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.