Nonlinear Functional Analysis

Gerald Teschl

Gerald Teschl Institut für Mathematik Nordbergstraße 15 Universität Wien 1090 Wien, Austria

E-mail address: Gerald.Teschl@univie.ac.at
URL: http://www.mat.univie.ac.at/~gerald/

1991 Mathematics subject classification. 46-01, 47H10, 47H11, 58Fxx, 76D05

Abstract. This manuscript provides a brief introduction to nonlinear functional analysis.

We start out with calculus in Banach spaces, review differentiation and integration, derive the implicit function theorem (using the uniform contraction principle) and apply the result to prove existence and uniqueness of solutions for ordinary differential equations in Banach spaces.

Next we introduce the mapping degree in both finite (Brouwer degree) and infinite dimensional (Leray-Schauder degree) Banach spaces. Several applications to game theory, integral equations, and ordinary differential equations are discussed. Moreover, we give a brief discussion of monotone operators.

Finally we consider partial differential equations, by proving existence and uniqueness for solutions of the stationary Navier-Stokes equation.

Keywords and phrases. Mapping degree, fixed-point theorems, differential equations, Navier–Stokes equation.

Typeset by L^{A}TEX and Makeindex. Version: June 23, 2004 Copyright \begin{matrix}c \\ ◯ \end{matrix} 1998-2004 by Gerald Teschl ii

Preface

The present manuscript was written for my course Nonlinear Functional Analysis held at the University of Vienna in Summer 1998 and 2001. It is supposed to give a brief introduction to the field of Nonlinear Functional Analysis with emphasis on applications and examples. The material covered is highly selective and many important and interesting topics are not covered.

It is available from

         http://www.mat.univie.ac.at/~gerald/ftp/book-nlfa/

Acknowledgments

I’d like to thank Volker Enß for making his lecture notes available to me.

Gerald Teschl

Vienna, Austria February 2001 iv Preface

Contents

Preface  iii
1 Analysis in Banach spaces  1
1.1 Differentiation and integration in Banach spaces . . . . . . . . . . . 1
1.2 Contraction principles . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Ordinary differential equations . . . . . . . . . . . . . . . . . . . . . 8
2 The Brouwer mapping degree  11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Definition of the mapping degree and the determinant formula … 13
2.3 Extension of the determinant formula . . . . . . . . . . . . . . . . . 17
2.4 The Brouwer fixed-point theorem . . . . . . . . . . . . . . . . . . . 24
2.5 Kakutani’s fixed-point theorem and applications to game theory . . 25
2.6 Further properties of the degree . . . . . . . . . . . . . . . . . . . . 29
2.7 The Jordan curve theorem . . . . . . . . . . . . . . . . . . . . . . . 31
3 The Leray–Schauder mapping degree  33
3.1 The mapping degree on finite dimensional Banach spaces . . . . . . 33
3.2 Compact operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 The Leray–Schauder mapping degree . . . . . . . . . . . . . . . . . 35
3.4 The Leray–Schauder principle and the Schauder fixed-point theorem 37
3.5 Applications to integral and differential equations . . . . . . . . . . 39
4 The stationary Navier–Stokes equation  43
4.1 Introduction and motivation . . . . . . . . . . . . . . . . . . . . . . 43
4.2 An insert on Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . 44
4.3 Existence and uniqueness of solutions . . . . . . . . . . . . . . . . . 50

5 Monotone operators  53
5.1 Monotone operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 The nonlinear Lax–Milgram theorem . . . . . . . . . . . . . . . . . 55
5.3 The main theorem of monotone operators . . . . . . . . . . . . . . 57
Bibliography  61
Glossary of notations  63
Index  65

Chapter 1

Analysis in Banach spaces

1.1 Differentiation and integration in Banach spaces

We first review some basic facts from calculus in Banach spaces.

Let X and Y be two Banach spaces and denote by C(X, Y) the set of continuous functions from X to Y and by L(X, Y) ⊂ C(X, Y) the set of (bounded) linear functions. Let U be an open subset of X. Then a function F : U → Y is called differentiable at x ∈ U if there exists a linear function dF (x) ∈ L(X, Y) such that

F (x + u) = F (x) + dF (x) u + o(u),  (1.1)

where o, O are the Landau symbols. The linear map dF (x) is called derivative of F at x. If F is differentiable for all x ∈ U we call F differentiable. In this case we get a map

\begin{aligned}dF : & U & → & L(X, Y) \\ & x & ↦ & dF (x) \end{aligned} .  (1.2)

If dF is continuous, we call F continuously differentiable and write F ∈ C^{1}(U, Y).

Let Y = ∏_{j=1}^{m} Y_{j} and let F : X → Y be given by F = (F_{1}, …, F_{m}) with F_{j} : X → Y_{i}. Then F ∈ C^{1}(X, Y) if and only if F_{j} ∈ C^{1}(X, Y_{j}), 1 ≤ j ≤ m, and in this case dF = (dF_{1}, …, dF_{m}). Similarly, if X = ∏_{i=1}^{m} X_{i}, then one can define the partial derivative ∂_{i}F ∈ L(X_{i}, Y), which is the derivative of F considered as a function of the i-th variable alone (the other variables being fixed). We have dF v = ∑_{i=1}^{n} ∂_{i}F v_{i}, v = (v_{1}, …, v_{n}) ∈ X, and F ∈ C^{1}(X, Y) if and only all partial derivatives exist and are continuous.

In the case of X = ℝ^{m} and Y = ℝ^{n} ,the matrix representation of dF with respect to the canonical basis in ℝ^{m} and ℝ^{n} is given by the partial derivatives ∂_{i}F_{j}(x) and is called Jacobi matrix of F at x.

We can iterate the procedure of differentiation and write F ∈ C^{r}(U, Y), r ≥ 1, if the r-th derivative of F, d^{r}F (i.e., the derivative of the (r − 1)-th derivative of F), exists and is continuous. Finally, we set C^{∞}(U, Y) = ⋂_{r∈ℕ} C^{r}(U, Y) and, for notational convenience, C^{0}(U, Y) = C(U, Y) and d^{0}F = F.

It is often necessary to equip C^{r}(U, Y) with a norm. A suitable choice is

|F | = max_{0≤j≤r} sup_{x∈U} |d^{j}F (x)|.  (1.3)

The set of all r times differentiable functions for which this norm is finite forms a Banach space which is denoted by C_{b}^{r}(U, Y).

If F is bijective and F, F^{−1} are both of class C^{r}, r ≥ 1, then F is called a diffeomorphism of class C^{r}.

Note that if F ∈ L(X, Y), then dF (x) = F (independent of x) and d^{r}F (x) = 0, r > 1.

For the composition of mappings we note the following result (which is easy to prove).

Lemma 1.1 (Chain rule) Let F ∈ C^{r}(X, Y) and G ∈ C^{r}(Y, Z), r ≥ 1. Then G ∘ F ∈ C^{r}(X, Z) and

d(G ∘ F)(x) = dG(F (x)) ∘ dF (x),  x ∈ X.  (1.4)

In particular, if λ ∈ Y^{∗} is a linear functional, then d(λ∘F) = dλ∘dF = λ∘dF. In addition, we have the following mean value theorem.

Theorem 1.2 (Mean value) Suppose U ⊆ X and F ∈ C^{1}(U, Y). If U is convex, then

|F (x) − F (y)| ≤ M|x − y|,  M = max_{0≤t≤1} |dF ((1 − t)x + ty)|.  (1.5)

Conversely, (for any open U) if

|F (x) − F (y)| ≤ M|x − y|,  x, y ∈ U,  (1.6)

then

sup_{x∈U} |dF (x)| ≤ M.  (1.7)

Proof. Abbreviate f(t) = F ((1 − t)x + ty), 0 ≤ t ≤ 1, and hence df(t) = dF ((1 − t)x + ty)(y − x) implying |df(t)| ≤ M̃ = M|x − y|. For the first part it suffices to show

ϕ(t) = |f(t) − f(0)| − (M̃ + δ)t ≤ 0  (1.8)

for any δ > 0. Let t_{0} = max{t ∈ [0, 1]|ϕ(t) ≤ 0}. If t_{0} < 1 then

ϕ(t_{0} + ε) = |f(t_{0} + ε) − f(t_{0}) + f(t_{0}) − f(0)| − (M̃ + δ)(t_{0} + ε)
≤ |f(t_{0} + ε) − f(t_{0})| − (M̃ + δ)ε + ϕ(t_{0})
≤ |df(t_{0})ε + o(ε)| − (M̃ + δ)ε
≤ (M̃ + o(1) − M̃ − δ)ε = (−δ + o(1))ε ≤ 0,  (1.9)

for ε ≥ 0, small enough. Thus t_{0} = 1.

To prove the second claim suppose there is an x_{0} ∈ U such that |dF (x_{0})| = M + δ, δ > 0. Then we can find an e ∈ X, |e| = 1 such that |dF (x_{0})e| = M + δ and hence

Mε ≥ |F (x_{0} + εe) − F (x_{0})| = |dF (x_{0})(εe) + o(ε)|
≥ (M + δ)ε − |o(ε)| > Mε  (1.10)

since we can assume |o(ε)| < εδ for ε > 0 small enough, a contradiction. □

As an immediate consequence we obtain

Corollary 1.3 Suppose U is a connected subset of a Banach space X. A mapping F ∈ C^{1}(U, Y) is constant if and only if dF = 0. In addition, if F_{1,2} ∈ C^{1}(U, Y) and dF_{1} = dF_{2}, then F_{1} and F_{2} differ only by a constant.

Next we want to look at higher derivatives more closely. Let X = ∏_{i=1}^{m} X_{i}, then F : X → Y is called multilinear if it is linear with respect to each argument.

It is not hard to see that F is continuous if and only if

|F | = sup ; x:∏_{i=1}^{m} |x_{i}|=1 |F (x_{1}, …, x_{m})| < ∞.  (1.11)

If we take n copies of the same space, the set of multilinear functions F : X^{n} → Y will be denoted by L^{n}(X, Y). A multilinear function is called symmetric provided its value remains unchanged if any two arguments are switched. With the norm from above it is a Banach space and in fact there is a canonical isometric isomorphism between L^{n}(X, Y) and L(X, L^{n−1}(X, Y)) given by F : (x_{1}, …, x_{n}) ↦ F (x_{1}, …, x_{n}) maps to x_{1} ↦ F (x_{1}, .). In addition, note that to each F ∈ L^{n}(X, Y) we can assign its polar form F ∈ C(X, Y) using F (x) = F (x, …, x), x ∈ X. If F is symmetric it can be reconstructed from its polar form using

F (x_{1}, …, x_{n}) = \frac{1}{n!}∂_{t_{1}} ⋯ ∂_{t_{n}}F (∑_{i=1}^{n} t_{i}x_{i})|_{t_{1}=···=t_{n}=0}.  (1.12)

Moreover, the r-th derivative of F ∈ C^{r}(X, Y) is symmetric since,

d^{r}F_{x}(v_{1}, …, v_{r}) = ∂_{t_{1}} ⋯ ∂_{t_{r}}F (x + ∑_{i=1}^{r} t_{i}v_{i})|_{t_{1}=···=t_{r}=0},  (1.13)

where the order of the partial derivatives can be shown to be irrelevant.

Now we turn to integration. We will only consider the case of mappings f : I → X where I = [a, b] ⊂ ℝ is a compact interval and X is a Banach space. A function f : I → X is called simple if the image of f is finite, f(I) = {x_{i}}_{i=1}^{n}, and if each inverse image f^{−1}(x_{i}), 1 ≤ i ≤ n is a Borel set. The set of simple functions S(I, X) forms a linear space and can be equipped with the sup norm. The corresponding Banach space obtained after completion is called the set of regulated functions R(I, X).

Observe that C(I, X) ⊂ R(I, X). In fact, consider f_{n} = ∑_{i=0}^{n−1} f(t_{i})χ_{[t_{i},t_{i+1})} ∈ S(I, X), where t_{i} = a+i\frac{b−a}{n} and χ is the characteristic function. Since f ∈ C(I, X) is uniformly continuous, we infer that f_{n} converges uniformly to f.

For f ∈ S(I, X) we can define a linear map ∫ : S(I, X) → X by

∫_{a}^{b} f(t)dt = ∑_{i=1}^{n} x_{i}μ(f^{−1}(x_{i})),  (1.14)

where μ denotes the Lebesgue measure on I. This map satisfies

∫_{a}^{b} f(t)dt ≤ |f|(b − a).  (1.15)

and hence it can be extended uniquely to a linear map ∫ : R(I, X) → X with the same norm (b − a). We even have

∫_{a}^{b} f(t)dt ≤ ∫_{a}^{b} |f(t)|dt.  (1.16)

In addition, if λ ∈ X^{∗} is a continuous linear functional, then

λ(∫_{a}^{b} f(t)dt) = ∫_{a}^{b} λ(f(t))dt,  f ∈ R(I, X).  (1.17)

We use the usual conventions ∫_{t_{1}}^{t_{2}} f(s)ds = ∫_{a}^{b} χ_{(t_{1},t_{2})}(s)f(s)ds and ∫_{t_{2}}^{t_{1}} f(s)ds = − ∫_{t_{1}}^{t_{2}} f(s)ds.

If I ⊆ ℝ, we have an isomorphism L(I, X) ≡ X and if F : I → X we will write Ḟ (t) in stead of dF (t) if we regard dF (t) as an element of X. In particular, if f ∈ C(I, X), then F (t) = ∫_{a}^{t} f(s)ds ∈ C^{1}(I, X) and Ḟ (t) = f(t) as can be seen from

|∫_{a}^{t+ε}f(s)ds − ∫_{a}^{t}f(s)ds − f(t)ε| = |∫_{t}^{t+ε}(f(s) − f(t))ds| ≤ |ε| sup_{s∈[t,t+ε]}|f(s) − f(t)|.

(1.18)

This even shows that F (t) = F (a) + ∫_{a}^{t}(Ḟ (s))ds for any F ∈ C^{1}(I, X).

1.2 Contraction principles

A fixed point of a mapping F : C ⊆ X → C is an element x ∈ C such that F (x) = x. Moreover, F is called a contraction if there is a contraction constant θ ∈ [0, 1) such that

|F (x) − F (x̃)| ≤ θ|x − x̃|,  x, x̃ ∈ C.  (1.19)

Note that a contraction is continuous. We also recall the notation F^{n}(x) = F (F^{n−1}(x)), F^{0}(x) = x.

Theorem 1.4 (Contraction principle) Let C be a closed subset of a Banach space X and let F : C → C be a contraction, then F has a unique fixed point \overline{x} ∈ C such that

|F^{n}(x) − \overline{x}| ≤ \frac{θ^{n}}{1 − θ}|F (x) − x|,  x ∈ C.  (1.20)

Proof. If x = F (x) and x̃ = F (x̃), then |x − x̃| = |F (x) − F (x̃)| ≤ θ|x − x̃| shows that there can be at most one fixed point.

Concerning existence, fix x_{0} ∈ C and consider the sequence x_{n} = F^{n}(x_{0}). We have

|x_{n+1} − x_{n}| ≤ θ|x_{n} − x_{n−1}| ≤ ⋯ ≤ θ^{n}|x_{1} − x_{0}|  (1.21)

and hence by the triangle inequality (for n > m)

|x_{n} − x_{m}| ≤ ∑_{j=m+1}^{n} |x_{j} − x_{j−1}| ≤ θ^{m} ∑_{j=0}^{n−m−1} θ^{j}|x_{1} − x_{0}|

≤ \frac{θ^{m}}{1 − θ}|x_{1} − x_{0}|.  (1.22)

Thus x_{n} is Cauchy and tends to a limit \overline{x}. Moreover,

|F (\overline{x}) − \overline{x}| = lim_{n→∞} |x_{n+1} − x_{n}| = 0  (1.23)

shows that \overline{x} is a fixed point and the estimate (1.20) follows after taking the limit n → ∞ in (1.22). □

Next, we want to investigate how fixed points of contractions vary with respect to a parameter. Let U ⊆ X, V ⊆ Y be open and consider F : \overline{U} × V → U. The mapping F is called a uniform contraction if there is a θ ∈ [0, 1) such that

|F (x, y) − F (x̃, y)| ≤ θ|x − x̃|, x, x̃ ∈ \overline{U}, y ∈ V.  (1.24)

Theorem 1.5 (Uniform contraction principle) Let U, V be open subsets of Banach spaces X, Y, respectively. Let F : \overline{U} × V → U be a uniform contraction and denote by \overline{x}(y) ∈ U the unique fixed point of F (., y). If F ∈ C^{r}(U × V, U), r ≥ 0, then \overline{x}(.) ∈ C^{r}(V, U).

Proof. Let us first show that \overline{x}(y) is continuous. From

|\overline{x}(y + v) − \overline{x}(y)| = |F (\overline{x}(y + v), y + v) − F (\overline{x}(y), y + v)
+ F (\overline{x}(y), y + v) − F (\overline{x}(y), y)|
≤ θ|\overline{x}(y + v) − \overline{x}(y)| + |F (\overline{x}(y), y + v) − F (\overline{x}(y), y)|  (1.25)

we infer

|\overline{x}(y + v) − \overline{x}(y)| ≤ \frac{1}{1 − θ}|F (\overline{x}(y), y + v) − F (\overline{x}(y), y)|  (1.26)

and hence \overline{x}(y) ∈ C(V, U). Now let r = 1 and let us formally differentiate \overline{x}(y) = F (\overline{x}(y), y) with respect to y,

d \overline{x}(y) = ∂_{x}F (\overline{x}(y), y)d \overline{x}(y) + ∂_{y}F (\overline{x}(y), y).  (1.27)

Considering this as a fixed point equation T (x′, y) = x′, where T (., y) : L(Y, X) → L(Y, X), x′ ↦ ∂_{x}F (\overline{x}(y), y)x′ +∂_{y}F (\overline{x}(y), y) is a uniform contraction since we have |∂_{x}F (\overline{x}(y), y)| ≤ θ by Theorem 1.2. Hence we get a unique continuous solution \overline{x}′(y). It remains to show

\overline{x}(y + v) − \overline{x}(y) − \overline{x}′(y)v = o(v).  (1.28)

Let us abbreviate u = \overline{x}(y + v) − \overline{x}(y), then using (1.27) and the fixed point property of \overline{x}(y) we see

(1 − ∂_{x}F (\overline{x}(y), y))(u − \overline{x}′(y)v) =

= F (\overline{x}(y) + u, y + v) − F (\overline{x}(y), y) − ∂_{x}F (\overline{x}(y), y)u − ∂_{y}F (\overline{x}(y), y)v

= o(u) + o(v)  (1.29)

since F ∈ C^{1}(U ×V, U) by assumption. Moreover, |(1−∂_{x}F (\overline{x}(y), y))^{−1}| ≤ (1−θ)^{−1} and u = O(v) (by (1.26)) implying u − \overline{x}′(y)v = o(v) as desired.

Finally, suppose that the result holds for some r − 1 ≥ 1. Thus, if F is C^{r}, then \overline{x}(y) is at least C^{r−1} and the fact that d \overline{x}(y) satisfies (1.27) implies \overline{x}(y) ∈ C^{r}(V, U). □

As an important consequence we obtain the implicit function theorem.

Theorem 1.6 (Implicit function) Let X, Y, and Z be Banach spaces and let U, V be open subsets of X, Y, respectively. Let F ∈ C^{r}(U × V, Z), r ≥ 1, and fix (x_{0}, y_{0}) ∈ U × V. Suppose ∂_{x}F (x_{0}, y_{0}) ∈ L(X, Y) is an isomorphism. Then there exists an open neighborhood U_{1} × V_{1} ⊆ U × V of (x_{0}, y_{0}) such that for each y ∈ V_{1} there exists a unique point (ξ(y), y) ∈ U_{1} × V_{1} satisfying F (ξ(y), y) = F (x_{0}, y_{0}). Moreover, the map ξ is in C^{r}(V_{1}, Z) and fulfills

dξ(y) = −(∂_{x}F (ξ(y), y))^{−1} ∘ ∂_{y}F (ξ(y), y).  (1.30)

Proof. Using the shift F → F − F (x_{0}, y_{0}) we can assume F (x_{0}, y_{0}) = 0. Next, the fixed points of G(x, y) = x − (∂_{x}F (x_{0}, y_{0}))^{−1}F (x, y) are the solutions of F (x, y) = 0. The function G has the same smoothness properties as F and since |∂_{x}G(x_{0}, y_{0})| = 0, we can find balls U_{1} and V_{1} around x_{0} and y_{0} such that |∂_{x}G(x, y)| ≤ θ < 1. Thus G(., y) is a uniform contraction and in particular, G(U_{1}, y) ⊂ U_{1}, that is, G : U_{1} × V_{1} → U_{1}. The rest follows from the uniform contraction principle. Formula (1.30) follows from differentiating F (ξ(y), y) = 0 using the chain rule. □

Note that our proof is constructive, since it shows that the solution ξ(y) can be obtained by iterating x − (∂_{x}F (x_{0}, y_{0}))^{−1}F (x, y).

Moreover, as a corollary of the implicit function theorem we also obtain the inverse function theorem.

Theorem 1.7 (Inverse function) Suppose F ∈ C^{r}(U, Y), U ⊆ X, and let dF (x_{0}) be an isomorphism for some x_{0} ∈ U. Then there are neighborhoods U_{1}, V_{1} of x_{0}, F (x_{0}), respectively, such that F ∈ C^{r}(U_{1}, V_{1}) is a diffeomorphism.

Proof. Apply the implicit function theorem to G(x, y) = y − F (x). □

1.3 Ordinary differential equations

As a first application of the implicit function theorem, we prove (local) existence and uniqueness for solutions of ordinary differential equations in Banach spaces.

The following lemma will be needed in the proof.

Lemma 1.8 Suppose I ⊆ ℝ is a compact interval and f ∈ C^{r}(U, Y). Then f_{∗} ∈ C^{r}(C_{b}(I, U), C_{b}(I, Y)), where

(f_{∗}x)(t) = f(x(t)).  (1.31)

Proof. Fix x_{0} ∈ C_{b}(I, U) and ε > 0. For each t ∈ I we have a δ(t) > 0 such that |f(x) − f(x_{0}(t))| ≤ ε/2 for all x ∈ U with |x − x_{0}(t)| ≤ 2δ(t). The balls B_{δ(t)}(x_{0}(t)), t ∈ I, cover the set {x_{0}(t)}_{t∈I} and since I is compact, there is a finite subcover B_{δ(t_{j})}(x_{0}(t_{j})), 1 ≤ j ≤ n. Let |x − x_{0}| ≤ δ = min_{1≤j≤n} δ(t_{j}). Then for each t ∈ I there is t_{i} such that |x_{0}(t) − x_{0}(t_{j})| ≤ δ(t_{j}) and hence |f(x(t)) − f(x_{0}(t))| ≤ |f(x(t)) − f(x_{0}(t_{j}))| + |f(x_{0}(t_{j})) − f(x_{0}(t))| ≤ ε since |x(t) − x_{0}(t_{j})| ≤ |x(t) − x_{0}(t)| + |x_{0}(t) − x_{0}(t_{j})| ≤ 2δ(t_{j}). This settles the case r = 0.

Next let us turn to r = 1. We claim that df_{∗} is given by (df_{∗}(x_{0})x)(t) = df(x_{0}(t))x(t). Hence we need to show that for each ε > 0 we can find a δ > 0 such that

sup_{t∈I} |f_{∗}(x_{0}(t) + x(t)) − f_{∗}(x_{0}(t)) − df(x_{0}(t))x(t)| ≤ εδ  (1.32)

whenever |x − x_{0}| ≤ δ. By assumption we have

|f_{∗}(x_{0}(t) + x(t)) − f_{∗}(x_{0}(t)) − df(x_{0}(t))x(t)| ≤ εδ(t)  (1.33)

whenever |x(t) − x_{0}(t)| ≤ δ(t). Now argue as before. It remains to show that df_{∗} is continuous. To see this we use the linear map

λ : \begin{matrix}C_{b}(I, L(X, Y)) & → & L(C_{b}(I, X), C_{b}(I, Y)) \\ T & ↦ & T_{∗}x \end{matrix} ,  (1.34)

where (T_{∗}x)(t) = T (t)x(t). Since we have

|T_{∗}x| = sup_{t∈I} |T (t)x(t)| ≤ sup_{t∈I} |T (t)||x(t)| ≤ |T ||x|,  (1.35)

we infer |λ| ≤ 1 and hence λ is continuous. Now observe df_{∗} = λ ∘ (df)_{∗}. The general case r > 1 follows from induction. □

Now we come to our existence and uniqueness result for the initial value problem in Banach spaces.

Theorem 1.9 Let I be an open interval, U an open subset of a Banach space X and Λ an open subset of another Banach space. Suppose F ∈ C^{r}(I × U × Λ, X), then the initial value problem

ẋ(t) = F (t, x, λ),  x(t_{0}) = x_{0}, (t_{0}, x_{0}, λ) ∈ I × U × Λ,  (1.36)

has a unique solution x(t, t_{0}, x_{0}, λ) ∈ C^{r}(I_{1} × I_{2} × U_{1} × Λ_{1}, X), where I_{1,2}, U_{1}, and Λ_{1} are open subsets of I, U, and Λ, respectively. The sets I_{2}, U_{1}, and Λ_{1} can be chosen to contain any point t_{0} ∈ I, x_{0} ∈ U, and λ_{0} ∈ Λ, respectively.

Proof. If we shift t → t − t_{0}, x → x − x_{0}, and hence F → F (. + t_{0}, . + x_{0}, λ), we see that it is no restriction to assume x_{0} = 0, t_{0} = 0 and to consider (t_{0}, x_{0}) as part of the parameter λ (i.e., λ → (t_{0}, x_{0}, λ)). Moreover, using the standard transformation ẋ = F (τ, x, λ), τ̇ = 1, we can even assume that F is independent of t. We will also replace U by a smaller (bounded) subset such that F is uniformly continuous with respect to x on this subset.

Our goal is to invoke the implicit function theorem. In order to do this we introduce an additional parameter ε ∈ ℝ and consider

ẋ = εF (x, λ), x ∈ D^{r+1} = {x ∈ C_{b}^{r+1}((−1, 1), U)|x(0) = 0},  (1.37)

such that we know the solution for ε = 0. The implicit function theorem will show that solutions still exists as long as ε remains small. At first sight this doesn’t seem to be good enough for us since our original problem corresponds to ε = 1. But since ε corresponds to a scaling t → εt, the solution for one ε > 0 suffices. Now let us turn to the details.

Our problem (1.37) is equivalent to looking for zeros of the function

\begin{aligned}G : & D^{r+1} × Λ × (−ε_{0}, ε_{0}) & → & C_{b}^{r}((−1, 1), X) \\ & (x, λ, ε) & ↦ & ẋ − εF (x, λ) \end{aligned} .  (1.38)

Lemma 1.8 ensures that this function is C^{r}. Now fix λ_{0}, then G(0, λ_{0}, 0) = 0 and ∂_{x}G(0, λ_{0}, 0) = T, where T x = ẋ. Since (T^{−1}x)(t) = ∫_{0}^{t} x(s)ds we can apply the implicit function theorem to conclude that there is a unique solution x(λ, ε) ∈ C^{r}(Λ_{1} × (−ε_{0}, ε_{0}), D^{r+1}). In particular, the map (λ, t) ↦ x(λ, ε)(t/ε) is in C^{r}(Λ_{1}, C^{r+1}((−ε, ε), X)) ↪ C^{r}(Λ × (−ε, ε), X). Hence it is the desired solution of our original problem. □

The Brouwer mapping degree

2.1 Introduction

Many applications lead to the problem of finding all zeros of a mapping f : U ⊆ X → X, where X is some (real) Banach space. That is, we are interested in the solutions of

f(x) = 0, x ∈ U.  (2.1)

In most cases it turns out that this is too much to ask for, since determining the zeros analytically is in general impossible.

Hence one has to ask some weaker questions and hope to find answers for them. One such question would be ”Are there any solutions, respectively, how many are there?”. Luckily, this questions allows some progress.

To see how, lets consider the case f ∈ H(ℂ), where H(ℂ) denotes the set of holomorphic functions on a domain U ⊂ ℂ. Recall the concept of the winding number from complex analysis. The winding number of a path γ : [0, 1] → ℂ around a point z_{0} ∈ ℂ is defined by

n(γ, z_{0}) = \frac{1}{2πi} ∫_{γ} \frac{dz}{z − z_{0}} ∈ ℤ.  (2.2)

It gives the number of times γ encircles z_{0} taking orientation into account. That is, encirclings in opposite directions are counted with opposite signs.

In particular, if we pick f ∈ H(ℂ) one computes (assuming 0 ∉ f(γ))

n(f(γ), 0) = \frac{1}{2πi} ∫_{γ} \frac{f′(z)}{f(z)}dz = ∑_{k} n(γ, z_{k})α_{k},  (2.3)

where z_{k} denote the zeros of f and α_{k} their respective multiplicity. Moreover, if γ is a Jordan curve encircling a simply connected domain U ⊂ ℂ, then n(γ, z_{k}) = 0 if z_{k} ∉ U and n(γ, z_{k}) = 1 if z_{k} ∈ U. Hence n(f(γ), 0) counts the number of zeros inside U.

However, this result is useless unless we have an efficient way of computing n(f(γ), 0) (which does not involve the knowledge of the zeros z_{k}). This is our next task.

Now, lets recall how one would compute complex integrals along complicated paths. Clearly, one would use homotopy invariance and look for a simpler path along which the integral can be computed and which is homotopic to the original one. In particular, if f : γ → ℂ\{0} and g : γ → ℂ\{0} are homotopic, we have n(f(γ), 0) = n(g(γ), 0) (which is known as Rouchés theorem).

More explicitly, we need to find a mapping g for which n(g(γ), 0) can be computed and a homotopy H : [0, 1] × γ → ℂ\{0} such that H(0, z) = f(z) and H(1, z) = g(z) for z ∈ γ. For example, how many zeros of f(z) = \frac{1}{2}z^{6} + z − \frac{1}{3} lie inside the unit circle? Consider g(z) = z, then H(t, z) = (1 − t)f(z) + t g(z) is the required homotopy since |f(z) − g(z)| < |g(z)|, |z| = 1, implying H(t, z) ≠ 0 on [0, 1] × γ. Hence f(z) has one zero inside the unit circle.

Summarizing, given a (sufficiently smooth) domain U with enclosing Jordan curve ∂U, we have defined a degree deg(f, U, z_{0}) = n(f(∂U), z_{0}) = n(f(∂U) − z_{0}, 0) ∈ ℤ which counts the number of solutions of f(z) = z_{0} inside U. The invariance of this degree with respect to certain deformations of f allowed us to explicitly compute deg(f, U, z_{0}) even in nontrivial cases.

Our ultimate goal is to extend this approach to continuous functions f : ℝ^{n} → ℝ^{n}. However, such a generalization runs into several problems. First of all, it is unclear how one should define the multiplicity of a zero. But even more severe is the fact, that the number of zeros is unstable with respect to small perturbations. For example, consider f_{ε} : [−1, 2] → ℝ, x ↦ x^{2} + ε. Then f_{ε} has no zeros for ε > 0, one zero for ε = 0, two zeros for 0 < ε < 1, one for 1 < ε < \sqrt{2}, and none for ε > \sqrt{2}. This shows the following facts.

1. Zeros with f′ ≠ 0 are stable under small perturbations.

2. The number of zeros can change if two zeros with opposite sign change (i.e., opposite signs of f′) run into each other.

3. The number of zeros can change if a zero drops over the boundary.

Hence we see that we cannot expect too much from our degree. In addition, since it is unclear how it should be defined, we will first require some basic properties a degree should have and then we will look for functions satisfying these properties.

2.2 Definition of the mapping degree and the determinant formula

To begin with, let us introduce some useful notation. Throughout this section U will be a bounded open subset of ℝ^{n}. For f ∈ C^{1}(U, ℝ^{n}) the Jacobi matrix of f at x ∈ U is f′(x) = (∂_{x_{i}}f_{j}(x))_{1≤i,j≤n} and the Jacobi determinant of f at x ∈ U is

J_{f}(x) = det f′(x).  (2.4)

The set of regular values is

RV(f) = {y ∈ ℝ^{n}|∀x ∈ f^{−1}(y) : J_{f}(x) ≠ 0}.  (2.5)

Its complement CV(f) = ℝ^{n}\RV(f) is called the set of critical values. We set C^{r}(\overline{U}, ℝ^{n}) = \{f ∈ C^{r}(U, ℝ^{n})|d^{j}f ∈ C(\overline{U}, ℝ^{n}), 0 ≤ j ≤ r\} and

D_{y}^{r}(\overline{U}, ℝ^{n}) = \{f ∈ C^{r}(\overline{U}, ℝ^{n})|y ∉ f(∂U)\}, D_{y}(\overline{U}, ℝ^{n}) = D_{y}^{0}(\overline{U}, ℝ^{n}), y ∈ ℝ^{n}.

(2.6)

Now that these things are out of the way, we come to the formulation of the requirements for our degree.

A function deg which assigns each f ∈ D_{y}(\overline{U}, ℝ^{n}), y ∈ ℝ^{n}, a real number deg(f, U, y) will be called degree if it satisfies the following conditions.

(D1). deg(f, U, y) = deg(f − y, U, 0) (translation invariance).

(D2). deg(1l, U, y) = 1 if y ∈ U (normalization).

(D3). If U_{1,2} are open, disjoint subsets of U such that y ∉ f(\overline{U}\(U_{1} ∪ U_{2})), then deg(f, U, y) = deg(f, U_{1}, y) + deg(f, U_{2}, y) (additivity).

(D4). If H(t) = (1−t)f + tg ∈ D_{y}(\overline{U}, ℝ^{n}), t ∈ [0, 1], then deg(f, U, y) = deg(g, U, y) (homotopy invariance).

Before we draw some first conclusions form this definition, let us discuss the properties (D1)–(D4) first. (D1) is natural since deg(f, U, y) should have something to do with the solutions of f(x) = y, x ∈ U, which is the same as the solutions of f(x) − y = 0, x ∈ U. (D2) is a normalization since any multiple of deg would also satisfy the other requirements. (D3) is also quite natural since it requires deg to be additive with respect to components. In addition, it implies that sets where f ≠ y do not contribute. (D4) is not that natural since it already rules out the case where deg is the cardinality of f^{−1}(U). On the other hand it will give us the ability to compute deg(f, U, y) in several cases.

Theorem 2.1 Suppose deg satisfies (D1)–(D4) and let f, g ∈ D_{y}(\overline{U}, ℝ^{n}), then the following statements hold.

(i). We have deg(f, ∅, y) = 0. Moreover, if U_{i}, 1 ≤ i ≤ N, are disjoint open subsets of U such that y ∉ f(\overline{U}\ ⋃_{i=1}^{N} U_{i}), then deg(f, U, y) = ∑_{i=1}^{N} deg(f, U_{i}, y).

(ii). If y ∉ f(U), then deg(f, U, y) = 0 (but not the other way round). Equivalently, if deg(f, U, y) ≠ 0, then y ∈ f(U).

(iii). If |f(x) − g(x)| < dist(y, f(∂U)), x ∈ ∂U, then deg(f, U, y) = deg(g, U, y). In particular, this is true if f(x) = g(x) for x ∈ ∂U.

Proof. For the first part of (i) use (D3) with U_{1} = U and U_{2} = ∅. For the second part use U_{2} = ∅ in (D3) if i = 1 and the rest follows from induction. For (ii) use i = 1 and U_{1} = ∅ in (ii). For (iii) note that H(t, x) = (1 − t)f(x) + t g(x) satisfies |H(t, x) − y| ≥ dist(y, f(∂U)) − |f(x) − g(x)| for x on the boundary. □

Next we show that (D.4) implies several at first sight much stronger looking facts.

Theorem 2.2 We have that deg(., U, y) and deg(f, U, .) are both continuous. In fact, we even have

(i). deg(., U, y) is constant on each component of D_{y}(\overline{U}, ℝ^{n}).

(ii). deg(f, U, .) is constant on each component of ℝ^{n}\f(∂U).

Moreover, if H : [0, 1] × \overline{U} → ℝ^{n} and y : [0, 1] → ℝ^{n} are both continuous such that H(t) ∈ D_{y(t)}(U, ℝ^{n}), t ∈ [0, 1], then deg(H(0), U, y(0)) = deg(H(1), U, y(1)).

Proof. For (i) let C be a component of D_{y}(\overline{U}, ℝ^{n}) and let d_{0} ∈ deg(C, U, y). It suffices to show that deg(., U, y) is locally constant. But if |g −f| < dist(y, f(∂U)), then deg(f, U, y) = deg(g, U, y) by (D.4) since |H(t) − y| ≥ |f − y| − |g − f| > 0, H(t) = (1 − t)f + t g. The proof of (ii) is similar. For the remaining part observe, that if H : [0, 1] × \overline{U} → ℝ^{n}, (t, x) ↦ H(t, x), is continuous, then so is H : [0, 1] → C(\overline{U}, ℝ^{n}), t ↦ H(t), since \overline{U} is compact. Hence, if in addition H(t) ∈ D_{y}(\overline{U}, ℝ^{n}), then deg(H(t), U, y) is independent of t and if y = y(t) we can use deg(H(0), U, y(0)) = deg(H(t) − y(t), U, 0) = deg(H(1), U, y(1)). □

Note that this result also shows why deg(f, U, y) cannot be defined meaningful for y ∈ f(∂D). Indeed, approaching y from within different components of ℝ^{n}\f(∂U) will result in different limits in general!

In addition, note that if Q is a closed subset of a locally pathwise connected space X, then the components of X\Q are open (in the topology of X) and pathwise connected (the set of points for which a path to a fixed point x_{0} exists is both open and closed).

Now let us try to compute deg using its properties. Lets start with a simple case and suppose f ∈ C^{1}(U, ℝ^{n}) and y ∉ CV(f) ∪ f(∂U). Without restriction we consider y = 0. In addition, we avoid the trivial case f^{−1}(y) = ∅. Since the points of f^{−1}(0) inside U are isolated (use J_{f}(x) ≠ 0 and the inverse function theorem) they can only cluster at the boundary ∂U. But this is also impossible since f would equal y at the limit point on the boundary by continuity. Hence f^{−1}(0) = {x^{i}}_{i=1}^{N}. Picking sufficiently small neighborhoods U(x^{i}) around x^{i} we consequently get

deg(f, U, 0) = ∑_{i=1}^{N} deg(f, U(x^{i}), 0).  (2.7)

It suffices to consider one of the zeros, say x^{1}. Moreover, we can even assume x^{1} = 0 and U(x^{1}) = B_{δ}(0). Next we replace f by its linear approximation around 0. By the definition of the derivative we have

f(x) = f′(0)x + |x|r(x),  r ∈ C(B_{δ}(0), ℝ^{n}), r(0) = 0.  (2.8)

Now consider the homotopy H(t, x) = f′(0)x + (1 − t)|x|r(x). In order to conclude deg(f, B_{δ}(0), 0) = deg(f′(0), B_{δ}(0), 0) we need to show 0 ∉ H(t, ∂B_{δ}(0)). Since J_{f}(0) ≠ 0 we can find a constant λ such that |f′(0)x| ≥ λ|x| and since r(0) = 0 we can decrease δ such that |r| ≤ λ. This implies H(t, x) ≥ ||f′(0)x| − (1 − t)|x||r(x)|| ≥ λδ − δ|r| > 0 for x ∈ ∂B_{δ}(0) as desired.

In order to compute the degree of a nonsingular matrix we need the following lemma.

Lemma 2.3 Two nonsingular matrices M_{1,2} ∈ GL(n) are homotopic in GL(n) if and only if sgn det M_{1} = sgn det M_{2}.

Proof. We will show that any given nonsingular matrix M is homotopic to diag(sgn det M, 1, …, 1), where diag(m_{1}, …, m_{n}) denotes a diagonal matrix with diagonal entries m_{i}.

In fact, note that adding one row to another and multiplying a row by a positive constant can be realized by continuous deformations such that all intermediate matrices are nonsingular. Hence we can reduce M to a diagonal matrix diag(m_{1}, …, m_{n}) with (m_{i})^{2} = 1. Next,

( \begin{matrix}± cos(πt) & ∓ sin(πt) \\ sin(πt) & cos(πt), \end{matrix} )  (2.9)

shows that diag(±1, 1) and diag(∓1, −1) are homotopic. Now we apply this result to all two by two subblocks as follows. For each i starting from n and going down to 2 transform the subblock diag(m_{i−1}, m_{i}) into diag(1, 1) respectively diag(−1, 1). The result is the desired form for M.

To conclude the proof note that a continuous deformation within GL(n) cannot change the sign of the determinant since otherwise the determinant would have to vanish somewhere in between (i.e., we would leave GL(n)). □

Using this lemma we can now show the main result of this section.

Theorem 2.4 Suppose f ∈ D^{1}(\overline{U}, ℝ^{n}) and y ∉ CV(f), then a degree satisfying (D1)–(D4) satisfies

deg(f, U, y) = ∑_{x∈f^{−1}(y)} sgn J_{f}(x),  (2.10)

where the sum is finite and we agree to set ∑_{x∈∅} = 0.

Proof. By the previous lemma we obtain

deg(f′(0), B_{δ}(0), 0) = deg(diag(sgnJ_{f}(0), 1, …, 1), B_{δ}(0), 0)  (2.11)

since det M ≠ 0 is equivalent to Mx ≠ 0 for x ∈ ∂B_{δ}(0). Hence it remains to show deg(f′(0), B_{δ}(0), 0) = sgnJ_{f}(0).

If sgnJ_{f}(0) = 1 this is true by (D2). Otherwise we can replace f′(0) by M_{−} = diag(−1, 1, …, 1).

Now let U_{1} = {x ∈ ℝ^{n}||x_{i}| < 1, 1 ≤ i ≤ n}, U_{2} = {x ∈ ℝ^{n}|1 < x_{1} < 3, |x_{i}| < 1, 2 ≤ i ≤ n}, U = {x ∈ ℝ^{n}| − 1 < x_{1} < 3, |x_{i}| < 1, 2 ≤ i ≤ n}, and abbreviate y_{0} = (2, 0, …, 0). On U consider two continuous mappings M_{1,2} : U → ℝ^{n} such that M_{1}(x) = M_{−} if x ∈ U_{1}, M_{1}(x) = 1l−y_{0} if x ∈ U_{2}, and M_{2}(x) = (1, x_{2}, …, x_{n}).

Since M_{1}(x) = M_{2}(x) for x ∈ ∂U we infer deg(M_{1}, U, 0) = deg(M_{2}, U, 0) = 0. Moreover, we have deg(M_{1}, U, 0) = deg(M_{1}, U_{1}, 0) + deg(M_{1}, U_{2}, 0) and hence deg(M_{−}, U_{1}, 0) = − deg(1l − y_{0}, U_{2}, 0) = − deg(1l, U_{2}, y_{0}) = −1 as claimed. □

Up to this point we have only shown that a degree (provided there is one at all) necessarily satisfies (2.10). Once we have shown that regular values are dense, it will follow that the degree is uniquely determined by (2.10) since the remaining values follow from point (iv) of Theorem 2.1. On the other hand, we don’t even know whether a degree exists. Hence we need to show that (2.10) can be extended to f ∈ D_{y}(\overline{U}, ℝ^{n}) and that this extension satisfies our requirements (D1)–(D4).

2.3 Extension of the determinant formula

Our present objective is to show that the determinant formula (2.10) can be extended to all f ∈ D_{y}(\overline{U}, ℝ^{n}). This will be done in two steps, where we will show that deg(f, U, y) as defined in (2.10) is locally constant with respect to both y (step one) and f (step two).

Before we work out the technical details for these two steps, we prove that the set of regular values is dense as a warm up. This is a consequence of a special case of Sard’s theorem which says that CV(f) has zero measure.

Lemma 2.5 (Sard) Suppose f ∈ C^{1}(U, ℝ^{n}), then the Lebesgue measure of CV(f) is zero.

Proof. Since the claim is easy for linear mappings our strategy is as follows. We divide U into sufficiently small subsets. Then we replace f by its linear approximation in each subset and estimate the error.

Let CP(f) = {x ∈ U|J_{f}(x) = 0} be the set of critical points of f. We first pass to cubes which are easier to divide. Let {Q_{i}}_{i∈ℕ} be a countable cover for U consisting of open cubes such that \overline{Q_{i}} ⊂ U. Then it suffices to prove that f(CP(f) ∩ Q_{i}) has zero measure since CV(f) = f(CP(f)) = ⋃_{i} f(CP(f) ∩ Q_{i}) (the Q_{i}’s are a cover).

Let Q be any of these cubes and denote by ρ the length of its edges. Fix ε > 0 and divide Q into N^{n} cubes Q_{i} of length ρ/N. Since f′(x) is uniformly continuous on Q we can find an N (independent of i) such that

|f(x) − f(x̃) − f′(x̃)(x − x̃)| ≤ ∫_{0}^{1} |f′(x̃ + t(x − x̃)) − f′(x̃)||x̃ − x|dt ≤ \frac{ερ}{N} (2.12)

for x̃, x ∈ Q_{i}. Now pick a Q_{i} which contains a critical point x̃_{i} ∈ CP(f). Without restriction we assume x̃_{i} = 0, f(x̃_{i}) = 0 and set M = f′(x̃_{i}). By det M = 0 there is an orthonormal basis {b^{i}}_{1≤i≤n} of ℝ^{n} such that b^{n} is orthogonal to the image of M. In addition, there is a constant C_{1} such that Q_{i} ⊆ \{∑_{i=1}^{n−1} λ_{i}b^{i}| |λ_{i}| ≤ C_{1} \frac{ρ}{N} \} (e.g., C_{1} = n2^{(n/2)}) and hence there is a second constant (again independent of i) such that

MQ_{i} ⊆ \{∑_{i=1}^{n−1} λ_{i}b^{i}| |λ_{i}| ≤ C_{2} \frac{ρ}{N} \}  (2.13)

(e.g., C_{2} = nC_{1} max_{x∈\overline{Q}} |f′(x)|). Next, by our estimate (2.12) we even have

f(Q_{i}) ⊆ \{∑_{i=1}^{n} λ_{i}b^{i}| |λ_{i}| ≤ (C_{2} + ε)\frac{ρ}{N} , |λ_{n}| ≤ ε\frac{ρ}{N} \}  (2.14)

and hence the measure of f(Q_{i}) is smaller than \frac{C_{3}ε}{N^{n}}. Since there are at most N^{n} such Q_{i}’s, we see that the measure of f(Q) is smaller than C_{3}ε. □

Having this result out of the way we can come to step one and two from above.

Step 1: Admitting critical values

By (v) of Theorem 2.1, deg(f, U, y) should be constant on each component of ℝ^{n}\f(∂U). Unfortunately, if we connect y and a nearby regular value ỹ by a path, then there might be some critical values in between. To overcome this problem we need a definition for deg which works for critical values as well. Let us try to look for an integral representation. Formally (2.10) can be written as deg(f, U, y) = ∫_{U} δ_{y}(f(x))J_{f}(x)dx, where δ_{y}(.) is the Dirac distribution at y. But since we don’t want to mess with distributions, we replace δ_{y}(.) by ϕ_{ε}(.−y), where {ϕ_{ε}}_{ε>0} is a family of functions such that ϕ_{ε} is supported on the ball B_{ε}(0) of radius ε around 0 and satisfies ∫_{ℝ^{n}} ϕ_{ε}(x)dx = 1.

Lemma 2.6 Let f ∈ D_{y}^{1}(\overline{U}, ℝ^{n}), y ∉ CV(f). Then

deg(f, U, y) = ∫_{U} ϕ_{ε}(f(x) − y)J_{f}(x)dx  (2.15)

for all positive ε smaller than a certain ε_{0} depending on f and y. Moreover, supp(ϕ_{ε}(f(.) − y)) ⊂ U for ε < dist(y, f(∂U)).

Proof. If f^{−1}(y) = ∅, we can set ε_{0} = dist(y, f(\overline{U})), implying ϕ_{ε}(f(x) − y) = 0 for x ∈ \overline{U}.

If f^{−1}(y) = {x^{i}}_{1≤i≤N}, we can find an ε_{0} > 0 such that f^{−1}(B_{ε_{0}}(y)) is a union of disjoint neighborhoods U(x^{i}) of x^{i} by the inverse function theorem. Moreover, after possibly decreasing ε_{0} we can assume that f|_{U(x^{i})} is a bijection and that J_{f}(x) is nonzero on U(x^{i}). Again ϕ_{ε}(f(x) − y) = 0 for x ∈ \overline{U}\ ⋃_{i=1}^{N} U(x^{i}) and hence

∫_{U} ϕ_{ε}(f(x) − y)J_{f}(x)dx = ∑_{i=1}^{N} ∫_{U(x^{i})} ϕ_{ε}(f(x) − y)J_{f}(x)dx

= ∑_{i=1}^{N} sgn(J_{f}(x)) ∫_{B_{ε_{0}}(0)} ϕ_{ε}(x̃)dx̃ = deg(f, U, y),  (2.16)

where we have used the change of variables x̃ = f(x) in the second step. □

Our new integral representation makes sense even for critical values. But since ε depends on y, continuity with respect to y is not clear. This will be shown next at the expense of requiring f ∈ C^{2} rather than f ∈ C^{1}.

The key idea is to rewrite deg(f, U, y^{2}) − deg(f, U, y^{1}) as an integral over a divergence (here we will need f ∈ C^{2}) supported in U and then apply Stokes theorem. For this purpose the following result will be used.

Lemma 2.7 Suppose f ∈ C^{2}(U, ℝ^{n}) and u ∈ C^{1}(ℝ^{n}, ℝ^{n}), then

(div u)(f)J_{f} = divD_{f}(u),  (2.17)

where D_{f}(u)_{j} is the determinant of the matrix obtained from f′ by replacing the j-th column by u(f).

Proof. We compute

divD_{f}(u) = ∑_{j=1}^{n} ∂_{x_{j}}D_{f}(u)_{j} = ∑_{j,k=1}^{n} D_{f}(u)_{j,k},  (2.18)

where D_{f}(u)_{j,k} is the determinant of the matrix obtained from the matrix associated with D_{f}(u)_{j} by applying ∂_{x_{j}} to the k-th column. Since ∂_{x_{j}}∂_{x_{k}}f = ∂_{x_{k}}∂_{x_{j}}f we infer D_{f}(u)_{j,k} = −D_{f}(u)_{k,j}, j ≠ k, by exchanging the k-th and the j-th column. Hence

divD_{f}(u) = ∑_{i=1}^{n} D_{f}(u)_{i,i}.  (2.19)

Now let J_{f}^{(i,j)}(x) denote the (i, j) minor of f′(x) and recall ∑_{i=1}^{n} J_{f}^{(i,j)}∂_{x_{i}}f_{k} = δ_{j,k}J_{f}. Using this to expand the determinant D_{f}(u)_{i,i} along the i-th column shows

divD_{f}(u) = ∑_{i,j=1}^{n} J_{f}^{(i,j)}∂_{x_{i}}u_{j}(f) = ∑_{i,j=1}^{n} J_{f}^{(i,j)} ∑_{k=1}^{n}(∂_{x_{k}}u_{j})(f)∂_{x_{i}}f_{k}

= ∑_{j,k=1}^{n}(∂_{x_{k}}u_{j})(f) ∑_{i=1}^{n} J_{f}^{(i,j)}∂_{x_{j}}f_{k} = ∑_{j=1}^{n}(∂_{x_{j}}u_{j})(f)J_{f} (2.20)

as required. □

Now we can prove

Lemma 2.8 Suppose f ∈ C^{2}(\overline{U}, ℝ^{n}). Then deg(f, U, .) is constant in each ball contained in ℝ^{n}\f(∂U), whenever defined.

Proof. Fix ỹ ∈ ℝ^{n}\f(∂U) and consider the largest ball B_{ρ}(ỹ), ρ = dist(ỹ, f(∂U)) around ỹ contained in ℝ^{n}\f(∂U). Pick y^{i} ∈ B_{ρ}(ỹ) ∩ RV(f) and consider

deg(f, U, y^{2}) − deg(f, U, y^{1}) = ∫_{U}(ϕ_{ε}(f(x) − y^{2}) − ϕ_{ε}(f(x) − y^{1}))J_{f}(x)dx (2.21)

for suitable ϕ_{ε} ∈ C^{2}(ℝ^{n}, ℝ) and suitable ε > 0. Now observe

(div u)(y) = ∫_{0}^{1} z_{j}∂_{y_{j}}ϕ(y + tz)dt
= ∫_{0}^{1}(\frac{d}{dt}ϕ(y + t z))dt = ϕ_{ε}(y − y^{2}) − ϕ_{ε}(y − y^{1}), (2.22)

where

u(y) = z ∫_{0}^{1} ϕ(y + t z)dt, ϕ(y) = ϕ_{ε}(y − y^{1}), z = y^{2} − y^{1},  (2.23)

and apply the previous lemma to rewrite the integral as ∫_{U} divD_{f}(u)dx. Since the integrand vanishes in a neighborhood of ∂U it is no restriction to assume that ∂U is smooth such that we can apply Stokes theorem. Hence we have ∫_{U} divD_{f}(u)dx = ∫_{∂U} D_{f}(u)dF = 0 since u is supported inside B_{ρ}(ỹ) provided ε is small enough (e.g., ε < ρ − max{|y^{i} − ỹ|}_{i=1,2}). □

As a consequence we can define

deg(f, U, y) = deg(f, U, ỹ),  y ∉ f(∂U), f ∈ C^{2}(\overline{U}, ℝ^{n}),  (2.24)

where ỹ is a regular value of f with |ỹ − y| < dist(y, f(∂U)).

Remark 2.9 Let me remark a different approach due to Kronecker. For U with sufficiently smooth boundary we have

deg(f, U, 0) = \frac{1}{|S^{n−1}|} ∫_{∂U} Df̃(x)dF = \frac{1}{|S^{n}|} ∫_{∂U} \frac{1}{|f|^{n}} D_{f}(x)dF, f̃ = \frac{f}{|f|}, (2.25)

for f ∈ C_{y}^{2}(\overline{U}, ℝ^{n}). Explicitly we have

deg(f, U, 0) = \frac{1}{|S^{n−1}|} ∫_{∂U} ∑_{j=1}^{n}(−1)^{j−1} \frac{f_{j}}{|f|^{n}} df_{1} ∧⋯∧df_{j−1} ∧df_{j+1} ∧⋯∧df_{n}. (2.26)

Since f̃ : ∂U → S^{n−1} the integrand can also be written as the pull back f̃^{∗}dS of the canonical surface element dS on S^{n−1}.

This coincides with the boundary value approach for complex functions (note that holomorphic functions are orientation preserving).

Step 2: Admitting continuous functions

Our final step is to remove the condition f ∈ C^{2}. As before we want the degree to be constant in each ball contained in D_{y}(\overline{U}, ℝ^{n}). For example, fix f ∈ D_{y}(\overline{U}, ℝ^{n}) and set ρ = dist(y, f(∂U)) > 0. Choose f^{i} ∈ C^{2}(\overline{U}, ℝ^{n}) such that |f^{i} − f| < ρ, implying f^{i} ∈ D_{y}(\overline{U}, ℝ^{n}). Then H(t, x) = (1 − t)f^{1}(x) + tf^{2}(x) ∈ D_{y}(\overline{U}, ℝ^{n}) ∩ C^{2}(U, ℝ^{n}), t ∈ [0, 1], and |H(t) − f| < ρ. If we can show that deg(H(t), U, y) is locally constant with respect to t, then it is continuous with respect to t and hence constant (since [0, 1] is connected). Consequently we can define

deg(f, U, y) = deg(f̃, U, y),  f ∈ D_{y}(\overline{U}, ℝ^{n}),  (2.27)

where f̃ ∈ C^{2}(\overline{U}, ℝ^{n}) with |f̃− f| < dist(y, f(∂U)).

It remains to show that t ↦ deg(H(t), U, y) is locally constant.

Lemma 2.10 Suppose f ∈ C_{y}^{2}(\overline{U}, ℝ^{n}). Then for each f̃ ∈ C^{2}(\overline{U}, ℝ^{n}) there is an ε > 0 such that deg(f + t f̃, U, y) = deg(f, U, y) for all t ∈ (−ε, ε).

Proof. If f^{−1}(y) = ∅ the same is true for f + t g if |t| < dist(y, f(\overline{U}))/|g|. Hence we can exclude this case. For the remaining case we use our usual strategy of considering y ∈ RV(f) first and then approximating general y by regular ones.

Suppose y ∈ RV(f) and let f^{−1}(y) = {x^{i}}_{j=1}^{N}. By the implicit function theorem we can find disjoint neighborhoods U(x^{i}) such that there exists a unique solution x^{i}(t) ∈ U(x^{i}) of (f + t g)(x) = y for |t| < ε_{1}. By reducing U(x^{i}) if necessary, we can even assume that the sign of J_{f+t g} is constant on U(x^{i}). Finally, let ε_{2} = dist(y, f(U\ ⋃_{i=1}^{N} U(x^{i})))/|g|. Then |f + t g| > 0 for |t| < ε_{2} and ε = min(ε_{1}, ε_{2}) is the quantity we are looking for.

It remains to consider the case y ∈ CV(f). pick a regular value ỹ ∈ B_{ρ/3}(y), where ρ = dist(y, f(∂U)), implying deg(f, U, y) = deg(f, U, ỹ). Then we can find an ε̃ > 0 such that deg(f, U, ỹ) = deg(f + t g, U, ỹ) for |t| < ε̃. Setting ε = min(ε̃, ρ/(3|g|)) we infer ỹ − (f + t g)(x) ≥ ρ/3 for x ∈ ∂U, that is |ỹ − y| < dist(ỹ, (f + t g)(∂U)), and thus deg(f + t g, U, ỹ) = deg(f + t g, U, y). Putting it all together implies deg(f, U, y) = deg(f + t g, U, y) for |t| < ε as required. □

Now we can finally prove our main theorem.

Theorem 2.11 There is a unique degree deg satisfying (D1)-(D4). Moreover, deg(., U, y) : D_{y}(\overline{U}, ℝ^{n}) → ℤ is constant on each component and given f ∈ D_{y}(\overline{U}, ℝ^{n}) we have

deg(f, U, y) = ∑_{x∈f̃^{−1}(y)} sgn Jf̃(x)  (2.28)

where f̃ ∈ D_{y}^{2}(\overline{U}, ℝ^{n}) is in the same component of D_{y}(\overline{U}, ℝ^{n}), say |f − f̃| < dist(y, f(∂U)), such that y ∈ RV(f̃).

Proof. Our previous considerations show that deg is well-defined and locally constant with respect to the first argument by construction. Hence deg(., U, y) : D_{y}(\overline{U}, ℝ^{n}) → ℤ is continuous and thus necessarily constant on components since ℤ is discrete. (D2) is clear and (D1) is satisfied since it holds for f̃ by construction. Similarly, taking U_{1,2} as in (D3) we can require |f − f̃| < dist(y, f(\overline{U}\(U_{1} ∪ U_{2})). Then (D3) is satisfied since it also holds for f̃ by construction. Finally, (D3) is a consequence of continuity. □

To conclude this section, let us give a few simple examples illustrating the use of the Brouwer degree.

First, let’s investigate the zeros of

f(x_{1}, x_{2}) = (x_{1} − 2x_{2} + cos(x_{1} + x_{2}), x_{2} + 2x_{1} + sin(x_{1} + x_{2})).  (2.29)

Denote the linear part by

g(x_{1}, x_{2}) = (x_{1} − 2x_{2}, x_{2} + 2x_{1}).  (2.30)

Then we have |g(x)| = \sqrt{5}|x| and |f(x)−g(x)| = 1 and hence h(t) = (1−t)g+t f = g + t(f − g) satisfies |h(t)| ≥ |g| − t|f − g| > 0 for |x| > 1/\sqrt{5} implying

deg(f, B_{5}(0), 0) = deg(g, B_{5}(0), 0) = 1.  (2.31)

Moreover, since J_{f}(x) = 5 + 3 cos(x_{1} + x_{2}) + sin(x_{1} + x_{2}) > 1 we see that f(x) = 0 has a unique solution in ℝ^{2}. This solution has even to lie on the circle |x| = 1/\sqrt{5} since f(x) = 0 implies 1 = |f(x) − g(x)| = |g(x)| = \sqrt{5}|x|.

Next let us prove the following result which implies the hairy ball (or hedgehog) theorem.

Theorem 2.12 Suppose U contains the origin and let f : ∂U → ℝ^{n}\{0} be continuous. If n is odd, then there exists a x ∈ ∂U and a λ ≠ 0 such that f(x) = λx.

Proof. By Theorem 2.15 we can assume f ∈ C(\overline{U}, ℝ^{n}) and since n is odd we have deg(−1l, U, 0) = −1. Now if deg(f, U, 0) ≠ −1, then H(t, x) = (1−t)f(x)−tx must have a zero (t_{0}, x_{0}) ∈ (0, 1) × ∂U and hence f(x_{0}) = \frac{t_{0}}{1−t_{0}} x_{0}. Otherwise, if deg(f, U, 0) = −1 we can apply the same argument to H(t, x) = (1−t)f(x)+tx. □

In particular this result implies that a continuous tangent vector field on the unit sphere f : S^{n−1} → ℝ^{n} (with f(x)x = 0 for all x ∈ S^{n}) must vanish somewhere if n is odd. Or, for n = 3, you cannot smoothly comb a hedgehog without leaving a bald spot or making a parting. It is however possible to comb the hair smoothly on a torus and that is why the magnetic containers in nuclear fusion are toroidal.

Another simple consequence is the fact that a vector field on ℝ^{n}, which points outwards (or inwards) on a sphere, must vanish somewhere inside the sphere.

Theorem 2.13 Suppose f : B_{R}(0) → ℝ^{n} is continuous and satisfies

f(x)x > 0,  |x| = R.  (2.32)

Then f(x) vanishes somewhere inside B_{R}(0).

Proof. If f does not vanish, then H(t, x) = (1 − t)x + tf(x) must vanish at some point (t_{0}, x_{0}) ∈ (0, 1) × ∂B_{R}(0) and thus

0 = H(t_{0}, x_{0})x_{0} = (1 − t_{0})R^{2} + t_{0}f(x_{0})x_{0}.  (2.33)

But the last part is positive by assumption, a contradiction. □

2.4 The Brouwer fixed-point theorem

Now we can show that the famous Brouwer fixed-point theorem is a simple consequence of the properties of our degree.

Theorem 2.14 (Brouwer fixed point) Let K be a topological space homeomorphic to a compact, convex subset of ℝ^{n} and let f ∈ C(K, K), then f has at least one fixed point.

Proof. Clearly we can assume K ⊂ ℝ^{n} since homeomorphisms preserve fixed points. Now lets assume K = B_{r}(0). If there is a fixed-point on the boundary ∂B_{r}(0)) we are done. Otherwise H(t, x) = x − t f(x) satisfies 0 ∉ H(t, ∂B_{r}(0)) since |H(t, x)| ≥ |x| − t|f(x)| ≥ (1 − t)r > 0, 0 ≤ t < 1. And the claim follows from deg(x − f(x), B_{r}(0), 0) = deg(x, B_{r}(0), 0) = 1.

Now let K be convex. Then K ⊆ B_{ρ}(0) and, by Theorem 2.15 below, we can find a continuous retraction R : ℝ^{n} → K (i.e., R(x) = x for x ∈ K) and consider f̃ = f ∘ R ∈ C(\overline{B_{ρ}(0)}, \overline{B_{ρ}(0)}). By our previous analysis, there is a fixed point x = f̃(x) ∈ conv(f(K)) ⊆ K. □

Note that any compact, convex subset of a finite dimensional Banach space (complex or real) is isomorphic to a compact, convex subset of ℝ^{n} since linear transformations preserve both properties. In addition, observe that all assumptions are needed. For example, the map f : ℝ → ℝ, x ↦ x + 1, has no fixed point (ℝ is homeomorphic to a bounded set but not to a compact one). The same is true for the map f : ∂B_{1}(0) → ∂B_{1}(0), x ↦ −x (∂B_{1}(0) ⊂ ℝ^{n} is simply connected for n ≥ 3 but not homeomorphic to a convex set).

It remains to prove the result from topology needed in the proof of the Brouwer fixed-point theorem.

Theorem 2.15 Let X and Y be Banach spaces and let K be a closed subset of X. Then F ∈ C(K, Y) has a continuous extension F ∈ C(X, Y) such that F (X) ⊆ conv(F (K)).

Proof. Consider the open cover {B_{ρ(x)}(x)}_{x∈X\K} for X\K, where ρ(x) = dist(x, X\K)/2. Choose a (locally finite) partition of unity {ϕ_{λ}}_{λ∈Λ} subordinate to this cover and set

F (x) = ∑_{λ∈Λ} ϕ_{λ}(x)F (x_{λ}) for x ∈ X\K,  (2.34)

where x_{λ} ∈ K satisfies dist(x_{λ}, suppϕ_{λ}) ≤ 2dist(K, suppϕ_{λ}). By construction, F is continuous except for possibly at the boundary of K. Fix x_{0} ∈ ∂K, ε > 0 and choose δ > 0 such that |F (x) − F (x_{0})| ≤ ε for all x ∈ K with |x − x_{0}| < 4δ. We will show that |F (x) − F (x_{0})| ≤ ε for all x ∈ X with |x − x_{0}| < δ. Suppose x ∉ K, then |F (x) − F (x_{0})| ≤ ∑_{λ∈Λ} ϕ_{λ}(x)|F (x_{λ}) − F (x_{0})|. By our construction, x_{λ} should be close to x for all λ with x ∈ suppϕ_{λ} since x is close to K. In fact, if x ∈ suppϕ_{λ} we have

|x − x_{λ}| ≤ dist(x_{λ}, suppϕ_{λ}) + d(suppϕ_{λ}) ≤ 2dist(K, suppϕ_{λ}) + d(suppϕ_{λ}), (2.35)

where d(suppϕ_{λ}) = sup_{x,y∈suppϕ_{λ}} |x−y|. Since our partition of unity is subordinate to the cover {B_{ρ(x)}(x)}_{x∈X\K} we can find a x̃ ∈ X\K such that suppϕ_{λ} ⊂ B_{ρ(x̃)}(x̃) and hence d(suppϕ_{λ}) ≤ ρ(x̃) ≤ dist(K, suppϕ_{λ}). Putting it all together we have |x − x_{λ}| ≤ 3dist(x_{λ}, suppϕ_{λ}) and hence

|x_{0} − x_{λ}| ≤ |x_{0} − x| + |x − x_{λ}| ≤ 4dist(x_{λ}, suppϕ_{λ}) ≤ 4|x − x_{0}| ≤ 4δ (2.36)

as expected. By our choice of δ we have |F (x_{λ}) − F (x_{0})| ≤ ε for all λ with ϕ_{λ}(x) ≠ 0. Hence |F (x) − F (x_{0})| ≤ ε whenever |x − x_{0}| ≤ δ and we are done. □

Note that the same proof works if X is only a metric space.

Finally, let me remark that the Brouwer fixed point theorem is equivalent to the fact that there is no continuous retraction R : B_{1}(0) → ∂B_{1}(0) (with R(x) = x for x ∈ ∂B_{1}(0)) from the unit ball to the unit sphere in ℝ^{n}.

In fact, if R would be such a retraction, −R would have a fixed point x_{0} ∈ ∂B_{1}(0) by Brouwer’s theorem. But then x_{0} = −f(x_{0}) = −x_{0} which is impossible. Conversely, if a continuous function f : B_{1}(0) → B_{1}(0) has no fixed point we can define a retraction R(x) = f(x) + t(x)(x − f(x)), where t(x) ≥ 0 is chosen such that |R(x)|^{2} = 1 (i.e., R(x) lies on the intersection of the line spanned by x, f(x) with the unit sphere).

Using this equivalence the Brouwer fixed point theorem can also be derived easily by showing that the homology groups of the unit ball B_{1}(0) and its boundary (the unit sphere) differ (see, e.g., [9] for details).

2.5 Kakutani’s fixed-point theorem and applications to game theory

In this section we want to apply Brouwer’s fixed-point theorem to show the existence of Nash equilibria for n-person games. As a preparation we extend Brouwer’s fixed-point theorem to set valued functions. This generalization will be more suitable for our purpose.

Denote by CS(K) the set of all nonempty convex subsets of K.

Theorem 2.16 (Kakutani) Suppose K is a compact convex subset of ℝ^{n} and f : K → CS(K). If the set

Γ = {(x, y)|y ∈ f(x)} ⊆ K^{2}  (2.37)

is closed, then there is a point x ∈ K such that x ∈ f(x).

Proof. Our strategy is to apply Brouwer’s theorem, hence we need a function related to f. For this purpose it is convenient to assume that K is a simplex

K = ⟨v_{1}, …, v_{m}⟩,  m ≤ n,  (2.38)

where v_{i} are the vertices. If we pick y_{i} ∈ f(v_{i}) we could set

f^{1}(x) = ∑_{i=1}^{m} λ_{i}y_{i},  (2.39)

where λ_{i} are the barycentric coordinates of x (i.e., λ_{i} ≥ 0, ∑_{i=1}^{m} λ_{i} = 1 and x = ∑_{i=1}^{n} λ_{i}v_{i}). By construction, f^{1} ∈ C(K, K) and there is a fixed point x^{1}. But unless x^{1} is one of the vertices, this doesn’t help us too much. So lets choose a better function as follows. Consider the k-th barycentric subdivision and for each vertex v_{i} in this subdivision pick an element y_{i} ∈ f(v_{i}). Now define f^{k}(v_{i}) = y_{i} and extend f^{k} to the interior of each subsimplex as before. Hence f^{k} ∈ C(K, K) and there is a fixed point

x^{k} = ∑_{i=1}^{m} λ_{i}^{k}v_{i}^{k} = ∑_{i=1}^{m} λ_{i}^{k}y_{i}^{k}, y_{i}^{k} = f^{k}(v_{i}^{k}),  (2.40)

in the subsimplex ⟨v_{1}^{k}, …, v_{m}^{k}⟩. Since (x^{k}, λ_{1}^{k}, …, λ_{m}^{k}, y_{1}^{k}, …, y_{m}^{k}) ∈ K^{2m+1} we can assume that this sequence converges to (x^{0}, λ_{1}^{0}, …, λ_{m}^{0}, y_{1}^{0}, …, y_{m}^{0}) after passing to a subsequence. Since the subsimplices shrink to a point, this implies v_{i}^{k} → x^{0} and hence y_{i}^{0} ∈ f(x^{0}) since (v_{i}^{k}, y_{i}^{k}) ∈ Γ → (v_{i}^{0}, y_{i}^{0}) ∈ Γ by the closedness assumption. Now (2.40) tells us

x^{0} = ∑_{i=1}^{m} λ_{i}^{k}y_{i}^{k} ∈ f(x^{0})  (2.41)

since f(x^{0}) is convex and the claim holds if K is a simplex.

If K is not a simplex, we can pick a simplex S containing K and proceed as in the proof of the Brouwer theorem. □

If f(x) contains precisely one point for all x, then Kakutani’s theorem reduces to the Brouwer’s theorem.

Now we want to see how this applies to game theory.

An n-person game consists of n players who have m_{i} possible actions to choose from. The set of all possible actions for the i-th player will be denoted by Φ_{i} = {1, …, m_{i}}. An element φ_{i} ∈ Φ_{i} is also called a pure strategy for reasons to become clear in a moment. Once all players have chosen their move φ_{i}, the payoff for each player is given by the payoff function

R_{i}(φ),  φ = (φ_{1}, …, φ_{n}) ∈ Φ = ∏_{i=1}^{n} Φ_{i}  (2.42)

of the i-th player. We will consider the case where the game is repeated a large number of times and where in each step the players choose their action according to a fixed strategy. Here a strategy s_{i} for the i-th player is a probability distribution on Φ_{i}, that is, s_{i} = (s_{i}^{1}, …, s_{i}^{m_{i}}) such that s_{i}^{k} ≥ 0 and ∑_{k=1}^{m_{i}} s_{i}^{k} = 1. The set of all possible strategies for the i-th player is denoted by S_{i}. The number s_{i}^{k} is the probability for the k-th pure strategy to be chosen. Consequently, if s = (s_{1}, …, s_{n}) ∈ S = ∏_{i=1}^{n} S_{i} is a collection of strategies, then the probability that a given collection of pure strategies gets chosen is

s(φ) = ∏_{i=1}^{n} s_{i}(φ),  s_{i}(φ) = s_{i}^{k_{i}}, φ = (k_{1}, …, k_{n}) ∈ Φ  (2.43)

(assuming all players make their choice independently) and the expected payoff for player i is

R_{i}(s) = ∑_{φ∈Φ} s(φ)R_{i}(φ).  (2.44)

By construction, R_{i}(s) is continuous.

The question is of course, what is an optimal strategy for a player? If the other strategies are known, a best reply of player i against s would be a strategy \overline{s_{i}} satisfying

R_{i}(s\\overline{s}_{i}) = max_{s̃_{i}∈S_{i}} R_{i}(s\s̃_{i})  (2.45)

Here s\s̃_{i} denotes the strategy combination obtained from s by replacing s_{i} by s̃_{i}. The set of all best replies against s for the i-th player is denoted by B_{i}(s). Explicitly, \overline{s}_{i} ∈ B(s) if and only if \overline{s}_{i}^{k} = 0 whenever R_{i}(s\k) < max_{1≤l≤m_{i}} R_{i}(s\l) (in particular B_{i}(s) ≠ ∅).

Let s, \overline{s} ∈ S, we call \overline{s} a best reply against s if \overline{s}_{i} is a best reply against s for all i. The set of all best replies against s is B(s) = ∏_{i=1}^{n} B_{i}(s).

A strategy combination \overline{s} ∈ S is a Nash equilibrium for the game if it is a best reply against itself, that is,

\overline{s} ∈ B(\overline{s}).  (2.46)

Or, put differently, \overline{s} is a Nash equilibrium if no player can increase his payoff by changing his strategy as long as all others stick to their respective strategies. In addition, if a player sticks to his equilibrium strategy, he is assured that his payoff will not decrease no matter what the others do.

To illustrate these concepts, let us consider the famous prisoners dilemma. Here we have two players which can choose to defect or cooperate. The payoff is symmetric for both players and given by the following diagram

\begin{matrix}R_{1} & d_{2} & c_{2} \\ d_{1} & 0 & 2 \\ c_{1} & −1 & 1 \end{matrix}  \begin{matrix}R_{2} & d_{2} & c_{2} \\ d_{1} & 0 & −1 \\ c_{1} & 2 & 1 \end{matrix}  (2.47)

where c_{i} or d_{i} means that player i cooperates or defects, respectively. It is easy to see that the (pure) strategy pair (d_{1}, d_{2}) is the only Nash equilibrium for this game and that the expected payoff is 0 for both players. Of course, both players could get the payoff 1 if they both agree to cooperate. But if one would break this agreement in order to increase his payoff, the other one would get less. Hence it might be safer to defect.

Now that we have seen that Nash equilibria are a useful concept, we want to know when such an equilibrium exists. Luckily we have the following result.

Theorem 2.17 (Nash) Every n-person game has at least one Nash equilibrium.

Proof. The definition of a Nash equilibrium begs us to apply Kakutani’s theorem to the set valued function s ↦ B(s). First of all, S is compact and convex and so are the sets B(s). Next, observe that the closedness condition of Kakutani’s theorem is satisfied since if s^{m} ∈ S and \overline{s}^{m} ∈ B(s^{n}) both converge to s and \overline{s}, respectively, then (2.45) for s^{m}, \overline{s}^{m}

R_{i}(s^{m}\s̃_{i}) ≤ R_{i}(s^{m}\\overline{s}_{i}^{m}),  s̃_{i} ∈ S_{i}, 1 ≤ i ≤ n,  (2.48)

implies (2.45) for the limits s, \overline{s}

R_{i}(s\s̃_{i}) ≤ R_{i}(s\\overline{s}_{i}),  s̃_{i} ∈ S_{i}, 1 ≤ i ≤ n,  (2.49)

by continuity of R_{i}(s). □

2.6 Further properties of the degree

We now prove some additional properties of the mapping degree. The first one will relate the degree in ℝ^{n} with the degree in ℝ^{m}. It will be needed later on to extend the definition of degree to infinite dimensional spaces. By virtue of the canonical embedding ℝ^{m} ↪ ℝ^{m} × {0} ⊂ ℝ^{n} we can consider ℝ^{m} as a subspace of ℝ^{n}.

Theorem 2.18 (Reduction property) Let f ∈ C(\overline{U}, ℝ^{m}) and y ∈ ℝ^{m}\(1l + f)(∂U), then

deg(1l + f, U, y) = deg(1l + f_{m}, U_{m}, y),  (2.50)

where f_{m} = f|_{U_{m}}, where U − M is the projection of U to ℝ^{m}.

Proof. Choose a f̃ ∈ C^{2}(U, ℝ^{m}) sufficiently close to f such that y ∈ RV(f̃). Let x ∈ (1l+f̃)^{−1}(y), then x = y−f(x) ∈ ℝ^{m} implies (1l+f̃)^{−1}(y) = (1l+f̃_{m})^{−1}(y). Moreover,

J_{1l+f̃}(x) = det(1l + f̃′)(x) = det ( \begin{matrix}δ_{ij} + ∂_{j}f̃_{i}(x) & ∂_{j}f̃_{j}(x) \\ 0 & δ_{ij} \end{matrix} )
= det(δ_{ij} + ∂_{j}f̃_{i}) = J_{1l+f̃_{m}}(x)  (2.51)

shows deg(1l+f, U, y) = deg(1l+f̃, U, y) = deg(1l+f̃_{m}, U_{m}, y) = deg(1l+f_{m}, U_{m}, y) as desired. □

Let U ⊆ ℝ^{n} and f ∈ C(\overline{U}, ℝ^{n}) be as usual. By Theorem 2.2 we know that deg(f, U, y) is the same for every y in a connected component of ℝ^{n}\f(∂U). We will denote these components by K_{j} and write deg(f, U, y) = deg(f, U, K_{j}) if y ∈ K_{j}.

Theorem 2.19 (Product formula) Let U ⊆ ℝ^{n} be a bounded and open set and denote by G_{j} the connected components of ℝ^{n}\f(∂U). If g ∘ f ∈ D_{y}(U, ℝ^{n}), then

deg(g ∘ f, U, y) = ∑_{j} deg(f, U, G_{j}) deg(g, G_{j}, y),  (2.52)

where only finitely many terms in the sum are nonzero.

Proof. Since f(\overline{U}) is is compact, we can find an r > 0 such that f(\overline{U}) ⊆ B_{r}(0). Moreover, since g^{−1}(y) is closed, g^{−1}(y) ∩ B_{r}(0) is compact and hence can be covered by finitely many components {G_{j}}_{j=1}^{m}. In particular, the others will have deg(f, U, G_{k}) = 0 and hence only finitely many terms in the above sum are nonzero.

We begin by computing deg(g ∘ f, U, y) in the case where f, g ∈ C^{1} and y ∉ CV(g ∘ f). Since (g ∘ f)′(x) = g′(f(x))f′(x) the claim is a straightforward calculation

deg(g ∘ f, U, y) = ∑_{x∈(g∘f)^{−1}(y)} sgn(J_{g∘f}(x))
= ∑_{x∈(g∘f)^{−1}(y)} sgn(J_{g}(f(x)))sgn(J_{f}(x))
= ∑_{z∈g^{−1}(y)} sgn(J_{g}(z)) ∑_{x∈f^{−1}(z)} sgn(J_{f}(x))
= ∑_{z∈g^{−1}(y)} sgn(J_{g}(z)) deg(f, U, z)

and, using our cover {G_{j}}_{j=1}^{m},

deg(g ∘ f, U, y) = ∑_{j=1}^{m} ∑_{z∈g^{−1}(y)∩G_{j}} sgn(J_{g}(z)) deg(f, U, z)

= ∑_{j=1}^{m} deg(f, U, G_{j}) ∑_{z∈g^{−1}(y)∩G_{j}} sgn(J_{g}(z))  (2.53)

= ∑_{j=1}^{m} deg(f, U, G_{j}) deg(g, G_{j}, y).  (2.54)

Moreover, this formula still holds for y ∈ CV(g ∘ f) and for g ∈ C by construction of the Brouwer degree. However, the case f ∈ C will need a closer investigation since the sets G_{j} depend on f. To overcome this problem we will introduce the sets

L_{l} = {z ∈ ℝ^{n}\f(∂U)| deg(f, U, z) = l}.  (2.55)

Observe that L_{l}, l > 0, must be a union of some sets of {G_{j}}_{j=1}^{m}.

Now choose f̃ ∈ C^{1} such that |f(x) − f̃(x)| < 2^{−1}dist(g^{−1}(y), f(∂U)) for x ∈ \overline{U} and define K̃_{j}, L̃_{l} accordingly. Then we have U_{l} ∩ g^{−1}(y) = Ũ_{l} ∩ g^{−1}(y) by Theorem 2.1 (iii). Moreover,

deg(f ∘ g, U, y) = deg(f̃∘ g, U, y) = ∑_{j} deg(f, U, K̃_{j}) deg(g, K̃_{j}, y)
= ∑_{l>0} l deg(g, Ũ_{l}, y) = ∑_{l>0} l deg(g, U_{l}, y)
= ∑_{j} deg(f, U, G_{j}) deg(g, G_{j}, y)  (2.56)

which proves the claim. □

2.7 The Jordan curve theorem

In this section we want to show how the product formula (2.52) for the Brouwer degree can be used to prove the famous Jordan curve theorem which states that a homeomorphic image of the circle dissects ℝ^{2} into two components (which necessarily have the image of the circle as common boundary). In fact, we will even prove a slightly more general result.

Theorem 2.20 Let C_{j} ⊂ ℝ^{n}, j = 1, 2, be homeomorphic compact sets. Then ℝ^{n}\C_{1} and ℝ^{n}\C_{2} have the same number of connected components.

Proof. Denote the components of ℝ^{n}\C_{1} by H_{j} and those of ℝ^{n}\C_{2} by K_{j}. Let h : C_{1} → C_{2} be a homeomorphism with inverse k : C_{2} → C_{1}. By Theorem 2.15 we can extend both to ℝ^{n}. Then Theorem 2.1 (iii) and the product formula imply

1 = deg(k ∘ h, H_{j}, y) = ∑_{l} deg(h, H_{j}, G_{l}) deg(k, G_{l}, y)  (2.57)

for any y ∈ H_{j}. Now we have

⋃_{i} K_{i} = ℝ^{n}\C_{2} ⊆ ℝ^{n}\h(∂H_{j}) = ⋃_{l} G_{l}  (2.58)

and hence fore every i we have K_{i} ⊆ G_{l} for some l since components are maximal connected sets. Let N_{l} = {i|K_{i} ⊆ G_{l}} and observe that we have deg(k, G_{l}, y) = ∑_{i∈N_{l}} deg(k, K_{i}, y) and deg(h, H_{j}, G_{l}) = deg(h, H_{j}, K_{i}) for every i ∈ N_{l}. Therefore,

1 = ∑_{l} ∑_{i∈N_{l}} deg(h, H_{j}, K_{i}) deg(k, K_{i}, y) = ∑_{i} deg(h, H_{j}, K_{i}) deg(k, K_{i}, H_{j})

(2.59)

By reversing the role of C_{1} and C_{2}, the same formula holds with H_{j} and K_{i} interchanged.

Hence

∑_{i} 1 = ∑_{i} ∑_{j} deg(h, H_{j}, K_{i}) deg(k, K_{i}, H_{j}) = ∑_{j} 1  (2.60)

shows that if the number of components of ℝ^{n}\C_{1} or ℝ^{n}\C_{2} is finite, then so is the other and both are equal. Otherwise there is nothing to prove. □

The Leray–Schauder mapping degree

3.1 The mapping degree on finite dimensional Banach spaces

The objective of this section is to extend the mapping degree from ℝ^{n} to general Banach spaces. Naturally, we will first consider the finite dimensional case.

Let X be a (real) Banach space of dimension n and let ϕ be any isomorphism between X and ℝ^{n}. Then, for f ∈ D_{y}(\overline{U}, X), U ⊂ X open, y ∈ X, we can define

deg(f, U, y) = deg(ϕ ∘ f ∘ ϕ^{−1}, ϕ(U), ϕ(y))  (3.1)

provided this definition is independent of the basis chosen. To see this let ψ be a second isomorphism. Then A = ψ ∘ ϕ^{−1} ∈ GL(n). Abbreviate f^{∗} = ϕ ∘ f ∘ ϕ^{−1}, y^{∗} = ϕ(y) and pick f̃^{∗} ∈ C_{y}^{1}(ϕ(\overline{U}), ℝ^{n}) in the same component of D_{y}(ϕ(\overline{U}), ℝ^{n}) as f^{∗} such that y^{∗} ∈ RV(f^{∗}). Then A ∘ f̃^{∗} ∘ A^{−1} ∈ C_{y}^{1}(ψ(U), ℝ^{n}) is the same component of D_{y}(ψ(\overline{U}), ℝ^{n}) as A ∘ f^{∗} ∘ A^{−1} = ψ ∘ f ∘ ψ^{−1} (since A is also a homeomorphism) and

J_{A∘f̃^{∗}∘A^{−1}}(Ay^{∗}) = det(A)Jf̃^{∗}(y^{∗}) det(A^{−1}) = Jf̃^{∗}(y^{∗})  (3.2)

by the chain rule. Thus we have deg(ψ ∘ f ∘ ψ^{−1}, ψ(U), ψ(y)) = deg(ϕ ∘ f ∘ ϕ^{−1}, ϕ(U), ϕ(y)) and our definition is independent of the basis chosen. In addition, it inherits all properties from the mapping degree in ℝ^{n}. Note also that the reduction property holds if ℝ^{m} is replaced by an arbitrary subspace X_{1} since we can always choose ϕ : X → ℝ^{n} such that ϕ(X_{1}) = ℝ^{m}.

Our next aim is to tackle the infinite dimensional case. The general idea is to approximate F by finite dimensional operators (in the same spirit as we approximated continuous f by smooth functions). To do this we need to know which operators can be approximated by finite dimensional operators. Hence we have to recall some basic facts first.

3.2 Compact operators

Let X, Y be Banach spaces and U ⊂ X. An operator F : U ⊂ X → Y is called finite dimensional if its range is finite dimensional. In addition, it is called compact if it is continuous and maps bounded sets into relatively compact ones. The set of all compact operators is denoted by C(U, Y) and the set of all compact, finite dimensional operators is denoted by F(U, Y). Both sets are normed linear spaces and we have F(U, Y) ⊆ C(U, Y) ⊆ C(U, Y).

If U is compact, then C(U, Y) = C(U, Y) (since the continuous image of a compact set is compact) and if dim(Y) < ∞, then F(U, Y) = C(U, Y). In particular, if U ⊂ ℝ^{n} is bounded, then F(\overline{U}, ℝ^{n}) = C(\overline{U}, ℝ^{n}) = C(\overline{U}, ℝ^{n}).

Now let us collect some results to be needed in the sequel.

Lemma 3.1 If K ⊂ X is compact, then for every ε > 0 there is a finite dimensional subspace X_{ε} ⊆ X and a continuous map P_{ε} : K → X_{ε} such that |P_{ε}(x) − x| ≤ ε for all x ∈ K.

Proof. Pick {x_{i}}_{i=1}^{n} ⊆ K such that ⋃_{i=1}^{n} B_{ε}(x_{i}) covers K. Let {ϕ_{i}}_{i=1}^{n} be a partition of unity (restricted to K) subordinate to {B_{ε}(x_{i})}_{i=1}^{n}, that is, ϕ_{i} ∈ C(K, [0, 1]) with supp(ϕ_{i}) ⊂ B_{ε}(x_{i}) and ∑_{i=1}^{n} ϕ_{i}(x) = 1, x ∈ K. Set

P_{ε}(x) = ∑_{i=1}^{n} ϕ_{i}(x)x_{i},  (3.3)

then

|P_{ε}(x) − x| = | ∑_{i=1}^{n} ϕ_{i}(x)x − ∑_{i=1}^{n} ϕ_{i}(x)x_{i}|  (3.4)

≤ ∑_{i=1}^{n} ϕ_{i}(x)|x − x_{i}| ≤ ε.
□

This lemma enables us to prove the following important result.

Theorem 3.2 Let U be bounded, then the closure of F(U, Y) in C(U, Y) is C(U, Y).

Proof. Suppose F_{N} ∈ C(U, Y) converges to F. If F ∉ C(U, Y) then we can find a sequence x_{n} ∈ U such that |F (x_{n}) − F (x_{m})| ≥ ρ > 0 for n ≠ m. If N is so large that |F − F_{N}| ≤ ρ/4, then

|F_{N}(x_{n}) − F_{N}(x_{m})| ≥ |F (x_{n}) − F (x_{m})| − |F_{N}(x_{n}) − F (x_{n})| − |F_{N}(x_{m}) − F (x_{m})|
≥ ρ − 2\frac{ρ}{4} = \frac{ρ}{2}  (3.5)

This contradiction shows \overline{F(U, y)} ⊆ C(U, Y). Conversely, let K = \overline{F (U)} and choose P_{ε} according to Lemma 3.1, then F_{ε} = P_{ε} ∘ F ∈ F(U, Y) converges to F. Hence C(U, Y) ⊆ \overline{F(U, y)} and we are done. □

Finally, let us show some interesting properties of mappings 1l + F, where F ∈ C(U, Y).

Lemma 3.3 Let U be bounded and closed. Suppose F ∈ C(U, Y), then 1l + F is proper (i.e., inverse images of compact sets are compact) and maps closed subsets to closed subsets.

Proof. Let A ⊆ U be closed and y_{n} = (1l + F)(x_{n}) ∈ (1l + F)(A). Since {y_{n} − x_{n}} ⊂ F^{−1}({y_{n}}) we can assume that y_{n} − x_{n} → z after passing to a subsequence and hence x_{n} → x = y − z ∈ A. Since y = x + F (x) ∈ (1l + F)(A), (1l + F)(A) is closed.

Next, let U be closed and K ⊂ X be compact. Let {x_{n}} ⊆ (1l + F)^{−1}(K). Then we can pass to a subsequence y_{n_{m}} = x_{n_{m}} + F (x_{n_{m}}) such that y_{n_{m}} → y. As before this implies x_{n_{m}} → x and thus (1l + F)^{−1}(K) is compact. □

Now we are all set for the definition of the Leray–Schauder degree, that is, for the extension of our degree to infinite dimensional Banach spaces.

3.3 The Leray–Schauder mapping degree

For U ⊂ X we set D_{y}(\overline{U}, X) = \{F ∈ C(\overline{U}, X)|y ∉ (1l + F)(∂U)\} and F_{y}(\overline{U}, X) = \{F ∈ F(\overline{U}, X)|y ∉ (1l+F)(∂U)\}. Note that for F ∈ D_{y}(\overline{U}, X) we have dist(y, (1l+ F)(∂U)) > 0 since 1l + F maps closed sets to closed sets.

Abbreviate ρ = dist(y, (1l + F)(∂U)) and pick F_{1} ∈ F(\overline{U}, X) such that |F − F_{1}| < ρ implying F_{1} ∈ F_{y}(\overline{U}, X). Next, let X_{1} be a finite dimensional subspace of X such that F_{1}(U) ⊂ X_{1}, y ∈ X_{1} and set U_{1} = U ∩ X_{1}. Then we have F_{1} ∈ F_{y}(\overline{U_{1}}, X_{1}) and might define

deg(1l + F, U, y) = deg(1l + F_{1}, U_{1}, y)  (3.6)

provided we show that this definition is independent of F_{1} and X_{1} (as above). Pick another operator F_{2} ∈ F(\overline{U}, X) such that |F − F_{2}| < ρ and let X_{2} be a corresponding finite dimensional subspace as above. Consider X_{0} = X_{1} + X_{2}, U_{0} = U ∩ X_{0}, then F_{i} ∈ F_{y}(\overline{U_{0}}, X_{0}), i = 1, 2, and

deg(1l + F_{i}, U_{0}, y) = deg(1l + F_{i}, U_{i}, y), i = 1, 2,  (3.7)

by the reduction property. Moreover, set H(t) = 1l + (1 − t)F_{1} + t F_{2} implying H(t) ∈, t ∈ [0, 1], since |H(t) − (1l + F)| < ρ for t ∈ [0, 1]. Hence homotopy invariance

deg(1l + F_{1}, U_{0}, y) = deg(1l + F_{2}, U_{0}, y)  (3.8)

shows that (3.6) is independent of F_{1}, X_{1}.

Theorem 3.4 Let U be a bounded open subset of a (real) Banach space X and let F ∈ D_{y}(\overline{U}, X), y ∈ X. Then the following hold true.

(i). deg(1l + F, U, y) = deg(1l + F − y, U, 0).

(ii). deg(1l, U, y) = 1 if y ∈ U.

(iii). If U_{1,2} are open, disjoint subsets of U such that y ∉ f(\overline{U}\(U_{1} ∪ U_{2})), then deg(1l + F, U, y) = deg(1l + F, U_{1}, y) + deg(1l + F, U_{2}, y).

(iv). If H : [0, 1]×\overline{U} → X and y : [0, 1] → X are both continuous such that H(t) ∈ D_{y(t)}(U, ℝ^{n}), t ∈ [0, 1], then deg(1l + H(0), U, y(0)) = deg(1l + H(1), U, y(1)).

Proof. Except for (iv) all statements follow easily from the definition of the degree and the corresponding property for the degree in finite dimensional spaces. Considering H(t, x) − y(t), we can assume y(t) = 0 by (i). Since H([0, 1], ∂U) is compact, we have ρ = dist(y, H([0, 1], ∂U) > 0. By Theorem 3.2 we can pick H_{1} ∈ F([0, 1] × U, X) such that |H(t) − H_{1}(t)| < ρ, t ∈ [0, 1]. this implies deg(1l+H(t), U, 0) = deg(1l+H_{1}(t), U, 0) and the rest follows from Theorem 2.2. □

In addition, Theorem 2.1 and Theorem 2.2 hold for the new situation as well (no changes are needed in the proofs).

Theorem 3.5 Let F, G ∈ D_{y}(U, X), then the following statements hold.

(i). We have deg(1l + F, ∅, y) = 0. Moreover, if U_{i}, 1 ≤ i ≤ N, are disjoint open subsets of U such that y ∉ (1l + F)(\overline{U}\ ⋃_{i=1}^{N} U_{i}), then deg(1l + F, U, y) =

∑_{i=1}^{N} deg(1l + F, U_{i}, y).

(ii). If y ∉ (1l + F)(U), then deg(1l + F, U, y) = 0 (but not the other way round). Equivalently, if deg(1l + F, U, y) ≠ 0, then y ∈ (1l + F)(U).

(iii). If |f(x) − g(x)| < dist(y, f(∂U)), x ∈ ∂U, then deg(f, U, y) = deg(g, U, y). In particular, this is true if f(x) = g(x) for x ∈ ∂U.

(iv). deg(1l + ., U, y) is constant on each component of D_{y}(\overline{U}, X).

(v). deg(1l + F, U, .) is constant on each component of X\f(∂U).

3.4 The Leray–Schauder principle and the Schauder fixed-point theorem

As a first consequence we note the Leray–Schauder principle which says that a priori estimates yield existence.

Theorem 3.6 (Leray–Schauder principle) Suppose F ∈ C(X, X) and any solution x of x = tF (x), t ∈ [0, 1] satisfies the a priori bound |x| ≤ M for some M > 0, then F has a fixed point.

Proof. Pick ρ > M and observe deg(1l + F, B_{ρ}(0), 0) = deg(1l, B_{ρ}(0), 0) = 1 using the compact homotopy H(t, x) = tF (x). Here 0 ∉ H(t, ∂B_{ρ}(0)) due to the a priori bound. □

Now we can extend the Brouwer fixed-point theorem to infinite dimensional spaces as well.

Theorem 3.7 (Schauder fixed point) Let K be a closed, convex, and bounded subset of a Banach space X. If F ∈ C(K, K), then F has at least one fixed point. The result remains valid if K is only homeomorphic to a closed, convex, and bounded subset.

Proof. Since K is bounded, there is a ρ > 0 such that K ⊆ B_{ρ}(0). By Theorem 2.15 we can find a continuous retraction R : X → K (i.e., R(x) = x for x ∈ K) and consider F̃ = F ∘ R ∈ C(\overline{B_{ρ}(0)}, \overline{B_{ρ}(0)}). The compact homotopy H(t, x) = tF̃(x) shows that deg(1l + F̃, B_{ρ}(0), 0) = deg(1l, B_{ρ}(0), 0) = 1. Hence there is a point x_{0} = F̃(x_{0}) ∈ K. Since F̃(x_{0}) = F (x_{0}) for x_{0} ∈ K we are done. □

Finally, let us prove another fixed-point theorem which covers several others as special cases.

Theorem 3.8 Let U ⊂ X be open and bounded and let F ∈ C(\overline{U}, X). Suppose there is an x_{0} ∈ U such that

F (x) − x_{0} ≠ α(x − x_{0}),  x ∈ ∂U, α ∈ (1, ∞).  (3.9)

Then F has a fixed point.

Proof. Consider H(t, x) = x − x_{0} − t(F (x) − x_{0}), then we have H(t, x) ≠ 0 for x ∈ ∂U and t ∈ [0, 1] by assumption. If H(1, x) = 0 for some x ∈ ∂U, then x is a fixed point and we are done. Otherwise we have deg(1l − F, U, 0) = deg(1l − x_{0}, U, 0) = deg(1l, U, x_{0}) = 1 and hence F has a fixed point. □

Now we come to the anticipated corollaries.

Corollary 3.9 Let U ⊂ X be open and bounded and let F ∈ C(\overline{U}, X). Then F has a fixed point if one of the following conditions holds.

1. U = B_{ρ}(0) and F (∂U) ⊆ \overline{U} (Rothe).

2. U = B_{ρ}(0) and |F (x) − x|^{2} ≥ |F (x)|^{2} − |x|^{2} for x ∈ ∂U (Altman).

3. X is a Hilbert space, U = B_{ρ}(0) and ⟨F (x), x⟩ ≤ |x|^{2} for x ∈ ∂U (Krasnosel’skii).

Proof. (1). F (∂U) ⊆ \overline{U} and F (x) = αx for |x| = ρ implies |α|ρ ≤ ρ and hence (3.9) holds. (2). F (x) = αx for |x| = ρ implies (α − 1)^{2}ρ^{2} ≥ (α^{2} − 1)ρ^{2} and hence α ≤ 0. (3). Special case of (2) since |F (x) − x|^{2} = |F (x)|^{2} − 2⟨F (x), x⟩ + |x|^{2}. □

3.5 Applications to integral and differential equations

In this section we want to show how our results can be applied to integral and differential equations. To be able to apply our results we will need to know that certain integral operators are compact.

Lemma 3.10 Suppose I = [a, b] ⊂ ℝ and f ∈ C(I × I × ℝ^{n}, ℝ^{n}), τ ∈ C(I, I), then

F : \begin{matrix}C(I, ℝ^{n}) & → & C(I, ℝ^{n}) \\ x(t) & ↦ & F (x)(t) = ∫_{a}^{τ(t)} f(t, s, x(s))ds \end{matrix}  (3.10)

is compact.

Proof. We first need to prove that F is continuous. Fix x_{0} ∈ C(I, ℝ^{n}) and ε > 0. Set ρ = |x_{0}| + 1 and abbreviate \overline{B} = \overline{B_{ρ}(0)} ⊂ ℝ^{n}. The function f is uniformly continuous on Q = I ×I ×\overline{B} since Q is compact. Hence for ε_{1} = ε/(b−a) we can find a δ ∈ (0, 1] such that |f(t, s, x) − f(t, s, y)| ≤ ε_{1} for |x − y| < δ. But this implies

|F (x) − F (x_{0})| = sup_{t∈I} |∫_{a}^{τ(t)} f(t, s, x(s)) − f(t, s, x_{0}(s))ds|

≤ sup_{t∈I} ∫_{a}^{τ(t)} |f(t, s, x(s)) − f(t, s, x_{0}(s))|ds
≤ sup_{t∈I}(b − a)ε_{1} = ε,  (3.11)

for |x−x_{0}| < δ. In other words, F is continuous. Next we note that if U ⊂ C(I, ℝ^{n}) is bounded, say |U| < ρ, then

|F (U)| ≤ sup_{x∈U} |∫_{a}^{τ(t)} f(t, s, x(s))ds| ≤ (b − a)M,  (3.12)

where M = max |f(I, I, \overline{B_{ρ}(0)})|. Moreover, the family F (U) is equicontinuous. Fix ε and ε_{1} = ε/(2(b − a)), ε_{2} = ε/(2M). Since f and τ are uniformly continuous on I × I × \overline{B_{ρ}(0)} and I, respectively, we can find a δ > 0 such that |f(t, s, x) − f(t_{0}, s, x)| ≤ ε_{1} and |τ(t)−τ(t_{0})| ≤ ε_{2} for |t−t_{0}| < δ. Hence we infer for |t−t_{0}| < δ

|F (x)(t) − F (x)(t_{0})| = |∫_{a}^{τ(t)} f(t, s, x(s))ds − ∫_{a}^{τ(t_{0})} f(t_{0}, s, x(s))ds|

≤ ∫_{a}^{τ(t_{0})} |f(t, s, x(s)) − f(t_{0}, s, x(s))|ds + |∫_{τ(t_{0})}^{τ(t)} |f(t, s, x(s))|ds|
≤ (b − a)ε_{1} + ε_{2}M = ε.  (3.13)

This implies that F (U) is relatively compact by the Arzelà-Ascoli theorem. Thus F is compact. □

As a first application we use this result to show existence of solutions to integral equations.

Theorem 3.11 Let F be as in the previous lemma. Then the integral equation

x − λF (x) = y,  λ ∈ ℝ, y ∈ C(I, ℝ^{n})  (3.14)

has at least one solution x ∈ C(I, ℝ^{n}) if |λ| ≤ ρ/M(ρ), where M(ρ) = (b − a) max_{(s,t,x)∈I×I×\overline{B_{ρ}(0)}} |f(s, t, x − y(s))| and ρ > 0 is arbitrary.

Proof. Note that, by our assumption on λ, λF maps B_{ρ}(y) into itself. Now apply the Schauder fixed-point theorem. □

This result immediately gives the Peano theorem for ordinary differential equations.

Theorem 3.12 (Peano) Consider the initial value problem

ẋ = f(t, x), x(t_{0}) = x_{0},  (3.15)

where f ∈ C(I, ℝ^{n}) and I ⊂ ℝ is an interval containing t_{0}. Then (3.15) has at least one local solution x ∈ C^{1}([t_{0} − ε, t_{0} + ε], ℝ^{n}), ε > 0. For example, any ε satisfying εM(ε, ρ) ≤ ρ, ρ > 0 with M(ε, ρ) = max |f([t_{0} − ε, t_{0} + ε], \overline{B_{ρ}(x_{0})})| works. In addition, if M(ε, ρ) ≤ M̃(ε)(1 + ρ), then there exists a global solution.

Proof. For notational simplicity we make the shift t → t − t_{0}, x → x − x_{0}, f(t, x) → f(t + t_{0}, x + t_{0}) and assume t_{0} = 0, x_{0} = 0. In addition, it suffices to consider t ≥ 0 since t → −t amounts to f → −f.

Now observe, that (3.15) is equivalent to

x(t) − ∫_{0}^{t} f(s, x(s))ds,  x ∈ C([−ε, ε], ℝ^{n})  (3.16)

and the first part follows from our previous theorem. To show the second, fix ε > 0 and assume M(ε, ρ) ≤ M̃(ε)(1 + ρ). Then

|x(t)| ≤ ∫_{0}^{t} |f(s, x(s))|ds ≤ M̃(ε) ∫_{0}^{t}(1 + |x(s)|)ds  (3.17)

implies |x(t)| ≤ exp(M̃(ε)ε) by Gronwall’s inequality. Hence we have an a priori bound which implies existence by the Leary–Schauder principle. Since ε was arbitrary we are done. □

42 Chapter 3. The Leray–Schauder mapping degree

The stationary Navier–Stokes equation

4.1 Introduction and motivation

In this final chapter we turn to partial differential equations. In fact, we will only consider one example, namely the stationary Navier–Stokes equation. Our goal is to use the Leray–Schauder principle to prove an existence and uniqueness result for solutions.

Let U (≠ ∅) be an open, bounded, and connected subset of ℝ^{3}. We assume that U is filled with an incompressible fluid described by its velocity field v_{j}(t, x) and its pressure p(t, x), (t, x) ∈ ℝ × U. The requirement that our fluid is incompressible implies ∂_{j}v_{j} = 0 (we sum over two equal indices from 1 to 3), which follows from the Gauss theorem since the flux trough any closed surface must be zero.

Rather than just writing down the equation, let me give a short physical motivation. To obtain the equation which governs such a fluid we consider the forces acting on a small cube spanned by the points (x_{1}, x_{2}, x_{3}) and (x_{1} + Δx_{1}, x_{2} + Δx_{2}, x_{3} + Δx_{3}). We have three contributions from outer forces, pressure differences, and viscosity.

The outer force density (force per volume) will be denoted by K_{j} and we assume that it is known (e.g. gravity).

The force from pressure acting on the surface through (x_{1}, x_{2}, x_{3}) normal to the x_{1}-direction is pΔx_{2}Δx_{3}δ_{1j}. The force from pressure acting on the opposite surface is −(p + ∂_{1}pΔx_{1})Δx_{2}Δx_{3}δ_{1j}. In summary, we obtain

− (∂_{j}p)ΔV,  (4.1)

where ΔV = Δx_{1}Δx_{2}Δx_{3}.

The viscosity acting on the surface through (x_{1}, x_{2}, x_{3}) normal to the x_{1}- direction is −ηΔx_{2}Δx_{3}∂_{1}v_{j} by some physical law. Here η > 0 is the viscosity constant of the fluid. On the opposite surface we have ηΔx_{2}Δx_{3}∂_{1}(v_{j} + ∂_{1}v_{j}Δx_{1}). Adding up the contributions of all surface we end up with

ηΔV ∂_{i}∂_{i}v_{j}.  (4.2)

Putting it all together we obtain from Newton’s law

ρΔV \frac{d}{dt}v_{j}(t, x(t)) = ηΔV ∂_{i}∂_{i}v_{j}(t, x(t)) − (∂_{j}p(t, x(t)) + ΔV K_{j}(t, x(t)), (4.3)

where ρ > 0 is the density of the fluid. Dividing by ΔV and using the chain rule yields the Navier–Stokes equation

ρ∂_{t}v_{j} = η∂_{i}∂_{i}v_{j} − ρ(v_{i}∂_{i})v_{j} − ∂_{j}p + K_{j}.  (4.4)

Note that it is no restriction to assume ρ = 1.

In what follows we will only consider the stationary Navier–Stokes equation

0 = η∂_{i}∂_{i}v_{j} − (v_{i}∂_{i})v_{j} − ∂_{j}p + K_{j}.  (4.5)

In addition to the incompressibility condition ∂_{j}v_{j} = 0 we also require the boundary condition v|_{∂U} = 0, which follows from experimental observations.

In summary, we consider the problem (4.5) for v in (e.g.) X = \{v ∈ C^{2}(\overline{U}, ℝ^{3})| ∂_{j}v_{j} = 0 and v|_{∂U} = 0}.

Our strategy is to rewrite the stationary Navier–Stokes equation in integral form, which is more suitable for our further analysis. For this purpose we need to introduce some function spaces first.

4.2 An insert on Sobolev spaces

Let U be a bounded open subset of ℝ^{n} and let L^{p}(U, ℝ) denote the Lebesgue spaces of p integrable functions with norm

|u|_{p} = (∫_{U} |u(x)|^{p}dx)^{1/p} .  (4.6)

In the case p = 2 we even have scalar product

⟨u, v⟩_{2} = ∫_{U} u(x)v(x)dx  (4.7)

and our aim is to extend this case to include derivatives.

Given the set C^{1}(U, ℝ) we can consider the scalar product

⟨u, v⟩_{2,1} = ∫_{U} u(x)v(x)dx + ∫_{U}(∂_{j}u)(x)(∂_{j}v)(x)dx.  (4.8)

Taking the completion with respect to the associated norm we obtain the Sobolev space H^{1}(U, ℝ). Similarly, taking the completion of C_{0}^{1}(U, ℝ) with respect to the same norm, we obtain the Sobolev space H_{0}^{1}(U, ℝ). Here C_{0}^{r}(U, Y) denotes the set of functions in C^{r}(U, Y) with compact support. This construction of H^{1}(U, ℝ) implies that a sequence u_{k} in C^{1}(U, ℝ) converges to u ∈ H^{1}(U, ℝ) if and only if u_{k} and all its first order derivatives ∂_{j}u_{k} converge in L^{2}(U, ℝ). Hence we can assign each u ∈ H^{1}(U, ℝ) its first order derivatives ∂_{j}u by taking the limits from above. In order to show that this is a useful generalization of the ordinary derivative, we need to show that the derivative depends only on the limiting function u ∈ L^{2}(U, ℝ). To see this we need the following lemma.

Lemma 4.1 (Integration by parts) Suppose u ∈ H_{0}^{1}(U, ℝ) and v ∈ H^{1}(U, ℝ), then

∫_{U} u(∂_{j}v)dx = − ∫_{U}(∂_{j}u)v dx.  (4.9)

Proof. By continuity it is no restriction to assume u ∈ C_{0}^{1}(U, ℝ) and v ∈ C^{1}(U, ℝ). Moreover, we can find a function ϕ ∈ C_{0}^{1}(U, ℝ) which is 1 on the support of u. Hence by considering ϕv we can even assume v ∈ C_{0}^{1}(U, ℝ).

Moreover, we can replace U by a rectangle K containing U and extend u, v to K by setting it 0 outside U. Now use integration by parts with respect to the j-th coordinate. □

In particular, this lemma says that if u ∈ H^{1}(U, ℝ), then

∫_{U}(∂_{j}u)ϕdx = − ∫_{U} u(∂_{j}ϕ) dx,  ϕ ∈ C_{0}^{∞}(U, ℝ).  (4.10)

And since C_{0}^{∞}(U, ℝ) is dense in L^{2}(U, ℝ), the derivatives are uniquely determined by u ∈ L^{2}(U, ℝ) alone. Moreover, if u ∈ C^{1}(U, ℝ), then the derivative in the Sobolev space corresponds to the usual derivative. In summary, H^{1}(U, ℝ) is the space of all functions u ∈ L^{2}(U, ℝ) which have first order derivatives (in the sense of distributions, i.e., (4.10)) in L^{2}(U, ℝ).

Next, we want to consider some additional properties which will be used later on. First of all, the Poincaré-Friedrichs inequality.

Lemma 4.2 (Poincaré-Friedrichs inequality) Suppose u ∈ H_{0}^{1}(U, ℝ), then

∫_{U} u^{2}dx ≤ d_{j}^{2} ∫_{U}(∂_{j}u)^{2}dx,  (4.11)

where d_{j} = sup{(x_{j} − y_{j})^{2}|(x_{1}, …, x_{n}), (y_{1}, …, y_{n}) ∈ U}.

Proof. Again we can assume u ∈ C_{0}^{1}(U, ℝ) and we assume j = 1 for notational convenience. Replace U by a set K = [a, b] × K̃ containing U and extend u to K by setting it 0 outside U. Then we have

u(x_{1}, x_{2}, …, x_{n})^{2} = (∫_{a}^{x_{1}} 1 · (∂_{1}u)(ξ, x_{2}, …, x_{n})dξ)^{2}
≤ (b − a) ∫_{a}^{b}(∂_{1}u)^{2}(ξ, x_{2}, …, x_{n})dξ,  (4.12)

where we have used the Cauchy-Schwarz inequality. Integrating this result over [a, b] gives

∫_{a}^{b} u^{2}(ξ, x_{2}, …, x_{n})dξ ≤ (b − a)^{2} ∫_{a}^{b}(∂_{1}u)^{2}(ξ, x_{2}, …, x_{n})dξ  (4.13)

and integrating over K̃ finishes the proof. □

Hence, from the view point of Banach spaces, we could also equip H_{0}^{1}(U, ℝ) with the scalar product

⟨u, v⟩ = ∫_{U}(∂_{j}u)(x)(∂_{j}v)(x)dx.  (4.14)

This scalar product will be more convenient for our purpose and hence we will use it from now on. (However, all results stated will hold in either case.) The norm corresponding to this scalar product will be denoted by |.|.

Next, we want to consider the embedding H_{0}^{1}(U, ℝ) ↪ L^{2}(U, ℝ) a little closer. This embedding is clearly continuous since by the Poincaré-Friedrichs inequality we have

|u|_{2} ≤ \frac{d(U)}{\sqrt{n}}|u|,  d(U) = sup\{|x − y| |x, y ∈ U\}.  (4.15)

Moreover, by a famous result of Rellich, it is even compact. To see this we first prove the following inequality.

Lemma 4.3 (Poincaré inequality) Let Q ⊂ ℝ^{n} be a cube with edge length ρ. Then

∫_{Q} u^{2}dx ≤ \frac{1}{ρ^{n}} (∫_{Q} udx)^{2} + \frac{nρ^{2}}{2} ∫_{Q}(∂_{k}u)(∂_{k}u)dx  (4.16)

for all u ∈ H^{1}(Q, ℝ).

Proof. After a scaling we can assume Q = (0, 1)^{n}. Moreover, it suffices to consider u ∈ C^{1}(Q, ℝ).

Now observe

u(x) − u(x̃) = ∑_{i=1}^{n} ∫_{x^{i−1}}^{x^{i}}(∂_{i}u)dx_{i},  (4.17)

where x^{i} = (x̃_{1}, …, x̃_{i}, x_{i+1}, …, x_{n}). Squaring this equation and using Cauchy– Schwarz on the right hand side we obtain

u(x)^{2} − 2u(x)u(x̃) + u(x̃)^{2} ≤ (∑_{i=1}^{n} ∫_{0}^{1} |∂_{i}u|dx_{i})^{2} ≤ n ∑_{i=1}^{n} (∫_{0}^{1} |∂_{i}u|dx_{i})^{2}

≤ n ∑_{i=1}^{n} ∫_{0}^{1}(∂_{i}u)^{2}dx_{i}.  (4.18)

Now we integrate over x and x̃, which gives

2 ∫_{Q} u^{2}dx − 2 (∫_{Q} u dx)^{2} ≤ n ∫_{Q}(∂_{i}u)(∂_{i}u)dx  (4.19)

and finishes the proof. □

Now we are ready to show Rellich’s compactness theorem.

Theorem 4.4 (Rellich’s compactness theorem) Let U be a bounded open subset of ℝ^{n}. Then the embedding

H_{0}^{1}(U, ℝ) ↪ L^{2}(U, ℝ)  (4.20)

is compact.

Proof. Pick a cube Q (with edge length ρ) containing U and a bounded sequence u^{k} ∈ H^{1}(U, ℝ). Since bounded sets are weakly compact, it is no restriction to assume that u^{k} is weakly convergent in L^{2}(U, ℝ). By setting u^{k}(x) = 0 for x ∉ U we can also assume u^{k} ∈ H^{1}(Q, ℝ). Next, subdivide Q into N subcubes Q_{i} with edge lengths ρ/N. On each subcube (4.16) holds and hence

∫_{U} u^{2}dx = ∫_{Q} u^{2}dx = ∑_{i=1}^{N^{n}} \frac{N}{ρ} (∫_{Q_{i}} udx)^{2} + \frac{nρ^{2}}{2N^{2}} ∫_{U}(∂_{k}u)(∂_{k}u)dx  (4.21)

for all u ∈ H^{1}(U, ℝ). Hence we infer

|u^{k} − u^{ℓ}|_{2}^{2} ≤ ∑_{i=1}^{N^{n}} \frac{N}{ρ} (∫_{Q_{i}}(u^{k} − u^{ℓ})dx)^{2} + \frac{nρ^{2}}{2N^{2}} |u^{k} − u^{ℓ}|^{2}.  (4.22)

The last term can be made arbitrarily small by picking N large. The first term converges to 0 since u^{k} converges weakly and each summand contains the L^{2} scalar product of u^{k} − u^{ℓ} and χ_{Q_{i}} (the characteristic function of Q_{i}). □

In addition to this result we will also need the following interpolation inequality.

Lemma 4.5 (Ladyzhenskaya inequality) Let U ⊂ ℝ^{3}. For all u ∈ H_{0}^{1}(U, ℝ) we have

|u|_{4} ≤ ^{4} \sqrt{8}|u|_{2}^{1/4}|u|^{3/4}.  (4.23)

Proof. We first prove the case where u ∈ C_{0}^{1}(U, ℝ). The key idea is to start with U ⊂ ℝ^{1} and then work ones way up to U ⊂ ℝ^{2} and U ⊂ ℝ^{3}.

If U ⊂ ℝ^{1} we have

u(x)^{2} = ∫ ^{x} ∂_{1}u^{2}(x_{1})dx_{1} ≤ 2 ∫ |u∂_{1}u|dx_{1}  (4.24)

and hence

max_{x∈U} u(x)^{2} ≤ 2 ∫ |u∂_{1}u|dx_{1}.  (4.25)

Here, if an integration limit is missing, it means that the integral is taken over the whole support of the function.

If U ⊂ ℝ^{2} we have

∫∫ u^{4}dx_{1}dx_{2} ≤ ∫ max_{x} u(x, x_{2})^{2}dx_{2} ∫ max_{y} u(x_{1}, y)^{2}dx_{1}

≤ 4 ∫∫ |u∂_{1}u|dx_{1}dx_{2} ∫∫ |u∂_{2}u|dx_{1}dx_{2}

≤ 4( ∫∫ u^{2}dx_{1}dx_{2})^{2/2}( ∫∫ (∂_{1}u)^{2}dx_{1}dx_{2})^{1/2}( ∫∫ (∂_{2}u)^{2}dx_{1}dx_{2})^{1/2}

≤ 4 ∫∫ u^{2}dx_{1}dx_{2} ∫∫ ((∂_{1}u)^{2} + (∂_{2}u)^{2})dx_{1}dx_{2}  (4.26)

Now let U ⊂ ℝ^{3}, then

∫∫∫ u^{4}dx_{1}dx_{2}dx_{3} ≤ 4 ∫ dx_{3} ∫∫ u^{2}dx_{1}dx_{2} ∫∫ ((∂_{1}u)^{2} + (∂_{2}u)^{2})dx_{1}dx_{2}

≤ 4 ∫∫ max_{z} u(x_{1}, x_{2}, z)^{2}dx_{1}dx_{2} ∫∫∫ ((∂_{1}u)^{2} + (∂_{2}u)^{2})dx_{1}dx_{2}dx_{3}

≤ 8 ∫∫∫ |u∂_{3}u|dx_{1}dx_{2}dx_{3} ∫∫∫ ((∂_{1}u)^{2} + (∂_{2}u)^{2})dx_{1}dx_{2}dx_{3}  (4.27)

and applying Cauchy–Schwarz finishes the proof for u ∈ C_{0}^{1}(U, ℝ).

If u ∈ H_{0}^{1}(U, ℝ) pick a sequence u_{k} in C_{0}^{1}(U, ℝ) which converges to u in H_{0}^{1}(U, ℝ) and hence in L^{2}(U, ℝ). By our inequality, this sequence is Cauchy in L^{4}(U, ℝ) and converges to a limit v ∈ L^{4}(U, ℝ). Since |u|_{2} ≤ ^{4} \sqrt{|U|}|u|_{4} (∫ 1 · u^{2}dx ≤

\sqrt{∫ 1 dx ∫ u^{4}dx}), u_{k} converges to v in L^{2}(U, ℝ) as well and hence u = v. Now take the limit in the inequality for u_{k}. □

As a consequence we obtain

|u|_{4} ≤ (\frac{8d(U)}{\sqrt{3}})^{1/4} |u|,  U ⊂ ℝ^{3},  (4.28)

and

Corollary 4.6 The embedding

H_{0}^{1}(U, ℝ) ↪ L^{4}(U, ℝ),  U ⊂ ℝ^{3},  (4.29)

is compact.

Proof. Let u_{k} be a bounded sequence in H_{0}^{1}(U, ℝ). By Rellich’s theorem there is a subsequence converging in L^{2}(U, ℝ). By the Ladyzhenskaya inequality this subsequence converges in L^{4}(U, ℝ). □

Our analysis clearly extends to functions with values in ℝ^{n} since H_{0}^{1}(U, ℝ^{n}) = ⊕_{j=1}^{n}H_{0}^{1}(U, ℝ).

4.3 Existence and uniqueness of solutions

Now we come to the reformulation of our original problem (4.5). We pick as underlying Hilbert space H_{0}^{1}(U, ℝ^{3}) with scalar product

⟨u, v⟩ = ∫_{U}(∂_{j}u_{i})(∂_{j}v_{i})dx.  (4.30)

Let X be the closure of X in H_{0}^{1}(U, ℝ^{3}), that is,

X = \overline{\{v ∈ C^{2}(\overline{U}, ℝ^{3})|∂_{j}v_{j} = 0 and v|_{∂U} = 0\}} = \{v ∈ H_{0}^{1}(U, ℝ^{3})|∂_{j}v_{j} = 0\}. (4.31)

Now we multiply (4.5) by w ∈ X and integrate over U

∫_{U} (η∂_{k}∂_{k}v_{j} − (v_{k}∂_{k})v_{j} + K_{j})w_{j} dx = ∫_{U}(∂_{j}p)w_{j} dx = 0.  (4.32)

Using integration by parts this can be rewritten as

∫_{U} (η(∂_{k}v_{j})(∂_{k}w_{j}) − v_{k}v_{j}(∂_{k}w_{j}) − K_{j}w_{j})dx = 0.  (4.33)

Hence if v is a solution of the Navier-Stokes equation, then it is also a solution of

η⟨v, w⟩ − a(v, v, w) − ∫_{U} Kw dx = 0, for all w ∈ X ,  (4.34)

where

a(u, v, w) = ∫_{U} u_{k}v_{j}(∂_{k}w_{j}) dx.  (4.35)

In other words, (4.34) represents a necessary solubility condition for the Navier-Stokes equations. A solution of (4.34) will also be called a weak solution of the Navier-Stokes equations. If we can show that a weak solution is in C^{2}, then we can read our argument backwards and it will be also a classical solution. However, in general this might not be true and it will only solve the Navier-Stokes equations in the sense of distributions. But let us try to show existence of solutions for (4.34) first.

For later use we note

a(v, v, v) = ∫_{U} v_{k}v_{j}(∂_{k}v_{j}) dx = \frac{1}{2} ∫_{U} v_{k}∂_{k}(v_{j}v_{j}) dx
= −\frac{1}{2} ∫_{U}(v_{j}v_{j})∂_{k}v_{k} dx = 0,  v ∈ X .  (4.36)

We proceed by studying (4.34). Let K ∈ L^{2}(U, ℝ^{3}), then ∫_{U} Kw dx is a linear functional on X and hence there is a K̃ ∈ X such that

∫_{U} Kw dx = ⟨K̃, w⟩,  w ∈ X .  (4.37)

Moreover, the same is true for the map a(u, v, .), u, v ∈ X, and hence there is an element B(u, v) ∈ X such that

a(u, v, w) = ⟨B(u, v), w⟩,  w ∈ X .  (4.38)

In addition, the map B : X^{2} → X is bilinear. In summary we obtain

⟨ηv − B(v, v) − K̃, w⟩ = 0,  w ∈ X ,  (4.39)

and hence

ηv − B(v, v) = K̃.  (4.40)

So in order to apply the theory from our previous chapter, we need a Banach space Y such that X ↪ Y is compact.

Let us pick Y = L^{4}(U, ℝ^{3}). Then, applying the Cauchy-Schwarz inequality twice to each summand in a(u, v, w) we see

|a(u, v, w)| ≤ ∑_{j,k} ( ∫_{U}(u_{k}v_{j})^{2}dx)^{1/2}( ∫_{U}(∂_{k}w_{j})^{2}dx)^{1/2}

≤ |w| ∑_{j,k} ( ∫_{U}(u_{k})^{4}dx)^{1/4}( ∫_{U}(v_{j})^{4}dx)^{1/4} = |u|_{4}|v|_{4}|w|. (4.41)

Moreover, by Corollary 4.6 the embedding X ↪ Y is compact as required. Motivated by this analysis we formulate the following theorem.

Theorem 4.7 Let X be a Hilbert space, Y a Banach space, and suppose there is a compact embedding X ↪ Y. In particular, |u|_{Y} ≤ β|u|. Let a : X^{3} → ℝ be a multilinear form such that

|a(u, v, w)| ≤ α|u|_{Y} |v|_{Y} |w|  (4.42)

and a(v, v, v) = 0. Then for any K̃ ∈ X, η > 0 we have a solution v ∈ X to the problem

η⟨v, w⟩ − a(v, v, w) = ⟨K̃, w⟩, w ∈ X . (4.43)

Moreover, if 2αβ|K̃| < η^{2} this solution is unique.

Proof. It is no loss to set η = 1. Arguing as before we see that our equation is equivalent to

v − B(v, v) + K̃ = 0,  (4.44)

where our assumption (4.42) implies

|B(u, v)| ≤ α|u|_{Y} |v|_{Y} ≤ αβ^{2}|u||v|  (4.45)

Here the second equality follows since the embedding X ↪ Y is continuous.

Abbreviate F (v) = B(v, v). Observe that F is locally Lipschitz continuous since if |u|, |v| ≤ ρ we have

|F (u)−F (v)| = |B(u−v, u)−B(v, u−v)| ≤ 2α ρ |u−v|_{Y} ≤ 2αβ^{2} ρ|u−v|. (4.46)

Moreover, let v_{n} be a bounded sequence in X. After passing to a subsequence we can assume that v_{n} is Cauchy in Y and hence F (v_{n}) is Cauchy in X by |F (u) − F (v)| ≤ 2α ρ|u − v|_{Y}. Thus F : X → X is compact.

Hence all we need to apply the Leray-Schauder principle is an a priori estimate. Suppose v solves v = tF (v) + tK̃, t ∈ [0, 1], then

⟨v, v⟩ = t a(v, v, v) + t⟨K̃, v⟩ = t⟨K̃, v⟩.  (4.47)

Hence |v| ≤ |K̃| is the desired estimate and the Leray-Schauder principle yields existence of a solution.

Now suppose there are two solutions v_{i}, i = 1, 2. By our estimate they satisfy |v_{i}| ≤ |K̃| and hence |v_{1} − v_{2}| = |F (v_{1}) − F (v_{2})| ≤ 2αβ^{2} |K̃||v_{1} − v_{2}| which is a contradiction if 2αβ^{2} |K̃| < 1. □

Hence we have found a solution v to the generalized problem (4.34). This solution is unique if 2(\frac{2d(U)}{\sqrt{3}})^{3/2}|K|_{2} < η^{2}. Under suitable additional conditions on the outer forces and the domain, it can be shown that weak solutions are C^{2} and thus also classical solutions. However, this is beyond the scope of this introductory text.

Monotone operators

5.1 Monotone operators

The Leray–Schauder theory can only be applied to compact perturbations of the identity. If F is not compact, we need different tools. In this section we briefly present another class of operators, namely monotone ones, which allow some progress.

If F : ℝ → ℝ is continuous and we want F (x) = y to have a unique solution for every y ∈ ℝ, then f should clearly be strictly monotone increasing (or decreasing) and satisfy lim_{x→±∞} F (x) = ±∞. Rewriting these conditions slightly such that they make sense for vector valued functions the analogous result holds.

Lemma 5.1 Suppose F : ℝ^{n} → ℝ^{n} is continuous and satisfies

lim_{|x|→∞} \frac{F (x)x}{|x|} = ∞.  (5.1)

Then the equation

F (x) = y  (5.2)

has a solution for every y ∈ ℝ^{n}. If F is strictly monotone

(F (x) − F (y))(x − y) > 0,  x ≠ y,  (5.3)

then this solution is unique.

Proof. Our first assumption implies that G(x) = F (x) − y satisfies G(x)x = F (x)x − yx > 0 for |x| sufficiently large. Hence the first claim follows from Theorem 2.13. The second claim is trivial. □

Now we want to generalize this result to infinite dimensional spaces. Throughout this chapter, X will be a Hilbert space with scalar product ⟨., ..⟩. An operator F : X → X is called monotone if

⟨F (x) − F (y), x − y⟩ ≥ 0,  x, y ∈ X,  (5.4)

strictly monotone if

⟨F (x) − F (y), x − y⟩ > 0,  x ≠ y ∈ X,  (5.5)

and finally strongly monotone if there is a constant C > 0 such that

⟨F (x) − F (y), x − y⟩ ≥ C|x − y|^{2},  x, y ∈ X.  (5.6)

Note that the same definitions can be made if X is a Banach space and F : X → X^{∗}.

Observe that if F is strongly monotone, then it automatically satisfies

lim_{|x|→∞} \frac{⟨F (x), x⟩}{|x|} = ∞.  (5.7)

(Just take y = 0 in the definition of strong monotonicity.) Hence the following result is not surprising.

Theorem 5.2 (Zarantonello) Suppose F ∈ C(X, X) is (globally) Lipschitz continuous and strongly monotone. Then, for each y ∈ X the equation

F (x) = y  (5.8)

has a unique solution x ∈ X.

Proof. Set

G(x) = x − t(F (x) − y),  t > 0,  (5.9)

then F (x) = y is equivalent to the fixed point equation

G(x) = x.  (5.10)

It remains to show that G is a contraction. We compute

|G(x) − G(x̃)|^{2} = |x − x̃|^{2} − 2t⟨F (x) − F (x̃), x − x̃⟩ + t^{2}|F (x) − F (x̃)|^{2}
≤ (1 − 2\frac{C}{L} (Lt) + (Lt)^{2})|x − x̃|^{2},  (5.11)

where L is a Lipschitz constant for F (i.e., |F (x) − F (x̃)| ≤ L|x − x̃|). Thus, if t ∈ (0, \frac{2C}{L}), G is a contraction and the rest follows from the contraction principle. □

Again observe that our proof is constructive. In fact, the best choice for t is clearly t = \frac{C}{L} such that the contraction constant θ = 1 − (\frac{C}{L})^{2} is minimal. Then the sequence

x_{n+1} = x_{n} − (1 − (\frac{C}{L})^{2})(F (x_{n}) − y), x_{0} = x,  (5.12)

converges to the solution.

5.2 The nonlinear Lax–Milgram theorem

As a consequence of the last theorem we obtain a nonlinear version of the Lax– Milgram theorem. We want to investigate the following problem:

a(x, y) = b(y),  for all y ∈ X,  (5.13)

where a : X^{2} → ℝ and b : X → ℝ. For this equation the following result holds.

Theorem 5.3 (Nonlinear Lax–Milgram theorem) Suppose b ∈ L(X, ℝ) and a(x, .) ∈ L(X, ℝ), x ∈ X, are linear functionals such that there are positive constants L and C such that for all x, y, z ∈ X we have

a(x, x − y) − a(y, x − y) ≥ C|x − y|^{2}  (5.14)

and

|a(x, z) − a(y, z)| ≤ L|z||x − y|.  (5.15)

Then there is a unique x ∈ X such that (5.13) holds.

Proof. By the Riez theorem there are elements F (x) ∈ X and z ∈ X such that a(x, y) = b(y) is equivalent to ⟨F (x) − z, y⟩ = 0, y ∈ X, and hence to

F (x) = z.  (5.16)

By (5.14) the operator F is strongly monotone. Moreover, by (5.15) we infer

|F (x) − F (y)| = sup_{x̃∈X,|x̃|=1} |⟨F (x) − F (y), x̃⟩| ≤ L|x − y|  (5.17)

that F is Lipschitz continuous. Now apply Theorem 5.2. □

The special case where a ∈ L^{2}(X, ℝ) is a bounded bilinear form which is strongly continuous, that is,

a(x, x) ≥ C|x|^{2},  x ∈ X,  (5.18)

is usually known as (linear) Lax–Milgram theorem.

The typical application this theorem is the existence of a unique weak solution of the Dirichlet problem for elliptic equations

∂_{i}A_{ij}(x)∂_{j}u(x) + b_{j}(x)∂_{j}u(x) + c(x)u(x) = f(x), x ∈ U,
u(x) = 0, x ∈ ∂U,  (5.19)

where U is a bounded open subset of ℝ^{n}. By elliptic we mean that all coefficients A, b, c plus the right hand side f are bounded and a_{0} > 0, where

a_{0} = inf_{e∈S^{n},x∈U} e_{i}A_{ij}(x)e_{j}, b_{0} = − inf_{x∈U} b(x), c_{0} = inf_{x∈U} c(x).  (5.20)

As in Section 4.3 we pick H_{0}^{1}(U, ℝ) with scalar product

⟨u, v⟩ = ∫_{U}(∂_{j}u)(∂_{j}v)dx  (5.21)

as underlying Hilbert space. Next we multiply (4.5) by v ∈ H_{0}^{1} and integrate over U

∫_{U} (∂_{i}A_{ij}(x)∂_{j}u(x) + b_{j}(x)∂_{j}u(x) + c(x)u(x))v(x) dx = ∫_{U} f(x)v(x) dx. (5.22)

After a partial integration we can write this equation as

a(v, u) = f(v),  v ∈ H_{0}^{1},  (5.23)

where

a(v, u) = ∫_{U} (∂_{i}v(x)A_{ij}(x)∂_{j}u(x) + b_{j}(x)∂_{j}u(x)v(x) + c(x)u(x))dx
f(v) = ∫_{U} f(x)v(x) dx,  (5.24)

We call a solution of (5.23) a weak solution of the elliptic Dirichlet problem (5.19).

By a simple use of the Cauchy-Schwarz and Poincaré-Friedrichs inequalities we see that the bilinear form a(u, v) is bounded. To be able to apply the (linear) Lax–Milgram theorem we need to show that it satisfies a(u, u) ≥ ∫ |∂_{j}u|dx.

Using (5.20) we have

a(u, u) ≥ ∫_{U} (a_{0}|∂_{j}u|^{2} − b_{0}|u||∂_{j}u| + c_{0}|u|^{2}),  (5.25)

where −b_{0} = inf b(x), c_{0} = inf c(x) and we need to control the middle term. If b_{0} ≤ 0 there is nothing to do and it suffices to require c_{0} ≥ 0.

If b_{0} > 0 we distribute the middle term by means of the elementary inequality

|u||∂_{j}u| ≤ \frac{ε}{2}|u|^{2} + \frac{1}{2ε}|∂_{j}u|^{2}  (5.26)

which gives

a(u, u) ≥ ∫_{U} ((a_{0} − \frac{b_{0}}{2ε})|∂_{j}u|^{2} + (c_{0} − \frac{εb_{0}}{2} )|u|^{2}).  (5.27)

Since we need a_{0} − \frac{b_{0}}{2ε} > 0 and c_{0} − \frac{εb_{0}}{2} ≥ 0, or equivalently \frac{2c_{0}}{b_{0}} ≥ ε > \frac{b_{0}}{2a_{0}} , we see that we can apply the Lax–Milgram theorem if 4a_{0}c_{0} > b_{0}^{2}. In summary, we have proven

Theorem 5.4 The elliptic Dirichlet problem (5.19) has a unique weak solution u ∈ H_{0}^{1}(U, ℝ) if a_{0} > 0, b_{0} ≤ 0, c_{0} ≥ 0 or 4a_{0}c_{0} > b_{0}^{2}.

5.3 The main theorem of monotone operators

Now we return to the investigation of F (x) = y and weaken the conditions of Theorem 5.2. We will assume that X is a separable Hilbert space and that F : X → X is a continuous monotone operator satisfying

lim_{|x|→∞} \frac{⟨F (x), x⟩}{|x|} = ∞.  (5.28)

In fact, if suffices to assume that F is weakly continuous

lim_{n→∞}⟨F (x_{n}), y⟩ = ⟨F (x), y⟩,  for all y ∈ X  (5.29)

whenever x_{n} → x.

The idea is as follows: Start with a finite dimensional subspace X_{n} ⊂ X and project the equation F (x) = y to X_{n} resulting in an equation

F_{n}(x_{n}) = y_{n},  x_{n}, y_{n} ∈ X_{n}.  (5.30)

More precisely, let P_{n} be the (linear) projection onto X_{n} and set F_{n}(x_{n}) = P_{n}F (x_{n}), y_{n} = P_{n}y (verify that F_{n} is continuous and monotone!).

Now Lemma 5.1 ensures that there exists a solution u_{n}. Now chose the subspaces X_{n} such that X_{n} → X (i.e., X_{n} ⊂ X_{n+1} and ⋃_{n=1}^{∞} X_{n} is dense). Then our hope is that u_{n} converges to a solution u.

This approach is quite common when solving equations in infinite dimensional spaces and is known as Galerkin approximation. It can often be used for numerical computations and the right choice of the spaces X_{n} will have a significant impact on the quality of the approximation.

So how should we show that x_{n} converges? First of all observe that our construction of x_{n} shows that x_{n} lies in some ball with radius R_{n}, which is chosen such that

⟨F_{n}(x), x⟩ > |y_{n}||x|,  |x| ≥ R_{n}, x ∈ X_{n}.  (5.31)

Since ⟨F_{n}(x), x⟩ = ⟨P_{n}F (x), x⟩ = ⟨F (x), P_{n}x⟩ = ⟨F (x), x⟩ for x ∈ X_{n} we can drop all n’s to obtain a constant R which works for all n. So the sequence x_{n} is uniformly bounded

|x_{n}| ≤ R.  (5.32)

Now a well-known result there exists a weakly convergent subsequence. That is, after dropping some terms, we can assume that there is some x such that x_{n} ⇀ x, that is,

⟨x_{n}, z⟩ → ⟨x, z⟩,  for every z ∈ X.  (5.33)

And it remains to show that x is indeed a solution. This follows from

Lemma 5.5 Suppose F : X → X is weakly continuous and monotone, then

⟨y − F (z), x − z⟩ ≥ 0 for every z ∈ X  (5.34)

implies F (x) = y.

Proof. Choose z = x ± tw, then ∓⟨y − F (x ± tw), w⟩ ≥ 0 and by continuity ∓⟨y −F (x), w⟩ ≥ 0. Thus ⟨y −F (x), w⟩ = 0 for every w implying y −F (x) = 0. □

Now we can show Theorem 5.6 (Browder, Minty) Suppose F : X → X is weakly continuous, monotone, and satisfies

lim_{|x|→∞} \frac{F (x)x}{|x|} = ∞.  (5.35)

Then the equation

F (x) = y  (5.36)

has a solution for every y ∈ X. If F is strictly monotone then this solution is unique.

Proof. Abbreviate y_{n} = F (x_{n}), then we have ⟨y − F (z), x_{n} − z⟩ = ⟨y_{n} − F_{n}(z), x_{n} − z⟩ ≥ 0 for z ∈ X_{n}. Taking the limit implies ⟨y − F (z), x − z⟩ ≥ 0 for every z ∈ X_{∞} = ⋃_{n=1}^{∞} X_{n}. Since X_{∞} is dense, ⟨y − F (z), x − z⟩ ≥ 0 for every z ∈ X by continuity and hence F (x) = y by our lemma. □

Note that in the infinite dimensional case we need monotonicity even to show existence. Moreover, this result can be further generalized in two more ways. First of all, the Hilbert space X can be replaced by a reflexive Banach space if F : X → X^{∗}. The proof is almost identical. Secondly, it suffices if

t ↦ ⟨F (x + ty), z⟩  (5.37)

is continuous for t ∈ [0, 1] and all x, y, z ∈ X, since this condition together with monotonicity can be shown to imply weak continuity.

60 Chapter 5. Monotone operators

Bibliography

[1] M. Berger and M. Berger, Perspectives in Nonlinearity, Benjamin, New York, 1968.

[2] L. C. Evans, Weak Convergence Methods for nonlinear Partial Differential Equations, CBMS 74, American Mathematical Society, Providence, 1990.

[3] S.-N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer, New York, 1982.

[4] K. Deimling, Nichtlineare Gleichungen und Abbildungsgrade, Springer, Berlin, 1974.

[5] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.

[6] J. Franklin, Methods of Mathematical Economics, Springer, New York 1980.

[7] O.A. Ladyzhenskaya, The Boundary Values Problems of Mathematical Physics, Springer, New York, 1985.

[8] N. Lloyd, Degree Theory, Cambridge University Press, London, 1978.

[9] J.J. Rotman, Introduction to Algebraic Topology, Springer, New York, 1988.

[10] M. Ru̇žička, Nichtlineare Funktionalanalysis, Springer, Berlin, 2004.

[11] E. Zeidler, Applied Functional Analysis: Applications to Mathematical Physics, Springer, New York 1995.

[12] E. Zeidler, Applied Functional Analysis: Main Principles and Their Applications, Springer, New York 1995.

62 Bibliography 63 64 Glossary of notations

Glossary of notations

B_{ρ}(x)  … ball of radius ρ around x
conv(.)  … convex hull
C(U, Y)  … set of continuous functions from U to Y, 1
C^{r}(U, Y)  … set of r times continuously differentiable functions, 2
C_{0}^{r}(U, Y)  … functions in C^{r} with compact support, 45
C(U, Y)  … set of compact functions from U to Y, 34
CP(f)  … critical points of f, 13
CS(K)  … nonempty convex subsets of K, 26
CV(f)  … critical values of f, 13
deg(D, f, y) … mapping degree, 13, 22
det  … determinant
dim  … dimension of a linear space
div  … divergence
dist(U, V)  = inf_{(x,y)∈U×V} |x − y| distance of two sets
D_{y}^{r}(U, Y)  … functions in C^{r}(\overline{U}, Y) which do not attain y on the boundary.
dF  … derivative of F, 1
F(X, Y)  … set of compact finite dimensional functions, 34
GL(n)  … general linear group in n dimensions
H(ℂ)  … set of holomorphic functions, 11
H^{1}(U, ℝ^{n}) … Sobolev space, 45
H_{0}^{1}(U, ℝ^{n}) … Sobolev space, 45
inf  … infimum
J_{f}(x)  = det f′(x) Jacobi determinant of f at x, 13
L(X, Y)  … set of bounded linear functions, 1
L^{p}(U, ℝ^{n}) … Lebesgue space of p integrable functions, 44
max  … maximum
n(γ, z_{0})  … winding number
O(.)  … Landau symbol, f = O(g) iff lim sup_{x→x_{0}} |f(x)/g(x)| < ∞
o(.)  … Landau symbol, f = o(g) iff lim_{x→x_{0}} |f(x)/g(x)| = 0

∂U  … boundary of the set U
∂_{x}F (x, y) … partial derivative with respect to x, 1
RV(f)  … regular values of f, 13
R(I, X) … set of regulated functions, 4
S(I, X) … set of simple functions, 4
sgn  … sign of a number
sup  … supremum
supp  … support of a functions

Index

Arzelà-Ascoli theorem, 40

Best reply, 27 Brouwer fixed-point theorem, 24

Chain rule, 2 Characteristic function, 4 Compact operator, 34 Contraction principle, 5 Critical values, 13

Derivative, 1

partial, 1 Diffeomorphism, 2 Differentiable, 1 Differential equations, 8 Distribution, 46

Elliptic equation, 56 Embedding, 48 Equilibrium

Nash, 28

Finite dimensional operator, 34 Fixed-point theorem

Altman, 38 Brouwer, 24 contraction principle, 5 Kakutani, 26 Krasnosel’skii, 38 Rothe, 38 Schauder, 37 Functional, linear, 5

Galerkin approximation, 58 Gronwall’s inequality, 41

Holomorphic function, 11 Homotopy, 12 Homotopy invariance, 13

Implicit function theorem, 7 Integral, 4 Integration by parts, 45 Inverse function theorem, 8

Jordan curve theorem, 31

Kakutani’s fixed-point theorem, 26

Ladyzhenskaya inequality, 48 Landau symbols, 1 Lax–Milgram theorem, 55 Leray–Schauder principle, 37

Mean value theorem, 2 monotone, 54

operator, 53 strictly, 54 strongly, 54

Multilinear function, 3

Nash equilibrium, 28 Nash theorem, 28 Navier–Stokes equation, 44 \underline{Index 67}

stationary, 44 n-person game, 27

Payoff, 27 Peano theorem, 40 Poincaré inequality, 47 Poincaré-Friedrichs inequality, 46 Prisoners dilemma, 28 Proper, 35

Reduction property, 29 Regular values, 13 Regulated function, 4 Rellich’s compactness theorem, 48 Rouchés theorem, 12

Sard’s theorem, 17 Simple function, 4 Stokes theorem, 19 Strategy, 27 Symmetric multilinear function, 3

Uniform contraction principle, 6

Weak solution, 50, 56 Winding number, 11