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ZERMELO AND FIXED-POINT THEOREMS. TACL’2009 , AMSTERDAM , JULY 7-11, 2009 Janusz Czelakowski University of Opole , Institute of Mathematics and Informatics , Poland [email protected] J. Czelakowski , Zermelo and fixed-point theorems , TACL’2009 , AMSTERDAM, JULY 7-11, 2009.

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ZERMELO AND FIXED-POINT THEOREMS

TACL’2009, AMSTERDAM, JULY 7-11, 2009

JanuszCzelakowski

University of Opole, Institute of Mathematics and Informatics, Poland

J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

1. Zermelo’s Fixed-Point Theorem.

A poset P = (P, ) is inductive if every chain in P has a supremum. Every inductive poset possesses the least element 0 – the supremum of the empty chain.

Zermelo’s Fixed-Point Theorem in the standard formulation states the following:

Version I.

Every expansive mapping from an inductive poset

P = (P, ) toP has a fixed-point.

(is expansive if for every a P it is the case that

a  (a).)

The above theorem is equivalent to the following statement:

J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

Version II.

Let P = (P, ) be a non-empty poset in which every non-empty chain has a supremum. Every expansive mapping fromPtoP has a fixed-point. 

Zermelo’s Fixed-Point Theorem (in both versions) is provable in ZF. Yet another version of Zermelo’s Fixed-Point Theorem is provable in ZFC by a straightforward application of Zorn’s Lemma:

J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

Version III.

LetP = (P, ) be a non-empty poset in which every non-empty chain has an upper bound. Then everyexpansive mapping from P toP has a fixed-point.

In fact, every maximal element in P is a fixed-point of .

J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

2. Some Applications ofZermelo’s Fixed-Point Theorem.

The theory of fixed-points splits into two, to a large extent autonomous and conceptually independent, areas of research. Each of these fields is determined by the specific choice of underlying mathematical models:

(1) the theory of fixed-points conducted in the framework of order-complete partially ordered sets

(see e.g. Gunter and Scott [1990]).

(2) the theory of fixed-points developed in the setting ofcomplete metric spaces (see e.g. Goebel [2001], Goebel and Kirk [1990], Kirk and Sims [2001]).

J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

Zermelo’sTheorem (in all versions) belongs to the first of the mentioned branches (the so called order-oriented theory of fixed points).

(1) is developed in the envoirment formed by the class of posets endowed with appropriate mappings. The properties of these mappings as e.g. monotonicity, expansivity, or various forms of order-continuity are strictly conjoined with the order.

(1) offers other non-trivial results as e.g. the famous Tarski Fixed–Point Theorem, Fujimoto Fixed-Point Theorem, etc. for mappings or relations defined on ordered sets. All these theorems to a large extent exploit various completenesspropertiesof theunderlyingposets.

J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

On the other hand,(2) contains an array of results that are anchored in the class of complete metric spaces augmented with suitably selected continuous mappings, and not in ordered structures with derived mappings.

We mention in this context the names of Brouwer, Schauder, Kakutani, Nielsen, Caristi or Banach.

They all have essentially contributed to the development of the metric-oriented theory of fixed-points.

J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

In the talk we shall underlie the fundamental role Zermelo’s Theorem (in the three versions) plays not only in the order theory of fixed-points but also in the metric theory of fixed-points.

We shall show how to derive Caristi and Banach Fixed-Point Theorems from Zermelo’s Theorem in a straightforward way using the Principle of Countable Choice.

These facts show the logical dimension of Zermelo’s Theorem as a tool enabling a uniform presentation of the theory of fixed-points from the order-theoretical standpoint.

We shall also show how derive Brouwer Fixed-Point Theorem from Zermelo’s one.

J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

Let  be a partial order on a topological space X.  is said to be compatible with the topology of X if the relation  is a closed set in the product topology on X X.

If (X, d ) is a metric space, then  is compatible with the metric topology if and only if for any two convergent sequences (in the metric d) {x n} and

{y n}of elements of X, x ny n for all n implies that

lim n x n lim n y n.

J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

R stands for the set of real numbers.

t : XR is upper-bounded if the set { t (x ) : xX } has an upper bound in R (or, equivalently,

sup{t (x) : xX }exists inR ).

For a given t : XR we define the following binary relation ton the space X :

(*) xty if and only if d (x, y) t (y ) t (x ),

for all x, yX.

J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

The following observation is crucial.

Theorem 1. Assume ZF + the Principle of Countable Choice (AC). Let (X, d ) be metric space with a metricd. Lett : XRbe a function. Then:

(A) tis n partial order on X.

(B) t is a monotone mapping from the poset (X, t) to

(R, ), i.e., xty implies t (x) t (y), for all

x, yX.

J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

(C) Suppose the metric space (X, d ) is complete.

If t : XRis continuous and upper-bounded,

then in the poset (X, t) every non-empty chain has a supremum.

(D) If t : XRis continuous, then the order tis compatible with the metric topology (X, d ). 

Note. The above definition of the order tin (X, d) is due to Caristi [1976]. It seems, however, that conditions (C )-(D) have not been explicitly considered in the literature. 

J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

Theorem 2. (Caristi’s Fixed-Point Theorem).

Assume ZF + AC. Let (X, d ) be a complete metric space. Let t : XRbe a continuous and upper-boundedfunction. Then every function  : XXwhich satisfies the following condition:

(**) d (x,  (x )) t ( (x ))-t (x ) for all x X,

has a fixed-point.

J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

Proof.

Zermelo’s Fixed Point-Theorem in Version II is applied here.

Define the poset (X, l ) as above. Since t is continuous, Theorem 1 implies that every non-empty chain in (X, l ) has a supremum. But condition (**), defined by Caristi, actually states that  is expansive in the sense of the poset (X, l ) (cf. (*)). Zermelo’s Theorem implies that  has a fixed-point. 

J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

Let (X, d ) be a metric space. A mapping  : XX is called a contraction if there exists a positive real number k, 0 k  1, such that

d((x),  (y )) k d(x, y )

for all x, y X.

It is clear that every contraction is continuous (in the metric d).

The following observation is known:

J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

Proposition 3.Let : XX be a contraction of

(X, d ). Definet : XR as follows:

t (x ) := d (x,  (x )) / 1 – k,

for all x X . The function tis continuous. Moreover

(1) sup{t (x ) : xX }  0,

(2) d (x,  (x )) t ( (x)) t (x ) for all x X . 

Theorem 4. (Banach Fixed Point Theorem).

Assume ZF + AC. Let (X, d ) be a complete metric space. Every contraction  : XX has a unique fixed-point.

J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

Banach Theorem is directly derivable from Zermelo’s Fixed-Point Theorem in Version II, Theorem 1 and Proposition 3.

Let t : XRbe defined as in Proposition 3. As t is continuous and upper bounded, Theorem 1 implies that in the poset(X, t ) every non-empty chain has asupremum. In turn, (2) states that the mapping  is expansive in the sense of

(X,t ). Zermelo’s Theorem implies that  has a fixed-point. (This fact also follows from Caristi’s Theorem.) The uniqueness of the fixed-point trivially follows from the fact that  is a contraction.

J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

3. Zermelo and Brouwer.

Brouwer Fixed-Point Theorem can be derived from Zermelo’s Fixed- Point Theorem.

Let k be a positive integer. I k denotes the k-th power of the unit interval I, where I := {x R : 0 x  1}. The elements of I k are k–tuples of real numbers from I.

k denotes the standard order relation in I k. Thus for

a = (a 1,…, ak), b = (b 1,…, bk) I k

a k b means that a 1b 1 … akbk.

J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

The poset (I k, k ) is a complete lattice.

Let X I k and a = (a 1,…, ak). Then a = sup(X) in

(I k, k ) iff ai = sup(pi [X ]) in the standard order  on I,for i = 1,…, k.

Generally, for k  2, if X is a closed set in I k, thensup(X) need not be an element of X.

J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

Theorem 5. Assume AC. Letk be a positive integer. Let f = (f 1,…, fk) be a continuous function from I k to I k. Then there exists a non-empty closed subset A I ksatisfying the following conditions:

(1) Every non-empty chain in the poset

A := (A, k ) has a supremum.

(2) f maps A to A.

(3) f is expansive in the poset A.

J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

Note. The crucial fact in the above theorem is the non-emptiness of A.

Moreover, itmay happen that the set A with the above properties is a singleton.

In the one-dimensional case, let f : I I be a continuous function such that f (0) = 0 and f (x) x for all x  0.

Then A = {0}because x f (x) only if x = 0. 

J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

Theorem 6. (Brouwer Fixed-Point Theorem).

Let k be a positive integer. Let f = (f 1,…, fk) be a continuous function from I k to I k. Then fhas a fixed- point.

Proof.

Let A be defined as in Theorem 5. Zermelo’s Fixed-Point Theorem (Version 2) when applied to (1)-(3) implies that f has a fixed-point. 

J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

It should be noted that Brouwer Fixed-Point Theorem implies Theorem 5.While the existence of a set A satisfying conditions (1)-(3) of Theorem 5 can be easily established, the critical property is that of its non-emptiness.

Brouwer Fixed-Point Theorem implies that there exists such a non-empty set – we take as A the singleton containing a fixed-point of f .

J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

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J. Czelakowski, Zermelo and fixed-pointtheorems,TACL’2009, AMSTERDAM, JULY 7-11, 2009

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