Fixed Point Theorems John Hillas University of Auckland

Fixed Point Theorems

John Hillas University of Auckland

Contents

Chapter 1. Fixed Point Theorems 1 1. Deﬁnitions 1 2. The Contraction Mapping Theorem 5 3. Brouwer’s Theorem 6 4. Kakutani’s Theorem 6 5. Schauder’s Theorem 7 6. The Fan-Glicksberg Theorem 8 Bibliography 11

i CHAPTER 1

Fixed Point Theorems

These notes were written as a reference for the Intensive Course on the Theory of Strategic Equilibrium, given at SUNY at Stony Brook in the summer of 1990. They are not complete and are not intended, nor are they suitable, as a substitute for a good book on the topic. I have found the part of Franklin (1980) on fixed point theorems to be very good for the intuition behind many of the results covered in these notes. It also contains three different proofs of Brouwer’s Theorem, which is not proved here. I have borrowed heavily from a number of sources. Most of the proof of Schaud- er’s Theorem comes from Franklin (1980). The proof of Kakutani’s Theorem is a mixture of Franklin (1980) and Kakutani (1941). The proof of the Fan-Glicksberg Theorem is from Glicksberg (1952). Many of the definitions in the following section come from Franklin (1980), Kelley (1955), and Zeidler (1986).

1. Definitions Before starting to discuss the fixed point theorems that are the subject of these notes we state for reference a number of relevant definitions. These will, at best, serve as a revision. If you are not familiar with these concepts you should consult a good text. A standard text for much of the material is Kelley (1955). The concept of a topological vector space (also called a topological linear space) is covered in most books on functional analysis, for example Rudin (1973). The material on correspondences or multivalued functions is less standard in the mathematics liter- ature, though it is of central importance in economics and game theory. Some of this material is covered in Berge (1963). The theory of correspondences is covered in great detail in Klein and Thompson (1984). Definition 1. A topological space is a pair (X, τ) where X is a set and τ is a collection of subsets of X satisfying (i) both X and belong to τ, and (ii) the intersection∅ of a finite number and the union of an arbitrary number of sets in τ belong to τ. The sets in τ are called open sets and τ is called the topology. An open neighbourhood of a point x X is an open set containing x. The complement of an open set is called a closed∈ set.

Definition 2. Let (X, τ) and (X0, τ 0) be two topological spaces. The function 1 f : X X0 is said to be continuous if for all sets A τ 0 the set f − (A) = x X f(→x) A τ. We express this by saying the inverse∈ images of open sets{ are∈ open.| ∈ } ∈

Definition 3. A collection of open sets Oα α A is called an open cover of { } ∈ the set M if M α AOα. A set M is compact if every open cover of M has a ⊂ ∪ ∈ ﬁnite subcover, i.e., a ﬁnite subcollection of Oα α A say O1,...,Ok such that M k O . { } ∈ { } ⊂ ∪i=1 i

1 2 1. FIXED POINT THEOREMS

Definition 4. A topological space (X, τ) is called a Hausdorff space if for any two points x, y X with x = y there are open neighbourhoods of x and y say U(x) and U(y) such that∈ U(x) 6 U(y) = . ∩ ∅ Just as we can speak of convergent sequences in a metric space (see below) we can define convergent sequences in a topological space. It happens that this is not the most useful notion of convergence for a topological space. A somewhat more general notion is the concept of Moore-Smith convergence defined below. Definition 5. Sequential Convergence. Let (X, τ) be a topological space. A sequence (xn) in X is a function from N the natural numbers to X. We say that almost all x lie in M if and only if there is N such that x M for n n ∈ all n N. Also (xn) is frequently in M if and only if for every m there is n m such that≥ x M. ≥ n ∈ A sequence (xn) converges to x if and only if every neighbourhood of x • contains almost all x . We write x x. n n → A point x is a cluster point of (xn) if and only if this sequence is frequently • in every neighbourhood of x. A function f : X Y is sequentially continuous at x if f(xn) f(x) • whenever x x. → → n → The set M is sequentially closed if whenever xn M for all n and xn x • then x M ∈ → ∈ Unfortunately the concepts of “sequentially continuous” and “sequentially closed” do not coincide with the concepts of “continuous” and “closed.” We can however characterize these concepts in terms of Moore-Smith sequences. Definition 6. A set A is directed if and only if there is a relation defined on certain pairs (α, β) with α, β A such that for all elements of A ∈ (i) α α, (ii) if α β and β γ then α γ, and (iii) for α, β A there exists δ A such that α δ and β δ. ∈ ∈ Definition 7. Let X be a topological space and A a directed set. A Moore- Smith sequence (xα) is a function from A to X. We say that almost all x lie in M if and only if there is γ such that x M α α ∈ for all γ α. Also (xα) is frequently in M if and only if for every γ there is α such that γ α such that x M. (Recall the definition for sequences above.) α ∈ A section of the Moore-Smith sequence (xα) is a set xα α γ for some fixed γ. { | } Let (xα) and (yβ) be Moore-Smith sequences with index sets A and B respec- tively. Then (yβ) is a Moore-Smith subsequence of (xα) if and only if every section of (xα) contains almost all yβ. Convergence and cluster points are defined as for sequences.

Remark 1. Moore-Smith sequences are sometimes called nets. (For example, in Kelley (1955).)

Proposition 1. A topological space is Hausdorﬀ if and only if each Moore- Smith sequence converges to at most one point.

Proposition 2. Let X be a topological space. The set M X is compact if and only if every Moore-Smith sequence in M has a cluster point⊂ in M. 1. DEFINITIONS 3

Definition 8. A metric space is a pair (X, d) where X is a set and d : X X [0, ) is a function satisfying × → ∞ (i) d(x, y) = d(y, x), (ii) d(x, z) d(x, y) + d(y, z), and (iii) d(x, y)≤ = 0 if and only if x = y. The function d is called a metric. Property (ii) is called the triangle inequality.

Definition 9. The open -ball about the point x in the metric space (X, d) denoted B (x) is the set y X d(x, y) < . A set A X is open if for any { ∈ | } ⊂ x A there is some > 0 such that B(x) A. A set is closed if its complement is∈ open. ⊂ A sequence of points x , x ,... in X is said to converge to x X if for any 1 2 ∈ > 0 there is some integer N such that for all n > N xn B(x). The point x is called the limit of the sequence. Any sequence with a limit∈ is called a convergent sequence. This allows an alternate deﬁnition of a closed set. A set is closed if every convergent sequence contained in the set converges to a point in the set.

Definition 10. A sequence x1, x2,... in a metric space (X, d) is called a Cauchy sequence if limp,q d(xp, xq) = 0. →∞

Definition 11. A metric space (X, d) is called a complete metric space if every Cauchy sequence in (X, d) converges (to a limit in X).

Definition 12. Let (X, d) be a metric space. A function f : X X is called a contraction if there is some 0 θ < 1 such that for all x, y X d(→f(x), f(y)) θd(x, y). ≤ ∈ ≤

Definition 13. A (real) linear space or vector space is a set X together with two operations, addition and scalar multiplication such that for all x, y X and ∈ all α R both x + y and αx are in X, and for all x, y, z X and all α, β R the following∈ properties are satisﬁed ∈ ∈ (i) x + y = y + x, (ii) (x + y) + z = x + (y + z), (iii) (α + β)x = αx + βx, (iv) α(x + y) = αx + αy, (v) α(βx) = (αβ)x, (vi) there exists 0 X such that for all x0 X x0 + 0 = x0, and (vii) there exists w∈ X such that x + 0 = x∈. ∈ This linear structure allows us to deﬁne the notion of a convex set. Definition 14. A subset S of a vector space X is said to be convex if for all x, y S and all λ [0, 1] we have λx + (1 λ)y S. ∈ ∈ − ∈ Definition 15. Let X be a vector space. A function : X [0, ) is k · k → ∞ called a norm if and only if for all x, y X and all α R ∈ ∈ (i) αx = α x , (ii) kx +ky | |kxk+ y , and (iii) kx =k 0 ≤ if kandk onlyk k if x = 0. k k The pair (X, ) is then called a normed vector space. Property (ii) is again called the triangle inequality.k · k 4 1. FIXED POINT THEOREMS

A normed vector space (X, ) defines a metric space (X, d) with d defined by d(x, y) = x y . Thus we alreadyk · k have all the objects we defined for metric spaces automaticallyk − definedk for normed vector spaces. The following definition identifies an important class of normed vector spaces. Definition 16. A Banach space is a complete normed vector space. We see that a Banach space puts all of the topological structure, all of the metric structure, and indeed more, on a vector space. It is possible to put the topological structure on a vector space without going as far as putting on all of the structure of a Banach space. The following definition does exactly this. Of course we want the topological structure to “respect” the linear structure of the vector space. Definition 17. A topological vector space is a linear space X together with a topology τ under which both addition and scalar multiplication are continuous functions.

Definition 18. A topological vector space is said to be locally convex if every neighbourhood of zero contains a convex neighbourhood of zero.

Definition 19. A correspondence F between the sets X and Y , written F : X Y , is a function from the set X to the set of all subsets of Y (written ) such that for all x XF (x) = . P ∈ 6 ∅ The notion of continuity that we shall be using for correspondences is called upper hemi-continuity. It is most satisfactorily defined by defining a topology on the space of all subsets of the space Y . This is done in great detail in Klein and Thompson [1984]. When we confine our attention to the case that the space Y is compact and that F takes on as values only closed sets there is an easier way to make the definition. Definition 20. The graph of the correspondence F : X Y denoted Gf(F ) is the set (x, y) X Y y F (x) . { ∈ × | ∈ } Definition 21. Let X be a topological space. Let Y be a compact topological space. Let F : X Y be a closed valued correspondence. Then F is upper hemi-continuous if and only if Gf(F ) is closed in X Y . × Finally we give a number of definitions having to do with simplices which are used in the proof of Kakutani’s Theorem

Definition 22. An n-dimensional simplex deﬁned by the n+1 points v0, v1, . . . , vn p in R , p n, is denoted v0, v1, . . . , vn and is deﬁned to be the set ≥ h i n n p x R x = θjvj θj = 1, θj 0 . { ∈ | ≥ } j=0 j=0 X X The simplex is said to be nondegenerate if the n vectors v1 v0, . . . , vn v0 n − − are linearly independent. If x = j=0 θjvj the numbers θ0, . . . , θn are called the barycentric coordinates of x. P The barycenter of the simplex v0, v1, . . . , vn is the point having barycentric coordinates θ = θ = = θ = 1/h(n + 1). i 0 1 ··· n Definition 23. Barycentric Subdivision of a Simplex. A 0-dimensional simplex— which is a single point—is itself its subdivided simplex. A 1-dimensional simplex x0, x1 is subdivided into two simplices of the same dimension x0, y and y, x1 whereh iy is the barycenter of x , x . h i h i h 0 1i 2. THE CONTRACTION MAPPING THEOREM 5

Now suppose that we have defined the subdivision of all simplices of dimension less than k and that for dimension ` < k an `-dimensional simplex is subdivided into (` + 1)! simplices of the same dimension. (This is clearly true for ` = 0 and ` = 1 defined above.) Thus all (k 1)-dimensional faces of a k-dimensional simplex v , v , . . . , v are assumed to have− been subdivided to k! simplices of dimension k h 0 1 ki − 1. Let y be the barycenter of v0, v1, . . . , vk and let y0, y1, . . . , yk 1 be a simplex h i h − i obtained by subdividing a (k 1)-dimensional face of v0, v1, . . . , vk . Since there are k + 1 faces of dimension k −1, and each face is subdividedh to k! simplices,i there − are (k + 1)! simplices such as y0, y1, . . . , yk 1, y . Now v0, v1, . . . , vk is divided h − i h i into these (k + 1)! simplices for the following reason. Any point of v0, v1, . . . , vk which neither is the barycenter nor lies on any proper face is an a segmenth joiningi the barycenter to a point of some simplex such as y0, y1, . . . , yk 1 and so belongs h − i to some y0, y1, . . . , yk 1, y . An mh times iterative− applicationi of barycentric subdivision to a given simplex v0, v1, . . . , vk gives rise to the mth barycentric subdivision by which the simplex decomposesh toi derived simplices of order m. The following Proposition confirms the intuition that the larger m becomes the smaller the derived simplices become. Proposition 3. Let T (m) be any derived simplex of order m in the mth barycen- (m) tric subdivision of S = v0, v1, . . . , vk . Then their diameters δ(T ) and δ(S) satisfy h i k m δ(T (m)) δ(S). ≤ k + 1 Remark 2. The previous Definition and Proposition are taken from Nikaido (1968).

2. The Contraction Mapping Theorem Theorem 1 (Banach 1922). Let (M, d) be a complete metric space. Let f : M M be a contraction. That is, for any x, y M we have d(f(x), f(y)) θd(x,→ y) where 0 θ < 1. Then f has a unique fixed∈ point in M, i.e., there is≤ a ≤ unique point x∗ M such that x∗ = f(x∗). ∈ Proof. Uniqueness. Suppose that there were two fixed points x and y. Then d(x, y) = d(f(x), f(y)) θd(x, y) ≤ or (1 θ)d(x, y) 0. − ≤ Thus, since (1 θ) > 0, we have d(x, y) = 0 or x = y. Any two fixed points are identical. − Existence. Choose any x M. Define the sequence x , x ,... by 0 ∈ 1 2 xn+1 = f(xn). We first show that this sequence is a Cauchy sequence. Note that n d(xn, xn+1) = d(f(xn 1), f(xn)) θd(xn 1, xn) θ d(x0, x1). − ≤ − ≤ · · · ≤ Now, for any p and q such that p < q the triangle inequality implies that

d(xp, xq) d(xp, xp+1) + + d(xq 1, xq) ≤ ··· − p q 1 d(x0, x 1)(θ + + θ − ) ≤ − p ··· q p 1 d(x0, x 1)θ (1 + + θ − − ) ≤ − p ··· 1 < d(x , x 1)θ (1 θ)− . 0 − − 6 1. FIXED POINT THEOREMS

Now this last term goes to zero as p goes to infinity. Thus the sequence (xn) is a Cauchy sequence and, since M is a complete metric space this sequence converges, say xn x. We now show that such an x is a fixed point. Consider d(x, f(x)). By the triangle→ inequality d(x, f(x)) d(x, x ) + d(x , f(x)) ≤ n+1 n+1 = d(x, xn+1) + d(f(xn), f(x)) d(x, x ) + θd(x , x). ≤ n+1 n and both these last two terms go to zero as n goes to infinity. Thus d(x, f(x)) = 0. (It is less than any strictly positive number.) And so x = f(x), as required.

3. Brouwer’s Theorem Theorem 2 (Brouwer 1910). Let f : Bn Bn be a continuous function from →n the n-ball to itself. Then there is some x∗ B such that x∗ = f(x∗), i.e., there is a ﬁxed point. ∈ The following generality is costless.

Theorem 3. Let M Rn be a compact convex set. Let f : M M be a ⊂ → continuous function. Then there is some x∗ M such that x∗ = f(x∗). ∈

4. Kakutani’s Theorem Theorem 4 (Kakutani 1941). Let M Rn be a compact convex set. Let F : ⊂ M M be an upper hemi-continuous convex valued correspondence. Then there is some x∗ M such that x∗ F (x∗). ∈ ∈ Proof. We first prove Kakutani’s Theorem for the case that M is a nondegenerate simplex in Rn. We then generalize to the case of compact convex subsets of Rn. Let M be the simplex defined by the n + 1 points v0, v1, . . . , vn. That is M = v0, v1, . . . , vn . Now form the mth barycentric subdivision of M. We define the continuoush functioni f m : M M as follows: If x is the vertex of any cell of the subdivision let f m(x) = y for→ some y F (x). For any other x we define f m(x) by extending the function in a linear manner∈ inside each cell. That is, if x is in the cell x , x , . . . , x and x = n θ x with n θ = 1, θ 0 then f m(x) is h 0 1 ni j=0 j j j=0 j j ≥ defined to be n θ f m(x ). j=0 j j P P Now we apply Brouwer’s Theorem to obtain a fixed point of the map f m, say xm. If xm is aP vertex of one of the cells in the subdivision then we are done since xm = f m(xm) F (xm). If xm is not a vertex of one of the cells then let the cell in ∈ m m m m m m which it does lie be x0 , x1 , . . . , xn , and let θ0 , θ1 , . . . , θn be the barycentric coordinates of xm relativeh to that cell.i Thus n m m m x = θj xj , j=0 X and n m m m m m (1) x = f (x ) = θj yj j=0 X m m m m where yj = f (xj ) F (xj ) for all j = 0, 1, . . . , n. Now we choose a subsequence ∈ m mk mk of m say m such that x k x∗, θ θ , and y y . Also, since → ∞ k → ∞ → j → j j → j 5. SCHAUDER’S THEOREM 7

mk the cells shrink to points as mk each of the vertices of the cell containing x mk → ∞ also converges to x∗, i.e., x x∗. Thus from (1) j → n

x∗ = θjyj. j=0 X

Also since F is upper hemi-continuous y F (x∗). Since F (x∗) is convex and x∗ j ∈ is a convex combination of the y ’s this implies that x∗ F (x∗) as required. j ∈ Now what happens if M is not a simplex? We take some simplex M 0 containing M and a retraction ψ : M 0 M (i.e., a continuous function taking M 0 to M that → leaves all points of M fixed.) Then F 0 : M 0 M 0 defined by F 0(x) = F (ψ(x)) is clearly an upper hemi-continuous correspondence and a fixed point of F 0 clearly lies in M and so is also a fixed point of F .

Remark: This proof is essentially the one originally given by Kakutani. An- other way of presenting the proof is by showing that for any convex valued upper hemi-continuous correspondence there is a continuous function whose graph is close to the graph of the correspondence. In the proof we have given we have essentially constructed such a function.

5. Schauder’s Theorem Theorem 5 (Schauder 1930). Let M be a nonempty convex subset of a Banach space X. Let N be a compact subset of M. Let f : M N be a continuous function. → Then there is some x∗ M such that x∗ = f(x∗). ∈ Proof. For any ﬁxed > 0 cover N with a collection of -balls. Since N is compact we may take a ﬁnite subset that also covers N, say the balls centred at y1, y2, . . . , yn. Thus every point in N is within of at least some yj. Now let

n n M = θ y θ = 1, θ 0 . { i i | i i ≥ } i=1 i=1 X X Clearly M M since M is convex. Also M lies in the finite dimensional ⊂ linear subspace of X spanned by y1, y2, . . . , yn. There is a natural identification of this space with the Euclidean space of the same dimension and M then becomes a compact convex subset of this space to which we may apply Brouwer’s Theorem. We now define a continuous function p : N M that for any y N chooses → ∈ a point in M that approximates y to within , i.e., p(y) y < for all y in N. We do so in the following way. First let k − k

0 if y y ϕ (y) = i i y y otherwise.k − k ≥ − k i − k Clearly, each ϕ is continuous and for all y N there is at least one i such that i ∈ ϕi(y) > 0. Now let

ϕi(y) θi(y) = n j=1 ϕj(y) and P n

p(y) = θi(y)yi. i=1 X 8 1. FIXED POINT THEOREMS

This function maps N into M . Recall that θ (y) = 0 unless y y < . Thus i k i − k n p (y) y = θ (y)(y y) k − k k i i − k i=1 nX θ (y) (y y) ≤ i k i − k i=1 Xn

< θi(y) = . i=1 X We can now define the continuous function f : M M by → f (x) = p (f(x)) for x M . ∈ As we said earlier we can apply Brouwer’s Theorem to obtain a fixed point of this function, say x = f (x ). Let y = f(x ). Since y N and N is compact we may ∈ take a sequence 0 such that y converges to a limit, say y∗, in N. Now k → k x = f(x) = p(f(x)) = p(y). And so x y = p (y ) y < . k − k k − k Therefore x y∗, and, since f is continuous f(y∗) = y∗. k → The following is an immediate corollary to Schauder’s Theorem. Theorem 6 (Tychonoff 1935). Let M be a nonempty compact convex subset of a Banach space. Let f : M M be a continuous function. Then there is some → x∗ M such that x∗ = f(x∗). ∈ 6. The Fan-Glicksberg Theorem Theorem 7 ((Fan, 1952; Glicksberg, 1952)). Let M be a nonempty compact convex subset of a convex Hausdorff topological vector space. Let F : M M be an upper hemi-continuous convex valued correspondence. Then there is some x∗ M such that x∗ F (x∗). ∈ ∈ Proof. First fix V a closed neighbourhood of 0. Now since M is compact we can find a finite set y , y , . . . , y M such that { 1 2 n} ⊂ n M ( y + V ). ⊂ { i} i=1 [ Now let n n M = θ y θ = 1, θ 0 . V { i i | i i ≥ } i=1 i=1 X X Let F (x) = (F (x) + V ) M . V ∩ V Now FV is clearly a convex valued correspondence from MV to MV . Moreover F is upper hemi-continuous since if x x, y F (x ), and y y then V δ → δ ∈ V δ δ → y (F (x ) + V ) M . δ ∈ δ ∩ V Thus there exist zδ F (xδ) and vδ V such that yδ = zδ + vδ MV . Since z F (x ) M and∈M is compact z∈ has a cluster point z M. Also∈ z F (x) δ ∈ δ ⊂ { δ} ∈ ∈ since xδ x and F is upper hemi-continuous. Since vδ = yδ zδ and yδ y, vδ must have→ v = y z as a cluster point and , since v V −and V closed→ v { V}. − δ ∈ ∈ 6. THE FAN-GLICKSBERG THEOREM 9

Thus y = z + v F (x) + V . Also y M since y F (x ) M , y y, and ∈ ∈ V δ ∈ V δ ⊂ V δ → MV is closed. Thus y (F (x) + V ) M = F (x). ∈ ∩ V V Now, as we did in proving Schauder’s Theorem, we can identify MV with a subset of Euclidean space and this allows us to apply Kakutani’s Theorem to ﬁnd a ﬁxed point xV MV such that xV FV (xV ). Now the closed∈ neighbourhoods of∈ 0 are naturally ordered by set inclusion so that x is a directed system and so has a cluster point x. Now form { V } ∆0 = (V,U) x U, U a neighbourhood of x { | V ∈ } which is a directed set, and let

x = x for (V,U) ∆0 V,U V ∈ so that

(2) xV,U x −→∆0

(3) V (V,U) ∆0 such that V V . ∀ 0 ∃ ∈ ⊂ 0 Then we have zV,U such that x z V and z F (x ) V,U − V,U ∈ V,U ∈ V,U from xV F (xV ) + V , so that zV,U ∆0 x by (3). But then x F (x) since F is upper hemi-continuous.∈ → ∈ 10 1. FIXED POINT THEOREMS Bibliography Claude Berge: (1963): Topological Spaces, Macmillan, New York. 1 Kim C. Border: (1985): Fixed Point Theorems with Applications to Economics and Game Theory, Cambridge University Press, Cambridge. Ky Fan: (1952): “Fixed-Point and Minimax Theorems in Locally Convex Linear Spaces,” Proceedings of the National Academy of Sciences, USA, 38, 121–126. 7 Joel Franklin: (1980): Methods of Mathematical Economics: Linear and Non- linear Programming, Fixed Point Theorems, Springer-Verlag, New York. 1, 1, 1, 1 I. L. Glicksberg: (1952): “A Further Generalization of the Kakutani Fixed- Point Theorem with applications to Nash Equilibrium Points,” Proceedings of the American Mathematics Society, 3, 170–174. 1, 7 Shizuo Kakutani: (1941): “A Generalization of Brouwer’s Fixed-Point Theo- rem,” Duke Mathematics Journal, 8, 457–459. 1 John L. Kelley: (1955): General Topology, Springer-Verlag, New York. 1, 1, 1 Erwin Klein and Anthony C. Thompson: (1984): Theory of Correspondences: Including Applications to Mathematical Economics, John Wiley & Sons, New York. 1 Hukukane Nikaido: (1968): Convex Structures and Economic Theory, Academic Press, New York. 2 Walter Rudin: (1973): Functional Analysis, McGraw Hill, New York. 1 Eberhard Zeidler: (1986): Nonlinear Functional Analysis and its Applications I: Fixed Point Theorems, Springer-Verlag, New York. 1