Brouwers fixed point theorem with sequentially at most one fixed point

It is well known that Brouwer's fixed point theorem can not be constructively proved 1 . Sperner's lemma which is used to prove Brouwer's theorem, however, can be constructively proved. Some authors have presented an approximate version of Brouwer's theorem using Sperner's lemma. See [8] and [9]. Thus, Brouwer's fixed point theorem is constructively, in the sense of constructive mathematics á la Bishop, proved in its approximate version.

Also in [8] Dalen states a conjecture that a uniformly continuous function f from a simplex to itself, with property that each open set contains a point x such that x = f (x), which means |x -f (x)| > 0, and also at every point x on the boundaries of the simplex x = f (x), has an exact fixed point. In this note we present a partial answer to Dalen’s conjecture.

Recently [2] showed that the following theorem is equivalent to Brouwer’s fan theorem.

Each uniformly continuous function ϕ from a compact metric space X into itself with at most one fixed point and approximate fixed points has a fixed point. By reference to the notion of sequentially at most one maximum in [1] we require a stronger condition that a function ϕ has sequentially at most one fixed point, and will show the following result.

Each uniformly continuous function ϕ from a compact metric space X into itself with sequentially at most one fixed point and approximate fixed points has a fixed point, without the fan theorem. In [7] Orevkov constructed a computably coded continuous function f from the unit square to itself, which is defined at each computable point of the square, such that f has no computable fixed point. His map consists of a retract of the computable elements of the square to its boundary followed by a rotation of the boundary of the square. As pointed out by Hirst in [5], since there is no retract of the square to its boundary, his map does not have a total extension.

In the next section we present our theorem and its proof. In Section 3, as an application of the theorem we consider the mini-max theorem of two-person zerosum games.

Let p be a point in a compact metric space X, and consider a uniformly continuous function ϕ from X into itself. According to [8] and [9] ϕ has an approximate fixed point. It means For each ε > 0 there exists p ∈ X such that |p -ϕ(p)| < ε.

The notion that ϕ has at most one fixed point is defined as follows;

Definition 1 (At most one fixed point). For all p, q ∈ X, if p = q, then ϕ(p) = p or ϕ(q) = q.

Next by reference to the notion of sequentially at most one maximum in [1], we define the notion that ϕ has sequentially at most one fixed point as follows;

Definition 2 (Sequentially at most one fixed point). All sequences (p n ) n≥1 , (q n ) n≥1 in X such that |ϕ(p n )p n | -→ 0 and |ϕ(q n )q n | -→ 0 are eventually close in the sense that |p nq n | -→ 0. Now we show the following lemma.

Lemma 1. Let ϕ be a uniformly continuous function from a compact metric space X into itself. Assume inf p∈X |p -ϕ(p)| = 0. If the following property holds, For each δ > 0 there exists ε > 0 such that if p, q ∈ X, |ϕ(p)-p| < ε and |ϕ(q) -q| < ε, then |p -q| ≤ δ, then, there exists a point r ∈ X such that ϕ(r) = r, that is , we have a fixed point of ϕ.

Since δ > 0 is arbitrary, (p n ) n≥1 is a Cauchy sequence in X, and converges to a limit r ∈ X. The continuity of ϕ yields |ϕ(r) -r| = 0, that is, ϕ(r) = r.

Next we show the following theorem.

Theorem 1. Each uniformly continuous function ϕ from a compact metric space X into itself with sequentially at most one fixed point and approximate fixed points has a fixed point Proof. Choose a sequence (r n ) n≥1 in X such that |ϕ(r n )r n | -→ 0. In view of Lemma 1 it is enough to prove that the following condition holds.

For each δ > 0 there exists ε > 0 such that if p, q ∈ X, |ϕ(p)-p| < ε and |ϕ(q) -q| < ε, then |p -q| ≤ δ.

Assume that the set

is nonempty and compact 2 . Since the mapping (p, q) -→ max(|ϕ(p)-p|, |ϕ(q)-q|) is uniformly continuous, we can construct an increasing binary sequence (λ n ) n≥1 such that

It suffices to find n such that λ n = 1. In that case, if

Computing N such that |p Nq N | < δ, we must have λ N = 1. We have completed the proof.

Consider a two person zero-sum game. There are two players A and B. Player A has m alternative pure strategies, and the set of his pure strategies is denoted by S A = {a 1 , a 2 , . . . , a m }. Player B has n alternative pure strategies, and the set of his pure strategies is denoted by S B = {b 1 , b 2 , . . . , b n }. m and n are finite natural numbers. The payoff of player A when a combination of players’ strategies is (a i , b j ) is denoted by M (a i , b j ). Since we consider a zero-sum game, the payoff of player B is equal to -M (a i , b j ). Let p i be a probability that A chooses his strategy a i , and q j be a probability that B chooses his strategy b j . A mixed strategy of A is represented by a probability distribution over S A , and is denoted by p = (p 1 , p 2 ,

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