1. Introduction
The Bolzano theorem for continuous functions
, which states that
f has a zero in if , was first proved in 1817 by Bolzano [
1] and, independently and differently in 1821 by Cauchy [
2]. Its various proofs are not very long, and depend only upon the order and completeness properties of
. A consequence of the Bolzano theorem applied to
is that
, continuous, has a fixed point in if and . This is the case if
.
As is the closed ball of center 0 and radius R in , a natural question is to know if, denoting the closed ball of center 0 and radius , any continuous mapping such that has a fixed point, and, in particular, if any continuous mapping has a fixed point. The answer is yes, and the first result, usually called the Rothe fixed point theorem (FPT), is more correctly referred as the Birkhoff–Kellog FPT, and the second one as the Brouwer FPT.
Many different proofs of those results have been given since the first published one of the Brouwer FPT by Hadamard in 1910 [
3]. Brouwer’s original proof [
4], published in 1912, was topological and based on some fixed point theorems on spheres proved with the help of the topological degree introduced in the same paper. The Birkhoff–Kellogg FPT was first proved by Birkhoff and Kellogg in 1922 [
5]. Its standard name Rothe FPT refers to its extension to Banach spaces by Rothe [
6] in 1937.
The existing proofs use ideas from various areas of mathematics such as algebraic topology, combinatorics, differential topology, analysis, algebraic geometry, and even mathematical economics. A survey and a bibliography can be found in [
7]. Even for
, they cease to be elementary and/or can be technically complicated. The aim of this paper is to survey recent results on some elementary approaches to the Birkhoff–Kellogg and Brouwer FPT, and on how to deduce from them in a simple and systematic way other fixed point and existence theorems for mappings in
. Recall that these results, combined with basic facts of functional analysis, are fundamental in obtaining useful extensions to some classes of mappings in infinite-dimensional normed spaces.
After recalling the simple concept of curvilinear integral in
, we first propose in
Section 2 an elementary proof of the Birkhoff–Kellogg FPT for
, based upon such integrals. As the extension to arbitrary
n, using differential
forms in
, leads to very cumbersome computations, we adopt in
Section 3 a variant given in [
8], using differential
n-forms, which in dimension
n happens to be significantly simpler than the direct extension of the approach of
Section 2.
The generalizations of the Birkhoff–Kellogg and Brouwer FPT to a closed ball in
and their homeomorphic images are stated in
Section 4. After the concepts of retract and retraction are introduced, the Leray–Schauder–Schaefer FPT on a closed ball is deduced from the Brouwer FPT, whose statement is also extended to retracts of a closed ball in
. Finally, the equivalence of the Birkhoff–Kellogg and Brouwer FPT on a closed ball is established.
The Brouwer FPT and retractions are then used in
Section 5 to prove, in a very simple and unified way inspired by the approach of [
9], several conditions for the existence of zeros continuous mappings in
, namely the Poincaré–Bohl theorem on a closed ball, the Hadamard theorem on a compact convex set, the Poincaré–Miranda theorem on a closed
n-interval, and the Hartman–Stampacchia theorem on variational inequalities.
Finally, in
Section 6, following the method introduced in [
10], simple versions of the Cauchy integral theorem provide criterions for the existence of zeros of a holomorphic function in same spirit of the approach in
Section 2. They allow very simple proofs of the Hadamard and Poincaré–Miranda theorems and of the Birkhoff–Kellogg and Brouwer FPT for holomorphic functions.
2. A Proof the Birkhoff–Kellogg Theorem on a Closed Disc Based on Curvilinear Integrals
Let be open and nonempty and let denote the usual inner product in . Given and of class , we consider the corresponding curvilinear integral defined by where denotes the derivative with respect to t.
The following result is fundamental for our proof of the Birkhoff–Kellogg FPT on a closed disc.
Lemma 1. Ifis of classand such thatand ifis of classand such thatfor all, thenis constant on.
Proof. It suffices to prove that
for all
. We have, with differentiation under integral sign easily justified and the use of assumptions, the Schwarz theorem and the fundamental theorem of calculus, and omitting the arguments
for the sake of brevity
□
Let , with the Euclidian norm. We prove the Birkhoff–Kellogg FPT on a closed disc.
Theorem 1. Any continuous mappingsuch thathas a fixed point in.
Proof. Assume that
T has no fixed point in
. Then,
for all
, and, as
,
for all
. Similarly,
for all
. As
T is continuous, there exists
such that
and
for all
. From the Weierstrass approximation theorem, there is a polynomial
such that
for all
Consequently, letting
and
we have, for all
,
and
. Hence, there exists an open neighborhood
of
such that
and
for all
. If
then
If
is a parametric representation of
, so that
, it follows from Lemma 1 that the integrals
are constant for
. Hence, noticing that
,
However, as
is constant and
,
a contradiction. □
A direct consequence is the Brouwer FPT on a closed disc.
Corollary 1. Any continuous mappinghas a fixed point in.
3. A Proof of the Birkhoff–Kellogg Theorem on a Closed n-Ball Based on Differential n-Forms
The argument used in
Section 2 for mappings in
can be extended to mappings in
, using the basic properties of differential
k-forms in
. For
, the differential 1-forms and differential
-forms coincide, and it is the last ones that are requested for extending the proof of Theorem 1 to arbitrary
n. We leave to the motivated reader the work to write down this extension of the first approach and to realize that this generalization to dimension
n of Lemma 1 is very cumbersome and lengthy. Fortunately a similar approach based on differential
n-forms instead of
-forms has been introduced in [
8], which, for
, has the same length and technicality as the one used in
Section 2, but keeps its simplicity for arbitrary
n. We describe it in this section.
For
open, bounded and nonempty, we need the concept of differential
-forms and
n-forms and suppose that the reader is familiar with the notions, notations and properties of differential
k-forms
on
D, wedge products, pull backs, exterior differentials and the Stokes–Cartan theorem for differential forms with compact support [
11]. All the functions involved in differential forms are supposed to be of class
. We associate to the functions
the
differential 1-form in D, and the
differential -form
where
means that the corresponding term is missing. We associate also to
the
differential n-form . For example, given the function
with partial derivatives
, its
differential is the differential 1-form
.
Let
be open, bounded and nonempty,
. For each fixed
,
is well defined. To shorten the notations, we write
for
. We define the
derivative with respect to of
by
so that
Furthermore,
On the other hand,
where
denotes the Jacobian of
at
and
The following two results replace Lemma 1 in
Section 2. The first one shows that the differential
n-form
is exact in
, i.e., is the exterior differential of a
-differential form in
.
Lemma 2. For eachwe have Corollary 2. If, Δ is open, bounded and verify , then is independent of λ on
Proof. Using Lemma 2, the assumption and Stokes–Cartan theorem, we get
□
Let with the Euclidian norm. We now show that Proposition 2 allows a simple proof of the Birkhoff–Kellogg FPT on a closed n-ball, quite similar to that of Theorem 1.
Theorem 2. Any continuous mappingsuch thathas a fixed point in.
Proof. Assume that
T has no fixed point in
Then,
for
and for
we have
Thus,
for all
. On the other hand, for
we have
and hence
for all
. By continuity, there exists
such that
for all
. Let
be a polynomial such that
and define
and
by
and
so that
and
for all
. Let
with supp
, the open ball of center 0 and radius
, and
Then, by Proposition 2 with
, we get
and
a contradiction. □
The Brouwer FPT on a closed n-ball is a special case.
Corollary 3. Any continuous mappinghas a fixed point in.
4. Fixed Points, Homeomorphisms and Retractions in
Now, if , if there exists a homeomorphism , and if is continuous, is continuous, has a fixed point by Theorem 3, and is a fixed point of T. Consequently, we have a Brouwer FPT for homeomorphic images of a closed n-ball.
Theorem 3. Ifis homeomorphic to, any continuous mappinghas a fixed point in K.
For example, K can be any closed n-interval , or an n-simplex
Remark 1. In Theorem 3, the boundedness assumption on K cannot be omitted: a translationinwithhas no fixed point. The closedness assumption on K cannot be omitted as well:has no fixed point in. Theorem 3 does not hold for any closed bounded set: a nontrivial rotation of the closed annulushas no fixed point in A.
We now introduce concepts and results due to Borsuk [
12] which provide another class of sets on which the Brouwer FPT holds and simple proofs of various equivalent formulations of this theorem. We say that
is a
retract of
V if there exists a continuous mapping
such that
on
U (
retraction of V in U). For example,
is a retract of
, with a retraction
r given by
Similarly, for any , is a retract of .
Remark 2. The Brouwer FPT onimplies the Birkhoff–Kellogg FPT on. Indeed, ifis continuous,, and r is given by (1), thenis continuous and, by the Brouwer FPT 3, has a fixed point. If,and, a contradiction. Thus,and. Thus, the two statements are equivalent. Remark 3. The Brouwer FPT has for immediate topological consequence the well-knownno-retraction theorem, stating thatis not a retract ofin. We do not repeat here the simple proof of this result and the proof of Brouwer FPT from the no-retraction theorem.
An easy consequence of Theorem 3 is the
Leray–Schauder–Schaefer fixed point theorem, a special case of a more general result obtained in 1934 by Leray and Schauder [
13]. The proof given here is due to Schaefer [
14].
Theorem 4. Any continuous mappingsuch thatfor allhas a fixed point in.
Proof. Let
be the retraction of
onto
defined in Equation (
1). Theorem 3 implies the existence of
such that
. If
, then
, so that
and
with
, a contradiction with the assumption. Hence,
and
. □
Remark 4. If , it is clear that the assumption of Theorem 4 is satisfied. Thus the Leray–Schauder–Schaefer FPT implies the Birkhoff–Kellogg FPT, and hence the two statements are quivalent.
The Brouwer FPT holds for retracts of a closed ball.
Theorem 5. Ifis a retract of, any continuous mappinghas a fixed point.
Proof. Let for some retraction . Then, has a fixed point . Hence, , and . □
If
is non- empty, closed and convex, the
orthogonal projection on C of
, defined by
, is a retraction of
onto
C [
15]. Consequently,
C is a retract of any , giving a
Brouwer FPT on compact convex sets.
Corollary 4. Ifis compact and convex, any continuous mappinghas a fixed point in C.
5. Zeros of Continuous Mappings in
The first theorem on the existence of a zero for a mapping from
into
was first stated and proved for
mappings by Bohl [
16] in 1904, and extended to continuous mappings by Hadamard in 1910 [
3], under the name
Poincaré–Bohl theorem. It is a reformulation of the Leray–Schauder–Schaefer FPT Theorem 4.
Theorem 6. Any continuous mappingsuch thatfor alland for allhas a zero in.
Proof. Define the continuous mapping
by
. For
, we have, by assumption,
By Theorem 4, T has a fixed point in , which is a zero of f. □
In 1910, two years before the publication of [
4], Hadamard, informed by a letter from Brouwer of the statement of his fixed point theorem, published a simple proof based on the Kronecker index (a forerunner of the Brouwer topological degree) in an appendix to an introductory analysis book of Tannery [
3]. Hadamard’s proof consisted in showing that Brouwer’s assumption implies that the condition
holds for all
, where
denotes the usual inner product in
. This condition implies the existence of a zero of
, because the assumption of the Poincaré–Bohl theorem 6 is satisfied. Hadamard’s reasoning using the Kronecker index does not depend upon the special structure
of the mapping in the inner product. Hence, it is natural (although not usual) to call
Hadamard theorem the statement of existence of a zero for a continuous mapping
, when
is replaced by
in the inequality above, a statement which became in the year 1960 a key ingredient in the theory of monotone operators in reflexive Banach spaces. Using convex analysis, we give an extension to compact convex sets.
Let
be compact and convex and
be the orthogonal projection of
x on
C [
15]. Recall that
is characterized by the condition
For
, the set
is nonempty and called the
normal cone to
C at
x, and its elements
are called the
outer normals to
C at
x. The relation in Equation (
2) shows that, for each
,
. It can also be shown that each
is the orthogonal projection of some
, so that
The
Hadamard theorem on a convex compact set follows in a similar way as Theorem 6 from the Brouwer FPT 3.
Theorem 7. Ifis a compact and convex, any continuoussuch thatfor alland allhas a zero in C.
Proof. Let
be defined by
. Then, for all
,
and
T maps
into itself. By Theorem 3, there exists
such that
. If
, the assumption implies that
a contradiction. Thus,
,
and
. □
Corollary 5. Any continuous mappingsuch thatfor allhas a zero in.
Proof. For each , , and we apply Theorem 7. □
Remark 5. As shown when mentioning Hadamard’s contribution, Theorem 5 implies the Brouwer FPT, and even the Birkhoff–Kellopg FPT, on. Consequently, those statements are equivalent.
Some twenty years before the publication of Brouwer’s paper [
4], Poincaré [
17] stated in 1883 a theorem about the existence of a zero of a continuous mapping
when, for each
,
takes opposite signs on the opposite faces of
PPoincaré’s proof just told that the result was a consequence of the Kronecker index, which is correct but sketchy. The statement, forgotten for a while, was rediscovered by Cinquini [
18] in 1940 with an inconclusive proof, and shown to be equivalent to the Brouwer FPT on
P one year later by Miranda [
19]. Many other proofs have been given since, and we again refer to [
7,
20] for a more complete history, variations and references, and to [
21,
22,
23] for useful generalizations to more complicated sets than closed
n-intervals. Here, we obtain the
Poincaré–Miranda theorem on a closed n-interval as a special case of Theorem 7.
Corollary 6. Any continuous mappingsuch thatfor allandfor allhas a zero in P.
Proof. If x is in the (relative) interior of the face , then , where is the orthonormal basis in , and the assumption of Theorem 7 becomes , i.e., . Similarly, if x is in the (relative) interior to the face , then , and the assumption of Theorem 7 becomes . Of course, and also belong to the respective normal cones for and respectively, and if, say, then for all , and . In general, when x belongs to the intersection of several faces of P, will be made of the linear combination of the corresponding to the indices of the faces, with a negative coefficient for a face having symbol − and positive coefficient for a face having symbol +, so that, using the assumption, for all and all . The result follows from Theorem 7. □
Remark 6. Corollary 6 implies the Brouwer FPT on P. Indeed, ifis continuous, and if we set, then, asfor all, we have, forsuch that,and, forsuch that,. Thus f has at least one zero in P, which is a fixed point of T. Consequently, the two statements are equivalent.
Remark 7. Both the Hadamard theorem onand the Poincaré–Miranda theorem can be seen as distinct n-dimensional generalizations of the Bolzano theorem to closed ball and n-intervals respectively.
Remark 8. Using the Brouwer degree, it is easy to obtain the conclusion of the Hadamard Theorem 7 for a compact convex neighborhood of 0 under the weaker condition that for eachthere existssuch that. No proof based only upon the Brouwer FPT seems to be known.
If
is a compact convex set and
is of class
, then
g reaches its minimum on
C at some
for which
so that, dividing both members by
and letting
, we obtain
where
denotes the gradient of
g. For example, if
is fixed and
is defined by
, the minimization problem corresponds to the definition of
, and, as
, the inequality above is just Equation (
2). In 1966, Hartman and Stampacchia [
24] proved that the existence of such a
still holds when
is replaced by an arbitrary continuous function
. When
C is a simplex, the same result was proved independently the same year by Karamardian [
25]. We give here a proof, due to Brezis (see [
26]) and based upon Brouwer’s FPT, of the
Hartman–Stampacchia theorem on variational inequalities.
Theorem 8. Ifis compact, convex andcontinuous, there existssuch thatfor all.
Proof. The Brouwer FPT on
C (Corollary 4) applied to the continuous mapping
implies the existence of
such that
Taking
in Equation (
2) and using Equation (
3), one gets
which is the requested inequality. □
Remark 9. The conclusion of Theorem 8 is called avariational inequality. In the terminology of the theory of convex sets [15], the conclusion of Theorem 8 means that there existssuch that eitherorandis asupporting hyperplanefor C passing throughi.e., C is entirely contained in one of the two closed half-spaces determined by H. Remark 10. The Brouwer FPT on C (Corollary 4) also follows from the Hartman–Stampacchia theorem. Indeed, ifis continuous andis given by Theorem 8 applied to, then, takingin the variational inequality, we obtainso that. Hence, the two statements are equivalent.
Remark 11. Ifandare two distinct solutions of the variational inequality, thenand hence. Consequently, the variational inequality has a unique solution if f satisfies the conditionfor all, i.e., if
f is
strictly monotone on
C.
6. A Direct Approach for Holomorphic Functions in
The assumption of the Bolzano theorem for a continuous function
can be, without loss of generality, be written
or, equivalently,
In 1982, Shih [
27] proposed a version of the Bolzano theorem for a complex function
f holomorphic on a suitable bounded open neighborhood
of 0 and continuous on
. He showed that
f has a unique zero in
when
on
, using the Rouché theorem applied to
and
for a suitable real
. As
, Shih’s condition is just Hadamard’s one in Theorem 5 with strict inequality sign. Following the approach introduced in [
10], we show in this section that, when the (non strict) Hadamard condition holds on the boundary of a ball, the existence of a zero of a holomorphic function results in a very simple way from an immediate consequence of the Cauchy integral theorem. The same is true for a Poincaré–Miranda theorem on a rectangle, giving another extension of the Bolzano theorem to complex functions. The Brouwer’s FPT for holomorphic functions on a closed ball or a closed rectangle follow immediately.
We suppose the reader familiar with the concepts of
holomorphic function f,
piecewise cycle , and
integral of f along [
28]. We denote by
the open disc of center 0 and radius
in
, and by
the corresponding closed disc. Let
be the standard
-cycle whose image is
. The
Cauchy integral theorem on a circle is proved here in a simple way, reminiscent of Cauchy’s proof in 1825 [
29], reworked by Falk in 1883 [
30], and similar in spirit to the proof of Lemma 2.
Proposition 1. Ifis continuous onand holomorphic on, then.
Proof. Define
by
, so that
and
is the constant zero mapping. To show that
is constant in
, we have (with differentiation under the integral sign easily justified and
denoting the derivative with respect to
z)
By continuity,
is constant in
, and hence
□
Let
be the corresponding closed rectangle in
, and let us introduce the continuous mapping
of class
on
defined by
whose image
. We state and prove the
Cauchy’s integral theorem on the boundary of a rectangle.
Proposition 2. Ifis continuous on P and holomorphic on int, then.
Proof. It is entirely similar to that of Proposition 1. If we define by , the integral has to be decomposed into four integrals over , , , and , respectively, of , and each integral has to be differentiated with respect to separately. The details are left to the reader. □
Propositions 1 and 2 immediately imply the following simple theorem for the existence of a zero of f.
Proposition 3. Any function(respectively,) holomorphic on(respectively, int), continuous on(respectively, P), different from zero on(respectively,) and such thathas a zero in(respectively, P). Proof. It is entirely similar in both cases and we prove it for . If f has no zero in , then is holomorphic on and continuous on . By Proposition 1, , a contradiction to the assumption. □
Proposition 3 provides a very simple proof of the Hadamard theorem for a holomorphic function on .
Theorem 9. Any functionholomorphic on, continuous onand such thatfor all, has a zero in.
Proof. For each integer
, define
by
Each
has the regularity properties of
f and is such that, for any
,
, so that
for all
, and
By Proposition 3, for each
,
has a zero
in
, and, by the Bolzano–Weierstrass theorem, a subsequence
of
converges to some
such that
. □
The Birkhoff–Kellog FPT for a holomorphic function on a disc is a direct consequence of Theorem 9.
Corollary 7. Any functioncontinuous on, holomorphic onand such thathas a fixed point in.
Proof. For each , one has . □
Example 1. For any integer, the mapping T defined byis such that for,. There is no uniqueness as T has the fixed points 0 and 1 in .
Let , , and be the opposite vertical and horizontal sides of P, respectively. Proposition 3 provides a Poincaré–Miranda theorem for a holomorphic function on a rectangle.
Theorem 10. Any functioncontinuous on P, holomorphic on intand such thatfor all,for all,for allandfor allhas a zero in P.
Proof. For each integer
, the function
defined on
P by
is such that
for
,
for
,
for
, and
for
. Hence,
for each
. Let
be the cycle defined by Equation (
4). By the assumptions,
For
, Proposition 3 implies the existence of
such that
. Using the Bolzano–Weierstrass theorem, a subsequence
converges to some
such that
. □
Example 2. Let the holomorphic functionbe defined by. Taking, one hasand f has a zero in. A direct consequence of Theorem 10 is the Birkhoff–Kellogg FPT for a holomorphic function on a rectangle.
Corollary 8. Any functioncontinuous on P, holomorphic on int, and such thathas a fixed point in P.
Proof. Define by for all . The assumption is equivalent to and for all , and, hence, if , , if , , if , , and if , . Thus, by Theorem 10, f has a zero in P and T a fixed point in P. □