Variations on the Brouwer Fixed Point Theorem: A Survey

: This paper surveys some recent simple proofs of various ﬁxed point and existence theorems for continuous mappings in R n . The main tools are basic facts of the exterior calculus and the use of retractions. The special case of holomorphic functions is considered, based only on the Cauchy integral theorem.


Introduction
The Bolzano theorem for continuous functions f : [a, b] ⊂ R → R, which states that f has a zero in [a, b] if f (a) f (b) ≤ 0, 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 R. As [−R, R] is the closed ball of center 0 and radius R in R, a natural question is to know if, B R denoting the closed ball B R ⊂ R n of center 0 and radius R > 0, any continuous mapping T : B R → R n such that T(∂B R ) ⊂ B R has a fixed point, and, in particular, if any continuous mapping T : B R → B R 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 n = 2, 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 R n . 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 R 2 , we first propose in Section 2 an elementary proof of the Birkhoff-Kellogg FPT for n = 2, based upon such integrals. As the extension to arbitrary n, using differential (n − 1) forms in R n , 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 R n 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 R n . 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 R n , 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.

A Proof the Birkhoff-Kellogg Theorem on a Closed Disc Based on Curvilinear Integrals
Let D ⊂ R 2 be open and nonempty and let ·, · denote the usual inner product in R 2 .
The following result is fundamental for our proof of the Birkhoff-Kellogg FPT on a closed disc.
Proof. It suffices to prove that ∂ λ b a f (Φ(t, λ), ∂ t Φ(t, λ) dt = 0 for all λ ∈ [0, 1]. 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 (t, λ) for the sake of brevity Let B R := {x ∈ R 2 : |x| ≤ R}, with |x| the Euclidian norm. We prove the Birkhoff-Kellogg FPT on a closed disc. Theorem 1. Any continuous mapping T : B R → R 2 such that T(∂B R ) ⊂ B R has a fixed point in B R .

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 R 2 can be extended to mappings in R n , using the basic properties of differential k-forms in R n . For n = 2, the differential 1-forms and differential (n − 1)-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 (n − 1)-forms has been introduced in [8], which, for n = 2, 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 D ⊂ R n open, bounded and nonempty, we need the concept of differential (n − 1)-forms and n-forms and suppose that the reader is familiar with the notions, notations and properties of differential k-forms (1 ≤ k ≤ n) 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 C 2 . We associate to the functions f j : D → R (j = 1, . . . , n) the differential 1-form ω f := ∑ n j=1 f j dx j in D, and the differential (n − 1)-form where dx j means that the corresponding term is missing. We associate also to g : D → R n the differential n-form µ g = g dx 1 ∧ . . . ∧ dx n . For example, given the function w : D → R with partial derivatives ∂ j w, its differential dw := ∑ n j=1 (∂ j w) dx j is the differential 1-form ω ∇w . Let ∆ ⊂ R n be open, bounded and nonempty, F : is well defined. To shorten the notations, we write F j for F j (·, λ). We define the derivative with respect to λ of F * ω f by On the other hand, where J F(·,λ) (y, λ) denotes the Jacobian of F(·, λ) at (y, λ) ∈ ∆ × [0, 1], and The following two results replace Lemma 1 in Section 2. The first one shows that the differential n-form ∂ λ (F * µ g ) is exact in ∆, i.e., is the exterior differential of a (n − 1)-differential form in ∆.
Proof. We have Proof. Using Lemma 2, the assumption and Stokes-Cartan theorem, we get Let B R := {x ∈ R n : |x| ≤ R} with |x| 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. Proof. Assume that T has no fixed point in B R . Then, , the open ball of center 0 and radius δ/2, and B R w(y) dy = 1. Then, by Proposition 2 with ∆ = B R , we get The Brouwer FPT on a closed n-ball is a special case.

Fixed Points, Homeomorphisms and Retractions in
is continuous, has a fixed point x * by Theorem 3, and h(x * ) ∈ K is a fixed point of T. Consequently, we have a Brouwer FPT for homeomorphic images of a closed n-ball. Theorem 3. If K ⊂ R n is homeomorphic to B R , any continuous mapping T : K → K has a fixed point in K.

Remark 1.
In Theorem 3, the boundedness assumption on K cannot be omitted: a translation x → x + a in R n with a = 0 has no fixed point. The closedness assumption on K cannot be omitted as well: T : (0, 1) → (0, 1), x → x 2 has no fixed point in (0, 1). Theorem 3 does not hold for any closed bounded set: a nontrivial rotation of the closed annulus A = {x ∈ R 2 : r 1 ≤ |x| ≤ r 2 } has 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 U ⊂ V ⊂ R n is a retract of V if there exists a continuous mapping r : V → U such that r = I on U (retraction of V in U). For example, B R is a retract of R n , with a retraction r given by Similarly, for any 0 < R 1 ≤ R 2 , B R 1 is a retract of B R 2 . Remark 3. The Brouwer FPT has for immediate topological consequence the well-known no-retraction theorem, stating that ∂B R is not a retract of B R in R n . 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 mapping T : B R ⊂ R n → R n such that x = λT(x) for all (x, λ) ∈ ∂B R × (0, 1) has a fixed point in B R .
Proof. Let r : R n → B R be the retraction of R n onto B R defined in Equation (1). Theorem 3 implies the existence of x * ∈ B R such that |T(x * )| < 1, a contradiction with the assumption. Hence, |T(x * )| ≤ R and x * = T(x * ). Proof. Let U = r(B R ) for some retraction r : B R → U. Then, T • r : B R → U ⊂ B R has a fixed point x * ∈ U. Hence, x * = r(x * ), and x * = T(x * ).
If C ⊂ R n is non-empty, closed and convex, the orthogonal projection p C (x) on C of x ∈ R n , defined by |p C (x) − x| = min y∈C |y − x|, is a retraction of R n onto C [15]. Consequently, C is a retract of any B R ⊃ C, giving a Brouwer FPT on compact convex sets. Corollary 4. If C ⊂ R n is compact and convex, any continuous mapping T : C → C has a fixed point in C.

Zeros of Continuous Mappings in R n
The first theorem on the existence of a zero for a mapping from B R into R n was first stated and proved for C 1 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 mapping f : B R → R n such that f (x) = µx for all x ∈ ∂B R and for all µ < 0 has a zero in B R .
Proof. Define the continuous mapping T : By Theorem 4, T has a fixed point x * in B R , 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 x, x − T(x) ≥ 0 holds for all x ∈ ∂B R , where ·, · denotes the usual inner product in R n . This condition implies the existence of a zero of I − T, 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 I − T 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 f : B R → R n , when x − T(x) is replaced by f (x) 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 C ⊂ R n be compact and convex and p C : R n → C be the orthogonal projection of x on C [15]. Recall that p C (x) is characterized by the condition (2) For x ∈ ∂C, the set is nonempty and called the normal cone to C at x, and its elements ν are called the outer normals to C at x. The relation in Equation (2) shows that, for each x ∈ C, x − p(x) ∈ N p(x) \ {0}. It can also be shown that each x ∈ ∂C is the orthogonal projection of some z ∈ C, so that N x = {z ∈ R n \ C : p(z) = x}. The Hadamard theorem on a convex compact set follows in a similar way as Theorem 6 from the Brouwer FPT 3.

Theorem 7.
If C ⊂ R n is a compact and convex, any continuous f : C → R n such that ν, f (x) ≥ 0 for all x ∈ ∂C and all ν ∈ N x has a zero in C.
Proof. Let T : R n → R n be defined by T = p C − f • p C . Then, for all x ∈ R n , and T maps B R into itself. By Theorem 3, there exists x * ∈ B R such that x * = p C (x * ) − f (p C (x * )). If x * ∈ C, the assumption implies that a contradiction. Thus, x * ∈ C, x * = p C (x * ) and f (x * ) = 0.

Corollary 5.
Any continuous mapping f : B R → R n such that x, f (x) ≥ 0 for all x ∈ ∂B R has a zero in B R .

Remark 5.
As shown when mentioning Hadamard's contribution, Theorem 5 implies the Brouwer FPT, and even the Birkhoff-Kellopg FPT, on B R . 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 f : P = [−R 1 , R 1 ] × · · · × [−R n , R n ] → R n when, for each i = 1, . . . , n, f i takes opposite signs on the opposite faces of P P − i := {x ∈ P : x i = −R i }, P + i := {x ∈ P : x i = R i } (i = 1, . . . , n).
Poincaré'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 mapping f : P → R n such that f i (x) ≤ 0 for all x ∈ P − i and f i (x) ≥ 0 for all x ∈ P + i (i = 1, . . . , n) has a zero in P.
Proof. If x is in the (relative) interior of the face P − i , then N x = {−λe i : λ > 0}, where (e 1 , e 2 , . . . , e n ) is the orthonormal basis in R n , and the assumption of Similarly, if x is in the (relative) interior to the face P + i , then N x = {λe i : λ > 0}, and the assumption of Theorem 7 becomes f i (x) ≥ 0. Of course, −λe i and λe i (λ > 0) also belong to the respective normal cones for x ∈ P − i and P + i respectively, and if, say, x ∈ P − i ∩ P + j then ν = −λe i + µe j ∈ N x for all λ, µ > 0, and ν, f (x) = −λ f i (x) + µ f k (x) ≥ 0. In general, when x belongs to the intersection of several faces of P, N x will be made of the linear combination of the e i 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, ν, f (x) ≥ 0 for all x ∈ ∂P and all ν ∈ N x . The result follows from Theorem 7. Remark 6. Corollary 6 implies the Brouwer FPT on P. Indeed, if T : P → P is continuous, and if we set 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 on B R and 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 each x ∈ ∂C, there exists ν ∈ N x such that ν, f (x) ≥ 0. No proof based only upon the Brouwer FPT seems to be known.
If C ⊂ R n is a compact convex set and g : C → R is of class C 1 , then g reaches its minimum on C at some x * ∈ C for which so that, dividing both members by λ and letting λ → 0+, we obtain ∇g(x * ), v − x * ≥ 0 for all v ∈ C, where ∇g denotes the gradient of g. For example, if u ∈ R n is fixed and g : C → R is defined by g(x) = (1/2)|x − u| 2 , the minimization problem corresponds to the definition of p C (u), and, as ∇g(x) = x − u, the inequality above is just Equation (2). In 1966, Hartman and Stampacchia [24] proved that the existence of such a x * still holds when ∇g is replaced by an arbitrary continuous function f : C → R n . 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. If C ⊂ R n is compact, convex and f : C → R n continuous, there exists x * ∈ C such that Proof. The Brouwer FPT on C (Corollary 4) applied to the continuous mapping p C • (I − f ) : C → C implies the existence of x * ∈ C such that Taking (2) and using Equation (3), one gets which is the requested inequality.
Remark 9. The conclusion of Theorem 8 is called a variational inequality. In the terminology of the theory of convex sets [15], the conclusion of Theorem 8 means that there exists x * ∈ C such that either f (x * ) = 0 or f (x * ) = 0 and H := {y ∈ R n : f (x * ), y − x * = 0} is a supporting hyperplane for C passing through x * i.e., C is entirely contained in one of the two closed half-spaces determined by H.
. Hence, the two statements are equivalent.
Remark 11. If x * and x # are two distinct solutions of the variational inequality, then Consequently, the variational inequality has a unique solution if f satisfies the condition f (x) − f (y), x − y > 0 for all x = y ∈ C, i.e., if f is strictly monotone on C.

A Direct Approach for Holomorphic Functions in C
The assumption of the Bolzano theorem for a continuous function f : [−R, R] → R can be, without loss of generality, be written f (−R) ≤ 0 ≤ f (R) or, equivalently, x f (x) ≥ 0 for |x| = R. In 1982, Shih [27] proposed a version of the Bolzano theorem for a complex function f holomorphic on a suitable bounded open neighborhood Ω ⊂ C of 0 and continuous on Ω. He showed that f has a unique zero in Ω when [z f (z)] > 0 on ∂Ω, using the Rouché theorem applied to f (z) and g(z) = αz for a suitable real α. As [z f (z)] = z · f (z) + z · f (z), 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.
Proposition 2. If f : P → C is continuous on P and holomorphic on int P, then ρ f (z) dz = 0.
Propositions 1 and 2 immediately imply the following simple theorem for the existence of a zero of f . Proposition 3. Any function f : B R → C (respectively, f : P → C) holomorphic on B(R) (respectively, int P), continuous on B R (respectively, P), different from zero on ∂B R (respectively, ∂P) and such that has a zero in B(R) (respectively, P).