THEOREM AND CONSEQUENCES 81

Let {f(i,U),j : j & J(i, U)} be a continuous partition of unity subordinated to thisfinite covering, so that the continuous functions f(i,U),j : K " [0, 1] (j & J(i, U))are such that f(i,U),j(x) = 0 for x %& x(i,U),j +Ui and

Bj"J(i,U) f(i,U),j(x) = 1 for all

x & K. Choose y(i,U),j & T (x(i,U),j) arbitrarily and denote by C(i,U) the convex hullof {y(i,U),j : j & J(i, U)}. Clearly, C(i,U) is a compact, convex and finite-dimensionalsubset of K. Finally, define the continuous mapping f(i,U) : C(i,U) " C(i,U) by

f(i,U)(x) ="

j"J(i,U)

f(i,U),j(x)y(i,U),j .

Since the topology induced on any finite-dimensional subspace of E by its locallyconvex topology coincides with the usual Euclidian one, Brouwer’s fixed point the-orem 8.2.1 implies the existence of u(i,U) & C(i,U) such that

u(i,U) = f(i,U)(u(i,U)) ="

j"J(i,U)

f(i,U),j(u(i,U))y(i,U),j . (9.24)

Since K is compact, there is a cluster point u0 & K of the net {u(i,U) : (i, U) &N#1U}, which means that, for any neighborhood V of u0, and any (m, W ) & N#1U ,there exists (i, U) & N# 1 U with (i, U) ( (m, W ) such that u(i,U) & V.

We prove now that u0 & T (u0). If u0 %& T (u0), since T (u0) is compact, thereexists by separation an open (in E), convex neighborhood W of T (u0) such that u0 %&W. On the other hand, since T is upper semi-continuous, there exists a neighborhoodV of u0 with V .W = , such that T (V .K) 'W. Since V ! u0 is a neighborhoodof 0, and the map [x, y] 2" x + y is continuous, there exists m & N# and U & Usuch that 1

mU + 1mU = Um + Um ' V ! u0, i.e. u0 + Um + Um ' V. Now,

u0 + Um is a neighborhood at u0 and, by definition of a cluster point, there is some(i, W ) ( (m, U) such that u(i,W ) & u0 + Um ' (u0 + Um) + Um ' V. Moreover,(i, W ) ( (m, U) implies that Wi ' Um, and hence

u(i,W ) + Wi ' u0 + Um + Wi ' u0 + Um + Um ' V. (9.25)

Because of f(i,W ),j(x) = 0 if x %& x(i,W ),j + Wi, we deduce from (9.24) that

u(i,W ) = f(i,W )(u(i,W )) ="

j"J(i,W ):u(i,W )"x(i,W ),j+Wi

f(i,W ),j(u(i,W ))y(i,W ),j . (9.26)

Since Wi is circled, one has Wi = !Wi and hence u(i,W ) & x(i,W ),j +Wi is equivalentwith x(i,W ),j & u(i,W ) + Wi ' V, so that x(i,W ),j & K . V. It follows that y(i,W ),j &T (x(i,W ),j) ' T (K . V ) ' W. By convexity of W, it follows from (9.26) thatu(i,W ) &W, which contradicts u(i,W ) & V and V .W = ,.

Ky Fan’s fixed point theorem has important special cases obtained by special-izing E or T.

The first one, corresponding to E = Rn, is Kakutani’s fixed point theorem,proved by Kakutani in 1941 [190], and called by the Nobel Prize in Economics G.Debreu ‘the most powerful tool for proofs of existence in economics’ [88].


Corollary 9.4.1 Let K ' Rn be non-empty, compact and convex and T : K " 2K

be upper semi-continuous and such that T (x) is non empty, closed and convex foreach x & K. Then there exists some u0 & K such that u0 & T (u0).

The second special case corresponds to the case of a single-valued mappingin a locally convex topological vector space. It was first proved in 1935 by A.N.Tychonov [391] and is called Tychonov’s fixed point theorem. Notice that, fora single-valued mapping, the upper semicontinuity reduces to classical continuity.

Corollary 9.4.2 Any continuous mapping f : K " K of a non-empty, compactand convex subset K of a locally convex topological vector space E has at least onefixed point.

The special case of Tychonov’s fixed point theorem where E is a normed vectorspace was first proved by J. Schauder [348] in 1930 and is called Schauder’s fixedpoint theorem. It is most useful in the study of nonlinear di!erential equations.

Corollary 9.4.3 Any continuous mapping f : K " K of a non-empty, compactand convex subset K of a normed vector space E has at least one fixed point.

9.5 Applications to game theory

A n-person-game is a set of n players, each player i having at his disposal afinite set of pure strategies $i,j (1 ) j ) Ni), and a pay-o! function &i

from the set of all n-uples of pure strategies into the real numbers, such that&i($1,i1 ,$2,i2 , . . . ,$n,in) represents the pay-o! of the ith player if player 1 choosesstrategy $1,i1 , player 2 choses strategy $2,i2 , and so on. More generally, a mixed

strategy for the player i is a convex combination of pure strategiesBNi

j=1 %i,j$i,j

where 0 ) %i,j ) 1 etBNi

j=1 %i,j = 1. This can be seen as the possibility for the

player i to chose randomly his jth strategy with some probability %i,j . Consider theBNi

j=1 %i,j$i,j as the points of a Ni-simplex Ki whose vertices are the $i,j , which canbe seen as a convex compact subset of a real vector space. The pay-o! function hasa unique extension to the n-uples of mixed strategies pi = (%i,1, . . . ,%i,Ni), which islinear in the mixed strategy of each player, and which can be written, fi(p1, . . . , pn)(i = 1, . . . , n).

The following concept of equilibrium was introduced in 1950 by J. Nash [294,295], and called to-day a Nash equilibrium. Nash shared the Nobel Prize ofEconomics in 1994 for this theory.

Definition 9.5.1 (p#1, . . . , p#n) is an equilibrium for the n-person-game with mixed

strategies if, for each i = 1, 2, . . . , n, one has

fi(p#1, . . . , p

#n) = max

x"Ki

fi(p#1, . . . , p

#i!1, x, p#i+1, . . . , p

#n).

The following existence theorem, due to J. Nash [294, 295] gives su$cient con-ditions upon the fi in order that an equilibrium exists.

9.5. APPLICATIONS TO GAME THEORY 83

Theorem 9.5.1 Let Ei (1 ) i ) n) be locally convex topological vector spaces,Ki ' Ei be non empty compact and convex and fi : K1 1 K2 1 . . . 1 Kn "R be continuous functions such that, for any fixed 1 ) i ) n and any fixed(p1, . . . , pi!1, pi+1, . . . , pn) & K11 . . .1Ki!11Ki+11 . . .1Kn, the mapping Ki "R, p 2" fi(p1, . . . , pi!1, p, pi+1, . . . , pn) is concave. Then there exists (p#1, . . . , p

#n) &

K1 1 . . .1Kn such that, for each i = 1, 2, . . . , n,

fi(p#1, . . . , p

#i!1, p

#i , p

#i+1, . . . , p

#n) = max

x"Ki

fi(p#1, . . . , p

#i!1, x, p#i+1, . . . , p

#n). (9.27)

Proof. The following notation will be used throughout the proof : for any i =1, 2, . . . , n, let

#Ki = K1 1 . . .1Ki!1 1Ki+1 1 . . .1Kn

Tpi = (p1, . . . , pi!1, pi+1, . . . , pn).

The basic idea is to apply Ky Fan’s fixed point theorem 9.4.1 in the followingframework : E = E1 1E2 1 . . .1En, K = K1 1K1 1 . . .1Kn and T : K " 2K isdefined as follows. For 1 ) i ) n,

Ti : #Ki " 2Ki, Tpi 2" {p & Ki : Ti(p1, . . . , pi!1, p, pi+1, . . . , pn)

= maxx"Ki

Ti(p1, . . . , pi!1, x, pi+1, . . . , pn)}.

and T (p1, . . . , pn) = (T1( Tp1), . . . , Tn(#pn)). Since E is locally convex and K is nonempty, convex and compact, it remains to verify that, for any (p1, . . . , pn) & K,T (p1, . . . , pn) is a non empty, closed and convex subset of K, and that T is up-per semi-continuous on K. Indeed, for any i = 1, 2, . . . , n, Ti(Tpi) is non-empty(Weierstrass theorem), convex (from the concavity assumption) and closed (fromthe continuity of fi). According to Lemma 9.4.3, the upper semicontinuity of T isequivalent to the closedness of the graph

G(T ) = {[(p1, . . . , pn), (y1, . . . , yn)] : pi & Ki, yi & Ti(Tpi) (i = 1, 2, . . . , n)}.

Let [(p(k)1 , . . . , p(k)

n ), (y(k)1 , . . . , y(k)

n )] be a sequence in G(T ) such that p(k)i " pi & Ki,

y(k)i " yi & Ki for any i = 1, 2, . . . , n. It su$ces to show that yi & Ti(Tpi), i.e., by

definition of Ti, that

fi(p1, . . . , pi!1, y, pi+1, . . . , pn) ( fi(p1, . . . , pi!1, x, pi+1, . . . , pn) (9.28)

for each x & Ki (i = 1, 2, . . . , n). The condition y(k)i & T (

7p(k)

i ) is equivalent to

fi(p(k)1 , . . . , p(k)

i!1, y(k)i , p(k)

i+1, . . . , p(k)n ) ( fi(p

(k)1 , . . . , p(k)

i!1, x, p(k)i+1, . . . , p

(k)n ) (9.29)

for any x & Ki (i = 1, 2, . . . , n), and it su$ces to let k " - in (9.29) to ob-tain (9.28). From Ky Fan’s fixed point theorem 9.4.1, there exists (p#1, . . . , p

#n) &

K1 1 . . . 1Kn such that p#i & Ti( Tp#i ) (i = 1, 2, . . . , n), so that, by definition of Ti,(p#1, . . . , p

#n) is a Nash equilibrium.


A consequence of Nash’s theorem is a result stated and proved in 1928 by J. vonNeumann [395, 396], and called von Neumann’s minimax theorem for zerosum 2-person-games.

Corollary 9.5.1 Let Ki be compact convex subsets of locally convex topologicalvector spaces Ei (i = 1, 2) and f : K11K2 " R be a continuous mapping such that

(1) for each p2 & K2, p1 2" f(p1, p2) is concave on K1;

(2) for each p1 & K1 p2 2" f(p1, p2) is convex on K2.

Then there exists (p#1, p#2) such that

f(p#1, p#2) = max

p1"K1

-min

p2"K2

f(p1, p2)

.= min

p2"K2

-max

p1"K1

f(p1, p2)

.. (9.30)

Proof. Applying Nash equilibrium theorem 9.5.1 with n = 2 and

f1(p1, p2) = f(p1, p2), f2(p1, p2) = !f(p1, p2) ((p1, p2) & K1 1K2,

we obtain (p#1, p#2) & K1 1K2) such that

f1(p#1, p

#2) = max

p1"K1

f(p1, p#2), f2(p

#1, p

#2) = max

p2"K2

f(p#1, p2)

and hence, in term of f,

f(p#1, p#2) = max

p1"K1

f(p1, p#2), (9.31)

f(p#1, p#2) = min

p2"K2

f(p#1, p2). (9.32)

From (9.31), one gets

f(p#1, p#2) ( min

p2"K2

-max

p2"K2

f(p1, p2)

.,

while from (9.32), one gets

f(p#1, p#2) ) max

p1"K1

-min

p2"K2

f(p1, p2)

.,

Thus

minp2"K2

-max

p2"K2

f(p1, p2)

.) f(p#1, p

#2) ) max

p1"K1

-min

p2"K2

f(p1, p2)

..

Since the inequality

maxp1"K1

-min

p2"K2

f(p1, p2)

.) min

p2"K2

-max

p2"K2

f(p1, p2)

.

is obvious, the result follows.

9.5. APPLICATIONS TO GAME THEORY 85

Remark 9.5.1 Analyzing the proof of Nash’s equilibrium theorem 9.5.1, it is easilyseen that the assumption of concavity of fi with respect to the ith variable can bereplaced by an assumption of quasi-concavity, i.e., for any * & R, the set

{x & Ki : fi(p1, . . . , pi!1, x, pi+1, . . . , pn) ( *}

is convex. This implies that, in von Neumann’s theorem 9.30, it su$ces to assumethat f is quasi-concave with respect to the first variable and quasi-convex with respectto the second one.

Remark 9.5.2 In [190], Kakutani deduced von Neumann’s theorem 9.30, when E1

and E2 have finite dimension, from the following intersection lemma, originallygiven by von Neumann [396], which is of independent interest.

Lemma 9.5.1 Let K ' Rm and L ' Rn be non-empty, compact and convex, andU ' K 1 L and V ' K 1 L be closed and such that

(i) for each x0 & K, Ux0 := {y & L : (x0, y) & U} is closed, convex and nonempty;

(ii) for each y0 & L, Vy0 := {x & K : (x, y0) & V } is closed, convex and nonempty.

Then U . V %= ,.

Proof. Define 2 : K 1 L " 2K'L by 2(x, y) = Vy 1 Ux for each (x, y) & K 1 L.Clearly, K 1 L is compact and convex in Rm 1 Rn and, for each (x, y) & K 1 L,2(x, y) is a closed, convex and non-empty subset of K 1 L. Moreover, it is easilyseen that G(2) is closed and hence 2 is upper semi-continuous by Lemma 9.4.3.Then, from Kakutani fixed point theorem 9.4.1, there exists (x0, y0) & K 1 L suchthat (x0, y0) & 2(x0, y0) = Vy0 1 Ux0 , which means that (x0, y0) & U . V.

To deduce directly the minimax theorem from the topological lemma, it su$ces,with von Neumann, to take S = K, T = L,

V = {(x, y) & K 1 L : f(x, y) ) minv"L

f(x, v)},

W = {(x, y) & K 1 L : f(x, y) ( maxu"K

f(u, y)}.

Then there exists (x#, y#) & V .W, i.e. such that

maxx"K

f(x, y#) ) f(x#, y#) ) miny"L

f(x#, y),

which immediately implies that

miny"L

maxx"K

f(x, y) ) f(x#, y#) ) maxx"K

miny"L

f(x, y).

As the opposed inequality

maxx"K

miny"L

f(x, y) ) miny"L

maxx"K

f(x, y)

always holds, the result follows.

Chapter 10

Equations in reflexiveBanach spaces

10.1 Monotone mappings

Corollary 8.4.1 is at the basis of some results due to F.E. Browder [43] and to G.J.Minty [285] for the existence of zeros of some mappings between a reflexive Banachspace and its dual, which verify some monotonicity conditions, and which can beused to obtain general existence theorem for some nonlinear elliptic boundary valueproblems.

Let X be a real Banach space, X# its dual space with the pairing betweenx# & X# and x & X denoted by 6x#, x7X,X$ or, if no confusion appears, by 6x#, x7.

Definition 10.1.1 If T is a (in general nonlinear) mapping with domain D(T ) inX and range R(T ) in X#, T is said to be

(i) monotone if, for all u and v in D(T ), one has

6T (u)! T (v), u! v7 ( 0;

(ii) strongly coercive if

6Tu, u7#u# " +- as #u# " -;

(iii) demicontinuous at u0 & D(T ) if, for each sequence (un) in D(T ) withun " u0 one has T (un) 1 T (u0), where 1 denotes the weak convergence inX#.

(iv) hemicontinuous at u0 & int D(T ) if, for any y and z in X,

6T (u0 + ty), z7 " 6T (u0), z7 as t" 0 + .

87

88 CHAPTER 10. EQUATIONS IN REFLEXIVE BANACH SPACES

Notice that, since u0 & int D(T ), u0+ty & D(T ) for any y & X if t > 0 is su$cientlysmall.

The first explicit definition of monotone mappings from a Banach space to its dualwas given in 1960 by R.I. Kacurovskii [189]. In 1963, F.E. Browder introducedthe concepts of weak-coercivity conditions, demicontinuity and of hemicontinuityrespectively in [43], [41] and [42].

The following simple result, usually called Minty’s trick [285] is useful in thestudy of monotone hemicontinuous mappings.

Lemma 10.1.1 If X is a Banach space, F : D(F ) ' X " X# is hemicontinuousat u0 & int D(T ), and if

6F (v), v ! u07 ( 0 (10.1)

for all v & B[u0, -] and some - > 0, then F (u0) = 0.

Proof. Let w & X and take v = u0 + tw with t > 0 su$ciently small so thatu0 + tw & B[u0, -]. We deduce from (10.1) that

6F (u0 + tw), w7 ( 0

and hence, if t" 0+, using hemicontinuity,

6F (u0), w7 ( 0.

The result follows by taking w = !F (u0).

It is clear that, if X has finite dimension, T is demicontinuous at u0 & D(T )if and only if T is continuous at u0. It is clear also that, for an arbitrary reflexiveBanach space X, the demicontinuity of T at u0 & int D(T ) implies its hemiconti-nuity at u0. It is more surprising that, when T is monotone, the converse is true.This result is due to the e!orts of R.T. Rockafellar [331], T. Kato [197] and F.E.Browder [44].

Lemma 10.1.2 Let X be a reflexive Banach space, T : D(T ) ' X " X# bemonotone and hemicontinuous at u0 & int D(T ). Then T is demicontinuous at u0.

Proof. We first prove that T is locally bounded at u0, i.e. that #T (u)#X$ ) c onBu0(r) ' D(T ) for some c > 0 and some r > 0. By translation, we can assumewithout loss of generality that u0 = 0. Let B(-) ' D(T ) and z & B(-). Then, bymonotonicity,

6T (x), x! z7 ( 6T (z), x! z7 ( !n

for all x & B(-) and some n & N. If we define

Mn := {z & B(-) : 6T (x), x! z7 ( 0 for all x & B(-)},

10.2. BROWDER-MINTY’S THEOREM 89

then Mn is closed and B(-) =H

n"NMn. From Baire’s theorem, one of the Mn,

say Mp has a non-empty interior, i.e. we can find r > 0 and z0 & B(-) such thatz0 + B(R) 'Mp. In other words,

6T (x), x! z0 ! y7 ( !p (10.2)

for all x & B(-) and y & B(R). Since !z0 &Mq for some q & N, we have also

6T (x), x + z07 ( !q (10.3)

for all x & B(-), and hence, by adding (10.2) and (10.3),

6T (x), 2x! y7 ( !(p + q) (10.4)

for all x & B(-) and y & B(R). Now, if x & B(r/4) and z & B(r/2), theny = 2x! z & B(R) and (10.4) gives

6T (x), z7 ( !(p + q)

for x & B(r/4) and z & B(r/2). Consequently, for all x & B(r/4),

#T (x)#X$ =2

rsup{6F (x), z7 : z & B(r/2)} ) 2

r(p + q)

and T is locally bounded at u0. Now, let un " u0. By the local boundedness ofT, we may assume that (T (un)) is bounded, and as X is reflexive, we may assume,going if necessary to a subsequence, that T (un) 1 y for some y & X#. Therefore

0 ) 6T (un)! T (v), un ! v7 " 6y ! T (v), u0 ! v7

for every v & Bu0(-) ' D(T ). Using Lemma 10.1.1 with F (v) = T (v) ! y, weget y = T (u0). As y is the same for any converging subsequence of (T (un)) theconclusion follows.

10.2 Browder-Minty’s theorem

To prove Browder-Minty’s theorem, we need a variant of Corollary 8.4.1.

Lemma 10.2.1 Let Y be a finite-dimensional real normed vector space, and h :Y " Y # be continuous and strongly coercive. Then h is onto.

Proof. Let n = dim Y , j : (Rn, # ·#)" (Y, # ·#Y ) be an isomorphism and j# : Y # "(Rn)# its adjoint. We identify (Rn)# and Rn through the standard relation

6f, x7Rn,(Rn)$ = 6%f, x7 (x & Rn)


for any f & (Rn)# and the corresponding element %f & Rn. Define now g : Rn " Rn

by g = % * j# * h * j. Clearly, g is continuous and, for each x & Rn \ {0}, we have

6g(x), x7#x# =

6(% * j# * h * j)(x), x7#x# =

6(j# * h * j)(x), x7Rn ,(Rn)$

#x#

=6h(j(x)), j(x)7Y,Y $

#x# .

Now, the norms #x# and #j(x)#Y are clearly equivalent in Y, and hence the coer-civity assumption implies that

6g(x), x7#x# " +- as #x# " +-.

From Corollary 8.4.1, g : Rn " Rn is onto, which immediately implies that h : Y "Y # is onto as well.

We now state and prove Browder-Minty’s theorem.

Theorem 10.2.1 Let X be a real reflexive Banach space and T : X " X# amonotone, hemicontinuous and strongly coercive operator. Then T is onto.

Proof. First observe that it is su$cient to prove that equation T (u) = 0 has asolution, because if T is monotone, hemicontinuous and strongly coercive, and ify & X#, the same is true for 1T defined by 1T (u) = T (u) ! f . Let us denote byF the family of all finite-dimensional vector subspaces of X, and, for F & F , letiF : F " X be the canonical injection and i#F : X# " F # its adjoint, so that

6i#F (x#), u7F,F$ = 6x#, iF (u)7X,X$ = 6x#, u7X,X$ for all x# & F #, u & X.

Define TF : F " F # by TF = i#F *T *iF . It is easy to check that TF is hemicontinous,and hence, using Lemma 10.1.2, continuous on F. On the other side, for any x, y & F,we have

6TF (x), y7F,F$ = 6i#F (T (iF (x))), y7F,F$ = 6T (iF (x)), iF (y)7X,X$

= 6T (x), y7X,X$ . (10.5)

Relation (10.5) immediately implies that TF is monotone and strongly coercive onF. Then, using Lemma 10.2.1,

WF := {uF & F : TF (uF ) = 0} %= , (F & F).

We now prove thatH

F"F WF is bounded. From the strong coercivity, there existsr > 0 such that, for any F & F , and u & F,

6TF (u), u7X,X$ = 6T (u), u7X,X$ ( #u#X ( R if #u#X ( r,

10.2. BROWDER-MINTY’S THEOREM 91

and hence #uF#X < r when uF & F is such that TF (uF ) = 0. ThusH

F"F WF 'B(R). For F0 & F , let

VF0 :=Q

F"F ,F.F0

WF .

Clearly VF0 ' B(R) and, B(R) being compact for the weak topology $(X, X#),

we have VF0

+(X,X$) ' B(R). Moreover, the family {VF+(X,X$)

: F & F} has thefinite intersection property. Indeed, if (Fi)i"I is a finite family in F , and F0 &F is such that F0 0 Fi for all i & I, then VF0 '

Ri"I VFi and, consequently

VF0

+(X,X$) 'R

i"I VFi

+(X,X$). We conclude that

RF"F VF

+(X,X$) %= ,. Let u0 &R

F"F VF+(X,X$)

, and let us prove that T (u0) = 0. Choose u & X arbitrary and

F0 & F such that F0 9 u. Since u0 & VF0

+(X,X$), VF0 is bounded and X is reflexive,

it follows from Proposition 7.2 in [47] that there exists a sequence (un) in VF0 suchthat un 1 u0. By the monotonicity of T, we have

6T (u)! T (un), u ! un7X,X$ ( 0, (10.6)

and, by definition of VF0 , we have

6T (un), u! un7X,X$ = 6TF0(un), u! un7F0,F$

0

= 60, u! un7F0,F0$= 0.

Combined with (10.6), this gives

6T (u), u! un7X,X$ ( 0

and hence, letting n"-,

6T (u), u! u07X,X$ ( 0

for any u & X. Lemma 10.1.1 with F = T implies that T (u0) = 0.

The following result was first proved independently by M.M. Vainberg [392] andE.H. Zarantonello [406] in 1960 for Lipschitzian mappings.

Corollary 10.2.1 If X is a reflexive Banach space and T : X " X# is hemiconti-nous and strongly monotone, i.e. if

6T (u)! T (v), u! v7 ( *#u! v#2 (10.7)

for all u, v & X and some * > 0, then T : X " X# is a bijection and T!1 : X# " Xis Lipschitzian.

Proof. T : X " X# is obviously monotone, and it follows from (10.7) that it isstrongly coercive. Hence, from Browder-Minty’s theorem 10.2.1, T is onto. Con-dition (10.7) immediately implies that T : X " X# is one-to-one and that, forw, z & X#,

#T!1(w) ! T!1(z)#2 ) *!16w ! z, T!1(w) ! T!1(z)7) #w ! z#X$#T!1(w)! T!1(z)#,

so that T!1 is Lipschitzian with Lipschitz constant *!1.


10.3 A monotone quasilinear Dirichlet problem

We give in this section a simple application of Browder-Minty’s theorem to somequasilinear Dirichlet problem on a bounded open subset % of Rn. This example istreated in [60] under more restrictive assumptions. Let * : Rn " Rn be a continuousmapping such that there exist M > 0 having the property that

#*(/)# )M#/# (/ & Rn). (10.8)

Let H1(%) be the Sobolev space of functions u & L2(%) whose distributional partialderivatives also belong to L2(%), with the usual inner product

!

!u(x)v(x) dx +

!

!6+u(x),+v(x)7 dx,

Let H10 (%) be the closure in H1(%) of the space of infinitely di!erentiable func-

tions having compact support in %, endowed with the equivalent inner productP!6+u(x),+v(x)7 dx, and let f & H!1(%) (the dual space of H1

0 (%)) be given. Weconsider the problem of finding weak solutions for the quasilinear elliptic Dirichletproblem

!div *(+u) = f, u & H10 (%). (10.9)

More explicitely, a weak solution of problem (10.9) is a function u & H10 (%) such

that!

!6*(+u(x)),+v(x)7 dx = 6f, v7H (10.10)

for all v & H10 (%), where 6·, ·7H denotes the duality map between H!1(%) and H1

0%).Notice that, because of condition (10.8), we have, for u, v & H1

0 (%)

|6*(+u),+v7| ) #*(+u)##+v# )M#+u##+v# & L1(%),

and hence the integral in (10.10) is well defined and problem (10.9) makes sense.

Theorem 10.3.1 If * satisfies condition (10.8) and if there exists m > 0 such that

6*(/) ! *(/$), / ! /$7 ( m#/ ! /$#2 (10.11)

for all /, /$ & Rn, then problem (10.9) has a unique solution.

Proof. Let us introduce the operator T : H10 (%)" H!1(%) by the relation

6T (u), v7H =

!

!6*(+u(x)),+v(x)7 dx (10.12)

10.3. A MONOTONE QUASILINEAR DIRICHLET PROBLEM 93

for all v & H10 (%). The operator T is well defined, because, if u & H1

0 (%), using(10.8) and Cauchy-Schwarz inequality,

|6T (u), v7H | )!

!#*(+u(x))##+v(x)# dx

) M

!

!#+u(x)##+v(x)# dx )M#+u#L2#+v#L2

= M#u#H10#v#H1

0

for all v & H10 (%), and hence T (u) & H!1(%) and #T (u)#H!1 ) M#u#H1

0. We now

show that T is strongly monotone. If u, v & H10 (%), then, using assumption (10.11),

6T (u)! T (v), u! v7H =

!

!6*(u(x)) ! *(v(x)),+u(x) !+v(x)7 dx

( m

!

!#+(u(x)! v(x))#2 dx = m#u! v#2H1

0. (10.13)

The inequality (10.13) implies that T is strongly coercive, because

6T (u), u7H#u#H1

0

=6T (u)! T (0), u7H

#u#H10

+6T (0), u7H#u#H1

0

( m#u#H10! #T (0)#

" +- as #u#H10"-.

We now show that T is hemicontinuous at any u & H10 (%). For any positive sequence

(tk) converging to zero and any h, v & H10 (%), one has

6T (u + tkh)! T (u), v7H (10.14)

=

!

!6*(+u(x) + tk+h(x)) ! *(+u(x)),+v(x)7 dx.

Set, for k = 1, 2, . . . ,

gk(x) = 6*(+u(x) + tk+h(x)) ! *(+(x)),+v(x)7.

Clearly, each gk & L1(%), (fk) converges to 0 almost everywhere in %, and, assuming,without loss of generality, that tk ) 1 for all k = 1, 2, . . . , we have

#*(+u(x) + tk+h(x)) ! *(+u(x))# ) M#+u(x) + tk+h(x)# + M#+u(x)#) M [2#+u(x)#+ #+h(x)#]

and hence

|gk(x)| )M [2+u(x)#+ #+h(x)#]#+v(x)# (k = 1, 2, . . .)

with a right-hand member belonging to L1(%). Hence, Lebesgue’s dominated con-vergence theorem applied to (10.14) implies that

limk(%

6T (u + tkh), v7H = 6T (u), v7H .

The existence of a unique solution follows then from Browder-Minty’s theorem 10.2.1and Corollary 10.2.1, with, in addition the Lipschitz character of T!1.


10.4 A non-monotone quasilinear Dirichlet prob-

lem

We show in this section when monotonicity is lacking, some compactness argumentscan be used to obtain existence results from finite-dimensional approximations. Thecorresponding method has been first introduced by Vishik and called the compacte-ness method by J.L. Lions [253]. The example we give is treated in a somewhatdi!erent way in [60]. Let % ' Rn be an open bounded set and * : % 1 R " R

a Caratheodory function, i.e. for a.e. x & %, *(x, ·) is continuous and for anys & R, *(·, s) is measurable. Furthermore, let us assume that there exist constants0 < m )M such that

m ) *(x, s) )M (10.15)

for a.e. x & % and all s & R. Given f & H!1(%), we are interested in finding weaksolutions of the following Dirichlet problem for a quasilinear elliptic equation

!div (*(x, u))+u) = f, u & H10 (%) (10.16)

i.e. finding u & H10 (%) such that

!

!*(x, u(x))6+u(x),+v(x)7 dx = 6f, v7H (10.17)

for all v & H10 (%). We use the same notations as in the previous section.

Theorem 10.4.1 If condition (10.15) holds, problem (10.16) has at least one so-lution.

Proof. Let V be a finite-dimensional subspace of H10 (%) endowed with the H1

0 -norm,and V # its dual. Define the mappings H : V 1 [0, 1]" V # by

6H(u,"), v7H =

!

!*(x,"u)6+u(x),+v(x)7 dx ! 6f, v7H

for all v & V. Because of condition (10.15), H is well defined. Let us show now that

{u & V : H(u,") = 0 for some " & [0, 1]} ' B(m!1#f#H!1). (10.18)

Indeed, if H(u,") = 0 for some (u,") & V 1 [0, 1], then

0 = 6H(u,"), u7H ( m#u#2H10! #f#H!1#u#H1

0,

which implies that #u#H10) m!1#f#H!1. Consequently, for any R > m!1#f#H!1 ,

we have

H(u,") %= 0 if (u,") & !BV (R)1 [0, 1]. (10.19)

10.4. A NON-MONOTONE QUASILINEAR DIRICHLET PROBLEM 95

Now, if (u,") & BV

(R)1 [0, 1], we have

|6H(u,"), v7V )M#u#H10#v#H1

0+ #f#H1#v#H1

0) (MR + #f#H!1)#v#H1

0,

for all v & H10 (%), and hence

H(BV

(R)1 [0, 1]) ' BV $

(MR + #f#H!1). (10.20)

We now show that H is continuous on BV

(R)1 [0, 1]. Let (un,"n) & BV

(R)1 [0, 1]converge to (u,") in V 1 [0, 1], i.e. in H1

0 (%)1 [0, 1]. Since (H(un,"n)) is boundedbecause of (10.20), to prove that H(un,"n)" H(u,"), it is su$cient to show thatH(u,") is the unique cluster point of (H(un),"n). Let g & V # be such a clusterpoint, ("nk) a subsequence of ("n), and (unk) a subsequence of (un) such thatH(unk ,"nk)" g in V #. Since unk " u in H1

0 (%), it follows that unk " u in L2(%),and hence, going if necessary to a subsequence, we may assume that unk " u a.e.in %. This implies that *(x,"nk unk)" *(x,"u) a.e. in %, and hence, for any v & V,

*(x,"nk unk)+v " *(x,"u)+v

in L2(%). On the other hand, !iunk " !iu in L2(%). We conclude that

6H(unk ,"nk), v7H

=

!

![*(x,"nk unk)6+unk ,+v(x)7] dx

"!

!*(x,"u(x))6+u(x),+v(x)7 dx = 6H(u,"), v7H .

Thus g = H(u,"). All those properties allow us to apply the homotopy invarianceproperty (3.4.2) and obtain

dB[H(·, 1), B(R), 0] = dB[H(·, 0), B(R)]. (10.21)

But H(u, 0) = 0 is equivalent to the linear problem!

!*(x, 0)6+u(x),+v(x)7 dx = 6f, v7H

for all v & V, whose solution is unique because of the boundedness of the set of itspossible solutions. Consequently, dB[H(·, 0), B(R), 0] = ±1, and, from (10.21) andthe existence property 3.4.1 of degree, there exists u & BV (R) which satisfies

!

!*(x, u(x))6+u(x),+v(x)7 dx = 6f, v7H , #u#H1

0) m!1#f#H!1 (10.22)

for all v & V.Now, it is well known that one can write H1

0 (%) =H

k/1Vk where Vk ' Vk+1

(k ( 1) and Vk has dimension k. Consequently, given any v & H10 (%), there exists


a sequence (vk) with vk & Vk which converges to v. On the other hand, by (10.22)applied to V = Vk, there exists, for each k ( 1, some uk & Vk such that

!

!*(x, uk(x))6+uk(x),+w(x)7 dx = 6f, w7H , #uk#H1

0) m!1#f#H!1

for all w & Vk. In particular, taking w = vk introduced above,!

!*(x, uk(x))6+uk(x),+vk(x)7 dx = 6f, vk7H , #uk#H1

0) m!1#f#H!1 (10.23)

for all k ( 1. The estimate in (10.23) implies that, going if necessary to subse-quences, we can assume that there exists u & H1

0 (%) such that uk 1 u in H10 (%),

uk " u in L2(%) and uk " u a.e. in %. As (*(·, uk(·))) is bounded, and +vk " +vstrongly in L2(%), one can let k "- in (10.23) to obtain

!

!*(x, u(x))6+u(x),+v(x)7 dx = 6f, v7H ,

and, v & H10 (%) being arbitrary, u is a weak solution of (10.16).

10.5 Homogeneous Navier-Stokes equations

Let % ' Rn, with n ( 2, be a bounded open set with Lipschitz boundary, letf = (f1, . . . , fn) & L2(%) 1 . . . 1 L2(%) := L2

n(%). Similar notations will beused for other products of n vector spaces. We are looking for a vector functionu = (u1, . . . , un) and a scalar function p on % (representing the velocity and thepressure of the fluid respectively) satisfying the following homogeneous station-ary Navier-Stokes equations with Dirichlet boundary conditions, with ( areal coe$cient (representing the coe$cient of kinematic viscosity of the fluid) :

!(#u +n"

i=1

(ui!i)u + grad p = f in %, (10.24)

div u = 0 in % (10.25)

u = 0 on & := !%. (10.26)

In a scalar form, the system (10.24) can be rewritten as

!(n"

i=1

!2iiuk +

n"

i=1

ui!iuk + !kp = fk in % (k = 1, 2, . . . , n).

Denote by

V = {u & Dn(%) : div u = 0} (10.27)

10.6. VARIATIONAL FORMULATION 97

the space of test functions, and let V be the closure of V in the Sobolev spaceH1

0,n(%) of (u1, . . . , un) such that u & L2n(%) and !iu & L1

n(%) for all i = 1, 2, . . . , n,with the norm

#u#2H10,n

=n"

i,j=1

!

!|!iuj(x)|2 dx =

n"

j=1

#uj#2H10 (!).

The following result is proved in [383].

Proposition 10.5.1 Let % ' Rn be open with a Lipschitzian boundary. Then

V = {u & H10,n(%) : div u = 0}. (10.28)

If f, p and u are smooth functions satisfying (10.24)-(10.26), then, taking the scalarproduct of (10.24) in L1

n(%) with some v & V , integrating by parts and usingconditions (10.25) and (10.26), we get

(6u, v7H10,n(!) + b(u, u, v) = 6f, v7L2

n(!) (v & V), (10.29)

where

6u, v7H10,n

(%) =n"

i,j=1

!

!!iuj(x)!ivj(x) dx, (10.30)

and b is defined by

b(u, v, w) =n"

i,j=1

!

!ui(!ivj)wj dx. (10.31)

Consequently, if f, u and p are smooth functions satisfying (10.24)-(10.26) then, forany v & V , equality (10.29) holds. Continuity and density arguments then show that(10.29) is satisfied for any v & V. Of course a smooth function u satisfying (10.24)-(10.26) is a smooth function in V . To prove the converse, we need a condition fora vector-valued distribution to be a gradient given in [87] or in [383].

Proposition 10.5.2 Let % ' Rn be open and f = (f1, . . . , fn) with fi & D$(%)(i = 1, 2, . . . , n). A necessary and su!cient condition in order that f = grad p forsome p & D$(%) is that 66f, v77 = 0 for any v & V , where 66·, ·77 denotes the dualitypairing between D(%) and D$(%).

By using Proposition 10.5.2, we deduce from (10.28) the existence of a distributionp such that (10.24) is satisfied, (10.25) and (10.26) being also satisfied since u & V.

10.6 Variational formulation

Because of the fact that, for u and v in V, the expression b(u, u, v) needs not tomake sense, the variational formulation of problem (10.24)-(10.26) cannot be just :


find u & V such that (10.29) holds for any v & V. Some preparation is required toget the correct variational formulation, and, in particular, we need to recall someresults on Sobolev spaces H1

0 (%) = H10,1(%). Again, let % ' Rn be an open bounded

subsets with Lipschitzian boundary. Then the following result is true.

Proposition 10.6.1 (i) If n = 2, H10 (%) ' Lq(%) with compact injection for

any 1 ) q < +-;

(ii) If n ( 3, H10 (%) ' Lq(%) with continuous injection if 1 ) q ) 2n/(n! 2) and

compact injection if 1 ) q < 2n/(n! 2). In particular, H10 (%) ' L2(%) with

compact injection.

In particular, under the assumptions of continuous embedding, there exists c > 0such that

#u#Lq(!) ) c#u#H10(!) (10.32)

for all u & H10 (%) (Friedrichs-Poincare’s inequality).

Corollary 10.6.1 If n = 2, 3, 4, one has H10 (%) ' Ln(%) with continuous embed-

ding, i.e. there exists c > 0 such that

#u#Ln(!) ) c#u#H10(!)

for all u & H10 (%).

Proof. Follows from Proposition 10.6.1 by noticing that L2n/(n!2)(%) ' Ln(%) withcontinuous injection if 2n/(n! 2) ( n, i.e. if n ) 4.

Skipping from now one (%) from the spaces notations, consider the space H10,n.

Lnn equipped with the norm

#u#H10,n0Ln

n= #u#H1

0,n+ #u#Ln

n,

Notice that

V ' Dn ' H10,n . Ln

n,

and define the space 1V as the closure of V in H10,n . Ln

n.The following result is an immediate consequence of Proposition 10.6.1 and

Corollary 10.6.1

Proposition 10.6.2 For each integer n ( 2, one has 1V ' V and 1V = V forn = 2, 3, 4.

We can now precise the domain where b is defined.

Lemma 10.6.1 For any n ( 2, the form b is defined and continuous on H10,n 1

H10,n 1 (H1

0,n . Lnn).

10.6. VARIATIONAL FORMULATION 99

Proof. Let us first suppose that n ( 3 and let (u, v, w) & H10,n1H1

0,n1 (H10,n.Ln

n).

Hence !ivj & L2, wj & Ln, and, by Proposition 10.6.1 we have ui & L2n/(n!2)

(i, j = 1, 2, . . . , n). As n!22n + 1

2 + 1n = 1, we get ui(!ivj)wj & L1 and, using Holder’s

inequality

DDDD!

!ui(x)(!ivj(x))wj(x) dx

DDDD ) #ui#L2n/(n!2)#!ivj#L2#wj#Ln . (10.33)

Using again Proposition 10.6.1 we get

#ui#L2n/(n!2) ) c#ui#H10) c#u#H1

0,n.

Now, trivially

#!ivj#L2 ) #v#H10,n

, #wj#Ln ) #w#Ln ) #w#H10,n0Ln

n

which gives, by summing

|b(u, v, w)| ) ,#u#H10,n#v#H1

0,n#w#H1

0,n0Lnn, (10.34)

for some , > 0 depending only upon n and Friedrichs-Poincare’s constant. Whenn = 2, we have H1

0,2 ' Lq2 for any 1 ) q < - with continuous injection. Conse-

quently, for u, v & H10,2 and w & H1

0,2 . L22 = H1

0,2, we have

ui & H10 ' L4, !ivj & L2, wj & H1

0 ' L4 (i, j = 1, 2),

14 + 1

2 + 14 = 1, and, using Holder’s inequality,

DDDD

!

!ui(x)(!iwj(x))wj(x) dx

DDDD ) #ui#L4#!ivj#L2#wj#L4. (10.35)

To conclude, the procedure used for n ( 3 applies if we replace inequalities (10.33)by inequalities (10.35).

Because V is a closed subspace of H10,n and 1V a closed subspace of H1

0,n . Lnn,

which coincides with V for n = 2, 3, 4, Lemma 1 has the following direct corollary.

Corollary 10.6.2 The form b is trilinear and continuous on V 1 V 1 1V . If n =2, 3, 4, it is trilinear and continuous on V 1 V 1 V.

We need the following further properties of b.

Lemma 10.6.2 One has

(a) b(u, v, v) = 0 for all u & V and v & H10,n . Ln

n;

(b) b(u, v, w) = !b(u, w, v) for all u & H10,n, v, w & H1

0,n . Lnn.


Proof. (a) The result is true for u, v & V , because, integrating by parts, we have,for all i, j = 1, . . . , n,!

!ui(x)(!ivj(x))vj(x) dx =

!

!ui(x)!i(v

2j (x)/2) dx = !1

2

!

!(!iui(x))v2

j (x) dx,

and hence

b(u, v, v) = !1

2

!

!(div u(x))#v(x)#2 dx = 0.

If u & V and v & H10,n . Ln

n, the result follows by density of V in V (with respectto the H1

0,n-norm) and in H10,n .Ln

n, (with respect to the induced norm), and fromthe continuity of b.

(b) It is a consequence of (a), namely, for u & V , v, w & H10,n . Ln

n, one has

0 = b(u, v + w, v + w)

= b(u, v, v) + b(u, v, w) + b(u, w, v) + b(u, w, w)

= b(u, v, w) + b(u, w, v).

Now we are able to give an appropriate variational formulation of problem(10.24)- (10.26) : find u & V such that

(6u, v7H10,n

+ b(u, u, v) = 6f, v7L2n

for all v & 1V . (10.36)

Remember that 1V ' V and that 1V = V for n = 2, 3, 4. The variational formulationof problem (10.24)-(10.26) was first introduced in 1933 by J. Leray [247] for dimen-sions 2 and 3 in a somewhat di!erent way prefigurating Leray-Schauder’s theory,in which an a priori estimate for the possible solutions was obtained. His reasoningwas clarified by E. Hopf [175] in 1950, who used Galerkin’s method. A variant ofLeray’s existence theorem was obtained by 0.A. Ladyzhenskaia [234] in 1959 forn = 2 and 3, and by Shinbrot [365] for n = 4. Finally, H. Fujita [140] gave a proofbased upon Hopf’s Galerkin’s method, which, modified by R. Finn [125] was validfor all dimensions n. See [125] for more details on the history of the problem.

10.7 Existence of a solution

The existence theorem for (10.36) goes as follows. It is another example of appli-cation of the compactness method.

Theorem 10.7.1 For each f & Lnn, problem (10.36) has at least one solution.

Proof. Step 1. Galerkin approximation of the space 1V . The space 1V being sepa-rable as subspace of H1

0,n, there exists a sequence (wm) (m = 1, 2, . . .) of linearly

10.7. EXISTENCE OF A SOLUTION 101

independent elements which is total in 1V . Because V is dense in 1V , this system canbe chosen in V . Consequently, if Xm = span [w1, . . . , wm],

Hm/1 Xm is dense in 1V .

Of course, for any m = 1, 2, . . . , one has

Xm ' V ' 1V ' V ' H10,n.

Step 2. The operators Pm : Xm " Xm. Let us consider in Xm the innerproduct induced by that of V , namely 6u, v7Xm = 6u, v7H1

0,n. Define the operator

Pm : Xm " Xm as follows : for each u & Xm, Pm(u) is the unique element of Xm

which satisfies

6Pm(u), v7Xm = 6Pm(u), v7H10,n

= (6u, v7H10,n

+ b(u, u, v)! 6f, v7L2n

(10.37)

for all v & Xm. The existence and uniqueness of Pm(u) is justified by the fact that,for any fixed u & Xm, the linear form on Xm

v 2" (6u, v7H10,n

+ b(u, u, v)! 6f, v7L2n

is continuous, and Riesz’ theorem applies. The continuity of Pm follows from thefact that if u and u$ are two elements of Xm, then, for any v & Xm, one has

6Pm(u)! Pm(u$), v7Xm = (6u! u$, v7H10,n

+ b(u, u, v)! b(u$, u$, v).

From the continuity of b it follows that if u " u$ in Xm, then Pm(u) 1 Pm(u$) inXm, and hence Pm(u)" Pm(u$) as Xm is finite-dimensional.

Step 3. Existence of a solution for Pm(u) = 0It follows from Lemma 10.6.2 that

6Pm(u), u7Xm = (#u#2H10,n! 6f, u7L2

n( (#u#2H1

0,n! c#f#L2

n#u#H1

0,n

where c is the constant of the continuous embedding of H10,n into L2

n. Consequently,

6Pm(u), u7H10,m( 0 if #u#Xm = #u#H1

0,n( R ( c

(#f#L2

n.

and it follows from Poincare-Bohl’s existence theorem 8.4.2 that there exists um &Xm such that #um#Xm ) R and Pm(um) = 0, or, equivalently

6Pm(um), v7Xm = (6um, v7H10,m

+ b(um, um, v)! 6f, v7L2n

= 0 (10.38)

for any v & Xm.Step 4. Boundedness of (um) in V. We deduce in particular from (10.38) and

Lemma 10.6.2 that

0 = 6Pm(um), um7Xm = (#um#2H10,n! 6f, um7L2

n= 0,

so that, reasoning like above,

#um#H10,n) c

(#f#L2

n(10.39)


for all m = 1, 2, . . . . Then, going if necessary to a subsequence, we can assume,using the reflexivity of H1

0,n and the compact injection of H10,n into L2

n (Proposition10.6.1) that there exists some u & V such that um 1 u in V and un " u in L2

n. Toshow that b(um, um, v) " b(u, u, v) for any v & V , using the fact that, by Lemma10.6.2

b(um, um, v) = !b(um, v, um) = !n"

i,j=1

!

!um,i(x)um,j(x)!ivj(x) dx,

it is su$cient to show that

n"

i,j=1

!

!um,i(x)um,j(x)!ivj(x) dx"

n"

i,j=1

!

!ui(x)uj(x)!ivj(x) dx,

which is easy because !ivj & L% for all 1 ) i, j ) n.

Step 5. u is a solution of (10.36). Let v & 1V . SinceH

m/1 Xm is dense in 1V ,

there is a sequence (vk) inH

m/1 Xm such that vk " v in 1V . Because of

#vk ! v#eV = #vk ! v#H10,n0Ln

n= #vk ! v#H1

0,n+ #vk ! v#Ln

n,

the strong convergence of vk to v in 1V implies that vk " v strongly in V and inL2

n. Furthermore, we may assume that for any k & N there exists mk ( k such thatvk & Xmk . Consequently, it follows from (10.38) that

(6umk , vk7H10,n

+ b(umk , umk , vk) = 6f, vk7L1n,

which, letting k "-, gives (10.36).

10.8 Uniqueness conditions

We now state and prove a result of R. Finn [126] showing that the solution of (10.36)is unique if ( is su$ciently large or f su$ciently small.

Theorem 10.8.1 If 2 ) n ) 4 and if

(2 > c,#f#L2n

(10.40)

where c is the constant in Friedrichs-Poincare’s inequality (10.32) and , is definedin (10.34), there is a unique solution for (10.36).

Proof. First we give an estimate, valid for any n ( 2, for any solution u whoseexistence was proved in Theorem 10.7.1. Indeed, it follows from (10.39) that

#u#V ) lim supm(%

#um#V )c

(#f#L2

n, (10.41)

10.8. UNIQUENESS CONDITIONS 103

Now let n ) 4, so that V = 1V , and assume that u & V and u$ & V are two solutionsof

(6u, v7H10,n

+ b(u, u, v) = 6f, v7L2n

(10.42)

for any v & 1V = V. Then, by substraction of the two equations, we get

(6u! u$, v7H10,n

+ b(u, u, v)! b(u$, u$, v) = 0

for any v & V, Now, a simple computation gives

b(u, u, v)! b(u$, u$, v) = b(u, u! u$, v) + b(u! u$, u$, v),

so that, by taking v = u! u$ we obtain, using Lemma 10.6.2,

(#u! u$#H10,n

= !b(u! u$, u$, u! u$) = b(u! u$, u! u$, u$). (10.43)

Now, because n ) 4, the function b is continuous on V 1 V 1 V, so that, using(10.41), we obtain

|b(u! u$, u! u$, u$)| ) ,#u! u$#2H10,n#u$#H1

0,n) ,

c

(#f#L2

n#u! u$#2H1

0,n. (10.44)

Combining (10.43) and (10.44) we derive

2( ! ,c

(#f#L2

n

3#u! u$#2H1

0,n) 0,

and conclude that u = u$ if (10.40) holds.

Chapter 11

First order di!erenceequations

11.1 Periodic solutions

Let n ( 2 be a fixed integer. For (x1, · · · , xn) & Rn, define the first order di!erenceoperator (#x1, · · · ,#xn!1) & Rn!1 by

#xm := xm+1 ! xm, (1 ) m ) n! 1).

Let fm : Rn " R (1 ) m ) n!1) be continuous functions. We study the existenceof solutions for the periodic boundary value problem

#xm + fm(x1, · · · , xn) = 0 (1 ) m ) n! 1), x1 = xn. (11.1)

Let

Un!1 = {x & Rn : x1 = xn} 5 R

n!1, (11.2)

as an element of Un!1 can be characterized by the coordinates x1, · · · , xn!1. Therestriction L : Rn!1 " Rn!1 of D to Rn!1 is given by

(Lx)m = xm+1 ! xm (1 ) m ) n! 2), (Lx)n!1 = x1 ! xn!1, (11.3)

or, in matrix form, by the circulant matrix

5

UUUUUU6

!1 1 0 · · · · · · 00 !1 1 0 · · · 0· · · · · · · · · · · · · · · · · ·· · · · · · · · · · · · · · · · · ·0 · · · · · · 0 !1 11 0 · · · · · · 0 !1

8

VVVVVV9. (11.4)

105

106 CHAPTER 11. FIRST ORDER DIFFERENCE EQUATIONS

If we define F : Rn!1 " Rn!1 by

Fm(x1, · · · , xn!1) = fm(x1, x2, · · · , xn!1, x1) (1 ) m ) n! 1),

problem (11.1) is equivalent to study the zeros of the continuous mapping H :Rn!1 " Rn!1 defined by

Hm(x) = (Lx)m + Fm(x) (1 ) m ) n! 1). (11.5)

11.2 Bounded nonlinearities

Let us first consider the linear periodic problem

#xm + *xm = 0 (1 ) m ) n! 1), x1 = xn, (11.6)

where * & R. The solutions of the system are given by

xm = (1! *)m!1x1, (1 ) m ) n! 1),

and hence (11.6) has a solution if and only if

x1 = (1! *)n!1x1.

This immediately implies the following

Lemma 11.2.1 Problem (11.6) has only the trivial solution if n is even and * %= 0or if n is odd and * %& {0, 2}. When * = 0, the solutions are of the form xm =c (1 ) m ) n ! 1), and when n is odd and * = 2, they have the form xm =(!1)m!1c (1 ) m ) n! 1), with c & R arbitrary.

Let

bm : Rn " R, (x1, · · · , xn) 2" bm(x1, · · · , xn) (1 ) m ) n! 1) (11.7)

be continuous and bounded, and consider the semilinear periodic problem

#xm + *xm + bm(x1, · · · , xn) = 0 (1 ) m ) n! 1), x1 = xn. (11.8)

If L is defined like above and B : Rn!1 " Rn!1 by

Bm(x1, · · · , xn!1) = bm(x1, x2, · · · , xn!1, x1) (1 ) m ) n! 1),

then problem (11.8) is equivalent to the semilinear problem in Rn!1

Lx + *x + B(x) = 0. (11.9)

An immediate consequence of Lemma 11.2.1 and Corollary 8.1.2 is the followingexistence result.

Theorem 11.2.1 Assume n odd and * %= 0 or n even and * %& {0, 2} and assumethat the functions bm in (11.7) are continuous and bounded. Then problem (11.8)has at least one solution and, for all su!ciently large R,

dB[L + *I + B, B(R), 0] = ±1.

11.3. UPPER AND LOWER SOLUTIONS 107

11.3 Upper and lower solutions

Let fm : R " R (1 ) m ) n ! 1) be continuous functions, and let us consider theperiodic problem

#xm + fm(xm) = 0 (1 ) m ) n! 1), x1 = xn. (11.10)

Definition 11.3.1 * = (*1, · · · ,*n) (resp. , = (,1, · · · ,,n)) is called a lowersolution (resp. upper solution) for (11.10) if

*1 ( *n (resp. ,1 ) ,n),

and the inequalities

#*m + fm(*m) ( 0 (resp. #,m + fm(,m) ) 0) (11.11)

hold for all 1 ) m ) n ! 1. Such a lower or upper solution will be called strict ifthe inequality (11.11) is strict for all 1 ) m ) n! 1.

The basic theorem for the method of upper and lower solutions goes as follows.The proof given here is taken from [19] and modeled on the corresponding one fordi!erential equations in [267].

Theorem 11.3.1 If (11.10) has a lower solution * = (*1, · · · ,*n) and an uppersolution , = (,1, · · · ,,n) such that *m ) ,m (1 ) m ) n), then (11.10) has asolution x = (x1, · · · , xn) such that *m ) xm ) ,m (1 ) m ) n). Moreover, if *and , are strict, then *m < xm < ,m (1 ) m ) n! 1).

Proof. I. A modified problem.Let .m : R !" R (1 ) m ) n! 1) be the continuous functions defined by

.m(x) =

*+

,

,m if x > ,m

x if *m ) x ) ,m

*m if x < *m.(11.12)

We consider the modified problem

#xm ! xm + fm * .m(xm) + .m(xm) = 0 (1 ) m ) n! 1), x1 = xn, (11.13)

and show that if x = (x1, · · · , xn) is a solution of (11.13) then *m ) xm ) ,m (1 )m ) n), and hence x is a solution of (11.10). Suppose by contradiction that thereis some 1 ) i ) n such that *i!xi > 0 so that *m!xm = max1+j+n(*j !xj) > 0.If 1 ) m ) n! 1, then

*m+1 ! xm+1 ) *m ! xm,

which gives

#*m ) #xm = xm ! *m ! fm(*m) ) xm ! *m + #*m < #*m,


a contradiction. Now the condition *1 ( *n shows that the maximum is reached atm = n only if it is reached also at m = 1, a case already excluded. Analogously wecan show that xm ) ,m (1 ) m ) n). We remark that if *,, are strict, the samereasoning gives *m < xm < ,m (1 ) m ) n! 1).

II. Solution of the modified problem.We use Brouwer degree to study the zeros of the continuous mapping G : Rn!1 "Rn!1 defined by

Gm(x) = (Lx)m ! xm + fm * .m(xm) + .m(xm) (1 ) m ) n! 1). (11.14)

As seen above, L! I : Rn!1 " Rn!1 is invertible. On the other hand the mappingwith components fm *.m +.m (1 ) m ) n!1) is bounded on Rn!1. Consequently,Theorem 11.2.1 implies the existence of R > 0 such that, for all - > R, one has

|dB [G, B(-), 0]| = 1, (11.15)

and, in particular, G has a zero x & B(-). Hence, x is a solution of (11.13), whichmeans that *m ) xm ) ,m (1 ) m ) n) and x is a solution of (11.10). Moreoverif *,, are strict, then *m < xm < ,m (1 ) m ) n! 1).

Remark 11.3.1 Suppose that * (resp. ,) is a strict lower (resp. upper) solutionsof (11.10). Define the open set

%&' = {(x1, · · · , xn!1) & Rn!1 : *m < xm < ,m (1 ) m ) n! 1)}.

If - is large enough, then, using the additivity-excision property of Brouwer degree,we have

|dB [G,%&' , 0]| = |dB [G, B(-), 0]| = 1.

On the other hand, if H is the continuous mapping defined in (11.5), H is equal toG on %&' , and then

|dB[H,%&' , 0]| = 1. (11.16)

A simple but useful consequence of Theorem 11.3.1, goes as follows.

Corollary 11.3.1 Assume that there exists numbers * ) , such that

fm(*) ( 0 ( fm(,) (1 ) m ) n).

Then problem (11.10) has at least one solution with * ) xm ) , (1 ) m ) n! 1).

Proof. Just observe that (*, · · · ,*) is a lower solution and (,, · · · ,,) an upper so-lution for (11.10).

11.4. SPECTRUM OF THE LINEAR PART 109

Example 11.3.1 For each p > 0, am > 0 and bm & R (1 ) m ) n!1) the problem

#xm ! am|xm|pxm + bm = 0 (1 ) m ) n! 1), x1 = xn

has at least one solution.

Remark 11.3.2 When ,m ) *m (1 ) m ) n ! 1), one can try to repeat theargument of Theorem 11.3.1 by defining

+m(x) =

*+

,

*m if x > *m

x if ,m ) x ) *m

,m if x < ,m,

and considering the modified problem

(Lx)m + xm + fm * +m(xm)! +m(xm) = 0 (1 ) m ) n! 1), x1 = xn. (11.17)

As L + I is invertible, the degree argument still gives the existence of at least onesolution for (11.17). If one tries to show that, say, ,m ) xm (1 ) m ) n! 1), andassume by contradiction that ,i ! xi > 0 for some 1 ) i ) n, so that ,m ! xm =max1+j+n(,j ! xj) > 0, one has for 1 ) m ) n! 1

,m+1 ! xm+1 ) ,m ! xm,

which gives

#,m + fm(,m) ) #xm + fm * .m(xm) = !xm + ,m > 0,

implying no contradiction with #,m + fm(,m) ) 0. This is in contrast with theordinary di!erential equation case, for which the method of upper and lower solu-tions works independently of their order. The reason of this di!erence comes fromthe fact that a local extremum is characterized by an equality (vanishing of the firstderivative) in the di!erential case and by two inequalities (with only one usable inthe argument) in the di!erence case. We show now that, for di!erence equations,the existence theorem based upon lower and upper solutions does not work whenthe upper solution is smaller than the lower solution.

11.4 Spectrum of the linear part

The construction of the counter-example proving the last assertion above is madeclear by analyzing the spectral properties of the first order di!erence operator withperiodic boundary conditions.

Definition 11.4.1 An eigenvalue of the first order di"erence operator with peri-odic boundary conditions is any " & C such that the problem

#xm = "xm (1 ) m ) n! 1), x1 = xn (11.18)

has a nontrivial solution.


Explicitly, system (11.18) can be written as

x1 ! xn = 0

x2 ! (1 + ")x1 = 0

. . . . . . . . . . . . . . . (11.19)

xn ! (1 + ")xn!1 = 0

or5

UUUUUU6

!1! " 1 0 · · · · · · 00 !1! " 1 0 · · · 0· · · · · · · · · · · · · · · · · ·· · · · · · · · · · · · · · · · · ·0 · · · · · · 0 !1! " 11 0 · · · · · · 0 !1! "

8

VVVVVV9

5

UUUUUU6

x1

x2

··

xn!2

xn!1

8

VVVVVV9= 0.

Looking for solutions of the form xm = eim, (1 ) m ) n ! 1) for some ) & R, oneeasily finds that ) must verify the equations

" = !1 + ei,, ei(n!1), = 1,

which gives the n! 1 distinct eigenvalues

"k = !1 + e2k"in!1 (0 ) k ) n! 2), (11.20)

with the corresponding eigenvectors &k (0 ) k ) n ! 2) having components &km =

e2km"i

n!1 (1 ) m ) n). In particular, "0 = 0 is always a real eigenvalue, and all theother eigenvalues have negative real part. If n = 2, 0 is the unique eigenvalue; ifn > 2 is even, 0 is the unique real eigenvalue; if n is odd, "n!1

2= !2 is the unique

nonzero real eigenvalue.

Remark 11.4.1 The µk = "k +1 are the eigenvalues of the (permutation, unitary,circulant) matrix

5

UUUUUU6

0 1 0 · · · · · · 00 0 1 0 · · · 0· · · · · · · · · · · · · · · · · ·· · · · · · · · · · · · · · · · · ·0 · · · · · · 0 0 11 0 · · · · · · 0 0

8

VVVVVV9.

11.5 Reversing the order of upper and lower solu-

tions

For n ( 2 odd and " = !2, system (11.19) becomes

x1 ! xn = 0

11.5. REVERSING THE ORDER OF UPPER AND LOWER SOLUTIONS 111

x2 + x1 = 0

. . . . . . . . . (11.21)

xn + xn!1 = 0

and has the eigenvector with components &(n!1)/2m = (!1)m!1 (1 ) m ) n). The

adjoint system

x1 + x2 = 0

. . . . . . . . .

xn!1 + xn = 0 (11.22)

!x1 + xn = 0

has the nontrivial solution with components 3m = (!1)m!1 (1 ) m ) n). Asbm = +nm (1 ) m ) n) (Kronecker symbol) is not orthogonal to the kernel of theadjoint system (11.22), the problem

x1 ! xn = 0

x2 + x1 = 0

. . . . . . . . .

xn!1 + xn!2 = 0

xn + xn!1 = 1

has no solution, or, equivalently the problem

#xm + 2xm = 0 (1 ) m ) n! 2), #xn!1 + 2xn!1 = 1, x1 = xn (11.23)

has no solution. However, * = (1, 1, · · · , 1) is a lower solution and , = (0, 0, · · · , 0)is an upper solution of (11.23) such that ,m ) *m (1 ) m ) n).

If now n > 2 is even, the problem

#xm + 2xm = 0 (1 ) m ) n! 3), #xn!2 + 2xn!2 = 1,

#xn!1 = 0, x1 = xn (11.24)

is of course equivalent to the problem

#xm + 2xm = 0 (1 ) m ) n! 3), #xn!2 + 2xn!2 = 1, x1 = xn!1.

As n!1 is odd, it follows from the counter-example (11.23) that problem (11.24) hasno solution. However * = (1, 1, · · · , 1) is a lower solution and , = (0, 0, · · · , 0) is anupper solution of (11.24) such that ,m ) *m (1 ) m ) n). Those counter-exampleswere first given in [20].

For n = 2, problem (11.10) is equivalent to the unique scalar equation

f1(x1) = 0


and, in this case, the validity of the method of upper and lower solutions, indepen-dently of their order, follows from its equivalence with Bolzano’s theorem appliedto the real function f1.

Remark 11.5.1 Notice that, in contrast to the periodic problem for di!erenceequations, whose eigenvalues are in the left half-plane, all the eigenvalues "k = 2k"i

T

(k & Z) of the di!erential operator ddt with periodic boundary conditions on [0, T ]

are on the imaginary axis. This explains the indi!erence of the result with respectto the order of the lower and the upper solution.

11.6 Ambrosetti-Prodi type multiplicity result

Let f1, · · · , fn!1 : R " R be continuous functions, s & R. Consider the problem,with n ( 2,

#xm + fm(xm) = s (1 ) m ) n! 1), x1 = xn, (11.25)

with the coercivity condition

fm(x)"- as |x|"- (1 ) m ) n! 1). (11.26)

When n = 2, problem (11.25) is equivalent to the scalar equation

f1(x1) = s (11.27)

and, under condition (11.26) with m = 1, it is clear that there exists s1 (= minR f1)such that for s < s1, equation (11.27) has no solution, for s = s1, equation (11.27)has at least one solution, and for s > s1, equation (11.27) has at least two solutions.We show that a similar result holds for any n ( 2. Problems of this type wereinitiated by Ambrosetti- Prodi for second order semilinear Dirichlet problems andthe results given here, which can be found in [19], are modeled on the approachgiven in [268] for periodic solutions of first order ordinary di!erential equations.

Lemma 11.6.1 Let b & R. If condition (11.26) holds, there is - = -(b) > 0 suchthat each possible solution x of (11.25) with s ) b is such that #x# < -.

Proof. Let s ) b and (x1, · · · , xn) be a solution of (11.25). We see that

n!1"

m=1

fm(xm) = (n! 1)s ) (n! 1)b. (11.28)

From (11.26),

(: c & R)(;u & R)(;m & {1, . . . , n! 1) : fm(u) ( c, (11.29)

and

(;R > 0)(: r$ > 0)(;u & R : |u| ( r$)(;m & {1, . . . , n! 1} :

fm(u) ( R ! (n! 2)c. (11.30)

11.6. AMBROSETTI-PRODI TYPE MULTIPLICITY RESULT 113

If r =<

n! 1r$ and if x & Rm!1 is such that #x# ( r, then, for at least onej & {1, . . . , n! 1}, one has |xj | ( r$, so that, using (11.29) and (11.30),

n!1"

m=1

fm(xm) =n!1"

m=1

[fm(xm)! c] + (n! 1)c ( fj(xj)! c + (n! 1)c

( R! (n! 2)c! c + (n! 1)c = R.

Consequently,Bn!1

m=1 fm(xm) " +- if #x# " -, and the result follows from(11.28).

Theorem 11.6.1 If the functions fm (1 ) m ) n! 1) satisfy (11.26), there existss1 & R such that (9s) has zero, at least one or at least two solutions according tos < s1, s = s1, s > s1.

Proof. Let

Sj = {s & R : (11.25) has at least j solutions } (j ( 1).

(a) S1 %= !.Take s# > max1+m+n!1 fm(0) and use (11.26) to find R#

! < 0 such that

min1+m+n!1

fm(R#!) > s#.

Then * with *j = R#! < 0 (1 ) j ) n) is a strict lower solution and , with ,j = 0

(1 ) j ) n) is a strict upper solution for (11.25) with s = s#. Hence, using Theorem11.3.1, s# & S1.(b) If 1s & S1 and s > 1s then s & S1.Let 1x = (1x1, · · · , 1xn) be a solution of (11.25) with s = 1s, and let s > 1s. Then1x is a strict upper solution for (11.25). Take now R! < min1+m+n 1xm such thatmin1+m+n!1 fm(R!) > s. It follows that * with *j = R! (1 ) j ) n) is a strictlower solution for (11.25), and hence, using Theorem 11.3.1, s & S1.(c) s1 = inf S1 is finite and S1 0 ]s1,-[.Let s & R and suppose that (11.25) has a solution (x1, · · · , xn). Then (11.28) holds,from where we deduce that s ( c, with c & R such that fm(x) ( c for all x & R,and 1 ) m ) n ! 1. To obtain the second part of claim (c) S1 0 ]s1,-[ we apply(b).(d) S2 0 ]s1,-[.We reformulate (11.25) to apply Brouwer degree theory. Consider the continuousmapping G : R1 Rn!1 " Rn!1 defined by

Gm(s, x) = (Lx)m + fm(xm)! s (1 ) m ) n! 1).

Then (x1, · · · , xn) is a solution of (11.25) if and only if (x1, · · · , xn!1) & Rn!1 isa zero of G(s, ·). Let s3 < s1 < s2. Using Lemma 11.6.1 we find - > 0 such thateach possible zero of G(s, ·) with s & [s3, s2] is such that max1+m+n!1 |xm| < -.


Consequently, dB [G(s, ·), B(-), 0] is well defined and does not depend upon s &[s3, s2]. However, using (c), we see that G(s3, x) %= 0 for all x & Rn!1. This impliesthat dB [G(s3, ·), B(-), 0] = 0, so that dB [G(s2, ·), B(-), 0] = 0 and, by excisionproperty, dB[G(s2, ·), B(-$), 0] = 0 if -$ > -. Let s & ]s1, s2[ and Tx = (Tx1, · · · , Txn)be a solution of (11.25) (using (c)). Then Tx is a strict upper solution of (11.25)with s = s2. Let R < min1+j+n Txj be such that min1+m+n!1 fm(R) > s2. Then(R, · · · , R) & Rn is a strict lower solution of (11.25) with s = s2. Consequently,using Remark 11.3.1, (11.25) with s = s2 has a solution in %Rbx and

|dB[G(s2, ·),%Rbx, 0]| = 1.

Taking -$ su$ciently large, we deduce from the additivity property of Brouwerdegree that

|dB [G(s2, ·), B(-$) \ %Rbx, 0]| = |dB [G(s2, ·), B(-$), 0]! dB[G(s2, ·),%Rbx, 0]|= |dB [G(s2, ·),%Rbx, 0]| = 1,

and (11.25) with s = s2 has a second solution in B(-$) \ %Rbx.(e) s1 & S1.Taking a decreasing sequence ($k)k"N in ]s1,-[ converging to s1, a correspondingsequence (xk

1 , · · · , xkn) of solutions of (11.25) with s = $k and using Lemma 11.6.1,

we obtain a subsequence (xjk1 , · · · , xjk

n ) which converges to a solution (x1, · · · , xn)of (11.25) with s = s1.

Example 11.6.1 There exists s1 & R such that the periodic problem for Verhulstequation with harvesting

#xm + xm(xm ! 1) = s (1 ) m ) n! 1), x1 = xn

has no solution if s < s1, at least one solution if s = s1 and at least two solutionsif s > s1.

Similar arguments allow to prove the following result.

Theorem 11.6.2 If the functions fm satisfy condition

fm(x)" !- as |x|"- (1 ) m ) n! 1). (11.31)

then there is s1 & R such that (11.25) has zero, at least one or at least two solutionsaccording to s > s1, s = s1 or s < s1.

11.7 One-sided bounded nonlinearities

The nonlinearity in Ambrosetti-Prodi type problems is bounded from below andcoercive or bounded from above and anticoercive. In this section, we consider

11.7. ONE-SIDED BOUNDED NONLINEARITIES 115

nonlinearities which are bounded from below or above but have di!erent limits at+- and !-.

Let n ( 2 be an integer and fm : R" R continuous functions (1 ) m ) n! 1).Consider the problem

#xm + fm(xm) = 0 (1 ) m ) n! 1), x1 = xn. (11.32)

Let H : Rn!1 " Rn!1 be the continuous mapping defined in (11.5), whose zeroscorrespond to the solutions of (11.32). It is easy to check that the linear mappingL defined in (11.3) is such that

N(L) = {(c, · · · , c) & Rn!1 : c & R},

R(L) = {(y1, · · · , yn!1) & Rn!1 :

n!1"

m=1

ym = 0},

The projectors P : Rn!1 " Rn!1, Q : Rn!1 " Rn!1 defined by

P (x1, · · · , xn!1) = (x1, · · · , x1),

Q(y1, · · · , yn!1) =

E1

n! 1

n!1"

m=1

ym, · · · , 1

n! 1

n!1"

m=1

ym

F

are such that N(Q) = R(L), R(P ) = N(L). Let us finally define F : Rn!1 " Rn!1

by

F (x1, · · · , xn!1) = (f1(x1), · · · , fn!1(xn!1)),

so that H = L + F. To study the zeros of G using Brouwer degree, we use Theorem8.6.1. The following Lemma, taken from [19], adapts to di!erence equations anargument of Ward [398] for ordinary di!erential equations.

Lemma 11.7.1 If the functions fm (1 ) m ) n ! 1), are all bounded from belowor all bounded from above, and if for some R > 0

n!1"

m=1

fm(xm) %= 0 whenever min1+j+n!1

xj ( R or max1+j+n!1

xj ) !R, (11.33)

then there exists - ( R such that, for each " & ]0, 1] and each possible zero x ofL + "F, one has max1+j+n!1 |xj | < -.

Proof. Let (", x) & ]0, 1]1Rn!1 be a possible zero of L + "N. It is a solution of theequivalent system

n!1"

m=1

fm(xm) = 0, #xm + "fm(xm) = 0, x1 = xn, (1 ) m ) n! 1). (11.34)


On the other hand, if we assume that each fm (1 ) m ) n! 1) is bounded frombelow, say by c, we have, for all 1 ) m ) n! 1,

|fm(u)| ) fm(u) + 2|c| (u & R). (11.35)

Hence, using (11.34) and (11.35), we obtain

n!1"

m=1

|#xm| = "n!1"

m=1

|fm(xm)| )n!1"

m=1

|fm(xm)| (11.36)

)n!1"

m=1

fm(xm) + 2(n! 1)|c| = 2(n! 1)|c|. (11.37)

We deduce

max1+m+n!1

xm ) min1+m+n!1

xm +n!1"

m=1

|#xm| (11.38)

) min1+m+n!1

xm + 2(n! 1)|c|. (11.39)

Using (11.34) and assumption (11.33), we obtain min1+m+n!1 xm < R and !R <max1+m+n!1 xm. Combined with (11.38), this gives

![R + 2(n! 1)|c|] < min1+m+n!1

xm ) max1+m+n!1

xm < R + 2(n! 1)|c|.

It follows that we can take any - ( R+2(n!1)|c|. If the fm are bounded from above,it su$ces to consider the equivalent problem !Lx!F (x) = 0 with all function !fm

bounded from below, as !L has the same null-space and range than L.

Define & : R" R by

&(x) =1

n! 1

En!1"

m=1

fm(x)

F

The following result is taken from [19].

Theorem 11.7.1 Suppose that the functions fm (1 ) m ) n! 1) satisfy the con-ditions of Lemma 11.7.1 and that

&(!-)&(-) < 0. (11.40)

Then, problem (11.32) has at least one solution.

Proof. The first assumption of Theorem 8.6.1 is satisfied for D = B(-)1 [0, 1]. Now,N(L) 5 R 5 R(Q), and

QF (c, · · · , c) = &(c) (c & R).

Hence the second assumption of Theorem 8.6.1 is satisfied, and, furthermore,

|dB [QF, B(-) .N(L), 0]| = |dB[&, ]! -, -[, 0]| = 1,

so that the first assumption of Theorem 8.6.1 holds as well.


Example 11.7.1 Given positive am and bm (1 ) m ) n!1), the periodic problem

#xm + am exp xm ! bm = 0 (1 ) m ) n! 1), x1 = xn, (11.41)

related to the study of positive periodic solutions of Verhulst equation with variablecoe$cients has at least one solution.

Chapter 12

Second order di!erenceequations

12.1 Dirichlet problem

For a fixed integer n ( 2 and (x0, · · · , xn) & Rn+1, define (#2x0, · · · ,#2xn!2) &Rn!1 by

#2xm!1 = xm+1 ! 2xm + xm!1, (1 ) m ) n! 1)).

If fm : Rn+1 " R (1 ) m ) n! 1) are continuous functions, we study the existenceof solutions for the Dirichlet boundary value problem

#2xm!1 + fm(x0, x1, . . . , xn) = 0 (1 ) m ) n! 1), x0 = 0 = xn. (12.1)

We have

V n!1 := {x & Rn+1 : x0 = 0 = xn} 5 R

n!1, (12.2)

as its elements can be associated to the coordinates (x1, · · · , xn!1) and correspondto the elements of Rn+1 of the form (0, x1, · · · , xn!1, 0). The restriction L : Rn!1 "Rn!1 of D2 to Rn!1 is well defined in terms of (x1, · · · , xn!1) by

(Lx)1 = !2x1 + x2,

(Lx)m = xm+1 ! 2xm + xm!1 (2 ) m ) n! 1), (12.3)

(Lx)n!1 = !2xn!1 + xn!2,

or, in matrix form, by the symmetric Jacobi matrix

L =

5

UUUU6

!2 1 0 · · · · · · 01 !2 1 0 · · · 0· · · · · · · · · · · · · · · · · ·0 · · · 0 1 !2 10 · · · · · · 0 1 !2

8

VVVV9. (12.4)

119

120 CHAPTER 12. SECOND ORDER DIFFERENCE EQUATIONS

We define the continuous mapping F : Rn!1 " Rn!1 by

Fm(x1, . . . , xn!1) = fm(0, x1, . . . , xn!1, 0) (1 ) m ) n! 1), (12.5)

and the continuous mapping G : Rn!1 " Rn!1 by G = L + F. It is clear that thesolutions of (12.1) correspond to the zeros of G in Rn!1.

12.2 Spectrum of the linear part

The linear Dirichlet eigenvalue problem

#2xm!1 ! "xm = 0 (1 ) m ) n! 1), x0 = 0 = xn, (12.6)

where " & R, is equivalent to the linear eigenvalue problem in Rn!1

5

UUUUUU6

!2! " 1 0 · · · · · · 01 !2! " 1 0 · · · 0· · · · · · · · · · · · · · · · · ·· · · · · · · · · · · · · · · · · ·0 · · · · · · 1 !2! " 10 · · · · · · 0 1 !2! "

8

VVVVVV9

5

UUUUUU6

x1

x2

··

xn!2

xn!1

8

VVVVVV9= 0. (12.7)

If we search solutions of (12.6) of the form xm = sinm) (m = 0, 1, · · · , n), with) & ]0, 2%[ \{%}, we see that we must have

sin(m + 1)) ! (" + 2) sinm) + sin(m! 1)) = 0 (1 ) m ) n! 1),

or, equivalently

sinm)[2 cos ) ! "! 2] = 0 (1 ) m ) n! 1),

which holds if

" = 2 cos ) ! 2. (12.8)

The boundary condition x0 = 0 is satisfied for all ), and the boundary conditionxn = sinn) = 0 gives

) =k%

n(1 ) k ) n! 1),

which, together with (12.8) implies

"k = 2 cosk%

n! 2 = !4 sin2 k%

2n(1 ) k ) n! 1). (12.9)

This immediately leads to the following

12.3. BOUNDED NONLINEARITIES 121

Lemma 12.2.1 Problem (12.6) has only the trivial solution if and only if

" %= 2 cosk%

n! 2 (1 ) k ) n! 1).

If " = "k for some 1 ) k ) n! 1, the corresponding eigenvector &k & Rn!1 has theform

2sin

k%

n, sin

2k%

n, . . . , sin

(n! 1)k%

n

3.

In particular, the eigenvector &1 associated to the eigenvalue

"1 = 2Lcos

%

n! 1

M= !4 sin2 %

2n, (12.10)

given by

&1 =

2sin

%

n, sin

2%

n, · · · , sin (n! 1)%

n

3(12.11)

has all its components positive. Furthermore, as the &k constitute a system ofeigenvectors of the symmetric matrix L, they satisfy the orthogonality conditions6&j ,&k7 = 0 for j %= k, where 6·, ·7 denotes the inner product in Rn!1.

12.3 Bounded nonlinearities

Let bm : Rn+1 " R, (x0, x1, . . . , xn) 2" bm(x0, x1, · · · , xn) (1 ) m ) n ! 1) becontinuous and bounded, and consider the semilinear Dirichlet problem

#2xm!1 ! "xm + bm(x0, x1, · · · , xn) = 0 (1 ) m ) n! 1), x0 = 0 = xn.(12.12)

If L is defined in (12.4) and B : Rn!1 " Rn!1 by

Bm(x1, · · · , xn!1) = bm(0, x1, x2, · · · , xn!1, 0) (1 ) m ) n! 1),

problem (12.12) is equivalent to the semilinear problem in Rn!1

Lx! "x + B(x) = 0. (12.13)

An immediate consequence of Lemma 12.2.1 and Corollary 8.1.2 is the followingexistence result.

Theorem 12.3.1 Assume that " %= 2 cos k"n ! 2 (1 ) k ) n ! 1), and that the

functions bm are continuous and bounded. Then problem (12.12) has at least onesolution and, for all su!ciently large R,

dB[L! "I + B, B(R), 0] = ±1.

In particular, the result is true for any " & ]!-,!2] / [0, +-[.


12.4 Upper and lower solutions

Let fm : R " R (1 ) m ) n! 1) be continuous functions. We study the existenceof solutions for the Dirichlet boundary value problem

#2xm!1 + fm(xm) = 0 (1 ) m ) n! 1), x0 = 0 = xn. (12.14)

Definition 12.4.1 * = (*0, · · · ,*n) (resp. , = (,0, · · · ,,n)) is called a lowersolution (resp. upper solution) for (12.14) if

*0 ) 0, *n ) 0 (resp. ,0 ( 0, ,n ( 0) (12.15)

and the inequalities

#2*m!1 + fm(*m) ( 0 (resp. #2,m!1 + fm(,m) ) 0) (1 ) m ) n! 1)(12.16)

hold. Such a lower or upper solution will be called strict if the inequalities (12.16)are strict.

The proof of the following result, taken from [20], is modeled upon the one givenin [267] for the case of second order ordinary di!erential equations.

Theorem 12.4.1 If (12.14) has a lower solution * = (*0, · · · ,*n) and an uppersolution , = (,0, · · · ,,n) such that *m ) ,m (1 ) m ) n! 1), then (12.14) has asolution x = (x0, · · · , xn) such that *m ) xm ) ,m (1 ) m ) n ! 1). Moreover, if* and , are strict, then *m < xm < ,m (1 ) m ) n! 1).

Proof. I. A modified problem.Let .m : R" R (1 ) m ) n! 1) be the continuous functions defined by

.m(x) =

*+

,

,m, x > ,m

x, *m ) x ) ,m

*m, x < *m.

We consider the modified problem

#2xm!1 ! xm + fm * .m(xm) + .m(xm) = 0 (1 ) m ) n! 1),

x0 = 0 = xn, (12.17)

and show that if x = (x0, · · · , xn) is a solution of (12.17) then *m ) xm ) ,m

(1 ) m ) n ! 1), and hence x is a solution of (12.14). Suppose by contradictionthat there is some 1 ) i ) n ! 1 such that *i ! xi > 0 so that *m ! xm =max1+j+n!1(*j ! xj) > 0. Hence

#2(*m!1 ! xm!1) = (*m+1 ! xm+1)! 2(*m ! xm) + (*m!1 ! xm!1) ) 0,

and

#2*m!1 ) #2xm!1 = !Fm(xm) + (xm ! .m(xm)) = !fm(*m) + (xm ! *m)

< !fm(*m) ) #2*m!1,

12.4. UPPER AND LOWER SOLUTIONS 123

a contradiction. Analogously we can show that xm ) ,m (1 ) m ) n ! 1). Thesame reasoning shows that, if *,, are strict, then *m < xm < ,m (1 ) m ) n! 1).

II. Solution of the modified problem.We use Brouwer degree to study the zeros of the continuous mapping G : Rn!1 "Rn!1 defined by

Gm(x) = (Lx)m ! xm + fm * .m(xm) + .m(xm) (1 ) m ) n! 1). (12.18)

As seen above, L! I : Rn!1 " Rn!1 is invertible. On the other hand the mappingwith components fm *.m +.m (1 ) m ) n!1) is bounded on Rn!1. Consequently,Theorem 11.2.1 implies the existence of R > 0 such that, for all - > R, one has

|dB[G, B(-), 0]| = 1, (12.19)

and, in particular, G has a zero x & B(-). Hence, x is a solution of (12.17), whichmeans that *m ) xm ) ,m (1 ) m ) n ! 1), and x is a solution of (12.16).Moreover if *,, are strict, then *m < xm < ,m (1 ) m ) n! 1).

Remark 12.4.1 Suppose that * (resp. ,) is a strict lower (resp. upper) solutionof (12.16). As we have already seen, (12.16) admits at least one solution x suchthat *m < xm < ,m (1 ) m ) n! 1). Define the open set

%&,' = {(x1, · · · , xn!1) & Rn!1 : *m < xm < ,m (1 ) m ) n! 1)}.

If - is large enough, using the additivity-excision property of Brouwer degree, wehave

|d[G,%&,' , 0]| = |d[G, B(-), 0]| = 1.

On the other hand, if we define the continuous mapping 1G : Rn!1 " Rn!1 by

1Gm(x) = (Lx)m + fm(xm) (1 ) m ) n! 1), (12.20)

1G is equal to G on %&,', and then

|d[ 1G,%&,', 0]| = 1. (12.21)

A simple but useful consequence of Theorem 12.4.1, goes as follows.

Corollary 12.4.1 Assume that there exists numbers * ) , such that

fm(*) ( 0 ( fm(,) (1 ) m ) n! 1).

Then problem (12.16) has at least one solution with * ) xm ) , (1 ) m ) n! 1).

Proof. Just observe that (0,*, · · · ,*, 0) is a lower solution and (0,,, · · · ,,, 0) anupper solution for (12.16).

Example 12.4.1 For each p > 0, am > 0 and bm & R (1 ) m ) n!1) the problem

#2xm!1 ! am|xm|pxm + bm = 0 (1 ) m ) n! 1), x1 = xn



12.5 Ambrosetti-Prodi type multiplicity results

In this section we are interested in problems of the type

#2xm!1 ! "1xm + fm(xm) = s&1m (1 ) m ) n! 1),

x0 = 0 = xn, (12.22)

where n ( 2 is fixed, f1, · · · , fn!1 : R " R are continuous, s & R, "1 is given in(12.10), &1 is given in (12.11) and

fm(x)"- as |x|"- (1 ) m ) n! 1). (12.23)

We prove an Ambrosetti-Prodi type result for (12.22) first proved in [20], which isreminiscent of a multiplicity theorem for second order di!erential equations withDirichlet boundary conditions given in [59].

We need an identity for finite sequences satisfying the Dirichlet boundary con-ditions, which is a type of summation by parts.

Lemma 12.5.1 If (x0, · · · , xn) & Rn+1 and (y0, · · · , yn) & Rn+1 are such that x0 =0 = xn and y0 = 0 = yn, the identity

n!1"

m=1

xmD2ym =n!1"

m=1

ym#2xm!1 (12.24)

holds.

Proof. We have

n!1"

m=1

xm(ym+1 ! 2ym + ym!1)!n!1"

m=1

ym(xm+1 ! 2xm + xm!1)

=n!2"

m=1

xmym+1 +n!1"

m=2

xmym!1 !n!2"

m=1

ymxm+1 !n!1"

m=2

ymxm!1 = 0

The next lemma provides a priori bounds for the possible solutions of (12.22).

Lemma 12.5.2 Let b & R. Then there is - = -(b) > 0 such that any possiblesolution x of (12.22) with s ) b belongs to the open ball B(-).

Proof. Let s & [a, b] and (x0, · · · , xn) & Rn+1 be a solution of (12.22), or, equivalentlylet (x1, · · · , xn!1) be a solution of

(Lx)m ! "1xm + fm(xm) = s&1m.

12.5. AMBROSETTI-PRODI TYPE MULTIPLICITY RESULTS 125

Multiplying each equation by &1m and adding, we obtain

s

:n!1"

m=1

N&1

m

O2;

=n!1"

m=1

$&1

m(Lx)m ! "1&1mxm

%+

n!1"

m=1

&1mfm(xm).

But, using Lemma 12.5.1,

n!1"

m=1

$&1

m(Lx)m ! "1&1mxm

%=

n!1"

m=1

Wxm

$(L&1)m ! "1&

1m

%X= 0,

so that

n!1"

m=1

&1mfm(xm) = s#&1#2 ) b#&1#2. (12.25)

Using (12.23), there exists c & R such that

fm(u) ( c for all u & R and all m & {1, . . . , n! 1}, (12.26)

and

(;R > 0)(: r$ > 0)(;u & R : |u| ( r$)(;m & {1, . . . , n! 1}) :

&1mfm(u) ( R! c

5

6"

k 1=m

&1k

8

9 . (12.27)

Let r =<

n! 1r$ and x & Rn!1 be such that #x# ( r. Then for at least one j onehas |xj | ( r$, so that, using (12.26) and (12.27), one obtains

n!1"

m=1

&1mfm(xm) =

n!1"

m=1

&1m[fm(xm)! c] + c

:n!1"

m=1

&1m

;

( &1j [fj(xj)! c] + c

:n!1"

m=1

&1m

;

( R ! c

5

6"

k 1=j

&1k

8

9! c&1j + c

:n!1"

m=1

&1m

;= R.

HenceBn!1

m=1 &1mfm(xm) " +- if #x# " -, and the result follows from (12.25).

Theorem 12.5.1 If the functions fm (1 ) m ) n ! 1) satisfy (12.23), then thereis s1 & R such that (12.22) has zero, at least one or at least two solutions accordingto s < s1, s = s1 or s > s1.


Proof. Let

Sj = {s & R : (12.22) has at least j solutions } (j ( 1).

(a) S1 %= !.

Take s# > max1+m+n!1fm(0)$1

mand use (12.23) to find R#

! < 0 such that

fm(R#!&1

m) > s#&1m (1 ) m ) n! 1).

Then * with *0 = 0 = *n and *j = R#!&1

j < 0 (1 ) j ) n ! 1) is a strict lowersolution and , with ,j = 0 (1 ) j ) n) is a strict upper solution for (12.22) withs = s#. Hence, using Theorem 12.4.1, s# & S1.(b) If Ts & S1 and s > Ts then s & S1.Let Tx = (Tx0, Tx1, · · · , 1xn) be a solution of (12.22) with s = Ts, and let s > Ts. Then Txis a strict upper solution for (12.22). Take now R! < min1+m+n!1

bxm$1

msuch that

fm(R!&1m) > s&1

m (1 ) m ) n ! 1). It follows that * with *0 = 0 = *n and*j = R!&1

j (1 ) j ) n! 1) is a strict lower solution for (12.22), and hence, usingTheorem 12.4.1, s & S1.(c) s1 = inf S1 is finite and S1 0 ]s1,-[.Let s & R and suppose that (12.22) has a solution (x0, x1, · · · , xn). Then (12.25)holds, from where we deduce that, with c defined in (12.26), s ( c

Bn!1m=1 &1

m. Toobtain the second part of claim (c), we apply (b).(d) S2 0 ]s1,-[.We reformulate (12.22) to apply Brouwer degree theory. Consider the continuousmapping G : R1 Rn!1 " Rn!1 defined by

Gm(s, x) = (Lx)m ! "1xm + fm(xm)! s&1m (1 ) m ) n! 1).

Then (x0, · · · , xn) is a solution of (12.22) if and only if (x1, · · · , xn!1) & Rn!1 isa zero of G(s, ·). Let s3 < s1 < s2. Using Lemma 12.5.2 we find - > 0 such thateach possible zero of G(s, ·) with s & [s3, s2] is such that max1+m+n!1 |xm| < -.Consequently, the Brouwer degree dB[G(s, ·), B(-), 0] is well defined and does notdepend upon s & [s3, s2]. However, using (c), we see that G(s3, x) %= 0 for all x &Rn!1. This implies that dB [G(s3, ·), B(-), 0] = 0, so that dB [G(s2, ·), B(-), 0] = 0and, by excision property, dB [G(s2, ·), B(-$), 0] = 0 if -$ > -. Let s & ]s1, s2[ andTx = (Tx1, · · · , Txn) be a solution of (12.22) (using (c)). Then Tx is a strict upper solutionof (12.22) with s = s2. Let R < min1+m+n

bxm$1

mbe such that fm(R&1

m) > s2&1m

(1 ) m ) n! 1). Then (0, R&11, · · · , R&1

n!1, 0) & Rn+1 is a strict lower solution of(12.22) with s = s2. Consequently, using Remark 12.4.1, (12.22) with s = s2 has asolution in %R$1,bx and

|dB[G(s2, ·),%R$1,bx, 0]| = 1.

Taking -$ su$ciently large, we deduce from the additivity property of Brouwerdegree that

|dB[G(s2, ·), B(-$)\%R$1,bx, 0]|= |dB[G(s2, ·), B(-$), 0]! dB[G(s2, ·),%R$1,bx, 0]|= |dB[G(s2, ·),%R$1,bx, 0]| = 1,


and (12.22) with s = s2 has a second solution in B(-$) \ %R$1,bx.(e) s1 & S1.Taking a decreasing sequence ($k)k"N in ]s1,-[ converging to s1, a correspondingsequence (xk

0 , xk1 , · · · , xk

n) of solutions of (12.22) with s = $k and using Lemma12.5.2, we obtain a subsequence (xjk

0 , xjk1 , · · · , xjk

n ) which converges to a solution(x0, x1, · · · , xn) of (12.22) with s = s1.

Similar arguments allow to prove the following result.

Theorem 12.5.2 If the functions fm satisfy condition

fm(x)" !- as |x|"-, (2 ) m ) n! 1). (12.28)

then there is s1 & R such that (12.22) has zero, at least one or at least two solutionsaccording to s > s1, s = s1 or s < s1.

Example 12.5.1 There exists s1 & R such that the problem

#2xm!1 ! "1xm + |xm|1/2 = s&1m (1 ) m ) n! 1), x0 = 0 = xn

has no solution if s < s1, at least one solution if s = s1 and at least two solutionsif s > s1.

Example 12.5.2 There exists s1 & R such that the problem

#2xm!1 ! "1xm ! exp x2m = s&1

m (1 ) m ) n! 1), x0 = 0 = xn

has no solution if s > s1, at least one solution if s = s1 and at least two solutionsif s < s1.

12.6 One-sided bounded nonlinearities

Let n ( 2 be a fixed integer, and fm : R" R continuous functions (2 ) m ) n!1).Consider the problem

#2xm!1 ! "1xm + fm(xm) = 0 (1 ) m ) n! 1), x0 = 0 = xn.(12.29)

We define the continuous mapping G : Rn!1 " Rn!1 by

Gm(x) = (Lx)m ! "1xm + fm(xm) (1 ) m ) n! 1), (12.30)

where L is given in (12.3), so that (0, x1, · · · , xn!1, 0) is a solution of (12.29) if andonly if (x1, · · · , xn!1) & Rn!1 is a zero of G. We also define F : Rn!1 " Rn!1 by

F (x) = (f1(x1), · · · , fn!1(xn!1)),

and set L1 := L! "1I. We have

N(L1) = {c&1 : c & R},


and, by the properties of symmetric matrices,

R(L1) = {y & Rn!1 : 6y,&17 = 0}.

Consider the projector P : Rn!1 " Rn!1 defined by

P (x) = 6x,&17 &1

#&1#2 ,

so that N(P ) = R(L1), R(P ) = N(L1).

We will need the following inequality. For x = (x1, · · · , xn!1), define

x := 6x,&17 &1

||&1||2 = Px, 1x = x! x, (12.31)

so that

xm =

En!1"

m=1

xm sinm%

n

F&1

m

||&1||2 (1 ) m ) n! 1), 61x,&17 = 0.

Notice that, with L defined in (12.3) or (12.4),

(Lx)m ! "1xm = (Lx)m ! "1xm + (L1x)m ! "11xm

= (L1x)m ! "11xm. (12.32)

Lemma 12.6.1 If x = (x1, · · · , xn!1), then there exists a constant cn > 0, whichdepends only on n, such that

max1+m+n!1

|1xm| ) cn

:n!1"

m=1

|(L1x)m ! "11xm|&1m

;. (12.33)

Proof. The application

(x0, · · · , xn) 2"n!1"

m=1

|(Lx)m ! "1xm|&1m

defines a norm on the subspace V = {x & Rn!1 : 6x,&17 = 0}, because L ! "1I :V " V is one-to-one. It is therefore equivalent to the norm max1+m+n!1 |xm|, andinequality (12.33) holds.

We now obtain a priori estimates for the possible zeros of L1+"F when " & ]0, 1].

Lemma 12.6.2 If all functions fm (1 ) m ) n ! 1) are bounded from below orare bounded from above, and if, for some R > 0, one has

n!1"

m=1

fm(xm)&1m %= 0 whenever min

1+j+n!1xj ( R or max

1+j+n!1xj ) !R(12.34)

then there exists - > R such that each possible zero (", x) & ]0, 1]1Rn!1 of L1 +"Fis such that #x# < -.


Proof. Assume first that each fm is bounded from below by c. Let (", x) & [0, 1]1Rn!1 be such that L1x + "F (x) = 0. Applying P and (I ! P ) to the equation, weobtain

PF (x) = 0, L1x + "(I !Q)F (x) = 0,

or, equivalently

n!1"

m=1

fm(xm)&1m = 0, (12.35)

(L1x)m ! "11xm + "fm(xm) = 0 (1 ) m ) n! 1). (12.36)

We can then repeat the reasoning of the proof of Lemma 12.5.2 to obtain that

n!1"

m=1

|fm(xm)|&1m ) (|c|! c)

n!1"

m=1

&1m := R1.

Hence

n!1"

m=1

|(L1x)m ! "11xm|&1m = "

n!1"

m=1

|fm(xm)|&1m ) R1.

Consequently, using Lemma 12.6.1, we obtain

max1+m+n!1

|(1x)m| ) cnR1 := R2.

Then, by (12.34) and (12.35), there exists 1 ) k ) n ! 1 and 1 ) l ) n ! 1such that xk < R and xl > !R. Consequently, xk = xk ! 1xk < R + R2 andxl = xl ! 1xl > !R!R2. Therefore, for each 1 ) m ) n! 1,

xm =xk

&1k

&1m < (R + R2) max

1+m+n!1

&1m

&1k

:= R3,

(x)m =(x)l

&1l

&1m > !(R + R2) max

1+m+n!1

&1m

&1k

:= !R3.

Consequently #x# < - for some - > 0.In the case where the fm are bounded from above, if su$ces to write the problem

1L1x + 1F (x) = 0,

with 1L1 = !L1 and 1F = !F to reduce it to a problem with 1F bounded from below,noticing that L1 and 1L1 have the same kernel and the same range.


Let & : R" R be the continuous function defined by

&(u) =n!1"

m=1

fm(u&1m)&1

m. (12.37)

The following theorem is taken from [20].

Theorem 12.6.1 If the functions fm (1 ) m ) n ! 1) satisfy the conditions ofLemma 12.6.2 and if

&(!R)&(R) < 0, (12.38)

then, problem (12.29) has at least one solution.

Proof. The first assumption of Theorem 8.6.1 applied to L1 + F is satisfied forD = B(-)1 [0, 1]. Now,

PF (u&1) = &(u)&1 (u & R).

Hence the second assumption of Theorem 8.6.1 is satisfied, and, furthermore,

|dB [QF, B(-) .N(L), 0]| = |dB[&, ]! -, -[, 0]| = 1,

so that the first assumption of Theorem 8.6.1 holds as well.

Example 12.6.1 The problem

#2xm!1 ! "1xm + exp xm ! tm = 0 (1 ) m ) n! 1), x0 = 0 = xn,

has at least one solution if and only ifBn!1

m=1 tm&1m > 0.

The necessity follows from summing both members of the equation from 1 ton ! 1 after multiplication by &1

m, and the su$ciency from Theorem 12.6.1, ifwe observe that there exists R > 0 such that the function & defined by &(u) =Bn!1

m=1

$exp(u&1

m)! tm%&1

m is such that &(u) > 0 for u ( R and &(u) < 0 foru ) !R.

Example 12.6.2 If g : R " R is a continuous function bounded from below orfrom above and (t1, · · · , tn!1) & Rn!1 such that

!- < lim supu(!%

g(u) <

Bn!1m=1 tm&1

mBn!1m=1 &1

m

< lim infu(+%

g(u) < +-, (12.39)

then the problem

#2xm!1 ! "1xm + g(xm)! tm = 0 (1 ) m ) n! 1), x0 = 0 = xn,

has at least one solution.Condition (12.39) is a Landesman-Lazer-type condition for di!erence equations. Itis easily shown to be necessary if lim supu(!% g(u) < g(v) < lim infu(+% g(u) forall v & R.

Chapter 13

Bifurcation

13.1 Eigenvalues of couples of linear mappings

Let (L, A) be a couple of linear mappings L : Rn " Rn and A : Rn " Rn. Thefollowing results and concepts have been introduced in [240].

Definition 13.1.1 µ & C is an eigenvalue for (L, A) is N(L! µA) %= {0}.

When A = I, an eigenvalue for (L, I) is nothing but a classical eigenvalue of L.When N(L) . N(A) %= {0}, any µ & C is an eigenvalue for (L, 0). In order to beable to define the multiplicity of an eigenvalue for (L, A), we make the followingassumption of non-degeneracy :

(ND) : The set $(L, A) of eigenvalues for (L, A) is not C.

Definition 13.1.2 Assume that (L, A) satisfies condition (ND) and let µ0 & C \$(L, A). The spectral operator Aµ0 associated to (L, A) and µ0 is the linear operator

A0 := (L! µ0A)!1A.

Proposition 13.1.1 Assume that condition (ND) holds and let µ0 & C \ $(L, A).Then µ & C is an eigenvalue for (L, A) if and only µ ! µ0 is a (classical) charac-teristic value of A0, i.e. if and only if N(I ! (µ! µ0)A0) %= {0}.

Proof. We have

L! µA = L! µ0A! (µ! µ0)A = (L! µ0A)[I ! (µ! µ0)A0] (13.1)

and, as L!µ0A : Rn " Rn is an isomorphism (being one-to-one), N(L!µA) %= {0}if and only if N(I ! (µ! µ0)A0) %= {0}, i.e. if and only if µ! µ0 is a characteristicvalue of A0.

131

132 CHAPTER 13. BIFURCATION

We now show that the (algebraic) multiplicity of a characteristic value of A0

does not depend upon the choice of µ0 & C \ $(L, A). Recall that the algebraicmultiplicity (B(4) of a characteristic value 4 of a linear mapping B : Rn " Rn isthe dimension of N [(I!4B)n

0 ], where n0 is the smallest nonnegative integer n suchthat

N [(I ! 4B)n+1] = N [(I ! 4B)n].

In other words,

(B(4) = dimQ

n/1

N [(I ! 4B)n].

Lemma 13.1.1 If assumption (ND) holds, and if µ1 and µ2 are in C \ $(L, A),then

[I ! (µ1 ! µ2)A2][I ! (µ! µ1)A1] = I ! (µ! µ2)A2 (13.2)

Proof. We have

I ! (µ! µ1)A1 = I ! (µ! µ1)(L ! µ1)!1A

= I ! (µ! µ1)[L! µ2A! (µ1 ! µ2)A]!1A

= I ! (µ! µ1)[I ! (µ1 ! µ2)(L ! µ2A)!1A]!1(L ! µ2A)!1A

= I ! (µ! µ1)[I ! (µ1 ! µ2)A2]!1A2,

and hence

(I ! (µ1 ! µ2)A2)(I ! (µ! µ1)A1)

= (I ! (µ1 ! µ2)A2){I ! (µ! µ1)[I ! (µ1 ! µ2)A2]!1A2}

= I ! (µ1 ! µ2)A2 ! (µ! µ1)A2 = I ! (µ! µ2)A2.

Proposition 13.1.2 If assumption (ND) holds, µ0 and µ1 belong to C \ $(L, A),and if µ & $(L, A), then

(A0(µ! µ0) = (A1(µ! µ1).

Proof. We have, using (13.2) and the trivial commutation identity

[I ! (µ1 ! µ2)A2]!1[I ! (µ! µ2)A2] = [I ! (µ! µ2)A2][I ! (µ1 ! µ2)A2]

!1

[I ! (µ! µ1)A1]n = {[I ! (µ1 ! µ2)A2]

!1[I ! (µ! µ2)A2]}n

= [I ! (µ1 ! µ2)A2]!n[I ! (µ! µ2)A2]

n.

Hence, for any positive integer n,

N [I ! (µ! µ1)A1]n = N [I ! (µ! µ2)A2]

n,


13.1. EIGENVALUES OF COUPLES OF LINEAR MAPPINGS 133

Proposition 13.1.2 justifies the following definition, given in [240].

Definition 13.1.3 If Assumption (ND) holds, the multiplicity m(µ) of the eigen-value µ of (L, A) is defined by

m(µ) = (A0(µ! µ0) (13.3)

where (A0(µ ! µ0) is the (algebraic) multiplicity of the characteristic value µ ! µ0

of A0 = (L ! µ0A)!1A for some µ0 & C \ $(L, A).

When L = I, one can take µ0 = 0 and recover the multiplicity of the characteristicvalue µ of A. When A = I, there always exists µ0 & C \ $(L, I) = C \ $(L) and theidentity

(L ! µ0I)!1(L ! µI) = (L! µI)(L! µ0I)!1 = I ! (µ! µ0)(L ! µ0I)!1

easily implies that, for all integers n ( 1,

[I ! (µ! µ0)(L! µ0I)!1]n = (L! µ0I)!n(L! µ0I)n,

and hence m(µ) reduces to the usual (algebraic) multiplicity of the eigenvalue µ ofL.

We now give two special cases in which ,(µ) can be easily computed. The firstone can be found in [240].

Proposition 13.1.3 If Assumption (ND) holds and if µ & $(L, A), then

m(µ) = dim N(L! µA) (13.4)

is and only if

A[N(L ! µA)] .R[(L! µA)] = {0}. (13.5)

Proof. Let µ0 & C \ $(L, A). We know that

N(L! µA) = N [I ! (µ! µ0)A0]

and hence, using classical results of linear algebra,

m(µ) = dim N(L! µA) 4 N [I ! (µ! µ0)A0] .R[I ! (µ! µ0)A0] = {0}4 N(L! µA) .R[(L! µ0I)!1(L! µA)] = {0}4 (L ! µ0A)[N(L ! µA)] .R(L! µA) = {0}= (µ! µ0)A[N(L! µA)] .R(L! µA) = {0},



The following result is essentially due to Laloux [238].

Proposition 13.1.4 Assume that Assumption (ND) holds and that LA = AL.Then, for any µ & $(L, A), one has

m(µ) = dimQ

n/1

N(L! µA)n. (13.6)

Proof. If follows immediately from LA = AL that, for any µ, ( & C one has

(L! µA)(L ! (A) = (L! (A)(L ! µA),

and hence, if µ0 & C \ $(L, A),

(L! µ0A)!1(L! µA) = (L! µA)(L ! µ0A)!1.

Consequently, for any positive integer n,

[I ! (µ! µ0)A0]n = [(L! µ0A)!1(L! µA)]n = (L! µ0)

!n(L ! µA)n.

Consequently,

dim N [I ! (µ! µ0)A0]n = dim N [(L! µA)n],

for any integer n ( 1, and (13.6) follows.

13.2 A Leray-Schauder formula for Brouwer index

Let L, A : Rn " Rn be linear mappings such that Assumption (ND) holds, withµ0 & C \ $(L, A). This implies that the set of eigenvalues $(L, A) is equal, up to atranslation, to the set of characteristic values of A0 = (L! µ0A)!1A, and hence isfinite. Hence, we can assume, without loss of generality, that µ0 & R \ $(L, A). Inparticular, the Brouwer index iL(L! µ0A, 0) is well defined.

Lemma 13.2.1 If B : Rn " Rn is linear and I !B invertible, then

iB[I !B, 0] = (!1)m (13.7)

where m is the sum the multiplicities of the characteristic values of B contained in]0, 1[ .

Proof. It follows from formula 5.3 that

iB[I !B, 0] = (!1)m"

where m$ is the sum of the multiplicities of the negative characteristic values ofI !B. Now, as

det[I ! "(I !B)] = det[(1! ")I%B ] = (1! ")n det

-I ! "

"! 1B

.,

" is an characteristic value of I !B if and only if %1!% is a characteristic value of B

and %1!% < 0 if and only if " & ]0, 1[ . So, m$ = m.

13.2. A LERAY-SCHAUDER FORMULA FOR BROUWER INDEX 135

The following result is essentially due to Leray and Schauder [251].

Theorem 13.2.1 Assume that condition (ND) holds for (L, A) and let µ1 < µ2 beelements of R \ $(L, A). then

iB[L! µ2A, 0] = (!1)miB[L! µ1A, 0], (13.8)

where m is the sum of the multiplicities of the eigenvalues of (L, A) contained in]µ1, µ2[ .

Proof. From formula (13.1), we have

L! µ2A = (L ! µ1A)[I ! (µ2 ! µ1)A1]

where all mappings are isomorphisms, and hence

iB[L! µ2A, 0] = iB[L! µ1A, 0] · iB[I ! (µ2 ! µ1)A1, 0].

On the other hand, using (13.7),

iB[I ! (µ2 ! µ1)A1, 0] = (!1)m"

where m$ is the sum of the multiplicities of the characteristic values of (µ2! µ1A1)contained in ]0, 1[ and hence the sum of the multiplicities of the characteristic valuesof A1 contained in ]0, µ2!µ1[. But the eigenvalues of (L, A) are obtained by addingµ1 to the characteristic values of A1, and their multiplicity is the multiplicity of thecorresponding characteristic value of A1. Hence, m$ is the sum m of the multiplicitiesof the eigenvalues of (L, A) contained in ]µ1, µ2[ .

Corollary 13.2.1 Assume that Assumption (ND) holds for (L, A) and let ( &$(L, A) . R. If iB[L ! ((!)A, 0] (resp. iB[L ! ((+)A, 0]) denotes the value ofiB[L ! (( ! ')A, 0] (resp. iB[L ! (( + ')A, 0]) for ' > 0 such that [( ! ', ( + '] .$(L, A) = {(}, then

iB[L! ((+)A, 0] = (!1)m(-)iL[L! ((!)A, 0] (13.9)

where m(() denotes the multiplicity of (.

Proof. Recall that $(L, A) is finite and hence any eigenvalue is isolated, which in-sures the existence of ' above. On the other hand by Theorem 13.2.1, iB[L! (( !')A, 0] and iB[L!((+')A, 0] are independent of ' as long as [(!', (+'].$(L, A) ={(}, and the result follows from formula (13.8).


13.3 Bifurcation points

Let U ' R be an open nonempty interval and V ' Rn an open neighborhood of 0,and F : U 1 V " Rn be continuous and such that

F (µ, 0) = 0 for all µ & U. (13.10)

Consequently, equation

F (µ, x) = 0 (13.11)

has, for any µ & U, the trivial solution x = 0. We are interested in finding values ofµ for which a nontrivial solution exists, i.e. in the bifurcations of the solution set.

The following definition can be traced to H. Poincare in his studies of the equi-librium of rotating fluid masses [322], where he used for the first time Kronecker’sindex in bifurcation theory.

Definition 13.3.1 The point (µ0, 0) & U1V is a bifurcation point for the solutionsof (13.11) with respect to U 1 {0} (briefly, a bifurcation point for (13.11)) if anyneighborhood of (µ0, 0) in U 1 V contains at least one zero (µ, x) of F such thatx %= 0.

Remark 13.3.1 If we take F (µ, x) = Lx! µAx where L, A : Rn " Rn are linear,we see at once that any (µ0, 0), where µ0 is an eigenvalue for (L, A), is a bifurcationpoint of Lx!µAx = 0. That is why bifurcation problems are sometimes also callednonlinear eigenvalue problems.

The following preliminary result is used in bifurcation theory.

Lemma 13.3.1 Let K ' U be a nonempty compact interval such that K 1 {0}Kcontains no bifurcation point for (13.11). Then there exists r > 0 such that for any(µ, x) & K 1 (V .B(r)) satisfying (13.11), one has x = 0.

Proof. If it is not the case, there exists a sequence (µn, xn) contained in K 1 (V .B(1/n)) such that

F (µn, xn) = 0 and xn %= 0 (n = 1, 2, . . .).

Going if necessary to a subsequence, we may assume that µn " µ0 & K and hence(µ0, 0) is a bifurcation point for (13.11), a contradiction.

The following fundamental su$cient condition for the existence of a bifurcationpoint is essentially due to M.A. Krasnosel’skii [210, 212, 213]. It show how Brouwer’sindex allows to detect bifurcation points.

Theorem 13.3.1 Assume that µ1 < µ2 are two points of U such that iB[F (µ1, ·), 0]and iB[F (µ2, ·), 0] are defined and di"erent. Then there exists µ0 & [µ1, µ2] suchthat (µ0, 0) is a bifurcation point for (13.11).

13.4. BIFURCATION THROUGH LINEARIZATION 137

Proof. We prove the contrapositive form : if (13.11) has no bifurcation point on[µ1, µ2]1 {0}, then

iB[F (·µ1, ·), 0] = iB[F (µ2, ·), 0].

From Lemma 13.3.1 with K = [µ1, µ2], there exists r > 0 such that B(r) ' V and

F (µ, 0) = 0, (µ, x) & [µ1, µ2]1B(r)8 x = 0.

Define the homotopy F : B(r) 1 [0, 1]" Rn by

F(x,") = F (x, (1! ")µ1 + "µ2)

so that, for any " & [0, 1] and any (x") & !B(r) 1 [0, 1], one has F(x,") %= 0. Thehomotopy invariance property 3.4.2 and the definition of the Brouwer index implythat

iB[F (µ1, ·), 0] = iB[F(·, 0)] = iB[F(·, 1)]

= iB[F (·µ2, ·), 0].

Remark 13.3.2 Altough quite general and widely applied, the above su$cientcondition is not necessary, in the class of continuous mappings, for the existence of abifurcation point. In fact, if F (µ, x) = x!µx with x & R2, ((0, 0), 1) is a bifurcationpoint. On the other hand, for any µ %= 1, iB[F (µ, ·), 0] = sign (1! µ)2 = 1.

13.4 Bifurcation through linearization

Assume now that F can be written as follows

F (µ, x) = Lx! µAx + R(µ, x) (13.12)

where L, A : Rn " Rn are linear and R : U 1 V " Rn is continuous and such that

limx(0

R(µ, x)

#x# = 0 (13.13)

uniformly on compact subintervals of V.We first prove a necessary conditions for the existence of a bifurcation point

(µ0, 0) for equation

Lx! µAx + R(µ, x) = 0. (13.14)

Proposition 13.4.1 If (µ0, 0) is a bifurcation point for equation (13.14) with Rsatisfying (13.13), then µ0 is an eigenvalue for (L, A).


Proof. We prove the contrapositive proposition. Let µ0 & U \ $(L, A) Then

#Lx! µAx + R(µ, x)# = #(L! µ0A)x + (µ0 ! µ)Ax + R(µ, x)#( #(L! µ0A)x# ! |µ0 ! µ|#A##x# ! #R(µ, x)#.

By our assumptions, there exists * > 0 such that

#(L! µ0A)x# ( *#x# for all x & Rn.

Consequently, if we first take '0 > 0 such that [µ0 ! '0, µ0 + '0] ' U and

'0#A# )*

3,

and then r > 0 such that

#R(µ, x)# ) *

3#x# for all (µ, x) & [µ0 ! '0, µ0 + '0]1B(r),

we get, for (µ, x) & [µ0 ! '0, µ0 + '0]1B(r),

#Lx! µAx + R(µ, x)# ( *#x# ! *

3#x# ! *

3#x# =

*

3#x#

and (µ0, 0) is not a bifurcation point for (13.14).

Remark 13.4.1 The condition of Proposition 13.4.1 is not su$cient for (µ0, 0) tobe a bifurcation point of (13.14). Indeed, taking L = A = I in R2, and R(x1, x2) =(!x3

2, x31), we see that 1 is an eigenvalue for (I, I) but the system of equations

(1! µ)x1 ! x32 = 0, (1! µ)x2 + x3

1 = 0 (13.15)

does not admit (1, (0, 0)) as a bifurcation point, because it has no bifurcation point.Indeed, if (x1, x2, µ) is a solution of (13.15), then, multiplying the first equationby !x1, the second one by x2 and adding, we obtain x4

2 + x41 = 0, and hence

x1 = x2 = 0.

We prove now a su$cient condition for the existence of a bifurcation point for(13.14) essentially due to M.A. Krasnosel’skii [210, 212, 213].

Theorem 13.4.1 If condition (ND) holds for (L, A) and condition (13.13) for R,and if µ0 & U is an eigenvalue for (L, A) having odd multiplicity m(µ0), then (µ0, 0)is a bifurcation point for (13.14). Moreover, there exists r0 > 0 such that, for anyr & ]0, r0], equation (13.14) has at least one solution (µr , xr) such that #xr# = rand µr " 0 if r " 0.

Proof. The eigenvalue µ0 being isolated, there exists '0 > 0 such that [µ0! '0, µ0 +'0] ' U and [µ0 ! '0, µ0 + '0] . $(L, A) = {µ0}. Consequently,

N [L! (µ0 ! '0)A] = {0} = N [L! (µ0 + '0)A].

13.4. BIFURCATION THROUGH LINEARIZATION 139

Therefore, using Theorem 5.2.1 and Corollary 13.2.1, we obtain

iB[L! (µ0 + '0)A + R(µ0 + '0, ·), 0] = iB[L! (µ0 + '0)A, 0]

= (!1)m(µ0)iB[L! (µ0 ! '0)A, 0] = !iB[L! (µ0 ! '0)A, 0]

= !iB[L! (µ0 + '0)A + R(µ0 + '0, ·), 0] (13.16)

with all those indices having absolue value one. Hence all conditions of Theorem13.3.1 hold, and (0, µ0) is a bifurcation point for (13.14).

Now, let r0 > 0 be such that B(r0) ' V and such that x = 0 is the uniquesolution of the equations

Lx! (µ0 ± '0)Ax + R(µ0 ± '0, x) = 0

located in B(r0). Such a r0 exists by Lemma 13.3.1. Thus, if r & ]0, r0], we have,using (13.16),

dB[L! (µ0 ! '0)A + R(µ0 ! '0, ·), B(r), 0]

= iL[L! (µ0 ! '0)A + R(µ0 ! '0, ·), 0]

= !iL[L! (µ0 + '0)A + R(µ0 + '0, ·), 0]

= !dB[L! (µ0 ! '0)A + R(µ0 ! '0, ·), B(r), 0]

with all numbers having absolute value 1, so that the two degrees are unequal. Fromthe contrapositive version of the homotopy invariance property 3.4.2, there mustexist µr & [µ0 ! '0, µ0 + '0] and xr & !B(r) such that

Lxr ! µrAxr + R(µr, xr) = 0.

Consequently, letting yr = xr)xr) , we have

Lyr ! µrAyr + #xr#!1R(µr, xr) = 0. (13.17)

Hence, we can find a convergent subsequence (µrk , yrk) with limit (µ#, y#) & [µ0 !'0, µ0 + '0]1 !B(1) when rk " 0. From (13.17), we obtain

Ly# ! µ#Ay# = 0,

i.e. µ# & [µ0 ! '0, µ0 + '0] is an eigenvalue for (L, A). From our construction, thisimplies that µ# = µ0. Thus, any convergent subsequence of (µr) converges to µ0,and the proof is complete.

Remark 13.4.2 The proof of Theorem 13.4.1 also shows that every convergentsubsequence of (yr) has for limit an element of N(L! µ0A).

A direct consequence of Theorem 13.4.1, of Corollary 13.2.1 and of 13.1.3 is thefollowing

Corollary 13.4.1 If µ0 is an eigenvalue for (L, A) such that


(i) A(N(L ! µ0A)) .R(L! µ0A) = {0}

(i) dim N(L! µ0A) is odd,

then the conclusion of Theorem 13.4.1 holds.

A direct consequence of Theorem 13.4.1, of Corollary 13.2.1 and of 13.1.4 is thefollowing

Corollary 13.4.2 If LA = AL and µ0 is an eigenvalue for (L, A) such that

dimQ

n/1

N [(L! µ0A)n] is odd,

then the conclusion of Theorem 13.4.1 holds.

13.5 Global structure of bifurcation branches

The results of the previous sections give conditions to detect bifurcation points andsome information about the set of solutions in the neighborhood of the correspond-ing bifurcation point. We now study the global structure, first obtained in 1971 byP. Rabinowitz [324, 325], following an approach introduced in 1976 by J. Ize [181](see also [297]). We keep the notations and assumptions of the previous sections,and consider the problem (13.14) with µ0 be an eigenvalue for (L, A) with oddmultiplicity.

We need a preliminary lemma. Let 1X = 1X 1 R, with the norm #(x, ()# =(#x#2 + |(|2)1/2, and define, for any r > 0, the continuous mapping 1Fr : U 1 (V !µ0)" 1X by

1Fr(x, () = (F (µ0 + (, x), #x#2 ! r2).

Lemma 13.5.1 There exists (0 > 0 and r0 > 0 such that, for r & ]0, r0],

dB [ 1Fr, B([r2 + (20 ]1/2), 0] = iB[L! (µ0!)A, 0]! iB[L! (µ0+)A, 0].

Proof. Let (0 > 0 be such that [µ0!(0, µ0+(0] ' V and [µ0!(0, µ0+(0].$(L, A) ={µ0}. By proceeding like in the proof of Proposition 13.4.1, there exists r0 > 0 suchthat B(r0) ' U and such that, for any " & [0, 1], the only solution of the equations

Lx! (µ0 ± (0)Ax + R(µ0, x) = 0

such that #x# ) r0 is x = 0. For any r & ]0, r0], let us define the continuous mapping1Fr : V 1 (v ! µ0)1 [0, 1]" 1X by

1Fr(x, (,") = (Lx! (µ0 + ()A + "R(µ0 + (, x),"(#x#2 ! r2) + (1! ")((20 ! (2)).

If 1Fr(x, (,") = 0 for some " & [0, 1] and (x, () & !B([r2 + (20 ]1/2), we have

#x#2 ! r2 = (20 ! (2, "(#x#2 ! r2) + (1! ")((2

0 ! (2),

13.5. GLOBAL STRUCTURE OF BIFURCATION BRANCHES 141

and hence

#x# = r and ( = ±(0,

which is impossible by the choice of (0 and r0. Hence the homotopy invarianceproperty of degree 3.4.2 implies that, for any r & ]0, r0],

dB [ 1Fr, B([r2 + (20 ]1/2), 0] = dB [ 1Fr(·, ·, 1), B([r2 + (2

0 ]1/2), 0]

= dB [ 1Fr(·, ·, 0), B([r2 + (20 ]1/2), 0]. (13.18)

Now 1Fr(x, (, 0) = (Lx! (µ0 + ()A, (20 ! (2) has only the two isolated zeros (0,!(0)

and (0, (0), which, together with Proposition 5.1.1, implies that

dB[ 1Fr(·, ·, 0), B([r2 + (20 ]1/2), 0]

= iB[ 1Fr(·, ·, 0), (0,!(0)] + iB[ 1Fr(·, ·, 0), (0, (0)] (13.19)

= iL[ 1Fr(·, ·! (0, 0), (0, 0)] + iB[ 1Fr(·, · + (0, 0), (0, 0)].

Now,

1Fr(x( ! (0, 0) = (Lx! (µ0 ! (0 + ()Ax, 2(0( ! (2)

and hence, by Theorem 5.2.1, we have

iB[ 1Fr(·, · + (0, 0), (0, 0)] = iB[ 1G, (0, 0)] (13.20)

where 1G : 1X " 1X is defined by

1G(x, () = (Lx! (µ0 ! (0)Ax, 2(0().

Defining now 1G : 1X 1 [0, 1]" 1X by

1G(x, (,") = (Lx! (µ0 ! (0)Ax, (2(0(1! ") + ")()),

and noticing that, for any " & [0, 1], the only zero of 1G(·, ·,") is (0, 0), we obtain,by the homotopy invariance property 3.4.2,

iB[ 1G, (0, 0)] = iB[1G(·, ·, 1), (0, 0)] (13.21)

with 1G(x, (, 1) = (Lx ! (µ0 ! (0)Ax, (). From the definition of degree, we obtain,where I is identity on R,

iB[1G(·, ·, 1), (0, 0)] = iB[L! (µ0 ! (0)A, 0] · iB[I, 0] = iB[L! (µ0 ! (0)A, 0].(13.22)

Using relations (13.18) to (13.22), and proceeding similarly for the second term in(13.19), we finish the proof.


We can now state and prove Rabinowitz’ result on the global structure of thebifurcation branch emanating from (0, µ0). Let us denote by S the closure of theset

{(µ, x) & U 1 V : x %= 0 and F (µ, x) = 0}

and by C the connected component of S containing (0, µ0, 0). The global bifurcationtheorem provides information about the structure of C.

Theorem 13.5.1 Let µ0 be an eigenvalue for (L, A) with odd multiplicity. Theneither

(i) C is not compact in U 1 V (in case U 1 V = R 1 Rn, this means that C isunbounded in R1 Rn), or

(ii) C contains a finite number of points (µj , 0) with µj & U an eigenvalue for(L, A) (j = 1, 2, . . . , m). Furthermore, the number of those eigenvalues µj

having odd multiplicity is even.

Proof. Suppose C is compact in U1V. The set of eigenvalues for (L, A) being finite,C contains a finite number of points of the form (µj , 0) (j = 0, 1, . . . , m) with µj aneigenvalue for (L, A). C being a connected compact subset of U 1V, Corollary 7.2.1implies the existence of a bounded set D, open in U1V (and hence in R1Rn) suchthat, on !D, F has no zero (µ, x) with x %= 0, and such that D contains no otherpoint of the form (µ, 0), with µ an eigenvalue for (L, A), than the (µj , 0). For anyr > 0, define the continuous mapping Fr : D " Rn 1R by

Fr(x, µ) = (F (µ, x), #x#2 ! r2),

Since Fr(x, µ) = 0 implies #x# = r, dB [Fr,D, 0] is well defined. Furthermore, given0 < r1 < r2, let us introduce the homotopy F : D 1 [0, 1]" Rn 1 R by

F(x, µ,") = (F (µ, x), #x#2 ! (1! ")r21 ! "r2

2).

Again, F(x, µ,") %= 0 for any (x, µ,") & !D 1 [0, 1], and, using the homotopyinvariance property, we obtain

dB [Fr1 ,D, 0] = dB[F (0, ·, ·),D, 0] = dB[F (0, ·, ·),D, 1] = dB [Fr2 ,D, 0],

i.e. dB[Fr,D, 0] is independent of r. But, if we take r > 0 such that D ' B(r)1R,then Fr has no zero in D, which implies, by the existence property 3.4.1 that

dB[Fr ,D, 0] = 0 for any r > 0. (13.23)

Now, by Proposition 13.4.1 and Lemma 13.3.1, for any ' > 0, there exists r*>0 suchthat µ & U \

Hmj=0[µj ! ', µj + '], x & D . B(r) and F (µ, x) = 0 imply x = 0.

Consequently, if (x, µ) is a zero of Fr such that 0 < r ) r*, then µ & [µj ! ', µj + ']for some j & {0, 1, . . . , m}, so that, using the additivity-excision property of degree,

13.6. AN APPLICATION TO DIFFERENCE EQUATIONS 143

the definition of Brouwer’s index, and Lemma 13.5.1, we obtain, by taking ' and rsu$ciently small

0 = dB[Fr,D, 0] =m"

j=0

{iB[L! (µj!)A, 0]! iB[L! (µj+)A, 0]}.

Since, by Corollary 13.2.1, one has

iB[L! (µj+)A, 0] = (!1)m(µj)iB[L! (µj!)A, 0],

the only contributions in the sum above come from the µj having an odd multiplicity,there must be an even number of them.

Example 13.5.1 The example of F : R1 R" R given by

F (µ, x) = x! µx + x2

shows that the first alternative of Theorem 13.5.1 may occur, with the unboundedbranch x = µ! 1 emanating from the simple eigenvalue µ = 1. The example, givenin [149], of F : R1 R2 " R2 given by

F (µ, x1, x2) = (x1 ! µx1 + x32, 2x2 ! µx2 ! x3

1),

for which µ1 = 1 and µ2 = 2 are simple eigenvalues and which has a pair of non-trivial solutions given by

(x1, x2) =?±(µ! 1)1/8(2! µ)3/8,±(2! µ)1/8(µ! 1)3/8

@(µ & [1, 2])

shows that the alternative (ii) may occur. An appropriate combination of the ex-amples shows that (i) and (ii) can occur simultaneously.

13.6 An application to di!erence equations

Keeping the notations of Chapter 12, let us consider the two-point boundary valueproblem

#2xm!1 ! µxm + fm(µ, xm) = 0 (m = 1, . . . , n! 1), x0 = 0 = xn, (13.24)

where n ( 2 is a fixed integer and each fm : R2 " R is continuous (m = 1, . . . , n!1)and such that

limy(0

fm(µ, u)

|y| = 0 (13.25)

uniformly on compact µ-intervals. As shown in Chapter 12, this problem can bewritten as the equation in V n!1 5 Rn!1

Lx! µx + f(µ, x) = 0 (13.26)


where L is the Jacobi matrix given in (12.4), x = (x1, . . . , xn!1) and f(µ, x) =(f1(µ, x1), . . . , fn!1(µ, xn!1)).

As shown in Lemma 12.2.1, the linear problem

Lx! µx = 0 (13.27)

has nontrivial solutions if and only if

µ = µk := 2 cosk%

n! 2 (k = 1, . . . , n! 1)

and the corresponding eigenvectors vk are multiple of

&k :=

2sin

k%

n, sin

2k%

n, . . . , sin

(n! 1)k%

n

3.

and the sequence of components of each &k changes sign exactly k ! 1 times. Sodefine

S+k = {v = (v1, . . . , vk) & V n!1 : v1 > 0 and (v1, . . . , vk) changes sign k ! 1 times},(13.28)

S!k = !S+

k , Sk = S+k / S!

k (k = 1, . . . , n! 1).(13.29)

It is clear that S+k , S+

! and Sk are open in V n!1, that *Sk = Sk for any * & R\{0},and that any v & Sk, provides a unique w & S+

k such that #w# = 1.Let S denote the closure of the set of (µ, x) such that Lx!µx+ f(µ, x) = 0 and

x %= 0.

Lemma 13.6.1 If µk & $(L), there exists a neighborhood Nk of (µk, 0) in R1V n!1

such that if (µ, x) & S .Nk, then x & Sk.

Proof. If it were not the case, there would exist a sequence (*j , xj) in S convergingto (µk, 0) and such that xj %& Sk. Then,

L

2xj

#xj#

3! *j

xj

#xj#+

f(*j , xj)

#xj#= 0 (j & N), (13.30)

and, going if necessary to a subsequence, we can assume that ( xj

)xj) ) converges to

some v with #v# = 1. Consequently, going to the limit in (13.30) and using (13.25),we obtain

Lv ! µkv = 0, #v# = 1,

and hence v & Sk. Consequently, there exists j0 such that xj

)xj) & Sk for j ( j0,

which implies that xj & Sk for j ( j0, a contradiction.

We can now describe the structure of the bifurcation branches of (13.26).

Theorem 13.6.1 If condition (13.25) holds, then, for each k & {1, . . . , n ! 1},equation (13.26) possesses a connected component of solutions Ck in R1V n!1 withCk ' (R1 Sk) / {(µk, 0), and Ck is unbounded in R1 V n!1.

13.6. AN APPLICATION TO DIFFERENCE EQUATIONS 145

Proof. First notice that each eigenvalue µk has multiplicity one. Hence, for eachk = 1, 2, . . . , n! 1, Theorem 13.5.1 implies the existence of a connected componentCk of S satisfying the alternative of the conclusion of this theorem. By Lemma13.6.1, there exists a neighborhood Nk of (µk, 0) such that

Ck .Nk ' (R1 Sk) / {(µk, 0)}. (13.31)

If

Ck ' (R1 Sk) / {µk, 0}, (13.32)

then, by Theorem 13.5.1, it is either unbounded in this set, or it contains (µj , 0)for some j %= k. But, by Lemma 13.6.1 again, any nontrivial solution of (13.26)near (µj , 0) lies in R 1 Sj . Since Sj . Sk = , when j %= k, the validity of (13.32)shows that Ck cannot contain (µj , 0) for j %= k, and hence must be unbounded in(R1 Sk) / {(µk, 0)}.

To verify (13.32), notice that (13.31) shows that (??) holds for that part of Ck

in Nk. If we assume it does not hold globally, there is some (µ, x) & !(R 1 Sk)C)such that (µ, x) %= (µk, 0). Hence x & !Sk. Now, x & !Sk if and only if there existsat least one j & {2, . . . , n! 2}

Chapter 14

Poincare’s operator

14.1 Initial value and periodic problems

Let T > 0 be fixed, f : R 1 Rn " Rn, (t, y) 2" f(t, y) be locally Lipschitzian in yand continuous. Then it is well known that for each (s, x) & R1Rn, there exists aunique solution y(t; s, x) of the Cauchy problem

y$(t) = f(t, y(t)), y(s) = x,

which is defined on a maximal interval ]5!(s, x), 5+(s, x)[, for some

!- ) 5!(s, x) < s < 5+(s, x) ) +-.

Moreover, y is continuous on the set

G = {(t, s, x) & R1 R1 Rn : 5!(s, x) < t < 5+(s, x)}

on which it is defined, and G is open.A T-periodic solution of the di!erential equation

y$(t) = f(t, y(t)) (14.1)

is a solution of (14.1) defined over [0, T ] and such that

y(0) = y(T ).

If we assume in addition that the function f is T-periodic with respect to t, i.e.such that

f(t, y) = f(t + T, y)

for all t & R, then a T-periodic solution of (14.1) can be continued as a solutiondefined over R and such that

y(t) = y(t + T )

147

148 CHAPTER 14. POINCARE’S OPERATOR

for all t & R. As already observed by Poincare [323] at the end of the XIXth century,y(t; 0, x) will be a T-periodic solution of (14.1) if and only if x & Rn is such that5+(0, x) > T and

x = y(T ; 0, x),

i.e. x is a fixed point of the Poincare operator PT defined by

PT (x) = y(T ; 0, x).

14.2 Bounded perturbations of some linear sys-

tems

Let A : R" L(Rn, Rn) and h : R" Rn be continuous and T-periodic and considerthe linear system

y$ = A(t)y + h(t). (14.2)

By the variation of constants formula, the solution of eq. (14.2) such that y(0) = xis given by

y(t) = Y (t)x +

! t

0Y (t)Y !1(s)h(s) ds, (14.3)

where Y (t) is the fundamental matrix solution of equation y$ = A(t)y, i.e. thesolution of the problem

Y $ = A(t)Y, Y (0) = I.

The homogeneous equation

y$ = A(t)y (14.4)

has a non-trivial T-periodic solution if and only if one can find x %= 0 such that

[Y (T )! I]x = 0,

i.e. if and only if det [I ! Y (T )] = 0. On the other hand, the initial conditions x ofthe possible T-periodic solutions of eq. (14.2) are the solutions of the linear systemin Rn

[I ! Y (T )]x =

! T

0Y (T )Y !1(s)h(s) ds, (14.5)

and such a non-homogeneous linear system will have a solution for any continuousT-periodic h if and only if det [I ! Y (T )] %= 0. In other words, we have proved thefollowing result.

14.2. BOUNDED PERTURBATIONS OF SOME LINEAR SYSTEMS 149

Proposition 14.2.1 The linear di"erential system (14.2) has a T-periodic solutionfor each continuous T-periodic h if and only if the homogeneous system (14.4) onlyhas the trivial T-periodic solution, in which case the T-periodic solution of (14.2)is unique.

Let now g : R1Rn " Rn be continuous, T-periodic in t, locally Lipschitzian inx, and consider the nonlinear di!erential system

y$ = A(t)y + g(t, y). (14.6)

The following result is a nonlinear extension of Proposition 14.2.1.

Theorem 14.2.1 If system (14.4) only has the trivial T-periodic solution and ifthere exists M > 0 such that

#g(t, x)# )M for all (t, x) & R1 Rn, (14.7)

then system (14.6) has at least one T-periodic solution.

Proof. It follows from condition (14.7) and classical results on the Cauchy problemthat all the solutions of eq. (14.6) exist over R. Using the variation of constantsformula (14.3), we see that the solution y(t; 0, x) of eq. (14.6) such that y(0) = xsatisfies the integral equation

y(t; 0, x) = Y (t)x +

! t

0Y (t)Y !1(s)g(s, y(s; 0, x)) ds, (t & R).

Hence, y(t; 0, x) is a T-periodic solution of eq. (14.6) if and only if x verifies thenonlinear equation in Rn

x = [I ! Y (T )]!1

! T

0Y (T )Y !1(s)g(s, y(s; 0, x)) ds := G(x). (14.8)

G : Rn " Rn is continuous and such that, for all x & Rn,

#G(x)# ) #[I ! Y (T )]!1Y (T )#M! T

0#Y !1(s)# ds := R,

and hence G continuously maps Rn into the closed ball B(r) ' Rn. By Brouwer’sfixed point theorem 8.1.3, G has a fixed point in B(r).

Remark 14.2.1 The conclusion of Theorem 14.2.1 may not hold if the linear sys-tem (14.4) has a non-trivial solution, as show by the elementary example

y$ = 1,

which corresponds to n = 1, A(t) > 0 and g(t, y) > 1, and which obviously has noT-periodic solution for any T > 0.


14.3 Stampacchia’s method

Guido Stampacchia has introduced in 1947 [374, 375] an interesting approach tostudy boundary value problems for ordinary di!erential systems, based upon finite-dimensional techniques. Like in the previous section, let A : [0, T ] " L(Rn, Rn)and g : [0, T ]1 Rn " Rn be continuous, and let B & L(Rn, Rn), C & L(Rn, Rn),and d & Rn. Consider the di!erential system

y$(t) = A(t)y(t) + g(t, y(t)) (t & [0, T ]) (14.9)

with the two-point boundary condition

By(0) + Cy(T ) = d. (14.10)

Notice that, for B = !C = I, d = 0 (14.10) reduces to the periodic boundaryconditions, and for B = C = I, d = 0 to the anti-periodic boundary condition.

Using the variation of constants formula, if Y denotes the fundamental matrixsolution of the linear homogeneous system

y$(t) = A(t)y(t),

the solution of system (14.9) such that y(0) = a & Rn satisfies the nonlinear integralequation

y(t) = Y (t)a +

! t

0Y (t)Y !1(s)g(s, y(s)) ds, (14.11)

for all t ( 0 for which it exists. Stampacchia’s idea, which is an adaptation of anearlier idea of L. Tonelli [388] for proving the existence of a solution to Cauchy’sproblem, consists in defining the sequence of mappings yk : [!T

k , T ] 1 Rn " Rn

(k ( 1) as follows

yk(t, a) = (14.12)

*+

,

a if !Tk ) t ) 0

Y (t)a +P t0 Y (t)Y !1(s)g(s, a) ds if 0 < t ) k!1T

Y (t)a +P t0 Y (t)Y !1(s)g(s, yk(s! k!1T, a)) ds if (j ! 1)k!1T < t ) jk!1T,

where j = 2, . . . , k. By construction, each yk is continuous on [!k!1T, T ]1Rn, andsatisfies the identity

yk(t, a) = Y (t)a + Y (t)

! t

0Y !1(s)g(s, yk(s! k!1T, a)) ds (t & [0, T ]). (14.13)

Furthermore,

y$k(t, a) = Y $(t)a + Y $(t)

! t

0Y !1(s)g(s, yk(s! k!1T, a)) ds (14.14)

+ Y (t)Y !1(t)g(t, yk(t! k!1T, a)) = A(t)yk(t, a) + g(t, yk(t! k!1T, a)).

We can now prove the following existence theorem.

14.3. STAMPACCHIA’S METHOD 151

Theorem 14.3.1 Assume that the following conditions hold.

(i) f : [0, T ] 1 Rn " Rn is bounded, namely there exists M > 0 such that#f(t, y)# )M for all (t, y) & [0, T ]1 Rn.

(ii) B + CY (T ) is nonsingular.

Then problem (14.9)-(14.10) has at least one solution.

Proof. Let (yk) be the sequence of functions defined by Stampacchia’s algorithm(14.12). We first show that for each k ( 1, there exists at least one ak & Rn suchthat

Byk(0, ak) + Cyk(T, ak) = d. (14.15)

Using formula (14.12), condition (14.15) can be written

Bak + C

:Y (T )ak +

! T

0Y (T )Y !1(s)g(s, yk(s! k!1T, ak)) ds

;= d,

or, using Assumption (ii),

ak = (B + CY (T ))!1

:d!

! T

0Y (T )Y !1(s)g(s, yk(s! k!1T, ak)) ds

;. (14.16)

The mapping Fk : Rn " Rn defined by the right-hand member of (14.16) is contin-uous, and it follows easily from Assumption (i) that it is bounded, by a constantR > 0 independent of k. So, Brouwer fixed point theorem 8.1.3 implies the existenceof at least one fixed point a#

k & B(R).Next, we prove that the family {yk(·, a#

k) : k = 1, 2, . . .} is relatively com-pact in C([0, T ], Rn), or equivalently, using Ascoli-Arzela’s theorem, is boundedand equicontinuous. The boundedness is a direct consequence of Stampacchia’sconstruction of yk and of Assumption (i). For the equicontinuity, it follows from(14.14) and Assumption (i) that there exists S > 0 such that #y$

k(t, a#k)# ) S for

all k = 1, 2, . . . . Thus, going ifbjj necessary to a subsequence, we can assume that(yk(·, a#

k)) converges uniformly on [0, T ] to some y# & C([0, T ], Rn). In particular,a#

k = yk(0, a#k)" y#(0) if k "-. From relations (14.15), we deduce that

By#(0) + Cy#(T ) = d,

i.e. that y? satisfies the boundary condition (14.10). On the other hand, usingidentity (14.13) and Lebesgue dominated convergence theorem, we obtain

y#(t) = Y (t)y#(0) +

! t

0Y !1(s)g(s, y#(s)), ds (t & [0, T ]),

which means that y# is of class C1 and is solution of (14.9).


Remark 14.3.1 Elementary considerations of linear algebra and the variation ofconstant formula immediately imply that assumption (ii) holds if and only if thelinear boundary value problem

y$(t) = A(t)y(t), By(0) + Cy(T ) = 0 (14.17)

only has the trivial solution.

Remark 14.3.2 When B = !C = I and d = 0, Theorem 14.3.1 generalizes Theo-rem 14.2.1 by suppression the assumption of Lipschitz continuity upon g.

Corollary 14.3.1 For any continuous and bounded g : [0, T ]1 Rn " Rn, and anyd & Rn, the problem

y$(t) = g(t, y(t)), y(0) + y(T ) = d

has at least one solution. This is in particular the case for the anti-periodic problemy(0) + y(T ) = 0.

Corollary 14.3.2 For any continuous and bounded g : [0, T ]1 Rn " Rn, and anya %= 0, the problem

y$(t) = ay(t) + g(t, y(t)), y(0) = y(T )


14.4 The method of lower and upper solutions

Let f : [0, T ] 1 R " R be continuous, c & R and consider the periodic boundaryvalue problem

x$$ + cx$ = f(t, x), x(0) = x(T ), x$(0) = x$(T ). (14.18)

The following concept, which can be traced to Oskar Perron, was introduced inthe present setting by G. Scorza-Dragoni [354] and by M. Nagumo [291]. See [86]for the development of many aspect of lower and upper solutions.

Definition 14.4.1 We say that * & C2([0, T ], R) (resp. , & C2([0, T ], R)) is alower solution (resp. upper solution) for problem (14.18) if

*$$(t) + c*$(t) ( f(t,*(t)) (t & [0, T ]), *(0) = *(T ), *$(0) ( *$(T ). (14.19)

(resp.

,$$(t) + c,$(t) ) f(t,,(t)) (t & [0, T ]), ,(0) = ,(T ), ,$(0) ) ,$(T ).) (14.20)

The lower (resp. upper) solution is called strict if strict inequalities hold in thedi"erential inequality part of the definition.

14.4. THE METHOD OF LOWER AND UPPER SOLUTIONS 153

The existence of an ordered couple of lower and upper solutions implies theexistence and localization of a solution. For periodic problems, this result was firstproved by H.W. Knobloch [204] in 1963 when f is locally Lipschitzian in x. Theproof given here is taken from [267].

Theorem 14.4.1 If problem (14.18) admits a lower solution * and an upper so-lution , such that *(t) ) ,(t) for all t & [0, T ], then it has at least one solution xsuch that

*(t) ) x(t) ) ,(t) (t & [0, T ]). (14.21)

Furthermore, if * and , are strict, then *(t) < x(t) < ,(t) for all t & [0, T ].

Proof. We first introduce a modified problem which coincides with (14.18) when*(t) ) x(t) ) ,(t) (t & [0, T ]). Define . & C([0, T ]1 R) by

.(t, x) =

*+

,

*(t) if x < *(t)x if *(t) ) x ) ,(t),(t) if x > ,(t).

It is clear that . is bounded. Consider the modified problem

x$$ + cx$ ! x = f(t, .(t, x))! .(t, x), x(0) = x(T ), x$(0) = x$(T ). (14.22)

We show that any possible solution x of (14.22) satisfies (14.21), and hence is asolution of (14.18). We show the first inequality, the second one being similar. Itthe left inequality does not hold, then max[0,T ](*!x) > 0, and hence, if it is reachedat t# & ]0, T [, one has

*(t#) > x(t#), *$(t#) = x$(t#),

and

*$$(t#) ) x$$(t#) = !cx$(t#) + x(t#) + f(t, .(t#, x(t#))) ! .(t#, x(t#))

= !c*$(t#) + x(t#) + f(t#,*(t#))! *(t#)

< !c*$(t#) + f(t,*(t#)),

a contradiction with the definition of lower solution. If max[0,T ](* ! x) is reachedat 0 or T, then, by periodicity, it is reached at 0 and T, so that

*$(0)! x$(0) ) 0 ) *$(T )! x$(T ) ) *$(0)! x$(0),

i.e.

*$(0)! x$(0) = 0 = *$(T )! x$(T ).

Hence,

*$$(0) ) x$$(0) = !cx$(0) + x(0) + f(t, .(0, x(0)))! .(0, x(0))

= !c*$(0) + x(0) + f(0,*(0))! *(0)

< !c*$(0) + f(t! 0,*(0)),


a contradiction with the definition of lower solution. The case of strict lower andupper solutions is treated similarly.

Hence, it remains to prove that the modified problem (14.22) has at least onesolution. If we write it in system form by letting y1 = x, y2 = x$, we get

y$1 = y2, y$

2 = !cy2 + y1 + f(t, .(t, y1))! .(t, y1),

y1(0)! y1(T ) = 0, y$1(0)! y$

1(T ) = 0. (14.23)

which is of the type (14.9)-(14.10) with n = 2,

A(t) =

20 11 !c

3, g(t, y1, y2) =

20

f(t, .(t, y1))! .(t, y1)

3,

B = !C = I, and d = 0. Furthermore, g is bounded over [0, T ]1R2, and the linearproblem

y$1 = y2, y$

2 = !cy2, y1(0)! y1(T ) = 0, y$1(0)! y$

1(T ) = 0

has only the trivial solution, as matrix

20 11 !c

3

has real nonzero eigenvalues. Hence, the existence of a solution follows from Stam-pacchia’s theorem 14.3.1.

14.5 Strict lower and upper solutions

Consider problem (14.18) and assume that f is locally Lipschitzian with respect tox. The following result can be traced, in di!erent versions, to [266] and [305].

Theorem 14.5.1 Assume that (14.18) admits an ordered couple of strict lower andupper solutions * and ,, let M > 0 be defined by

M = max(t,u)"[0,T ]'R

|f(t, .(t, u))! .(t, u)|. (14.24)

and let PT be Poincare’s operator associated to problem (14.18). Then,

dB[I ! PT , ]*(0),,(0)[1 ]! TM, TM [ , 0] = !1. (14.25)

Proof. Necessarily, *(t) < ,(t) for all t & [0, T ] and the solutions of the modifiedproblem (14.22) are all contained in the set of T-periodic functions such that

*(t) < x(t) < ,(t) (t & [0, T ]).

14.5. STRICT LOWER AND UPPER SOLUTIONS 155

In particular, their initial positions are such that *(0) < x(0) < ,(0). If x isconstant, that is all we need. If x is not constant, multiplying both members of(14.22) by x$$ and integrating the result over [0, T ] gives, using T-periodicity

! T

0|x$$(t)|2 dt +

! T

0|x$(t)|2 dt =

! T

0[f(t, .(t, x(t)))! .(t, x(t))]x$$(t) dt

and hence, using Cauchy-Schwarz inequality,

E! T

0|x$$(t)|2 dt

F1/2

<<

TM

where M is given by (14.24). Consequently, using the fact that x$ must vanish atsome 5 & [0, T ],

|x$(t)| )<

T

E! T

0|x$$(t)|2 dt

F1/2

< TM

for all t & [0, T ]. In particular, the initial velocities are such that |x$(0)| < TM.Therefore, if we consider any bounded open set D in R2 containing ]*(0),,(0)[1 ]!TM, TM [ the Brouwer degree dB[I ! PT , D, 0], where PT is Poincare’s operatorassociated to (14.22), is well defined. To compute it, let us consider the family ofT-periodic problems

x$$ + cx$ ! x = "[f(t, .(t, x)) ! .(t, x)], x(0) = x(T ), x$(0) = x$(T ), (14.26)

indexed by " & [0, 1], which reduce to (14.22) for " = 1. It can be written, equiva-lently, as

x(t) = "

! T

0G(t, s)[f(s, .(s, x(s))) ! .(s, x(s))] ds,

where G(t, s) is the Green matrix of the associated linear problem

x$$ + cx$ ! x = h(t), x(0) = x(T ), x$(0) = x$(T ),

so that, we also have

x$(t) = "

! T

0G$

t(t, s)[f(s, .(s, x(s))) ! .(s, x(s))] ds,

and consequently,

|x(t)| < &1M, |x$(t)| < &2M,

where the &j only depend upon the Green matrix G. In other words, the initialconditions (x(0), x$(0)) of any possible solutions of (14.26) are contained in the


open bounded set ] ! &1M,&1M [1 ]! &1M,&2M [ . If PT (·,") denotes Poincare’soperator associated to (14.26), the homotopy invariance property 3.4.2 of Brouwerdegree implies that, for any bounded open set D 0 ]!&1M,&1M [1 ]!&1M,&2M [one has

dB[I ! PT (·, 1), D, 0] = dB[I ! PT (·, 0), D, 0].

Now, for " = 0, (14.26) reduces to the linear problem

x$$ + cx$ ! x = 0, x(0) = x(T ), x$(0) = x$(T ), (14.27)

and hence PT (·, 0) is linear. Consequently, if "1 = ! c2 !

Yc2

4 + 1 and "2 = ! c2 +

Yc2

4 + 1 are the characteristic values of (14.27), elementary computations show that

the solution of Cauchy’s problem

x$$ + cx$ ! x = 0, x(0) = y1, x$(0) = y2

is given by

x(t) = ("1 ! "2)!1$(!"2y1 + y2)e

%1t + ("1y1 ! y2)e%2t%,

so that

x$(t) = ("1 ! "2)!1$"1(!"2y1 + y2)e

%1t + "2("1y1 ! y2)e%2t%,

and[I ! PT (·, 0)](y1, y2) = ("1 ! "2)

!1·

·2

"1 ! "2 + "2e%1T ! "1e%2T !e%1T + e%2T

!"1"2(!e%1T + e%2T ) "1 ! "2 ! "1e%1T + "2e%2T

32y1

y2

3

Hence

det [I ! PT (·, 0)] = 1! e%1T ! e%2T + e(%1+%2)T = (1! e%1T )(1! e%2T ),

and, as "1 < 0 < "2, we have

dB[I ! PT (·, 0), D, 0] = iB[I ! PT (·, 0)] = sign (1! e%1T )(1! e%2T ) = !1.

Now, as (14.18) and (14.22) coincide on the set of C1 functions u such that *(t) )u(t) ) ,(t), Poincare’s operator PT of (14.18) and PT (·, 1) coincide on a neighbor-hood of the set of initial conditions of solutions x of (14.18) such that *(t) ) x(t) ),(t) for all t & [0, T ], and hence

dB[I ! PT , ]*(0),,(0)[1 ]! TM, TM [ , 0] = !1.

14.6. KRASNOSEL’SKII-PEROV’S EXISTENCE THEOREM 157

14.6 Krasnosel’skii-Perov’s existence theorem

One obstacle in applying the Poincare method lies in the fact that the maximal in-terval of existence of y(t; 0, x) can be di$cult to estimate and that y(t; 0, x) is rarelyexplicitely known, so that the same is true for PT . To overcome the second di$culty,one can use the homotopy invariance of degree. One way of doing this leads to aninteresting existence theorem whose special cases can be traced to I. Berstein andA. Halanay [24] and which is due, in its general form, to M.A. Krasnosel’skii andA.I. Perov [216], [217] (see also [214]).

Theorem 14.6.1 Assume that there exists an open bounded set G ' Rn such thatthe following conditions hold :

1. For each x & G, the solution y(t; 0, x) of (14.1) exists at least over [0, T ].

2. For each " & ]0, 1] and each x & !G, one has y("T ; 0, x) %= x.

3. For each x & !G, one has f(0, x) %= 0.

ThendB[I ! PT , G, 0] = (!1)ndB[f(0, ·), G, 0].

If we assume moreover that

4. dB[f(0, ·), G, 0] %= 0,

then equation (14.1) has at least one T-periodic solution y such that y(0) & G.

Proof. It follows from Assumption 3 that

* := minx"!G

#f(0, x)# > 0,

and hence, by continuity of f, there exists "0 & ]0, 1[ such thatCCCCC

1

T

! T

0f("s, y("s, 0, x)) ds! f(0, x)

CCCCC

=

CCCCC1

T

! T

0[f("s, y("s, 0, x)) ! f(0, x)] ds

CCCCC< *

whenever " & [0,"0]. Hence, by Rouche’s theorem 3.4.2,

dB[(1/T )

! T

0f("s, y("s, 0, ·)) ds, G, 0] = dB[f(0, ·), G, 0] (14.28)

for " & [0,"0]. On the other hand, for all " & ]0, 1], we have

(1/"T )[x! y("T, 0, x)] = !(1/"T )

! %T

0f(t, y(t, 0, x)) dt

= !(1/T )

! T

0f("s, y("s, 0, x)) ds,


and hence, for " & ["0, 1],

dB[(1/T )[I ! y("T, 0, ·)], G, 0] = dB[!(1/T )

! T

0f("s, y("s, 0, x)), G, 0]. (14.29)

Therefore, it follows from Asumption 2, the homotopy invariance property 3.4.2,(14.28) and (14.29) that

dB [I ! PT , G, 0] = dB[I ! y("0T, 0, ·), G, 0] = dB[(1/T )(I ! y("0T, 0, ·)), G, 0]

= dB[!(1/T )

! T

0f("0s, y("0s, 0, x)), G, 0]

= (!1)ndB[(1/T )

! T

0f("0s, y("0s, 0, x)), G, 0]

= (!1)ndB[f(0, ·), G, 0].

If Assumption 4 also holds, then, by the existence theorem 3.4.1, I!PT has as leastone zero in G.

14.7 Degree of gradient mappings

The following consequence of Theorem 14.6.1 is an extension of a result of M.A.Krasnosel’skii [214] due to H. Amann [11]. For a function V : % ' Rn " R, andc & R, we denote by V c and V c respectively the sets {x & % : V (x) < c} and{x & % : V (x) ) c}.

Theorem 14.7.1 Let % & Rn be open and V & C1(%, R),with gradient +V locallyLipschitzian. Assume that there exists , & R such that V ' ' % is compact, and* < ,, r > 0 and x0 & % such that

V & ' Bx0(r) ' V ' ,

and +V (x) %= 0 for every x & V ' \ V &. Then

dB [+V, V ' , 0] = 1.

Proof. We consider the associated gradient system

y$(t) = !+V (y(t)), (14.30)

and observe that for each c ) ,, the set V c = {x & Rn : V (x) ) c} is bounded(as contained in V ') and is positively invariant for (14.30). Indeed, if y(t; 0, x) isthe solution of (14.30) such that y(0; 0, x) = x, then, denoting by 6·, ·7 the innerproduct in Rn, we have

d

dtV (y(t; 0, x)) = 6+V (y(t; 0, x)), y$(t; 0, x)7 = !#+V (y(t; 0, x))#2 ) 0, (14.31)

14.7. DEGREE OF GRADIENT MAPPINGS 159

so that if x & V c, then V [y(t; 0, x)] ) V (x) ) c, i.e. y(t; 0, x) & V c for t &[0, 5+(0, x)[. As V c is bounded, this implies that 5+(0, x) = +- (see e.g. [71]).Notice also that if x & !V ' i.e. if V (x) = ,, then, by (14.31),

d

dtV [y(t; 0, x)]|t=0 = !#+V (x)#2 < 0,

and hence V [y(t; 0, x)] < V (x) for all t > 0. Consequently, y(t; 0, x) %= x for allt > 0. It then follows from Theorem 14.6.1 with G = V ' that, for all t > 0, one has

dB [I ! y(t; 0, ·), V ' , 0] = (!1)ndB [!+V, V ' , 0] = dB [+V, V ', 0]. (14.32)

We shall now show that dB[I ! y(t; 0, ·), V ' , 0] = 1 for t su$ciently large. To thise!ect, if

. = minV #\V $

#+V (x)#2,

then . > 0. If x & !V ', then, by the invariance of V ' , y(t; 0, x) & V ' for all t ( 0.If y(t; 0, x) & V ' \ V & for t & [0, 5 ], then

,!* ( V (x)!V [y(5 ; 0, x)] =

! 0

#

d

dtV [y(t; 0, x)] dt =

! #

0#+V [y(t; 0, x))#2 dt ( .5.

Consequently, for 5 > '!&. , we necessarily have y(5 ; 0, x) & V &, and hence y(5 ; 0, x)

& B(x0; r). Thus y(5 ; 0, ·) maps !V ' into Bx0(r). Consequently, for each " & [0, 1]and each x & !V ' , we have

#(1! ")(x ! x0) + "(x ! y(5 ; 0, x))# = #x! x0 ! "(y(5 ; 0, x) ! x0)#( #x! x0# ! #y(5 ; 0, x)! x0# > 0,

as #x! x0# > r for x & !V '. The homotopy invariance theorem 3.4.2 then impliesthat

dB [I ! y(5 ; 0, ·), V ' , 0] = dB[I ! x0, V' , 0] = 1.

The following consequence of Theorem 14.7.1 is a result of M.A. Krasnosel’skii[214] for the Brouwer degree of the gradient of a coercive real function.

Corollary 14.7.1 Let V & C1(Rn, R), with gradient +V locally Lipschitzian, besuch that +V (x) %= 0 for some r0 > 0 and all x & Rn with #x# ( r0. If V is coercive,i.e. if

V (x)" +- as #x# " -,

then dB [+V, B(r), 0] = 1 for all r ( r0.

Proof. Let * = max)x)+r0V (x). By the coercivity, V & is bounded and hence there

exists r > 0 such that V & ' B(r). One then takes , > max)x)+r V (x) and all theassumptions of Theorem 14.7.1 with x0 = 0 are satisfied. Thus, dB[+V, V ' , 0] = 1,and the result follows from the excision theorem 3.4.1.


Another consequence of Theorem 14.7.1 was first derived by E. Rothe [334].

Corollary 14.7.2 Let U be an open neighborhood of x0 & Rn and V & C1(U, R),with gradient +V locally Lipschitzian. If x0 is an isolated critical point of V atwhich V has a local minimum, then for all su!ciently small - > 0 one has

dB[+V, Bx0(-), 0] = 1.

Proof. Without loss of generality, we can take U = B(-), x0 = 0, V (0) = 0, with- > 0 such that V (x) > 0 for all x & U. Let us fix 0 < r1 < r2 < - and let, = minr1+)x)+r2

V (x), so that , > 0. As V ' is an open neighbourhood of 0, there

exists r > 0 such that B(r) ' V '. If we take * = 12 minr+)x)+r2

V (x), then * < ,and all conditions of Theorem 14.7.1 are satisfied. The result follows from thistheorem and from the excision theorem 3.4.1.

Remark 14.7.1 All the above results hold when +V is only assumed to be contin-uous. This can be shown using a partition of unity argument in the proofs above.We shall use freely this fact in the sequel.

A further consequence of Theorem 14.6.1 concerns an autonomous system of theform

x$(t) = f(x(t)), (14.33)

where f : Rn " Rn is locally Lipschitzian and such that f(0) = 0. Recall thatthe equilibrium 0 is said to be stable if for each 6 > 0 we can find + > 0 suchthat #y(t; 0, z)# ) 6 for each t ( 0 and z & Rn such that #z# ) +, and (uniformly)asymptotically stable if it is stable and attractive, i.e. there exists , > 0 suchthat for each 7 > 0 one can find T > 0 such that #y(t; 0, z)# ) 7 whenever t ( Tand #z# ) ,. See [337] for details.

Theorem 14.7.2 If 0 is an isolated zero of f and is asymptotically stable, thenthere exists - > 0 such that

dB [!f, B(-), 0] = 1.

Proof. We first notice that the stability of 0 implies that all solutions of su$cientlysmall initial condition exist on [0, +-[ and we claim that there exists some -0 &]0,,] such that, for each - & ]0, -0], each z with #z# = - and each t > 0 we havey(t; 0, z) %= z. If it is not the case, that, for each integer k ( 1, such that 1/k ) ,,we can find some -k & ]0, 1/k], some zk with #zk# = -k and some tk > 0 suchthat y(tk; 0, zk) = zk, i.e. such that y(t; 0, zk) is tk-periodic. But, this implies thaty(t; 0, zk) %" 0 for t " +- and contradicts the attractivity of 0. Consequently, forall z & B(-0) \ {0}, all " & ]0, 1], and for all T > 0, we have y("T ; 0, z) %= z. Hence,by Theorem 14.6.1,

dB[!f, B(-), 0] = dB [I ! y(T ; 0, ·), B(-), 0]

14.7. DEGREE OF GRADIENT MAPPINGS 161

for all T > 0 and all - & ]0, -0]. For T > 0 associated to 7 = -/2 in the definition ofattractivity, we have #y(T ; 0, z)# ) -/2 for all z & B(-), and hence

#z ! "y(T ; 0, z)# ( -/2

whenever |z| = - and " & [0, 1], so that, by Rouche’s theorem 3.4.2

dB[I ! y(T ; 0, ·), B(-), 0] = dB [I, B(-), 0] = 1,


For more delicate results on the degree when 0 is only assumed to be stable, thereader may consult [385, 386].

Chapter 15

Stability and index ofperiodic solutions

15.1 Introduction

Topological degree techniques have been widely applied in the study of the exis-tence and the multiplicity of solutions of nonlinear boundary value problems, andin particular of periodic solutions of nonlinear di!erential equations. Much less hasbeen done using those techniques in the study of the stability of those solutions.Concentrating on the periodic case, we can quote some pioneering studies for ar-bitrary systems ([214]), [276]) and for second order equations ([290], [205], [263]).Among the more recent results, we may quote [269] (for periodic solutions of firstorder scalar equations), [303], [304], [305], [306], [307], [243], [244]. We shall analyzebriefly here some of the results of R. Ortega taken from [303] and [304]. The readercan consult the original papers, as well as the survey [308] for more details andfurther developments.

15.2 Planar periodic systems

We consider the following two-dimensional system

x$ = f(t, x), (15.1)

where the continuous mapping f : R 1 R2 " R is T-periodic with respect to t forfixed x and of class C1 with respect to x for fixed t. We denote by x(t; x0) thesolution of (15.1) such that x(0;x0) = x0 and by PT Poincare’s mapping PT : x0 2"x(T ; x0). This mapping is defined for the (open) subset of R2 made of the x0 suchthat x(.; x0) is defined over [0, T ]. Recall that x(.; x0) is a T-periodic solution of(15.1) if and only if x0 is a fixed point of PT . If x is an isolated T-periodic solutionof (15.1), then x(0) is an isolated fixed point of PT and hence the Brouwer index

ind[PT , x(0)] := dB[I ! PT , Bx0(r), 0],

163

164 CHAPTER 15. STABILITY AND INDEX OF PERIODIC SOLUTIONS

where r > 0 is su$ciently small, is well defined, called the index of period T ofx, and denoted by .T (x). Similarly, we need to consider the mapping P2T : x0 2"x(2T ; x0) and if x is an isolated 2T-periodic solution of (15.1), we define the indexof period 2T of x by

.2T (x) = ind[P2T , x(0)] = dB[I ! P2T , Bx(0)(r), 0].

Notice that every T-periodic solution x of (15.1) is also a 2T-periodic solution and,if isolated as a 2T-periodic solution, is also isolated as a T-periodic solutions. Itadmits both indices .T and .2T .

Definition 15.2.1 A T-periodic solution x of (15.1) is called nondegenerate ofperiod T (resp. of period 2T) if the linearized equation

y$ = fx(t, x(t))y (15.2)

has no nontrivial T-periodic (resp. 2T-periodic) solution.

If x is T-periodic and nondegenerate of period 2T , an implicit function argumentshows that x is isolated for periods T and 2T and the linearization formula for degreetogether with the Leray-Schauder formula for linear mappings show that

|.T (x)| = |.2T (x)| = 1.

More explicitely, if x is a solution of (15.1) with initial value x0, then from theidentity

x$(t; x0) = f(t, x(t; x0))

we deduce

[x$x0

(t; x0)]$ = f $

x(t, x(t; x0))x$x0

(t; x0),

x$x0

(0;x0) = I,

and hence x$x0

(t, x0) is the fundamental matrix Y (t) of the variational system (15.2)associated to the solution x(t; x0). In particular, Y (T ) = x$

x0(T ; x0) = P $

T (x0) isthe monodromy matrix of equation (15.2) when x(t; x0) is T-periodic. Now, ifµ1 and µ2 are the characteristic multipliers of (15.2), i.e. the eigenvalues of amonodromy matrix, we have, by linear algebra

det [I ! P $T ](x0) = (1! µ1)(1! µ2).

Consequently, if x(t, x0) is isolated (period T) and if µj %= 1 (j = 1, 2), Theorem5.2.1 implies that, for su$ciently small r > 0,

.T (x0) = iB[I ! PT , Bx0(r), 0] = sign JI!PT (x0) = sign det [I ! P $T ](x0)

= sign[(1! µ1)(1! µ2)]. (15.3)

15.2. PLANAR PERIODIC SYSTEMS 165

Now, for each x0 & R2, we have

P2T (x0) = x(2T ; x0) = x[T ; x(T ; x0)] = PT (PT (x0)),

and henceP $

2T (x0) = P $T (PT (x0)) * P $

T (x0),

which implies in particular, as x(t; x0) is T-periodic, that

P $2T [x0] = P $

T [x0] * P $T [x0] (15.4)

As P $T (x0) = Y (T ), we get

det[I ! P $2T (x0)] = det{[I ! P $

T [x0] * P $T [x0]} = det[I ! Y (T ) * Y (T )]

= (1! µ21)(1! µ2

2).

Consequently, if x is a T-periodic solution which is isolated of period 2T and ifµj %= ±1 (j = 1, 2), then

.2T (x) = sign[(1! µ21)(1! µ2

2)]. (15.5)

Theorem 15.2.1 Assume that condition

div f(t, x) < 0, (15.6)

holds for all (t, x), and let x be a T-periodic solution of (15.1) which is nondegen-erate of period 2T . Then .2T (x) = 1 (resp. ! 1) if and only if x is uniformlyasymptotically stable (resp. is unstable).

Proof. Let µ1 and µ2 be the characteristic multipliers of the variational equation(15.2) considered as an equation with T-periodic coe$cients. Recall that if Y (t) isthe fundamental matrix of (15.2), then µ1 and µ2 are the eigenvalues of the matrixY (T ), which implies that the identity

µ2 ! (µ1 + µ2)µ + µ1µ2 = det[Y (T )! µI],

is valid for all µ & C, and that µ2 = µ1 whenever µ1 is complex. In particular,using Liouville’s formula and assumption (15.6), we have

µ1µ2 = detY (T ) = exp

E! T

0div fx(s, x(s)) ds

F& ]0, 1[. (15.7)

The nondegeneracy condition implies that µi %& {!1, 1}, (i = 1, 2). Condition (15.7)implies that if say µ1 is not real, then µ2 = µ1 and |µ1|2 = |µ2|2 = µ1µ2 < 1. Thus,in any case,

|µi| %= 1, (i = 1, 2). (15.8)


Thus, if one of the µi (and hence the other one too) is not real, we have |µi| <1, i = 1, 2, and

.2T (x) = sign|(1! µ21)|2 = 1.

Now, x is uniformly asymptotically stable if and only if |µi| < 1, (i = 1, 2), andhence, in the case of non real multipliers, the statement about the uniform stabilityis proved. If the multipliers are real, then .2T = 1 if and only if µ2

1 and µ22 are

both smaller or both larger than one, and the second case is excluded by relation(15.7), so that the statement about the uniform stability is proved also in this case.Finally, x is unstable if and only if one multiplier, say µ1 has absolute value greaterthan one, which can only arrive if they are real. Thus, this is equivalent to the factthat µ2

1 > 1 so that necessarily, by relation (15.7), µ22 < 1 and, by (15.5), those two

conditions are equivalent to .2T = !1.

15.3 The case of a second order di!erential equa-

tion

We consider in this section the second order di!erential equation

u$$ + cu$ + g(t, u) = 0, (15.9)

with c > 0 and g : R1R" R a continuous function, T-periodic with respect to thefirst variable and having a continuous partial derivative with respect to the secondvariable. Such an equation is equivalent to the planar system

x$1 = x2, x$

2 = !cx2 ! g(t, x1), (15.10)

This system satisfies condition (15.6), because

div (x2,!cx2 ! g(t, x1)) = !c < 0

for all (x1, x2) & R2. The aim of this section is to characterize the stability of aT-periodic solution u of (15.9) (i.e. of the associated T-periodic solution x = (u, u$)of (15.10)) in terms of its index .T (u) := .T (x) only. If * and , are real functionsdefined over [0, T ], we shall write * @ , if *(t) ) ,(t) for all t & [0, T ] and thestrict inequality holds on a subset of positive Lebesgue measure.

The following lemma is proved in [277].

Lemma 15.3.1 Assume that the real function , is continuous over [t0, t0 +T ] andsuch that

, @ %2

T 2.

Then the two-point boundary value problem

w$$(t) + ,(t)w = 0, w(t0) = 0 = w(t0 + %2/T 2)

only has the trivial solution.

15.3. THE CASE OF A SECOND ORDER DIFFERENTIAL EQUATION 167

Lemma 15.3.2 Assume that c > 0 and that the real function * is continuous,T-periodic and such that

*@ %2

T 2+

c2

4.

Then the linear equation

v$$ + cv$ + *(t)v = 0 (15.11)

does not admit negative characteristic multipliers.

Proof. Setting

v(t) = expL! c

2tM

w(t),

equation (15.11) becomes

w$$ + [*(t) ! c2

4]w = 0 (15.12)

and the corresponding solutions v and w obviously have the same zeros. If (15.11)admits a negative characteristic multiplier µ, then it has a nontrivial solution v suchthat

v(t + T ) = µv(t)

for all t & R. If t is such that v(t) %= 0, then v has opposite signs at t and t + T andtherefore must vanish at some t0 & ]t, t + T [, and hence also at t0 + T . Thus w is asolution of equation (15.12) satisfying the two-point boundary conditions

y(t0) = y(t0 + T ) = 0.

But "2

T 2 is the first eigenvalue of the two-point boundary value problem on an intervalof length T for the operator !w$$ and Lemma 15.3.1 together with the assumptionover * imply that w must be identically zero, a contradiction.

Theorem 15.3.1 . Assume that u is an isolated T-periodic solution of (15.9) suchthat condition

g$u(t, u(t)) ) %2

T 2+

c2

4(15.13)

holds for all t & R. Then u is uniformly asymptotically stable (resp. unstable) ifand only if .T (u) = 1 (resp. .T (u) = !1).

Proof. If g$u(t, u(t)) = "2

T 2 + c2

4 for all t & R, the variational equation

v$$ + cv$ + g$u(t, u(t))v = 0, (15.14)


has constant coe$cients and the characteristic exponents have negative real parts,which immediately implies that u is uniformly asymptotically stable and .T (u) =.2T (u) = 1. We can therefore assume that

g$u(·, u(·))@ %2

T 2+

c2

4.

Lemma 15.3.2 implies that the variational equation (15.14) has no negative char-acteristic multipliers. We already know from the proof of Theorem 15.2.1 that.T (u) = .2T (u) = 1 and u is uniformly asymptotically stable if the characteristicmultipliers are not real. If the µi are real, they are both positive and

sign[(1! µ21)(1! µ2

2)] = sign[(1! µ1)(1! µ2)(1 + µ1)(1 + µ2)]

= sign[(1! µ1)(1! µ2)],

so that .T (u) = .2T (u) and the result follows from Theorem 15.2.1

Remark 15.3.1 The following example from [303] shows that the assumption inTheorem 15.3.1 is optimal. Let , be a T-periodic function defined by

,(t) =

Z!#2

1 + c2

4 if t & ]0, a[,

#22 + c2

4 if t & ]a, T [,

for some 0 < a < T and #1, #2 > 0. We consider the equation

u$$ + cu$ + #(t)u = 0.

If the assumption of Theorem 15.3.1 does not hold for this equation, then #22 > "2

T 2

and we can fix a so that #2(T ! a) = %. Direct integration of the equation showsthat one can select #1 in such a way that the trivial solution is unstable and hasindex 1. Hence, "2

T 2 + c2

4 cannot be replaced by any greater constant. One couldobject that in this example the function # is not continuous but this can be ofcourse easily overcame by an approximation argument.

Remark 15.3.2 Using a class of systems introduced by Smith [369] and for whichMassera’s convergence theorem holds, Ortega [305] has obtained the following par-tial improvement of Theorem 15.3.1 :

Proposition 15.3.1 Let u be a T-periodic solution of (15.9) such that

g$u(t, u(t)) <c2

4

for all t & R. Then the conclusion of Theorem 15.3.1 holds.

Remark 15.3.3 Combining degree techniques with the theory of center manifolds,Ortega [306] has obtained the following improvement of Theorem 15.3.1:

Proposition 15.3.2 Let x be a T-periodic solution of (15.1) which is isolated ofperiod 2T . Then .2T (x) = 1 (resp. %= 1) if and only if x is uniformly asymptoticallystable (resp. is unstable).

He has also obtained a version of this result in Rn.

15.4. SECOND ORDER EQUATIONS WITH CONVEX NONLINEARITY 169

15.4 Second order equations with convex nonlin-

earity

We consider in this section the second order parametric equation

u$$ + cu$ + g(t, u) = s, (15.15)

where c > 0, s is a real parameter and the continuous function g : R 1 R " R isT-periodic with respect to the first variable and has a continuous partial derivativewith respect to the second variable. Assume moreover that g is strictly convex,namely that

[g$u(t, u)! g$u(t, v)](u ! v) > 0, (15.16)

for all t & R and all u %= v in R. To insure that condition (15.13) holds, we assumealso that

limu(+%

g$u(t, u) ) %2

T 2+

c2

4. (15.17)

Finally, let us assume that the coercitivity condition

lim|u|(%

g(t, u) = +- (15.18)

introduced in [109] holds uniformly in t & [0, T ]. Following [303], we complete theexistence discussion given in [109] by results on the stability of the correspondingT-periodic solutions. The following lemma is used in the proof.

Lemma 15.4.1 Consider the di"erential operator L& defined by

L&(u) = u$$ + cu$ + *(.)u,

acting on T-periodic functions, where * & L1(R/TZ) satisfies condition

*@ %2

T 2+

c2

4. (15.19)

Then the following conclusions are true.

(a) For each real µ each possible T-periodic solution u of equation L&(u) = µ iseither trivial or di"erent from zero for each t & [0, T ].

(b) Let *1 and *2 be functions in L1(R/TZ) satisfying (15.19) and such that*1 @ *2. Then equations

L&i(u) = 0, (i = 1, 2)

cannot admit nontrivial T-periodic solutions simultaneously.


Proof. If a nontrivial T-periodic solution u of equation L&(u) = µ vanishes at somet0, then it vanishes also at t0 + T , and, by the same reasoning as in the proof ofLemma 15.3.2, the function w defined by w(t) = exp(! c

2 t)u(t) is a solution of thedi!erential equation

w$$ + [*(t) ! c2

4]w = 0

which vanishes at t0 and t0 + T , a contradiction with Lemma 15.3.1. To prove (b),we first notice that conclusion (a) obviously holds also for the adjoint operator L#

&

defined by L#&(u) = u$$! cu$ + *(.)u. If both equations L&i(u) = 0 admit nontrivial

T-periodic solutions u1 and u2 respectively, then the adjoint equation L#&1

(u) = 0admits a nontrivial T-periodic solution &. By the part (a) of the Lemma, we canassume without loss of generality that u2 > 0 and & > 0 on [0, T ]. Moreover,

L&1(u2) = L&2(u2) + (*1 ! *2)u2 = (*1 ! *2)u2,

so that, using the Fredholm alternative, we get

0 =

! T

0[*1(t)! *2(t)]u2(t)&(t) dt,

which is a contradiction with the assumption *1 @ *2.

The following lemma is essentially Lemma 2 of [109].

Lemma 15.4.2 Assume that condition (15.18) holds. Then there exists an increas-ing function M(s) such that each possible T-periodic solution u of (15.15) satisfiesthe inequality

|u(t)| + |u$(t)| < M(|s|), (t & R).

Proof. By assumption (15.18), there exists . & R such that

g(t, u) ( . (15.20)

for all (t, u) & [0, T ]1 R. Let u be a possible T-periodic solution of (15.15). Then,integrating both members of (15.15) over [0, T ], we get

s = g(·, u(·)) :=1

T

! T

0g(t, u(t)) dt ( .. (15.21)

Furthermore,

! T

0|u$$(t) + cu$(t)| dt =

! T

0|g(t, u(t))! s| dt (15.22)

)! T

0|g(t, u(t))! .| dt + T |. ! s| (15.23)

=

! T

0[g(t, u(t))! .] dt + T |. ! s| (15.24)

= T (s! .) + T |. ! s| = 2T (s! .)+. (15.25)


Consequently, if 5 & [0, T ] is such that u$(5) = 0, we deduce from (15.22) that, forall t & [0, T ],

|u$(t)| =

DDDD! t

#e!c(t!+)[u$$($) + cu$($)] d$

DDDD (15.26)

)! T

0|u$$($) + cu$($)| d$ ) 2T (s! .)+, (15.27)

which immediately implies that, for all t & [0, T ],

|u(t)! u(0)| ) 2T 2(s! .)+. (15.28)

Now, it follows from condition (15.18) that there exists an increasing function R(s)of s such that

f(t, u) > s whenever t & [0, T ] and |u| ( R(s). (15.29)

Consequently, if |u(0)| ( R(s) + 2T 2(s! .)+, then, for all t & [0, T ],

|u(t)| = |u(0) + u(t)! u(0)| ( |u(0)|! |u(t)! u(0)| ( R(s),

so that

g(·, u(·)) > s,

a contradiction with (15.21). Hence, |u(0)| < R(s)+2T 2(s!.)+, and, consequently,we have, for all t & [0, T ],

|u(t)| ) |u(0)| + |u(t)! u(0)| < R(s) + 4T 2(s! .)+. (15.30)

and it su$ces to take M(s) = R(s) + (4T 2 + 2T )(s! .)+.

We also need the following preliminary result.

Lemma 15.4.3 Assume that the function g satisfies conditions (15.17) and (15.18).Then, given s1, s2 & R and u1, u2 be T-periodic solutions of (15.15) for s = s1, s =s2 respectively such that u1 %= u2. Then either u1 > u2 or u1 < u2 on [0, T ].

Proof. The di!erence v = u1 ! u2 satisfies an equation of the form

L&(v) = s1 ! s2,

for some T-periodic function * which, because of conditions (15.16) and (15.17)satisfies the conditions of Lemma 15.4.1. The conclusion follows from this lemma.

We can now state and prove the main result of this section.

Theorem 15.4.1 Assume that the function g satisfies conditions (15.16), (15.17)and (15.18). Then there exists s0 & R such that the following conclusions hold.


(i) If s > s0, equation (15.15) has exactly two T-periodic solutions, one uniformlyasymptotically stable and another unstable.

(ii) If s = s0, equation (15.15) has a unique T-periodic solution which is notasymptotically stable.

(iii) If s < s0, every solution of equation (15.15) is unbounded.

Proof. We first follow the arguments of [109]. If s# is such that g(t, 0) ) s# forall t & [0, T ], then 0 is an upper solution for (15.15) with T-periodic boundaryconditions and s = s#. From condition (15.18), there exists R > 0 such thatg(t,!R) ( s# for all t & R, and hence !R is a lower solution for (15.15) withT-periodic boundary conditions and s = s#. Hence, by Theorem 14.4.1, equation(15.15) with s = s# has at least one T-periodic solution taking values in [!R, 0].In other words, the set S of s & R such that (15.15) has at least one T-periodicsolution is non-empty. On the other hand, if u is a T-periodic solution of (15.15),then g(·, u(·)) ( ., which shows that S is bounded below by .. Now, S is an intervalunbounded from above, because if s1 belong to S, u1 is a corresponding T-periodicsolution of (15.15) with s = s1, and if s1 < s, then

u$$1 + cu$

1 + g(t, u1) = s1 < s,

so that u1 is an upper solution. Now it follows again from assumption (15.18)that one can find R ( !min[0,T ] u1 so that f(t,!R) ( s for all t & [0, T ], (15.15)with T-periodic boundary conditions. Using again Theorem (14.4.1), we obtain theexistence of a T-periodic solution lying between !R and u1(t). Let s0 := inf S,so that s0 ( .. A limiting and compactness argument using the conclusion (a) ofLemma 15.4.2 implies that s0 & S. Thus, s ( s0 equation (15.15) has at least oneT-periodic solution, and for s < s0, equation (15.15) has no T-periodic solution.

We now show that equation (15.15) has at most two T-periodic solutions, sothat they are necessarily isolated. Otherwise, let ui, (i = 1, 2, 3) be three di!erentT-periodic solutions. By Lemma 15.4.3, they can be ordered, say u1 < u2 < u3.Setting v1 = u2 ! u1, v2 = u3 ! u2, we see immediately that vi satisfies equationL&i(vi) = 0 with *i = [g(t, ui+1)! g(t, ui)]/vi, (i = 1, 2). The strict convexity of gimplies that *1 < *2 on [0, T ]. Using Lemma 15.4.1, we obtain that either v1 or v2

must be zero, a contradiction.Let now u0 be a T-periodic solution of equation (15.15) for s = s0. It follows

from the continuity of the index that u0 is degenerate and that .T (u0) = 0, because,if it was not the case, equation (15.15) would have a T-periodic solutions for alls & ]s0 ! 6, s0[ for some 6 > 0. Notice now that if u1 is another T-periodic solutionof equation (15.15) for some s1 ( s0, then, by Lemma 15.4.3, either

u1 < u0 and hence g$u(t, u1) < g$u(t, u0) on [0, T ],

oru1 > u0 and hence g$u(t, u1) > g$u(t, u0) on [0, T ].


By the second part of Lemma 15.4.1, we conclude that u1 is nondegenerate, andhence that s1 > s0. So we have proved that, for s = s0 equation (15.15) has a uniqueT-periodic solution u0 which satisfies .T (u0) = 0, and hence cannot be uniformlyasymptotically stable by Theorem 15.3.1.

A consequence of the above reasoning is the existence, for s > s0, of exactlytwo T-periodic solutions of (15.15) and they are nondegenerate. Given s1 > s0, ByLemma 15.4.1, the Poincare operator P #

T,s associated to equation (15.15) (writtenas a planar system) has no fixed point on the boundary of the nonempty subset% = {(x1, x2 & domPT : |x1| + |x2| < M0}, with M0 = M(max(|s0|, |s1|), for eachs & [s0, s1]. Hence, by the homotopy invariance property 3.4.2,

dBI ! P #T,s1

,%, 0] = dB[I ! P #T,s0

,%, 0] = .T (u0) = 0. (15.31)

Now, if u1 denotes the T-periodic solution whose existence has been proved using themethod of lower and upper solutions, u1 is non-degenerate and hence .T (u1) = ±1.Therefore, by (15.31) and Proposition 5.1.1, there must exist a second T-periodicsolution u#

1 such that.T (u#

1) = !.T (u1).

Thus, by Theorem 15.3.1, equation (15.15) has exactly one uniformly asymptoticallystable and one unstable T-periodic solution for each s > s0.

Finally, let u be a bounded solution of equation (15.15) for some s < s0.The equation easily implies that u$ is also bounded and given u+ > supu(t) (

inf u(t) > u! with |u±| large enough, let us consider the associated truncatedequation

u$$ + cu$ + g#(t, u) = s. (15.32)

where the function (bounded and Lipschitzian in u) g# is defined by

g#(t, u) =

*+

,

g(t, u) if u! < u < u+,g(t, u!) if u ) u!,g(t, u+) if u ( u+.

By construction, u is also a solution of this equation (for which the Cauchy problemis globally uniquely solvable) with |u| + |u$| bounded. A theorem of Massera [259]implies then that equation (15.32) should have a T-periodic solution, a contradictionwith Lemma 15.4.2 and the nonexistence of T-periodic solution for equation (15.15)when s < s0. Thus, for those values of s, equation (15.15) has no bounded solution.

Remark 15.4.1 Theorem 15.4.1 can be considered as a partial extension to a sec-ond order di!erential equation of the result of [271] which gives a complete qualita-tive picture of the set of solutions for first order scalar equations with convex andcoercive nonlinearities.

Remark 15.4.2 More results on equation 15.15, including the existence of somesubharmonics of order 2 can be found in [304].


15.5 Periodic solutions of pendulum-type equati-

ons

In this section, we study periodic problems of the form

u$$ + cu$ + f(t, u) = p(t) + s, u(0) = u(T ), u$(0) = u$(T ), (15.33)

where c ( 0, s & R, f : [0, T ]1 R " R is continuous, 2%-periodic with respect tou, such that f $

u exists and is continuous for any u & R, p : [0, T ]" R is continuousand has mean value

p :=1

T

! T

0p(t) dt

equal to zero. A special case is the forced pendulum equation, for which g(t, u) =a sinu. T-periodic solutions of (15.33) which di!er by a multiple of 2% will be iden-tified.

Because of the boundedness and regularity of f, solutions of the Cauchy problemfor the di!erential equation in (15.33) exist over R and are unique, so that Poincare’soperator PT is well defined everywhere. On the other hand, it follows from theinvariance of (15.33) under the transformation u " u + 2% and the uniquess ofCauchy problem that

PT (u0 + 2%, v0) = PT (u0, v0) (15.34)

for all (u0, v0) & R2. Again, let S denote the set of s & R such that equation (15.33)has at least one T-periodic solution. If (15.33) has a T-periodic solution u, then,integrating both members of (15.33) over [0, T ] we obtain, if we set

fL(t) := minu"R

f(t, u), fL(t) := maxu"R

f(t, u), (15.35)

fL ) s ) fM ,

and hence S ' [fL, fM ] is bounded.

Proposition 15.5.1 S is not empty.

Proof. Let a = u(0) and v(t) = u(t)! a, so that v(0) = 0. As u$ = v$, the search ofT-periodic solutions of (15.33) is equivalent to the search of a, v(t) such that

v$$ + cv$ + f(t, a + v) = p(t) + s, v(0) = 0 = v(T ), v$(0) = v$(T ). (15.36)

If we let u$(0) = v$(0) = v0, it is easy to see that the solution v(t; v0, s) of thedi!erential equation in (15.36) such that v(0) = 0, v$(0) = v0 verifies the identities

v$(t) = v0 +

! t

0e!c(t!#)[p(5) + s! f(5, a + v(5, v0, s))] d5 (15.37)

v(t) = v0t +

! t

0

! +

0e!c(+!#)[p(5) + s! f(5, a + v(5, v0, s)] d5 d$.

15.5. PERIODIC SOLUTIONS OF PENDULUM-TYPE EQUATIONS 175

Consequently, for a given a, v will be a solution of problem (15.36) if v0 and s aresuch that

! T

0e!c(T!#)[p(5) + s! f(5, a + v(5, v0, s))] d5 = 0 (15.38)

v0T +

! T

0

! +

0e!c(+!#)[p(5) + s! f(5, a + v(5, v0, s)] d5 d$ = 0.

It follows from the boundedness of f that there exists s# > 0 such that the left-hand member of the first equation in (15.38) is positive when s ( s# and all v0 & R,and is negative when s ) s# and all v0 & R. On the other hand, for the samereason, there exists v# > 0 such that the left-hand member of the second equationin (15.38) is positive when v0 ( v# and s & [!s#, s#], and negative when v0 ) !v#

and s & [!s#, s#]. Consequently, Poincare-Miranda’s theorem implies the existenceof (Ts, Tv0) & [!s#, s#] 1 [!v#, v#] such that conditions (15.38) hold, i.e. such thatv(t; Tv, Ts) solves problem (15.36) with s = Ts. In other words, equation (15.33) withs = Ts has at least one T-periodic solution, or, equivalently, Ts & S.

Let us now prove that S is a closed interval.

Proposition 15.5.2 S is a closed (possibly degenerate) interval.

Proof. If S contains only one point, there is nothing to prove. If it is not the case,let s1 < s2 be elements of S, and u1, u2 be corresponding T-periodic solutions of(15.33) with s = s1 and s = s2 respectively. If s1 ) s ) s2, then

u$$1 + cu$

1 + f(t, u1) = p(t) + s1 ) p(t) + s,

u$$2 + cu$

2 + f(t, u2) = p(t) + s2 ( p(t) + s,

which shows that u1 is an upper solution and u2 is a lower solution for equation(15.33) with T-periodic boundary conditions. The same is true for u2+2k% (k & Z),and hence, by a suitable choice of k, we can assume that u0(t) = u2(t)+2k% ) u1(t)for all t & [0, T ]. Consequently, Theorem 14.4.1 implies the existence of at least oneT-periodic solution between u0 and u1, i.e. s & S.

To show that S is closed, we first need some a priori estimates on the possibleT-periodic solutions of (15.33). If u is such a solution and 5 & [0, T ] is such thatu$(5) = 0, integrating the equation implies that, for all t & [0, T ],

u$(t) =

! t

#e!c(t!+)[p($) + s! f($, u($))] d$

and hence

|u$(t)| )M1

for some M1 > 0 depending only upon c, p, s, T and a bound upon |f |. Consequently,for all t & [0, T ],

|u(t)| ) |u(0)| +! t

0|u$($)| d$ ) |u(0)| + TM1,


and, as we identify the solutions di!ering from a multiple of 2%, we can restrict u(0)in such that way that |u(0)| < 2%, so that, for all t & [0, T ],

|u(t)| < 2% + TM1 := M2.

From those a priori estimates, a limiting and compactness arguments easily impliesthat S is closed.

The following lemma [303] is useful in computing the Brouwer degree of I !PT

for pendulum-like equations.

Lemma 15.5.1 Let B = ]*!,*+[1 ],!,,+[ be an open rectangle and h : B " R2

be continuous and such that

h(*!, x2) = h(*+, x2) for all x2 & [,!,,+], (15.39)

h1(x1,,+) < 0 < h1(x1,,!) for all x1 & [*!,*+]. (15.40)

Then dB [h, B, 0] = 0.

Proof. Define the mapping g : [!1, 1]1 [!1, 1]" R2 by

g(y1, y2) = h{(1/2)[*+ ! *!)y1 + *+ + *!], (1/2)[(,+ ! ,!)y2 + ,+ + ,!]},

so that, as easily checked,

dB [h, B, 0] = dB[g, ]! 1, 1[ 2, 0],

and define the mapping g# : [!1, 1]2 " R2 by

g#(y1, y2) = g(!y1, y2).

By definition of degree,

dB[g, ]! 1, 1[ 2, 0] = !dB[g#, ]! 1, 1[ 2, 0].

On the other hand, it is easy to see that, by assumptions,

(1! ")g(y1, y2) + "g#(y1, y2) %= 0

whenever (y1, y2) & ![!1, 1]2, and hence, by the homotopy invariance property3.4.2, we have

dB[g, ]! 1, 1[ 2, 0] = dB[g#, ]! 1, 1[ 2, 0].

Hence dB[g, ]! 1, 1[ 2, 0] = 0.

15.5. PERIODIC SOLUTIONS OF PENDULUM-TYPE EQUATIONS 177

Somes simple estimates on Poincare’s operator are necessary to conclude.

Lemma 15.5.2 There exist - > 0 such that, for any (x0, y0) & R2 verifying |y0| >-, one has

y0[x0 ! x(T ; x0, y0)] > 0.

Proof. We know that the solution x(t; x0, y0) of (15.33) such that x(0) = x0, x$(0) =y0 satisfies the identities

x$(t; x0, y0) = y0 +

! t

0e!c(t!#)[p(5) + s! f(5, x(5 ; x0, y0))] d5,

and hence

x(T ; x0, y0)! x0 = y0T +

! T

0

! t

0e!c(t!#)[p(5) + s! f(5, x(5 ; x0, y0))] d5 dt.

Letting

M = max(t,x,s)"[0,T ]'R'[fL,fM ]

|p(t) + s!+f(t, x)|,

we obtain

y20T ! |y0|M ) y0x(T ; x0, y0)! x0


Proposition 15.5.3 Assume that the set of solutions of (15.33) is finite and givenby u1, . . . , um. Then

m"

j=1

.T (uj) = 0.

Proof. Let * %= uj(0) (j = 1, 2, . . . , m), and consider the rectangle R = ]*,* +2%[1 ]!-, -[ , where - > 0 is given by Lemma 15.5.2. It follows from Lemma 15.5.1that dB [I ! PT , R, 0] = 0, and the result follows from Proposition 5.1.1.

This result provides some information about the values of the indices of solutions.Let S = [s1, s2].

Corollary 15.5.1 Assume that the set of solutions of (15.33) is finite and givenby u1, . . . , um.

(i) If s = s1 or s = s2, then .T (uj) = 0 (j = 1, 2, . . . , m).

(ii) If s1 < s2 and s & ]s1, s2[ , there exist k, l & {1, 2, . . . , m such that .T (uk) < 0and .T (ul) = 1.


Proof. (i) If some index were not zero, the stability of the index under small per-turbations would imply the existence of solutions of (15.33) for s0 ! ' or for s1 + 'and ' > 0 su$ciently small, a contradiction.

(ii) As observed in the previous section, if s1 < s < s2, a T-periodic solution u1 of(15.33) with s = s1 is a strict upper solution for (15.33) with T-periodic boundaryconditions, and a T-periodic solution u2 of (15.33) with s = s2 is a strict lowersolution for (15.33) with T-periodic boundary conditions. By a suitable addition ofsome 2k% to one of them, one can assume that they are ordered, and hence Theorem14.5.1 implies that the index of at least one of the solution must be negative. Onthe other hand, Proposition 15.5.3 implies then that one of them must be positive.

Chapter 16

Guiding functions

16.1 Definition and preliminaries

We have already considered, in Chapter 14, the concept of a gradient system

y$(t) = !+V (y(t)),

where V & C1(Rn, R) has a locally Lipschitzian gradient +V. The T-periodic solu-tions of this gradient system are particularly simple because, if y is such a T-periodicsolution, then, we have

! T

0#y$(t)#2 dt = !

! T

06+V (y(t)), y$(t)7 dt = !

! T

0

d

dtV (y(t)) dt

= V (y(T ))! V (y(0)) = 0.

Consequently, each possible T-periodic solution of a gradient system is a constant,and hence a zero of +V. This observation and the analogy with Lyapunov’s secondmethod in stability theory (see e.g. [337]), has led M.A. Krasnosel’skii and hisschool to introduce the important concept of guiding or directing function inthe study of periodic solutions of ordinary di!erential equations (see e.g. [214],[219], [265], [337] for references).

Let f : R 1 Rn " Rn, (t, y) 2" f(t, y) be T-periodic in t and continuous. Weconsider the system of di!erential equations (14.1). The following concept wasintroduced by M.A. Krasnosel’skii and A.I. Perov in [216], [217].

Definition 16.1.1 We say that V & C1(Rn, R) is a guiding function for (14.1)if there exists -0 > 0 such that

6+V (x), f(t, x)7 > 0 (16.1)

for all t & R and all x & Rn such that #x# ( -0.

179

180 CHAPTER 16. GUIDING FUNCTIONS

So, for large x, the qualitative behavior of the vector field f is similar to that of+V. By (16.1), we have necessarily

+V (x) %= 0 (16.2)

for all x & Rn with #x# ( -0, and hence the Brouwer degree dB [+V, B(r), 0] isdefined and constant for all r ( -0. It is called the index at infinity of V anddenoted by iB[V,-]. It follows from Corollary 14.7.1 that if V is a coercive guidingfunction, then

iB[V,-] = 1. (16.3)

By (16.1) and 16.2), we have also, for all " & [0, 1], t & R and all x & Rn with#x# ( -0,

6+V (x), (1! ")+V (x) + "f(t, x)7 > 0,

and hence, by the homotopy invariance theorem 3.4.2, we get

iB[V,-] = dB[+V, B(r), 0] = dB[f(t, ·), B(r), 0] (16.4)

for all t & R and all r ( -0.

The existence of a guiding function implies some a priori localization of thepossible T-periodic solutions of eq. (14.1).

Lemma 16.1.1 If eq. (14.1) admits a guiding function V, and if we set

R0 = max)x)+(0

V (x), (16.5)

then any possible T-periodic solution y of eq. (14.1) is such that

V (y(t)) ) R0 for all t & R. (16.6)

Proof. Let y be a possible T-periodic solution of eq. (14.1). If the conclusion of thelemma does not hold, there exists t0 & R such that V (y(t0)) > R0. Consequently,

V (y(5)) = maxt"R

V (y(t)) > R0,

which implies that

#y(5)# > -0, 0 =d

dt[V (y(5))] = 6+V (y(5)), y$(5)7 = 6+V (y(5)), f(5, y(5))7,

a contradiction the the definition of a guiding function.

The following lemma can be found in [214] and [219].

Lemma 16.1.2 Assume that f is locally Lipschitzian with respect to y, that 5+(s, x)= +- for all the solutions of (14.1) with initial conditions y(s) = x, and that (14.1)admits a guiding function V. Then, for each t, s & R with t > s, there exists some-1 = -1(t, s) ( -0 such that for each x & Rn with #x# > -1(t, s) and each 5 & ]s, t],one has x %= y(5 ; s, x).

16.2. SYSTEMS WITH CONTINUABLE SOLUTIONS 181

Proof. For t > s, let

-1 = -1(t, s) = max{#y(5 ;$, x)# : s ) $ ) 5 < t, #x# ) -0},

so that -1 ( -0. We claim that this -1 satisfies the conclusion of Lemma 16.1.2. Ifit is not the case, there will exist x0 & Rn with #x0# > -1, and 50 & ]s, t] such that

y(50; s, x0) = x0. (16.7)

By the definition of -1, we have necessarily

#y(5 ; s, x0)# > -0

for all 5 & [s, 50]. Hence, letting v(5) = V (y(5 ; s, x0)), we get

v$(5) = 6+V (y(5 ; s, x0)), y$(5 ; s, x0)7 = 6+V (y(5 ; s, x0)), f(5, y(5 ; s, x0))7 > 0,

whenever 5 & [s, 50]. Therefore,

V (y(50; s, x0)) = v(50) > v(s) = V (x0),

a contradiction with (16.7).

The following Corollary is an easy consequence of Lemma 16.1.2.

Corollary 16.1.1 Assume that f is locally Lipschitzian with respect to y, T-pe-riodic with respect to t, that 5+(s, x) = +- for all the solutions of (14.1) withinitial conditions y(s) = x and that (14.1) admits a guiding function V. Then, thereexists some -1 = -1(T, 0) ( -0 such that for each x & Rn with #x# > -1 and each5 & ]0, T ], one has x %= y(5 ; 0, x).

16.2 Systems with continuable solutions

We can now state and prove an existence theorem for T-periodic solutions due toM.A. Krasnosel’skii and V.V. Strygin [218] (see also [214]-[219]).

Theorem 16.2.1 Assume that f is locally Lipschitzian with respect to y, T-perio-dic with respect to t, and that the following conditions hold :

1. (14.1) admits a guiding function V.

2. iB[V,-] %= 0.

3. 5+(s, x) = +- for each solution of (14.1) with initial conditions y(s) =x, (s & R, x & Rn).

Then (14.1) has at least one T-periodic solution y such that V (y(t)) ) R0 for allt & R and R0 defined in (16.5).


Proof. Let -1 = -1(T, 0) be given by Corollary 16.1.1, - > -1, and G = B(-) ' Rn.By Corollary 16.1.1, for each x & !G, we have y("T ; 0, x) %= x for all " & ]0, 1].By the existence of the guiding function, we have f(t, x) %= 0 for all t & R and allx & !G, and moreover

dB [f(0, ·), G, 0] = dB[+V, G, 0] = iB[V,-] %= 0,

by Assumption 3. The result then follows from Theorem 14.6.1 and Lemma 16.1.1.

The following consequence of Theorem 16.2.1, due to M.A. Krasnosel’skii andA.I. Perov [216], has assumptions which are more easy to verify in practice.

Corollary 16.2.1 Assume that f is locally Lipschitzian with respect to y, T-peri-odic with respect to t. If (14.1) has a guiding function V such that V or !V iscoercive, then (14.1) has as least one T-periodic solution y such that V (y(t)) ) R0

for all t & R and R0 defined in (16.5).

Proof. Assume first that !V is coercive. Then, letting W = !V, we see that

6+W (x), f(t, x)7 < 0,

for all t & R and all x & Rn with #x# ( -0. Hence, by a reasoning made in the proofof Theorem 14.7.1, we see that for each c ( -0, the sets W c = {x & Rn : W (x) ) c}are bounded (by the coercivity of W ) and positively invariant for the solutions of(14.1). Consequently, Assumption 3 of Theorem 16.2.1 holds. Moreover,

iB[V,-] = dB[+V, B(-0), 0] = (!1)ndB [+W, B(-0), 0] = (!1)n %= 0.

Thus all the assumptions of Theorem 16.2.1 are satisfied and (14.1) has a T-periodicsolution. If V is coercive, then letting u(t) = y(!t), we see that y is a T-periodicsolution of (14.1) if and only if u is a T-periodic solution of

u$(t) = !f(!t, u(t)). (16.8)

For this equation, W = !V is a guiding function such that !W is coercive, andthe first part of the proof implies the existence of a T-periodic solution u.

Example 16.2.1 Let p : R " R be a polynomial of odd degree and h : R " R acontinuous T-periodic function. Then equation

y$(t) = p(y(t)) + h(t)

has at least one T-periodic solution.Indeed, if

p(x) =2m+1"

j=0

ajxj ,

16.2. SYSTEMS WITH CONTINUABLE SOLUTIONS 183

with a2m+1 %= 0, let us take

V (x) =a2m+1

2x2.

Then V or !V is coercive, and

a2m+1x[p(x) + h(t)]

= a22m+1x

2m+2[1 +2m"

j=0

a!22m+1ajx

j!2m!1 + a!22m+1h(t)x!2m!1] > 0

for |x| su$ciently large. Thus V is guiding function for the above equation and theexistence follows.

As another application, we can prove the following existence theorem due to G.Gustafson and K. Schmitt [158].

Theorem 16.2.2 Assume that f is continuous, locally Lipschitzian in x, T-periodicin t, and that there exists R > 0 such that, for all t & R and all x & Rn with #x# = R,one has

6x, f(t, x)7 ( 0. (16.9)

Then eq. (14.1) has at least one T-periodic solution y such that #y(t)# ) R for allt & R.

Proof. Let us define g : R1 Rn " Rn by

g(t, x) = f(t, x) if #x# ) R

g(t, x) =

21! R

#x#

3x + f

2t,

R

#x#x3

if #x# ( R.

Then g is continuous, T-periodic in t and locally Lipschitzian in x. Furthermore,for all t & R and x & Rn with #x# > R, we have

6x, g(t, x)7 =

21! R

#x#

3#x#2

+#x#R

[R

#x#x, f

2t,

R

#x#x3\

> 0,

so that V (x) = )x)2

2 is a coercive guiding function for equation

y$ = g(t, y) (16.10)

if we choose any -0 > R. By Corollary 16.2.1, eq. (16.10) has at least one T-periodicsolution y such that #y(t)# ) -0 for all t & R and all -0 > R, and hence such that#y(t)# ) R for all t & R. Thus, y is a T-periodic solution of eq. (14.1).


Remark 16.2.1 The conclusion of Theorem 16.2.2 also holds if condition (16.9) isreplaced by

6x, f(t, x)7 ) 0 whenever t & R, #x# = R. (16.11)

An interesting consequence of Theorem 16.2.2 for the scalar equation

y$ = g(t, y) (16.12)

with g : R 1 R " R continuous, T-periodic in t, locally Lipschitzian in y, is thefollowing result.

Corollary 16.2.2 Assume that there exists A < B such that, for all t & R, onehas

g(t, A) ) 0 ) g(t, B) or g(t, A) ( 0 ( g(t, B). (16.13)

Then eq. (16.12) has at least one T-periodic solution such that, for all t & R,

A ) y(t) ) B.

Proof. Letting

u(t) = y(t)! A + B

2, f(t, u) = g

2t, u +

A + B

2

3,

we see that y is a T-periodic of eq. (16.12) if and only if u is a T-periodic solutionof equation

u$ = f(t, u). (16.14)

Now, conditions (16.13) become

f

2t,

A!B

2

3) 0 ) f

2t,

B !A

2

3or f

2t,

A! B

2

3( 0 ( f

2t,

B !A

2

3,

i.e. the one-dimensional version of (16.9) or of (16.11) with R = B!A2 .

As a special case which is useful in mathematical demography, let us considerthe Kolmogorov equation

u$ = uh(t, u), (16.15)

where h : R 1 R " R is continuous, T-periodic in t, locally Lipschitzian in u. Weare interested in the positive T-periodic solutions of eq. (16.15), and hence we set

u(t) = exp y(t),

which transforms eq. (16.15) into

y$ = h(t, exp y). (16.16)

16.3. SYSTEMS WITH NON-CONTINUABLE SOLUTIONS 185

Corollary 16.2.3 Assume that the following conditions hold.

1. h(t, 0) > 0 for all t & R.

2. h(t, C) ) 0 for some C > 0.

Then eq. (16.15) has at least one positive T-periodic solution.

Proof. It su$ces to prove that eq. (16.16) has a T-periodic solution. By assumption2, we have h(t, expB) ) 0 for B = log C and all t & R. By assumption 1 and T-periodicity, we have h(t, 0) ( * > 0 for all t & R and some * > 0, and hence thereexist A < B such thath(t, exp A) ( */2 > 0 for all t & R. The result follows fromCorollary 16.2.2.

In particular, the conditions of Corollary 16.2.3 are satisfied for Verhulst equa-tion with T-periodic continuous positive coe$cients a and b

u$ = u[a(t)! b(t)u].

Notice that, in this special case, the transformation u = 1v transforms the problem

into the first order linear di!erential equation

v$ = !a(t)v + b(t),

for which the (unique) positive T-periodic can be computed explicitely.Further applications of the method of guiding functions can be found in [214,

219].

16.3 Systems with non-continuable solutions

We now describe a generalization of Theorem 16.2.1, which can be found in [215],and does not require the global extendability of the solutions of the Cauchy problem.Its proof requires the introduction of the following concepts. Consider the value

*(-0) = inf0+t0+T ; )x0)+(0

[5+(t0, x0)! 5!(t0, x0)]. (16.17)

By a classical result (see e.g. [231]), 5+ ! 5! is a lower semicontinuous function,and hence the infimum is reached. Therefore, this value is equal to either +- orsome positive number. Without special notice we shall use the equalities

*(-0) = inf#+t0+T+# ; )x0)+(0

[5+(t0, x0)! 5!(t0, x0)], 5 & R.

and*(-0) = inf

t0"R; )x0)+(0

[5+(t0, x0)! 5!(t0, x0)],

which follow from the T-periodicity of f with respect to t.


Definition 16.3.1 A solution y(t) of (14.1) is called a (T, -0, -1)-solution of (14.1)(-1 > -0) if it is defined on some closed bounded interval [t1, t2] and

(i) #y(t1)# = #y(t2)# = maxt1+t+t2 #y(t)# = -1,

(ii) #y(t0)# ) -0 for some t0 & (t1, t2),

(iii) t2 ! t1 ) T.

Remark 16.3.1 According to the periodicity of the function f(t, x) we can assumewithout loss of generality that the values t0, t1 t2 satisfy the inequalities

0 ) t1 ) T, 0 ) t1 < t0 < t2 ) 2T.

Remark 16.3.2 If (14.1) has some (T, -0, -1)-solution y(t) (t1 ) t ) t2) then(14.1) has a (T, -0, -)-solution for any - & ]-0, -1[. Indeed, consider the values

t1,( = max {t & [t1, t0], #y(t)# = -} , t2,( = min {t & [t0, t2], #y(t)# = -} .

The function y(t) considered on the interval [t1,(, t2,(] is the (T, -0, -)-solution of(14.1).

Lemma 16.3.1 Assume that f is T-periodic in t, locally Lipschitzian in y andcontinuous. If T < *(-0), then there exist -# > -0 such that equation (14.1) has no(T, -0, -1)-solution for -1 ( -#.

Proof. Due to Remark 16.3.2, it is su$cient to prove the existence of -# > -0 suchthat the system (14.1) has no (T, -0, -#)-solution. If it is not the case, we can finda sequence (-n) with -n > -0 and -n "- such that, for every n the system(14.1)has a (T, -0, -n)-solution yn(t) (tn1 ) t ) tn2 ). It means that

#yn(tn1 )# = #yn(tn2 )# = maxtn1 +t+tn

2

#yn(t)# = -n,

and #yn(tn0 )# ) -0 for some tn0 & ]tn1 , tn2 [ . Due to Remark 16.3.1 we can assume that

0 ) tn1 ) T, 0 ) tn1 < tn0 < tn2 ) 2T, tn2 ! tn1 ) T.

Without loss of generality (consider if necessary some subsequences), we can sup-pose that all the sequences (tn1 ), (tn0 ), (tn2 ) and (yn(tn0 )) converge to some limitst#1, t

#0, t

#2, y

#0 . Obviously, 0 ) t#1 ) T, t#1 ) t#0 ) t#2 ) 2T, t#2 ! t#1 ) T, #y#

0# ) -0.Consider the solution y#(t) = y(t; t#0, y

#0) of (14.1). Since by assumption the inequal-

ityT < 5+(t#0, y

#0)! 5!(t#0, y

#0)

holds, then the solution y#(t) is defined at least on one of the closed intervals[t#1, t

#0], [t#0, t

#2], say the interval [t#1, t

#0]. Now,

yn(tn1 ) = y(tn1 ; tn0 , yn(tn0 )), y#(tn1 ) = y(tn1 ; t#0, y#0),

16.3. SYSTEMS WITH NON-CONTINUABLE SOLUTIONS 187

and the classical theorem on the continuity of solutions with respect to initial values(see e.g. [71]) implies that for su$ciently large n the estimate

#yn(tn1 )! y#(tn1 )# < 1

is valid; therefore-n = #yn(tn1 )# < 1 + #y#(t#1)# <-,

so thatlim sup

n(%-n ) 1 + #x#(t#1)# <-.

This contradiction proves the lemma.

We now prove a result on the existence of a priori estimates for the periodicsolutions of (14.1).

Lemma 16.3.2 Assume that the conditions of Lemma 16.3.1 hold and that (14.1)has a guiding function V. Then each possible T -periodic solution y(t) of system(14.1) satisfies the a priori estimate

#y(t)# < -#, t & R.

Proof. Let the statement of the lemma be false. Then (14.1) has a T -periodicsolution y(t) such that

max0+t+T

#y(t)# = r ( -#. (16.18)

Consequently, either

min0+t+T

#y(t)# ( -0, (16.19)

or for some t0 & [0, T ]

#y(t0)# < -0. (16.20)

Let (16.19) be valid. Put

v(t) = V [y(t)], t & R. (16.21)

Due to the T -periodicity of y(t), the real function (16.21) is also T -periodic. Onthe other hand

v$(t) = (f [t, y(t)], V $[y(t)]) > 0,

which contradicts the periodicity of v(t). Let (16.20) be valid for some t0 & [0, T ].Put

t1 = max {t < t0 : #y(t)# = -#} , t2 = min {t > t0 : #y(t)# = -#}

Since y(t) is T -periodic, t2 ! t1 ) T . Therefore y(t) is a (T, -0, -#)-solution of(14.1). But Lemma 16.3.1 implies that (14.1) has no (T, -0, -#)-solutions. Thiscontradiction proves the lemma.


We can now state and prove the following existence theorem ([215]).

Theorem 16.3.1 Assume that f is T-periodic with respect to T, locally Lipschitzi-an with respect to y and continuous. Assume that (14.1) has a guiding function Vsuch that iB[V,-] %= 0 and that T < *(-0). Then equation (14.1) has at least oneT-periodic solution y with maxt"R #y(t)# < -#.

Proof. Let,(r) = max

0+t+T, |y|+r#f(t, y)#, 0 ) r <-.

This function is non-decreasing and since

6f(t, y),+V (y)7 > 0, #y# ( -0

its values are positive for r ( -0. Put

&(y) =

*+

,

1 if 0 ) #y# ) -#linear if -# ) #y# ) 1 + -#

(1 + ,(#y#))!1 if 1 + -# ) #y#.

Consider the auxiliary equation

dy

dt= &(y)f(t, y). (16.22)

Since the right-hand side of (16.22) is uniformly bounded we see that all the solutionsof (16.22) are infinitely continuable both to the left and to the right. Since, &(y) > 0for all y & Rn, the function V (y) is a guiding function for system (16.22), with thesame-0, and the index at infinity of the guiding function V (y) is di!erent from 0.Therefore Theorem 16.2.1 implies the existence of at least one T -periodic solutionof (16.22). Let us show that every T -periodic solution y(t) of (16.22) is a T -periodicsolution of (14.1), i.e. let us prove the estimate

#y(t)# ) -#, t & R. (16.23)

We shall use constructions which has been already used in the proof of Lemma16.3.2. Assume (16.23) is not valid. If

min0+t+T

#y(t)# ( -0

then the T -periodic scalar-valued function w(t) = V [y(t)] (t & R) satisfies theestimate

dw(t)

dt= &[y(t)] 6f(t, y(t)),+V [y(t)]7 > 0,

which is impossible. Now, if #y(t0)# < -0, for some t0 & [0, T ] then, for somet1 < t1 with t2 ! t1 < T, y(t) (t1 ) t ) t2) is a (T, -0, -#)-solution of (16.22).Therefore #y(t)# ) -# for t & [t1, t2] and y(t) is a (T, -0, -#)-solution of system(14.1). This contradicts Lemma 16.3.1. Hence (16.23) is proved, which implies thatevery T -periodic solution of (16.22) is a T -periodic solution of (14.1).

16.4. GENERALIZED GUIDING FUNCTIONS 189

16.4 Generalized guiding functions

In this section, we extend Theorem 16.3.1 by weakening the definition of a guidingfunction. The following concept was introduced in [265].

Definition 16.4.1 We say that V & C1(Rn, R) is a generalized guiding func-tion for (14.1) if there exists some -0 > 0 such that

6f(t, x),+V (x)7 ( 0,

for all t & R and all x & Rn such that #x# ( -0.

We want to replace in Theorem 16.3.1 the assumption of the existence of a guidingfunction by that of a generalized guiding function V such that +V (x) %= 0 for#x# ( -0. A natural idea to prove such a result would be to consider the perturbedproblems

y$ = f(t, y) + 6+V (y),

for which V is a guiding function, as immediately checked. But +V is not neces-sarily locally Lipschitzian, and hence it is necessary to replace it by another locallyLipschitzian function having the same qualitative behavior.

Lemma 16.4.1 Let V & C1(Rn, R) be a generalized guiding function for (14.1)such that +V (x) %= 0 whenever #x# ( -0. Then there exists a locally Lipschitzianfunction g(x) : Rn " Rn such that

6+V (x), g(x)7 > 0, #x# ( -0. (16.24)

Proof. The continuity of+V implies that for every x, #x# ( -0 there is a r(x) & ]0, 1]such that

6+V (y),+V (x)7 > 0, y & B(x; r(x)). (16.25)

The open balls Bx(r(x)) form a cover of the closed set F = {x : #x# ( -0} ' Rn.Let us choose from this cover a countable subcover

Bxi(r(xi)), i = 1, 2, . . . (16.26)

such that every point x belongs to a finite number only of balls (16.26). Define forevery i = 1, 2, . . . the function ,i : Rn " R, x 2" dist(x, Rn \Bxi(r(xi))). Each ,i islocally Lipschitzian. Put, for x & Rn,

g(x) =%"

i=1

,i(x)+V (xi), x & Rn. (16.27)

In this sum, for every x, a finite number of values ,i(x) only di!er from 0. Thefunction (16.27) is obviously locally Lipschitzian on Rn, and

6+V (x), g(x)7 =%"

i=1

,i(x) 6+V (x),+V (xi)7

="

{i:x"Bxi (r(xi))}

,i(x) 6+V (x),+V (xi)7 > 0.


We can now state and prove the main result of this section, which is taken from[215].

Theorem 16.4.1 Assume that f is T-periodic with respect to T, locally Lipschitzi-an with respect to y and continuous. Assume that (14.1) has a generalized guidingfunction V such that +V (x) %= 0 whenever #x# ( -0 and iB[V,-] %= 0. If T < *(-0),then equation (14.1) has at least one T-periodic solution x with maxt"R #x(t)# ) -#.

Proof. Let g be given by Lemma 16.4.1 and consider the family of equations

y$ = f(t, y) + 6g(y), (16.28)

with 6 ( 0. Let ]5!(s, y, 6), 5+(s, y, 6)[ denote the maximal existence interval ofthe solution of the Cauchy problem y(s) = x, for equation (16.28). We know (seee.g. [231]) that 5+ ! 5! is lower semicontinuous. In particular, 5+ ! 5! reaches itsinfimum on each compact set. We first claim that there exist 6# > 0 such that

T < inf0+s+T ;)x)+(0;0+/+/$

[5+(s, x, 6)! 5!(s, x, 6)].

If it is not the case, then, for each positive integer k, we have

T ( inf0+s+T ;)x)+(0;0+/+1/k

[5+(s, x, 6)! 5!(s, x, 6)] = 5+(sk, xk, 6k)! 5!(sk, xk, 6k),

for some sk & [0, T ], #xk# ) -0, 6k & [0, 1/k]. Going if necessary to a subsequence,we can assume that sk " s# and xk " x# for some s# & [0, T ] and #x## ) -0.Therefore, by lower semicontinuity,

T ( lim infk(%

[5+(sk, xk, 6k)! 5!(sk, xk, 6k)] ( 5+(s#, x#, 0)! 5!(s#, x#, 0) > T,

a contradiction. Let (6n) be a sequence in ]0, 6#] which converges to zero. For eachpositive integer n, equation (16.28) with 6 = 6n satisfies the conditions of Theorem16.3.1 and hence has at least one T-periodic solution yn(t), such that

#yn(t)# ) -#, t & R.

Since

#y"

n(t)# ) max0+t+T, )y)+($

#f(t, y)#+ 6# max)y)+($

#g(y)#, t & R; n = 1, 2, . . . ,

the sequence (y"

n) is bounded and hence the sequence (yn) is compact in C = C[0, T ].Therefore one can choose a subsequence (ynk) (k = 1, 2, . . .) converging in C tosome function y. Every function ynk is T-periodic and satisfies the integral equation

ynk(t) = ynk(0) +

! t

0f [s, ynk(s)] ds + 6nk

! t

0g[ynk(s)]ds.

Consequently, letting k going to infinity, we see that y is T-periodic and is a solutionof

y(t) = y(0) +

! t

0f [s, y(s)] ds,

and hence a T-periodic solution of the di!erential equation (14.1).

16.4. GENERALIZED GUIDING FUNCTIONS 191

Remark 16.4.1 The above proof implies the estimate #y(t)# ) -# (t & R). Thisestimate does not mean that there is an a priori estimate of all the T-periodicsolutions of equation (14.1). Under assumptions of Theorem 16.3.1, T -periodicsolutions of (14.1) can have an arbitrary large norm in C. In those cases theycannot be constructed by the procedure used above in the proof of Theorem 16.4.1.A very simple example is given by the equation y$ = 0.

The application of Theorem 16.4.1 requires precise estimates of the value *(-0).An answer to this problem makes use of theorems on di!erential inequalities (seee.g. [214],[236]). For example, let M(u) (u ( 0) be a positive and continuousfunction. Natural examples of M(u) are the functions

a + bup (p ( 1), a + bu ln (1 + u), a + beu, a + beu2

, (16.29)

etc. Denote by u(t) (t ( 0) a solution of Cauchy problem

du

dt= M(u), u(0) = -2

0.

The solution u(t) increases (since M(u) > 0). Let [0, m(-0)[ be the maximal intervalwhere u(t) is defined. Assume that

6f(t, x), x7 ) 1

2M(#x#2), #x# ( -0. (16.30)

Consider some solution y(t) of (14.1) satisfying

y(t0) = x0, #x0# = -0, #y(t)# ( -0, (t ( t0).

Assumption (16.30) (for t > t#) implies

d

dt#y(t)#2 = 2

[dy

dt(t), y(t)

\= 2 6f [t, y(t)], y(t)7 )M

N#y(t)#2

O.

Therefore the estimate#y(t)#2 ) u(t! t0)

holds for t & [t0, t0 + m(-0)[. We have proved the following result.

Proposition 16.4.1 The estimate (16.30) guarantees an estimate

5+(t0, x0) ( t0 + m(-0)

for every t0 & R and #x0# ( -0.

The next statement is dual to Proposition 16.4.1 and can be proved analogously.Consider one function M1 of type (16.29). Let the function M1(u) (u ( 0) bepositive and continuous. Denote by u1(t) (t ( 0) a solution of Cauchy problem

du1

dt= M1(u), u1(0) = -2

0.

Let [0, m1(-0)[ be the maximal interval where u1(t) is defined.


Proposition 16.4.2 The estimate

6f(t, x), x7 ( !1

2M1(#x#2), #x# ( -0 (16.31)

guarantees an estimate5!(t0, x0) ) t0 !m1(-0)

for every t0 & R and #x0# ( -0.

Condition (16.30) implies *(-0) ( m(-0), condition (16.31) implies *(-0) ( m1(-0).If both conditions hold then

*(-0) ( m(-0) + m1(-0),

In this case the assumption over T of Theorem 16.4.1 can be rewritten as

T < m(-0) + m1(-0).

A suitable choice of functions M(u) and M1(u) makes possible the obtention ofsharp estimates for the period T .

If we cannot estimate the value of *(-0), then Theorem 16.4.1 has the followingmeaning : the existence of a generalized guiding function of non-zero index at infinityguarantees the existence of T -periodic solutions for su!ciently small T > 0, i.e. ofoscillations with su$ciently high frequency.

As a special case of Theorem 16.4.1, we can obtain the following improvementof Corollary 16.2.1, which was first proved in [265]. Its proof is similar to that ofCorollary 16.2.1.

Corollary 16.4.1 Assume that f is locally Lipschitzian with respect to y, T-pe-riodic with respect to t. If (14.1) has a generalized guiding function V such that+V (x) %= 0 whenever #x# ( -0 and V or !V is coercive, then (14.1) has as leastone T-periodic solution.

Chapter 17

Computing degree indimension two

17.1 The Kronecker index on a closed simple curve

Let D ' R be an open bounded set such that !D is a closed Jordan curve of classC2, and let f & C1(D, R) be such that 0 %& f(!D). Recall that the Kronecker indexof f on !D is defined by

iK [f, !D] =

!

!D#f#!2[f1 df2 ! f2 df1].

So, if !D has the parametric representation & : [0, 2%] " R2, with &(0) = &(2%)and &(s) %= &(s$) for s %= s$ in [0, 2%[, we have

iK [f, !D] =1

2%

! 2"

0

f1[&(s)] ddsf2[&(s)]! f2[&(s)] d

dsf1[&(s)]

f21 [&(s)] + f2

2 [&(s)]ds

=1

2%

! 2"

0

F1(s)F $2(s)! F2(s)F $

1(s)

F 21 (s) + F 2

2 (s)ds, (17.1)

if we write

Fj(s) = fj(&(s)), (j = 1, 2).

Theorem 17.1.1 Assume that F1 has a finite number of zeros

0 ) s1 < s2 < . . . < sN < 2%,

and that

F $1(sk) %= 0, (1 ) k ) N).

193

194 CHAPTER 17. COMPUTING DEGREE IN DIMENSION TWO

Then

iK [f, !D] = !1

2

N"

k=1

sign F2(sk)sign F $1(sk). (17.2)

Proof. For 6 > 0 su$ciently small and 1 ) k ) N, we have

! sk+1!/

sk+/

F1(s)F $2(s)! F2(s)F $

1(s)

F 21 (s) + F 2

2 (s)ds =

! sk+1!/

sk+/

-arctan

F2(s)

F1(s)

.$ds

= arctanF2(sk+1 ! 6)

F1(sk+1 ! 6)! arctan

F2(sk + 6)

F1(sk + 6). (17.3)

Now,

lim/(0

arctanF2(sk+1 ! 6)

F1(sk+1 ! 6)= lim

/(0arctan

F2(sk+1 ! 6)

F1(sk+1 ! 6)! F1(sk+1)

= !%

2sign F2(sk+1) sign F $

1(sk+1), (17.4)

lim/(0

arctanF2(sk + 6)

F1(sk + 6)= lim

/(0arctan

F2(sk + 6)

F1(sk + 6)! F1(sk)

=%

2sign F2(sk) sign F $

1(sk).

Consequently, using (17.1) and (17.4), we obtain, in the case where 0 < s1 (the cases1 = 0 is similar)

2%iK(f, !D) = lim/(0

-! s1!/

0

F1(s)F $2(s)! F2(s)F $

1(s)

F 21 (s) + F 2

2 (s)ds

+N!1"

k=1

! sk+1!/

sk+/

F1(s)F $2(s)! F2(s)F $

1(s)

F 21 (s) + F 2

2 (s)ds

+

! 2"

sN+/

F1(s)F $2(s)! F2(s)F $

1(s)

F 21 (s) + F 2

2 (s)ds

.

= lim/(0

/arctan

F2(s1 ! 6)

F1(s1 ! 6)! arctan

F2(0)

F1(0)

+N!1"

k=1

-arctan

F2(sk+1 ! 6)

F1(sk+1 ! 6)! arctan

F2(sk + 6)

F1(sk + 6)

.

+ arctanF2(2%)

F1(2%)! arctan

F2(sN + 6)

F1(sN + 6)

0

= !%

2

N+1"

k=0

sign F2(sk+1) sign F $1(sk+1)

! %

2

N"

k=1

sign F2(sk) sign F $1(sk).

17.1. THE KRONECKER INDEX ON A CLOSED SIMPLE CURVE 195

Hence,

iK [f, !D] = !1

2

N"

k=1

sign F2(sk) sign F $1(sk).

Example 17.1.1 Let

f : R2 " R

2, (x1, x2) 2" (x21 ! x2

2, 2x1x2), (17.5)

and !D := !B(1), with parametrization &(s) = (cos s, sin s). Then

F1(s) = cos2 s! sin2 s = cos 2s, F2(s) = 2 cos s sin s = sin 2s,

F!11 ({0}) . [0, 2%[=

/%

4,3%

4,5%

4,7%

4

0, F2(s)F

$1(s) = !2 sin2 2s,

and hence iK [f, !B(1)] = ! 12 (!4) = 2.

Remark 17.1.1 Formula (17.2) can be interpreted geometrically, if we rememberthat iK [f, !D] is the total number of anticlockwise rotations of f * & around zero.By continuity, this algebraic number of rotations of f * &(s) when s varies from 0to 2% is equal to the di!erence between the number of zeros of F1 at which F2 > 0and F $

1 < 0 and the number of zeros of F1 at which F2 > 0 and F $1 > 0. In other

words,

iK [f, !D] = !"

k:F2(sk)>0

sign F $1(sk) = !

"

k:F2(sk)>0

sign [F $1(sk)F2(sk)]

= !"

k:F2(sk)>0

sign F $1(sk) sign F2(sk). (17.6)

But this algebraic number of rotations is as well equal to the di!erence between thenumber of zeros of F1 at which F2 < 0 and F $

1 > 0 and the number of zeros of F1

at which F2 < 0 and F $1 < 0. In other words

iK [f, !D] ="

k:F2(sk)<0

sign F $1(sk) = !

"

sk:F2(sk)<0

sign [F $1(sk)F2(sk)]

= !"

k:F2(sk)<0

sign F $1(sk) sign F2(sk). (17.7)

Therefore, by adding both members of those two formulas, we get

iK [f, !D] = !1

2

N"

k=1

sign [F2(sk)F $1(sk)] = !1

2

N"

k=1

sign F2(sk) sign F $1(sk),

which is (17.2).


Remark 17.1.2 If we write F1(sk ! 0) (resp. F1(sk + 0)) for F1(sk ! ') (resp.F1(sk + ') for su$ciently small ' > 0, formula (17.2) can be written as well

iK [f, !D] =1

2

N"

k=1

sign F2(sk) sign [F1(sk ! 0)! F1(sk + 0)]

=1

4

N"

k=1

sign F2(sk)[ sign F1(sk ! 0)! sign F1(sk + 0)], (17.8)

as F1(sk) = 0 (1 ) k ) N).

Remark 17.1.3 Let 0 ) r1 be the smallest zero of F2 in [0, 2%[. Then !D can beparametrized as well on [r1, r1 + 2%] and F+

2 and F!2 are open subsets contained in

[r1, r1 + 2%]. Hence, formulas (17.6) and (17.7) can be respectively rewritten as

iK [f, !D] = !dB[F1, F+2 , 0], iK [f, !D] = dB [F1, F

!2 , 0],

so that

iK [f, !D] =1

2{dB[F1, F

!2 , 0]! dB [F1, F

+2 , 0]} (17.9)

Example 17.1.2 For the mapping (17.5) and D = B(1), we have

F+2 = ]0,%/2[/ ]%, 3%/2[, F!

2 = ]%/2,%[/ ]3%/2, 2%[,

and hence

iK [f, !B(1)] = !1

2{dB[cos 2·, ]0,%/2[, 0] + dB [cos 2·, ]%, 3%/2[ 0]}

+1

2{dB[cos 2·, ]%/2,%[, 0] + dB[cos 2·, ]3%/2, 2%[ 0]}

= !1

2(!1! 1) +

1

2(1 + 1) = 2.

17.2 Factorized multilinear mappings

Definition 17.2.1 A multilinear mapping f : R2 " R2 is an application of theform

f1(x1, x2) =m"

i=0

aixm!i1 xi

2, f2(x1, x2) =n"

j=1

bjxn!j1 xj

2, (17.10)

where the ai and bj are real numbers. f is said to be nondegenerate if f!1({0}) ={0}, and degenerate in the opposite case.

17.2. FACTORIZED MULTILINEAR MAPPINGS 197

The zeros of f1 are located on the straight lines through the origin whose angularcoe$cients are the real solutions of the equation

f1(1, k) :=m"

i=0

aiki = 0, (17.11)

(including the solution k = +- if the degree of f1(1, k) is smaller than m, i.e. ifam = 0), and the zeros of f2 are located on the straight lines through the originwhose angular coe$cients are the solutions of the equation

f2(1, k) :=n"

j=0

bjkj = 0, (17.12)

(including the solution k = +- if the degree of f2(1, k) is smaller than n, i.e. ifbn = 0). f is therefore nondegenerate if the two equations (17.11) and (17.12) haveno common root (including k = +-). For a nondegenerate multilinear mapping f,the Brouwer degree dB [f, B(r), 0] exists for each r > 0, and is independent of r.

Let us assume f nondegenerate and write it in a factorized form

f1(x1, x2) = K1

em4

i=1

(x2 ! kix1)pi , f2(x1, x2) = K2

en4

j=1

(x2 ! ljx1)qj , (17.13)

where the ki (1 ) i ) 1m) (resp. lj (1 ) j ) 1n)) are the distinct roots of equation(17.11) (resp. (17.12)), with respective multiplicities pi and qj . If some of the rootsare equal to +-, the corresponding factor will be taken equal to !x1. If we defineg : R2 " R2 by

g1(x1, x2) :=em4

i=1

(x2 ! kix1)pi ,

g2(x1, x2) :=en4

j=1

(x2 ! ljx1)qj ,

then

dB[f, B(r), 0] = sign K1 sign K2 dB [g, B(r), 0],

and it su$ces to compute the last degree.

One can neglect several factors of g without changing its degree. The followinglemmas as well as the final theorem have been proved in 1954 by J. Cronin [79](seen also [81]).

Lemma 17.2.1 One can suppress the pairs of factors (x2 ! ki1x1)(x2 ! ki2x1) inwhich ki1 = ki2 without changing dB[g, B(r), 0].


Proof. In this case,

(x2 ! ki1x1)(x2 ! ki2x1) = x22 + |ki1 |2x2

1,

and pi1 = pi2 . Using the homotopy G defined by

G1(x1, x2,") :=4

i1=ii1 ,ii2

(x2 ! kix1)pi [" + (1! ")(x2 ! ki1x1)

pi1 (x2 ! ki2x1)pi2 ]

G2(x1, x2,") := g2(x1, x2), " & [0, 1],

one easily shows that G(·,")!1({0}) = {0} for each " & [0, 1]. Theorem 3.4.2 impliesthat

dB [g, B(r), 0] = dB[G(·, 0), B(r), 0] = dB[G(·, 1), B(r), 0].

Lemma 17.2.2 One can suppress the factors (x2! ki1x1)pi1 , where ki1 is real andpi1 is even, without changing dB[g, B(r), 0].

Proof. One uses in this case the homotopy G defined by

G1(x1, x2,") :=4

i1=ii1

(x2 ! kix1)pi [" + (1! ")(x2 ! ki1x1)

pi1 ],

G2(x1, x2,") := g2(x1, x2), " & [0, 1].

Lemma 17.2.3 One can suppress the pairs of factors (x2 ! ki1x1), (x2 ! ki2x1)such that ki1 < ki2 and there exists no lj or ki3 verifying the inequalities

ki1 < lj < ki2 or ki1 < ki3 < ki2

without changing dB[g, B(r), 0].

Proof. To show it, one uses the homotopy G defined by

G1(x1, x2,") :=

&

'4

i1=i1,i2

(x2 ! kix1)pi

(

) (x2 ! ki1) [x2 ! [(1! ")ki2 + "ki1 ]x1] ,

G2(x1, x2,") := g2(x1, x2), " & [0, 1],

which, for " = 1, reduces the problem to the case of Lemma 17.2.2.

Lemma 17.2.4 One can suppress the pairs of factors (x2 ! k1x1), (x2 ! k emx1),where k1 and k em are the smallest and the largest among the numbers

k1, . . . , k em, l1, . . . , len

(uniquely in the case where both belong to the ki) without changing dB[g, B(r), 0].

17.2. FACTORIZED MULTILINEAR MAPPINGS 199

Proof. One uses in this case the homotopy G defined by

G1(x1, x2,") :=

:em!14

i=1

(x2 ! kix1)pi

;H(x1, x2,")[(1! "x2 ! k emx1],

G2(x1, x2,") := g2(x1, x2), " & [0, 1],

where

a. if k1 < 0, H(x1, x2,") = (1 ! ")x2 ! k1x1, which reduces the problem, for" = 1, to the case of Lemma 17.2.2;

b. if k1 = 0, H(x1, x2,") = x2!"(!')x1, with ' > 0, which reduces the problemto case a when " = 1;

c. if k1 > 0, H(x1, x2,") = x2 ! (1! ")k1x1, which reduces the problem to caseb when " = 1.

Lemma 17.2.5 One can suppress the factors listed above with the ki replaced bythe lj . without changing dB[g, B(r), 0].

If all the factors of g1 or of g2 are included in the five types listed above, thendB[g, B(r), 0] = 0, as g is homotopic to a mapping having one of its componentsconstant. If it is not the case, one is reduced to compute the degree of a mapping gwhose both components have the same number r of factors (x2! kix1), (x2! ljxi),where the ki and lj are such that either

k1 < l1 < k2 < l2 < . . . < kr < lr or l1 < k1 < l2 < k2 < . . . < lr < kr, (17.14)

and all the exponents pi and qj equal to one.

Theorem 17.2.1 If the nondegenerate mapping f given by formula (17.13) is suchthat

1m! 1n = 1 (mod 2), (17.15)

then dB [f, B(r), 0] = 0.

Proof. The operations given by the lemmas above always consist in suppressing aneven number of linear factors in one or another component of g. Hence, if condition(17.15) holds, one always obtains by reduction a component which is constant.

It remains to consider the case of a multilinear mapping h : R" R of the form

h1(x1, x2) =r4

i=1

(x2 ! kix1), h2(x1, x2) =r4

j=1

(x2 ! ljx1), (17.16)

where the ki and lj verify condition (17.14).


Theorem 17.2.2 If the nondegenerate mapping f given by formula (17.13) is re-duced to the form (17.16), then

dB[f, B(r), 0] = r sign (K1K2) or dB[f, B(r), 0] = !r sign (K1K2),

according to the first or the second condition holds in (17.14).

Proof. As f & C%(R, R) and is homogeneous,

dB[f, B(r), 0] = sign (K1K2) dB[h, B(r)] = sign (K1K2) iK [h, !B(r)]

= sign (K1K2) iK [h, !B(1)],

and we use formula 17.2 to compute iK [h, !B(1)]. Using the parametrization

x1 = cos s, x2 = sin s (s & [0, 2%]),

for !B(1), and letting

H1(s) := h1(cos s, sin s) =r4

i=1

(sin s! ki cos s),

H2(s) := h2(cos s, sin s) =r4

j=1

(sin s! lj cos s),

we see that the zeros of H1(s) are given by

sn = arctankn, sr+n = arctankn + % = sn + %, (1 ) n ) r),

so that

H2(sn) = (cos sn)rr4

j=1

(kn ! lj) (1 ) n ) r),

H2(sr+n) = (cos(sn + %))rr4

j=1

(kn ! lj) (1 ) n ) r).

Now,

H $1(s) =

r"

j=1

4

i1=j

(sin s! ki cos s)(cos s + kj sin s),

so that

H $1(sn) = (cos sn)r(1 + k2

n)4

i1=n

(kn ! ki) (1 ) n ) r),

H $1(sr+n) = (cos(sn + %))r(1 + k2

n)4

i1=n

(kn ! ki) (1 ) n ) r).

17.3. USING STURM SEQUENCES 201

Therefore,

sign [H2(sn)H $1(sn)] = sign

&

'r4

j=1

(kn ! lj)4

i1=n

(kn ! ki)

(

)

= sign [H2(sr+n)H $1(sr+n)] (1 ) n ) r).

So formula (17.2) gives

iK [h, !B(1)] = !r"

n=1

sign

&

'r4

j=1

(kn ! lj)4

i1=n

(kn ! ki)

(

)

= !r"

n=1

&

'r4

j=1

sign (kn ! lj)4

i1=n

sign (kn ! ki)

(

) .

Consequently, if the first condition in (17.14) holds, one has

iK [h, !B(1)] = !r"

n=1

(!1)r!n+1(!1)r!n = r,

and if the second condition holds, one has

iK [h, !B(1)] = !r"

n=1

(!1)r!n(!1)r!n = !r.

Example 17.2.1 Let us consider again the mapping (17.5). We have

K1 = !1, K2 = !2, r = 2, k1 = !1 < l1 = 0 < k2 = +1 < l2 = +-,

and dB[f, B(r), 0] = 2.

17.3 Using Sturm sequences

Let us now consider the nondegenerate multilinear mapping f : R2 " R2 definedby formula (17.10), and assume that, say,

m = n (mod 2) and m ( n, am %= 0. (17.17)

If the first condition is not satisfied, the Brouwer degree is zero. The last conditioncan be realized by permuting the components and/or the variables of f.


Proposition 17.3.1 Let the nondegenerate multilinear mapping f : R2 " R2 givenby formula (17.10) satisfy the assumptions (17.17). Let

k1 < k2 < . . . < kr (r ) m)

be the real roots of the equation

f1(1, y) := a0 + a1y + . . . + amym = 0.

Then

dB [f, B(1), 0] =1

2

r"

l=1

sign [f2(1, kl)][sign f1(1, kl ! 0)! sign f1(1, kl + 0)].(17.18)

Proof. The points of !B(1) where f1(x1, x2) = 0 verify the system of equations

x21 + x2

2 = 1, x2 = kix1 (1 ) i ) r), (17.19)

With the notations of the previous section, those points can be written

/l = (cos sl, sin sl), (1 ) l ) 2r),

where

sl = arctankl, sr+l = arctankl + % = sl + %, (1 ) l ) r),

so that

/r+l = !/l (1 ) l ) r), tan sl = tan sr+l = kl (1 ) l ) r).

Furthermore,

F1(s) = f1(cos s, sin s) = (cosm s) f1(1, tan s),

F2(s) = f2(cos s, sin s) = (cosn s) f2(1, tan s).

Therefore, using formula (17.8), we get

iK [f, !B(1)]

=1

2

r"

l=1

sign [cosn slf2(1, kl)]sign {cosm sl[f1(1, kl ! 0)! f1(1, kl + 0)]}

+1

2

r"

l=1

sign [cosn(sl + %)f2(1, kl)]sign {cosm(sl + %)[f1(1, kl ! 0)! f1(1, kl + 0)]}

=1

2

r"

l=1

sign [cosn+m slf2(1, kl)]sign [f1(1, kl ! 0)! f1(1, kl + 0)]

+1

2

r"

l=1

sign [cosn+m(sl + %)f2(1, kl)]sign [f1(1, kl ! 0)! f1(1, kl + 0)]

=r"

l=1

sign [f2(1, kl)]sign [f1(1, kl ! 0)! f1(1, kl + 0)]

=1

2

r"

l=1

sign [f2(1, kl)][sign f1(1, kl ! 0)! sign f1(1, kl + 0)],

17.3. USING STURM SEQUENCES 203

as m + n is even. Hence, formula (17.18) follows.

We show now that the right-hand member of formula (17.18) can be computedwithout an explicit knowledge of the roots kl. Let us write, for simplicity,

T0(y) := f1(1, y), T1(y) := f2(1, y). (17.20)

By the non-degeneracy of f, T0 and T1 have no common real roots. Using the algo-rithm introduced in 1836 by C. Sturm [377] to determine the number of real rootsof a polynomial in a given interval, define the polynomials T2(y), T3(y), . . . , Tp(y)by

T0(y) = '1(y)T1(y)! T2(y)

T1(y) = '2(y)T2(y)! T3(y)

. . . . . . . . . . . .

Ti!1(y) = 'i(y)Ti(y)! Ti+1(y) (17.21)

. . . . . . . . . . . .

Tp!2(y) = 'p!1(y)Tp!1(y)! Tp(y)

Tp!1(y) = 'p(y)Tp(y).

If Tp has a real root y0, then, by induction, T0(y0) = T1(y0) = 0, contradicting theassumptions, and hence Tp(y) has a constant sign, as T0 and T1 has no commonreal roots.

For each y & R, let S(y) denote the number of sign variations of the sequenceT0(y), T1(y), . . . , Tp(y).

Lemma 17.3.1 Let a < b be such that T0(a) %= 0 and T0(b) %= 0. Then

S(b)! S(a) =1

2

"sign T1(ki)[sign T0(ki ! 0)! sign T0(ki + 0)], (17.22)

where the sum is extended to all zeros ki of T0 located in [a, b].

Proof. By the continuity of the Ti, S(y) can only vary when y goes through a zerok0 of one of the polynomials Ti (0 ) i ) p ! 1). Let k0 be a zero of Ti for some1 ) i ) p ! 1. It follows from formulas (17.21) that Ti!1(k0) and Ti+1(k0) aredi!erent from zero (if not, this would imply, inductively, that T0(k0) = T1(k0) = 0),and that

Ti!1(k0)Ti+1(k0) < 0.

By continuity, this inequality holds in a neigbourhood of k0. Hence, because Ti(k0!0) has the sign of one of the Ti!1(k0 ! 0) and Ti+1(k0 + 0) and Ti(k0 + 0) has thesign of one of the Ti!1(k0 + 0) and Ti+1(k0 + 0), there is no change of the numberof variations of sign of the sequence T0(y), T1(y), . . . , Tl(y) when y crosses a zero ofTi, (k = 1, 2, . . . , p! 1). Les us consider now a zero k0 of T0. The variation of S(y)through k0 is given by

#S(k0) =1

2sign T1(k0)[sign T0(k0 ! 0)! sign T0(k0 + 0)].


Indeed, when T0 changes sign at k0, S(y) increases of 1 at k0 if sign T1(k0) (=sign T1(k0 ! 0)) and sign T0(k0 ! 0) are the same, and decreases of 1 if they areopposite. When T0 does not change sign at k0, S(y) does not change at k0.

Theorem 17.3.1 Let f be given by formula (17.10) be nondegenerate and satisfyconditions (17.17). Let the Ti(k) (0 ) i ) p) be defined by formulas (17.20) and(17.21). Then, for each r > 0,

dB [f, B(r), 0] = S(+-)! S(!-), (17.23)

where S(+-) (resp. S(!-)) denotes the number of sign variations of the sequenceof symbols

limk(+%

T0(k), limk(+%

T1(k), . . . , limk(+%

Tp(k)

(resp. limk(!%

T0(k), limk(!%

T0(k), . . . , limk(!%

Tp(k)),

or equivalently the number S(k) of sign variations of the sequence T0(k), T1(k),. . . , Tl(k) for positive (resp. negative) k with su!ciently large absolute value.

Proof. It is an immediate consequence of formula (17.18), Lemma 17.3.1 and theboundedness of the set of zeros of T0(k).

Remark 17.3.1 In (17.23), one can replace S(+-)! S(!-) by S(K)! S(!K),where K > max{|k| : T0(k) = 0}.

Example 17.3.1 We still consider the mapping (17.5). Then,

T0(y) = 1! y2, T1(y) = 2y, T2(y) = !1.

T0 T1 T2 S!- !- !- !1 0+- !- +- !1 2

Hence, for any r > 0,

dB[f, B(r), 0] = iK [f, !B(1)] = 2! 0 = 2.

17.4 Holomorphic functions

Let D ' C be an open bounded set and f : D ' C" C be holomorphic on an openneighbourhood U of D. By identifying C to R2, we can consider f as a mappingdefined on the closure of D seen as a bounded open subset of R2, but use freely thecomplex notations. So, for each y %& f(!D), dB [f, D, y] is well defined.

Proposition 17.4.1 If D ' C is bounded by a closed simple Jordan curve !D ofclass C2, and if y %& f(!D), then

dB[f, D, y] =1

2%i

!

!D

f $(z)

f(z)! ydz. (17.24)

17.4. HOLOMORPHIC FUNCTIONS 205

Proof. If g : D " C is defined by g(z) := f(z) ! y, then g is holomorphic on anopen neighbourhood of D and dB[f, D, y] = dB[g, D, 0], and it follows from (3.3)and (2.8)

dB[g, D, 0] = iK [g, !D] =1

2%i

!

!D

dg

g.

Now, using the Cauchy-Riemann equations

!1g(z) =1

i!2g(z) = g$(z).

we get

dg = !1g dx + !2g dy = g$ dx + ig$ dy = g$ dz,

and hence

dB [g, D, 0] =1

2%i

!

!D

g$(z)

g(z)dz.

Example 17.4.1 If f(z) = zn, n & N, and r > 0, we have f "(z)f(z) = n

z , and

dB[zn, B(r), 0] =1

2%i

!

!B(r)

n

zdz = n. (17.25)

An easy consequence of (17.25) is a proof of the fundamental theorem ofalgebra.

Proposition 17.4.2 Any non constant polynomial p : C" C has at least one zero.

Proof. We can write p(z) =Bn

j=0 ajzj with n ( 1, aj & C (j = 0, 1, . . . , n) andan %= 0, so that, without loss of generality, we can assume that an = 1. Now, forany " & [0, 1] and |z| = R > 0, we have

|P (z,")| :=

DDDDDDzn + "

5

6n!1"

j=0

ajzj

8

9

DDDDDD( |z|n !

n!1"

j=0

|aj ||z|j

= Rn

5

61!n!1"

j=0

|aj |Rj!n

8

9 > 0,

if we take R su$ciently large so thatBn!1

j=0 |aj |Rj!n < 1. For such a R, we have,from the homotopy invariance property 3.4.2 and (17.25),

dB [p, B(R), 0] = dB[P (·, 1), B(R), 0] = dB[P (·, 0), B(R), 0] = n ( 1,

and the result follows from the existence property 3.4.1.


Recall that if f : U ' C " C is holomorphic not identically zero and if a & U isa zero of f, the multiplicity of a is the smallest integer m such that f (m)(a) %= 0.Thus m ( 1 and Taylor’s theorem implies that m is finite. Taylor’s theorem alsoimplies that the zeros of f are isolated.

Proposition 17.4.3 Let f be holomorphic on an open neighborhood of the closureD of an open bounded set D ' C, and let y %& f(!D), Then dB[f, D, y] ( 0, and, ifdB[f, D, y] > 0, f!1(y) = {a1, · · · , an} is finite and non-empty, and, if mj denotesthe multiplicity of the zero aj of f(·)! y (1 ) j ) n), we have

dB [f, D, y] =n"

j=1

mj. (17.26)

Proof. Because f is holomorphic and f(·) ! y %= 0 on !D, f!1(y) is discrete andcompact, hence finite (possibly empty). If f!1(y) = ,, dB[f, D, y] = 0. If f!1(y) ={a1, · · · , an}, then, for each 1 ) j ) n, it follows from Taylor’s formula that

f(z)! y = (z ! aj)mj gj(z),

for some gj holomorphic on D and such that gj(aj) %= 0. Hence, for r > 0 su$cientlysmall, using formula (17.24), we get

dB[f, D, y] =n"

j=1

iB[f, aj ] =n"

j=1

dB [f, Baj (r), 0]

=n"

j=1

1

2%i

!

!Baj (r)

f $(z)

f(z)! ydz =

n"

j=1

!

!Baj (r)

-mj

z ! aj+

g$j(z)

gj(z)

.dz

=n"

j=1

mj .

Corollary 17.4.1 Let f be holomorphic on an open neighborhood of the closure Dof an open bounded set D ' C, and let y %& f(!D), Then dB[f, D, y] = 1 if and onlyif f!1(y) .D = {a} and f $(a) %= 0, i.e. if and only if f(·)! y has a unique zero inD which is of multiplicity one.

Proof. Su!ciency. If f!1(y).D = {a} and a has multiplicity one, formula (17.26)implies that dB[f, D, y] = 1.

Necessity. From formula (17.26), we have, if f!1(y) . D = {a1, . . . , an}, 1 =Bnj=1 mj , which implies n = 1 and m1 = 1, i.e. f $(a1) %= 0.

Remark 17.4.1 Proposition 17.4.3 shows that, for a holomorphic function, thedegree is an exact count of the number of zeros of f contained in D, if one take inaccount their multiplicity.

17.5. AN APPLICATION TO STABILITY AND CONTROL 207

17.5 An application to stability and control

If A & L(Rn, Rn), it is a standard result in the theory of ordinary di!erentialequations [74, 337] that the zero solution of the linear di!erential system

x$ = Ax

is asymptotically stable if and only if the eigenvalues of the matrix A are all locatedin the left-hand side half plane {z & C : Az < 0}. Similarly, it is a standard result inthe theory of di!erence equations [104] that the zero solution of the linear di!erencesystem

xk+1 = Axk (k & N)

is asymptotically stable if and only if the eigenvalues of the matrix A are all locatedin the open unit ball B(1) ' C. Now the eigenvalues are solutions of the character-istic equation det (A!zI) = 0, i.e. the zeros of a polynomial in z of degree n. Hencethe corresponding asymptotic stability problems are related to having informationof the location of the zeros of a polynomial in the complex plane. In the first case,the problem was first considered and solved, independently, by E.H. Routh [338]and A. Hurwitz [178] (to answer a question of A. Stodola). In the second case, theproblem was first considered by I. Schur [351] and his results completed by A. Cohn[72]. Notice that if the characteric equation is explicited as

zn +n!1"

j=1

ajzj = 0,

any possible solution is such that

|z|n )n!1"

j=1

|aj ||z|j,

and hence such that |z| ) R, where R > 0 is the positive root of equation

Rn !n!1"

j=1

|aj |Rj = 0. (17.27)

Thus the location problem for the roots is always a location problem in a su$cientlylarge open bounded set.

More generally, let % ' C be open and p : C" C be a polynomial.

Definition 17.5.1 We say that p is %-stable if p(z) %= 0 for z %& %.

When % = {z & C : Az < 0}, the %-stability is called the Routh-Hurwitzstability; when % = B(1) ' C, the %-stability is called the Schur-Cohn stability.


It follows imediately from Proposition 17.4.3 that p is %-stable if and only ifdB[p, B(-) \ %, 0] = 0 for any - > R, the positive root of equation (17.27).

For some applications, it is interesting to consider the corresponding propertiesfor a family of polynomials. Let ' ' Rm and P & C(C 1 ', C) be such that, foreach " & ', P (·,") is a polynomial on C.

Definition 17.5.2 We say that {P (·,")}%"" is %-stable if, for each " & ', P (·,")is %-stable, i.e. if P (z,") %= 0 for z %& % and " & '.

We now state and prove a more general version, given in [264], of a zero ex-clusion principle [13, 329].

Theorem 17.5.1 Let ' ' Rm and {P (·,")}%"" be a continuous family of polyno-mials on C. Assume that the following conditions hold:

1. ' is connected.

2. P (·,") has algebraic degree d ( 1 for each " & '.

3. P (·,"0) is %-stable for some "0 & '.

4. P (z,") %= 0 for each (z,") & !%1 '.

Then {P (·,")}%"" is %-stable.

Proof. By Assumption 2, we can assume, without loss of generality, that

P (z,") = zd +d"

k=1

ak(")zd!k,

where the ak : ' " C are continuous (1 ) k ) d). Hence, if P (z,") = 0 for some(z,") & C1 ', we have

|z|d )d"

k=1

*k|z|d!k,

where *k = max%"" |ak(")|, (1 ) k ) d). Consequently, |z| ) R, where R is thepositive root of equation

Rd !d"

k=1

*kRd!k = 0.

If %R := % . B(R + 1), the estimate for |z| above and Assumption 4 imply thatP (z,") %= 0 for each (z,") & !%R 1 '. Hence, by the connectedness of ' andhomotopy invariance property 3.4.2, dB [P (·,"),%R, 0] is independent of ". On theother hand, it follows from Proposition 17.4.3 that dB[P (·,"),%R, 0] is equal to thenumber of zeros of P (·,") in %R, counted with their multiplicities. By Assumption3 and the definition of %-stability, any possible zero of P (·,"0) lies in %, and hencein %R. Consequently, dB[P (·,"0),%R, 0] = d, and hence dB[P (·,"),%R, 0] = d foreach " & '. Thus all the zeros of P (·,") are in %R, hence in %, and each P (·,") is%-stable.


The following notion was introduced by Schur [352].

Definition 17.5.3 If p : C " C is a polynomial, the first Schur transform orparaconjugate of p is the polynomial p# : C" C defined by

p#(z) := p(!z). (17.28)

Notice that

p##(z) := (p#)#(z) = p#(!z) = p(z) = p(z),

and that, for c & C and another polynomial q over C, one has

(cp)#(z) = cp#(z), (p + q)#(z) = p#(z) + q#(z).

Define, with Zahreddine [404],

N(z) =1

2

$p(z) + (!1)dp#(z)

%, D(z) =

1

2

$p(z)! (!1)dp#(z)

%, (17.29)

so that

p(z) = N(z) + D(z), (17.30)

p#(z) = (!1)d[N(z)!D(z)]. (17.31)

N#(z) =1

2

$p#(z) + (!1)dp(z)

%= (!1)dN(z),

D#(z) =1

2

$p#(z)! (!1)dp(z)

%= (!1)d+1D(z).

Lemma 17.5.1 z0 is a common zero to N and D if and only if z0 and !z0 arezeros of p.

Proof. If z0 is a common zero to N and D, then, by (17.30) and (17.31), we havep(z0) = 0 = p(!z0). If z0 and !z0 are zeros of p, then p(z0) = 0 = p#(z0), and, by(17.29), we have N(z0) = D(z0) = 0.

Corollary 17.5.1 Any zero of p which lies on the imaginary axis is a common zeroof N and D.

Proof. For such a zero z0, one has z0 = !z0.

The resultant [258] of two polynomials p and q over C will be denoted by R[p, q].

Let us first consider the special case of the Routh-Hurwitz stability of a familyof polynomials. Let ' ' Rm be connected and let {P (·,")}%"" be a continuousfamily of polynomials on C such that P (·,") has degree d for each " & '. For each" & ', let

P (z,") = N(z,") + D(z,")

be the decomposition defined by (17.30). The following result, given in [264], gen-eralizes, with a simpler proof, a result of [404].


Theorem 17.5.2 The family {P (·,")}%"" is Routh-Hurwitz-stable if and only ifP (·,"0) is Routh-Hurwitz-stable for some "0 & ' and R[N(·,"), D(·,")] %= 0 foreach " & '.

Proof. Necessity. If R[N(·,"0), D(·,"0)] = 0 for some "0 & ', then, by a classicalresult [258], N(·,"0) and D(·,"0) have a common zero z0, so that, by Lemma 17.5.1,

P (z0,"0) = 0 = P (!z0,"0).

Now Az0 ( 0 if and only if A(!z0) ) 0, and hence P (·,"0) is not Routh-Hurwitz-stable.

Su!ciency. Assume now that R[N(·,"), D(·,")] %= 0 for each " & '. Then,for each " & ', N(·,") and D(·,") have no common zeros, and Corollary 17.5.1implies that P (·,") has no zero on the imaginary axis, which is the boundary of{z & C : Az < 0}. The conclusion follows then from Theorem 17.5.1.

We obtain immediately as a special case Zahreddine’s result. Let

p0(z) = zd +d"

k=1

a0kzd!k, p1(z) = zd +

d"

k=1

a1kzd!k

be two monic Routh-Hurwitz-stable polynomials. Let

pj(z) = Nj(z) + Dj(z), (j = 0, 1)

be their respective decompositions (17.30) and let

P (z,") = (1! ")p0 + "p1, (0 ) " ) 1),

be their convex combinations. It is immediate to check that if, for each " & [0, 1],P (z,") = N(z,") + D(z,") is the decomposition (17.30) of p(·,"), then

N(z,") = (1! ")N0(z) + "N1(z), D(z,") = (1! ")D0(z) + "D1(z)

(0 ) " ) 1).

Corollary 17.5.2 If p and q are Routh-Hurwitz-stable, their convex combinations(1 ! ")p + "q (" & [0, 1]), are Routh-Hurwitz-stable if and only if R[(1 ! ")N0 +"N1, (1! ")D0 + "D1] %= 0 for each " & ]0, 1[ .

Let us now introduce the second Schur transform of a polynomial and its prop-erties [351].

Definition 17.5.4 Given a monic polynomial p(z) = zd +Bd

k=1 akzd!k on C, thesecond Schur transform of p is the polynomial p# on C defined by

p#(z) := zdp

21

z

3, (17.32)


Notice that

p##(z) := (p#)#(z) = zdp#

21

z

3= zd 1

zdp(z) = p(z),

that p#(0) = 1 and that, for c & C and another monic polynomial q over C, one has

(cp)#(z) = cp#(z), (p + q)#(z) = p#(z) + q#(z).

Define, with Zahreddine [404],

H(z) =1

2

$p(z) + p#(z)

%, K(z) =

1

2

$p(z)! p#(z)

%, (17.33)

so that

p(z) = H(z) + K(z), (17.34)

p#(z) = H(z)!K(z). (17.35)

H#(z) =1

2

$p#(z) + p(z)

%= H(z),

K#(z) =1

2

$p#(z)! p(z)

%= !K(z),

Lemma 17.5.2 If z0 is a common zero to H and K then z0 %= 0, and z0 and 1z0

are zeros of p. If z0 %= 0 and 1z0

are zeros of p, then z0 is a common zero to H andK.

Proof. If z0 is a common zero to H and K, then, by (17.34) and (17.35), we have

0 = p(z0) = p#(z0), so that z0 %= 0 and pL

1z0

M= 0. If z0 %= 0 and 1

z0are zeros of p,

then p(z0) = 0 = p#(z0), and, by (17.33), we have H(z0) = K(z0) = 0.

Corollary 17.5.3 Any zero z0 of p which lies on the unit circle is a common zeroof H and K.

Proof. For such a zero z0, one has 0 %= z0 = 1z0

.

Let us now consider the Schur-Cohn-stability of a family of polynomials. Let' ' Rm be connected and let {P (·,")}%"" be a continuous family of polynomialson C such that P (·,") has degree d for each " & '. For each " & ', let

P (z,") = H(z,") + K(z,")

be the decomposition defined by (17.34). The following result, given in [271], gen-eralizes, with a simpler proof, a result of [404].


Theorem 17.5.3 The family {P (·,")}%"" is Schur-Cohn-stable if and only if poly-nomial P (·,"0) is Schur-Cohn-stable for some "0 & ' and R[H(·,"), K(·,")] %= 0for each " & '.

Proof. Necessity. If R[H(·,"0), K(·,"0)] = 0 for some "0 & ', then, by a classicalresult [258], H(·,"0) and K(·,"0) have a common zero z0, so that by Lemma 17.5.2,z0 %= 0 and

p(z0,"0) = 0 = p

21

z0,"0

3.

Now |z0| ( 1 if and only ifDDD 1z0

DDD ) 1, and hence P (·,"0) is not Schur-Cohn-stable.

Su!ciency. Assume now that R[H(·,"), K(·,")] %= 0 for each " & '. Then,for each " & ', K(·,") and H(·,") have no common zeros, which, by Corollary17.5.3, implies that P (·,") has no zero on !B(1). The conclusion follows then fromTheorem 17.5.1.

We obtain immediately as a special case a result of Zahreddine [404]. Let

p0(z) = zd +d"

k=1

a0kzd!k, p1(z) = zd +

d"

k=1

a1kzd!k

be two monic Schur-Cohn-stable polynomials. Let pj(z) = Hj(z) + Kj(z), (j =0, 1) be their respective decompositions (17.34) and let

p(z,") = (1! ")p0 + "p1, (0 ) " ) 1),

be their convex combinations. It is immediate to check that if, for each " & [0, 1],p(z,") = H(z,") + K(z,") is the decomposition (17.34) of p(·,"), then

H(z,") = (1! ")H0(z) + "H1(z), K(z,") = (1! ")K0(z) + "K1(z)

(0 ) " ) 1).

Corollary 17.5.4 Assume that p and q are Schur-Cohn-stable. Then their convexcombinations (1!")p + "q (" & [0, 1]), are Schur-Cohn-stable if and only if R[(1!")H0 + "H1, (1! ")K0 + "K1] %= 0 for each " & ]0, 1[ .

Chapter 18

Periodic solutions of someplanar systems

18.1 Autonomous systems

Let J denotes the 21 2 symplectic matrix

J =

20 !11 0

3.

Recall that

J2 = !I, JJT = JT J = I, 6Jv, v7 = 0 (v & R2). (18.1)

Let H & C1(R2, R) be positive on R2 \ {0}, positively homogeneous of degree 2,namely

H($u) = $2H(u) ($ ( 0, u & R2) (18.2)

and such that +H : R2 " R2 locally Lipschitzian. It follows from (18.2) that +His positively homogeneous of degree one, namely

+H($u) = $H(u) ($ ( 0, u & R2), (18.3)

It follows from (18.2) and (18.3) that

H(u) = 0 4 u = 0, +H(u) = 0 4 u = 0. (18.4)

Furthermore, for all u & R2 \ {0}, one has

*#u#2 ) H(u) = #u#2H(u/#u#) ) ,#u#2, (18.5)

where

0 < * := min)v)=1

H(v) ) max)v)=1

H(v) := ,. (18.6)

213

214 CHAPTER 18. PERIODIC SOLUTIONS OF SOME PLANAR SYSTEMS

Let us consider the corresponding planar Hamiltonian system

Ju$ = +H(u). (18.7)

The identity

0 = 6Ju$(t), u$(t)7 = 6+H(u(t)), u$(t)7 = [H(u(t))]$

satisfied by all solutions of (18.7) implies the existence of the first integral

H(u(t)) = E (E ( 0). (18.8)

Thus, (18.5) implies that any solution u of energy E is contained in the annuluscentered at 0 and of radii 2E/, and 2E/*. Furthermore, for each E > 0, the curveof equation H(u) = E surrounds the origin and intersects each half-line issued fromthe origin in a unique point, because, for each v & R2 with #v# = 1 the equation

H($v) = E 4 $2H(v) = E

has a unique positive solution $ = [E/H(v)]1/2. So, for E > 0, the curves H(u) = Eare closed curves around the origin, which therefore is a center.

If we fix a reference solution & : R" R2 of (18.7) such that

H(&(t)) = 1/2 (t & R), (18.9)

we have, using Euler’s identity

6J&$(t),&(t)7 = 6+H(&(t)),&(t)7 = 2H(&(t)) = 1 (t & R), (18.10)

and

6&$(t), J&(t)7 = 6JT J&$(t), J&7 = !6J&$(t),&(t)7 = !1. (18.11)

Hence, as the action of J on a vector means a rotation of %/2 in the counterclockwisedirection, the orbit of &(t) is described in the clockwise direction when t increases.It follows from the positive homogeneity of degree one of+H and of its autonomouscharacter that all solutions of (18.7) are given by

u(t) = A&(t + )) (A ( 0, ) & R). (18.12)

Therefore, if 5 > 0 denotes the time required for a solution of energy 1/2 to describeonce the orbit H(u) = 1/2, i.e. the smallest period of &, all solutions of (18.7) willhave the same period 5 , which means that 0 is an isochronous center.

18.2 Forced system : nonresonance

Keeping the notations and assumptions of Section 8.1 for H , let f : R " R2 be acontinuous function of period T > 0 (T-periodic function) and let us considerthe forced planar Hamiltonian system

Ju$ = +H(u) + f(t). (18.13)

18.2. FORCED SYSTEM : NONRESONANCE 215

Because of the fact that +H(u) is locally Lipschitzian and has a growth at mostlinear due to its positive homogeneity of degree one, all solutions u(t; u0) of (18.13)with initial condition u0 at t = 0 are uniquely defined for all t & R.

Let us assume that

T %= n5 for all positive integers n, (18.14)

so that N5 < T < (N + 1)5 for some nonnegative integer N. This implies that theonly T-periodic solution of system (18.7) is u > 0.

Lemma 18.2.1 There exists R > 0 such that, for each " & [0, 1], each possibleT-periodic solution of system

Ju$ = +H(u) + "p(t) (18.15)

is such that #u#% < R.

Proof. If it is not the case, there exist a sequence ("k) in [0, 1], and a sequence (uk)of T-periodic solutions of

Ju$k = +H(uk) + "kp(t) (k & N).

such that

limk(%

#uk#% = +-.

Hence, letting vk = uk)uk)#

and using the positive homogeneity of +H, we have

Jv$k = +H(vk) + "kp(t)

#uk#%(k & N).

Hence, for all k such that #uk#% ( 1, we have

#vk#% = 1, #v$k#% ) max)u)=1

#+H(u)#+ #f#% := M,

so that Ascoli-Arzela’s theorem implies, going if necessary to a subsequence, that(vk) converges uniformly on R to a T-periodic function v such that #v#% = 1.Furthermore, from the identity

Jvk(t) = Jvk(0) +

! t

0

-+H(vk(s)) + "k

p(s)

#uk#%

.ds (t & R, k & N)

we deduce that

Jv(t) = Jv(0) +

! t

0+H(v(s)) ds (t & R),

and hence v & C1(R) and

Jv$ = +H(v).

Thus v is a T-periodic solution of (18.7) with #v#% = 1, a contradiction to (18.14).


Now, let us consider the orbit of (18.7) defined by

H(u) = ,R2 (18.16)

where , is defined in (18.6) and R given by Lemma 18.2.1. If the open bounded setDR is defined by

DR := {u & R2 : H(u) < ,R2},

we have

!DR = {u & R2 : H(u) = ,R2}, B(R) ' DR.

.The following result is proved in a di!erent way in [127].

Theorem 18.2.1 For each T-periodic forcing f, problem (18.13) has at least oneT-periodic solution.

Proof. For each " & [0, 1] and each u0 & R2, let us denote by u(t; u0,") the uniquesolution of (18.13) equal to u0 at t = 0, and let PT : R2 1 [0, 1]" R2 be Poincare’soperator associated to the T-periodic solutions of (18.15) defined by

PT (u0,") = u(T ; u0,") (u0 & R2, " & [0, 1]).

PT (·, 1) is Poincare’s operator associated to the T-periodic solutions of (18.13) andPT (·, 0) Poincare’s operator associated to the T-periodic solutions of (18.7). FromLemma 18.2.1, it follows that, for any " & [0, 1], each possible fixed point u0 ofPT (·,") belongs to B(R) ' DR. Hence, for each " & [0, 1] and each u0 & !DR, onehas u0 %= PT (u0,"), and the homotopy invariance of Brouwer degree implies that

dB [PT (·, 1)! I, DR, 0] = dB[PT (·, 0)! I, DR, 0]. (18.17)

Now, as !DR is an orbit for (18.7), PT (u0, 0) & !DR when u0 & !DR, and, by thenonresonance condition (18.14), PT (u0, 0) %= u0 for each u0 & !DR. Furthermore,as DR is strictly starshaped with respect to 0, µPT (u0, 0) & µ!DR ' DR for eachµ & [0, 1[ and each u0 & !DR. Hence, using again homotopy invariance of Brouwerdegree, we get

dB[PT (·, 0)! I, DR, 0] = dB[!I, DR, 0] = 1,

so that, using (18.17),

dB [PT (·, 1)! I, DR, 0] = 1,

and PT (·, 1) has a fixed point.

18.3. FORCED SYSTEM : EQUIVALENT FORMULATION 217

18.3 Forced system : equivalent formulation

In order to obtain existence conditions for the T-periodic solutions of (18.13) whenthe nonresonance condition (18.14) does not hold, it is convenient to write (18.13)in an equivalent form.

Using the method of variations of constants, let us introduce the new unknownfunctions -(t) and )(t) by the relation

u(t) = -(t)&(t + )(t)) (t & R2) (18.18)

where -(t) > 0 and & is the particular solution of (18.7) introduced in Section 8.1.If u is a solution of (18.13), then, using the positive homogeneity of +H,

-$(t)J&(t + )(t)) + -(t)J&$(t + )(t))(1 + )$(t)) = +H(-(t)&(t + )(t))) + f(t).

Hence, using the definition of &, we get

-$(t)J&(t + )(t)) + -(t)J&$(t + )(t)))$(t) = f(t). (18.19)

Taking the inner product of both members of (18.19) respectively with &$(t + )(t))and &(t + )(t)), and using relations (18.1), we obtain, as long as -(t) > 0,

-$(t) = !6f(t),&$(t + )(t))7, )$(t) =1

-(t)6f(t),&(t + )(t))7. (18.20)

Let -0 > 0 and )0 & [0, 5 [ be given, and let -(t; -0, )0) and )(t; -0, )0) denotethe solutions of (18.20) such that

-(0; -0, )0) = -0, )(0; -0, )0) = )0.

Lemma 18.3.1 There exists C1 = C1(&, f) such that, for each t & [0, T ] one has

|-(t; -0, )0)! -0| ) C1T. (18.21)

In particular, if -0 ( C1T + 1, then -(t; -0, )0) ( 1 for all t & [0, T ].

Proof. It follows from the first equation in (18.20) that, for t & [0, T ],

|-(t; -0, )0)! -0| =

DDDD! t

06f(s),&$(s + )(s; -0, )0))7 ds

DDDD

)! T

0#f(s)##&$(s + )(s; -0, )0))# ds

) max[0,T ]#f#max

[0,# ]#&$#T := C1T.


Lemma 18.3.2 If -0 ( C1T + 1, then there exists C2 = C2(&, f) such that, foreach t & [0, T ], one has

|)(t; -0, )0)! )0| )C2T

-0 ! C1T. (18.22)

Proof. It follows from the second equation in (18.20) and from (18.21) that, fort & [0, T ],

|)(t; -0, )0)! )0| =

DDDD! t

0

1

-(t; -0, )0)6f(s),&(s + )(s; -0, )0))7 ds

DDDD

) 1

-0 ! C1T

! T

0#f(s)##&(s + )(s; -0, )0))# ds

) 1

-0 ! C1Tmax[0,T ]#f#max

[0,# ]#&#T :=

C2T

-0 ! C1T.

In order to apply Poincare’s operator in the search of T-periodic solutions of(18.13), given -0 ( C1T + 1 and )0 & [0, 5 [ , we have to estimate

-(T ; -0, )0) = -0 !! T

06f(s),&$(s + )(s; -0, )0))7 ds (18.23)

)(T ; -0, )0) = )0 +

! T

0

6f(s),&(s + )(s; -0, )0))7-(s; -0, )0)

ds.

Define the 5 -periodic mapping & : [0, 5 ] 5 S1 " R2 by

&1()0) := !! T

06f(s),&(s + )0)7 ds (18.24)

&2()0) :=

! T

06f(s),&$(s + )0)7 ds = !&$

1()0).

Lemma 18.3.3 For -0 ( C1T + 1 and )0 & [0, 5 ], we have

)(T ; -0, )0) = )0 +1

-0[!&1()0) + R1(-0, )0)]

-(T ; -0, )0) = -0 ! &2()0) + R2(-0, )0), (18.25)

where

lim(0(+%

Rj(-0, )0) = 0 (j = 1, 2), (18.26)

uniformly in )0 & [0, 5 ].

18.4. FORCED SYSTEM : RESONANCE 219

Proof. We have, using (18.23) and the definition of &2,

R2(-0, )0) =

! T

06f(s),&$(s + )(s; -0, )0))! &$(s + )0)7 ds,

and, by (18.22),

lim(0(+%

)(t; -0, )0) = )0

uniformly in t & [0, T ] and )0 & [0, 5 ]. Hence,

lim(0(+%

R2(-0, )0) =

! T

06f(s), lim

(0(+%[&$(s + )(s; -0, )0))! &$(s + )0)]7 ds = 0,

uniformly in )0 & [0, 5 ].On the other hand,

R1(-0, )0) = -0

! T

0

6f(s),&(s + )(s; -0, )0))7-(s; -0, )0)

ds!! T

06f(s),&(s + )0)7 ds

=

! T

0

-0

-(s; -0, )0)6f(s),&(s + )(s; -0, )0))! &(s + )0)7 ds

+

! T

0

--0

-(s; -0, )0)! 1

.6f(s),&(s + )0)7 ds.

Now, from Lemmas 18.3.1 and 18.3.2, we have

lim(0(+%

-0

-(s; -0, )0)= 1, lim

(0(+%)(s; -0, )0) = )0,

uniformly in t & [0, T ] and )0 & [0, 5 ], and the result follows in an analogous way.

18.4 Forced system : resonance

From now on, let us assume that there exists a positive integer N such that

T = N5. (18.27)

Each nonzero initial condition u0 & R2 can be univoquely written as

u0 = -0&()0)

for some -0 > 0 and )0 & [0, 5 [ . On the other hand, if u(t; u0) denotes the uniquesolution of (18.13) such that u(0;u0) = u0, and if PT defined by PT (u0) = u(T ; u0)is Poincare’s operator, then, with the notations of the previous section,

PT (-0&()0)) = u(T ; -0&()0)) = -(T ; -0, )0)&(T + )(T ; -0, )0))

= -(T ; -0, )0)&(N5 + )(T ; -0, )0)) = -(T ; -0, )0)&()(T ; -0, )0)),


so that the initial conditions of T-periodic solutions of (18.13) are determined bythe zeros (-0, )0) of the map ( : ]0, +-[1S1 " R2 defined by

((-0, )0) := -(T ; -0, )0)&()(T ; -0, )0))! -0&()0), (18.28)

where we have identified [0, 5 ], taking in account the 5 -periodicity, to S1.In order to study the Brouwer degree of this map, let us introduce, for each

-0 > 0, the open bounded sets %(0 ' R2 defined by

D(0 := {$&()0) : )0 & [0, 5 ], $ & [0, -0[ }, (18.29)

so that !D(0 = {-0&()0) : )0 & [0, 5 ]} is an orbit of (18.7), and is oriented clockwise.Consequently, if

U : ]0, +-[1S1 " R2, (-0, )0) 2" -0&()0),

then ( = (PT ! I) * U , D(0 = U([0, -0[1S1)

JU (-0, )0) = -0[&1()0)&$2()0)! &2()0)&

$1()0)]

= !-06J&$()0),&()0)7 = !-0 < 0.

Hence it follows from the definition of Brouwer degree that

dB [PT ! I, D(0 , 0] = !dB[(PT ! I) * U, U!1(D(0)] (18.30)

= !dB[(, [0, -0[1S1, 0]. (18.31)

We need a few lemmas to compute this last degree. The first one is due to M.A.Krasnosel’skii [209, 220].

Lemma 18.4.1 Let D ' R2 be open, bounded and such that !D is a smooth Jordancurve. Let g1, g2 & C1(D, R2) be such that [g1(x), g2(x)] is a base for each x & !D.Let f & C1(D, R2) be such that f(x) %= 0 on !D, and h : D " R2 be defined by

h(x) = f1(x)g1(x) + f2(x)g2(x).

Then

iK [f, !D] = (sign det G) · {iK [h, !D]! iK [g1, !D]}, (18.32)

where G is the 21 2-matrix given by

G =

2g11(x

0) g21(x

0)g12(x

0) g22(x

0)

3. (18.33)

Proof. By assumption, we have

(1! ")g1(x) + "g2(x) %= 0

18.4. FORCED SYSTEM : RESONANCE 221

for all " & [0, 1] and all x & !D, so that

iK [g1, !D] = dB [g1, D] = dB [g2, D] = iK [g2, !D]. (18.34)

Let x0 & !D be fixed, and let h0 : D " R2 be defined by

h0(x) = f1(x)g1(x0) + f2(x)g2(x0).

Let us denote by .(x) the positively oriented angle between h(x) and g1(x), by)(x) the positively oriented angle between h(x) and g1(x0), and by * the positivelyoriented angle between g1(x) and g1(x0). We have therefore

d) = d. + d*,

and hence, by definition of Kronecker index,

iK [h, !D] = iK [h0, !D] + iK [g1, !D]. (18.35)

On the other hand, in terms of the canonical basis (e1, e2) of R2, we have

h0(x) = f1(x)g1(x0) + f2(x)g2(x0)

= [f1(x)g11(x0) + f2(x)g2

1(x0)]e1 + [f1(x)g1

2(x0) + f2(x)g22(x

0)]e2

so that

h0(x) = Gf(x),

where G is given in (18.33). Consequently,

iK [h, !D] = dB [h, D, 0] = (sign det G) dB [f, D, 0] = (sign det G) iK [f, D],(18.36)

and the result follows from (18.35) and (18.36).

The following lemma computes dB[PT ! I, D(0 , 0] in terms of iK [&, S1] when -0

is large.

Lemma 18.4.2 Assume that

&()0) %= 0 ()0 & [0, 5 ]). (18.37)

Then, for all -0 > 0 su!ciently large, one has

dB [PT ! I, D(0 , 0] = 1! iK [&, S1]. (18.38)

Proof. By Lemma 18.3.3, we have

-(T ; -0, )0)&()(T ; -0, )0))

= [-0 ! &2()0) + R2(-0, )0)]&()0 + -!10 [!&1()0) + R1(-0, )0)])

= -0&()0) + -0[&()0 + -!10 [!&1()0) + R1(-0, )0)])! &()0)]

! [&2()0)!R2(-0, )0)]&()0 + -!10 [!&1()0) + R1(-0, )0)]).


Hence, using (18.26) and definition (18.28)

lim(0(+%

((-0, )0) = !&1()0)&$()0)! &2()0)&()0) := !)()0). (18.39)

As &()0) and &$()0) are linearly independent, assumption (18.37) implies that )()0)%= 0 for all )0 & [0, 5 ], and Rouche’s theorem together with (18.39) implies that, forall -0 > 0 su$ciently large,

dB [(, [0, -0[1S1, 0] = dB [!), [0, -0[1S1, 0] = dB[), [0, -0[1S1, 0](18.40)

= dB [), [0, 1[1S1, 0] = iK [), S1], (18.41)

taking in account the fact that ) does not depend upon -0. For each )0 & [0, 5 ],

6J&$()0),&()0)7 = 1.

In particular, when &1()0) = 0, &$1()0)&2()0) = 1 > 0, and, as &1 vanishes twice on

[0, 5 [ , we have, from formula (17.2),

iK [&, S1] = !1,

and hence

iK [&$, S1] = iK [J&$, S1] = iK [&, S1] = !1. (18.42)

Furthermore,

det

2&$

1()0) &1()0)&$

2()0) &2()0)

3= 6J&$()0),&()0)7 = +1. (18.43)

Therefore, using Lemma 18.4.1 with g1 = &$, g2 = &, f = & = (&1,&2), h = ), and(18.42), we obtain

iK [&, S1] = iK [), S1]! iK [&$, S1] = iK [), S1] + 1. (18.44)

Hence, for all su$ciently large -0 > 0, using (18.30) and (18.44),

dB [PT ! I, D(0 , 0] = !iK [(, S1] = !iK [), S1] = 1! iK [&, S1].

We can now state and prove an existence theorem for T-periodic solutions of(18.13).

Theorem 18.4.1 Assume that the 5-periodic function

&1 : [0, 5 [" R, )0 2"! T

06f(s),&(s + )0)7 ds

does not vanish or has more than four zeros, which are all simple. Then system(18.13) has at least one T-periodic solution.

18.5. ASYMMETRIC PIECEWISE-LINEAR OSCILLATORS 223

Proof. If &1()0) %= 0 for all )0 & [0, 5 ], or if all its zeros are simple, then &()0) =(&1()0),!&$

1()0)) %= 0 for all )0 & [0, 5 ]. Hence, by Lemma 18.4.2, we have, for all-0 > 0 su$ciently large,

dB [PT ! I, D(0 , 0] = 1! iK [&, S1]. (18.45)

If &1()0) %= 0 for all )0 & [0, 5 [ , then iK [&, S1] = 0 and, by (18.45), dB[PT !I, D(0 , 0] = 1, so that PT has a fixed point in %(0 . If &1 has zeros, all simple, theirnumber is finite, and, if we denote them by )0,k (1 ) k ) m), we have, using formula(17.2),

iK [&, S1] =1

2

m"

k=1

sign &$1()0,k)sign &$

1()0,k) =m

2.

This shows that m is even, and, furthermore, using (18.45),

dB [PT ! I, D(0 , 0] = 1! m

2%= 0

if m ( 4.

18.5 Asymmetric piecewise-linear oscillators

The free linear oscillator

x$$ + *x = 0 (* > 0)

can be generalized to the free asymmetric oscillator

x$$ + µx+ ! (x! = 0 (µ > 0, ( > 0) (18.46)

where x+ := max(x, 0), x! := max(!x, 0). The linear restoring force !*x isreplaced by the piecewise linear one !µx+ + (x!. Letting

x = u1, x$ = u2,

we obtain the equivalent Hamiltonian system

Ju$ = +H(u) (18.47)

where

H(u) =u2

2

2+ µ

(u+1 )2

2+ (

(u!1 )2

2(18.48)

is of class C1, nonnegative, positively homogeneous of degree two, and +H(u) ispositively homogeneous of degree one and globally Lipschitzian.


It easily follows from the energy integral

H(u) = E (E ( 0),

and from the fact the the system corresponds to the linear oscillator

u$$1 + µu1 = 0 (resp. u$$

1 + (u1 = 0)

in the half phase-plane {u1 ( 0} (resp. {u1 < 0}) that the orbits of (18.47) in thephase plane R2 are egg-shaped closed curves around the origin made by half-ellipsesof equation

u22

2+ µ

u21

2= E

in the right half plane and

u22

2+ (

u21

2= E

in the left half plane. Hence, the origin is an isochronous center with minimal period

5 = %

21<

µ+

1<(

3.

Given a fixed period T > 0 that, without loss of generality and for simplicityof notations we take equal to 2%, it follows that equation (18.46) or equivalentlysystem (18.47) has a 2%-periodic solution if and only if there exists a positive integerN such that 2% = N5, i.e. such that

21<

µ+

1<(

3=

2

N. (18.49)

This is a family indexed by N of hyperbolic-like curves independently introducedby Fucik [136] and Dancer [82], and usually called the Fucik curves. Their pointsare also called the positive Fucik eigenvalues of problem (18.46) or (18.47) forperiod 2%. Notice that the Fucik curves intersect the diagonal in the (µ, ()-planeat the point (n2, n2), and the n2 are the usual eigenvalues of the linear oscillatorx$$ + "x = 0 for the period 2%.

If one fixes the positive integer N, and (µ, () satisfies condition (18.49), one candefine the corresponding Fucik eigenfunction for (18.46) by

$(t) :=

Z 12µ sin(

<µ t) for t & [0, "2

µ ],

! 12-

sinL<

((t! "2µ)M

for t & [ "2µ , 2"

n ],(18.50)

The corresponding solution of (18.47) is given by

&(t) = ($(t),$$(t)), (18.51)

18.5. ASYMMETRIC PIECEWISE-LINEAR OSCILLATORS 225

and it is easy to check that its satisfies relations (18.9) and (18.10). They are2(%/N)-periodic. Then the non-trivial solutions of (18.46) are given by

x(t) = A$(t + )) (A > 0, ) & [0, 2%/N [ ),

and the corresponding non-trivial solutions of (18.47) are given by

u(t) = A&(t + )) (A > 0, ) & [0, 2%/N [ ).

Consider now the forced asymmetric oscillator

x$$ + µx+ ! (x! = e(t) (µ > 0, ( > 0), (18.52)

where e : R " R is continuous and 2%-periodic. With the change of variables(u1, u2) = (x, x$), it is equivalent to the forced Hamiltonian system

Ju$ = +H(u) + f(t) (18.53)

where H is given in (18.48) and

f(t) = (!e(t), 0). (18.54)

Then, with the notations of the previous section, the corresponding function &1()0)takes here the form &1 = #, where # is given by

#()0) :=

! 2"

0e(t)$(t + )0) dt () & [0, 2%/N [ ), (18.55)

and is (2%/N)-periodic. It was first introduced for (18.52) by Dancer [82] in 1976.The following existence result for 2%-periodic solutions of (18.52) in the non-

resonance case, first proved in a di!erent way by Fucik [136] in 1976, is a directconsequence of Theorem 18.2.1.

Theorem 18.5.1 If (µ, (), with (µ > 0, ( > 0), is not a Fucik eigenvalue, thenthen equation (18.52) has at least one 2%-periodic solution for each 2%-periodiccontinuous forcing term e(t).

Because of this result, the set of Fucik eigenvalues is also called the Fucik spectrumfor the 2%-periodic solutions of (18.52).

The existence result for 2%-periodic solutions of (18.52) in the resonant case wasfirst proved by Dancer [82] (when # does not change sign), and by Fabry and Fonda[106] (when # changes sign). It is a direct consequence of Theorem 18.4.1.

Theorem 18.5.2 If (µ, () satisfies (18.49) for some positive integer N , and if the2%-periodic forcing e(t) is such that the function

#()0) :=

! 2"

0e(t)$(t + )0) dt,

with $ given in (18.50), does not vanish or has at least four zeros, all simple, thenequation (18.52) has at least one 2%-periodic solution.

Chapter 19

Computing degree in higherdimensions

19.1 Cartesian products of mappings

Let n1 ( 1, . . . , nm ( 1 be integers withBm

j=1 nj = n, Dj ' Rnj be open bounded

sets, f j & C(Dj , Rnj ).C2(Dj , Rnj ) and zj be such that zj %& f j(!Dj) (1 ) j ) m).As an easy consequence, D = D1 1 . . .1Dm is an open bounded subset of Rn andthe mapping f defined by

f : D = D1 1 . . .1Dm " Rn, x 2" f(x) = (f1(x1), . . . , fm(xm)) (19.1)

belongs to C(D, Rn) . C1(D, Rn) and is such that z %& f(!D).

Lemma 19.1.1 Under the above assumptions, if z is a regular value for f in D,then

dB[f, D, z] =m4

j=1

dB[f j , Dj , zj]. (19.2)

Proof. As z is a regular value, f!1(z) is a finite set. Now, with x = (x1, . . . , xm),

f(x) = z 4 f1(x1) = z1, . . . , fm(xm) = zm,

and hence each (f j)!1(zj) is finite, namely

(f j)!1(zj) = {xj,1, . . . , xj,qj} (1 ) j ) m),

so that

f!1(z) = {(x1,l1 , . . . , xm,lm) : 1 ) l1 ) q1, . . . , 1 ) lm ) qm}.

227

228 CHAPTER 19. COMPUTING DEGREE IN HIGHER DIMENSIONS

Furthermore, an easy computation shows that, for x & D,

Jf (x) =m4

j=1

Jfj (xj),

so that, in particular, each zj must be a regular value for f j (1 ) j ) m). Conse-quently,

dB[f, D, z) =q1"

l1=1

. . .qm"

lm=1

sign Jf (x1,l1 , . . . , xm,lm)

=q1"

l1=1

sign Jf1(x1,l1) . . . [qm"

lm=1

sign Jfm(xm,lm)]

= dB[f1, D1, z1] . . . dB [fm, Dm, zm] =m4

j=1

dB[f j , Dj , zj].

The result is easily extended to the situation of f continuous and z %& f(!D)arbitrary.

Theorem 19.1.1 If f & C(D, Rn) is given by (19.1) and if zj %& !Dj (1 ) j ) m)then formula (19.2) holds.

Proof. Take z$ regular for f su$ciently close to z so that

dB[f, D, z] = dB[f, D, z$] =m4

j=1

dB [f j, Dj , z$j] =m4

j=1

dB[f j , Dj , zj].

As an example, let us consider the mapping p0 : Cm " Cm defined by

p0(z1, . . . , zm) = (p01(z1), . . . , p

0m(zm))

where the functions p0j : C" C are defined by

p0j(z) = zkj (1 ) j ) m) (19.3)

with kj ( 1 is an integer (1 ) j ) m). p can also be seen as a mapping from R2m

into R2m. We know from example 17.4.1 that, for each r > 0,

dB[p0j , B(r), 0] = kj (1 ) j ) m),

so that Theorem 19.1.1 implies that

dB[p0, B(r) 1 . . .1B(r), 0] =m4

j=1

kj . (19.4)

19.2. HOMOGENEOUS POLYNOMIALS MAPPINGS 229

Now, p0(z) = 0 if and only if z = 0, and hence, using excision property 3.4.1 wededuce from (19.4) and from the definition of Brouwer index 5.1.1 that, for anyopen bounded neighborhood D of 0 in Cm, and given by (19.3), we have

dB[p0, D, 0] = iB[p0, 0] =m4

j=1

kj (19.5)

19.2 Homogeneous polynomials mappings

We consider now the more general situation of mappings p : Cm " Cm where

each component p[kj ]j is a homogeneous polynomial in (z1, . . . , zn) of degree kj ( 1

(1 ) j ) m). We assume that p!1(0) = {0} and compute iB[p, 0]. The followingresults were first obtained in 1953 by J. Cronin [78] (see also [81]).

Proposition 19.2.1 Let the mapping p : Cm " Cm be such that each component

p[kj]j is a homogeneous polynomial in (z1, . . . , zn) of degree kj ( 1 (1 ) j ) m). If

p!1(0) = {0}, then, for any open bounded neighborhood D of 0, one has

dB[p, D, 0] = iB[p, 0] =m4

j=1

kj . (19.6)

Proof. Given m homogeneous polynomial p1, . . . pm : Cm " C of respective de-grees k1, . . . , km, let us order their coe$cients in a sequence (a1, . . . , ap) consid-ered as an element of Cp. The resultant R(a1, . . . , ap) of the polynomials pj (j =1, . . . , m) is a polynomial in (a1, a2, . . . , ap) with integer coe$cients, such thatR(a1, . . . , ap) %= 0 if and only if 0 is the unique common zero of the pj (j = 1, . . . , m).Let b1, b2, . . . , bp denote the corresponding sequence of the coe$cients of the poly-

nomials p[k1]1 , . . . , p[km]

m , and (c1, . . . , cp) the corresponding sequence of the coe$-cients of the polynomial p0

1, . . . , p0m defined in (19.3). By assumption and definition,

R(b1, . . . , bp) %= 0 and R(c1, . . . , cp) %= 0. Define the polynomial 2 : C" C by

2(t) = R[(1! t)b1 + tc1, . . . , (1! t)bp + tcp].

Since 2(0) = R(b1, . . . , bp) %= 0 and 2(1) = R(c1, . . . , cp) %= 0, the polynomial 2(t)is not identically zero, and hence has a finite number of zeros, di!erent from 0 and1. Consequently, there is a continuous path 5 : [0, 1] " C such that 5(0) = 0,5(1) = 1 and 2[5(")] %= 0 for all " & [0, 1]. Consequently, the homotopy P : Cm 1[0, 1] " Cm where P (z,") = (P1(z,"), . . . , Pm(z,")), and P1(z,"), . . . , Pm(z,")are the homogeneous polynomials of respective degrees k1, . . . , km associated to thesequence of coe$cients

(1! 5("))b1 + 5(")c1, . . . , (1! 5("))bp + 5(")cp


is such that P (z, 0) = p(z), P (z, 1) = p0(z), and

R[(1! 5("))b1 + 5(")c1, . . . , (1! 5("))bp + 5(")cp] %= 0

for all " & [0, 1]. In other words, [P (·,")]!1(0) = {0} for all " & [0, 1], and hence,using the homotopy invariance property 3.4.2 and (19.5), we get, for any openbounded neighborhood D of 0,

iB[p, 0] = dB [p, D, 0] = dB[P (·, 0), D, 0] = dB[P (·, 1), D, 0] = dB [p0, D, 0]

= iB[p0, 0] =m4

j=1

kj .

Combining Proposition 19.2.1 with Propositions 5.2.2 and 5.3.1, we obtain im-mediately the following results.

Proposition 19.2.2 Let U ' Cm be an open neighborhood of 0, and f & C(U, Cm)

be such that, for some mapping p : Cm " Cm whose each component p[kj ]j a is

homogeneous polynomial in (z1, . . . , zn) of degree kj ( 1 (1 ) j ) m), one has

limz(0

fj(z)! pj(z)

#z#kj= 0 (1 ) j ) m).

If p!1(0) = {0}, then

iB[f, 0] =m4

j=1

kj .

Proposition 19.2.3 Let f & C(Cm, Cm) be such that, for some mapping p : Cm "Cm whose each component p

[kj ]j is a homogeneous polynomial in (z1, . . . , zn) of

degree kj ( 1 (1 ) j ) m), one has

limz(%

fj(z)! pj(z)

#z#kj= 0 (1 ) j ) m).

If p!1(0) = {0}, then, for any z & Cm,

iB[f,-, z] =m4

j=1

kj .

We can complete Proposition 19.2.2 by a result, also due to Cronin [78, 81]which only requires the degree of f to be defined on some ball.

19.3. ORIENTATION PRESERVING MAPPINGS 231

Proposition 19.2.4 Let U ' Cm be an open neighborhood of 0, and f & C(U, Cm)

be such that, for some mapping p : Cm " Cm whose each component p[kj]j is a

homogeneous polynomial in (z1, . . . , zn) of degree kj ( 1 (1 ) j ) m), one has

limz(0

fj(z)! pj(z)

#z#kj= 0 (1 ) j ) m).

If 0 %& f(!B(R)) for some R > 0, , then

dB[f, B(R), 0] (m4

j=1

kj .

Proof. Let us first assume that the resultant R of p[k1]1 , . . . , p[km]

m is nonzero. ByProposition 19.2.2, if ' > 0 is su$ciently small, dB [f, B('), 0] =

Gmj=1 kj . Take

' ) R. Using Sard’s theorem, there exists y arbitrarily close to zero such that y isnot a critical value of f and

dB[f, B('), 0] = dB[f, B('), y], dB[f, B(R), 0] = dB[f, B(R), y].

Consequently, as the Jacobian of f is positive on f!1(y),

m4

j=1

kj = dB[f, B('), y] = #f!1(y) .B(') ) #f!1(y) .B(R)

= dB[f, B(R), y].

If the resultant R of p[k1]1 , . . . , p[km]

m is zero, then, there exist arbitrary small changes

of the coe$cients of those polynomials giving polynomials q[k1]1 , . . . , q[km]

m the resul-tant of which is nonzero. Using Rouche’s property 3.4.2, we can take them so thatthe mapping g obtained in replacing p in f by q is such that

dB[f, B(R), 0] = dB[g, B(R), 0] (m4

j=1

kj ,

by applying the first part of the proof to g.

19.3 Orientation preserving mappings

We shall consider in this section and the following ones some classes of mappingswhich always have a nonnegative Brouwer degree. They contain in particular, indimension two, the mappings associated to holomorphic functions of C into C, andconsidered in Chapter 17. We will see that they contain various classes of mappingsbetween higher dimensional spaces.

Definition 19.3.1 The linear mapping L : Rn " Rn is said to be orientation-preserving if det L ( 0.


The following definitions were introduced in 1970 by F.E. Browder [46] underthe name of mapping of strict analytic type, and renamed by the same authorstrictly orientation-preserving mappings in [47]. Let U ' Rn be open, andlet f : U " Rn.

Definition 19.3.2 We say that f is strictly orientation-preserving if the followingconditions hold :

(a) f & C2(U, Rn) and, for each x & U, its di"erential f $(x) is orientation pre-serving;

(b) The set {x & U : Jf (x) = 0} is nowhere-dense in U.

Again, the following class of mapping was first introduced in [46] under the nameof mappings of analytic type and renamed orientation-preserving mappingsin [47].

Definition 19.3.3 We say that f & C(U, Rn) is orientation-preserving if, for eachopen subset V with V ' U and compact, there exists a sequence of mappings (fk),with fk : V " Rn of strictly orientation-preserving, such that fk " f uniformly onV.

We prove some general properties of the degree of orientation-preserving map-pings first proved by F.E. Browder in [46, 47]. We begin by proving them for strictlyorientation-preserving mappings.

Lemma 19.3.1 Let U ' Rn be open, f : U " Rn a strictly orientation-preservingmapping, D & Rn an open bounded set such that D ' U , and z %& f(!D). Then :

(a) dB [f, D, z] ( 0.

(b) dB [f, D, z] > 0 if and only if z & f(D).

(c) If dB[f, D, z] = 1, f!1(z) .D is connected.

(d) If f(D) contains a point of some component C of Rn \ f(!D), C ' f(D).

Proof. (a) If follows from (3.1) and definitions 19.3.1 and 19.3.2 that dB[f, D, z] ( 0.(b) If dB[f, D, z] > 0, z & f(D) by the existence property 3.4.1. If z & f(D), let

C be the connected component of Rn\f(!D) containing z. By Definition 19.3.2, theclosed set {x & D : Jf (x) = 0} has empty interior, and hence, if x0 & D is such thatf(x0) = z, there exists a sequence (xn) in D converging to x0 such that Jf (xn) > 0for all integers n ( 1. By continuity, f(xn) " f(x0) = z, and hence there existsm & N# such that f(x0) & C for n ( m. By the inverse mapping theorem xm isan isolated solution of equation f(x) = f(xm), and there exists r > 0 such thatBr(xm) only contains that solution. Consequently, using the additivity property3.4.3 and the property 3.4.6, we get

dB[f, D, z] = dB[f, D, f(xm)]

= dB[f, Br(xm), f(xm)] + dB [f, D \ Br(xm), f(xm)]

( dB[f, Br(xm), f(xm)] = iB[f, xm] = sign Jf (xm) = 1.

19.3. ORIENTATION PRESERVING MAPPINGS 233

(c) Suppose that dB [f, D, z] = 1 and that f!1(z) . D is not connected. Thenthere exists two disjoint open sets N1 and N2 of V with disjoint closures such thatf!1(z) .D ' N1 /N2, each Nj containing points of f!1(z). By conclusion (b),

dB [f, N1, z] ( 1, dB[f, N2, z] ( 1,

and by the additivity property 3.4.3 of degree,

dB[f, D, z] = dB [f, N1, z] + dB [f, N2, z] ( 2,

a contradiction.(d) Suppose that for a given component C of Rn \ f(!D), we have a point y &

f(D).C. From conclusion (b), dB[f, D, y] > 0, and by property 3.4.6, dB [f, D, z] =dB[f, D, y] for any z & C, so that, by the existence property 3.4.1, C ' f(D).

We now extend the conclusions of Lemma 19.3.1 to orientation-preserving map-pings.

Theorem 19.3.1 Let U ' Rn be open, f & C(U, Rn) an orientation-preservingmapping, D & Rn an open bounded set such that D ' U , and z %& f(!D). Then :

(a) dB[f, D, z] ( 0.

(b) dB[f, D, z] > 0 if and only if z & f(D).

(c) If dB [f, D, z] = 1, f!1(z) .D is connected.

(d) If f(D) contains a point of some component C of Rn \ f(!D), C ' f(D).

Proof. Let the open subset V be such that D ' V ' V ' U and V is compact.By definition of f, there exists a sequence (fk) of mappings of V in Rn strictlyorientation-preserving and converging uniformly to f on V as k " -. So theconclusions of Lemma 19.3.1 are valid for the fk.

(a) Let z & Rn \ f(!D), d0 = minx"!D #f(x) ! z# and k0 su$ciently large sothat maxx"!D #fk(x) ! f(x)# < d0. Then, from Rouche’s property 3.4.2, we havedB[f, D, z] = dB[fk, D, z] ( 0 (k ( k0) and the result follows letting k "-.

(b) If dB[f, D, z] > 0, then z & f(D) by the existence property 3.4.1. If z & f(D),and if d0 = minx"!D #f(x)! z#, let x0 & D be such that f(x0) = z. Choose k0 solarge that for k > k0, one has maxx"D #fk(x)!f(x)# < d0

2 . Then, for each x & !D,one has

#fk(x) ! z# ( #f(x)! z# ! #fk(x) ! f(x)# >d0

2,

which implies that By(d0/2) is contained in the same component of Rn \ fk(!D) asy. On the other hand

#fk(x0)! z# = #fk(x0)! f(x0)# <d0

2,


and hence fk(x0) lies in the same component of Rn \fk(!D) as y. By conclusion (b)of Lemma 19.3.1, we see that dB[fk, D, z] > 0, and hence, by Rouche’s property, wehave, for all k > k0,

dB[f, D, z] = dB [fk, D, z] > 0.

(c) and (d) Follow exactly the same argument as the proof of (c) and (d) inLemma 19.3.1.

19.4 Monotone mappings

As a first example, let us denote by (·, ·) the usual inner product in Rn, and assumethat, for some open U ' Rn, f : U " X is monotone, i.e. that (f(x) ! f(y), x!y) ( 0 for any x, y & U.

Proposition 19.4.1 If U ' Rn is open and f : U " Rn is monotone, then f isorientation-preserving on U.

Proof. Let V be an open set with compact closure V ' U, then dist(V , !U) > 0,and there exists + > 0 such that, for any x & V, Bx(+) ' U. If t > 0 is su$cientlysmall and q is a C2 scalar nonnegative function, with support contained in Bx(r)for some 0 < r < +, and such that

PRn q(x) dx = 1, we define gt : V " Rn by

gt(u) =

!

Rn

[f(u! v) + t(u! v)]q(v) dv =

!

Bu(r)[f(w + tw)]q(u ! w) dw.

Hence, gt & C2(V, Rn), and, using the monotonicity of f,

(gt(x)! gt(y), x! y) =

!

Rn

([f(x! v)! f(y ! v) + t(x! y)], x! y)q(v) dv

( t

!

Rn

#x! y)#2q(v) dv = t#x! y#2.

Taking x = y + sv with s %= 0 su$ciently small, we get

(gt(y + sv)! gt(y), v)

s( t#v#2

for all s %= 0 su$ciently small and all v & Rn, and hence

((gt)$(y)v, v) ( t#v#2

for all y & V and v & Rn. Hence, all real eigenvalues of (gt)$(y) are positive and,as complex eigenvalue occur in conjugate pairs, Jgt(y), equal to the product of theeigenvalues of (gt)$(y), is positive for all t > 0 and u & Rn. Finally, it is standardthat gt " f uniformly on V when t" 0 + .

19.5. HOLOMORPHIC MAPPINGS 235

19.5 Holomorphic mappings

As a second example, let us consider the class of holomorphic mappings from Cm

into Cm, viewed as mappings from R2m into R2m. We first recall a few definitions[56]. Let U ' Cm, f : U " Cm and a & U.

Definition 19.5.1 We say that f is holomorphic at a if there exists a C-linearmapping L and a mapping r : (U ! a) \ {0}" Cm such that limh(0 r(h) = 0 and,for h & U ! a,

f(a + h) = f(a) + L(h) + #h#r(h). (19.7)

If it is the case, L is unique and noted f $(a) or df(a), the limits

!zj f(a) := limt"C;t(0

f(a + tej)! f(a)

t(1 ) j ) m) (19.8)

exist, are equal to f $(a)(ej) (j = 1, 2, . . . , m) and

f $(a)h =m"

j=1

hj!zj f(a). (19.9)

In the terminology of di!erentials, (19.9) gives, for f(z) = z the identity

dz =m"

j=1

hjej = h, i.e. dzj = hj (j = 1, 2, . . . , m),

and hence (19.9) can be written

df =m"

j=1

!zj f dzj. (19.10)

If we write zj = xj + iyj with xj , yj & R (j = 1, 2, . . . , m) it follows from (19.7)that f, considered as a mapping from R2m to R2m by taking the real and imaginaryparts of the variables and of the components of the function, is di!erentiable, and

df =m"

j=1

(!xj f dxj + !yj f dyj). (19.11)

Using the relations

dzj = dxj + idyj , dzj = dzj = dxj ! idyj

dxj =1

2(dzj + dzj), dyj =

1

2i(dzj ! dzj) (j = 1, 2, . . . , m),

we can write (19.11) as

df =m"

j=1

-1

2(!xj f ! i!yj f )dzj +

1

2(!xj f + i!yj f) dzj

.. (19.12)


This implies in particular that

df =m"

j=1

-1

2(!xj f ! i!yjf) dzj +

1

2(!xj f + i!yj f) dzj

.

=m"

j=1

-1

2(!xj f + i!yjf )dzj +

1

2(!xj f ! !yj f) dzj

.

= df, (19.13)

and that f is holomorphic on U if and only if

!zj f :=1

2(!xj f + i!yj f) = 0 (j = 1, 2, . . . , m).

The following result is fundamental. For f : U " Cm holomorphic, we denoteby Jf (z) = det f $(z) the (complex) Jacobian associated to the C-di!erential of f,by & : U ' R2m " R2m the associated real mapping defined by

&(x1, y1, . . . , xm, ym) (19.14)

= (u1(x1, . . . , ym), v1(x1, . . . , ym), . . . , um(x1, . . . , ym), vm(x1, . . . , ym)),

and by J$(z) the (real) Jacobian associated to the R-di!erential of &.

Lemma 19.5.1 If U ' Cm is open, and f : U " Cm is holomorphic in U,

J$(x1, y1, . . . , xm, ym) = |Jf (z)|2 (19.15)

for all z = (z1, . . . , zm) = (x1, y1, . . . , xm, ym) & U.

Proof. Using the rules of exterior calculus, we have

df1 $ . . . $ dfm =

5

6m"

j1=1

!zj1f1 dzj1

8

9 $ . . . $

5

6m"

jm=1

!zjmfm dzjm

8

9

=

&

'"

#=(j1,...,jm)

(sign () !zj1f1 . . .!zjm

fm

(

) dz1 $ . . . $ dzm

= Jf dz1 $ . . . $ dzm, (19.16)

where the sum is taken over all the permutations ( = {j1, . . . , jm} of {1, . . . , m}.Therefore,

df1 $ . . . $ dfm = df1 $ . . . $ dfm = Jf dz1 $ . . . $ dzm. (19.17)

Similarly,

du1 $ dv1 $ . . . $ dum $ dvm = J$ dx1 $ dy1 $ . . . $ dxm $ dym. (19.18)


On the other hand

dzj $ dzj = (dxj + idyj) $ (dxj ! idyj) = !2i dxj $ dyj (19.19)

dfj $ df j = (duj + idvj) $ (duj ! idvj) = !2i duj $ dvj (j = 1, 2, . . . , m).

Hence,

df1 $ df1 $ . . . $ dfm $ dfm = (!2i)m du1 $ dv1 $ . . . dum $ dvm,

or, using the same permutations in the left and right-hand members,

df1 $ . . . $ dfn $ df1 $ . . . $ dfm

= (!2i)mdu1 $ . . . $ dum $ dv1 $ . . . $ dvm,

and, using (19.16), (19.17), (19.18) and (19.19), this gives

Jf · Jf dz1 $ . . . dzm $ dz1 $ . . . $ dzm

= (!2i)mJ$ dx1 $ . . . $ dxm $ dy1 $ . . . $ dym,

and hence (19.15), if we notice that, from (19.19)

dz1 $ . . . $ dzm $ dz1 $ . . . $ dzm = (!2i)m dx1 $ . . . $ dxm $ dy1 $ . . . $ dym.

We need a second preliminary result to show that the holomorphic mappingsare orientation-preserving. We use the notations of Lemma 19.5.1.

Lemma 19.5.2 Let U ' Cm be connected and open, and f : U " Cm holomorphic.Given +0 > 0, there exists ' & ]0, d0[ such that, if f*(z) = f(z) + 'z, the set S* :={x & U : J$+*I(x) = 0} is nowhere dense in U.

Proof. By Lemma 19.5.1, it su$ces to show that the set of z & U such thatJf!(z) = 0 is nowhere dense in U. Because the partial C-derivatives of an holo-morphic functions are holomorphic (see e.g. [56]), Jf! is a holomorphic function ofz, and, for fixed z = z0 a polynomial in t whose top coe$cient is one. Hence, givenany +0 > 0, there is some ' & [0, +0[ such that Jf!(z0) %= 0, and hence Jf!(z0) > 0.Suppose that, for this value of ', S* is dense in a non-trivial open subset of U.Since S* is closed in U, then Jf! = 0 on a non-trivial open subset of U. Since Jf!

is analytic and U is connected, we must have Jf!(z) = 0 on U, a contradiction withJf!(z0) %= 0.

Proposition 19.5.1 Let U ' Cm be open, and f : U " Cm holomorphic. Thenthe associated mapping & : U ' R2m " R2m is orientation-preserving.

Proof. By Lemmas 19.5.1 and 19.5.2, on each connected component of U, thereexists a sequence (&k) of strictly orientation-preserving mappings (correspondingto the f*k for a suitable sequence 'k " 0) which converges uniformly to & on anybounded subset of the connected component.


Let U ' Cm be open, f : U " Cm be holomorphic, & : U ' R2m " R2m

the associated real mapping defined in (19.14), D open bounded such that D ' U,and y %& f(!D). Notice that we use the same notations for corresponding points andcorresponding sets in Cm and R2m. We write dB[f, D, y] for dB[&, D, y]. From Propo-sition 19.5.1 and Theorem 19.3.1, we know that dB [f, D, y] ( 0 and dB[f, D, y] > 0if and only if y & f(D). A more precise relation between the number of zeros of fand its topological degree has been obtained in 1973 by P. Rabinowitz [326] (seealso [328]). It requires proving some preliminary algebraic result.

Lemma 19.5.3 Given 81, . . . , 8k in Cm \{0}, one can find a basis of Cm such that,in this basis, all coordinates 8jl (1 ) j ) k; 1 ) j ) m) are di"erent from zero.

Proof. Let µ & Cm and M the matrix with equal lines all made of the coordinatesof µ. Choose µ su$ciently small so that the matrix I + M is invertible. Considerethe basis transformation defined by the matrix (I + M). If wj = 8j + M8j, i.e., forany 1 ) l ) m, wjl = 8jl + (8j |µ), where (·|·) denotes the standard inner productin Cm. If we define the hyperplanes Hjl ' Cm by Hjl := {z & Cm : (8j , z) = !8jl

(1 ) j ) k; 1 ) l ) m), then Cm \H

1+j+k;1+l+m Hjm is everywhere dense in Cm,and hence we can choose µ & Cm \

H1+j+k;1+l+m Hjm su$ciently small so that

I + M is invertible. With this choice, wjl %= 0 for all 1 ) j ) k and 1 ) l ) m.

Theorem 19.5.1 If U ' Cm is open,f : U " Cm holomorphic, D open boundedsuch that D ' U y %& f(!D), and dB [f, D, y] = k, then equation f(z) = y has atmost k distinct solutions in D.

Proof. The proof is by recurrence over k. For k = 0, it follows from assertion (b)of Theorem 19.3.1 that equation f(z) = y has no solution in D. Assume nowthat the result is true for k ! 1 and all f, D, y satisfying the assumptions, andsuppose that the result is not true for k, i.e. that there exists some f, D, y such thatdB[f, D, y] = k and f!1(y).D contains at least k+1 distinct elements 80, 81, . . . , 8k.By a translation of the origin of coordinates and a translation of f, which do notchange the Brouwer degree, we can assume without loss of generality that 80 = 0and y = 0. Also, because Brouwer degree is indepedent of the chosen basis in R2m

and hence in Cn, we can use Lemma 19.5.3 to choose a basis in which none of thecoordinates of 81, . . . 8k is zero, i.e. such that 8jl %= 0 (1 ) j ) k; 1 ) l ) m). Let usintroduce then the application g : U " Cm defined by

gl(z) = fl(z) + 'k4

j=0

(zl ! 8jl) (1 ) l ) m),

where ' > 0. g is holomorphic on U and g(8j) = 0 for all j = 0, 1, . . . , k. Fur-thermore, it follows from Rouche’s property 3.4.2 that, for ' > 0 su$ciently small,dB[g, D, y] = dB[f, D, y] = k. Now Jg(0) is a polynomial in ' with coe$cient ofthe highest power equal to (!1)km

G1+j+k;1+l+m 8jl %= 0. Consequently we can find

' > 0 arbitrarily small such that Jg(0) %= 0, which implies that 0 is an isolated zero


of g, and hence, using the additivity property 3.4.3, we get, for some su$cientlysmall + > 0,

k = dB[g, D, 0] = dB[g, B(+), 0] + dB [g, D \ B(+), 0]

= iB[g, 0] + dB[g, D \ B(+), 0] = 1 + dB [g, D \ B(+), 0],

and hence dB[g, D \ B(+), 0] = k ! 1. Using the recurrence assumption, g has atmost k ! 1 distinct zeros in D \ B(+), which contradicts the assumption that g athas least k + 1 distinct zeros in D.

Corollary 19.5.1 If U ' Cm is open,f : U " Cm holomorphic, D open boundedsuch that D ' U y %& f(!D), and dB[f, D, y] = 1, then equation f(z) = y has aunique solution in D.

Remark 19.5.1 The converse of Corollary 19.5.1 is no true under the sole hy-pothesis that the solution is unique. Indeed, for m = 1, and p ( 2 an integer, theequation zp = 0 has the unique solution z = 0 and, according to example 17.4.1,dB[zp, B(R), 0] = p > 1 for any R > 0.

Corollary 19.5.1 allows a sharpening of the conclusion in Brouwer’s fixedpoint theorem for holomorphic mappings.

Corollary 19.5.2 Let U ' Cm be open, f : U " Cm holomorphic and assume thatthere exists an open, bounded, convex set D such that D ' U and f(!D) ' D. Iff has no fixed point on !A (in particular if f(A) ' A), then f has a unique fixedpoint in A.

Proof. Because the degree is invariant by translation, we can assume, without lossof generality, that 0 & D. Let us introduce the homotopy H : D 1 [0, 1] " Cm

defined by H(z,") = z ! "f(z). By assumption, 0 %& H(!D, 1), and, for z & !Dand " & [0, 1[ , we have, using convexity, "g(z) & A, and hence H(z,") %= 0. By thehomotopy invariance property 3.4.2, we get

dB [I ! f, D, 0] = dB [H(·, 1), D, 0] = dB [H(·, 0), D, 0] = dB[I, D, 0] = 1,

and the result follows from Corollary 19.5.1.

Corollary 19.5.3 Let G ' Cm be open and bounded, f : G" Cm be holomorphic,y & Cm and 8 & G such that f(8) = y. Let us denote by C the connected componentof f!1(y) containing 8. Then either 8 is an isolated solution of equation f(z) = y(and hence C = {y}), or C meets any neighborhood of !G.

Proof. Assume there exists a neighborhood of !G which does not meet C. Then onecan find an open set V such that C ' V ' V ' G. Now, f!1(y) . !V and C aretwo disjoint compact disjoint subsets of f!1(y). V . If we apply Whyburn’s lemma7.2.1, we obtain two compact subsets K and K $ of f!1(y) . V such that K 0 C,


K $ 0 f!1(y).!V, K .K $ = , and K /K $ = f!1(y).V . (Indeed the existence of aconnected component connecting C to f!1(y).!V would contradict the maximalityof C.) Consequently, there exists an open set D such that D 0 K, D .K $ = , andD ' V. Thus we have C ' D ' D ' V ' D, f!1(y) . !D = ,. Hence dB[f, D, y]is well defined and, from assertion (b) of Theorem 19.3.1, positive. From Theorem19.5.1, f!1(y) .D contains only a finite number of points, so that 8 is an isolatedsolution of f(z) = b and C = {8}.

Corollary 19.5.4 Let G ' Cm be open and bounded, p < m and f : G " Cp

holomorphic. Let 8 & G and y & Cp be such that f(8) = y. Then the connectedcomponent C of f!1(y) containing 8 meets any neighborhood of !A.

Proof. After possible translations, we can assume, without loss of generality, that8 = 0 and y = 0. Define fj : G " C by fj = 0 (j = p + 1, . . . , m), and 1f =

(f1, . . . , fm), so that f and 1f have the same zeros. Arguing by contradiction, assumethat C does not meet some neighborhood of !D. It follows from Corollary 19.5.3that 0 is an isolated zero of 1f, and hence there is some + > 0 such that 0 % 1f(!B(+)).

Consequently, dB[ 1f, B(+), 0] is defined and equal to k > 0. Define g : G" Cm by

gj(z) =

/fj(z) + 'zj if 1 ) j ) p,0 if p + 1 ) j ) m.

For su$ciently small ' > 0, it follows from Rouche’s property 3.4.2 that

dB[g, B(+), 0] = dB[ 1f, B(+), 0] = k > 0.

Hence Theorem 19.5.3 implies that 0 is an isolated zero of g. But, on the otherhand, the jacobian of (g1, . . . , gp) with respect to z1, . . . , zm computed at 0 is apolynomial in ' whose coe$cient of the highest power is 1. Hence, for arbitrarysmall values of ', this Jacobian is di!erent from zero. Using the implicit functiontheorem for complex mappings, this implies, for each of those ', the existence of pfunctions &j(Tz) (1 ) j ) p), where Tz = (zp+1, . . . , zm), which are holomorphic nearTz = 0, such that &j(0) = 0 (j = 1, . . . , p), and

gj [&1(Tz), . . . ,&p(Tz), Tz] = 0 (1 ) j ) m)

on a suitable neighborhood of Tz = 0. But this contradicts the isolated character of0 as a zero of g.

Remark 19.5.2 Recall that an analytic manifold can be defined as the set ofcommon zeros of a family of holomorphic functions {fi}i"I on some open set U ofCm. So Corollaries 19.5.3 and 19.5.4 provide results on the structure of analyticmanifolds when #I ) m. In particular, for #I = m, a compact analytic manifoldonly has a finite number of points, a result which is also true if #I ( m, as shownfor example in [157].


The following improvement of Corollary 19.5.1 is also due to Rabinowitz [326](see also [328]).

Theorem 19.5.2 Let U ' Cm be open, f : U " Cm holomorphic, D ' C open,bounded and such that D ' U, and y %& f(!D). Then dB[f, D, y] = 1 if and only ifthe following conditions hold :

(i) There exists a unique 8 & D such that f(8) = y.

(ii) Jf (8) %= 0.

Proof. Su!ciency. If conditions (i) and (ii) holds, it follows from Corollary 5.2.1that dB [f, D, y] = sign Jf (8) = 1.

Necessity. If dB[f, D, y] = 1, Corollary 19.5.1 implies that condition (i) holds.We now prove condition (ii). Without loss of generality, after translations, we mayassume that 8 = 0 = y. Let us assume by contradiction that Jf (0) = 0. Then 0 isan eigenvalue of f $(0), and hence its Jordan canonical form contains 0 in its maindiagonal, say !zmfm(0) = 0. Without loss of generality, we may assume that 0 is asimple eigenvalue of f $(0) and that 2a := !2

zmfm(0) %= 0. Indeed, if it were not the

case, we could perturb f in the following way by letting

(f*)i =

/fi(z) + 'zi if 1 ) i ) m! 1,fm(z) + 'z2

m if i = m,

and choosing ' > 0 su$ciently small so that dB[f*, D, 0] = dB[f, D, 0] = +1 (sothat 0 remains the unique solution of f*(z) = 0 contained in D). We notice that,by construction,

!zi(f*)i(0) %= 0 (i = 1, . . . , m! 1), !zm(f*)m(0) = 0, !2zm

(f*)m(0) %= 0.

Thus f $(0) has the form 5

UUUU6

· 0L · 0

· ·· · · · ·0 . . . 0 0 0

8

VVVV9

with L a non-singular Jordan matrix of dimension (m ! 1) 1 (m ! 1). Set Tf(z) =

(f1(z), . . . , fm!1(z)), so that f = ( Tf, fm), and Tz = (z1, . . . , zm!1), so that z =(Tz, zm). Because of the holomorphy of f near 0, we can write for all z & B(-1), ($a multi-index),

fj(z) ="

|+|/1

fj+z+ (1 ) j ) m).

Hence, we have

Tf(z) = LTz + g(z),


for some mapping g such that #g(z)# )M1#z#2 for all z & B(-1), and

|fm(z)!"

|+|=2

fm+z+| )M2#z#3,

where M1 and M2 are positive constants. If we define the homotopy F (·,") by

TF (z,") = LTz + "g(z), Fm(z,") = (1! ")az2m + "

"

|+|/2

fm+,

we have

|Fm(z,")| ( |a||zm|2 !

DDDDDD

"

|+|=2

fm+z+ ! az2m

DDDDDD!M2#z#3,

Now,

DDDDDD

"

|+|=2

fm+z+ ! az2m

DDDDDD)M3#Tz#|zm| + M4#Tz#2,

where M3 and M4 are positive constants. Let * > 0 be su$ciently small so thatM3* + M4*2 ) |a|

4 , and define the subsets of Cm

P& := {z & Cm : #Tz# ) *|zm|}, Q& := {z & C

m : #Tz# ( *|zm|},

so that P& /Q& = Cm. If z & P& .B(-1), we have

DDDDDD

"

|+|=2

fm+z+ ! az2m

DDDDDD) |a|

4|zm|2.

If, furthermore, z & B(-2), with -2 = |a|4M2(1+&2) , we have

M2#z#3 )|a|4|zm|2,

and hence, for z & P& .B(min{-1, -2}), we have

|Fm(z,")| ( |a|2|zm|2 ( |a|

2(1 + *2)#z#2. (19.20)

We now estimate TF (z,"). For z & B(-1), we have

# TF (z,")| ( #LTz# ! #g(z)# (M5#Tz# !M1#z#2,


where M5 is a positive constant. If z & Q& . B(-3), where -3 = M5

2M1

q1+ 1

$2

, we

obtain

# TF (z,")# ( M5

2#Tz# ( M5

2Y

1 + 1&2

#z#. (19.21)

Define - := min{-1, -2, -3}. Because of P& / Q& = Cm, the estimates (19.20) and(19.21) imply that 0 %& F (!B(-)1[0, 1]), and using successively the excision property3.4.1, the homotopy invariance property 3.4.2, and property 3.4.6, we obtain

1 = dB[f, D, 0] = dB[f, B(-), 0] = dB [F (·, 1), B(-), 0] = dB [F (·, 0), B(-), 0]

= dB[F (·, 0), B(-), )], (19.22)

when #)# is su$ciently small. Taking ) = (0, . . . , 0,,), with , %= 0 su$cientlysmall, we see that the equation F (z, 0) = ), i.e. L1z = 0, az2

m = ,, has two distinctsolutions 8± = (0, . . . , 0,±

<,), with JF (·,0)(8±) = ±(det L) · (2a

<,) %= 0. Hence,

dB[F (·, 0), B(-), 0] = +2, a contradiction with (19.22).

A consequence of Theorem 19.5.2 is an injectivity condition first proved in1913 by G.R. Clements [70] in a di!erent way.

Corollary 19.5.5 Let U ' Cm be open, f : U " Cm holomorphic. A necessaryand su!cient condition for f to be one-to-one in a neighborhood of 8 & U is thatJf (8) %= 0.

Proof. Su!ciency. Follows directly from the local inversion theorem.Necessity. Let f be one-to-one on a neighborhood of 8, and let B0(-) be

contained in this neighborhood. Hence f(8) %& f(!B0(+)) and dB[f, B0(+), f(8)]is defined and strictly positive by Theorem 19.3.1. From Sard’s theorem, thereexists y arbitrary closed to f(8) such that dB [f, B0(+), y] = dB[f, B0(+), f(8)]and Jf (8$) %= 0, where {8$} = f!1(y) . B0(+). This last set is non-empty asdB[f, B0(+), y] > 0 and is a singleton because of the injectivity assumption. HencedB[f, B0(+), y] = 1, so that dB [f, B0(+), f(8)] = 1, and hence, using Theorem 19.5.2,Jf (8) %= 0.

Remark 19.5.3 The conclusion of Theorem 19.5.2 is obviously false in the realcase. For example, with the real function f(x) = x3, one has dB [f, ]! 1, 1[ , 0] = 1f is one-to-one and f $(0) = 0.

Remark 19.5.4 Theorem 19.5.2 implies that the Brouwer index of an isolated so-lution 8 of equation f(z) = y is greater or equal to two if Jf (8) = 0. This is a specialcase of Cronin’s proposition 19.2.4 when 0 is an isolated zero of f. Notice howeverthat the proof of Theorem 19.5.2 avoids the use of the resultant of polynomials.


19.6 Quaternionic monomials

Another class of orientation-preserving mappings is that of quaternionic mono-mials. We refer to [98] for the required definitions and properties of quaternions.For q = q0 + q1i + q2j + q3k & H, we write

1q = Bq = q1i + q2j + q3k, q = q0 !Bq, #q# =Nq20 + q2

1 + q22 + q2

3

O1/2.

Let J1/J2 be a partition of {1, 2, . . . , n} (one of the Ji can be empty) and, for each1 ) j ) n, let cj : H" H be defined by

cj(q) = q if j & J1, cj(q) = q if j & J2, (19.23)

and, given aj & H \ {0}, (0 ) j ) n), let f : H" H be the quaternionic monomialdefined by

f(q) = a0c1(q)a1c2(q)a2 . . . cn(q)an. (19.24)

If we identify H with R4, f : R4 " R4 is an homogeneous mapping vanishing only at0, and hence dB[f, B(R), 0] is well defined and independent of R > 0. The followinglemma [274] is useful for its computation.

Lemma 19.6.1 Let n ( 1 be an integer and pn : H " H, q 2" qn considered as amapping from R4 into R4. Then

Jpn(q) =

Zn4#q#4(n!1) if Bq = 0

n2#q#2(n!1)?3(qn)3q

@2if Bq %= 0.

(19.25)

Proof. . An elementary induction shows that the value at h & H of the totalderivative Dpn(q) of pn at q & H is given by

Dpn(q)h =n!1"

m=0

qmhqn!1!m. (19.26)

We prove the result by choosing an appropriate basis for the quaternions. If q isreal (i.e. 1q = 0), then by commutativity of q and h, we get Dpn(q)h = nqn!1h andhence

Jpn(q) = n4#q#4(n!1).

If q is not real, q generates a 2-dimensional subfield of H which can be identified toC, namely C = {a + bI : a, b & R}, where I = 1q/#1q#. From (19.26), it is immediateto check that C is invariant under the action of Dpn(q). The same is true for afixed complementary subspace D of H spanned by quaternions J and K such thatI, J, K verify the usual relations. By analogy with the discussion of Section 6, onehas Dpn(q)c = nqn!1c for c & C, and the Jacobian of the restriction of Dpn(q) on C

19.6. QUATERNIONIC MONOMIALS 245

is given by n2#q#2(n!1). On the other hand, for c & C and d & D, we have dc = cd,where c is the complex conjugate of c in C, and hence

Dpn(q)d =

En!1"

m=0

qmqn!1!m

Fd =

qn ! qn

q ! qd =B(qn)

Bqd.

So the Jacobian of the restriction of Dpn(q) on D is?3(qn)3q

@2. Consequently, for q

not real one has

Jpn(q) = n2#q#2(n!1)

-B(qn)

Bq

.2

.

Corollary 19.6.1 With the notations of Lemma 19.6.1, Jpn(q) ( 0 for all q & H,and Jpn(q) > 0 outside of the set N of n-measure zero defined by

/q & H :

2sin2 k%

n

3q20 =

2cos2

k%

n

3Nq21 + q2

2 + q23

O(1 ) k ) n! 1)

0. (19.27)

In particular, pn is a stricly orientation-preserving mapping.

Proof. . If Bq = 0, it follows from (19.25) that Jpn(q) > 0 for q %= 0. If Bq %= 0, onecan write q = #q#(cos ) + I sin )) with

I =1q#1q# , cos ) =

q0

#q# , sin ) =#1q##q# ,

so that qn = #q#n(cosn) + I sinn)). By (19.25), Jpn(q) = 0 if and only if Bqn = 0,i.e. if and only if ) = k"

n (1 ) k ) 2n! 1). The set of those zeros is contained inthe n! 1 cones/

q & H :

2sin2 k%

n

3q20 =

2cos2

k%

n

3Nq21 + q2

2 + q23

O(1 ) k ) n! 1)

0.

So N has n-measure zero.

We now compute dB[pm, B(R), 0] and dB [pm, B(R), 0], where m ( 1 is an inte-ger.

Lemma 19.6.2 For each R > 0, one has

dB [pm, B(R), 0] = m. (19.28)

Proof. By homotopy invariance, we have, for all ' > 0 su$ciently small

dB[pm, B(R), 0] = dB [pm, B(R), 'i].


Now, the equation qm = 'i has m roots, which are the m complex roots of i, namely

qj = '1/m

-cos

(4k + 1)%

2m+ i sin

(4k + 1)%

2m

., (0 ) k ) m! 1). (19.29)

None of those qj is real, and hence, using formula 19.25, we obtain

Jpm(qk) = m2'2 sin2 (4k + 1)%

2m> 0,

so that, using formula 3.4, we get formula (19.28).

Lemma 19.6.3 For each r > 0, one has

dB [pm, B(R), 0] = !m. (19.30)

Proof. By homotopy invariance, we have, for all ' > 0 su$ciently small

dB[pm, B(R), 0] = dB[pm, B(R),!'i].

Now,qm = !'i4 qm = 'i4 q = qj (0 ) k ) m! 1),

where qj is given in formula 19.29. Furthermore, if we define the linear mappingC : H" H, q 2" q, then det C = !1, and pm = pm * C, so that

Jpm(q) = Jpm(q) · det C = !Jpm(q),

and formula (19.30) follows from Corollary 3.1.1 and Lemma 19.6.2.

Theorem 19.6.1 If the cj are defined by formula (19.23) (1 ) j ) n), aj &H \ {0},and if f is defined by formula (19.24), then, for each R > 0, one has

dB[f, B(R), 0] = #J1 !#J2. (19.31)

Proof. . Let Aj : [0, 1]" H \ {0} be continuous and such that

Aj(0) = 1, Aj(1) = aj, (0 ) j ) n),

and let g : H" H be defined by

g(q) = c1(q)c2(q) . . . cn(q).

By the homotopy invariance of degree, we have

dB[f, B(R), 0] = dB[g, B(R), 0].

Notice now that, as qq = qq, we have cj(q)ck(q) = ck(q)cj(q) for all 1 ) j, k ) n.Hence, of we write m1 = #J1, m2 = #J2,

g(q) =

*+

,

qm1!m2#q#2m2 if m1 > m2

#q#n if m1 = m2

qm2!m1#q#2m1 if m1 < m2.

19.6. QUATERNIONIC MONOMIALS 247

Noticing that, on !B(R), one has

g(q) =

*+

,

qm1!m2R2m2 if m1 > m2

Rn if m1 = m2

qm2!m1R2m1 if m1 < m2,

and using the boundary dependence property and Corollary 3.1.1, we obtain

dB [g, B(R), 0] =

*+

,

dB[pm1!m2 , B(R), 0] if m1 > m2

0 if m1 = m2

dB[pm2!m1 , B(R), 0] if m1 < m2,

and the proof is complete.

As an application, we prove a slight extension of Eilenberg-Niven’s fundamen-tal theorem of algebra for quaternions [101] (see also [274] for a more elemen-tary proof based upon di!erential forms). Let n ( 1 be an integer and g : H " H

a continuous function such that

lim)q)(%

#g(q)##q#n = 0. (19.32)

Theorem 19.6.2 With the notations of (19.23), if a0, a1, . . . , an are non-zero qua-ternions, if #J1 %= J2, and if condition (19.32) holds, then equation

p(q) := a0c1(q)a1c2(q)a2 . . . cn(q)an + g(q) = 0 (19.33)

has at least one solution in H

Proof. By assumption,

, := #a0a1a2 . . . an# = #a0##a1# . . . #an# > 0,

and, by condition (19.32), there exists R > 0 such that

#g(q)# ) ,#q#n

2(19.34)

whenever #q# ( R. We prove that equation (19.33) has at least one solution inB(R) ' H by showing that dB[p, B(R), 0] %= 0. Let us define the homotopy F :B(R)1 [0, 1]" H by

F (q,") = a0c1(q)a1c2(q)a2 . . . cn(q)an + "g(q).

For (q,") & !B(R)1 [0, 1], we have using (19.34),

#a0c1(q)a1c2(q)a2 . . . cn(q)an + "g(q)# ( ,#q#n ! #g(q)# ( ,Rn

2> 0.

Hence, using the homotopy invariance property 3.4.2 and Theorem 19.6.1, we get

dB[p, B(R), 0] = dB [F (·, 1), B(R), 0] = dB [F (·, 0), B(R), 0] = dB[f, B(R), 0]

= #J1 !#J2 %= 0,

and the result follows from the existence property 3.4.1.


Corollary 19.6.2 If a0, a1, . . . , an are non-zero quaternions and if (19.32) holds,then equation

f(q) := a0qa1qa2 . . . qan + g(q) = 0

has at least one solution in H. This is in particular the case if g(q) is a polynomialin q of degree lower of equal to n.

Chapter 20

Degree of some symmetricmappings

20.1 Odd mappings

The simplest non-trivial symmetry in Rn is associated to the group Z2.

Definition 20.1.1 E ' Rn is said to be symmetric with respect to the origin ifE = !E, and f : E " Rn is said to be odd if f(!x) = !f(x) for all x & E.

The following result is due to Karel Borsuk [36], and the given proof is due toGromes [155].

Theorem 20.1.1 Let D ' Rn be an open bounded symmetric neighborhood of 0. Iff : D " Rn is continuous, odd, and such that 0 %& f(!D), then dB [f, D, 0] is odd.

Proof. We first show that we may assume that f & C(D, Rn) . C1(D, Rn) andJf (0) %= 0. For this, observe that if g & C(D) . C1(D) is a C1 approximation of f(to be fixed later), and h(x) = (1/2)[g(x)! g(!x)] is its odd part, let us choose +

which is not an eigenvalue of h$0. Then 1f := h! +I & C1(D, Rn).C(D, Rn), is odd,

and such that 1f $0 = h$

0! +I is invertible, so that J ef (0) %= 0. Now, if µ := min!D #f#,and g & C(D, Rn) .C1(D, Rn) chosen in such a way that maxD #f ! g# < µ/2, wehave, for all x & D,

#f(x)! h(x)# = (1/2)#f(x)! g(x)! [f(!x)! g(!x)]# < (1/2)(µ + µ) = µ,

and, by Rouche’s property 3.4.2, dB[f, D, 0] = dB[h, D, 0].Let now f & C(D, Rn).C1(D, Rn) be odd and such that Jf (0) %= 0. If we show

the existence of an odd mapping g & C(D, Rn) . C1(D, Rn) su$ciently close to fand such that 0 is a regular value for g, then Rouche’s property 3.4.2 and Corollary5.2.1 imply that

dB [f, D, 0] = dB[g, D, 0] = sign Jf (0) +"

01=x"g!1(0)

sign Jg(x). (20.1)

249

250 CHAPTER 20. DEGREE OF SOME SYMMETRIC MAPPINGS

Now {x & D\{0} : g(x) = 0} is symmetric with respect to the origin, and Jg(!x) =Jg(x), which shows that the right-hand member in (20.1) is an odd integer.

We construct such a mapping g by induction. For each integer 1 ) k ) n, wedefine

Dk := {x & D : xi %= 0 for some 1 ) i ) k},

and we introduce an odd mapping & & C1(R) such that &$(0) = 0 and &(t) = 0 ifand only if t = 0. We first consider the mapping f(x) = f(x)/&(x1) on the openbounded set D1 := {x & D : x1 %= 0}. Notice that

f$x = [&(x1)]

!2[&(x1)f$x ! &$(x1)f(x)p1] (20.2)

with p1(h) = h1. By Sard’s theorem, we can find a regular value y1 for f on D1

with #y1# as small as we want. In this way, if g1(x) := f(x)! &(x1)y1 on D1, andg1(x) = 0 for some x & D1, we obtain, using formula (20.2), that

f$x = [&(x1)]

!1[f $x ! y1p1],

i.e. that (g1)$x = &(x1)f$x, so that 0 is a regular value for g1 on D1. Suppose now

that we have found an odd gk & C(D, Rn) . C1(D, Rn), close to f on D, and suchthat 0 is a regular value of gk on Dk for some integer 1 ) k < n. Then we definegk+1 = gk(x) ! &(xk+1)yk+1 with #yk+1# small and such that 0 is a regular valuefor gk+1 on {x & D : xk+1 %= 0}. Of course, gk+1 & C(D, Rn) . C1(D, Rn) is oddand close to f on D. If x & Dk+1 and xk+1 = 0, then x & Dk , gk+1(x) = gk(x)and (gk+1)$x = (gk)$x, so that Jgk+1(x) %= 0 and 0 is a regular value for gk+1 onDk+1. So, finally, g := gn is odd, close to f in D, and such that 0 is a regularvalue of g on D \ {0}, since Dn = D \ {0}. By the induction step, we also haveg$0 = (g1)$0 = . . . = f $

0, and hence 0 is a regular value for g in D.

We immediately deduce an easy generalization.

Corollary 20.1.1 Let D ' Rn be an open bounded symmetric neigborhood of 0,and f & C(D, Rn) be such that 0 %& f(!D) and

f(!x) %= µf(x) for all µ ( 1 and x & !D. (20.3)

Then dB [f, D, 0] is odd, and f(D) contains a neighborhood of 0.

Proof. Let us define the homotopy F by

F (x,") = f(x)! "f(!x), " & [0, 1],

so that F (·, 1) is odd, F (·, 0) = f, and, by (20.3) with µ = 1/" and condition0 %& f(!D), we have 0 %& F (!D 1 [0, 1]), and hence

dB[f, D, 0] = dB[F (·, 1), D, 0] = 1 (mod 2).

The last part of the statement follows from Corollary 3.4.7.

20.1. ODD MAPPINGS 251

A special case of Corollary 20.1.1 for mapping having di!erent directions atantipodal points goes as follows.

Corollary 20.1.2 Let D ' Rn be an open bounded symmetric neigborhood of 0,and f & C(D, Rn) be such that 0 %& f(!D) and

f(!x)

#f(!x)# %=f(x)

#f(x)# for all x & !D. (20.4)

Then dB[f, D, 0] is odd, and f(D) contains a neighborhood of 0.

Proof. If one had f(!x) = µf(x), for some µ ( 1 and x & !D, then #f(!x)# =

µ#f(x)# and hence f(!x))f(!x)) = f(x)

)f(x)) , a contradiction to (20.4). Thus the resultfollows from Corollary 20.1.1.

A consequence of Corollary 20.1.2 is the simultaneous existence of a zero and ofa fixed point for mappings odd on !D.

Corollary 20.1.3 Let D ' Rn be an open bounded symmetric neigborhood of 0and f & C(D, Rn) be odd on !D. Then there exists x# & D such that f(x#) = 0,and 1x & D such that 1x = f(1x).

Proof. If 0 & f(!D), the first part of the result is proved. If not, it follows fromCorollary 20.1.2 that dB[f, D, 0] = 1 (mod 2) and, by the existence property 3.4.1f has a zero in D. Now, the mapping g & C(D, Rn) defined by g(x) = x! f(x) hasthe same properties than f and the first part of the proof applies.

Remark 20.1.1 Corollary 20.1.3 implies the no-retraction of a closed ball onto itsboundary, because if r : B(R)" !B(R) is such a retraction, !r would have a fixedpoint x# by Corollary 20.1.3. which should be on !B(R) and such that x# = !x#,which is contradictory.

We end this section by some other non-existence properties.

Definition 20.1.2 If X, Y are Hausdor" topological spaces, h & C(X, Y ) is callednullhomotopic if there exists a homotopy H & C(X1 [0, 1], Y ) such that H(·, 0) = hand H(·, 1) is constant.

The following characterization of nullhomotopic mapping on a sphere is of in-terest.

Proposition 20.1.1 Let Y be a Hausdor" topological space. Then h & C(Sn, Y )is nullhomotopic if and only if h has a continuous extension 1h & C(B(1), Y ), whereB(1) ' Rn+1

Proof. If h has a continuous extension 1h to B(1), the homotopy H & C(Sn1[0, 1], Y )

defined by H(x, t) = 1h(tx) shows that h is nullhomotopic. If h is nullhomotopic and


H & C(Sn 1 [0, 1], Y ) is a corresponding homotopy, one can obtain an extension1h & C(B(1), Y ) by

1h(x) =

/H(Sn, 0) if 0 ) #x# ) 1/2,

H(x/#x#, 2#x# ! 1) if 1/2 ) #x# ) 1.

Corollary 20.1.4 Let D ' Rn be an open bounded symmetric neighborhood of 0and g & C(!D, Rn \ {0}) be odd on !D. Then g is not null homotopic.

Proof. Let 1g & C(D, Rn) be a continuous extension of g. Then, by Corollary 20.1.2,dB[1g, D, 0] = 1 (mod 2). If g & C(D, Rn) is nullhomotopic, G is the correspondinghomotopy, 1G & C(D1 [0, 1], Rn) is a continuous extension of G, and g0 the constantvalue of 1G(·, 1) on !D, then, using homotopy invariance property 3.4.2,

1 (mod 2) = dB[1g, D, 0] = dB[ 1G(·, 0), D, 0] = dB[ 1G(·, 1), D, 0]

= dB[g0, D, 0] = 0,

a contradiction.

In particular, a sphere cannot be continuously deformed into a point.

Corollary 20.1.5 There exists no homotopy H & C(Sn 1 [0, 1], Sn) such thatH(·, 0) = I and H(·, 1) is constant on Sn.

Proof. Apply Corollary 20.1.4 to D = B(1) ' Rn+1 and g = I.

20.2 Borsuk-Ulam’s theorem

The following lemma is a consequence of Corollary 20.1.2 with n replaced by n +1.

Lemma 20.2.1 If D ' Rn+1 is an open, bounded symmetric neighborhood of 0,g & C(!D, Rn+1) is odd and g(!D) ' Rn, then there exists some x# & !D suchthat g(x#) = 0.

Proof. If 0 %& g(!D) and 1g & C(D, Rn) is an extension of g, then 1g(!x) = !1g(x) forall x & !D, and Corollary 20.1.2 implies that 1g(D) contains a neighborhood of 0 inRn+1, a contradiction with 1g(D) ' Rn.

The following non-existence result is of interest.

Corollary 20.2.1 If n > m, and there exists no odd mapping g & C(Sn, Sm).

Proof. As g(Sn) ' Sm ' Rm+1 and m + 1 < n + 1, Lemma 20.2.1 implies thatg(x#) = 0 for some x# & Sn, a contradiction with g(Sn) ' Sm.

20.3. ELLIPTIC DIFFERENTIAL OPERATORS 253

The following consequence of Lemma 20.2.1, called Borsuk-Ulam’s theorem,was conjectured by S. Ulam and proved by Borsuk [36].

Theorem 20.2.1 If D ' Rn+1 is an open, bounded symmetric neighborhood of 0,and f & C(!D, Rn+1) is such that f(!D) ' Rn, there exists x# & !D such thatf(x#) = f(!x#).

Proof. Apply Lemma 20.2.1 to the continuous odd mapping g defined on !D byf(x) = g(x)! g(!x).

In particular, for the Earth identified with S2, assuming that temperature andpressure continuously depend upon the position, there always exists two antipodalpoints having the same temperature and pression.

20.3 Elliptic di!erential operators

Let us deduce here an interesting consequence of Borsuk-Ulam’s theorem 20.2.1 forelliptic di!erential operators with constant coe$cients. Let n, m ( 1 be integers andP (!) =

B|&|+m c&!& be a di!erential operator with constant (real or com-

plex) coe"cients c&. Here * = (*1,*2, . . . ,*n) with *k & N (k = 1, 2, . . . , n),|*| =

Bnk=1 *k, and !& = !&1

1 !&22 . . . !&n

n . The symbol of P is the polynomial (alsodenoted by P )

P (/) :="

|&|+m

c&/&,

where /& = /&11 /&2

2 . . . /&nn . We say that P is of order m if there exists * such that

|*| = m and c& %= 0. If P is of order m, the polynomial Pm(/) :=B

|&|=m c&/& iscalled the principal symbol of P. Pm is a homogeneous polynomial of degree m,namely

Pm(t/) = tmPm(/) for all / & Rn and t & R.

The di!erential operator P (!) is said to be elliptic if Pm(/) %= 0 whenever / &Rn \ {0}.

Proposition 20.3.1 Let n, m ( 1 be integers. Then

1. If P (!) is elliptic with real coe!cients and n ( 2, then m is even.

2. If P (!) is elliptic and n ( 3, then m is even.

Proof. 1) Let f : !B(1) ' Rn " Rn!1 be defined by

f(/) = (Pm(/), 0, . . . , 0)


where Pm(/) is followed by n! 2 zeros. Clearly f is continuous and Borsuk-Ulam’stheorem 20.2.1 implies the existence of /# & !B(1) such that f(/#) = f(!/#), andhence such that

Pm(/#) = Pm(!/#) = (!1)mPm(/#).

Because Pm(/#) %= 0, m must be even.2) Let Qm = APm, Rm = BPm, so that, for all / & Rn and t & R we have

Qm(t/) = (1/2)[Pm(t/) + Pm(t/)] = (1/2)[tmPm(/) + tmPm(/)] = tmQm(/),

and similarly for Rm. Let us define the continuous mapping g : !B(1) ' Rn " Rn!1

by

g(/) = (Qm(/), Rm(/), 0, . . . , 0)

with Rm(/) followed by (n ! 3) zeros. By Borsuk-Ulam’s theorem 20.2.1, thereexists /# & !B(1) such that g(/#) = g(!/#), and hence such that

Qm(/#) = Qm(!/#) = (!1)mQm(/#),

Rm(/#) = Rm(!/#) = (!1)mRm(/#).

Consequently, Pm(/#) = (!1)mPm(/#), and, as Pm(/#) %= 0, m must be even.

Remark 20.3.1 For n = 1, the thesis fails. Indeed, taking P (!) = ddx , we have

P (/) = / = Pm(/) %= 0 for all / & R \ {0} and P (!) is elliptic.

Remark 20.3.2 For n = 2, and P not real, the thesis fails. Indeed, taking P (!) =!1 + i!2, we have P (/) = /1 + i/2 = P1(/) %= 0 for all / & R2 \ {0} and P (!) iselliptic.

20.4 Lusternik-Schnirel’mann’s covering theorem

Another interesting consequence of Borsuk-Ulam’s theorem with D = B(1) is Lus-ternik-Schnirelmann’s covering theorem proved in 1930 by L. Lusternik andL. Schnirelmann [256] and in 1933 by Borsuk [36]. We denote by * : Sn " Sn theantipodal mapping defined by *(x) = !x.

Corollary 20.4.1 In any closed covering {M1, . . . , Mn+1} of Sn by n + 1 subsetsof Sn, there is at least one Mi containing a pair of antipodal points.

Proof. If there is some closed covering {M1, . . . , Mn+1} of Sn with no Mi containinga pair of points x and !x, then we have Mi . *(Mi) = , for all i = 1, . . . , n + 1.Hence, Urysohn’s function gi & C(Sn, [0, 1]) defined by

gi(x) =dist(x, Mi)

dist(x, Mi) + dist(x,*(Mi))

20.5. MEASURE OF NON-COMPACTNESS OF THE UNIT SPHERE IN A BANACH SPACE255

is such that gi(x) = 0 if x & Mi and gi(x) = 1 if x & *(Mi) (i = 1, 2, . . . , n + 1).Hence, if we define g : Sn " Rn by g(x) = (g1(x), . . . , gn(x)), it follows from Borsuk-Ulam’s theorem 20.2.1 that there exists x# & Sn such that g(x#) = g(*(x#)), i.e.

g1(x#) = g1(*(x#)), . . . , gn(x#) = gn(*(x#)).

Consequently, we must have x# %& Mi (i = 1, 2, . . . , n) and *(x#) %& Mi (i =1, 2, . . . , n). Hence x# &Mn+1 / *(Mn+1), a contradiction.

Define an antipode preserving mapping f : Sn " Sk by the conditionf(x) = f(!x) for all x & Sn.

Corollary 20.4.2 An antipode preserving mapping f & C(Sn, Sn) is not nullho-motopic.

Proof. If it were the case, using Proposition 20.1.1, f would have a continuousextension 1f : B(1) ' Rn+1 " Sn. Let

Sn+1± =

Wx & Sn+1 : xn+1 ! 0

X,

% : Sn+1 " B(1), (x1, . . . , xn+2) 2" (x1, . . . , xn+1)

and define g : Sn+1 " Sn by

g(x) =

Z1f * %(x) if x & Sn+1

+ ,

! 1f * % * *(x) if x & Sn+1! ,

then g & C(Sn+1, Sn) is an odd mapping, a contradiction to Corollary 20.2.1.

Another way to formulate Corollary 20.4.2, using Proposition 20.1.1, is the fol-lowing one.

Corollary 20.4.3 If f & C(Sn, Sn) is antipode preserving, any continuous exten-

sion 1f : B(1) ' Rn+1 " Rn+1 of f to B vanishes at some x# & B(1).

Proof. If there exists a continuation 1f & C(B(1), Rn+1 \ {0}) of f to B(1) ' Rn+1,

then g = 1f/# 1f# & C(B(1), Sn) is a continuous extension of f, and, by Proposition20.1.1, f is nullhomotopic, a contradiction to Corollary 20.4.2.

20.5 Measure of non-compactness of the unit sphe-

re in a Banach space

A nice application of Lusternik-Schnirel’mann’s covering result 20.4.1, which is dueto M. Furi and A. Vignoli [141] and to R. Nussbaum [299] is a computation of Ku-ratowski’s measure of non-compactness of the unit sphere in an infinite-dimensionalBanach space.

Let (X, d) be a metric space and B be the collection of all bounded subsets ofX . In 1930, K. Kuratowski [229] has introduced as follows the concept of measureof non-compactness * : B " [0, +-[ . For any B & B,


*(B) := inf{' > 0 : B can be covered with a finite number of subsets of diametersmaller of equal to '}.

Let us denote by +(B) the diameter of B.We list here without proofs some fundamental properties of the mapping *

(notice that the first two ones, used in the proof of theorem 20.5.1below, are directconsequences of the definition of *) :

(i) *(B) ) +(B).

(ii) *(B) = 0 if and only if B is relatively compact.

(iii) If B1 ' B2 belong to B, then *(B1) ) *(B2).

(iv) for any B & B, *(B) = *(B).

(v) for any B1, B2 belonging to B, *(B1 /B2) = max{*(B1,*(B2)).

If we now assume that X is a real Banach space, we have the following supplemen-tary properties :

(vi) for any B & B, *(co B) = *(B).

(vii) for any B, C belonging to B, *(B + C) ) *(B) + *(C).

(viii) for any B & B and any " & R, *("B) = |"|*(B).

For the proofs and for further properties, see e.g. [89] or [254].Now, since a Banach space is finite-dimensional if and only if !B(1) is compact,

if follows from Property (ii) that *(!B(1)) = 0 if dim X < +-, while *(!B(1)) > 0and, according to Property (i), *(!B(1)) ) 2 if dim X = -. This result can beprecised as follows.

Theorem 20.5.1 If X be an infinite-dimensional real Banach space, *(!B(1)) =2.

Proof. Assume by contradiction that *(!B(1)) < 2. Then we can write !B(1) =S1 / S2 / . . . / Sn with +(Si) < 2 for all i = 1, 2, . . . , n. Taking Si instead of Si ifnecessary, we may assume that the Si are closed. Let Xn be a n-dimensional vectorsubspace of X. Then !B(1).Xn is covered with {S1 .Xn, S2 .Xn, . . . , Sn .Xn}.Lusternik-Schnirel’mann’s covering property 20.4.1 implies the existence of at leastone of the Si . Xn, say Sk . Xn for some k & {1, 2, . . . , n}, containing a pair ofantipodal points, i.e. x# & Sk . Sn with !x# & Sk . Sn. This implies that

2 = #x# ! (!x#)# ) +(Sk .Xn) ) +(Sk),

a contradiction.

Remark 20.5.1 By the properties (iv) and (v) of the measure of non-compactness,we deduce from Theorem 20.5.1 that X is a real infinite-dimensional Banach spaceif and only if

*(!B(1)) = *(B(1)) = *(B(1)) = 2.

20.6. GENUS 257

20.6 Genus

Let E be a real Banach space, and denote by *(E) the set of all closed subsetsof E which do not contain 0 and are symmetric with respect to 0. For example!B(R) and {!x, x} for x & E \ {0} belong to *(E). The concept of genus was firstintroduced in 1952 by M.A. Krasnosel’skii [211] (see also [213]) as an alternativeof the Lusternik-Schnirelmann category in the minimax approach of critical points.This genus measure the ‘size’ of some subsets of E which are symmetric with respectto the origin.

Definition 20.6.1 The genus .(A) of a non-empty set A & *(E) is the smallestinteger n & N# for which there exists an odd mapping g & C(A, Rn \ {0}). We write.(A) = +- if no such mapping exists, and .(,) = 0.

Example 20.6.1 If A = Bx(r)/B!x(r) with 0 < r < #x#, then .(A) = 1, becausethe mapping g : A" Rn \ {0} defined by

g(x) =

/!1 if x & B!x(r)+1 if x & Bx(r)

is continuous. More generally, any non connected set in *(E) has genus equal to 1.

Proposition 20.6.1 If D ' Rn is a bounded open symmetric neighborhood of 0,then .(!D) = n.

Proof. First, .(!D) ) n, because I & C(!D, Rn \ {0}) is odd. If .(!D) = m < n,then there exists an odd mapping g & C(!D, Rm \ {0}), a contradiction to Lemma20.2.1.

We give now the main properties of the genus. First, taking the image of aset in * by a continuous odd mapping does not decrease its genus.

Proposition 20.6.2 If A, B & *(E) and f & C(A, B) is odd, then .(A) ) .(B).

Proof. The result is trivial if .(B) = +-. If .(B) = n, there exists g & C(B, Rn \{0}) and odd, and hence g * f & C(A, Rn \ {0}) is odd, so that .(A) ) n.

The genus is not decreasing with respect to inclusion.

Proposition 20.6.3 If A, B & *(E) and A ' B, then .(A) ) .(B).

Proof. Take f = I in Proposition 20.6.2.

The genus is invariant under an odd homeomorphism.

Proposition 20.6.4 If h : A " B is an odd homeomorphism of A onto B, then.(A) = .(B).

Proof. A consequence of Proposition 20.6.2 with f = h and f = h!1.


The genus is subadditive with respect to union.

Proposition 20.6.5 If A, B & *(E), then .(A /B) ) .(A) + .(B).

Proof. If .(A) or .(B) is equal to +-, the result is trivial. If .(A) = n and.(B) = m, there exists odd mappings g & C(A, Rn \ {0}) and h & C(A, Rm \ {0}),and, by Tietze theorem, those mappings have continuous extensions 1g & C(E, Rn)

and 1h & C(E, Rm), which can be assumed to be odd by taking if necessary their odd

part. Then, the mapping f =L1g|A&B,1h|A&B

Mis odd and f & C(A/B, Rn+m\{0}),

because one of the components is always non zero on A/B. Hence .(A/B) ) n+m.

The genus of the di!erence of two sets is larger than the di!erence of their genus.

Corollary 20.6.1 If A, B & *(E) and .(B) < +-, then .(A \ B) ( .(A)!.(B).

Proof. The relation A ' (A \ B /B) and Propositions 20.6.3 and 20.6.5 imply that.(A) ) .(A \ B) + .(B).

The genus of a compact set is always finite.

Proposition 20.6.6 If A & *(E) is compact, then .(A) < +-.

Proof. The family of open sets

{Bx(#x# < 2) /B!x(#x#/2) : x & A}

is an open covering of A, and hence contains a finite subcovering

{Bxj(#xj#/2) /B!xj (#xj#/2) : j = 1, 2, . . . , m}.

Using Proposition 20.6.5 and Example 20.6.1, we get

.(A) )m"

j=1

.[Bxj (#xj#/2) /B!xj (#xj#/2)] = m.

Finally, the genus of some neighborhood of a compact A ' * is the same as thatof A.

Proposition 20.6.7 If A & *(E) is compact, there exists + > 0 such that ifN1(A) = {x & E : dist(x, A) ) +}, one has .(N1(A)) = .(A).

Proof. The result is trivial by Corollary 20.6.3 if .(A) = +-. If .(A) = n, thereexists an odd mapping g(A, Rn \ {0}). Let 1g & C(E, Rn) be an odd extension of g.A being compact and 1g non zero on A and uniformly continuous on A, there exists+ > 0 such that 1g(x) %= 0 for all x & N1(A). Hence 1g & C(N1(A), Rn \ {0}) and isodd, so that .(N1(A)) ) n. On the other hand, it follows from Proposition 20.6.3that n = .(A) ) .(N1(A)).

20.6. GENUS 259

The concept of genus allows a strenghtening of Lemma 20.2.1.

Theorem 20.6.1 Let n > m, D ' Rn+1 be an open, bounded and symmetricneighborhood of 0, g & C(!D, Rm+1) be an odd mapping, and A = {x & !D :g(x) = 0}. Then .(A) ( n!m.

Proof. If follows from Proposition 20.6.7 that there exists + > 0 such that .(A) =.(N1(A)). Furthermore, there exists '0 such that, for each ' & ]0, '0], one has

Z* := {x & !D : #g(x)# ) '} ' N1(A).

Indeed, if it is not the case, there exists a sequence (xn) in !D such that #g(xn)# )1/n and xn %& N1(A). Going if necessary to a subsequence, we can assume that(xn) converges to some x# %& int(N1(A)), and hence x# %& A and g(x#) = 0, acontradiction. Consequently, for any ' & ]0, '0], one has A ' Z* ' N1(A), andhence, from Propositions 20.6.3 and 20.6.7, .(A) ) .(Z*) ) .(N1(A)) = .(A), andhence

.(Z*) = .(A) (' & ]0, '0].

For any 7 > 0, let Y2 := {x & !D : #g(x)# ( 7}, and let p : Rm+1 " Sn be theprojection defined by p(x) = x/#x#. Then p*g & C(Y2, Sm) is odd, and hence, fromProposition 20.6.2 .(Y2) ) .(Sm) = m+1. Therefore, using Proposition 20.6.1 andCorollary 20.6.1, we deduce that

.(!D \ Y2) ( .(!D)! .(Y2) ( n + 1! (m + 1) = n!m.

But Z2 = !D \ Y2, and hence, for ' & ]0, '0], we get

.(A) = .(Z*) = .(!D \ Y*) ( n!m.

Theorem 20.6.1 implies a strenghtening of Borsuk-Ulam’s theorem 20.2.1.

Corollary 20.6.2 If D ' Rn+1 is an open, bounded symmetric neighborhood of 0,and f & C(!D, Rm+1) with n > m, then .({x & !D : f(x) = f(!x)}) ( n!m.

Proof. Take g(x) = f(x)! f(!x) in Theorem 20.6.1.

Finally, the following lemma will allow to prove a characterization of sets ofgenus n through sets of genus one.

Lemma 20.6.1 There exists n+1 closed antipodal sets Bi = Ci/(!Ci) ' Sn suchthat Ci . (!Ci) = , (1 ) i ) n + 1) and Sn =

Hn+1i=1 Bi.

Proof. . We proceed by recurrence over n. The case where n = 0 is trivial, becauseS0 = {1} / {!1}. If n = 1, one has S1 = B1 /B2 with

B1 = {(x, y) & R2 : x > 1/2}/ {(x, y) & R

2 : x < !1/2} = C1 / (!C1)

B2 = {(x, y) & R2 : y > 1/2} / {(x, y) & R

2 : y < !1/2} = C2 / !C2.


Let us assume that the result is true till order n ! 1, so Sn!1 =Hn

i=1 B$i, where

the B$i are n closed antipodal sets B$

i = C$i / (!C$

i) and C$i . (!C$

i) = ,. Denote by(x$, xn+1) the point (x1, . . . , xn, xn+1) of Rn+1 and identify Rn with the hyperplane{x & Rn+1 : xn+1 = 0}. Let Cn+1 := {(x$, xn+1) & Sn : xn+1 ( 1/4}, and denoteby Ci the sets

Ci =

*+

,(x$, xn+1) & Sn : xn+1 ) 1/2,x$

Y1! x2

n+1

& C$i

<=

> (i = 1, 2, . . . , n).

Each set Ci ' Sn is closed and Ci.(!Ci) = ,. Set Bi = Ci/(!Ci) (i = 1, 2, . . . , n+1). The family {Bi : i = 1, 2, . . . , n+1} is a covering of Sn through closed antipodalsets.

Lemma 20.6.2 If A & *(E) and .(A) = j for some integer j ( 1, there existsD1, . . . , Dj & *(E) with .(Di) = 1 (i = 1, 2, . . . , j) such that A '

Hji=1 Di.

Proof. As .(A) = j, there exists an odd mapping g & C(A, Rj \ {0}). From Lemma20.6.1, Sj!1 =

Hji=1 Bi, where the Bi are closed antipodal sets Bi = Ci/(!Ci) and

Ci . (!Ci) = , (i = 1, 2, . . . , j). Let p : Rj \ {0} " Sj!1 be the radial projectiondefined by p(x) = x/#x#, and let

Di := g!1 * p!1(Bi) = g!1 * p!1(Ci) / g!1 * p!1(!Ci) (i = 1, 2, . . . , j).

Each Di & *(E) and A 'Hj

i=1 Di. Each Di is made of two closed disjoint sets, andhence .(Di) ) 1 (i = 1, 2, . . . , j). But, from A '

HDi and the subadditivy property

20.6.5, we get .(A) = j )Bj

i=1 .(Di), and hence .(Di) = 1 (i = 1, 2, . . . , j).

Theorem 20.6.2 If A & *(E), then .(A) = n if and only if n is the smallestinteger such that one can find n sets A1, . . . , An & *(E) such that .(Aj) = 1(j = 1, 2, . . . , n) and A '

Hnj=1 Aj .

Proof. Su!ciency. If A 'Hn

j=1 Aj , then, by the subadditivy property 20.6.5, onegets .(A) ) n. If we had .(A) = j < n, then Lemma 20.6.2 would contradict theminimal character of n.

Necessity. If .(A) = n, Lemma 20.6.2 implies the existence of A1, . . . , An &*(E) such that .(Ai) = 1 (i = 1, 2, . . . , n) and A '

Hni=1 Ai. Now, we cannot have

A 'Hm

i=1 Bi with Bi & *(E) and .(Bi) = 1 for all i = 1, 2, . . . , m and some m < n,because then, by the subadditivity property 20.6.5, we would have .(A) ) m < n.

20.7 Isometric representations of S1 over Rn

Let G be a topological group. Recall that a representation of G over Rn is afamily {T (g)}g"G of linear operators T (g) : Rn " Rn such that

T (0) = I, T (g1 + g2) = T (g1)T (g2), (g, u)" T (g)u is continuous.

20.8. S1-EQUIVARIANT MAPPINGS 261

A subset E ' Rn is invariant (under the representation) if T (g)E = E for allg & G. A representation {T (g)}g"G is isometric if #T (g)x# = #x# for all g & Gand all x & Rn.

For example, an isometric representation of G = Z2 is given by T (e) = I,T (!e) = !I, where e is the unit element in Z2 and I the identity matrix in Rn.The invariant sets in Rn are the sets which are symmetric with respect to the origin.

We are be interested in the group S1 of rotations on the unit circle. Here weconsider S1 as the complex numbers of modulus one / = ei,. The following resultis classical (see e.g. [278]).

Lemma 20.7.1 Let {T ())},"S1 be an isometric representation of S1 over Rn. ThenT ()) has the matrix representation

diag{M1, . . . , Mk} (20.5)

where Mj is either of order 1 and Mj = 1, or is of order 2 and, for some n & N\{0}such that

2cos n) ! sin n)sin n) cos n)

3. (20.6)

DefineFix T := {u & R

n : T ())u = u for all ) & S1}.

20.8 S1-equivariant mappings

Let {T ())},"S1 be the isometric representation of S1 over Rn given by

T ())x = T ())(y, z1, . . . , zk) = T ())(y, z) =Ny, eim1,z1, . . . , e

imk,zk

O, (20.7)

where we identify R2 with C, m1, . . . , mk & N \ {0} and where y & Rm (m ( 0).Thus, S1 acts on Rn via

ei, 0 (y, z) = (y, eim1,z1, ·, eimk,zk).

Notice that x & Fix T if and only if x = (y, 0), so that dim Fix T = m. Let{U())},"S1 be the isometric representation of S1 over Rn given by

U())x =Ny, ein1,z1, . . . , e

ink,zk

O, (20.8)

where n1, . . . , nk & N \ {0} and where y & Rm. Let D ' Rn be an open boundedinvariant set, let f & C(D, Rn) . C1(D, Rn) be such that

0 %& f(!D), (20.9)

and, for all ) & S1 and x & !D,

[f * T ())](x) = [U()) * f ](x). (20.10)


To help in the notations, let us write, for r & N \ {0}, VrC= C for the representation

of S1 given by T ())w = eir,w, so that we can write

f : D ' Rm 1 Vm1 1 . . .1 Vmk " R

m 1 Vn1 1 . . .1 Vnk . (20.11)

We need a few preliminary lemmas.

Lemma 20.8.1 Let mj & N \ {0} (1 ) j ) k) and let p : Rm 1 Ck " Rm 1 Ck bedefined by

p(y, z1, · · · , zk) = (y, zm11 , · · · , zmk

k ) .

Then

dB [p, B(r), 0] = m1 · · ·mk. (20.12)

Proof. Using excision, the cartesian product property 19.1.1 and formula (17.25),we get, with B(R) ' Rm, Bj(r) ' C, (j = 1, . . . , k),

dB [p, B(r), 0] = dB[p, B(R)1B1(r) 1 . . .1Bk(r), 0]

= dB[I, B(R), 0]k4

j=1

dB[zmj , Bj(r), 0] =k4

j=1

mj.

The following result is a special case of the parametrized Sard theorem.

Lemma 20.8.2 Let % ' Rp, M ' Rs be open, and let G & C1(% 1M, Rp). If0 & Rp is a regular value of G (i.e. G$

(x,v) is onto for each (z, v) & G!1(0)), then,

for almost every v &M, 0 is a regular value of G(·, v).

Given n & N \ {0}, and g & C(D, Rn) . C1(D, Rn), let us define the mappingG : D 1 Ck'k " Rm 1 Ck by

G(y, z, v) := g(x)!k"

j=1

znj vj . (20.13)

Lemma 20.8.3 Under the assumptions above for g, 0 & Rm1Ck is a regular valueof the restriction of G to D0 := {(y, z) & D : z %= 0} for almost every v & Ck'k.

Proof. Set M = Ck'k, and apply Lemma 20.8.2 to G with p = n, and s = k2. Itsu$ces to check that F $

(x,v) is onto for each (x, v) & D01M, i.e. to be able to solve

the equation in (a, b) & Rn 1 Ck'k

F $x(x, v)a + F $

v(x, v)b = c

for each c & Rn. Taking a = 0, it remains to solve the equation

zn1 b1 + · · · + zn

k bn = !c,

which is always possible as z %= 0.


Let fT : D.Fix T " Rn denote the restriction of f to the fixed point set Fix Tof T. By the equivariance property, we have, if we write f = (f0, f1, · · · , fk), withf0 & Rm, fj & C (1 ) j ) k),

(f0(y, 0), f1(y, 0), · · · , fk(y, 0)) = (f0(y, 0), ein1,f1(y, 0), · · · , eink%fk(y, 0))

for all ) & R, so that, taking successively ) = %/nj , we get fj(y, 0) = 0 (1 ) j ) k),and hence fV (y) = f(y, 0) = (f0(y, 0), 0, · · · , 0), i.e.

fV : D . Fix T " Fix T. (20.14)

Theorem 20.8.1 Let D ' Rm 1 Vm1 1 · · · 1 Vmk be an open, bounded invariantsubset and f : D " Rm 1 Vn1 1 · · ·1 Vnk be continuous, equivariant and such that0 %& f(!D). Then

dB [f, D, 0] · m1 · · ·mk = dB [fT , D . Fix T, 0] · n1 · · ·nk. (20.15)

Proof. We reduce the proof to a simpler situation. For

n = n1 · · ·nk,

define the representations of S1 in Rn

V ())(y, z) = (y, ei,z1, · · · , ei,zk),

W ())(y, z) = (y, ein,z1, · · · , ein,zk).

Consider the maps

& : Rm 1 V k

1 " Rm 1 Vm1 1 · · ·1 Vmk , (y, z1, · · · , zr) 2" (y, zm1

1 , · · · , zmkr )

3 : Rm 1 Vn1 1 · · ·1 Vnk " R

m 1 V kn , (y, z1, · · · , zk) 2" (y, zn/n1

1 , · · · , zn/nk

k ).

We have, for all ) & S1,

[& * V ())](x) = (y, eim1,z1, · · · , eimk,zk) = [T ()) * &](x),

[3 * U())](x) = (y, (ein1,z1)n/n1 , · · · , (eink,zk)n/nk)

= (y, ein,zn/n1

1 , · · · , ein,zn/nk

k ) = [W ()) * 3](x),

so that & and 3 are equivariant. Hence,

1f := 3 * f * & : 1D " Rm 1 V k

n

is equivariant, where1D = &!1(D) ' R

m 1 V k1 .

Now, &!1(0) = {0}, 3!1(0) = {0}, so that the product formula (21.1) and formula(20.12) imply that

dB[ 1f, 1D, 0] = iB[3, 0] · dB[f, D, 0] · iB[&, 0] =nk

n· dB [f, D, 0] · m1 · · ·mk

= nk!1 · dB [f, D, 0] · m1 · · ·mk. (20.16)


Furthermore, as &|Fix V = I, 3|Fix U = I, we have

dB [ 1fV , 1D . Fix V, 0] = dB [fT , D . Fix T, 0]. (20.17)

Choose a neighbourhood U of 1f!1(0).Rm1{0} and ' > 0 such that U 1 B(') ' 1D.

Define Tf by

Tf(y, z) =

Z( 1fV (y), zn

1 , · · · , znk ) if (y, z) & U 1B(')

1f(y, z) if (y, z) & [ 1D . Fix V ] / ! 1D.

Here 1fV : 1D . Fix V " Fix V denotes the restriction as usual. Observe thatfor (y, z) & [ 1D . Fix V ] . U 1B(') we have z = 0 and by equivariance 1f(y, 0) =

( 1fV (y), 0). Thus Tf is well defined. Then extend Tf continuously over all of 1D,keeping the same notation. We have

dB[ 1f, 1D, 0] = dB[ Tf, 1D, 0]. (20.18)

Letµ := min

!D# 1f# = min

!D# Tf#,

and let fµ & C1( 1D, Rn) be such that

maxeD# Tf ! fµ# < µ.

Set

f(y, z) =

!

S1

/ 0 fµ(/!1 0 (u, z))d/.

f is equivariant, and, if (y, z) & [ 1D . Fix V ] / ! 1D, we have

#f(y, z)! Tf(y, z)# = #!

S1

/ 0 [fµ(y, z)! Tf(y, z)] d/#

= #!

S1

/ 0 [fµ(y, z)! 1f(y, z)] d/# < µ.

Hence

dB[f, 1D, 0] = dB[ Tf, 1D, 0]. (20.19)

and

dB[fV

, 1D . Fix V, 0] = dB[ 1fV , 1D . Fix V, 0]. (20.20)

Now, for v = (v1, · · · , vk) with vi & Ck ' Rm 1 Ck we define

F (y, z, v) = f(y, z)!k"

j=1

znj vj .


One has

F (V ())x, v) = [f * V ())](x) !k"

j=1

(ei,zj)nvj

= [W ()) * f ](x)!n"

j=1

ein,znj vj = [W ()) * F ](x, v),

so that that F (·, v) : 1D " Rm1V rn is equivariant and, by Lemma 20.8.3, F has 0 as

a regular value. Now the parametrized Sard theorem (Lemma 20.8.2) implies thatthere exists v & Ck'k arbitrarily close to 0 such that 0 is a regular value of F (·, v). Ifv is small enough the degrees of F (·, v) and f are the same. Also, restricted to Rm,F (·, v) and f coincide. But if 0 is a regular value of F (·, v), then F (·, v)!1(0) ' Rm.This is obvious since any (y, z) & F (·, v)!1(0) with z %= 0 yields a one-dimensionalmanifold {/ 0 (y, z) : / & S1} of zeroes contradicting the regularity condition. So

F (·, v)!1(0) = f!1

(0) . Rm. Now, in the neighbourhood U 1 B(') of F (·, v)!1(0)we have by construction

F (y, z, v) = [fV

(y), (zn1 , · · · , zn

k )!k"

j=1

znj vj ].

Applying excision, homotopy invariance and the product formula once more weobtain

dB[f, 1D, 0] = dB[F (·, v), 1D, 0]

= dB[fV

, 1D . Fix V, 0] · iB[(zn1 , · · · , zn

k )!k"

j=1

znj vj , 0]

= dB[fV

, 1D . Fix V, 0] · nk.

Consequently, using (20.16), (20.19) and (20.20),

dB[fT , D . Fix T, 0] · nk = nk!1 · dB[f, D, 0] · m1 · · ·mk,

which immediately gives formula (20.15).

Chapter 21

Leray’s product formula

21.1 The product formula

Let D ' Rn be open and bounded, and f : D " Rn continous. Since f(!D)is compact, Rn \ f(!D) has one unbounded connected component K% if n ( 2,and two unbounded connected components if n = 1, whose union will still denotedby K%. As f(D) is bounded as well, K% contains points y %& f(D), and hencedB[f, D, K%] = 0. Let now g : Rn " Rn be continuous and let y %& g(f(!D)).Hence dB[g * f, D, y] is well defined, and the aim is to express it in terms of somedegrees of f and of g. The following result is due to Leray [250].

Theorem 21.1.1 Let y %& g(f(!D)) and let Ki be the bounded connected compo-nents of Rn \ f(!D). Then

dB[g * f, D, y] ="

i

dB [f, D, Ki] dB[g, Ki, y], (21.1)

where only finitely many terms in the right-hand side are di"erent from zero.

Proof. Let us first show that finitely many terms only in the sum of (21.1) are nonzero. Let r > 0 be such that f(D) ' B(r). Since M := B(R) . g!1(y) is compactand contained in Rn \ f(!D) =

Hi Ki, the family of mutually disjoint open sets

{Ki (i %= -); K% . B(0, r + 1)} covers M, and hence finitely many of them, say(after renumbering) K1, . . . , Kp, Kp+1 = K% .B(0, r + 1) cover M. We only retainKj if Kj . M %= ,. Now, dB[f, D, Kp+1] = 0 because Kp+1 contains points notin f(D), and dB[g, Kj, y] = 0 for j ( p + 2 because, for these j, Kj ' B(r), butg!1(y) . Kj = ,. Thus the summation in (21.1) is finite. The remaining of theproof will be divided into the parts, namely f and g smooth, and then f and gcontinuous.

267

268 CHAPTER 21. LERAY’S PRODUCT FORMULA

21.2 Proof in the smooth case

Let f & C(D) . C1(D), g & C1(Rn), and assume first that y is a regular value ofg * f. We have, using Corollary 5.2.1,

dB [g * f, D, y] ="

x"(gf)!1(y)

sign Jg*f (x) ="

x"(gf)!1(y)

sign Jg(f(x)) sign Jf (x)

="

x"f!1(z); z"g!1(y)

sign Jg(z) sign Jf (x)

="

z"g!1(y)0f(D)

sign Jg(z)

&

'"

x"f!1(z)

sign Jf (x)

(

)

="

z"f(D)0g!1(y)

sign Jg(z) dB [f, D, z].

Now, in the last sum, we may replace z & f(D) by z & B(R) \ f(!D), sincedB[f, D, z] = 0 whenever f %& f(D). Consequently, as the Ki are mutually disjoint,we obtain

dB[g * f, D, y] =p"

i=1

"

z"Ki0g!1(y)

sign Jg(z)dB[f, D, z]

=p"

i=1

dB [f, D, Ki]

&

'"

z"Ki0g!1(y)

sign Jg(z)

(

)

="

i

dB [f, D, Ki] dB[g, Ki, y].

21.3 Proof in the continuous case

To deduce the continuous case from the smooth one, it is convenient to rearrangethe right-hand member of (21.1) as follows. Let, for each integer m,

Sm := {z & B(0, r + 1) \ f(!D) : dB[f, D, z] = m},Nm := {i & N : dB [f, D, Ki] = m}.

Clearly, Sm =H

i"NmKi, and hence, by the additivity of degree,

"

i

dB[f, D, Ki] dB[g, Ki, y] ="

m

m

:"

i"Nm

dB[g, Ki, y]

;="

m

m · dB [g, Sm, y].

21.3. PROOF IN THE CONTINUOUS CASE 269

It remains to show that

dB[g * f, D, y] ="

m

m · dB[g, Sm, y]. (21.2)

Since !Sm ' f(!D), and #g(f(x)) ! y# ( µ > 0 for all x & !D, we can findg0 & C1(Rn) such that

dB[g0 * f, D, y] = dB[g * f, D, y], dB [g0, Sm, y] = dB[g, Sm, y] (m & N). (21.3)

Furthermore, we may assume that M0 := B[0, r + 1] . g!10 (y) %= ,, because, if it

were empty, (21.3) implies that (21.2) is reduced to 0 = 0. As M0 is compact andy %& g0(f(!D)), we have dist[M0, f(!D)] > 0. Let us now choose f0 & C(D).C1(D)such that

maxD#f ! f0# < dist[M0, f(!D)], f0(D) ' B(0, r + 1),

and let us define

1Sm := {z & B(0, r + 1) \ f0(!D) : dB[f0, D, z] = m}.

We have Sm .M0 = 1Sm .M0, since z &M0 implies that

dist[z, f(!D)] ( dist[M0, f(!D)] > max #f ! f0#,

and hence dB[f0, D, z] = dB[f, D, z] by Rouche’s theorem. Consequently, both setsSm .M0 and 1Sm .M0 are contained in Sm . 1Sm, and hence, by excision,

dB[g0, Sm, y] = dB[g0, Sm . 1Sm, y] = dB [g0, 1Sm, y]. (21.4)

Using the result of the previous section, (21.3) and (21.4), we obtain

dB[g0 * f0, D, y] ="

m

m · dB [g0, 1Sm, y] ="

m

m · dB[g, Sm, y],

so that it remains to show that dB[g0 * f0, D, y] = dB[g0 * f, D, y]. Introducing thehomotopy

h(x,") = g0 * [f + "(f0 ! f)], " & [0, 1],

we observe that, if y & h(!D 1 [0, 1]), then f(x) + "[f0(x) ! f(x)] & M0 for some(x,") & !D 1 [0, 1], but we have, for all z &M0,

#z ! f(x)! "[f0(x) ! f(x)]# ( dist[M0, f(!D)]!max!#f ! f0# > 0.


21.4 Jordan-Brouwer’s separation theorem

C. Jordan’s famous theorem [187]says that a simple closed curve & in the planedivides it into two regions G1 and G2 such that & = !G1 = !G2 and G2 : R2 \ G1.Since & is homeomorphic to !B(1), and B(1) and R2 \ B(1) are the componentsof R2 \ !B(1), Jordan’s separation theorem may also be formulated as follows : if& ' R2 is homeomorphic to !B(1), then R2 \ & has precisely two components, abounded one and an unbounded one.

The extension to Rn of Jordan’s result is due to Brouwer [40] and called theJordan-Brouwer theorem. The following proof is due to Leray [250].

Theorem 21.4.1 Let n ( 2 and C1 ' Rn, C2 ' Rn be compact sets homeomor-phic to each other. Then Rn \ C1 and Rn \ C2 has the same number of connectedcomponents, and both numbers are either finite or countably infinite.

Proof. Let h : C1 " C2 be a homeomorphism. According to Dugundgi’s theorem,

h has a continuous extension 1h over Rn, and h!1 has a continuous extension ]h!1

over Rn. Let Kj denote the bounded components of Rn \ C1 and Li the boundedcomponents of Rn \ C2. They are open in Rn and such that

!Kj ' C1,

1h(!Kj) = h(!Kj) ' h(C1) = C2 (21.5)

Rn \ C2 ' R

n \ 1h(!Kj) = Rn \ h(!Kj),

and similar relations for the Li. Let j be fixed and let us denote by Gq the compo-nents of Rn \ h(!Kj). Clearly,

Q

i

Li = Rn \ C2 ' R

n \ h(!Kj) =Q

q

Gq. (21.6)

Now (21.6) implies that, for any i there is some q such that Li ' Gq. Indeed, letx & Li; it follows from (21.6) that x & Gq for some q. Now, Li is a connectedcomponent of Rn \ C2 ' Rn \ h(!Kj), so that Li is connected in Rn \ h(!Kj). Onthe other hand, Gq is a connected component of Rn \ h(!Kj), and x & Li .Gq. Wededuce that Li.Gq is connected in Rn \h(!Kj), and the maximality of Gq impliesthat Li ' Gq. In particular, the unique unbounded component L% of Rn \ C2 isincluded in the unique unbounded component G% of Rn \ h(!Kj). Let z & Kj

be arbitrarily chosen. Since Kj is open in Rn, ]h!11h|!Kj = I!Kj and z & Kj , the

Brouwer degree dB[]h!1, Kj , z] is well defined and, using Corollary 3.4.3, we get

dB []h!1, Kj, z] = dB[I!Kj , Kj, z] = dB [I, Kj, z] = 1.

On the other hand, using Theorem 21.1.1, one has

1 = dB []h!11h, Kj, z] ="

q

dB[1h, Kj, Gq] · dB[]h!1, Gq, z]. (21.7)

21.4. JORDAN-BROUWER’S SEPARATION THEOREM 271

Let us fix q arbitrarily and estimate dB[]h!1, Gq, z]. Let x & Gq be a possible solution

of equation ]h!1(x) = z. There exists some i such that x & Li and then Li ' Gq.Such an x cannot be in C2, because, if it would be the case, then

Kj 9 z = ]h!1(x) = h!1(x) & C1,

a contradiction. Kj being a connected component in Rn \ C1, we conclude thatx & Rn \ C2 =

Hi Li, so that, for some i, x & Li ' Gq. Let Nq := {i : Li ' Gq}.

Since the sets Li are open and pairwise disjoint, Theorem 3.4.1 (excision property)and Theorem 3.4.3 (additivity property) give

dB[]h!1, Gq,z ] ="

i"Nq

dB []h!1, Li, z],

which, together with (21.7), give

1 ="

q

dB [1h, Kj, Gq] · dB[]h!1, Li, z]. (21.8)

Now, for any i & Nq, Li is connected and included in Gq. Consequently,

dB[1h, Kj , Gq] = dB [1h, Kj, Li],

and (21.8) can be rewritten as

1 ="

q

"

i"Nq

dB[1h, Kj , Li] · dB[]h!1, Li, z]

="

q

"

i"Nq ;Li,Cq

dB[1h, Kj , Li] · dB[]h!1, Li, Kj ], (21.9)

since z & Kj = Rn \ C1 = Rn \ h!1(C2) ' Rn \ h!1(!Li). Now, since the Li arepairwise disjoint, the same is true for the Gq, and since, to any Li there exists a Gq

such that Li ' Gq, (21.9) can be rewritten as

1 ="

i

dB [1h, Kj , Li] · dB[]h!1, Li, Kj ] (21.10)

for every j. Repeating the above reasoning with fixed Li instead of Kj, we obtain

1 ="

j

dB [1h, Kj , Li] · dB[]h!1, Li, Kj ] (21.11)

for every i. Assume now that there are m components Li. Then

m =m"

i=1

1 =m"

i=1

&

'"

j

dB[1h, Kj , Li] · dB[]h!1, Li, Kj]

(

)

="

j

:m"

i=1

dB[1h, Kj , Li] · dB[]h!1, Li, Kj ]

;="

j

1,


which shows that there are also m components Kj. The same reasoning can bamade assuming that there are a finite number of components Kj. We conclude thatRn \ C1 and Rn \ C2 either have the same finite number of connected components,or both have countably infinitely many connected components.

Chapter 22

Invariance of domain andapplications

22.1 Invariance of domain

Let D ' Rn be open and bounded, f : D " Rn continuous, and z & Rn \ f(!D).Corollary 3.4.7 has interesting consequences for one-to-one continuous mappings.We first prove a preliminary result.

Lemma 22.1.1 Let % ' Rn be an open neighborhood of 0 and f : % " Rn becontinuous and such that f(0) = 0. If f is one-to-one on some closed ball B(R) ' %,then f(%) is a neighborhood of 0 = f(0).

Proof. The injectivity of f on B(R) implies that dB[f, B(R), 0] is well defined be-cause f(x) %= 0 = f(0) for each x & !B(R). In order to apply Lemma 3.4.7, let usshow that dB [f, B(R), 0] %= 0. Define the homotopy H : B(R)1 [0, 1]" Rn by

H(x,") = f

21

1 + "x

3! f

2!"

1 + "x

3(x & B(R), " & [0, 1]).

Notice that H(x, 0) = f(x) and H(x, 1) = fN

x2

O!f

N!x

2

Ois odd. Now, H(x,") %= 0

for any (x,") & !B(R)1 [0, 1], because, if H(x,") = 0 for such a (x,"), then

f

21

1 + "x

3= f

2!"

1 = "x

3,

and, f being one-to-one on B(R), this gives

1

1 + "x =

!"

1 + "x

and hence x = 0, a contradiction. Using the homotopy invariance property andBorsuk’s theorem 20.1.1, we get

dB[f, B(R), 0] = dB[H(·, 0), B(R), 0] = dB [H(·, 1), B(R), 0] = 1 (mod 2).

273

274 CHAPTER 22. INVARIANCE OF DOMAIN AND APPLICATIONS

Thus, from Lemma 3.4.7, f(%) 0 f(B(R)) is a neighborhood of 0.

We can now prove the important invariance of domain theorem.

Theorem 22.1.1 If % ' Rn is open, and f : % " Rn is continuous and one-to-one, then f(%) is open.

Proof. We show that f(%) is a neighborhood of each of its points. If y0 & f(%) and

x0 & % is such that f(x0) = y0, let us define 1% = %! x0 and 1f : 1%" Rn by

1f(x) = f(x)! f(x0).

The assumptions of Lemma 22.1.1 hold for 1f on 1%, and f(1%) is a neighborhood of0. Hence,

f(%) = f(1% + x0) = 1f(1%) + f(x0) = 1f(1%) + y0

is a neighborhood of y0.

The invariance of domain theorem implies the invariance of dimension.

Corollary 22.1.1 If m < n, there is no continuous and locally one-to-one mappingf : Rn " Rm.

Proof. Suppose such a map f exists, and define g : Rn " Rn by

g(x) = (f(x), 0Rn!m) .

Theorem 22.1.1 implies that g(Rn) ' Rm1 {0Rn!m} is open in Rn, a contradiction.

This last result can be completed by some information about the range of aone-to-one continuous mapping f : Rn " Rm when m < n.

Proposition 22.1.1 Let n < m be positive integers and f & C(Rn, Rm) be one-to-one. Then Rm \ f(Rn) is dense in Rm.

Proof. It is su$cient to prove that, for any positive k & N, Rm \ f(Ink ) is dense

in Rm, where Ik = [!k, k]. Indeed, assuming that Rm \ f(Ink ) is dense in Rm, it

follows from Rn =H%

k=1 Ink that Rm \ f(Rn) =

R%k=1[R

m \ f(Ink )]. As each f(In

k ) iscompact, Rm \ f(In

k ) is open and, according to our assumption, dense in Rm. FromBaire’s theorem,

R%k=1(R

m \ f(Ink )) is dense in Rm.

Let k & N be positive and assume that Rm\f(Ink ) is not dense in Rm. Then there

exists x0 & Rm and r0 > 0 such that Bx0(r0) ' f(Ink ). Setting K := f!1(Bx0(r0))

and g = f |K , we see that g : K " Bx0(r0) is a homeomorphism, as well of course asg!1 : Bx0(r0)" K. Now, h : B(1) ' Rm " Bx0(r0) defined by h(u) = r0u + x0 isa homeomorphism, and the same is true for g!1 * h : B(1) ' Rm " K ' Rn. Thiscontradicts Borsuk-Ulam’s theorem 20.2.1 according to there exists x# & !B(1) 'Rm such that g!1 * h(x#) = g!1 * h(!x#).

22.2. A SURJECTIVITY RESULT FOR ONE-TO-ONE COERCIVE MAPPINGS275

22.2 A surjectivity result for one-to-one coercive

mappings

Another consequence of the invariance of domain theorem is a surjectivity resultfor some one-to-one continuous coercive mappings.

Proposition 22.2.1 Let f : Rn " Rn be continuous, locally one-to-one and coer-cive, i.e. such that

#f(x)# " +- as #x# " +-. (22.1)

Then f is onto.

Proof. From Theorem 22.1.1, f(Rn) is open. We now show that f(Rn) is closed. Let(yn) be a sequence in f(Rn) converging to y0. If xn is such that f(xn) = yn (n & N),then (22.1) implies that (xn) is bounded. Going if necessary to a subsequence, wecan assume that xn " x0, which implies that f(xn) " f(x0), so that f(x0) = y0.Since Rn is connected and f(Rn) is both open and closed, f(Rn) = Rn.

We can compare Corollary 22.2.1 with Corollary 8.4.1 : no injectivity conditionis required in Corollary 8.4.1, but the strong coercivity condition (8.6) is strongerthan the coercivity condition (22.1) in Corollary 22.2.1.

22.3 Degree of one-to-one mappings

We now prove that the Brouwer degree of a one-to-one continuous map on an openbounded open set with respect to any point of its image has absolute value one.

Theorem 22.3.1 Let D ' Rn be an open bounded set and f : D " Rn be contin-uous and one-to-one. Then, for any z & f(D), one has dB [f, D, z] = ±1.

Proof. First, the one-to-one character of f and condition z & f(D) imply z %& f(!D),so that dB[f, D, z] is well defined. By the invariance of domain theorem 22.1.1,f(D) is open and f!1 : f(D) " D is a homeomorphism. Let r > 0 be such

that Bz(r) ' f(D). We denote by 1f the continuous extension of f to Rn. Since1f * f!1 = I on !Bz(r), we have

dB[ 1f * f!1, Bz(r), z] = dB[I, Bz(r), z] = 1. (22.2)

Since !Bz(r) and f!1(!Bz(r)) are homeomorphic, it follows from Jordan-Brouwertheorem 21.4.1 that Rn \ f!1(!Bz(r)) has two open connected components U1 andU2, with, say, U1 bounded and U2 unbounded. By Leray’s product theorem 21.1.1,it follows from (22.2) that

1 =2"

i=1

dB [f!1, Bz(r), Ui] · dB [ 1f, Ui, z]. (22.3)


As f!1(Bz(r)) is bounded, U2 contains points p %& f!1(Bz(r)), and hence

dB [f!1, Bz(r), U2] = dB[f!1, Bz(r), p] = 0.

Consequently, (22.3) can be rewritten as

1 = dB [f!1, Bz(r), U1] · dB[ 1f, U1, z], (22.4)

which implies in particular that dB [f!1, Bz(r), U1] %= 0. Consequently,

U1 ' f!1(Bz(r)) ' D \ f!1(!Bz(r)) ' Rn \ f!1(!Bz(r)). (22.5)

Since Bz(r) is connected and f!1 continuous, it follows from (22.5) that U1 =f!1(Bz(r)) ' D. Consequently, (22.4) becomes

1 = dB[f!1, B, f!1(Bz(r))] · dB[f, f!1(Bz(r)), z].

From this last equality, we deduce that

dB[f, f!1(Bz(r)), z] = ±1.

Finally, we conclude the proof by observing that

dB[f, f!1(Bz(r)), z] = dB[f, D, z],

a consequence of the excision theorem 3.4.1, taking in account that

z & f(f!1(Bz(r))) = Bz(r),

and, from the one-to-one character of f, z %& f(D \ f!1(Bz(r))).

22.4 Deformation of an elastic body

We now give, following Ph. Ciarlet [62], an application of Brouwer degree, and inparticular of the invariance of domain theorem 22.1.1 to mathematical elasticity.We first recall some used fundamental analytical tools.

Proposition 22.4.1 Let X and Y be topological spaces such that X is compact andY is separated. If f : X " Y is bijective and continuous, then f is a homeomor-phism.

If X and Y are two normed spaces, let Isom(X, Y ), or simply Isom(X) whenX = Y, denote the set of all continuous linear bijective mappings A : X " Y withcontinuous inverse A!1 : Y " X. The following result is a classical consequence ofthe implicit function theorem.

Proposition 22.4.2 Let X, Y be Banach space, % ' X open and f :& C1(%, Y ) amapping such that f $(x) & Isom(X, Y ) for each x & %. Then f(%) is open in Y. If,in addition, f : %" Y is one-to-one, then f : %" f(%) is a C1 -di"eomorphism.

22.4. DEFORMATION OF AN ELASTIC BODY 277

We also need the following simple consequences of the invariance of domaintheorem 22.1.1.

Proposition 22.4.3 Let A ' Rn have nonempty interior int A and f : A" Rn becontinuous. Then

(i) if f is locally one-to-one, f(int A) ' int f(A);

(ii) if f is one-to-one, f(int A) = int f(A).

Proof. (i) Let y & f(int A), so that y = f(x) for some x & int A, and let r > 0be such that Bx(r) ' A. It follows that y = f(x) & f [Bx(r)] ' f(A). Since f islocally one-to-one, we may assume that f is one-to-one on Bx(r). By the invarianceof domain theorem 22.1.1, f(Bx(r)) is open and, consequently, y & int f(A).

(ii) it su$ces to prove that int f(A) ' f(int A). Let y & int f(A) and r > 0such that By(r) ' f(A). Let x & A be such that y = f(x) & By(r) ' f(A). Then,x & f!1[By(r)] ' f!1[f(A)] = A, since f is one-to-one. By the continuity of f,f!1[By(r)] is open. Consequently, x & int A and y = f(x) & f(int A).

Let us recall some elementary facts concerning the deformation of an elasticbody. We assume that an origin 0 and an orthonormal basis {e1, e2, e3} have beenchosen in the three-dimensional Euclidian space, which is therefore identified withthe space R3.

Let % ' R3 be a domain (i.e. a bounded, open, connected subset) with a su$-ciently smooth bounday. We shall think of its closure % as representing the volumeoccupied by a body ‘before it is deformed’, and therefore called the referenceconfiguration.

Definition 22.4.1 A deformation of the reference configuration % is a vector field& : %" R3, where & is su!ciently smooth, one-to-one (except possibly at !%) andorientation-preserving.

The orientation-preserving character, i.e.

& & C1(%, R3) and J$(x) > 0 for all x & %

and the interior injectivity are two properties that the deformation & must possessin order to be physically acceptable as a deformation of an elastic body (see [62],section 5.4). The expresion ‘smooth enough’ means that, in a given definition,theorem, proof, etc., the smoothness of the involved deformations is such that allused arguments make sense. The reason why a deformation may loose the injectivityon the boundary !% of % is that ‘self-contact’ must be allowed (see [62], Chapter5, for more details).

Let us remark that, from the implicit function theorem, an orientation-preser-ving mapping is locally invertible, but local invertibility does not entail, in generalinjectivity. As an example, consider a rectangular rod % ' R3 of length 2% contained


in {x & R3 : x1 > 0}, with sides of the basis parallel to e1 and e2, symmetrical withrespect to the (e1, e2)-plane, and the mapping & : %" R3 defined by

&(x1, x2, x3) = (x1 cosx2, x1 sinx2, x3) .

Then J$(x) = x1 > 0 for all x & %, yet & is not one-to-one, since &(x1,%, x3) =&(x1,!%, x3), i.e. the injectivity is lost on the boundary. The deformation & hastransformed the rectilinear rod into an annular one. This example shows that it isnot unnecessary to discuss the injectiveness of orientation-preserving mappings.

22.5 Injectivity of some orientation-preserving map-

pings

The following result provides a class of one-to-one orientation-preserving mappings.

Theorem 22.5.1 Let % ' Rn be bounded, open, connected and such that int% =%. Let &0 : %" Rn be continuous and one-to-one, and let & & C(%, Rn).C1(%, Rn)be such that

J$(x) > 0 for all x & %, &(x) = &0(x) for all x & !%.

Then

a) &(%) = &0(%), &(%) = &0(%);

b) & : %" &(%) is a homeomorphism (in particular & is one-to-one on %);

c) & : %" &(%) is a C1-di"eomorphism.

Proof. a) Step 1 : &0(%) ' &(%), and, for any p & &0(%), there is a unique x & %such that p = &(x).

Let p & &0(%). By Theorem 22.3.1, dB [&0,%, p] = ±1, and, since &|!! = &0|!!,it follows from Corollary 3.4.3 that

dB [&,%, p] = dB [&0,%, p] = ±1 (22.6)

and hence, by the existence theorem 3.4.1, &!1(p) %= ,. Now, condition J$(x) > 0implies that

dB[&,%, p] ="

x"$!1(p)

sign J$(x) = #&!1(p) ( 1. (22.7)

By comparing (22.6) and (22.7), we obtain

dB [&,%, p] = 1 = #&!1(p).

Consequently, p & &(%) and equation &(x) = p has a unique solution.Step 2 : &(%) ' &0(%).

22.5. INJECTIVITY OF SOME ORIENTATION-PRESERVING MAPPINGS279

We prove, equivalently, that Rn \&0(%) ' Rn \&(%). Let p & Rn \&0(%). HencedB[&0,%, p] = dB[&,%, p] = 0, and hence, from the fact that dB[&,%, p] = #&!1(p)if p & &(%), we have p %& &(%). Now it follows from p %& &0(%) that p %& &0(!%) =&(!%), and hence that p %& &(%) / &(!%) = &(%).

Step 3 : &0(%) = &0(%).Indeed, since &0(%) ' &0(%), and &0(%) is compact, it follows that &0(%) '

&0(%). Conversely, let y & &0(%) and x & % such that y = &0(x), and let (xn) in %such that xn " x. Then &0(xn)" &0(x) = y, and hence y & &(%).

To sum up, we have established the following inclusions

&0(%) ' &(%) ' &(%) ' &0(%) ' &0(%),

which gives, by taking the closures,

&0(%) = &0(%) = &(%) = &(%),

and, using Proposition 22.4.3,

int&(%) = int&0(%) = &0(int%) = &0(%).

On the other hand, Proposition 22.4.2 implies that &(%) is open. Since int&(%) isthe largest open set contained in &(%), it follows that &(%) ' int&(%) ' &0(%).Together with the result in Step 1, we obtain &0(%) = &(%).

b) It su$ces to prove that & is one-to-one on %, and then Theorem 22.1.1 applies.Indeed, according to the result established in Step 1, for any p & &0(%), equation&0(x) = p has a unique solution. Since &0(%) = &(%), and & is one-to-one on %,the same is true for equation &(x) = p, and & is one-to-one on %. Since & = &0 on!% and &0 is one-to-one on %, & is one-to-one on !%. Finally, let x & !% and y & %and let us show that &(x) %= &(y). Indeed, &(x) = &0(x), &(y) & &(%) = &0(%),and therefore there exists z & % such that &(y) = &0(z). Consequently x %= z, sothat, using the injectivity of &0 on %, &(x) = &0(x) %= &0(z) = &(y).

c) The result follows from Proposition 22.4.1.

Corollary 22.5.1 Let D ' Rn be open and bounded, let u : D " Rn and & = I+u.

(a) If u is strictly non-expansive, i.e. such that

#u(x)! u(y)# < #x! y# for each x %= y in D. (22.8)

then, & is one-to-one on D, and, for any z & &(D),

dB [&, D, z] = ±1. (22.9)

(b) If D is convex, u & C1(D, Rn) and

supx"D

#u$(x)# < 1, (22.10)

then u is strictly non-expansive.


(c) If D ' Rn is a domain, there exists a constant c(D) > 0 such that anymapping u & C(D, Rn) . C(D, Rn) satisfying condition

supx"D

#u$(x)# < c(D)

is strictly non-expansive.

Proof. (a) It su$ces to prove that & = I +u is one-to-one on D and the second partfollows from Theorem 22.3.1. Let x %= y in D. then

#&(x)! &(y)# = #(x! y) + [u(x)! u(y)]# ( #x! y# ! #u(x)! u(y)# > 0.

(b) Let x %= y in D. Using the mean value inequality, we have, for some 5 & [0, 1],

#u(x)! u(y)# ) #u$(y + 5(x ! y))##x! y# ) supz"D

#u$(z)##x! y#,

and (22.10) implies that u is strictly non-expansive.(c) The proof is essentially based on the remark (see [62], p. 52, exercice 1.9)

that a domain D ' Rn has the following geometrical property : there exists anumber c(D) such that, given arbitrary points x %= y in D, there exists a finitenumber of points yk (1 ) k ) l + 1) such that

y1 = x, yl+1 = y, yk & D (2 ) k ) l),

]yk, yk+1[ = {yk + t(yk+1 ! yk) : 0 < t < 1} ' D,l"

k=1

#yk ! yk+1# ) 1

c(D)#x! y#. (22.11)

Now, if x %= y in D and supz"D #u$(z)# < c(D), we have, using the mean valueinequality, for some 5k & [0, 1] (1 ) k ) l),

#u(x)! u(y)# =

CCCCC

l"

k=1

[u(yk+1)! u(yk)]

CCCCC

)l"

k=1

#y$(yk + 5k(yk+1 ! yk))##yk+1 ! yk#

< c(D)l"

k=1

#yk+1 ! yk# ) #x! y#.

Remark 22.5.1 Since the hypotheses of points (b) and (c) entail the fulfilment ofcondition 22.8 (and, consequently the injectivity of & on D,) it follows, using theresult given by (a) that formula (22.9) holds in both situations.

Chapter 23

A history of Brouwer fixedpoint theorem

23.1 The discovery and the publication

The statement of Brouwer’s fixed point theorem concludes an article on continuousmappings between manifolds [39] by Luitzen Egbertus Jan Brouwer, published July25 1911 and dated from Amsterdam, July 1910. This work developed a previouspaper [38], sent by the same author to the same journal one month before, issuedFebruary 14 1911, and giving the first correct proof of the non-existence of anyhomeomorphism between Rm and Rp when p > m [77, 85, 186, 184]. To this aim,Brouwer implicitely introduced the concept of topological degree dB[g, M, N ](the future Brouwer degree) of a continuous mapping g between two oriented,compact, boundaryless manifolds M and N having the same finite dimension, andthe technique of simplicial approximation.

Brouwer developed this topological degree in the second paper (the statement ofthe main properties of degree is his Theorem 1), and his first application concernedthe study of vector fields on a sphere, and in particular, his Theorem 2, the so-calledhairy ball theorem :

Any continuous vector field on a sphere having even dimension has atleast one singular point.

Brouwer then computed, in his Theorem 3, the topological degree of mappingswithout fixed point of a sphere into itself :

The topological degree of a continuous mappings without fixed point of an-dimensional sphere into itself is equal to (!1)n+1,

which easily led him to a fixed point theorem on the sphere, already given byhim in 1909 for n = 2 [37] :

281

282 CHAPTER 23. A HISTORY OF BROUWER FIXED POINT THEOREM

Any continuous mapping of a n-dimensional sphere into itself, having atopological degree di"erent from (!1)n+1, has a fixed point.

Brouwer then used a rather complicated argument to deduce, without further com-ment, his fixed point theorem for a ball, whose assertion avoids any explicitreference to topological degree. The ideas of Brouwer’s proof are the following ones.If h denotes a homeomorphism from the Northern hemisphere (xn+1 ( 0) Sn

+ of thesphere Sn, then f1 = h!1 * f * h : Sn

+ " Sn+ is continuous. Brouwer extends f1 to

Sn by setting f1(s(x)) = f1(x) for each symmetrical image s(x) of x with respectto the equator plane {xn+1 = 0}. If f2 : Sn " Sn

+ denotes this extension, f2 has afixed point fixed point due to Theorem 3, because, f2 which is not onto, has degreezero and hence di!erent from (!1)n+1. This fixed point needs to belong to Sn

+ andits image by h is a fixed point of f. Brouwer does not give in [39], and will nevergive, any application of his fixed point theorem for the ball.

Brouwer’s paper [39] contains few references, all dealing with his earlier work,with the exception of the following preliminary note :

During the printing of this article, has appeared in volume two of the ‘Intro-

duction a la theorie des functions d’une variable’ of J. Tannery, a note by J.

Hadamard, ‘Sur quelques applications de l’indice de Kronecker’. The theorydeveloped there meets in various ways the following developments.

What is the meaning of this note ?

In 1910, Jules Tannery published the second volume of the second edition ofhis book ‘Introduction a la theorie des fonctions de variables reelles’ [381], for thetime a very modern introduction to analysis, having introduced Weierstrass’ rigorin France. This volume two ends with a Note of Jacques Hadamard, connected inthe following way to Tannery’s book contents :

The proof, following M. Ames, of Jordan’s theorem on closed curves without

double point is based upon the concept of order of a point or, equivalently onthe consideration of the variation of the argument. The generalization to the

case where the dimension is larger than two is given by Kronecker index. It

is a now classical notion, mainly since the publication of the Traite d’Analyse

of Mr. Picard (T. I, p. 123; T. II, p. 193). It has received new applications

in various recent works. My aim is to present here some of them. All the

following reasonings [...] only use the continuity of the considered functions.

Kronecker index was introduced and developed in 1869 by Leopold Kronecker [222],for systems of n + 1 C1 real functions of n real variables g0, g1, . . . , gn, such that 0is a regular value of g0, K := g!1

0 (] !-, 0]) is bounded, and the gj (j = 1, . . . , n)do not vanish together on !K. If we set g = (g1, . . . , gn), Kronecker shows that thenumber 4[g0, g] defined (in modern notations) by the integral

1

vol Sn!1

!

!K

n"

j=1

(!1)j!1 gjdg1 $ . . . $ dgj!1 $ dgj+1 $ . . . $ dgn

(g21 + . . . + g2

n)n/2, (23.1)

23.1. THE DISCOVERY AND THE PUBLICATION 283

is equal toB

x"g!1(0) Jg(x), when this sum exists, i.e. when g!1(0)is finite and the

Jacobian Jg of g does not vanish on g!1(0). Consequently, Kronecker’s integral(1) provides an ‘algebrical count’ of the number of zeros of g located in K, namedKronecker index. The special case where n = 2 was already considered by Au-gustin Cauchy, in terms of the rotation of g along the close curve !K, whenhe extended to regular planar mappings his index theory for holomorphic functions.Kronecker’s work was motivated by the papers of Sturm, Liouville, Cauchy, Hermiteand Sylvester on the number of real roots of algebraic equations [281], and by thework of Gauss on potential theory [146] and the linking of curves [147]. In [162],Hadamard extended Kronecker index to the case of g continuous on an orientedmanifold K. The concept of oriented manifold is defined in a precise way, and thecontinuous mapping is approximated through a polyhedral approximations. Thisgeneralized index 4[!K, g] is nothing but Brouwer degree dB[g/#g#, !K, Sn], de-fined in a di!erent way, of the mapping g/#g# : !K " Sn!1. One can find on p.472 of [162] the following statement :

The previous considerations also allow to prove, for the volume V ofthe sphere in any dimensions, Brouwer’s theorem : [...] Any perfect andcontinuous mapping of the volume V into itself leaves at least one pointinvariant, either inside, or on the boundary.

To prove the result, assuming without loss of generality that f : V " V has no fixedpoint on the sphere !V, Hadamard noticed that 6f, x ! f(x)7 > 0 for any x & !V,and he had shown previously that this implied 4[!V, I ! f ] = 4[!V, I] = 1, andhence the existence of a zero of I ! f in V. When he applied Kronecker index tothe proof of the unpublished results of Brouwer on vector fields and mappings of asphere into itself, Hadamard only mentioned Brouwer through his 1909 paper [37],

How can one explain that Brouwer’s fixed point theorem was already publishedand named more than one year before the publication of Brouwer’s paper ? Thishas been explained by H. Freudenthal [133], using documents discovered when pub-lishing the topological part of Brouwer’s Complete Work. Freudenthal found thecopies of two letters, sent at the very beginning of January 1910, respectively toDavid Hilbert, and to Hadamard, and written in Paris, where Brouwer spent Sea-sons holidays with his brother, a geologist, living on 6, rue de l’Abbe de l’Epee. Inthose letters, Brouwer anounced the main results of his future paper [39] dealingwith fixed points of mappings of a sphere or of a ball into itself :

Any single-valued (not necessarily one-to-one) continuous transforma-tion of the volume of a m-dimensional sphere into itself has at least oneinvariant point.

Hence Brouwer’s fixed point theorem has a famous godfather, Hadamard. We shownow that it has also famous ancestors.


23.2 Early equivalent formulations

In 1883 (Brouwer is two years old), Henri Poincare was studying the existence ofsymmetric periodic solutions of the three body problem. To do so, he shows thattheir initial conditions must satisfy some system of n equations in n unkowns. Toshow that a solution exists, Poincare generalizes as follows Bolzano’s intermediatevalue theorem :

Let /1, /2, . . . , /n be n continuous functions of n variables x1,x2, . . . ,xn; the variable xi is required to vary between the limits +ai and !ai.Assume that, for xi = ai, /i is constantly positive, and for xi = !ai

constantly negative; I claim that there exist a system of values x1, . . . , xn

for which all the /i vanish.

This result is anounced in 1883 in a note to the Comptes-Rendus [320] and developedin 1884 in an article published in the first volume of the Bulletin astronomique [321].In both cases, the proof given by Poincare was more than sketched; for example, hewrote in [321] :

Mr. Kronecker has given, in the Monatsberichte (1869), a formula providing

the number of the solutions of n equations in n unknowns, satisfying somegiven inequalities. We will make the following application of this formula.

Probably because most readers of [321] are astronomers, very likely unaware ofKronecker’s work, Poincare added the following commentary :

To help understanding how one can prove this theorem, assume that we only

have two variables x1 and x2, seen as the coordinates of a point in a plane.[...] This point is inside some square ABCD. [...] The curve !2 = 0 joins then

any point of the side AB to some point ofCD; similarly the curve !1 = 0,

joining a point of BC to some point of DA, must meet the first curve inside

the square.

Notice that Kronecker index is only defined for functions /j of class C1 and thatPoincare did not explain how the extension to continuous functions was made.Notice that Weierstrass’ approximation theorem of a continuous function throughpolynomials only dates from 1885 !

It is easy to see that Poincare’s theorem applied to the function g = I ! fgives Brouwer’s theorem on the closed n-interval

Gni=1[!ai, ai], because Brouwer’s

assumptions imply that xi ! fi(x) ) 0 when xi = !ai and xi ! fi(x) ( 0 whenxi = ai. But the remark is anachronic, Brouwer’s theorem having still to wait somethirty years before being stated !

In his further important work on periodic solutions, and in particular his memoircrowned by King Oscar’s Prize and developed in the book ‘Methodes nouvelles de lamecanique celeste’, Poincare replaces the use of his intermediate value theorem bythe more classical implicit function theorem, su$cient for the considered problems.That may explain why Poincare’s theorem has been rapidly forgotten. One of

23.2. EARLY EQUIVALENT FORMULATIONS 285

the few mathematicians to notice and mention it is Hadamard, the great connois-seur and follower of Poincare’s work. In his analysis of Poincare’s mathematicalcontributions published in 1921 [163], Hadamard writes, p. 263 :

To obtain the periodic solutions, the memoir of 1883 uses Kronecker’s theorem.This theorem is essentially the natural generalization, to the case of system

of equations in several unkowns, of the most elementary existing method todetect the roots of a single equation, the one based upon sign changes of its

left-hand member.

Hadamard returns to the subject in 1920, in a conference delivered at the RiceInstitute [164], p. 163-164 :

The first paper on the subject [of periodic solutions] is devoted by Poincareto the construction of a simple class of periodic trajectories. He again ap-

plies the remarkable theorem of Kronecker on systems of equations already

alluded to. As a matter of fact, however, this theorem is not necessary forthe conclusions stated in this case : they all derived from the classic theorem

on the existence of the solution for a system of n ordinary equations in n

unknowns, the Jacobian of which does not vanish. Poincare’s argument mayeven be said to imply a new proof of that theorem, which indeed would be

of only mediocre interest if Poincare had not met with a curious intermediate

proposition by which the existence of the required solutions can be assertedinside a certain cube whenever we know that each of the left-hand members

of the given equations has contrary signs on two of its opposite faces.

It is somewhat surprising that such a penetrating mind as Hadamard, who knewboth Poincare’s and Brouwer’s theorems, did not see their connection, which, asshown later, will only be made in 1941 !

In 1904, the Latvian mathematician Piers Bohl published a long paper on theasymptotic behavior of the trajectories of a mechanical system near an equilibrium[31]. To study the behavior of those trajectories, Bohl stated and proved someresults on systems of n functions of n variables, defined and continuous on someclosed n-interval K =

Gni=1[!ai, ai] of Rn :

I. There is no continuous mapping g : K " Rn without zeros and equalto identity on !K.

II. If g : K " Rn is continuous and equal to identity on !K, then,g(K) 0 K and g(int K) 0 int K.

As a consequence of those results, Bohl proved the following theorem :

III. If g : K " Rn is continuous and does not vanish, there existsu & !K and µ < 0 such that g(u) = µu.

He also proved the corresponding result for a ball B in Rn centered at the origin :


IV. If g : B " Rn is continuous and does not vanish, there exists u & !Band µ < 0 such that g(u) = µu.

Bohl’s proofs are essentially based upon the fact that the integral (23.1) vanisheswhen the functions g1, . . . , gn do not vanish simultaneously in K. Bohl does notquote Kronecker, but only the first volume of Emile Picard’s ‘Traite d’analyse’ of[317] mentioned above by Hadamard, and describing Kronecker’s results in dimen-sions two and three (vol. I, p. 83-87, 123-127, vol. II, p. 193-207). Like in Poincare’spaper, the used tools require the functions to be of class C1 instead of continuous,a fact also ignored by Bohl.

The contraposition of Bohl’s Theorem provides a special case of Poincare’s the-orem, not quoted by Bohl. The same theorem easily implies the non-retractiontheorem for a closed n-interval :

There exists no continuous mapping r : K " !K such that r is equal toidentity on !K.

But this remark is not explicit in Bohl, and the statement of a variant of TheoremsI to III with K replaced by B is missing as well. In 1931, in his fondamental paperon the theory of retracts [35], Karel Borsuk deduced from Brouwer’s fixed pointtheorem the following result :

If E ' Rn is compact and g : E " Rn is continuous and such thatg(x) = x for all x & !E, then E ' g(E).

In the special case where E is the closed n-interval K, this is nothing but Bohl’stheorem II, that Borsuk does not quote.

Recall that if F ' E are Hausdor! topological spaces, r : E " F is called aretraction of E onto F if r is continuous on E and equal to the identity on F.Borsuk’s theorem immediately implies the theorem of non-retraction of a ballonto its sphere :

There exists no continuous mapping r : B " !B such that r is theidentity on !B.

Indeed, if we take E = B, such a mapping should be such that !B = r(B) 0 B.Conversely, this non-retraction theorem implies Brouwer’s fixed point theorem onB. A direct proof is given in [7]. Kakutani [191] has shown in 1943 that, in aninfinite-dimensional Hilbert space,

The surface S = {x|#x# : 1} of B is a retract in B, i.e., there exists acontinuous mapping x$ = 3(x) of B onto S such that 3(x) > x on S.

The contraposition of Bohl’s theorem IV applied to the function g = I! f, withf : B " Rn continuous, immediately gives a fixed point theorem for a ball Bcentered at the origin :

If f : B " Rn is continuous and f(x) %= (x for any x & !B and any( > 1, f has a fixed point in B.

23.2. EARLY EQUIVALENT FORMULATIONS 287

Brouwer’s fixed point theorem is a special case of this result (again not explicitedby Bohl who nowhere deals with fixed points), as well as the following extension,usually and erroneously called, as we will see later, Rothe’s fixed point theorem,by reference to its extension to Banach spaces, given in 1937 by Erich Rothe [335] :

If f : B " Rn is continuous and f(!B) ' B, f has a fixed point in B.

The fixed point versioon of Bohl’s Theorem IV can alwo be written, replacing ( by1/" :

If f : B " Rn is continuous and x %= "f(x) for any x & !B and any" & ]0, 1[, f has a fixed point in B.

This is a special case of Leray-Schauder’s fixed point theorem, stated andproved in 1934 by Jean Leray et Jules Schauder [251] (who do not quote Bohl),in the more general frame of compact perturbations of identity in a Banach space.Brouwer, Rothe and Leray-Schauder’s theorems are indeed equivalent because, asshown essentially by Helmut Schaefer [345] in 1955, Leray-Schauder’s theorem fol-lows from Brouwer’s theorem (see Chapter 8).

If Bohl’s paper did not attract the attention of many mathematicians, he didnot escape to the attention of Hadamard, who quotes it in [162], through a resultthat he named Poincare-Bohl’s theorem :

Let S be a compact, boundaryless, oriented manifold of dimension n! 1and let g : S " Rn and h : S " Rn be continuous mappings which donot vanish. If 4[S, g] %= 4[S, h], there exists at least one x# & S suchthat g1

h1=

g2

h2= . . . =

gn

hn< 0.

If 4[S, g]/4[S, h] %= (!1)n, there exists at least one x# & S such that

g1

h1=

g2

h2= . . . =

gn

hn> 0.

The given reference to Poincare is [323], and Hadamard notices that

the assertions given by the two quoted authors are di!erent and di!er from the

one given here, in that they particularize (each in his own way) the choice ofthe functions g. But their reasonings (which, by the way, are di!erent) easily

extend to the case where those functions are arbitrary. The proof we give here

is Mr. Poincare’s one.

The contraposition of this result tells that if 6g(x), h(x)7 ( 0 for any x & S, then4[S, g] = 4[S, h]. Hadamard uses the special case of this result with g = I ! f,h = I and S = !B to prove in [162] Brouwer’s fixed point theorem. It seems thatBohl’s results passed to the posterity only through this ‘Poincare-Bohl’s theorem’(for example in the influential book ‘Topologie’ of Alexandro! and Hopf [6]). Thisis historically questionable, as Bohl never explicitely used any concept of index ordegree.


Consequently, Brouwer’s fixed point theorem is an easy, but not explicited, con-sequence of Poincare or Bohl’s theorems. We now return to the work followingBrouwer’s paper.

23.3 New proofs and infinite-dimensional exten-

sions

In 1922, two new proofs of Brouwer’s fixed point theorem were published in theUnited States. James W. Alexander’s one [5] only di!ers from Brouwer’s one bythe use of Kronecker index instead of Brouwer degree. The one of George D. Birkho!and Oliver D. Kellogg [28] is more original and based upon a continuation methodapplied to a family of equations

x! "p(x) = 0, " & [0, 1],

with p : Rn " Rn a polynomial, and upon the approximation of the continuousmapping f through such polynomials. Birkho! and Kellogg proved the followingversion of Brouwer’s fixed point theorem :

Let Rn denote a bounded connected region of real n-space containingan interior point O (the origin for a set of rectangular coordinatesx1, x2, . . . , xn) such that every half-ray originating in O contains butone boundary point of Rn. Let T denote a one-valued and continuoustransformation x$

1 = f1(x1, x2, . . . , xn), x$2 = f2(x1, x2, . . . , xn), . . . ,

x$n = fn(x1, x2, . . . , xn), or briefly, x$ = f(x), which transforms each

point of Rn into a point of Rn. Then there exists a point a of Rn whichis invariant under f, i.e. such that a = f(a).

In a note at the bottom of p. 100, the authors notice that

The theorem is stated and proved in a degree of generality su"cient for ourlater purposes. It will be seen that the proof holds if merely the boundary

points of Rn are transformed into points of Rn, and the theorem admits the

obvious extension to any region susceptible of continuous one to one mappingon a region Rn of the kind described.

It is, apparently, the first explicit appearance of the statement of the so-calledRothe’s fixed point theorem followed, in 1926, by a note of Rolin Wavre and A.Bruttin [399] (presented par Hadamard), dealing with the special case where n = 2,which does not mention Birkho! and Kellogg.

The main aim of Birkho! and Kellogg’s paper was to extend Brouwer’s fixedpoint theorem to the function spaces C([a, b]) and L2(a, b), and to apply thoseextensions to di!erential equations, and in particular to the existence of a solutionto boundary value problems of the form

y(n) = F (x, y, y$, . . . , y(n!1)),

23.4. A COMBINATORIAL APPROACH 289

! a

0

n!1"

j=0

pij(x)y(j)(x) dx +n!1"

j=0

m"

k=1

qijky(j)(xk) = ci,

(i = 1, 2, . . . , n; 0 ) x1 ) x2 ) . . . ) xm ) a).

Those problems are reduced to a nonlinear integral equation, whose right-handmember defines transformation of C([0, a]) into itself. A similar result was redis-covered in 1930 by Renato Caccioppoli [50], and also applied to some nonlineardi!erential equations. The extension of Brouwer’s fixed point theorem theorem toa compact convex subset of an arbitrary Banach space, unifying those results, wasobtained the same year by Schauder [348], and nowadays called Schauder’s fixedpoint theorem :

Any continuous mapping f : C " C of a compact convex subset C of aBanach space E has at least one fixed point.

In 1927 already, Schauder [346, 347] had given a slightly less general version andhad applied it to the Cauchy problem for ordinary di!erential equations, to somenonlinear Dirichlet problems, and to hyperbolic equations.

The extension of Schauder’s fixed point theorem to locally convex vector spacesE was made by Andrei N. Tychono! [391] in 1935. It partly answer problem 54raised by Schauder in the ‘Scottish book’ [260]. Brouwer’s fixed point theoremremains true in some not locally convex topological vector spaces, but the validityof its extension to an arbitrary topological vector space is still under discussion (see[312]).

In 1943, Shizuo Kakutani [191] gave an example of a fixed-point free continuousmapping of the closed unit ball of an infinite-dimensional Hilbert space into itself.Using his extension theorem, James Dugundji [96] characterized in 1951, the finite-dimensional normed vector spaces, in terms of the fixed point property :

Let L be a normed linear space, and S = {x|#x# ) 1}. A necessary andsu!cient condition that every continuous f : S " S have a fixed pointis that S be compact.

In 1955, Victor L. Klee[202] answered in the following way Problem 36 raised byStanislaw Ulam in the ‘Scottish book’ [260] :

For any non compact, closed convex subset K of a Banach space, thereexists a continuous mapping f : K " K without fixed points.

23.4 A combinatorial approach

In 1928, Emanuel Sperner [371] based a simplified proof of Brouwer’s invariance ofdimension theorem upon a combinatorial lemma which is named to-day Sperner’slemma [372] :


Consider a simplicial division of the n-simplex S in sub-simplices s1,s2, . . . , sp. To each vertex of any sk, let us attach an integer between 0and n in such a way that if the vertex belongs to the convex hull of thevertices pi0 , . . . , pik of S, the integer must belong to {i0, . . . , ik}. Then thenumber of sub-simplicices whose vertices are numbered by {0, 1, . . . , n}is odd.

In the special case where S is the triangle made of the convex hull of three notcolinear points A, B, C, any point P & S can be written in a unique way as P =*1A + *2B + *3C, for some *1,*2,*3 ( 0 and such that *1 + *2 + *3 = 1. Thusone can write as well P = (*1,*2,*3), and the *i are the barycentric coordinates ofP. In particular, A = (1, 0, 0), B = (0, 1, 0) and C = (0, 0, 1). For any non negativeinteger k, let Tk be the regular triangulation S made of the set of points of S whosebarycentric coordinates are all of the type i

k!1 , with 0 ) i ) k ! 1. Attach in anarbitrary way to each of those points one of the three numbers 0,1 or 2, with theonly restriction that A (resp. B, C) receives the number 0 (resp. 1, 2) and thatany point of AB (resp. BC, CA) can only receive one of the numbers 0 or 1 (resp.1 or 2, 0 or 2). Sperner’s lemma claims that the number of elementary triangles ofTk with vertices numbered by the three numbers 0,1 and 2 is odd (and in particularnot zero).

Sperner’s lemma was immediately presented by Witold Hurewicz at a meet-ing of the Polish Mathematical Society, and Broniclaw Knaster raised the ques-tion of deducing Brouwer’s fixed point theorem from it. A few weeks later, Ste-fan Mazurkiewicz gave a positive answer, unfortunately with an uncomplete proof,soon corrected by Kasimier Kuratowski [231]. The famous paper [203] was born[232, 231], deducing Rothe’s fixed point theorem for a n-simplex from the followingintersection theorem, a consequence of Sperner’s lemma :

Let S be a simplex of dimension n, with vertices p0, p1, . . . , pn. If theclosed sets A0, A1, . . . , An are such that any simplex of dimension k(0 ) k ) n) pi0pi1 . . . pik is contained in Ai0 /Ai1 / . . . /Aik , then

A0 .A1 . . . . .An %= ,.

A variant of this intersection theorem for a cube was given the same year byHurewicz [176] :

Let us denote by Hi (resp. Hi) (1 ) i ) n) the opposed faces xi = 0,resp. xi = 1, of the unit cube K = [0, 1]n and assume that the n closedsets B1, . . . , Bn are such that, for each 1 ) i ) n, there exists two setsUi and U i open in K \ Bi such that

K \ Bi = Ui / U i, Hi ' Ui, Hi ' U i, Ui . U i = , (i ) k ) n).

Then B1 . . . . .Bn %= 0.

It is easy to deduce Poincare’s theorem from Hurwitz’ one, but Hurwitz did not doit (and did not refer to Poincare). The theorem is reproduced in [177], p. 40-41.

23.5. GAME THEORY AND MATHEMATICAL ECONOMICS 291

Brouwer’s fixed point theorem can also be deduced directly from Sperner’slemma. Let us return to the simplex S made of the convex hull of points A, B et C,and let f : S " S be continuous. If P = (*1,*2,*3) is a point of the triangulationTk and if f(P ) = (*#

1,*#2,*

#3), let us attach to P the number i!1, with i the smallest

integer such that *#i ) *i. Such an integer necessarily exists because the sum of the

*i and the sum of the *#i are equal to one. It is easy to show that this numbering

satisfies the conditions of Sperner’s lemma, and there exists at least one elemen-tary triangle of Tk whose vertices Ak = (*k,1,*k,2,*k,3), Bk = (,k,1,,k,2,,k,3),et Ck = (.k,1, .k,2, .k,3), are respectively numbered by 0, 1 and 2 and hence suchthat *#

k,1 ) *k,1, ,#k,2 ) ,k,2, .#

k,3 ) .k,3. When k " -, the sequence of thosepoints contains a subsequence converging to an element P = (*1,*2,*3) such that*#

1 ) *1, *#2 ) *2, *#

3 ) *3. Those inequalities are indeed equalities, because thesum of the terms of each member is equal to one, and P is a fixed point of f.

There are many generalizations of Sperner’s lemma and of Knaster-Kuratowski-Mazurkiewicz’s theorem in the literature. The most important ones are due to KyFan ([110]-[122]). To-day refered as KKM methods, they have been the source ofa large number of papers, for which one can consult, for example [313, 314].

23.5 Game theory and mathematical economics

In 1928, John von Neumann [395] stated and proved a fundamental minimaxtheorem for the theory of zero-sum 2-person-games with mixed strategies :

Investigate continuous functions of two sets of variables f(/, 7) whichsatisfy the following conditions (K) : If f(/$, 7) ( A, f(/$$, 7) ( A,then f(/, 7) ( A for every 0 ) ) ) 1, / = )/$ + (1 ! ))/$$ (i.e. /p =)/$p + (1 ! ))/$$p , p = 1, 2, . . . , M). If f(/, 7$) ) A, f(/, 7$$) ) A, thenf(/, 7) ) A for every 0 ) ) ) 1, 7 = )7$ + (1 ! ))7$$ (i.e. 7q =)7$

q + (1 ! ))7$$q , q = 1, 2, . . . , N). For these functions we are going to

prove that max) min2 f(/, 7) = min2 max) f(/, 7), where max) is takenover the range /1 ( 0, . . . , /M ( 0, /1 + . . . + /M ) 1 and min2 is takenover the range 71 ( 0, . . . , 7N ( 0, 71 + . . . + 7N ) 1.

He reduces its proof to an extension of Brouwer’s fixed point theorem to a mappingwith closed graph from a compact interval of R into the set of compact intervals ofR.

In a seminar devoted to a model of economical equilibrium, given in 1932 at theUniversity of Princeton and published five years later [396], von Neumann is led tothe solution of a system of inequalities, and claims that the existence of a solution

cannot be proved by any qualitative argument. The mathematical proof is

only possible through a generalization of Brouwer’s fixed point theorem, i.e.

through the use of completely fundamental topological facts. This generaliza-tion of the fixed point theorem is also interesting in itself.

This von Neumann’s topological lemma, which also provides a proof of theminimax theorem, goes as follows :


Let Rm be the m-dimensionsional space of all points X = (x1, . . . , xn),Rn the n-dimensionsional space of all points Y = (y1, . . . , yn), Rm+n

the m + n-dimensionsional space of all points

(X, Y ) = (x1, . . . , xm, y1, . . . , yn).

A set (in Rm or Rn or Rm+n) which is non empty, convex, closed andbounded we call a set C. Let S0, T 0 be sets C in Rm and Rn respectivelyand let S01T 0 be the set of all (X, Y ) (in Rm+n) where the range of Xis S0 and the range of Y is T 0. Let V, W be two closed sets of S0 1 T 0.For every X in S0, let the set Q(X) of all Y with (X, Y ) in V be a setC; for each Y dans T 0, let the set P (Y ) of all X with (X, Y ) in W be aset C. Under the above assumptions, V, W have (at least) one point incommon.

In collaboration with the economist Oskar Morgenstern, von Neumann deve-loped those ideas in a monumental treatise [397], published in 1944, which willinspire generations of experts in game theory and economics. Motivated by a paperof J. Ville [394], the authors prove the minimax theorem without using fixed pointtheory :

This theorem occured and was proved first in the original publication of one

of the authors on the theory of games (1928). A slightly more general form ofthis min-max problem arises in another question of mathematical economics in

connection with the equations of production (1937). The proof of our theorem,

given in the first paper, made a rather involved use of some topology and offunctional calculus. The second paper contained a di!erent proof, which was

fully topological and connected the theorem with an important device of that

discipline : the so-called ‘fixed point theorem’ of L.E.J. Brouwer. This aspectwas further clarified and the proof simplified by S. Kakutani (1941). The first

elementary proof was given by J. Ville (1938). The proof we are going togive carries the elementarization initiated by J. Ville further, and seems to

be particularly simple. The key of the procedure is, of course, the connection

with the theory of convexity.

In 1941, Kakutani [190] showed that von Neumann’s results follow very easilyfrom an extension of Brouwer’s fixed point theorem to multivalued applications.Let K(E) denote the family of all closed convex subsets of a normed vector spaceE. A multivalued or point-to-set application " of E in K, i.e. an application" : E " K(E), is called upper semi-continuous if its graph is closed in E 1 E.Kakutani’s fixed point theorem goes as follows :

If x " "(x) is an upper semi-continuous point-to-set mapping of a r-dimensional closed simplex S into K(S), then there exists an x0 & S suchthat x0 & "(x0). [...] Theorem 1 is also valid even if S is an arbitrarybounded closed convex set in a Euclidian space.

23.6. POINCARE’S THEOREM : TOPOLOGY OR NOT TOPOLOGY 293

Kakutani deduced von Neumann’s intersection theorem from this result, and thenvon Neumann’s minimax theorem. Extensions of Brouwer’s fixed point theorem tomultivalued applications with not necessarily convex values have also been given.See [18, 73, 100, 151, 300] and their references. Even if Kakutani’s contribution isfinally not necessary to prove von Neumann’s results, it has played and still playsa basic role in other questions of mathematical economics, as emphasized by theNobel Prize Gerard Debreu in Kakutani’s Selected Works :

von Neumann’s topological lemma, which, through Kakutani’s corollary, had

a major influence in particular in economics and in game theory, was notnecessary to prove any of the results that von Neumann wanted to establish.

[...] However the formulation given by Kakutani is by far the most convenient

to use, and its proof is definitely the most attractive one. One of the first, andof the most important applications of Kakutani’s theorem was made by Nash

in his proof of the existence of a equilibrium for a finite game. It was followed

by several hundreds of applications in game theory and in economics. In thislast area, Kakutani’s theorem has been for more than three decades the main

tool to prove the existence of an economical equilibrium.

As mentioned by Debreu, Brouwer and Kakutani’s fixed point theorems havebeen used in 1951 by John Nash, in his PhD thesis at the University of Prince-ton, to prove a fundamental theorem on the equilibrium of non-cooperativegames [294, 295], for which he will share, in 1994, the Nobel Prize of Economics[224]. Such a n-person game with mixed strategies is described by n simplicesS1, . . . , Sn with possible di!erent finite dimensions and n continuous real functionsfi : S1 1 . . .1 Sn " R such that p 2" fi(p1, . . . , p, . . . , pn) is concave for each fixed(p1, . . . , pi!1, pi+1, . . . , pn) (1 ) i ) n). An equilibrium for such a game is any(p#1, . . . , pn$) & S1 1 . . .1 Sn such that, for each i = 1, 2, . . . , n,

fi(p#1, . . . , pn$) = max

x"Sifi(p

#1, . . . , p

#i!1, x, p#i+1, . . . , p

#n).

Nash’s theorem states that :

any non-cooperative n-person-game has an equilibrium.

Let us notice that Nash has also invented a game, nowadays called hex game,which can also be studied using Brouwer’s fixed point theorem [143].

23.6 Poincare’s theorem : topology or not topol-

ogy

In 1932, Adolf Hammerstein studied the two-point boundary value problem

y$$(x) = f(x, y(x), y$(x)), y(a) = A, y(b) = B, (23.2)

when |f(x, y, z)| ) c|y| for all x & [a, b], all y, z & R, and some su$ciently smallc > 0. Using Galerkin’s method, he reduced the search of approximate solutions tofinding the zeros of n continuous functions F3 in n unknowns *3 which :


are positive for "! = d, |"" | ! d, # "= $ and negative for "! = #d, |"" | ! d,

# "= $, from where it follows from the displacement theorem of Brouwer [40]

[Brouwnschen Vershiebungssatz (sic)], that the system of equations F! = 0($ = 1, 2, . . . , n) has at least one system of solutions "1 = "

(n)1 ,. . . , "n = "

(n)n .

One recognizes the statement of Poincare’s theorem, who is not quoted. Brouwer’sdisplacement lemma is an intermediate result introduced in 1911 by Brouwer [38],to prove the invariance of dimension theorem :

When a q-dimensional cube is transformed in a q-dimensional manifoldin such a way that the maximum displacement is smaller than half of theside of the cube, there exists a concentric and homothetic cube containedin the range of the transformation.

Hammerstein did not precise how to go from Brouwer’s displacement lemma toPoincare’s theorem. In 1935, without quoting [168], Michael Golomb [150] used thesame arguments, with almost the same words, to prove the existence of solutionsto some systems of nonlinear integral equations.

In 1940, using a shooting method to study a multi-point boundary value problemfor a nonlinear di!erential equation of order larger than two, Silvio Cinquini [63]was led to prove the existence of a solution of a nonlinear system of n equations inn unknowns satisfying, on a hypercube, the conditions of Poincare’s theorem (thathe did not quote). He ‘proved’ it by an argument identical to Poincare’s heuristicremark (see the notes of p. 66-67 in [63]). Namely, in the simplest case of a thirdorder di!erential equation

y$$$(x) = f(x, y(x), y$(x), y$$(x)), (23.3)

with boundary conditions

y(x1) = y(x2) = y(x3) = 0, (23.4)

with x1 < x2 < x3, and f smooth enough and bounded on [x1, x4]1R4, and denotingby y(x) = y(x; y$

1, y$$1 ) the solution of (23.3) with initial conditions y(x1) = 0,

y$(x1) = y$1, y$$(x1) = y$$

1 , Cinquini showed that there exists "1 > 0 such that

y(x2; y$1, y

$$1 ) > 0 if y$

1 > "1, y(x2; y$1, y

$$1 ) < 0 if y$

1 < !"1,

and deduced from Bolzano’s theorem the existence of y$1(y

$$1 ) & (!"1,"1) such that

y[x2; y$1(y

$$1 ), y$$

1 ] = 0. He then showed the existence of some "2 > 0 such that

y[x3; y$1(y

$$1 ), y$$

1 ] > 0 if y$$1 > "2, y[x3; y

$1(y

$$1 ), y$$

1 ] < 0 if y$$1 < !"2,

and got in the same way y$$1 & (!"2,"2) such that y[x3; y$

1(y$$1 ), y$$

1 ] = 0. Introducingthis y$$

1 in y$1(y

$$1 ), he obtained the initial conditions of a solution of (23.3)-(23.4).

In his rewiew of [63] for Zentralblatt fur Mathematik (vol. 22 (1940), p. 339),Carlo Miranda noticed the lack of validity of this proof, due to the non-uniquenessof the function y$

1(y$$1 ). To overcome the di$culty, he proved one year later, in

two pages, the equivalence between Poincare’s theorem (whose existence was alsounknown to Miranda) and Brouwer’s fixed point theorem. His paper started asfollows :


S. Cinquini, in an exchange of ideas that I had with him some time ago, has

informed me how the study of existence questions in boundary value problems

for nonlinear di!erential equations of order n, reduced finally to the validityof the following theorem :

I. If F1, F2, . . . , Fn are n functions of the variables (x1, x2, . . . , xn), con-tinuous in the hypercube |xi| ) L (i = 1, 2, . . . , n) and verifying theinequalities

Fi(x1, . . . , xi!1,!L, xi+1, . . . , xn) ( 0,

Fi(x1, . . . , xi!1, L, xi+1, . . . , xn) ) 0,

there exists at least one solution of the system :

Fi(x1, x2, . . . , xn) = 0, (i = 1, 2, . . . , n).

In the special case where n = 2, S. Cinquini has also indicated to me a proof

of this theorem; another proof, still for n = 2, was communicated to me by G.

Scorza. As those proofs do not seem to be easily extendible to the case of anarbitrary n, and as it does not seem that other persons have considered those

questions, I propose to show how theorem I can be easily deduced from the

famous Brouwer’s fixed point theorem [theorem II]. [...] I will further showhow theorem II is in turn a consequence of theorem I.

Miranda did not quote the papers of Hammerstein [168] and of Golomb [150].But Miranda’s result (quoted in a note p. 134 of [65]) did not completely satisfy

Cinquini, who still dreamed of a proof independent of topology for his results ondi!erential equations. In a note on p. 168-169 of a paper of 1941 [64], Cinquini(wrongly) claimed that the argument of [63] was valid for polynomial functions, andthat it was su$cient, to obtain the general case, to use Weierstrass’ approximationtheorem.

Giuseppe Scorza Dragoni, a student of Caccioppoli who had received earlier someundue criticisms from Cinquini, was convinced of the necessity of using topologyto obtain the claimed existence results. He started with Cinquini a long quarrel,feeding some Italian mathematical journals for some ten years. In a note on p. 204of [355], Scorza Dragoni listed the gaps in Cinquini’s ‘proof’ of [63], called ‘trivial’the remark reducing the continuous case to the polynomial one given in [64], andobserved with an undisguised satisfaction that Miranda’s proof to which Cinquinirefered in [65], implied that

in this moment, Cinquini renounces also, implicitely, to avoid the use of topol-

ogy. [...] The point was easily predictible a priori.

Cinquini’s statement for polynomial mappings was then proved by Luigi Brusotti[49] in 1942, using arguments from algebraic geometry, which should be reconsid-ered in the light of present rigor criterions. This work allowed Cinquini, in theintroduction of [66], to


answer in a definite way to a few claims contained in a recently published note

of G. Scorza-Dragoni [355], and to show in this way that, taking in account the

proof of prof. Brusotti [49] [...] all the existence theorems for boundary valueproblems for ordinary di!erential equations and systems of such equations

(of an arbitrary order) can be obtained by completely avoiding the use oftopology.

He expressed more clearly his opinion on p. 220, using this opportunity to scratchCaccioppoli as well :

Not only is it neither necessary to use the functional analytical considerationsdue to Birkho! and Kellogg (and rediscovered eigth years later by R. Cacciop-

poli, who was then forced to recognize the priority of the mentioned authors),nor necessary to use the topology of finite-dimensional spaces.

In this politically di$cult period, Cinquini added a little bit of nationalism :

But this is a matter of taste : we are pleased that our modest contribution

emphasizes the method of the school of Cesare Arzela; in contrast, of course,Mr. Scorza-Dragoni prefers to insist in valorizing the ideas of Birkho! and

Kellogg !

Cinquini ended his polemical arguments as follows :

We will not loose any time in the future in answering to Scorza-Dragoni, butwe want to notice that, taking in account our observation and Miranda’s result,

the Note of Brusotti provides an elementary proof of Brouwer’s theorem.

If the Second World War brought a welcome break in the polemics, nothing wasforgotten and, in 1946, a memoir of more than seventy pages of Scorza-Dragoni[356] answered point by point to Cinquini’s note [65]. About Cinquini’s conclusion,Scorza-Dragoni pertinently observed that

the fact that the proof of Brouwer’s theorem can be reduced to the case of

a polynomial mapping was already observed by Birkho! and Kellogg. Theproof they gave of Brouwer’s theorem is as elementary as the one Brusotti

gave of [Miranda’s] theorem I.

The same year, Scorza-Dragoni published three di!erent proofs of Miranda’s the-orem [357, 358, 359], respectively based upon Kronecker index, Birkho!-Kellog’smethod and a variant of Sperner’s lemma, and his student Giuseppe Zwirner gavea fourth one based upon Poincare-Bohl’s theorem [411]. One can consult[227] formore recent versions.

In front of this avalanche, Cinquini renounced in 1947 to his silence and insisted,on page 183 of [67], upon the fact that

Brusotti’s proof can be understood by anybody (including those ones who

do not know at all topology), because it neither uses topological concepts

(Kronecker index, etc.) nor topological statements (Poincare-Bohl’s theorem,etc.).


He mentioned, in a note on p. 176 of the same paper, the earlier use by Golomb[150] of Miranda’s theorem I :

Recently, I have been kindly informed that Proposition C [Miranda’s theoremI] had already been used in : M. Golomb, Zur Theorie der nichtlinearen Inte-

gralgleichungen etc.. (Math Zeitschrift, Bd 39 (1935) pp. 45-75, see p. 55).

There, Proposition C is mentioned as a consequence of Brouwer’s theorem.

Cinquini ended his paper with the unpublished proof communicated in 1940 toMiranda.

Scorza-Dragoni felt forced to anwer to [67] in 1949 [360], because

Cinquini has given new evidence of uncorrect polemics : he mutilates in fact

the sentences and the pages in an essential way and, after having changed the

meaning, he triumphantly answers. Cinquini again gives the impression thathis competence in each of the arguments in question has doubtful deepness.

Making fun of Cinquini’s confusion between Brouwer’s displacement and Brouwer’sfixed point theorems, Scorza Dragoni noticed that the unpublished proof of Mi-randa’s theorem for n = 2 given in [67] just reduced this result to a Lebesgue’stopological lemma in dimension theory.

Cinquini’s reaction [68] was just a new promise to stop answering to Scorza-Dragoni, and a ‘Postilla ad una nota di S. Cinquini’ of Scorza Dragoni [361] regis-tered this missed reply, and definitely ends the quarrel.

Miranda has reproduced his result in 1949, without further comments or ref-erences, in his interesting monograph [287]. After having developed, in chapterIII, following [6], the (local) topological degree dB[g,%, 0] of a continuous map-ping g : % " Rn, when % is open and bounded, and 0 %& g(!%), Miranda deducesBrouwer’s fixed point theorem by taking % = B, g = I ! f and noticing that0 %& (I ! "f)(!B) for each " & [0, 1[. The invariance of degree with respect tohomotopy implies that dB[I ! "f, int B, 0] = deg[I, int B, 0] = 1, which gives theexistence of a fixed point x% & int B of "f for each " & [0, 1[. A classical compact-ness argument implies the existence of a fixed point of f in B. This may be the firstappearance of this now classical proof. In the pages of [287] following his equivalenceproof, Miranda tells in a sober and diplomatic way the origin of his contribution,mentions the proofs of Cinquini, Brusotti, Scorza Dragoni and Zwirner and the ap-plications to boundary value problems by Cinquini, Zwirner and Stampacchia, butwithout including, in his rich bibliography, any of the polemical papers related tothe question.

The n-dimensional generalization of Bolzano’s intermediate value theorem hasbeen generally named Miranda’s theorem, until one of the authors of this bookcalled, in 1973, the attention to Poincare’s forgotten papers, and reestablished thetrue priority [264, 48]. One usually speaks now of Poincare-Miranda’s theorem.

The first direct application of Brouwer’s fixed point theorem to nonlinear dif-ferential equations appeared in 1943 in the United States, with a paper of SolomonLefschetz [245], immediately followed by a more general result of Norman Levinson


[252]. Their aim was to prove, when e has period T, the existence of a solution ofperiod T for some di!erential equations of the type

x$$ + f(x)x$ + g(x) = e(t), (23.5)

occuring in various questions of mechanics and electronics, The authors showed theexistence of a subset C of the phase plane (x, x$), homeomorphic to a closed disc,and such that any solution (x(t; x0, y0), x$(t; x0, y0)) of (23.5) with initial conditionsx(0) = x0, x$(0) = y0 located in C, remains in C for all t & [0, T ]. Brouwer’s fixedpoint theorem applied to Poincare’s operator

P : C " C, (x0, y0) 2" (x(T ; x0, y0), x$(T ; x0, y0))

implies the existence of a fixed point (x#0, y

#0) of P, giving the initial conditions of

a solution of period T for (23.5). When describing those results in 1951, GiovanniSansone [340] wrote :

Brouwer’s theorem has the highest importance in analysis : its proof and in-formations about the contributions of S. Cinquini, L. Brusotti, C. Miranda,

G. Scorza-Dragoni, G. Zwirner and on its applications to the study of bound-

ary value problems can be found in the excellent monograph of C. Miranda :Problemi di esistenza in analisi funzionale.

He reproduced the proof of Miranda’s equivalence result in his influential treatise onnonlinear di!erential equations [341], written in collaboration with Roberto Conti.

23.7 Variational inequalities

Brouwer’s fixed point theorem is closely related to the main result of the theoryof variationnal inequalities stated and proved in 1966 by Philip Hartman andGuido Stampacchia [169] (see also [37]) :

Let C be a compact convex set in En and B(u) a continuous map of Cinto En. Then there exists u0 & C such that (B(u0), v ! u0) ( 0 forv & C, where (·, ·) denotes the scalar product in En.

Hartman et Stampacchia reduced this result, except for the trivial case whereC is a singleton, to the case where 0 & int C. For !C regular, they used topologicaldegree, and reduced the general case, to the regular one using an approximationtheorem of Minkowski. When C is a simplex, Hartman-Stampacchia’s theorem wasindependently proved in 1966 by Stepan Karamardian [193].

Another proof based upon Brouwer’s fixed point theorem has been given byHaım Brezis (see [327]) : the projection mapping r : Rn " C is characterizedby the relation (x ! r(x), y ! r(x)) ) 0 for all x & Rn, y & C. Brouwer’s fixedpoint theorem applied to the continuous mapping f = r * (I ! g) : C " C givesa fixed point u0 & C. Taking x = u0 ! g(u0) in the inequality above, one gets

23.8. APPROXIMATION OF FIXED POINTS 299

(u0 ! g(u0) ! u0, y ! u0) ) 0 for all y & C, which is just Hartman-Stampacchia’sstatement.

The extensions of the theory of variational inequalies to infinite-dimensionalspaces have important applications to free boundary, obstacle, infiltration andelastoplastic problems [201]. The theory of variational inequalities is also relatedto the complementarity problem first considered in 1966 by Richard W. Cottle[75] :

Given a continuous mapping g : Rn+ " Rn, find x# & R such that

g(x#) & Rn+ et 6g(x#), x#7 = 0.

This problem unifies various questions coming from mathematical programming,game theory, economical equilibrium theory, mechanics and geometry. Karamardian[194, 195, 196] and Jorge J. More [288] have reduced to complementarity problemto a variational inequality. For more details, see [76].

23.8 Approximation of fixed points

For a long time, it was commonly admitted that Brouwer’s fixed point theorembelonged to non constructive mathematics because it did not provide any algorithmto get the fixed point [16, 200]. This dogma was broken by Herbert Scarf in 1967[342, 343, 344], when, motivated by some applications to mathematical economics,he proposed, for a continuous mapping f : S " S from a simplex into itself, asystematic algorithm to obtain sub-simplices $(+) of diameter + > 0, for which,roughly speaking, lim1(0 $(+) = x and x = f(x). Scarf’s method is based upon avariant of Sperner’s lemma. The original simplex is decomposed in a simplicial wayand the method provides a path along the vertices of the simplicial decomposition,ending to a fixed point of a piece-wise linear approximation of f. One can illustrateScarf’s algorithm in the case of the triangulation Tk of the simplex S made of theconvex hull of the three points A, B et C considered previously. If (*1,*2,*3) arethe barycentric coordinates of P and (*#

1,*#2,*

#3) the barycentric coordinates of

f(P ), let us associate to P the smallest j & {1, 2, 3} such that *j = 0 if it exists,and, otherwise, the smallest i & {1, 2, 3} such that *#

i ( *i. In this way, A isnumbered by 2, B and C by 1, the other elements of AB by 3, of BC by 1 and ofAC by 2. Consider the elementary triangles of Tk like rooms, any side numbered12 like a door and any other side like a wall. With the convention that a door canonly be crossed once, starting from the only door opened to the outside of S, andgoing through the possible elementary triangles having two doors, one necessarilyreaches a triangle having only one door, and hence numbered by 1,2 et 3. Theremaining of the reasoning is similar to the proof of Brouwer’s fixed point theoremusing Sperner’s lemma, except that, numerically, it is su$cient to obtain a fixedpoint with an error of 6 > 0, provided by taking k su$ciently large. This work hasinspired, in the last thirty years, under the name of simplicial methods, a largenumber of variants and generalizations. On can consult [8] and its references.

A di!erent algorithm was proposed in 1974 by R. Kellogg, T. Li et J. Yorke[199] (see also [367]), when f : C " C and the compact convex set C are regular.


The idea is linked to that connecting Brouwer’s fixed point theorem to the non-retraction theorem. For any x not belonging to the set of fixed points of f in C,denote by r(x), the intersection with !C of the half-line issued from f(x) and goingthrough x. If x0 is a regular value of r, the component * of r!1(x0) containing x0

is a curve s " x(s) which can be parametrized by its length. This curve * cannotcontain any other extremity than x0, and hence there exists x# & * \ * for whichr is not defined, which implies that f(x#) = x#. As r(x(s)) > x0 for all x(s) & *,x(s) is solution of the initial value problem

r$(x(s))dx(s)

ds= 0, x(0) = x0,

[dx

ds(s),

dx

ds(s)

\= 1,

on a maximal interval [0, s#[, s# ) -. An algorithm for the fixed point of f is ob-tained by solving approximatively the initial value problem, giving approximationsfor the singular point x# of this problem. Several variants of this continuationmethod for obtaining a fixed point have been developed, for example the methodof homotopic continuation [4, 61].

23.9 Elementary proofs and equivalent formula-

tions

Brouwer’s fixed point theorem is a result located at the crossing of several im-portant mathematical disciplines, which can and has be proved using techniquesfrom algebraic topology [2], analysis [97, 132, 297, 302, 327], di!erential topology[156, 182, 282], combinatorics [105, 153, 230, 231], algebraic geometry [49] andeven algebra [316]. It has many equivalent formulations, looking quite di!erent.and numerous and diverse applications, going from algebraic topology (proofs ofnon-retraction theorems, Jordan separation result [226, 257], invariance of domain[228]), to mathematical biology [401], via geometry of convex bodies [208], di!er-ential equations [296, 341], control theory [51, 167, 166], game theory [14, 102],nonlinear programming [288], decision theory [364], combinatorics [134, 135] andmathematical economy [14, 103]. Numerical analysis itself, after a long period ofignoration, is now more than interested [9, 350, 387].

The importance and increasing scope of the applications of Brouwer’s fixed pointtheorem have motivated the search of elementary proofs. Knaster-Kuratowski-Ma-zurkiewicz’s one surely fills this criterion, but is based upon combinatory consider-ations which can look unusual to some users. One already finds other ‘elementary’proofs in the years 1930, and Cinquini-Scorza Dragoni’s quarrel is akin to this prob-lematics.

A large number of proofs have been proposed until recently, which are based onmore or less complicated results of mathematical analysis. Those proofs are veryoften ‘elementary’ looking translations of arguments based upon the integration ofsome di!erential forms, already implicitely contained in the definition of Kroneckerindex. Some proofs are based upon the comparison of integrals, as done already byBohl [154, 192, 225, 283, 332, 97], other ones use the change of variables formula

23.9. ELEMENTARY PROOFS AND EQUIVALENT FORMULATIONS 301

for multiple integrals [15, 242, 246], di!erential forms [179, 180, 260, 274, 339, 389],arguments of di!erential topology [27, 282] or algebraic topology [33], simplicialapproximations [173] (corrected in [188]), polynomial approximations [223, 363], orcontinuation methods [255].

We have seen the equivalence of Brouwer’s fixed point theorem with a numberof other statements. This study can be pushed further, and we refer to [159, 313,314, 327, 390, 407, 408, 409].

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Contents

1 Introduction 3

2 Kronecker index 72.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 The planar case : winding number . . . . . . . . . . . . . . . . . . . 82.3 The case where f : !D ' Rn " Sn!1 . . . . . . . . . . . . . . . . . . 102.4 A minimum problem from Ginzburg-Landau’s theory . . . . . . . . . 122.5 An exact form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.6 An equivalent formula . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Brouwer degree 173.1 Smooth mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 An exact form and homotopy invariance . . . . . . . . . . . . . . . . 203.3 Continuous mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.5 The case where f : !D ' Rn " Sn!1 . . . . . . . . . . . . . . . . . . 26

4 Uniqueness of degree 294.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Reduction to the linear case . . . . . . . . . . . . . . . . . . . . . . . 294.3 Uniqueness for linear mappings . . . . . . . . . . . . . . . . . . . . . 31

5 Brouwer index 375.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.2 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.3 The Brouwer index at infinity . . . . . . . . . . . . . . . . . . . . . . 40

6 Degree in finite-dimensional vector spaces 436.1 Composition with linear isomorphisms . . . . . . . . . . . . . . . . . 436.2 Definitions and basic properties . . . . . . . . . . . . . . . . . . . . . 446.3 The Brouwer index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.4 Reduction formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.5 Computation of the index in degenerate case . . . . . . . . . . . . . 48

331

332 CONTENTS

7 Continuation theorems 537.1 Extended homotopy invariance property . . . . . . . . . . . . . . . . 537.2 Leray-Schauder continuation theorem . . . . . . . . . . . . . . . . . 547.3 The case of an unbounded set . . . . . . . . . . . . . . . . . . . . . . 56

8 Fixed points and zeros 598.1 Semilinear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 598.2 Brouwer’s fixed point theorem and extensions . . . . . . . . . . . . . 618.3 Non-existence theorems . . . . . . . . . . . . . . . . . . . . . . . . . 638.4 Zeros of continuous mappings . . . . . . . . . . . . . . . . . . . . . . 648.5 A Brouwer-Poincare-Bohl’s theorem . . . . . . . . . . . . . . . . . . 668.6 Perturbed non-invertible linear mappings . . . . . . . . . . . . . . . 68

9 Fixed points and variational inequalities 699.1 Browder’s fixed point theorem . . . . . . . . . . . . . . . . . . . . . . 699.2 Variational inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 709.3 Hemivariational inequalities . . . . . . . . . . . . . . . . . . . . . . . 739.4 Ky Fan fixed point theorem and consequences . . . . . . . . . . . . . 799.5 Applications to game theory . . . . . . . . . . . . . . . . . . . . . . . 82

10 Equations in reflexive Banach spaces 8710.1 Monotone mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . 8710.2 Browder-Minty’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 8910.3 A monotone quasilinear Dirichlet problem . . . . . . . . . . . . . . . 9210.4 A non-monotone quasilinear Dirichlet problem . . . . . . . . . . . . 9410.5 Homogeneous Navier-Stokes equations . . . . . . . . . . . . . . . . . 9610.6 Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . 9710.7 Existence of a solution . . . . . . . . . . . . . . . . . . . . . . . . . . 10010.8 Uniqueness conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 102

11 First order di!erence equations 10511.1 Periodic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10511.2 Bounded nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . 10611.3 Upper and lower solutions . . . . . . . . . . . . . . . . . . . . . . . . 10711.4 Spectrum of the linear part . . . . . . . . . . . . . . . . . . . . . . . 10911.5 Reversing the order of upper and lower solutions . . . . . . . . . . . 11011.6 Ambrosetti-Prodi type multiplicity result . . . . . . . . . . . . . . . 11211.7 One-sided bounded nonlinearities . . . . . . . . . . . . . . . . . . . . 114

12 Second order di!erence equations 11912.1 Dirichlet problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11912.2 Spectrum of the linear part . . . . . . . . . . . . . . . . . . . . . . . 12012.3 Bounded nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . 12112.4 Upper and lower solutions . . . . . . . . . . . . . . . . . . . . . . . . 12212.5 Ambrosetti-Prodi type multiplicity results . . . . . . . . . . . . . . . 12412.6 One-sided bounded nonlinearities . . . . . . . . . . . . . . . . . . . . 127

CONTENTS 333

13 Bifurcation 13113.1 Eigenvalues of couples of linear mappings . . . . . . . . . . . . . . . 13113.2 A Leray-Schauder formula for Brouwer index . . . . . . . . . . . . . 13413.3 Bifurcation points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13613.4 Bifurcation through linearization . . . . . . . . . . . . . . . . . . . . 13713.5 Global structure of bifurcation branches . . . . . . . . . . . . . . . . 14013.6 An application to di!erence equations . . . . . . . . . . . . . . . . . 143

14 Poincare’s operator 14714.1 Initial value and periodic problems . . . . . . . . . . . . . . . . . . . 14714.2 Bounded perturbations of some linear systems . . . . . . . . . . . . . 14814.3 Stampacchia’s method . . . . . . . . . . . . . . . . . . . . . . . . . . 15014.4 The method of lower and upper solutions . . . . . . . . . . . . . . . 15214.5 Strict lower and upper solutions . . . . . . . . . . . . . . . . . . . . . 15414.6 Krasnosel’skii-Perov’s existence theorem . . . . . . . . . . . . . . . . 15714.7 Degree of gradient mappings . . . . . . . . . . . . . . . . . . . . . . 158

15 Stability and index of periodic solutions 16315.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16315.2 Planar periodic systems . . . . . . . . . . . . . . . . . . . . . . . . . 16315.3 The case of a second order di!erential equation . . . . . . . . . . . . 16615.4 Second order equations with convex nonlinearity . . . . . . . . . . . 16915.5 Periodic solutions of pendulum-type equations . . . . . . . . . . . . 174

16 Guiding functions 17916.1 Definition and preliminaries . . . . . . . . . . . . . . . . . . . . . . . 17916.2 Systems with continuable solutions . . . . . . . . . . . . . . . . . . . 18116.3 Systems with non-continuable solutions . . . . . . . . . . . . . . . . 18516.4 Generalized guiding functions . . . . . . . . . . . . . . . . . . . . . . 189

17 Computing degree in dimension two 19317.1 The Kronecker index on a closed simple curve . . . . . . . . . . . . . 19317.2 Factorized multilinear mappings . . . . . . . . . . . . . . . . . . . . 19617.3 Using Sturm sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 20117.4 Holomorphic functions . . . . . . . . . . . . . . . . . . . . . . . . . . 20417.5 An application to stability and control . . . . . . . . . . . . . . . . . 207

18 Periodic solutions of some planar systems 21318.1 Autonomous systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 21318.2 Forced system : nonresonance . . . . . . . . . . . . . . . . . . . . . . 21418.3 Forced system : equivalent formulation . . . . . . . . . . . . . . . . . 21718.4 Forced system : resonance . . . . . . . . . . . . . . . . . . . . . . . . 21918.5 Asymmetric piecewise-linear oscillators . . . . . . . . . . . . . . . . . 223

334 CONTENTS

19 Computing degree in higher dimensions 22719.1 Cartesian products of mappings . . . . . . . . . . . . . . . . . . . . . 22719.2 Homogeneous polynomials mappings . . . . . . . . . . . . . . . . . . 22919.3 Orientation preserving mappings . . . . . . . . . . . . . . . . . . . . 23119.4 Monotone mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . 23419.5 Holomorphic mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 23519.6 Quaternionic monomials . . . . . . . . . . . . . . . . . . . . . . . . . 244

20 Degree of some symmetric mappings 24920.1 Odd mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24920.2 Borsuk-Ulam’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . 25220.3 Elliptic di!erential operators . . . . . . . . . . . . . . . . . . . . . . 25320.4 Lusternik-Schnirel’mann’s covering theorem . . . . . . . . . . . . . . 25420.5 Measure of non-compactness of the unit sphere in a Banach space . . 25520.6 Genus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25720.7 Isometric representations of S1 over Rn . . . . . . . . . . . . . . . . 26020.8 S1-equivariant mappings . . . . . . . . . . . . . . . . . . . . . . . . . 261

21 Leray’s product formula 26721.1 The product formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 26721.2 Proof in the smooth case . . . . . . . . . . . . . . . . . . . . . . . . . 26821.3 Proof in the continuous case . . . . . . . . . . . . . . . . . . . . . . . 26821.4 Jordan-Brouwer’s separation theorem . . . . . . . . . . . . . . . . . . 270

22 Invariance of domain and applications 27322.1 Invariance of domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 27322.2 A surjectivity result for one-to-one coercive mappings . . . . . . . . 27522.3 Degree of one-to-one mappings . . . . . . . . . . . . . . . . . . . . . 27522.4 Deformation of an elastic body . . . . . . . . . . . . . . . . . . . . . 27622.5 Injectivity of some orientation-preserving mappings . . . . . . . . . . 278

23 A history of Brouwer fixed point theorem 28123.1 The discovery and the publication . . . . . . . . . . . . . . . . . . . 28123.2 Early equivalent formulations . . . . . . . . . . . . . . . . . . . . . . 28423.3 New proofs and infinite-dimensional extensions . . . . . . . . . . . . 28823.4 A combinatorial approach . . . . . . . . . . . . . . . . . . . . . . . . 28923.5 Game theory and mathematical economics . . . . . . . . . . . . . . . 29123.6 Poincare’s theorem : topology or not topology . . . . . . . . . . . . . 29323.7 Variational inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 29823.8 Approximation of fixed points . . . . . . . . . . . . . . . . . . . . . . 29923.9 Elementary proofs and equivalent formulations . . . . . . . . . . . . 300

Brouwer Degree and Applications - sorbonne-universite · Inﬂuenced by Brouwer’s construction...

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Transcript of Brouwer Degree and Applications - sorbonne-universite · Inﬂuenced by Brouwer’s construction...