A generic approach to measuring the strength of completeness/compactness of various types of spaces and ordered structures
aa r X i v : . [ m a t h . GN ] M a y A BASIC FRAMEWORK FOR FIXED POINTTHEOREMS: BALL SPACES AND SPHERICALCOMPLETENESS
HANNA ´CMIEL, FRANZ-VIKTOR KUHLMANN AND KATARZYNAKUHLMANN
Abstract.
We systematically develop a general framework inwhich various notions of functions being contractive, as well asof spaces being complete, can be simultaneously encoded. Derivedfrom the notions of ultrametric balls and spherical completeness, avery simple structure is obtained which allows this encoding. Wegive examples of generic fixed point theorems which then can bespecialized to theorems in various applications which work withcontracting functions and some sort of completeness property ofthe underlying space. As examples of such applications we dis-cuss metric spaces, ultrametric spaces, ordered groups and fields,topological spaces, partially ordered sets and lattices. We charac-terize the particular properties of each of these cases and classifythe strength of their completeness property. We discuss operationson the sets of balls to determine when they lead to larger sets; ifso, then the properties of these larger sets of balls are determined.This process can lead to stronger completeness properties of theball spaces, or to ball spaces of newly constructed structures, suchas products. Further, the general framework makes it possibleto transfer concepts and approaches from one application to theother; as examples we discuss theorems analogous to the Knaster–Tarski Fixed Point Theorem for lattices and theorems analogous tothe Tychonoff Theorem for topological spaces. Finally, we presentsome generic multidimensional fixed point theorems and coinci-dence theorems.
Date : May 22, 2019.1991
Mathematics Subject Classification.
Primary 54A05, 54H25; Secondary03E75, 06A05, 06A06, 06B23, 06B99, 06F20, 12J15, 12J20, 13A18, 47H09, 47H10,54C10, 54C60, 54E50.
Key words and phrases. ball space, metric space, ultrametric space, topologicalspace, partially ordered set, lattice, ordered abelian group, ordered field, sphericallycomplete, complete, compact, fixed point theorem, multivalued fixed point theorem,coincidence theorem, Caristi–Kirk Fixed Point Theorem, Knaster–Tarski Theorem,Tychonoff Theorem.The first two authors are supported by a Polish Opus grant2017/25/B/ST1/01815.
Contents
1. Introduction 32. Zorn’s Lemma in the context of ball spaces 102.1. The case of S ball spaces 112.2. Posets of nests of balls 112.3. Proof of the fixed point theorems 123. Some facts about the hierarchy of ball spaces 143.1. A refinement of the hierarchy 143.2. Connection with posets 153.3. Singleton balls 163.4. Ball spaces of ultrametric type 163.5. Intersection closed ball spaces 164. Ball spaces and their properties in various applications 184.1. Ultrametric spaces 184.2. Metric spaces with metric balls 214.3. Metric spaces with Caristi–Kirk balls or Oettli–Th´eraballs 244.4. Ordered abelian groups and fields 284.5. Topological spaces 314.6. Partially ordered sets 344.7. Lattices 385. S ∗ ball spaces 405.1. Spherical closures in S ∗ ball spaces 415.2. Ball spaces induced on subsets of S ∗ ball spaces 416. Set theoretic operations on ball spaces 436.1. Subsets of ball spaces 436.2. Unions of two ball spaces on the same set 446.3. Closure under unions of balls 446.4. Closure under nonempty intersections of balls 456.5. Closure under finite unions and under intersections 476.6. The topology associated with an S c ball space 477. Shifting concepts between applications 487.1. Knaster–Tarski type theorems 487.2. Tychonoff type theorems 528. Other results related to fixed point theorems 568.1. Multivalued fixed point theorems 578.2. Coincidence theorems 58References 59 RAMEWORK FOR FIXED POINT THEOREMS 3 Introduction
Fixed Point Theorems (FPTs) can be divided into two classes: thosedealing with functions that are in some sense “contracting”, like Ba-nach’s FPT and its ultrametric variant (cf. [22], [25]), and those thatdo not use this property (explicitly or implicitly), like Brouwer’s FPT.In this paper, we will be concerned with the first class.Under which conditions do “contracting” functions have a fixed point?First of all, we have to say in which space we work, and we have tospecify what we mean by “contracting”. These specifications will haveto be complemented by a suitable condition on the space, in the sensethat it is “rich” or “complete” enough to contain fixed points for all“contracting” functions.In our paper [12], we have developed a general framework for thisprocedure. It presents a minimal setting in which the necessary con-ditions on the function and the space can be formulated. After havingproved generic FPTs in this setting, they can then be adapted andinterpreted in the settings of various applications. Let us briefly sketchthe basics of this approach.In all of the applications we will discuss in this paper, we look atfunctions f on a set X that has some additional structure. In order forFPTs to work, the set must have some completeness property underthis structure. In all of the applications, the structure gives rise tocollections B of distinguished subsets, explicitly or implicitly. If theyare suitably chosen, then the required completeness property is equiv-alent to a completeness property of the collections B , which we willintroduce below. This will show that working with the set X togetherwith (one or more) collections B constitutes a unifying framework forFPTs. This is because they provide the minimal structure that allowsthe definition of completeness properties which encode the requiredcompleteness properties in each of the applications we will discuss.A ball space ( X, B ) consists of a nonempty set X together with anonempty set B of distinguished nonempty subsets B of X . Note that B , a subset of the power set P ( X ), is partially ordered by inclusion; wewill write ( B , ⊆ ) when we refer to this partially ordered set (in short:poset). A nest of balls in ( X, B ) is a nonempty totally ordered subsetof ( B , ⊆ ). The basic completeness notion for ball spaces is inspired bythe corresponding notion for ultrametric spaces: a ball space ( X, B )is called spherically complete if every nest of balls has a nonemptyintersection.We will now give examples of generic FPTs for ball spaces; they willbe proved in Section 2.3. More such theorems and related results such HANNA ´CMIEL, FRANZ-VIKTOR AND KATARZYNA KUHLMANN as coincidence theorems and so-called attractor theorems are presentedin [12, 13, 14, 16]. In the present paper we will not discuss the unique-ness of fixed points; see the cited papers for this aspect. However, anexception will be made in Theorem 1.2, as this will be used later for aninteresting comparison with a topological fixed point theorem provenin [29].Given a function f on a set X , we will call a subset S of X f -closed if f ( S ) ⊆ S . An f -closed set S will be called f -contracting if it satisfies f ( S ) ( S unless it is a singleton. In the search for fixed points, it is apossible strategy to try to find f -closed singletons { a } because then thecondition f ( { a } ) ⊆ { a } implies that f ( a ) = a . The significance of thisidea is particularly visible in the case of Caristi–Kirk and Oettli–Th´eraball spaces discussed in Section 4.3. Theorem 1.1.
Take a spherically complete ball space ( X, B ) and afunction f : X → X . If each f -closed set contains an f -contractingball, then f has a fixed point in each f -closed set. The concept of ball spaces enables us to distinguish various levels ofspherical completeness, which then helps to relax the conditions on thefunctions under consideration. On the one hand, we can specify whatthe intersection of a nest really is, apart from being nonempty. On theother hand, we can consider intersections of more general collectionsof balls than just nests. A directed system of balls is a nonemptycollection of balls such that the intersection of any two balls in the col-lection contains a ball included in the collection. A centered systemof balls is a nonempty collection of balls such that the intersectionof any finite number of balls in the collection is nonempty. Note thatevery directed system is also a centered system (but in general, theconverse is not true).We introduce the following hierarchy of spherical completeness prop-erties: S : The intersection of each nest in ( X, B ) is nonempty. S : The intersection of each nest in ( X, B ) contains a ball. S : The intersection of each nest in ( X, B ) contains maximal balls. S : The intersection of each nest in ( X, B ) contains a largest ball. S : The intersection of each nest in ( X, B ) is a ball. S di : The same as S i , but with “directed system” in place of “nest”. S ci : The same as S i , but with “centered system” in place of “nest”.Note that S is just the property of being spherically complete. RAMEWORK FOR FIXED POINT THEOREMS 5
The strongest of these properties is S c ; we will abbreviate it as S ∗ as it will play a central role, enabling us to prove useful results aboutseveral important ball spaces that have this property (it is the “star”among the above properties).We have the following implications:(1) S ⇐ S d ⇐ S c ⇑ ⇑ ⇑ S ⇐ S d ⇐ S c ⇑ ⇑ ⇑ S ⇐ S d ⇐ S c ⇑ ⇑ ⇑ S ⇐ S d ⇐ S c ⇑ ⇑ ⇑ S ⇐ S d ⇐ S c = S ∗ The properties of the above hierarchy will be studied in detail inSection 3. In particular it is shown that under various conditions onthe ball spaces certain properties in the hierarchy become equivalent.For completeness, a refinement of it will be discussed in Section 3.1.However, it will not be used further in the present paper.We will now give an example showing how some of the above strongernotions of spherical completeness can be employed in general FPTs.
Theorem 1.2.
Take an S ball space ( X, B ) and a function f : X → X .1) Assume that each f -closed ball is a singleton or contains a smaller f -closed ball. Then f has a fixed point in each f -closed ball.2) Assume that the image f ( B ) of every f -closed ball B ∈ B is an f -contracting ball. Then f has a unique fixed point in each f -contractingball. If in addition X is an f -contracting ball, then f has a unique fixedpoint. Theorem 1.3.
Take an S ball space ( X, B ) and a function f : X → X . If every ball in B contains a fixed point or a smaller ball, then f has a fixed point in every ball. We can get around asking that the ball space be S by giving acondition on the intersection of nests; note that it is implicit in thiscondition that the ball space is spherically complete. Theorem 1.4.
Take a ball space ( X, B ) and a function f : X → X such that the intersection of every nest of balls in B contains a fixed HANNA ´CMIEL, FRANZ-VIKTOR AND KATARZYNA KUHLMANN point or a smaller ball B ∈ B . Then f admits a fixed point in everyball of B . A condition like “contains a fixed point or a smaller ( f -closed) ball”may appear a little unusual at first. However, a possible algorithmfor finding fixed points should naturally be allowed to stop when it hasfound one, so from this point of view the condition is quite natural. Wealso sometimes use a condition like “each f -closed ball is a singleton orcontains a smaller f -closed ball”. This implies “contains a fixed pointor a smaller f -closed ball” because in an f -closed singleton { a } theelement a must be a fixed point. But this condition is too strong: aswe will see below, there are cases where finding a ball with a fixed pointis easier and more natural than finding a singleton. One example arepartially ordered sets where the balls are taken to be sets of the form[ a, ∞ ).The assumptions of these theorems can be slightly relaxed by adapt-ing them to the given function f . Instead of talking about the in-tersections of all nests of balls, we need information only about theintersections of nests of f -closed balls. Trivially, if ∅ 6 = B ′ ⊆ B , thenalso ( X, B ′ ) is a ball space, and if ( X, B ) is spherically complete, thenso is ( X, B ′ ). This flexibility of ball spaces appeared already implicitlyin Theorem 1.2 where only f -closed balls are used; if nonempty, thesubset of all f -closed balls is also a ball space, and it inherits importantproperties from the (possibly) larger ball space. Tayloring the assump-tions on the ball space to the given function also comes in handy in thefollowing refinement of Theorem 1.2. In its formulation, the condition“spherically complete” does not appear explicitly anymore, but is im-plicitly present for the ball space that is chosen in dependence on thefunction f . Theorem 1.5.
Take a function f : X → X and assume that there isa ball space ( X, B f ) such that (B1) each ball in B f is f -closed, (B2) the intersection of every nest of balls in B f is a singleton orcontains a smaller ball B ∈ B f .Then f admits a fixed point in every ball in B f . At first glance, the conditions of these theorems may appear toostrong, or somewhat unusual. But the reader should notice that theirstrength lies in the fact that we can freely choose the ball space. Forexample, it does not have to be a topology, and in fact, for essentiallyall of our applications it should not be . This makes it possible to even
RAMEWORK FOR FIXED POINT THEOREMS 7 choose the balls relative to the given function, which leads to resultslike the theorem above.When uniqueness of fixed points is not required, then in certainsettings (such as ultrametric spaces, see Section 4.1) the condition thata function be “contracting” on all of the space can often be relaxed tothe conditions that the function just be “non-expanding” everywhereand “contracting” on orbits. Again, there is some room for relaxation,and this is why we will now introduce the following notion. For each i ∈ N , f i will denote the i -th iteration of f , that is, f x = x and f i +1 x = f ( f i x ). A function f will be called ultimately contractingon orbits if there is a function X ∋ x B x ∈ B such that for all x ∈ X , the following conditions hold: (SC1) x ∈ B x , (SC2) B fx ⊆ B x , and if x = f x , then B f i x ( B x for some i ≥ f i x ∈ B x for all i ≥ N of balls is an f -nest if N = { B x | x ∈ S } for some set S ⊆ X that is closed under f . Now we can state our sixthbasic theorem: Theorem 1.6.
Take a function f on a ball space ( X, B ) which is ul-timately contracting on orbits. If for every f -nest N in this ball spacethere is some z ∈ T N such that B z ⊆ T N , then for every x ∈ X , f has a fixed point in B x . A particularly elegant version of this approach can be given in thecase of Caristi–Kirk and Oettli–Th´era ball spaces (see Theorem 4.12in Section 4.3). These ball spaces are used in complete metric spaces.Usually, proofs of fixed point theorems in this setting work with Cauchysequences, while the use of metric balls is inefficient and complicated.For this reason, a ball spaces approach to metric spaces may seem point-less at first glance. However, it has turned out that ball spaces made upof Caristi–Kirk or Oettli–Th´era balls have a particularly strong prop-erty (cf. Proposition 4.11), which makes the ball space approach in thiscase exceptionally successful, as demonstrated in Section 4.3 and thepapers [2, 14].Interestingly, the exceptional strength of the Caristi–Kirk and Oettli–Th´era ball spaces is shared by the ball space made up of the sets [ a, ∞ )on partially ordered sets. It would be worthwhile to find more examplesof such strong ball spaces. HANNA ´CMIEL, FRANZ-VIKTOR AND KATARZYNA KUHLMANN
The proofs of our generic fixed point theorems are based on Zorn’sLemma. They will be given in Section 2 after first investigating therelation between partially ordered sets and ball spaces. In the presentpaper we are not interested in avoiding the use of the axiom of choice,nor is it our task to study its equivalence with certain fixed pointtheorems. For a detailed discussion of the case of Caristi–Kirk andOettli–Th´era ball spaces, see Remark 4.15.After having provided the basic setting and results, the task of prov-ing fixed point theorems is shifted to finding the suitable interpretationsof the notion of “ball space” in the various applications, together withthe suitable level(s) of spherical completeness. Here are a few exampleswhich will be worked out in Section 4.spaces balls completenesspropertyultrametric spaces all closed ultrametric balls sphericallycompletemetric spaces metric balls with radii completein suitable sets ofpositive real numberstotally ordered sets, symmetricallyordered abelian all intervals [ a, b ] with a ≤ b completegroups and fieldstopological spaces all nonempty closed sets compactposets intervals [ a, ∞ ) inductivelyorderedmetric spaces Caristi–Kirk balls or completeOettli–Th´era ballsHere the last entry, the second one for metric spaces, is differentfrom all the other ones. In all the other cases the table has to beread as saying that the completeness property of the given space isequivalent to the spherical completeness of one single associated ball space containing the indicated balls. But if we work with Caristi–Kirkballs or Oettli–Th´era balls, then the completeness of the metric space isequivalent to the spherical completeness of a whole variety of Caristi–Kirk ball spaces or Oettli–Th´era ball spaces that can be defined on it(see Section 4.3). While this may appear impracticable at first glance,it turns out that these types of balls offer a much better ball spacesapproach to metric spaces than the metric balls, as noted above. RAMEWORK FOR FIXED POINT THEOREMS 9
Not only the specialization of the general framework to particularapplications is important. It is also fruitful to develop the abstracttheory of ball spaces, in particular the behaviour of the various levelsof spherical completeness in the hierarchy (1) under basic operationson ball spaces.In Section 5 we study our strongest, the S ∗ ball spaces. Examplesare the topological spaces, where we take the balls to be the nonemptyclosed sets. These ball spaces allow the definition of what we callspherical closures of subsets. These help us to deal with ball spacestructures induced on subsets of the set underlying the ball space.In Section 6 we will consider set theoretic operations on ball spaces,such as adding unions or intersections of their balls. Products of ballspaces will be studied in Section 7.2. In the paper [1], we discussa notion of continuity for functions between ball spaces, as well asquotient spaces and category theoretical aspects of ball spaces.Further, the fact that a general framework links various quite dif-ferent applications can help to transfer ideas, approaches and resultsfrom one to the other. For instance, the Knaster–Tarski Theorem inthe theory of complete lattices presents a useful property of the set offixed points: they form again a complete lattice. In Section 7.1, usingour general framework and in particular the results from Section 5, wetransfer this result to other applications, such as ultrametric and topo-logical spaces. Similarly, in Section 7.2 the Tychonoff Theorem fromtopology is proven for ball spaces and then transferred to ultramet-ric spaces. To derive the topological Tychonoff Theorem from its ballspaces analogue, particular use is made of the results of Section 6.Finally, the last section of our paper is devoted to a quick discussionof two types of theorems that are related to fixed point theorems (andin fact are generalizations, as fixed point theorems can be deduced fromthem). First, we will present generic multivalued fixed point theoremsfor ball spaces. Multivalued fixed point theorems consider functions F from a nonempty set X to its power set P ( X ) and ask for criteria thatguarantee the existence of a fixed point x ∈ X in the sense that x ∈ F ( x ) . Multivalued ultrametric fixed point theorems have been successfullyapplied in logic programming (see [26, 5]).Second, we will present generic coincidence theorems for ball spaces.Coincidence theorems consider two or more functions f , . . . , f n froma nonempty set X to itself and ask for criteria that guarantee the existence of a coincidence point x ∈ X in the sense that f ( x ) = . . . = f n ( x ) . A number of coincidence theorems for ball spaces and ultrametricspaces have been proven in [16] (see also [24] for theorems on ultra-metric spaces).For both types of theorems we will use two approaches. Inspired bythe theory of strongly contractive ball spaces which we will developin connection with Caristi–Kirk and Oettli–Th´era ball spaces in Sec-tion 4.3, we will first employ criteria for the existence of singleton ballswith suitable properties. Thereafter, we will prove variants which workwith minimal balls instead.We hope that we have convinced the reader that the advantage of ageneral framework is (at least) threefold: • provide generic proofs of results which then only have to be special-ized to the various applications, • exhibit the underlying principles that make the theorems in thevarious applications work, • transfer concepts and results from one application to another.2. Zorn’s Lemma in the context of ball spaces
Consider a poset (
T, < ). By a chain in T we mean a nonemptytotally ordered subset of T . An element a ∈ T is said to be an upperbound of a subset S ⊆ T if b ≤ a for all b ∈ S . A poset is said to be inductively ordered if every chain has an upper bound.Zorn’s Lemma states that every inductively ordered poset containsmaximal elements. By restricting the assertion to the set of all ele-ments in the chain and above it, we obtain the following more preciseassertion: Lemma 2.1.
In an inductively ordered poset, every chain has an upperbound which is a maximal element in the poset.
Corollary 2.2.
In an inductively ordered poset, every element lies be-low a maximal element.
Take a ball space ( X, B ). If we order B by setting B < B if B ) B , then we obtain a poset ( B , < ). Under this transformation,nests of balls in B correspond to chains in the poset. A maximal elementin the poset ( B , < ) is a minimal ball , i.e., a ball that does not containany smaller ball. RAMEWORK FOR FIXED POINT THEOREMS 11
The case of S ball spaces. The following observation is straightforward:
Lemma 2.3.
The ball space ( X, B ) is S if and only if every chain in ( B , < ) has an upper bound. From this fact, one easily deduces the following result.
Proposition 2.4.
In an S ball space, every ball and therefore also theintersection of every nest contains a minimal ball. In view of Lemma 2.3 it is important to note that every S ball space( X, B ) can easily be turned into an S ball space by adding all singletonsubsets of X : we define B s := B ∪ {{ a } | a ∈ X } . The proof of the following result is straightforward.
Lemma 2.5.
The ball space ( X, B s ) is S if and only if ( X, B ) is S . However, in many situations the point is exactly to prove that agiven ball space admits singleton balls. This is in particular the casewhen we work with ball spaces that are adapted to a given function,as in Theorem 1.5. In such cases, instead of applying Zorn’s Lemmato chains of balls, one can work with chains of nests instead, as we willdiscuss in Section 2.2.2.2.
Posets of nests of balls.
A poset is called chain complete if every chain of elements has a leastupper bound.
Lemma 2.6.
For every ball space ( X, B ) , the set of all nests of balls,ordered by inclusion, is a chain complete poset. Proof: The union over a chain of nests of balls is again a nest ofballs, and it is the smallest nest that contains all nests in the chain. (cid:3)
This shows that in particular every chain of nests that contains agiven nest N has an upper bound. Hence Zorn’s Lemma shows: Corollary 2.7.
Every nest N of balls in a ball space is contained ina maximal nest. Proof of the fixed point theorems.
Take a ball space ( X, B ) and a function f : X → X . By B f we willdenote the collection of all f -closed balls in B , provided there exist any.From Corollary 2.7 we infer that every nest in ( X, B ) and every nestin ( X, B f ) is contained in a maximal nest.Under various conditions on f and on ( X, B ) or ( X, B f ), we have tomake sure that the intersections of such nests contain a fixed point for f . We observe:a) If S is an f -closed set, then f f ( S ) ⊆ f ( S ) since f ( S ) ⊆ S , hence f ( S ) is f -closed.b) The intersection over any collection of f -closed sets is again an f -closed set.Proof of Theorem 1.1: Take any f -closed set S . By the assumptionof the theorem we know that it contains an f -contracting ball B . Bydefinition, B is f -closed. By Corollary 2.7 there exists a maximal nest N in the set B f of all f -closed balls in B which contains the nest { B } .Then by b) above, T N is an f -closed set. By assumption, it containsan f -contracting ball B ′ . Suppose that B ′ is not a singleton. Then B ′ properly contains f ( B ′ ), which by a) above is an f -closed set. Againby assumption, it contains an f -contracting and hence f -closed ball B ′′ . Since B ′′ ⊆ f ( B ′ ) ( B ′ ⊆ T N , we find that N ∪ { B ′′ } is a largernest than N , which contradicts the maximality of N . This proves that B ′ is an f -closed singleton contained in S and thus, S contains a fixedpoint. (cid:3) Proof of Theorem 1.2: Assume that ( X, B ) is an S ball space andthat each f -closed ball is a singleton or contains a smaller f -closedball. Take some f -closed ball B ∈ B .1): As in the previous proof, choose a maximal nest N in B f whichcontains the nest { B } . Then T N is an f -closed set. As ( X, B ) isassumed to be an S ball space, it is also a ball, so T N ∈ B f . Bythe maximality of N , we have that N ∪ { T N } = N , i.e., T N is thesmallest ball in N . It must be a singleton, because otherwise, it wouldcontain a smaller f -closed ball giving rise to a nest properly containing N , which is impossible. Thus, T N is an f -closed singleton containedin B and therefore, B contains a fixed point.2): We will apply Theorem 2 of [12], which states that if ( Y, B ′ ) is aball space and f : Y → Y such that Y is an f -contracting ball, theimage f ( B ) of every f -contracting ball is again an f -contracting ball,and that so is the intersection of every nest of f -contracting balls, then f has a unique fixed point. RAMEWORK FOR FIXED POINT THEOREMS 13
We fix an f -closed ball B ; if X is itself a ball, then we take B = X .Then we set Y = B and take B ′ to be the collection of all f -closedballs from B that are contained in B . Then the first two of the aboveconditions hold by our choice of Y and by our assumption that theimage f ( B ) of every f -closed ball B ∈ B is again an f -contracting ball.To show that the third condition holds, take a nest of f -contractingballs. As ( X, B ) is assumed to be an S ball space, its intersection T N is a ball. By our observation b) above, it is f -closed. Supposethat T N were not f -contracting. Then f ( T N ) = T N . But thenby the assumption of our theorem, T N is f -contracting. We haveshown that the conditions of the theorem are satisfied, which yieldsthe desired fixed point. (cid:3) Proof of Theorem 1.3: Assume that ( X, B ) is an S ball space andthat every ball in B contains a fixed point or a smaller ball. Takea ball B ∈ B . By Proposition 2.4, B contains a minimal ball B .As B cannot contain a smaller ball, it must contain a fixed point byassumption, which then is also an element of B . (cid:3) Proof of Theorem 1.4: Assume that the intersection of every nest ofballs in ( X, B ) contains a fixed point or a smaller ball. Take a ball B ∈ B . As before, there exists a maximal nest N in B which containsthe nest { B } . Now T N cannot contain a smaller ball since this wouldcontradict the maximality of N . Hence by assumption, T N and thusalso B must contain a fixed point. (cid:3) Proof of Theorem 1.5: Assume that B f is a ball space of f -closed ballsand that the intersection of every nest of balls in B f is a singleton orcontains a smaller ball B ∈ B f . Take a ball B ∈ B f . As in the previousproofs, there exists a maximal nest N in B f which contains the nest { B } . The intersection T N cannot contain a smaller ball B ′ ∈ B f sincethis would contradict the maximality of N . Hence by assumption, T N must be a singleton. As it is also f -closed and contained in B , we haveproved that f has a fixed point in B . (cid:3) Proof of Theorem 1.6: Take a function f on the ball space ( X, B )which is ultimately contracting on orbits. For every x ∈ X , the set { B f i x | i ≥ } is an f -nest. The set of all f -nests is partially orderedin the following way. If N = { B x | x ∈ S } and N = { B x | x ∈ S } are f -nests with S and S are closed under f , then we define N ≤ N if S ⊆ S . Then the union over an ascending chain of f -nests is againan f -nest since the union over sets that are closed under f is againclosed under f . Hence by Corollary 2.2, for every x ∈ X there is amaximal f -nest N containing { B f i x | i ≥ } . By the assumption of Theorem 1.6, there is some z ∈ T N such that B z ⊆ T N . We wish toshow that z is a fixed point of f . If z = f z would hold, then by (SC2), B f i z ( B z ⊆ T N for some i ≥
1, and the f -nest N ∪ { B f k z | k ∈ N } would properly contain N . But this would contradict the maximalityof N . Hence, z ∈ T N ⊆ B x is a fixed point of f . (cid:3) Some facts about the hierarchy of ball spaces
A refinement of the hierarchy.
By considering stronger properties of directed and centered systems ofballs, we will now add further entries to the hierarchy (1).We will say that a centered system of balls is c ′ if the intersection of any finite number of balls in the system containsa ball, c ′′ if the intersection of any finite number of balls in the system containsa largest ball, c ′′′ if the intersection of any finite number of balls in the collection is aball.We will say that a directed system of balls is d ′ if the intersection of any finite number of balls in the system containsa ball which is again in the system, d ′′ if the intersection of any finite number of balls in the system containsa largest ball which is again in the system, d ′′′ if the intersection of any finite number of balls in the system is aball which is again in the system.For 1 ≤ i ≤ S d ′ i (or S d ′′ i , or S d ′′′ i ) ifit satisfies the definition of S di with “directed system” replaced by “ d ′ directed system” (or “ d ′′ directed system”, or “ d ′′′ directed system”,respectively). Again for 1 ≤ i ≤
5, we will say that a ball space is S c ′ i (or S c ′′ i , or S c ′′′ i ) if it satisfies the definition of S ci with “centeredsystem” replaced by “ c ′ centered system” (or “ c ′′ centered system”, or“ c ′′′ centered system”, respectively).By induction one shows that in the above definitions for d ′ and d ′′′ ,“any finite number of” can be replaced by “any two” without changingthe meaning. In particular, every directed system of balls is d ′ . Wealso note that every nest of balls is a d ′′′ directed system of balls. Thistogether with the obvious implications between the properties defined RAMEWORK FOR FIXED POINT THEOREMS 15 above gives us the following refinement of each row of the hierarchy (1):(2) S i ⇐ S d ′′′ i ⇐ S d ′′ i ⇐ S d ′ i = S di ⇑ ⇑ ⇑ S c ′′′ i ⇐ S c ′′ i ⇐ S c ′ i ⇐ S ci for 1 ≤ i ≤ Connection with posets.
In a poset, a set S of elements is bounded if and only if it has an upperbound. A poset is bounded complete if every nonempty bounded sethas a least upper bound. A directed system in a poset is a nonemptysubset which contains an upper bound for any two of its elements.A poset is called directed complete if every directed system has aleast upper bound. As every chain is a directed system, every directedcomplete poset is chain complete.The proof of the following observations is straightforward: Proposition 3.1.
1) A ball space ( X, B ) is S if and only if everychain in ( B , < ) has an upper bound.2) A ball space ( X, B ) is S d if and only if every directed system in ( B , < ) has an upper bound.3) A ball space ( X, B ) is S if and only if ( B , < ) is chain complete.4) A ball space ( X, B ) is S d if and only if ( B , < ) is directed complete. Let us point out that the intersection of a system of balls may notbe itself a ball, even if it is nonempty (but if it is a ball, then it isclearly the largest ball contained in all of the balls in the system). Forthis reason, in general, the properties S , S d , S and S d cannot betranslated into a corresponding property of ( B , < ). This shows thatball spaces have more expressive strength than the associated posetstructures.A proof of the following fact can be found in [4, p. 33]. See also [18]for generalizations. Proposition 3.2.
Every chain complete poset is directed complete.
This proposition together with Proposition 3.1 yields:
Corollary 3.3.
Every S ball space is an S d ball space. From Proposition 3.1 and Corollary 2.2 we obtain that for an S ball space, every element of ( B , < ) lies below a maximal element. Thisproves: Proposition 3.4.
In an S ball space, every ball contains a minimalball. In the next sections, we will give further criteria for the equivalenceof various properties in the hierarchy.3.3.
Singleton balls.
In many applications (e.g. metric spaces, ultrametric spaces, T topo-logical spaces) the associated ball spaces have the property that single-ton sets are balls. The following observation is straightforward: Proposition 3.5.
For a ball space in which all singleton sets are balls, S is equivalent to S , S d is equivalent to S d , and S c is equivalent to S c . Ball spaces of ultrametric type.
We will call a ball space ( X, B ) of ultrametric type if any two ballsin B with nonempty intersection are comparable by inclusion. We willsee in Section 3.4 (Proposition 3.6) that the ball spaces associated withclassical ultrametric spaces are of ultrametric type. Proposition 3.6.
In a ball space of ultrametric type, every centeredsystem of balls is a nest. For such a ball space, S i , S di and S ci areequivalent, for each ∈ { , . . . , } . If in addition, in this ball space allsingleton sets are balls, then S is equivalent to S c . Proof: The first assertion follows from the fact that in a ball spaceof ultrametric type, every two balls in a centered system have nonemptyintersection and therefore are comparable by inclusion, so the systemis a nest. From this, the second assertion follows immediately. Thethird assertion follows by way of Proposition 3.5. (cid:3)
Intersection closed ball spaces.
A ball space ( X, B ) will be called finitely intersection closed if B is closed under nonempty intersections of any finite collection ofballs, chain intersection closed or nest intersection closed if B is closed under nonempty intersections of nests of balls, and intersec-tion closed if B is closed under nonempty intersections of arbitrarycollections of balls.We deduce from Proposition 3.6: Proposition 3.7.
Every chain intersection closed ball space of ultra-metric type is intersection closed.
RAMEWORK FOR FIXED POINT THEOREMS 17
Proof: Every collection C of balls with nonempty intersection inan arbitrary ball space is a centered system. If the ball space is ofultrametric type, then by Proposition 3.6, C is a nest. If in additionthe ball space is chain intersection closed, then the intersection T C isa ball. Hence under the assumptions of the proposition, the ball spaceis intersection closed. (cid:3) The proofs of the following two propositions are straightforward:
Proposition 3.8.
Assume that the ball space ( X, B ) is finitely inter-section closed. Then by closing under finite intersections, every cen-tered system of balls can be expanded to a directed system of balls whichhas the same intersection. Hence for a finitely intersection closed ballspace, S di is equivalent to S ci , for ≤ i ≤ . Proposition 3.9.
For chain intersection closed ball spaces, the prop-erties S , S , S , S and S are equivalent. As can be expected, the intersection closed ball spaces are the strong-est when it comes to equivalence of the properties in the hierarchy.
Theorem 3.10.
For an intersection closed ball space, S is equivalentto S ∗ , so all properties in the hierarchy are equivalent. Proof: Since ( X, B ) is intersection closed, the intersection of anynest is a ball as soon as it is nonempty. This yields that S is equivalentto S and hence also to S S and S . The same holds if we replace“nest” by “directed system”, i.e., the S i by S di , and if we replace “nest”by “centered system”, i.e., the S i by S ci .In particular, S implies S and S c implies S c . From Corollary 3.3we know that S implies S d , and from Proposition 3.8 that S d implies S c . Consequently, S implies S c , which shows that all properties inthe hierarchy (1) are equivalent. (cid:3) A bounded system of balls is a nonempty collection of balls whoseintersection contains a ball. Note that a bounded system of balls is acentered system, but the converse is in general not true (not even anest of balls is necessarily a bounded system if the ball space is not S ). Lemma 3.11.
The poset ( B , < ) is bounded complete if and only ifthe intersection of every bounded system of balls in ( X, B ) contains alargest ball. In an intersection closed ball space, the intersection ofevery bounded system of balls is a ball. Ball spaces and their properties in various applications
In what follows, we will give the interpretation of various levels ofspherical completeness in our applications of ball spaces.4.1.
Ultrametric spaces. An ultrametric u on a set X is a function from X × X to a partiallyordered set Γ with smallest element 0, such that for all x, y, z ∈ X andall γ ∈ Γ, (U1) u ( x, y ) = 0 if and only if x = y , (U2) if u ( x, y ) ≤ γ and u ( y, z ) ≤ γ , then u ( x, z ) ≤ γ , (U3) u ( x, y ) = u ( y, x ) (symmetry).The pair ( X, u ) is called an ultrametric space . Condition (U2) is theultrametric triangle law.We set uX := { u ( x, y ) | x, y ∈ X } and call it the value set of ( X, u ).If uX is totally ordered, we will call ( X, u ) a classical ultrametricspace ; in this case, (U2) is equivalent to: (UT) u ( x, z ) ≤ max { u ( x, y ) , u ( y, z ) } .We will now introduce four ways of deriving a ball space from an ul-trametric space. A closed ultrametric ball is a set B α ( x ) := { y ∈ X | u ( x, y ) ≤ α } , where x ∈ X and α ∈ Γ. We obtain the ultrametric ball space ( X, B u ) from ( X, u ) by taking B to be the set of all such balls B α ( x ).It follows from symmetry and the ultrametric triangle law that everyelement in a ball is a center, meaning that(3) B α ( x ) = B α ( y ) if y ∈ B α ( x ) . Further,(4) B β ( y ) ⊆ B α ( x ) if y ∈ B α ( x ) and β ≤ α . If B and B ′ are any two ultrametric balls with nonempty intersectionin a classical ultrametric space, then B ⊆ B ′ or B ′ ⊆ B .A problem with the ball B α ( x ) can be that it may not contain anyelement y such that u ( x, y ) = α ; if it does, it is called precise . It istherefore convenient to work with the precise balls of the form B ( x, y ) := { z ∈ X | u ( x, z ) ≤ u ( x, y ) } , where x, y ∈ X . We obtain the precise ultrametric ball space ( X, B [ u ] ) from ( X, u ) by taking B to be the set of all such balls B ( x, y ).It follows from symmetry and the ultrametric triangle law that B ( x, y ) = B ( y, x ) RAMEWORK FOR FIXED POINT THEOREMS 19 and that(5) B ( t, z ) ⊆ B ( x, y ) if and only if t ∈ B ( x, y ) and u ( t, z ) ≤ u ( x, y ) . In particular, B ( t, z ) ⊆ B ( x, y ) if t, z ∈ B ( x, y ) . Two elements γ and δ of Γ are comparable if γ ≤ δ or γ ≥ δ .Hence if u ( x, y ) and u ( y, z ) are comparable, then B ( x, y ) ⊆ B ( y, z ) or B ( y, z ) ⊆ B ( x, y ). If u ( y, z ) < u ( x, y ), then in addition, x / ∈ B ( y, z )and thus, B ( y, z ) ( B ( x, y ). We note:(6) u ( y, z ) < u ( x, y ) = ⇒ B ( y, z ) ( B ( x, y ) . From (4), we derive:
Proposition 4.1.
In a classical ultrametric space ( X, u ) , any twoballs with nonempty intersection are comparable by inclusion. Hence ( X, B [ u ] ) and ( X, B u ) are ball spaces of ultrametric type. We define (
X, u ) to be spherically complete if its ultrametric ballspace ( X, B u ) is S . For this definition, it actually makes no differencewhether we work with B u or B [ u ] : Proposition 4.2.
The classical ultrametric ball space ( X, B u ) is S ifand only if the precise ultrametric ball space ( X, B [ u ] ) is. Proof: Since B [ u ] ⊆ B u , the implication “ ⇒ ” is clear. Now takea nest N of balls in B u . We may assume that it does not containa smallest ball since otherwise this ball equals the intersection overthe nest, which consequently is nonempty. Further, there is a coinitialsubnest ( B α ν ( x ν )) ν<κ such that κ is an infinite limit ordinal and µ <ν < κ implies that B α ν ( x ν ) ( B α µ ( x µ ). It follows that this subnest hasthe same intersection as N .For every ν < κ , also ν + 1 < κ and thus B α ν +1 ( x ν +1 ) ( B α ν ( x ν ).Hence there is y ν +1 ∈ B α ν ( x ν ) \ B α ν +1 ( x ν +1 ). It follows that u ( x ν +1 , y ν +1 ) > α ν +1 , and from (4) we obtain that B α ν +1 ( x ν +1 ) ⊆ B u ( x ν +1 ,y ν +1 ) ( x ν +1 ) = B ( x ν +1 , y ν +1 ) . Since x ν +1 , y ν +1 ∈ B α ν ( x ν ), we know that u ( x ν +1 , y ν +1 ) ≤ max { u ( x ν +1 , x ν ) , u ( x ν , y ν +1 ) } ≤ α ν , and again from (4) we obtain that B ( x ν +1 , y ν +1 ) = B u ( x ν +1 ,y ν +1 ) ( x ν +1 ) ⊆ B α ν ( x ν +1 ) = B α ν ( x ν ) . It follows that \ N = \ ν<κ B α ν ( x ν ) = \ ν<κ B ( x ν +1 , y ν +1 ) . Consequently, if B [ u ] is S , then this intersection is nonempty and wehave proved that also B u is S . (cid:3) Since uX contains the smallest element 0 := u ( x, x ), B u containsall singletons { x } = B ( x ). Therefore, each ultrametric ball space isalready S once it is S . The same is true for the precise ultrametricball space ( X, B [ u ] ) in place of ( X, B u ). However, these ball spaces willin general not be S or S because even if an intersection of a nest isnonempty, it will not necessarily be a ball of the form B α ( x ) or B ( x, y ),respectively.If ( X, u ) is a classical ultrametric space, then this problem can beremedied if we work with a larger set of ultrametric balls. Given x ∈ X and an initial segment S = ∅ of uX , we define: B S ( x ) = { y ∈ X | u ( x, y ) ∈ S } . Setting B u + := { B S ( x ) | x ∈ X and S a nonempty initial segment of uX } , we obtain what we will call the full ultrametric ball space ( X, B u + ).Note that X = B uX ( x ) ∈ B u + . We leave it to the reader to prove:(7) B S ( x ) = [ α ∈ S B α ( x ) = \ β ≥ S B β ( x )where β ≥ S means that β ≥ γ for all γ ∈ S , and the intersection overan empty index set is taken to be X .We have that B [ u ] ⊆ B u ⊆ B u + where the second inclusion holds because B α ( x ) = B S ( x ) for the initialsegment S = [0 , α ] of uX . Proposition 4.3.
Let ( X, u ) be a classical ultrametric space. Then thefollowing assertions hold.1) The intersection over every nest of balls in ( X, B u + ) is equal to theintersection over a nest of balls in ( X, B u ) and therefore, ( X, B u + ) ischain intersection closed.2) The ball space ( X, B u + ) is spherically complete if and only if ( X, B u ) is.3) If the intersection of two balls in B u + is nonempty, then these twoballs are comparable under inclusion. RAMEWORK FOR FIXED POINT THEOREMS 21
Proof: Assertions 1) and 2) are proven in [10, Theorem 1.1]. Inorder to prove assertion 3), take two balls B , B ∈ B u + . In view of(7) we can write them as B = \ β ≥ S B β ( x ) and B = \ β ≥ S B β ( y )with x, y ∈ X and nonempty initial segments S , S of uX . Take anelement z ∈ B ∩ B . Then by (3), B β ( x ) = B β ( z ) for all β ≥ S and B β ( y ) = B β ( z ) for all β ≥ S . Hence if, say, S ⊆ S , then B = \ β ≥ S B β ( z ) ⊆ \ β ≥ S B β ( z ) = B . (cid:3) From part 3) of the proposition it follows that every centered systemof balls in B u + is already a nest. Therefore, from parts 1) and 2) of theproposition, we obtain: Theorem 4.4.
Let ( X, u ) be a classical ultrametric space. Then the fullultrametric ball space ( X, B u + ) is intersection closed and all propertiesin the hierarchy are equivalent for ( X, B u + ) .If ( X, B u ) is spherically complete, then ( X, B u + ) is an S ∗ ball space. By [10, Theorem 1.2], assertions 1) and 2) of Proposition 4.3 also holdfor all ultrametric spaces (
X, u ) with countable narrow value sets uX ;the condition narrow means that all sets of mutually incomparableelements in uX are finite. On the other hand, it is shown in [10] thatthe condition “narrow” cannot be dropped in this case. It is howeveran open question whether the condition “countable” can be dropped.A large number of ultrametric fixed point and coincidence point the-orems have been proven by S. Prieß-Crampe and P. Ribenboim (seee.g. [22, 23, 24, 25, 27]). Using ball spaces, some of them have beenreproven and new ones have been proven in [12, 13, 16].4.2. Metric spaces with metric balls.
In metric spaces (
X, d ) we can consider the closed metric balls B α ( x ) := { y ∈ X | d ( x, y ) ≤ α } for x ∈ X and α ∈ R ≥ := { r ∈ R | r ≥ } . We set B d := { B α ( x ) | x ∈ X , α ∈ R ≥ } . The following theorem will be deduced from Theorem 4.6 below:
Theorem 4.5.
If the ball space ( X, B d ) is spherically complete, then ( X, d ) is complete. The converse is not true. Consider a rational function field k ( x )together with the x -adic valuation v x . Choose an extension of v x to avaluation v of the algebraic closure K of k ( x ). Then the value groupis Q . An ultrametric in the sense of Section 4.1 is obtained by setting,for instance, u ( a, b ) := e − v ( a − b ) . Take (
K, u ) to be the completion of ( K , u ). It can be shown that theballs B α i i − X j =1 x − j ! with α i = e i (2 ≤ i ∈ N )have empty intersection in K . Hence ( K, u ) is not spherically complete,that is, the ultrametric ball space induced by u on K is not sphericallycomplete. But this ultrametric is a complete metric.Note that from Theorem 4.20 below it follows that the ball space( X, B d ) is spherically complete if every closed metric ball in ( X, d ) iscompact under the topology induced by d , as the closed metric ballsare closed in this topology.In order to characterize complete metric spaces by spherical com-pleteness, we have to choose smaller induced ball spaces. For anysubset S of the set R > of positive real numbers, we define: B S := { B r ( x ) | x ∈ X , r ∈ S } . Theorem 4.6.
The following assertions are equivalent:a) ( X, d ) is complete,b) the ball space ( X, B S ) is spherically complete for some S ⊂ R > which admits as its only accumulation point,c) the ball space ( X, B S ) is spherically complete for every S ⊂ R > which admits as its only accumulation point. Proof: We note that every S ⊂ R > which admits 0 as its onlyaccumulation point is discretely ordered. Take a nest N of balls in B S .If N contains a smallest ball, then this ball is equal to T N , which ishence nonempty. So one only has to consider nests without a smallestball. If we take such a nest N = { B r i ( x i ) | i ∈ I } in B S , then the set { r i | i ∈ I } ⊆ S has no smallest element and therefore, 0 is a limitpoint also of this set.a) ⇒ c): Assume that ( X, d ) is complete and take a set S ⊂ R > which admits 0 as its only accumulation point. This implies that S is RAMEWORK FOR FIXED POINT THEOREMS 23 discretely ordered, hence every infinite descending chain in S with amaximal element can be indexed by the natural numbers.Take any nest N of closed metric balls in B S . If the nest contains asmallest ball, then its intersection is nonempty; so we assume that itdoes not. If B ∈ N , then N B := { B ′ ∈ N | B ′ ⊆ B } is a nest of ballswith T N = T N B ; therefore, we may assume from the start that N contains a largest ball. Then the radii of the balls in N form an infinitedescending chain in S with a maximal element, and 0 is their uniqueaccumulation point. Hence we can write N = { B r i ( x i ) | i ∈ N } with r j < r i for i < j , and with lim i →∞ r i = 0.For every i ∈ N and all j ≥ i , the element x j lies in B r i ( x i ) andtherefore satisfies d ( x i , x j ) ≤ r i . This shows that ( x i ) i ∈ N is a Cauchysequence. Since ( X, d ) is complete, it has a limit x in X . We have that d ( x i , x ) ≤ r i , so x lies in every ball B r i ( x i ). This proves that the nesthas nonempty intersection.c) ⇒ b): Trivial.b) ⇒ a): Assume that ( X, B S ) is spherically complete. Take anyCauchy sequence ( x n ) n ∈ N in X . By our assumptions on S , we canchoose a sequence ( s i ) i ∈ N in { s ∈ S | s < s } such that 0 < s i +1 ≤ s i .Now we will use induction on i ∈ N to choose an increasing sequence( n i ) i ∈ N of natural numbers such that the balls B i := B s i ( x n i ) form anest.Since ( x n ) n ∈ N is a Cauchy sequence, we have that there is n suchthat d ( x n , x m ) < s for all n, m > n . Once we have chosen n i − ,we choose n i > n i − such that d ( x n , x m ) < s i +1 for all n, m ≥ n i .We show that the so obtained balls B i form a nest. Take i ∈ N and x ∈ B i +1 = B s i +1 ( x n i +1 ). This means that d ( x n i +1 , x ) ≤ s i +1 . Since n i , n i +1 ≥ n i , we have that d ( x n i , x n i +1 ) < s i +1 . We compute: d ( x n i , x ) ≤ d ( x n i , x n i +1 ) + d ( x n i +1 , x ) ≤ s i +1 + s i +1 = 2 s i +1 ≤ s i Thus x ∈ B i and hence B i +1 ⊆ B i for all i ∈ N . The intersection ofthis nest ( B i ) i ∈ N contains some y , by our assumption. We have that y ∈ B i for all i ∈ N , which means that d ( x n i , y ) ≤ s i . Sincelim i →∞ s i = 0 , we obtain that lim i →∞ x n i = y, which proves that ( X, d ) is a complete metric space. (cid:3)
Proof of Theorem 4.5: Assume that ( X, B d ) is spherically complete.Then so is ( X, B ′ ) for every nonempty B ′ ⊂ B d . Taking B ′ = B S with S as in Theorem 4.6, we obtain that ( X, d ) is complete. (cid:3)
Remark 4.7.
Theorems 4.5 and 4.6 remain true if instead of the closedmetric balls the open metric balls B α ( x ) := { y ∈ X | d ( x, y ) < α } are used for the metric ball space.4.3. Metric spaces with Caristi–Kirk balls or Oettli–Th´era balls.
Consider a metric space (
X, d ). A function ϕ : X → R is lower semi-continuous if for every y ∈ X ,lim inf x → y ϕ ( x ) ≥ ϕ ( y ) . If ϕ is lower semicontinuous and bounded from below, we call it a Caristi–Kirk function on X . For a fixed Caristi–Kirk function ϕ we consider Caristi–Kirk balls of the form(8) B ϕx := { y ∈ X | d ( x, y ) ≤ ϕ ( x ) − ϕ ( y ) } , x ∈ X, and the corresponding Caristi–Kirk ball space ( X, B ϕ ) given by B ϕ := { B ϕx | x ∈ X } . These ball spaces and their underlying theory can be employed to provethe Caristi–Kirk Theorem in a simple manner (see below). We foundthe sets that we call Caristi–Kirk balls in a proof of the Caristi–KirkTheorem given by J.-P. Penot in [20].We say that a function φ : X × X → ( −∞ , + ∞ ] is an Oettli–Th´erafunction on X if it satisfies the following conditions:( a ) φ ( x, · ) : X → ( −∞ , + ∞ ] is lower semicontinous for all x ∈ X ;( b ) φ ( x, x ) = 0 for all x ∈ X ;( c ) φ ( x, y ) ≤ φ ( x, z ) + φ ( z, y ) for all x, y, z ∈ X ;( d ) there exists x ∈ X such that inf x ∈ X φ ( x , x ) > −∞ . This notion was, to our knowledge, first introduced by Oettli and Th´erain [19]. An Oettli–Th´era function φ yields balls of the form B φx := { y ∈ X | d ( x, y ) ≤ − φ ( x, y ) } , x ∈ X, which will be called Oettli–Th´era balls . If an element x satisfiescondition ( d ) above, then we will call it an Oettli–Th´era element for
RAMEWORK FOR FIXED POINT THEOREMS 25 φ in X . For a fixed Oettli–Th´era element x we define the associated Oettli–Th´era ball space to be ( B φx , B φx ), where B φx := { B φx | x ∈ B φx } . We observe that for a given Caristi–Kirk function ϕ : X → R , themapping φ ( x, y ) := ϕ ( y ) − ϕ ( x )is an Oettli–Th´era function. Furthermore, every Caristi–Kirk ball isalso an Oettli–Th´era ball.In general the balls defined above are not metric balls. However,when working in complete metric spaces they prove to be a more use-ful tool than metric balls. As observed in the previous section, thecompleteness of a metric space need not imply spherical completenessof the space of metric balls ( X, B d ). In case of Caristi–Kirk and Oettli–Th´era balls, completeness turns out to be equivalent to spherical com-pleteness, as shown in the following two propositions. Proposition 4.8.
Let ( X, d ) be a metric space. Then the followingassertions are equivalent:a) The metric space ( X, d ) is complete.b) Every Caristi–Kirk ball space ( X, B ϕ ) is spherically complete.c) For every continuous function ϕ : X → R bounded from below, theCaristi–Kirk ball space ( X, B ϕ ) is spherically complete. Proposition 4.9.
A metric space ( X, d ) is complete if and only ifthe Oettli–Th´era ball space ( B φx , B φx ) is spherically complete for everyOettli–Th´era function φ on X and every Oettli–Th´era element x for φ in X . The proofs of Proposition 4.8 and Proposition 4.9 can be found in[14, Proposition 3] and [2], respectively.To describe the properties of Caristi–Kirk and Oettli–Th´era balls,we introduce the following notion. A ball space ( X, B ) is a B x –ballspace if B = { B x | x ∈ X } . We call a B x –ball space contractive if forevery x, y ∈ X , the following conditions hold:(C1) x ∈ B x ,(C2) if y ∈ B x then B y ⊆ B x ,(C3) if B x is not a singleton, then there exists y ∈ B x such that B y ( B x .A B x –ball space ( X, B ) is strongly contractive if it satisfies (C1),(C2), and:(C3s) if y ∈ B x \ { x } , then B y ( B x . Then every strongly contractive ball space is contractive. On the otherhand, it will turn out that condition (C1), while present in many ap-plications, is not always necessary. Thus, we will call a B x –ball space( X, B ) weakly contractive if it just satisfies (C2) and (C3).The next proposition is proved in [2]. Proposition 4.10.
Every Caristi–Kirk ball space ( X, B ϕ ) and everyOettli–Th´era ball space ( B φx , B φx ) is strongly contractive. We will meet another strongly contractive ball space in the case ofpartially ordered sets; see Proposition 4.29.In general, a strongly contractive ball space ( X, B ) may not con-tain balls of the form { x } for every x ∈ X . Then we cannot applyLemma 2.5 to acquire the equivalence between properties S and S .However, the following lemma yields the existence of a “sufficient”amount of singleton balls to obtain this equivalence. Proposition 4.11.
In a weakly contractive B x –ball space, the inter-section of a maximal nest of balls is a singleton ball if it is nonempty. Proof: Let M be a maximal nest of balls and assume that a ∈ T M for some element a ∈ X . Since a ∈ B for every ball B ∈ M , we obtainfrom (C2) that B a ⊆ B for every B ∈ M and thus B a ⊆ T M . Thismeans that M ∪ { B a } is a nest of balls, so by maximality of M wehave that B a ∈ M . Consequently, B a = T M . Suppose that B a is nota singleton. Then by condition (C3) there is some element b such that B b ( B a whence B b / ∈ M . But then M ∪ { B b } is a nest which strictlycontains M . This contradiction to the maximality of M shows that B a is a singleton. (cid:3) Since by Corollary 2.7 every nest is contained in a maximal nest, thisproposition yields:
Theorem 4.12.
1) A weakly contractive B x –ball space is S if and only if it is S .2) In every spherically complete weakly contractive B x –ball space everyball B x contains a singleton ball. If in addition the ball space satisfies(C1), then there exists a ∈ B x such that B a = { a } .3) If ( X, B ) is a spherically complete contractive B x –ball space and f : X → X a function such that f ( x ) ∈ B x for every x ∈ X , thenevery ball B x contains a fixed point of f . A version of part 3) of this theorem (with “contractive” replaced by“strongly contractive”) together with Propositions 4.8 and 4.10 is used
RAMEWORK FOR FIXED POINT THEOREMS 27 in [2] to prove the
Caristi–Kirk Fixed Point Theorem (see also[14]):
Theorem 4.13.
Take a complete metric space ( X, d ) and a lower semi-continuous function ϕ : X → R which is bounded from below. If afunction f : X → X satisfies the Caristi condition d ( x, f x ) ≤ ϕ ( x ) − ϕ ( f x ) , for all x ∈ X , then f has a fixed point on X . Also in [2], the same tools (with Proposition 4.8 replaced by Propo-sition 4.9) are used to prove the following generalization:
Theorem 4.14.
Take a complete metric space ( X, d ) and φ an Oettli-Th´era function on X . If a function f : X → X satisfies d ( x, f x ) ≤ − φ ( x, f x ) , for all x ∈ X , then f has a fixed point on X . A variant of part 2) of Theorem 4.12 is used in [2] to give quick proofsof several theorems that are known to be equivalent to the Caristi–Kirk Fixed Point Theorem (see [19, 20, 21] for presentations of theseequivalent results and generalizations). In the Introduction, we alreadypointed out the importance of f -closed singletons for the existenceof fixed points of a given function f . The Caristi condition on f inthe Caristi–Kirk Fixed Point Theorem does not necessarily imply thatevery ball B x is f -closed, but simply that f ( x ) ∈ B x . If B x is asingleton, then this yields that x is a fixed point of f . This provesassertion 3) of the theorem, which in turn provides a quick proof of theCaristi–Kirk Fixed Point Theorem. The weak condition that f ( x ) ∈ B x together with the condition that the ball space be strongly contractiveimplies that (SC2) holds and enables us to drop the condition on f -nests in Theorem 1.6. Remark 4.15.
Assume that ( X, B ) is a B x –ball space which satisfies(C1) and (C2). Then we can define a partial ordering on X by setting x ≺ y : ⇔ B y ( B x . If ( X, B ) is strongly contractive, then the function x B x is injective,and X together with the reverse of the partial order we have definedis order isomorphic to B with inclusion, that is, the function x B x is an order isomorphism from ( X, ≺ ) onto ( B , < ) where the latter isdefined as in the beginning of Section 2.If the B x are the Caristi–Kirk balls defined in (8), then we have that x ≺ y ⇔ d ( x, y ) < ϕ ( x ) − ϕ ( y ) , which means that ≺ is the Brønsted ordering on X . The Eke-land Variational Principle (cf. [2]) states that if the metric space iscomplete, then ( X, ≺ ) admits maximal elements, or in other words, B admits minimal balls. The Brønsted ordering has been used in severaldifferent proofs of the Caristi–Kirk Fixed Point Theorem. However, atleast in the proofs that also define and use the Caristi–Kirk balls (suchas the one of Penot [20] in [14]), it makes more sense to use directlytheir natural partial ordering. But the main incentive to use the ballsinstead of the ordering is that it naturally subsumes the metric case inthe framework of fixed point theorems in several other areas of math-ematics which is provided by the general theory of ball spaces as laidout in the present paper (see also [12, 13, 16]).It has been shown that the Ekeland Variational Principle can beproven in the Zermelo Fraenkel axiom system ZF plus the axiom ofdependent choice DC which covers the usual mathematical induction(but not transfinite induction, which is equivalent to the full axiom ofchoice). Conversely, it has been shown in [3] that the Ekeland Varia-tional Principle implies the axiom of dependent choice.Several proofs have been provided for the Caristi–Kirk FPT thatwork in ZF+DC. Kozlowski has given a proof that is purely metric asdefined in his paper [9], which implies that the proof works in ZF+DC.The proofs of Proposition 4.8 in [14] and of Proposition 4.9 in [2] arepurely metric. The existence of singleton balls in Caristi–Kirk andOettli-Th´era ball spaces over complete metric spaces can also be showndirectly by purely metric proofs and this result can be used to give quickproofs of many princples that are equivalent to the Caristi–Kirk FPTin ZF+DC (cf. [2]). However, in other settings it may not be possible todeduce the existence in ZF+DC, so then the axiom of choice is needed.Therefore, in view of the number of possible applications even beyondthe scope as presented in this paper, we do not hesitate to use Zorn’sLemma for the proof.We should point out that proofs have been given that apparentlyprove the Caristi–Kirk FPT in ZF (see [17, 7]). This means that theCaristi–Kirk FPT and the Ekeland Variational Principle are equivalentin ZF+DC, but not in ZF. For the topic of axiomatic strength, see alsothe discussions in [6, 8, 9].4.4.
Ordered abelian groups and fields.
If (
K, < ) is an ordered abelian group or ordered field, then the distancefunction d ( x, y ) = | x − y | takes its values in the nonnegative part of K . But the ordering on K may not be archimedean, that is, ( K, < ) RAMEWORK FOR FIXED POINT THEOREMS 29 may not admit an order preserving embedding in the reals; in this case, d is not a metric. Nevertheless, we can form a meaningful ball spaceassociated with ( K, < ) by taking B cb to consist of all closed boundedintervals [ a, b ] with a, b ∈ K . More generally, the same can be done forany ordered set ( I, < ).The following fact is not hard to show. It was first proved in [28]for ordered fields, and then for any ordered sets in [15]. By a cut inan ordered set (
I, < ) we mean a partition (
C, D ) of I such that c < d for all c ∈ C , d ∈ D and C, D are nonempty. The cofinality of atotally ordered set is the least cardinality of all cofinal subsets, and the coinitiality of a totally ordered set is the cofinality of this set underthe reverse ordering.
Lemma 4.16.
The ball space ( I, B cb ) associated with the totally orderedset ( I, < ) is S if and only if every cut ( C, D ) in ( I, < ) is asymmetric,that is, the cofinality of C is different from the coinitiality of D . Totally ordered sets and ordered abelian groups or fields whose cutsare all asymmetric are called symmetrically complete . In [28] it hasbeen shown that arbitrarily large symmetrically complete ordered fieldsexist. With a different construction idea, this has been generalized in[15] to the case of ordered abelian groups and totally ordered sets, anda characterization of symmetrically complete ordered abelian groupsand fields has been given.In order to give an example of a fixed point theorem that can beproven in this setting, it is enough to consider symmetrically completeordered abelian groups, as the additive group of a symmetrically com-plete ordered field is a symmetrically complete ordered abelian group.The following is Theorem 21 of [12] (see also [15]).
Theorem 4.17.
Take an ordered abelian group ( G, < ) and a function f : G → G . Assume that every nonempty chain of closed boundedintervals in G has nonempty intersection and that f has the followingproperties:1) f is nonexpanding: | f x − f y | ≤ | x − y | for all x, y ∈ G , f is contracting on orbits: there is a positive rational number mn < with m, n ∈ N such that n | f x − f x | ≤ m | x − f x | for all x ∈ G .
Then f has a fixed point. As in the case of ultrametric spaces, all singletons in B cb are balls: { a } = [ a, a ]. So also here, ( I, B cb ) is S as soon as it is S . But againas in the case of ultrametric spaces, S does not necessarily imply S or even S . For example, consider a nonarchimedean ordered symmet-rically complete field. The set of infinitesimals is the intersection ofballs [ − a, a ] where a runs through all positive elements that are notinfinitesimals. This intersection is not a ball, nor is there a largest ballcontained in it.Further, we note: Lemma 4.18.
Assume that ( I, < ) is a totally ordered set and its as-sociated ball space ( I, B cb ) is an S d or S ball space. Then ( I, < ) is cutcomplete, that is, for every cut ( C, D ) in ( I, < ) , C has a largest or D has a smallest element. Proof: First assume that ( I, B cb ) is an S d ball space, and take acut ( C, D ) in I . If a, c ∈ C and b, d ∈ D , then max { a, c } ∈ C andmin { b, d } ∈ D and [ a, b ] ∩ [ c, d ] = [max { a, c } , min { b, d } ]. This showsthat { [ c, d ] | c ∈ C , b ∈ D } is a directed system in B cb . Hence its intersection is nonempty; if a iscontained in this intersetion, it must be the largest element of C or theleast element of D . Hence ( I, < ) is cut complete.Now assume that (
I, < ) is not cut complete; we wish to show that( I, B cb ) is not an S ball space. Take a cut ( C, D ) in I such that C has no largest element and D has no least element. Pick some c ∈ C .Then { [ c, d ] | d ∈ D } is a nest of balls in ( I, B cb ). Its intersection is the set { a ∈ C | c ≤ a } .Since C has no largest element, this set does not contain a maximalball. This shows that ( I, B cb ) is not an S ball space. (cid:3) It is a well known fact that the only cut complete densely orderedabelian group or ordered field is R . So we have: Proposition 4.19.
The associated ball space of the reals is S ∗ . For allother densely ordered abelian groups and ordered fields the associatedball space can at best be S . Proof: Take any centered system { [ a i , b i ] | i ∈ I } of intervals in R .We set a := sup i ∈ I a i and b := inf i ∈ I b i . Then \ i ∈ I [ a i , b i ] = [ a, b ] . RAMEWORK FOR FIXED POINT THEOREMS 31
We have to show that [ a, b ] = ∅ , i.e., a ≤ b . Suppose that a > b .Then there are i, j ∈ I such that a i > b j . But by assumption, [ a i , b i ] ∩ [ a j , b j ] = ∅ , a contradiction. We have now proved that the associatedball space of the reals is S ∗ .The second assertion follows from Lemma 4.18. (cid:3) Topological spaces. If X is a topological space on a set X , we will take its associated ballspace to be ( X, B ) where B consists of all nonempty closed sets. Sincethe intersections of arbitrary collections of closed sets are again closed,this ball space is intersection closed.The following theorem shows how compact topological spaces arecharacterized by the properties of their associated ball spaces; notethat we use “compact” in the sense of “quasi-compact”, that is, it doesnot imply the topology being Hausdorff. Theorem 4.20.
The following are equivalent for a topological space X :a) X is compact,b) the nonempty closed sets in X form an S ball space,c) the nonempty closed sets in X form an S ∗ ball space. Proof: a) ⇒ b): Assume that X is compact. Take a nest ( X i ) i ∈ I ofballs in ( X, B ) and suppose that T i ∈ I X i = ∅ . Then S i ∈ I X \ X i = X ,so { X \ X i | i ∈ I } is an open cover of X . It follows that thereare i , . . . , i n ∈ I such that X \ X i ∪ . . . ∪ X \ X i n = X , whence X i ∩ . . . ∩ X i n = ∅ . But since the X i form a nest, this intersectionequals the smallest of the X i j , which is nonempty. This contradictionproves that the nonempty closed sets in X form an S ball space.b) ⇒ c): This follows from Theorem 3.10.c) ⇒ a): Assume that the nonempty closed sets in X form an S ∗ ball space. Take an open cover Y i , i ∈ I , of X . Since S i ∈ I Y i = X ,we have that T i ∈ I X \ Y i = ∅ . As the ball space is S ∗ , this meansthat { X \ Y i | i ∈ I } cannot be a centered system. Consequently,there are i , . . . , i n ∈ I such that X \ Y i ∩ . . . ∩ X \ Y i n = ∅ , whence Y i ∪ . . . ∪ Y i n = X . (cid:3) The following two topological fixed point theorems were proven in[12, Theorem 11]. We will give their proofs here as they illustrateapplications of theorems 1.6 and 1.2.
Theorem 4.21.
Take a compact space X and a closed function f : X → X . Assume that for every x ∈ X with f x = x there is a closedsubset B of X such that x ∈ B and x / ∈ f ( B ) ⊆ B . Then f has a fixedpoint in B . Proof: For every x ∈ X we consider the following family of balls: B x := { B | B closed subset of X , x ∈ B and f ( B ) ⊆ B } . Note that B x is nonempty because it contains X . We define(9) B x := \ B x . We see that x ∈ B x and that f ( B x ) ⊆ B x . Further, B x is closed, beingthe intersection of closed sets. This shows that B x is the smallestmember of B x .For every B ∈ B x we have that f x ∈ B and therefore, B ∈ B fx .Hence we find that B fx ⊆ B x .Assume that f x = x . Then by hypothesis, there is a closed set B in X such that x ∈ B and x / ∈ f ( B ) ⊆ B . Since f is a closedfunction, f ( B ) is closed. Moreover, f ( f ( B )) ⊆ f ( B ) and f x ∈ f ( B ),so f ( B ) ∈ B fx . Since x / ∈ f ( B ), we conclude that x / ∈ B fx , whence B fx ( B x . We have now proved that f is ultimately contractingon orbits. Further, B ∈ B x , whence B x ⊂ B , f ( B x ) ⊂ f ( B ) andtherefore, x / ∈ f ( B x ). This shows that B x is the smallest of all closedsets B in X for which x ∈ B and x / ∈ f ( B ) ⊆ B .Take an f -nest N in { B x | x ∈ X } . Theorem 4.20 shows that T N is nonempty. Take any z ∈ T N . Choose an arbitrary B ∈ N . Then z ∈ B and thus, B ∈ B z . So we have that B z ⊆ B . Therefore, B z ⊆ T N . Our theorem now follows from Theorem 1.6. (cid:3) An interesting interpretation of the ball B x defined in (9) will be givenin Remark 5.3 below. Theorem 4.22.
Take a compact space X and a closed function f : X → X .1) If every nonempty closed and f -closed subset B of X contains aclosed f -contracting subset, then f has a fixed point in X .2) If every nonempty closed and f -closed subset B of X is f -contracting,then f has a unique fixed point in X .3) Assume that B ′ is a set of closed subsets of X such that f ( B ) ∈ B ′ for each B ∈ B ′ and B ′ is chain intersection closed. If every f -closedball B ∈ B ′ is f -contracting, then f has a unique fixed point in X . RAMEWORK FOR FIXED POINT THEOREMS 33
Proof: By Theorem 4.20 the ball space consisting of all nonemptyclosed subsets of the compact space X is S ∗ , hence also S . We willnow deduce our theorem from Theorem 1.2.1): Assume that every nonempty closed and f -closed subset B of X contains a closed f -contracting subset, that is, every f -closed ballcontains an f -contracting ball. We wish to show that each f -closed ball B is a singleton or contains a smaller f -closed ball. By assumption, B contains a closed f -contracting subset B ′ . We see that B ′ is an f -closedball. If B ′ = B , then we are done. Hence assume that B ′ = B , whichyields that B is f -contracting. If B is a singleton, then we are doneagain. Otherwise, we must have that f ( B ) ( B . By observation a)at the start of Section 2.3, f ( B ) is f -closed, and it is closed since f isassumed to be a closed function. As the f -closed ball f ( B ) is properlycontained in B , we are done also in this final case. Thus part 1) of ourtheorem follows from part 1) of Theorem 1.2.2): Now assume that every nonempty closed and f -closed subset B of X is f -contracting. We wish to show that the image f ( B ) of every f -closed ball B is an f -contracting ball. As before, f ( B ) is closed and f -closed. Therefore, it is f -contracting by assumption. Thus part 2)of our theorem follows from part 2) of Theorem 1.2.3): As mentioned in the beginning, the ball space of all nonemptyclosed subsets of X is S , hence also S , and thus the same holds for( X, B ′ ) (cf. Proposition 6.1 below). As B ′ is chain intersection closedby assumption, it follows from Proposition 3.9 that ( X, B ′ ) is S . Takean f -closed ball B ∈ B ′ . As before, f ( B ) is closed and f -closed. Byour assumptions, we also have that f ( B ) ∈ B ′ and then that f ( B ) is f -contracting. Thus part 3) of our theorem follows from part 2) ofTheorem 1.2, where B is taken to be B ′ . (cid:3) The condition that every f -closed ball is a singleton or contains asmaller f -closed ball appears to be quite strong. Yet there is a nat-ural example in the setting of topological spaces where this conditionis satisfied. In [29], Steprans, Watson and Just define the notion of J -contraction for functions f : X → X on topological spaces. We donot need the slightly lengthy definition here; instead, we use two impor-tant facts about J -contractions f on a connected compact Hausdorffspace X which the authors prove in the cited paper:(J1) If B is a closed subset of X with f ( B ) ⊆ B , then the restrictionof f to B is also a J -contraction ([29, Proposition 1, p. 552]);(J2) If f is onto, then | X | = 1 ([29, Proposition 4, p. 554]).The following is Theorem 4 of [29]: Theorem 4.23.
Take a connected compact Hausdorff space X and acontinuous J -contraction f : X → X . Then f has a unique fixed point. We show how to deduce this theorem from part 3) of Theorem 4.22.First, as f is a continuous function on the compact Hausdorff space X ,it is a closed function. We take B ′ to be the set of all closed connectedsubsets of X . Take any B ∈ B ′ . As f is a closed function, f ( B )is closed and since B is connected and f is continuous, f ( B ) is alsoconnected; hence f ( B ) ∈ B ′ . Further, the intersection of any chain ofclosed connected subsets of X is closed and connected, which showsthat B ′ is intersection closed.Finally, we have to show that every f -closed ball B ∈ B ′ is f -contracting. As B is closed in X , it is also compact Hausdorff, andit is connected as it is a ball in B ′ . By (J1), the restriction of f to B isalso a J -contraction. Therefore, we can replace X by B and apply (J2)to find that if f is onto then B is a singleton, i.e., B is f -contracting.Now Theorem 4.23 follows from part 3) of Theorem 4.22 as desired.It should be noted that J -contractions appear in a natural way inthe metric setting. The following results are Theorems 2 and 3 of [29]: Theorem 4.24.
Any contraction on a compact metric space is a J -contraction. Conversely, if f is a J -contraction on a connected com-pact metrizable space X , then X admits a metric under which f is acontraction. Partially ordered sets.
Take any nonempty partially ordered set (
T, < ). We will associatewith it two different ball spaces; first, the ball space of principal finalsegments, and then later the interval ball space.The ball space ( T, B pfs ), where B pfs := { [ a, ∞ ) | a ∈ T } is the setof all principal final segments [ a, ∞ ) := { c ∈ T | a ≤ c } . Thefollowing proposition gives the interpretation of spherical completenessfor this ball space: Proposition 4.25.
The following assertions are equivalent:a) the poset ( T, < ) is inductively ordered,b) the ball space ( T, B pfs ) is spherically complete,c) ( T, B pfs ) is an S ball space. Proof: We observe that { a i | i ∈ I } is a chain in T if and only if N = ([ a i , ∞ )) i ∈ I is a nest of balls in B pfs . RAMEWORK FOR FIXED POINT THEOREMS 35 a) ⇒ c): Take a nest N = ([ a i , ∞ )) i ∈ I . Since ( T, < ) is inductivelyordered, the chain { a i | i ∈ I } admits an upper bound a ∈ T . Then forall i ∈ I , a i ≤ a , whence [ a, ∞ ) ⊆ [ a i , ∞ ). Thus, [ a, ∞ ) ⊆ T N , whichproves that ( T, B pfs ) is an S ball space.c) ⇒ b): This holds by the general properties of the hierarchy.b) ⇒ a): Take a chain { a i | i ∈ I } in T . Since ( T, B pfs ) is sphericallycomplete, the intersection of the nest N = ([ a i , ∞ )) i ∈ I is nonempty. If a ∈ T N , then for all i ∈ I , a ∈ [ a i , ∞ ), whence a i ≤ a . Thus, a isan upper bound of { a i | i ∈ I } , which proves that ( T, < ) is inductivelyordered. (cid:3)
We leave it to the reader to show that ( T, B pfs ) is an S (or S d or S c )ball space if and only if every chain (or directed system, or centeredsystem, respectively) has minimal upper bounds.We will need the following observation: Lemma 4.26.
The equality [ a, ∞ ) = \ i ∈ I [ a i , ∞ ) holds if and only if a = sup i ∈ I a i . Further, T i ∈ I [ a i , ∞ ) is the (possiblyempty) set of all upper bounds for { a i | i ∈ I } . Proof: We have a ∈ T i ∈ I [ a i , ∞ ) if and only if a ∈ [ a i , ∞ ) andhence a ≥ a i for all i , which means that a is an upper bound for the a i . Hence, T i ∈ I [ a i , ∞ ) is the set of all upper bounds of the a i , andthis set is equal to [ a, ∞ ) if and only if a is the least upper bound. (cid:3) An element a in a poset is called top element if b ≤ a for all elements b in the poset, and bottom element if b ≥ a for all elements b in theposet. A top element is commonly denoted by ⊤ , and a bottom elementby ⊥ . A poset ( T, < ) is an upper semilattice if every two elementsin T have a least upper bound, and a complete upper semilattice if every nonempty set of elements in T has a least upper bound. Proposition 4.27. ( T, B pfs ) is finitely intersection closed if andonly if every nonempty finite bounded subset of T has a supremum.2) ( T, B pfs ) is intersection closed if and only if every nonempty boundedsubset of T has a supremum, i.e., ( T, < ) is bounded complete.If in addition ( T, < ) has a top element, then3) ( T, B pfs ) is finitely intersection closed if and only if ( T, < ) is anupper semilattice,4) ( T, B pfs ) is intersection closed if and only if ( T, < ) is a completeupper semilattice. Proof: 1), 2): Assume that ( T, B pfs ) is (finitely) intersection closedand take a nonempty (finite) subset { a i | i ∈ I } of T . If this set isbounded, then T i ∈ I [ a i , ∞ ) is nonempty, and thus by assumption it isequal to [ a, ∞ ) for some a ∈ T . By Lemma 4.26, this implies that a = sup i ∈ I a i , showing that { a i | i ∈ I } has a supremum.Now assume that every nonempty (finite) bounded subset of T hasa supremum. Take a nonempty (finite) set { [ a i , ∞ ) | i ∈ I } of balls in B pfs with nonempty intersection. Take b ∈ T i ∈ I [ a i , ∞ ). Then b is anupper bound of { a i | i ∈ I } . By assumption, there exists a = sup i ∈ I a i in T . Again by Lemma 4.26, this implies that T i ∈ I [ a i , ∞ ) = [ a, ∞ ).Hence, ( T, B pfs ) is (finitely) intersection closed.3) and 4) follow from 1) and 2), respectively, because if ( T, < ) has atop element, then every nonempty subset is bounded. (cid:3)
Now we can characterize chain complete and directed complete posetsby properties from our hierarchy:
Theorem 4.28.
Take a poset ( T, < ) . Then the following are equiva-lent:a) ( T, < ) is chain complete,b) ( T, < ) is directed complete,c) ( T, B pfs ) is an S ball space,d) ( T, B pfs ) is an S d ball space.If every finite bounded subset of T has a supremum, then the aboveproperties are also equivalent toe) ( T, B pfs ) is an S ∗ ball space. Proof: The equality of assertions a) and b) follows from Proposi-tion 3.2.b) ⇒ d): Assume that ( T, < ) is directed complete and take a directedsystem S = { [ a i , ∞ ) | i ∈ I } in B pfs . Then also { a i | i ∈ I } is a directedsystem in ( T, < ). By our assumption on (
T, < ) it follows that { a i | i ∈ I } has a supremum a in T . By Lemma 4.26, [ a, ∞ ) = T i ∈ I [ a i , ∞ ),which shows that the intersection of S is a ball.d) ⇒ c) holds by the general properties of the hierarchy.c) ⇒ a): Take a chain { a i | i ∈ I } in T . Since ( T, B pfs ) is an S ballspace, the intersection of the nest N = ([ a i , ∞ )) i ∈ I is a ball [ a, ∞ ). Itfollows by Lemma 4.26 that a is the least upper bound of the chain,which proves that ( T, < ) is chain complete.
RAMEWORK FOR FIXED POINT THEOREMS 37
If every finite bounded subset of T has a supremum, then by part1) of Proposition 4.27, ( T, B pfs ) is finitely intersection closed, hence byProposition 3.8, properties S d and S ∗ are equivalent. (cid:3) The ball space ( T, B pfs ) shares an important property with Caristi–Kirk and Oettli–Th´era ball spaces: Proposition 4.29.
The ball space ( T, B pfs ) is strongly contractive. Proof: We define B x := [ x, ∞ ) ∈ B pfs . Then x ∈ B x for every x ∈ T . If y ∈ B x , then x ≤ y and therefore[ y, ∞ ) ⊆ [ x, ∞ ); if in addition x = y , then x < y so that x / ∈ [ y, ∞ )and [ y, ∞ ) ( [ x, ∞ ). (cid:3) A function f on a poset ( T, < ) is increasing if f ( x ) ≥ x for all x ∈ T .The following result is an immediate consequence of Zorn’s Lemma, butcan also be seen as a corollary to Propositions 4.25 and 4.29 togetherwith part 3) of Theorem 4.12: Theorem 4.30.
Every increasing function f : X → X on an induc-tively ordered poset ( T, < ) has a fixed point. Note that this theorem implies the
Bourbaki-Witt Theorem , whichdiffers from it by assuming that every increasing chain in (
T, < ) evenhas a least upper bound.A function f on a poset ( T, < ) is called order preserving if x ≤ y implies f x ≤ f y . The following result is an easy consequence ofTheorem 4.30: Theorem 4.31.
Take an order preserving function f on a nonemptyposet ( T, < ) which contains at least one x such that f x ≥ x (in particu-lar, this holds when ( T, < ) has a bottom element). Assume that ( T, < ) is chain complete. Then f has a fixed point. Proof: Take S := { x ∈ T | f x ≥ x } 6 = ∅ . Then also S is chaincomplete. Indeed, if ( x i ) i ∈ I is a chain in S , hence also in T , then it hasa least upper bound z ∈ T . Since z ≥ x i and f is order preserving, wehave that f z ≥ f x i ≥ x i for all i ∈ I , so f z is also an upper bound for( x i ) i ∈ I . Therefore, f z ≥ z , showing that f z ∈ S . Now the existence ofa fixed point follows from Theorem 4.30. (cid:3) The second ball space we associate with posets will be particularlyuseful for the study of lattices. We define the interval ball space ( T, B iv ) of the poset ( T, < ) by taking B iv to consist of all closed inter-vals , that is, sets of the form [ a, b ] := { c ∈ T | a ≤ c ≤ b } for a, b ∈ T with a ≤ b , or of the form { c ∈ T | c ≤ a } or { c ∈ T | a ≤ c } for a ∈ T .Note that all closed intervals are of the form [ a, b ] if and only if T has atop element ⊤ and a bottom element ⊥ . Even if T does not have theseelements, we will still use the notation [ ⊥ , b ] for { c ∈ T | c ≤ b } and[ a, ⊤ ] for { c ∈ T | a ≤ c } . We also include [ ⊥ , ⊤ ] := T in B iv . Hence, B iv = { [ a, b ] | a ∈ T ∪ {⊥} , b ∈ T ∪ {⊤}} . Lemma 4.32.
We have that a = sup i ∈ I a i and b = inf i ∈ I b i if and onlyif [ a, b ] = \ i ∈ I [ a i , b i ] . Proof: We can write \ i ∈ I [ a i , b i ] = \ i ∈ I [ a i , ⊤ ] ∩ [ ⊥ , b i ] = \ i ∈ I [ a i , ⊤ ] ∩ \ i ∈ I [ ⊥ , b i ]Applying Lemma 4.26, we obtain that [ a, ⊤ ] = T i ∈ I [ a i , ⊤ ] if and onlyif a = sup i ∈ I a i . Applying Lemma 4.26 to L with the reverse order, weobtain that [ ⊥ , b ] = T i ∈ I [ ⊥ , b i ] if and only if b = inf i ∈ I b i . These twofacts together yield the assertion of our lemma. (cid:3) Lattices. A lattice is a poset in which every two elements have a supremum(least upper bound) and a infimum (greatest lower bound). It thenfollows that all finite sets in a lattice ( L, < ) have a supremum and aninfimum. A complete lattice is a poset in which all nonempty setshave a supremum and an infimum. Lemma 4.32 implies:
Proposition 4.33.
The ball space ( L, B iv ) associated to a lattice ( L, < ) is finitely intersection closed. The ball space ( L, B iv ) associated to acomplete lattice ( L, < ) is intersection closed. For a lattice (
L, < ), we denote by (
L, > ) the lattice endowed with thereverse order. We will now characterize complete lattices by propertiesfrom our hierarchy.
Theorem 4.34.
For a poset ( L, < ) , the following assertions are equiv-alent.a) ( L, < ) is a complete lattice,b) ( L, < ) and ( L, > ) are complete upper semilattices,c) the principal final segments of ( L, < ) and of ( L, > ) form S ∗ ballspaces, RAMEWORK FOR FIXED POINT THEOREMS 39 d) ( L, B iv ) is an S ∗ ball space and ( L, < ) admits a top and a bottomelement,e) ( L, B iv ) is an S ∗ ball space and every finite set in ( L, < ) has anupper and a lower bound. Proof: The equivalence of a) and b) follows directly from thedefinitions. The equivalence of b) and c) follows from part 4) of Propo-sition 4.27.a) ⇒ d): Assume that ( L, < ) is a complete lattice. Then it admitsa top element (supremum of all its elements) and a bottom element(infimum of all its elements). Take a centered system { [ a i , b i ] | i ∈ I } in ( L, B iv ). Then for all i, j ∈ I , [ a i , b i ] ∩ [ a j , b j ] = ∅ , so a i ≤ b j . Henceevery b j is an upper bound of { a i | i ∈ I } , and every a i is a lowerbound of { b j | j ∈ I } . Set a := sup i ∈ I a i and b := inf i ∈ I b i . It followsthat also b is an upper bound of { a i | i ∈ I } , and a is a lower bound of { b j | j ∈ I } . Therefore, a, b ∈ [ a i , b i ] for all i and thus, T i ∈ I [ a i , b i ] = ∅ .From Lemma 4.32 it follows that T i ∈ I [ a i , b i ] = [ a, b ] and hence is a ball.We have proved that ( L, B iv ) is an S ∗ ball space.d) ⇒ e): A top element is an upper bound and a bottom element alower bound for every set of elements.e) ⇒ a): Take a poset ( L, < ) that satisfies the assumptions of e),and any subset S ⊆ L . If S is a finite subset of S , then it has anupper bound b by assumption. Hence the balls [ a, ⊤ ], a ∈ S , have anonempty intersection, as it contains b . This shows that { [ a, ⊤ ] | a ∈ S } is a centered system of balls. Since ( L, B iv ) is an S ∗ ball space, itsintersection is a ball [ c, d ], where we must have d = ⊤ . By Lemma 4.26, c is the supremum of S .Working with the reverse order, one similarly shows that S has aninfimum since ( L, B iv ) is an S ∗ ball space. Hence, ( L, < ) is a completelattice. (cid:3)
For our next theorem, we will need one further lemma:
Lemma 4.35.
For a lattice ( L, < ) , the following are equivalent:a) ( L, < ) is a complete lattice,b) ( L, < ) and ( L, > ) are directed complete posets,c) ( L, < ) and ( L, > ) are chain complete posets. Proof: The implication a) ⇒ b) is trivial as every nonempty setin a complete lattice has a supremum and an infimum.b) ⇒ a): Take a nonempty subset S of L . Let S ′ be the closure of S under suprema and infima of arbitrary finite subsets of S . Then S ′ is a directed system in both ( L, < ) and (
L, > ). Hence by b), S ′ hasan infimum a and a supremum b . These are lower and upper bounds,respectively, for S . Suppose there was an upper bound c < b for S .Then there would be a supremum d of some finite subset of S such that d > c . But as c is also an upper bound of this finite subset, we musthave that d ≤ c . This contradiction shows that b is also the supremumof S . Similarly, one shows that a is also the infimum of S . This provesthat ( L, < ) is a complete lattice.b) ⇔ c) follows from Proposition 3.2. (cid:3) Now we can prove:
Theorem 4.36.
For a lattice ( L, < ) , the following are equivalent:a) ( L, < ) is a complete lattice,b) ( L, B iv ) is an S ball space,c) ( L, B iv ) is an S ∗ ball space. Proof: a) ⇒ c): This follows from Theorem 4.34.c) ⇒ b) holds by the general properties of the hierarchy.b) ⇒ a): By Lemma 4.35 it suffices to prove that ( L, < ) and (
L, > )are chain complete posets. Take a chain { a i | i ∈ I } in ( L, < ). Then { [ a i , ⊤ ] | i ∈ I } is a nest of balls in ( L, B iv ). Since ( L, B iv ) is an S ball space, the intersection of this nest is a ball [ a, b ] for some a, b ∈ L ;it must be of the form [ a, ⊤ ] since the intersection contains ⊤ . FromLemma 4.26 we infer that a = sup i ∈ I a i . This shows that ( L, < ) is achain complete poset. The proof for (
L, > ) is similar. (cid:3)
An example for a fixed point theorem that holds in complete latticesis the Knaster-Tarski Theorem, which we will discuss in Section 7.1.5. S ∗ ball spaces Take a ball space ( X, B ) and a subset Y ⊆ X . If there is at leastone ball B ∈ B such that Y ∩ B = ∅ , then with B ∩ Y := { B ∩ Y | B ∈ B} \ {∅} , ( Y, B ∩ Y ) is a ball space. We will now study how ( Y, B ∩ Y ) caninherit properties of spherical completeness from ( X, B ). For this it isimportant to know whether nests of balls in ( Y, B ∩ Y ) can be lifted tonests of balls in ( X, B ). As we will show, this can be done in S ∗ ballspaces. We will then apply our results in Section 7.1 to prove a genericKnaster–Tarski theorem which generalizes the original Knaster–TarskiTheorem. RAMEWORK FOR FIXED POINT THEOREMS 41
Spherical closures in S ∗ ball spaces. Throughout this and the next section, we consider an S ∗ ball space( X, B ). As before, if f : X → X is a function, then B f will denotethe collection of all f -closed balls in B . The following is a simple butuseful observation. It follows from the fact that the intersection overany collection of f -closed sets is again f -closed. Lemma 5.1.
Also ( X, B f ) is an S ∗ ball space, provided that B f = ∅ . For every nonempty subset S of a ball in B , we definescl B ( S ) := \ { B ∈ B | S ⊆ B } and call it the (spherical) closure of S in B . Lemma 5.2.
1) For every nonempty subset S of a ball in B , scl B ( S ) is the smallest ball in B containing S .2) If f : X → X is a function, then for every nonempty subset S ofan f -closed ball in B , scl B f ( S ) is the smallest f -closed ball containing S . Proof: 1) The collection of all balls containing the nonempty set S is a centered system. It is nonempty by our condition that S is asubset of a ball in B . The intersection of this system contains S , andsince ( X, B ) is S ∗ , it is a ball. As all balls containing S appear in thesystem, the intersection must be the smallest ball containing S .2) This follows from part 1) together with Lemma 5.1. (cid:3) Note that if X ∈ B , then we can drop the condition that S is the subsetof a ball (or an f -closed ball, respectively) in B . Remark 5.3.
The ball B x defined in (9) in the proof of Theorem 4.21is equal to scl B f ( { x } ), where B f is the set of all closed f -closed sets ofthe topological space under consideration.The proof of the following observation is straightforward: Lemma 5.4. If S ⊆ T are nonempty subsets of a ball in B , then scl B ( S ) ⊆ scl B ( T ) . Ball spaces induced on subsets of S ∗ ball spaces. Now we take Y ⊆ X and consider the ball space ( Y, B ∩ Y ). Lemma 5.5.
1) For each B ∈ B ∩ Y , scl B ( B ) ∩ Y = B .
2) The assignment
B ∩ Y ∋ B scl B ( B ) preserves inclusion in the strong sense that B ( B ⇐⇒ scl B ( B ) ( scl B ( B ) .
3) If ( B i ) i ∈ I is a centered system of balls in ( Y, B∩ Y ) , then (scl B ( B i )) i ∈ I is a centered system of balls in ( X, B ) with (10) \ i ∈ I B i = \ i ∈ I scl B ( B i ) ! ∩ Y .
Proof: 1): It follows from the definition of scl B ( B ) that B ⊆ scl B ( B ), so B ⊆ scl B ( B ) ∩ Y . Since B ∈ B ∩ Y , we can write B = B ′ ∩ Y for some B ′ ∈ B . Since scl B ( B ) is the smallest ball in X containing B ,it must be contained in B ′ and therefore, scl B ( B ) ∩ Y ⊆ B ′ ∩ Y = B .2): In view of Lemma 5.4, it suffices to show that B = B impliesscl B ( B ) = scl B ( B ). But this is a consequence of part 1) of this lemma.3): Take a centered system of balls ( B i ) i ∈ I in ( Y, B ∩ Y ). Then(scl B ( B i )) i ∈ I is a centered system of balls in ( X, B ) since B i ∩ . . . ∩ B i n = ∅ implies that scl B ( B i ) ∩ . . . ∩ scl B ( B i n ) = ∅ . By part 1), B i = scl B ( B i ) ∩ Y , whence \ i ∈ I B i = \ i ∈ I (scl B ( B i ) ∩ Y ) = \ i ∈ I scl B ( B i ) ! ∩ Y . (cid:3)
With the help of this lemma, we obtain:
Proposition 5.6.
Take an S ∗ ball space ( X, B ) and Y ⊂ X . Assumethat B ∩ Y = ∅ for every B ∈ B . Then also ( Y, B ∩ Y ) is an S ∗ ballspace. Proof: Take a centered system of balls ( B i ) i ∈ N in ( Y, B ∩ Y ).Then by part 3) of Lemma 5.5, (scl B ( B i )) i ∈ N is a centered system ofballs in ( X, B ) with T i ∈ I B i = (cid:0)T i ∈ I scl B ( B i ) (cid:1) ∩ Y . Since ( X, B ) isassumed to be S ∗ , T i ∈ I scl B ( B i ) is a ball in B . Therefore, T i ∈ I B i = (cid:0)T i ∈ I scl B ( B i ) (cid:1) ∩ Y = ∅ is a ball in B ∩ Y . (cid:3) In the special case considered in Section 7.1, the set Y is taken to bethe set Fix( f ) of fixed points of a given function f : X → X . If ( X, B )is an S ∗ ball space with B f = ∅ and every f -closed ball contains a fixed RAMEWORK FOR FIXED POINT THEOREMS 43 point, then it follows from Lemma 5.1 together with Proposition 5.6that also (Fix( f ) , B f ∩ Fix( f ))is an S ∗ ball space. However, we are more interested in the possiblyfiner ball space (Fix( f ) , B ∩
Fix( f )) . The following proposition gives a criterion for the two ball spaces to beequal:
Proposition 5.7.
Take an S ∗ ball space ( X, B ) and a function f : X → X . If B ∈ B ∩ Fix( f ) is such that scl B ( B ) is f -closed, then (11) scl B ( B ) = scl B f ( B ) . If this holds for every B ∈ B ∩ Fix( f ) , then (12) B f ∩ Fix( f ) = B ∩
Fix( f ) . Proof: Pick B ∈ B ∩ Fix( f ). By part 1) of Lemma 5.2, scl B ( B ) isthe smallest of all balls in B that contain B . Consequently, if scl B ( B )is f -closed, then it is also the smallest of all balls in B f that contain B . Then by part 2) of Lemma 5.2, it must be equal to scl B f ( B ).Since B = scl B ( B ) ∩ Fix( f ) by part 1) of Lemma 5.5, the equalityimplies that B = scl B f ( B ) ∩ Fix( f ) ∈ B f ∩ Fix( f ). If the equality(11) holds for all B ∈ B ∩ Fix( f ), then this implies (12). (cid:3) Corollary 5.8.
Take an S ∗ ball space ( X, B ) and a function f : X → X . Assume that f − ( B ) ∈ B for every B ∈ B that contains a fixedpoint. Then (12) holds. Proof: Pick B ∈ B ∩ Fix( f ). Since B := scl B ( B ) ∈ B , we havethat f − ( B ) ∈ B . Since B consists of fixed points, it is containedin f − ( B ). As B is the smallest ball containing B , it follows that B ⊆ f − ( B ) and thus f ( B ) ⊆ f ( f − ( B )) ⊆ B . Hence (11) holds,which by Proposition 5.7 implies that (12) holds. (cid:3) Set theoretic operations on ball spaces
Subsets of ball spaces.Proposition 6.1.
Take two ball spaces ( X, B ) and ( X, B ) on thesame set X such that B ⊆ B . If ( X, B ) is S (or S d or S c ), thenalso ( X, B ) is S (or S d or S c , respectively). This does in general nothold for any other property in the hierarchy. Proof: The first assertion holds since every nest (or directed system,or centered system) in B is also a nest (or directed system, or centeredsystem) in B . To prove the second assertion one constructs an S ∗ ballspace ( X, B ) and a nest (or directed system, or centered system) N such that the intersection T N ∈ B does not lie in N . Then to obtain B one removes all balls from B that lie in T N . (cid:3) Unions of two ball spaces on the same set.
The easy proof of the following proposition is left to the reader:
Proposition 6.2. If ( X, B ) and ( X, B ) are S ball spaces on the sameset X , then so is ( X, B ∪ B ) . The same holds with S or S in placeof S , and for all properties in the hierarchy if B is finite. Note that the assertion may become false if we replace S by S .Indeed, the intersection of a nest in B may properly contain a largestball which does not remain the largest ball contained in the intersectionin B ∪ B .It is also clear that in general infinite unions of S ball spaces on thesame set X will not again be S . For instance, ball spaces with justone ball are always S ; by a suitable infinite union of such spaces onecan build nests with empty intersection.For any ball space ( X, B ), we define: the ball space ( X, b B ) by setting: b B := B ∪ { X } . Taking B = B and B = { X } in Proposition 6.2, we obtain: Corollary 6.3.
A ball space ( X, B ) is S if and only if ( X, b B ) is S .The same holds for all properties in the hierarchy in place of S . Closure under unions of balls.
Take a ball space ( X, B ). By f-un( B ) we denote the set of all unions offinitely many balls in B .The following lemma is inspired by Alexander’s Subbase Theorem: Lemma 6.4. If S is a maximal centered system of balls in f-un( B ) (thatis, no subset of f-un( B ) properly containg S is a centered system), thenthere is a subset S of S which is a centered system in B and has thesame intersection as S . Proof: It suffices to prove the following: if B , . . . , B n ∈ B suchthat B ∪ . . . ∪ B n ∈ S , then there is some i ∈ { , . . . , n } such that B i ∈ S . RAMEWORK FOR FIXED POINT THEOREMS 45
Suppose that B , . . . , B n ∈ B\S . By the maximality of S this impliesthat for each i ∈ { , . . . , n } , S ∪ { B i } is not centered. This in turnmeans that there is a finite subset S i of S such that T S i ∩ B i = ∅ . Butthen S ∪ . . . ∪ S n is a finite subset of S such that \ ( S ∪ . . . ∪ S n ) ∩ ( B ∪ . . . ∪ B n ) = ∅ . This yields that B ∪ . . . ∪ B n / ∈ S , which proves our assertion. (cid:3) The centered systems of balls in a ball space form a poset underinclusion. Since the union of every chain of centered systems is again acentered system, this poset is chain complete. Hence by Corollary 2.2every centered system is contained in a maximal centered system.
Theorem 6.5. If ( X, B ) is an S c ball space, then so is ( X, f-un( B )) . Proof: Take a centered system S ′ of balls in f-un( B ). As shownbefore this theorem, there is a maximal centered system S of balls inf-un( B ) which contains S ′ . By Lemma 6.4 there is a centered system S of balls in B such that T S = T S ⊆ T S ′ . Since ( X, B ) is an S c ball space, we have that T S = ∅ , which yields that T S ′ = ∅ . Thisproves that ( X, f-un( B )) is an S c ball space. (cid:3) In [1] the following result is proven:
Theorem 6.6.
Take a symmetrically complete ordered field K and B tobe the set of all convex sets in K that are finite unions of closed boundedintervals and ultrametric balls. Then ( K, B ) is spherically complete. Closure under nonempty intersections of balls.
Take a ball space ( X, B ). We define:(a) ic( B ) to be the set of all nonempty intersections of arbitrarily manyballs in B ,(b) fic( B ) to be the set of all nonempty intersections of finitely manyballs in B ,(c) ci( B ) to be the set of all nonempty intersections of nests in B ,(d) di( B ) to be the set of all nonempty intersections of arbitrary di-rected systems of balls in B .Note that ( X, B ) is intersection closed if and only if ic( B ) = B , finitelyintersection closed if and only if fic( B ) = B , and chain intersectionclosed if and only if ci( B ) = B . If ( X, B ) is S , then ci( B ) = B .If ( X, B ) is S d , then di( B ) = B . If ( X, B ) is S ∗ , then ic( B ) = B (because an arbitrary set of balls that has a nonempty intersection isautomatically a centered system). We note the following observation:
Proposition 6.7.
Take an arbitrary ball space ( X, B ) . Then the ballspace ( X, ic( B )) is intersection closed, and the ball space ( X, fic( B )) isfinitely intersection closed. Proof: Take balls B i ∈ ic( B ), i ∈ I , and for every i ∈ I , balls B i,j ∈ B , j ∈ J i , such that B i = T j ∈ J i B i,j . Then \ i ∈ I B i = \ i ∈ I, j ∈ J i B i,j ∈ ic( B ) . If I is finite and B i ∈ fic( B ) for every i ∈ I , then every J i can be takento be finite and thus the right hand side is a ball in fic( B ). (cid:3) In view of these facts, we call ( X, ic( B )) the intersection closure of( X, B ), and ( X, fic( B )) the finite intersection closure of ( X, B ). Ifa chain intersection closed ball space ( X, B ′ ) is obtained from ( X, B )by a (possibly transfinite) iteration of the process of replacing B byci( B ), then we call ( X, B ′ ) a chain intersection closure of ( X, B ).Chain intersection closures are studied in [10] and conditions are givenfor ( X, ci( B )) to be the chain intersection closure of ( X, B ). As statedalready in Section 4.1 (cf. Theorem 4.4), this holds for classical ul-trametric spaces. By [10, Theorem 1.2], it also holds for ultrametricspaces with countable narrow value sets. Here is the essence of thecited Theorem 1.1: Theorem 6.8. If ( X, B ) is a ball space of ultrametric type, then ( X, ci( B )) is its intersection closure. Proof: By Proposition 3.6, every centered system of balls in B isa nest. Therefore, ic( B ) = ci( B ). (cid:3) Theorem 6.9. If ( X, B ) is an S c ball space, then its intersection clo-sure ( X, ic( B )) is an S ∗ ball space. Proof: Take a centered system { B i | i ∈ I } in ( X, ic( B )). Write B i = T j ∈ J i B i,j with B i,j ∈ B . Then { B i,j | i ∈ I, j ∈ J i } is a centeredsystem in ( X, B ): the intersection of finitely many balls B i ,j , . . . , B i n ,j n contains the intersection B i ∩ . . . ∩ B i n , which by assumption is non-empty. Since ( X, B ) is S c , T i B i = T i,j B i,j = ∅ . This proves that( X, ic( B )) is an S c ball space. Since ( X, ic( B )) is intersection closed,Theorem 3.10 now shows that ( X, ic( B )) is an S ∗ ball space. (cid:3) RAMEWORK FOR FIXED POINT THEOREMS 47
Closure under finite unions and under intersections.
From Theorems 6.5 and 6.9 we obtain:
Theorem 6.10.
Take any ball space ( X, B ) . If B ′ is obtained from B by first closing under finite unions and then under arbitrary nonemptyintersections, then:1) B ′ is closed under finite unions,2) B ′ is intersection closed,3) if in addition ( X, B ) is an S c ball space, then ( X, B ′ ) is an S ∗ ballspace. Proof: 1): Take S , . . . , S n ⊆ f-un( B ) such that T S i = ∅ for1 ≤ i ≤ n . Then (cid:16)\ S (cid:17) ∪ . . . ∪ (cid:16)\ S n (cid:17) = \ { B ∪ . . . ∪ B n | B i ∈ S i for 1 ≤ i ≤ n } . Since B i ∈ f-un( B ) for 1 ≤ i ≤ n , we have that also B ∪ . . . ∪ B n ∈ f-un( B ). This implies that ( T S ) ∪ . . . ∪ ( T S n ) ∈ B ′ .2): Since B ′ is an intersection closure, it is intersection closed.3): By Theorems 6.5 and 6.9, ( X, B ′ ) is an S ∗ ball space. (cid:3) The topology associated with an S c ball space. Take an S c ball space ( X, B ). Theorem 6.10 tells us that in a canonicalway we can associate with it an S ∗ ball space ( X, B ′ ) which is closedunder nonempty intersections and under finite unions. If we also add X and ∅ to B ′ , then we obtain the collection of closed sets for a topology.Every topology on a nonempty set X can be obtained in this way bystarting from the ball space associated with the topology. Theorem 6.11.
This associated topology is compact if and only if ( X, B ) is an S c ball space. Proof: One direction of the equivalence follows from Theorems 6.10and 4.20. The other direction follows from Proposition 6.1. (cid:3)
Example: the p -adics. The field Q p of p -adic numbers together with the p -adic valuation v p is spherically complete. (This fact can be used to prove the originalHensel’s Lemma via the ultrametric fixed point theorem, see [22], oreven better, via the ultrametric attractor theorem, see [11].) The as-sociated ball space is a classical ultrametric ball space and hence ofultrametric type. It follows from Proposition 3.6 that it is an S c ballspace. Hence by Theorem 6.11 the topology derived from this ballspace is compact. However, Q p is known to be locally compact, but not compact underthe topology induced by the p -adic metric. But this in this topologythe ultrametric balls B α ( x ) are basic open sets, whereas in the topol-ogy derived from the ultrametric ball space they are closed and theircomplements are the basic open sets. It follows that the balls B α ( x )are not open. It thus turns out that the usual p -adic topology on Q p isstrictly finer than the one we derived from the ultrametric ball space.7. Shifting concepts between applications
Knaster–Tarski type theorems.
In 1927 B. Knaster and A. Tarski proved a set-theoretical fixed pointtheorem by which every function on the family of all subsets of a givenset, which is increasing under inclusion, has at least one fixed point. In1955 Tarski generalized the result to the lattice-theoretical fixed pointtheorem which is now known as the
Knaster–Tarski Theorem (cf.[30, Theorem 1]). It states:
Theorem 7.1.
Let L be a complete lattice and f : L → L an order-preserving function. Then the set Fix( f ) of fixed points of f in L isalso a complete lattice. We prove an analogue for ball spaces ( X, B ) with a function f : X → X .As before, B f will denote the collection of all f -closed balls in B . Theorem 7.2.
Take an S ∗ ball space ( X, B ) and a function f : X → X . Assume that B contains an f -closed ball and every f -closed ball in B contains a fixed point or a smaller ball. Then every f -closed ball in B contains a fixed point, and (Fix( f ) , B f ∩ Fix( f )) is an S ∗ ball space. Proof: By Lemma 5.1, ( X, B f ) is an S ∗ ball space. Hence itfollows from our assumptions together with Theorem 1.3 that every f -closed ball B in B contains a fixed point, that is, B ∩ Fix( f ) = ∅ .By Proposition 5.6, it follows that (Fix( f ) , B f ∩ Fix( f )) is an S ∗ ballspace. (cid:3) In what follows, we will discuss some applications.
The case of lattices.
We show how to derive Theorem 7.1 from Theorem 7.2. We take acomplete lattice (
L, < ). By Theorem 4.36, the associated ball space( L, B iv ) is S ∗ . Take an order-preserving function f : L → L and con-sider the set B f iv of all f -closed balls in B iv , that is, all f -closed intervals RAMEWORK FOR FIXED POINT THEOREMS 49 [ a, b ]. Note that B f iv is nonempty since for each a ∈ L , [ ⊥ , a ] or [ a, ⊤ ]is f -closed. By Lemma 5.1, also ( L, B f iv ) is S ∗ .Take an f -closed interval [ a, b ]. Since f is order preserving, it followsthat a ≤ f ( a ) ≤ f ( b ) ≤ b . If f ( a ) = a or f ( b ) = b , then the intervalcontains a fixed point. If f ( a ) = a or f ( b ) = b , then [ f ( a ) , f ( b )]is an f -closed interval that is properly contained in [ a, b ]. We haveshown that the assumptions of Theorem 7.2 hold, so we obtain that(Fix( f ) , B f iv ∩ Fix( f )) is an S ∗ ball space.Next, we show that B f iv ∩ Fix( f ) is exactly the set of all intervals[ a, b ] Fix( f ) in the poset Fix( f ). Indeed, if a, b are fixed points, then[ a, b ] is an f -closed interval in L with [ a, b ] Fix( f ) = [ a, b ] ∩ Fix( f ) ∈B f iv ∩ Fix( f ). Conversely, if S = B ∩ Fix( f ) for some B ∈ B f iv , then thespherical closure scl B f iv ( S ) of S in the ball space ( L, B f iv ) is an f -closedinterval [ a, b ] in L , and it is contained in B . If a or b is not a fixed point,then [ f ( a ) , f ( b )] is an f -closed interval properly contained in [ a, b ]. Butas it also contains the set S of fixed points, this is a contradiction tothe definition of the spherical closure. Hence, a, b are fixed points. Wehave that S ⊆ [ a, b ] ∩ Fix( f ) = scl B f iv ( S ) ∩ Fix( f ) ⊆ B ∩ Fix( f ) = S ,so S = [ a, b ] ∩ Fix( f ) = [ a, b ] Fix( f ) .We have now shown that (Fix( f ) , { [ a, b ] Fix( f ) | a, b ∈ Fix( f ) } is an S ∗ ball space. Let us show that all finite sets S in Fix( f ) have anupper and a lower bound. Since ⊤ ∈ L , the f -closed intervals [ a, ⊤ ], a ∈ S , have a nonempty intersection. Since ( L, B f iv ) is an S ∗ ball space,their intersection is again an f -closed ball. By Theorem 7.2, it containsa fixed point, which consequently is an upper bound for S in Fix( f ).Similarly, one shows the existence of a lower bound. It now followsfrom Theorem 4.34 that Fix( f ) is a complete lattice. The ultrametric case.
Take a classical ultrametric space (
X, u ) and a function f : X → X .Then f is called nonexpanding if u ( f x, f y ) ≤ u ( x, y ) for all x, y ∈ X .Further, f is called contracting on orbits if u ( f x, f f x ) < u ( x, f x )for all x ∈ X such that x = f x .Now assume that ( X, u ) is spherically complete. Then by Theo-rem 4.4, the full ultrametric ball space ( X, B u + ) is S ∗ . Further, take afunction f : X → X . We need the following auxiliary result: Lemma 7.3. If f : X → X is nonexpanding, then every ball B ( x, f x ) is f -closed, and the same holds for every B ( x, y ) where x is a fixedpoint of f . Proof: Take z ∈ B ( x, f x ). Then u ( x, z ) ≤ u ( x, f x ) and since f isnonexpanding, u ( f x, f z ) ≤ u ( x, z ) ≤ u ( x, f x ). By the ultrametric tri-angle law, this yields that u ( x, f z ) ≤ u ( x, f x ), whence f z ∈ B ( x, f x ).Now assume that x is a fixed point of f , and take z ∈ B ( x, y ). Then u ( x, z ) ≤ u ( x, y ) and since f is nonexpanding, u ( x, f z ) = u ( f x, f z ) ≤ u ( x, z ) ≤ u ( x, y ), whence f z ∈ B ( x, y ). (cid:3) Now assume that f is both nonexpanding and contracting on or-bits. As before, we let B fu + denote the set of all f -closed balls in B u + .Lemma 7.3 shows that B fu + is nonempty, as it contains B ( x, f x ) foreach x ∈ X . Take B ∈ B fu + and x ∈ B . Then also f x ∈ B , hence B ( x, f x ) ⊆ B . If x = f x , then B contains a fixed point. Supposethat x = f x . Since f is contracting on orbits, we then have that B ( f x, f f x ) ( B ( x, f x ) ⊆ B , hence Lemma 7.3 shows that B ( f x, f f x )is an f -closed ball properly contained in B . Therefore, from Lemma 5.1and Theorem 7.2 we obtain that every f -closed ball in B fu + contains afixed point and (Fix( f ) , B fu + ∩ Fix( f ))is an S ∗ ball space.We observe: Lemma 7.4.
The ball space (Fix( f ) , B fu + ∩ Fix( f )) is equal to the fullball space of (Fix( f ) , u ) . Proof: For x, y ∈ Fix( f ), denote by B F ( x, y ) the smallest ball in(Fix( f ) , u ) that contains x and y .Take any ball B ∈ B fu + with B ∩ Fix( f ) = ∅ , and pick any element x ∈ B ∩ Fix( f ). Then B ∩ Fix( f ) = [ { B F ( x, y ) | y ∈ B ∩ Fix( f ) } . This shows in particular that all balls in B fu + ∩ Fix( f ) are balls in thefull ultrametric ball space of (Fix( f ) , u ).For the converse, consider any ball B F in the full ultrametric ballspace of (Fix( f ) , u ) and pick some x ∈ B F . Then B F can be writtenas B F = [ { B F ( x, y ) | y ∈ B F } = [ { B ( x, y ) ∩ Fix( f ) | y ∈ B F } = Fix( f ) ∩ [ { B ( x, y ) | y ∈ B F } . The second assertion of Lemma 7.3 shows that each ultrametric ball B ( x, y ) is f -closed. Therefore, S { B ( x, y ) | y ∈ B F } is an f -closedball in the full ultrametric ball space of ( X, u ). Hence B F ∈ B fu + ∩ Fix( f ). (cid:3) RAMEWORK FOR FIXED POINT THEOREMS 51
In fact, we could also have used Proposition 5.7. Indeed, it canbe seen from the second part of the above proof that the full ballspace of (Fix( f ) , u ) is equal to (Fix( f ) , B u + ∩ Fix( f )). Further, if B ∈B u + ∩ Fix( f ) and x ∈ B , thenscl B u + ( B ) = [ { B ( x, y ) | y ∈ scl B u + ( B ) } is a union of balls which by Lemma 7.3 are f -closed and is thus itself f -closed. This shows that the assumption of Proposition 5.7 is satisfiedand consequently, B u + ∩ Fix( f ) = B fu + ∩ Fix( f ) . We have now proved that Fix( f ) = ∅ and the full ultrametric ballspace of (Fix( f ) , u ) is S ∗ . It follows that (Fix( f ) , u ) is sphericallycomplete. So we obtain the following theorem: Theorem 7.5.
Take a spherically complete ultrametric space ( X, u ) and a nonexpanding function f : X → X which is contracting on orbits.Then every f -closed ultrametric ball contains a fixed point, Fix( f ) = ∅ ,and (Fix( f ) , u ) is again a spherically complete ultrametric space. The topological case.
Take a compact topological space X and ( X, B ) the associated ballspace formed by the collection B of all nonempty closed sets. If f : X → X is any function, then B f can be taken as the set of all nonemptyclosed and f -closed sets of a (possibly coarser) topology, as arbitraryunions and intersections of f -closed sets are again f -closed. From The-orem 4.20, Lemma 5.1 and Theorem 7.2, we obtain: Theorem 7.6.
Take a compact topological space X and a function f : X → X . Assume that every nonempty closed, f -closed set containsa fixed point or a smaller closed, f -closed set. Then the topology onthe set Fix( f ) of fixed points of f having B f ∩ Fix( f ) as its collectionof nonempty closed sets is itself compact. As we are rather interested in the topology on Fix( f ) induced bythe original topology of X , we ask for criteria on f which guaranteethat B f ∩ Fix( f ) = B ∩
Fix( f ). As an application of Corollary 5.8, weobtain: Corollary 7.7.
Take a compact topological space X and a continuousfunction f : X → X . Assume that every nonempty closed, f -closedset contains a fixed point or a smaller closed, f -closed set. Then theinduced topology on the set Fix( f ) of fixed points of f is itself compact. Tychonoff type theorems.
In [1] the notion of a continuous function between two ball spaces isintroduced. Further, it is shown that the category consisting of allball spaces together with the continuous functions as morphisms allowsproducts and coproducts. The products can be defined as follows.Assume that ( X j , B j ) j ∈ J is a family of ball spaces. We set X = Q j ∈ J X j and define the product ( X j , B j ) pr j ∈ J to be ( X, B pr ), where B pr := (Y i ∈ I B i ⊆ X | for some k ∈ I, B k ∈ b B k and ∀ i = k : B i = X i ) . Further, we define the topological product ( X j , B j ) tpr j ∈ J to be ( X, B tpr ),where B tpr := (Y j ∈ J B j | ∀ j ∈ J : B j ∈ b B j and B j = X j for almost all j ) . and the box product ( X j , B j ) bpr j ∈ J of the family to be ( X, B bpr ), where B bpr := (Y j ∈ J B j | ∀ j ∈ J : B j ∈ B j ) . Since the sets B i are nonempty, it follows that B 6 = ∅ , and as no ball inany B i is empty, it follows that no ball in B pr , B tpr and B bpr is empty.Note that B pr = b B pr and B tpr = b B tpr .The box product ( X j , b B j ) bpr j ∈ J is equal to ( X, b B bpr ), where b B bpr := (Y j ∈ J B j | ∀ j ∈ J : B j ∈ b B j ) . Note that B tpr and b B bpr coincide when J is finite. We also see that inall cases, B pr ⊆ B tpr ⊆ b B bpr . Hence if ( X, B tpr ) is spherically complete,then so is ( X, B pr ), and if ( X, b B bpr ) is spherically complete, then so are( X, B tpr ) and ( X, B pr ). The same holds with “ S d ” and “ S c ” in placeof “ S ”.We leave the proof of the following observations to the reader: Proposition 7.8.
The following equations hold: fic (cid:0) ( X j , B j ) pr j ∈ J (cid:1) = ( X j , fic( B j )) tpr j ∈ J ic (cid:0) ( X j , B j ) pr j ∈ J (cid:1) = ic (cid:16) ( X j , B j ) tpr j ∈ J (cid:17) = ( X j , ic( b B j )) bpr j ∈ J RAMEWORK FOR FIXED POINT THEOREMS 53
The following theorem presents our main results on the various prod-ucts.
Theorem 7.9.
The following assertions are equivalent:a) the ball spaces ( X j , B j ) , j ∈ J , are spherically complete,b) their box product is spherically complete,c) their topological product is spherically complete.d) their product is spherically complete.The same holds with “ S d ” and “ S c ” in place of “ S ”.The equivalence of a) and b) also holds for all other properties in thehierarchy, in place of “ S ”. Proof: Take ball spaces ( X j , B j ), j ∈ J , and in every B j take a setof balls { B i,j | i ∈ I } . Then we have:(13) \ i ∈ I Y j ∈ J B i,j = Y j ∈ J \ i ∈ I B i,j . If N = ( Q j ∈ J B i,j ) i ∈ I is a nest of balls in ( Q j ∈ J X j , b B pr ), then for every j ∈ J , also ( B i,j ) i ∈ I must be a nest in ( X j , b B j ).a) ⇒ b): Assume that all ball spaces ( X j , B j ), j ∈ J , are sphericallycomplete. Then for every j ∈ J , ( B i,j ) i ∈ I has nonempty intersection.By (13) it follows that T N 6 = ∅ . This proves the implication a) ⇒ b).b) ⇒ a): Assume that ( Q j ∈ J X j , B bpr ) is spherically complete. Take j ∈ J and a nest of balls N = ( B i ) i ∈ I in ( X j , B j ). For each i ∈ I ,set B i,j = B i and B i,j = B ,j for j = j where B ,j is an arbitraryfixed ball in B j . Then ( Q j ∈ J B i,j ) i ∈ I is a nest in ( Q j ∈ J X j , B bpr ). Byassumption, ∅ 6 = \ i ∈ I Y j ∈ J B i,j = \ i ∈ I B i ! × Y j = j ∈ J B ,j ! , whence T i ∈ I B i = ∅ . We have shown that for every j ∈ J , ( X j , B j ) isspherically complete. This proves the implication b) ⇒ a).a) ⇒ c): Assume that all ball spaces ( X j , B j ), j ∈ J , are sphericallycomplete. Then by Corollary 6.3, all ball spaces ( X j , b B j ), j ∈ J , arespherically complete. By the already proven implication a) ⇒ b), theirbox product ( X, b B pr ) is spherically complete. By our remark beforethe theorem, ( X, B tpr ) is spherically complete, too.c) ⇒ d): This has already been noted before the theorem.d) ⇒ a): Same as the proof of b) ⇒ a), where we now take B ,j = X j . A similar proof of the equivalence of a) and b) also holds for all otherproperties in the hierarchy. For the properties S , S , S and S , oneuses the fact that by definition, Q j ∈ J B j is a ball in B bpr if and only ifevery B j is a ball in B j and that1) Q j ∈ J B ′ j is a ball contained in Q j ∈ J B j if and only if every B j is aball contained in B j ,2) Q j ∈ J B ′ j is a maximal ball contained in Q j ∈ J B j if and only if every B j is a maximal ball contained in B j .For the transfer of the other properties, one observes the following:3) { Q j ∈ J B i,j | i ∈ I } is a centered system if and only if all sets { B i,j | i ∈ I } , j ∈ J , are.4) If { Q j ∈ J B i,j | i ∈ I } is a directed system, then so are { B i,j | i ∈ I } for all j ∈ J .5) Fix j ∈ J . If { B i,j | i ∈ I } is a directed system, then so is { Q j ∈ J B i,j | i ∈ I } when the balls are chosen as in the proof of b) ⇒ a). (cid:3) Example 7.10.
There are S ∗ ball spaces ( X j , B j ) , j ∈ N , such that theball space ( X, B tpr ) is not even S . Indeed, we choose a set Y with atleast two elements, and for every j ∈ N we take X j = Y and B j = { B } with ∅ 6 = B = Y . Then trivially, all ball spaces ( X j , B j ) are S ∗ . For all i, j ∈ N , define B i := B × B × . . . × B | {z } i times × Y × Y × . . . ∈ B tpr . Then N = { B i | i ∈ I } is a nest of balls in B tpr , but the intersection T N = Q j ∈ N B does not contain any ball in this ball space. Example 7.11.
There are S ∗ ball spaces ( X, B j ) , j = 1 , , such thatthe ball space ( X, B pr ) is not S c . Indeed, we choose again a set Y withat least two elements and take B = B = { B } with ∅ 6 = B = Y .Then as in the previous example, ( X j , B j ), j = 1 , S ∗ ball spaces.Further, B pr = { Y × Y, B × Y, Y × B } , which is a centered systemwhose intersection does not contain any ball. The ultrametric case.
If ( X j , u j ), j ∈ J are ultrametric spaces with value sets u j X j = { u j ( a, b ) | a, b ∈ X j } , and if B j = B γ j ( a j ) is an ultrametric ball in( X j , u j ) for each j , then Y j ∈ J B j = { ( b j ) j ∈ J | ∀ j ∈ J : u j ( a j , b j ) ≤ γ j } . RAMEWORK FOR FIXED POINT THEOREMS 55
This shows that the box product is the ultrametric ball space for theproduct ultrametric on Q j ∈ J X j which is defined as u prod (( a j ) j ∈ J , ( b j ) j ∈ J ) = ( u j ( a j , b j )) j ∈ J ∈ Y j ∈ J u j X j . The latter is a poset, but in general not totally ordered, even if all u j X j are totally ordered and even if J is finite. So the product ultrametric isa natural example for an ultrametric with partially ordered value set.If the index set J is finite and all u j X j are contained in some totallyordered set Γ such that all of them have a common least element 0 ∈ Γ,then we can define an ultrametric on the product Q j ∈ J X j which takesvalues in S j ∈ J u j X j ⊆ Γ as follows: u max (( a j ) j ∈ J , ( b j ) j ∈ J ) = max j u j ( a j , b j )for all ( a j ) j ∈ J , ( b j ) j ∈ J ∈ Q j ∈ J X j . We leave it to the reader to provethat this is indeed an ultrametric. The corresponding ultrametric ballsare the sets of the form { ( b j ) j ∈ J | ∀ j ∈ J : u j ( a j , b j ) ≤ γ } for some ( a j ) j ∈ J ∈ Q j ∈ J X j and γ ∈ S j ∈ J u j X j . Now the value set istotally ordered. It turns out that the collection of balls so obtained is a(usually proper) subset of the full ultrametric ball space of the productultrametric. Therefore, if all ( X j , u j ) are spherically complete, then sois ( Q j ∈ J X j , u max ) by Theorem 7.9 and Proposition 6.1. Theorem 7.12.
Take ultrametric spaces ( X j , u j ) , j ∈ J . Then theultrametric space ( Q j ∈ J X j , u prod ) is spherically complete if and only ifall ( X j , u j ) , j ∈ J , are spherically complete.If the index set J is finite and all u j X j are contained in some totallyordered set Γ such that all of them have a common least element, thenthe same also holds for u max in place of u prod . Proof: As was remarked earlier, the ultrametric ball space of theproduct ultrametric is the box product of the ultrametric ball spacesof the ultrametric spaces ( X j , u j ). Thus the first part of the theoremis a corollary to Theorem 7.9.To prove the second part of the theorem, it suffices to prove the con-verse of the implication we have stated just before the theorem. Assumethat the space ( Q j ∈ J X j , u max ) is spherically complete and choose any j ∈ J . Let N j = { B γ i ( a i,j ) | i ∈ I } be a nest of balls in ( X j , u j ).Further, for every j ∈ J \ { j } choose some element a j ∈ X j and for every i ∈ I set a i,j := a j and B i := { ( b j ) j ∈ J ∈ Y j ∈ J X j | u max (( a i,j ) j ∈ J , ( b j ) j ∈ J ) ≤ γ i } = { ( b j ) j ∈ J ∈ Y j ∈ J X j | ∀ j ∈ J : u j ( a i,j , b j ) ≤ γ i } . In order to show that N := { B i | i ∈ I } is a nest of balls in ( Q j ∈ J X j , u max ),we have to show that any two balls B i , B k , i, k ∈ I , have nonemptyintersection. Assume without loss of generality that γ i ≤ γ k . As { B γ i ( a i,j ) | i ∈ I } is a nest of balls, we have that a i,j ∈ B γ k ( a k,j ).It follows that u j ( a k,j , b j ) ≤ γ k , and since a i,j = a j = a k,j for every j ∈ J \ { j } ,( a i,j ) j ∈ J ∈ B i ∩{ ( b j ) j ∈ J ∈ Y j ∈ J X j | ∀ j ∈ J : u j ( a k,j , b j ) ≤ γ k } = B i ∩ B k . As ( Q j ∈ J X j , u max ) is assumed to be spherically complete, there is some( z j ) j ∈ J ∈ N ; it satisfies u j ( a i,j , z j ) ≤ γ i for all i ∈ I and all j ∈ J . Inparticular, taking j = j , we find that z j ∈ B γ i ( a i,j ) for all i ∈ I andthus, z j ∈ T N j . (cid:3) The topological case.
In which way does Tychonoff’s theorem follow from its analogue forball spaces? The problem in the case of topological spaces is that theproduct ball space we have defined, while containing only closed sets ofthe product, does not contain all of them, as it is not necessarily closedunder finite unions and arbitrary intersections. We have to close itunder these operations.If the topological spaces X i , i ∈ I , are compact, then their associatedball spaces ( X i , B i ) are S c . By Theorem 7.9 their topological product isalso S c . The product topology of the topological spaces X i is the closureof B tpr under finite unions and under arbitrary nonempty intersections,when ∅ and the whole space are adjoined. By Theorem 6.11, thistopology is compact.We have shown that Tychonoff’s Theorem follows from its ball spacesanalogue.8. Other results related to fixed point theorems
In this section, we will discuss two types of theorems that are relatedto fixed point theorems.
RAMEWORK FOR FIXED POINT THEOREMS 57
Multivalued fixed point theorems.
We take a function F froma nonempty set X to its power set P ( X ) and ask for criteria thatguarantee the existence of a fixed point x ∈ X in the sense that x ∈ F ( x ) . A very elegant approach to proving a generic multivalued fixed pointtheorem can be given by use of contractive ball spaces:
Theorem 8.1.
Take a spherically complete contractive ball space ( X, B ) and a function F : X → P ( X ) . Assume that B x ∩ F ( x ) = ∅ for all x ∈ X .
Then F admits a fixed point in X . Proof: By part 2) of Theorem 4.12, B contains a singleton ball B a = { a } . Since by hypothesis B a ∩ F ( a ) = ∅ , it follows that a ∈ F ( a ). (cid:3) This theorem together with Proposition 4.9 and 4.10 can be used toprove the following result:
Theorem 8.2.
Take a complete metric space ( X, d ) and an Oettli-Th´era function φ on X . If a function F : X → P ( X ) satisfies ∀ x ∈ X ∃ y ∈ F ( x ) : d ( x, y ) ≤ − φ ( x, y ) , then F has a fixed point on X . In [2] this theorem and its variants are proved using a version of part2) of Theorem 4.12 together with Proposition 4.9.The following is a slight generalization of Theorem 8.1, replacingthe existence of singletons by that of minimal balls. Here again, as inTheorems 1.5 and 1.6, the general condition on the ball space is adaptedto the given function: condition (C3) is replaced by a condition thatdepends on the function F . Theorem 8.3.
Take a nonempty set X and a function F : X → P ( X ) .Assume that ( X, { B x | x ∈ X } ) is a spherically complete ball space suchthat for all x, y ∈ X ,1) x ∈ B x and B x ∩ F ( x ) = ∅ ,2) if y ∈ B x , then B y ⊆ B x ,3) if x / ∈ F ( x ) , then there is some z ∈ B x such that B z ( B x .Then F admits a fixed point in X . Proof: A straightforward adaptation of the proof of Proposi-tion 4.11 shows that the intersection of a maximal nest of balls, ifnonempty, must be a minimal ball B a which consequently must satisfy a ∈ F ( a ). The assumption that the ball space is spherically completeguarantees that the intersection is nonempty. (cid:3) Coincidence theorems.
We take a nonempty set X and two ormore functions f , . . . , f n : X → X and ask for criteria that guaranteethe existence of a coincidence point x ∈ X in the sense that(14) f ( x ) = . . . = f n ( x ) . In order to obtain a generic coincidence theorem for ball spaces, onecan again use the idea of showing the existence of singleton balls withsuitable properties.
Theorem 8.4.
Take a spherically complete weakly contractive ball space ( X, B ) and functions f , . . . , f n : X → X . Assume that f ( x ) , . . . , f n ( x ) ∈ B x for all x ∈ X .
Then f , . . . , f n admit a coincidence point in X . Proof: By part 2) of Theorem 4.12, B contains a singleton ball B a . Since by hypothesis f ( a ) , . . . , f n ( a ) ∈ B a , it follows that f ( a ) = . . . = f n ( a ). (cid:3) As in the previous section, we prove a generalization that replacesthe existence of singletons by that of minimal balls.
Theorem 8.5.
Take a nonempty set X and functions f , . . . , f n : X → X . Assume that there is a B x –ball space B on X such that ( X, B ) isan S ball space and for all x ∈ X , if (14) does not hold, then there issome y ∈ X such that B y ( B x .Then f , . . . , f n admit a coincidence point in X . Proof: Let M be a maximal nest of balls in B (it exists by Corol-lary 2.7). Since ( X, B ) is an S ball space, there is a ball B x ⊆ T M .This means that M ∪ { B x } is a nest of balls, so by maximality of M we have that B x ∈ M . Consequently, B x = T M . Suppose that (14)does not hold. Then by hypothesis there is some element y ∈ X suchthat B y ( B x whence B y / ∈ M . But then M ∪ { B y } is a nest whichstrictly contains M . This contradiction to the maximality of M showsthat (14) must hold. (cid:3) Let us note that condition (14) can be replaced by any other conditionon x . In this way, a generic theorem is obtained that is neither a fixedpoint theorem nor a coincidence theorem but can be specialized to suchtheorems. This idea has been exploited in [16]. RAMEWORK FOR FIXED POINT THEOREMS 59
References [1] Bartsch, R. – Kuhlmann, F.-V. – Kuhlmann, K.:
Construction of ball spacesand the notion of continuity , submitted, arXiv:1810.09275 [2] B laszkiewicz, P. – ´Cmiel, H. – Linzi. A. – Szewczyk, P.:
Caristi–Kirk andOettli–Th´era ball spaces, and applications , submitted, arXiv:1901.03853 [3] Brunner, N.:
Topologische Maximalprinzipien (German) [Topological maximalprinciples], Z. Math. Logik Grundlag. Math. (1987), 135–139[4] Cohn, P. M.: Universal algebra , Harper and Row, New York, 1965[5] Hitzler, P. – Seda, A. K.:
The fixed-point theorems of Priess-Crampe andRibenboim in logic programming , Valuation theory and its applications, Vol. I(Saskatoon, SK, 1999), 219–235, Fields Inst. Commun. , Amer. Math. Soc.,Providence, RI, 2002[6] Jachymski, J. R.: Caristi’s fixed point theorem and selections of set-valuedcontractions , J. Math. Anal. Appl. (1998), 55–67[7] Jachymski, J. R.:
Order-theoretic aspects of metric fixed point theory , Hand-book of metric fixed point theory, 613641, Kluwer Acad. Publ., Dordrecht,2001[8] Kirk, W. A.:
Metric fixed point theory: a brief retrospective , Fixed Point The-ory Appl. 2015, article 215[9] Kozlowski, W. M.:
A purely metric proof of the Caristi fixed point theorem ,Bull. Aust. Math. Soc. (2017), 333–337[10] Kubis, W. – Kuhlmann, F.-V.: Chain intersection closures , submitted, arXiv:1810.05832 [11] Kuhlmann, F.-V.:
Maps on ultrametric spaces, Hensel’s Lemma, and differen-tial equations over valued fields , Comm. in Alg. (2011), 1730–1776[12] Kuhlmann, F.-V. – Kuhlmann, K.: A common generalization of metric andultrametric fixed point theorems , Forum Math. (2015), 303–327; and: Cor-rection to ”A common generalization of metric, ultrametric and topologicalfixed point theorems”, Forum Math. (2015), 329–330; alternative correctedversion available at: http://math.usask.ca/fvk/GENFPTAL.pdf [13] Kuhlmann, F.-V. – Kuhlmann, K.: Fixed point theorems for spaces with atransitive relation , Fixed Point Theory (2017), 663–672[14] Kuhlmann, F.-V. – Kuhlmann, K. – Paulsen, M.: The Caristi–Kirk FixedPoint Theorem from the point of view of ball spaces , Journal of Fixed PointTheory and Applications, open access (2018)[15] Kuhlmann, F.-V. – Kuhlmann, K. – Shelah, S.:
Symmetrically Complete Or-dered Sets, Abelian Groups and Fields , Israel J. Math. (2015), 261–290[16] Kuhlmann, F.-V. – Kuhlmann, K. – Sonallah, F.:
Coincidence Point Theo-rems for Ball Spaces and Their Applications , to appear in: Ordered AlgebraicStructures and Related Topics, CIRM, Luminy, France, October 12-16 2015,Contemporary Mathematics, AMS[17] Ma´nka, R.:
Some forms of the axiom of choice , Jbuch. Kurt-G¨odel-Ges. 1988,24–?34[18] Markowsky, G.:
Chain-complete posets and directed sets with applications , Al-gebra Universalis (1976), 53–68[19] Oettli, W. – Th´era, M.: Equivalents of Ekeland’s principle , Bull. Austral.Math. Soc. (1993), 385–392 [20] Penot, J.-P.: Fixed point theorems without convexity , Analyse non convexe(Proc. Colloq., Pau, 1977). Bull. Soc. Math. France M´em. (1979), 129–152[21] Penot, J.-P.: The drop theorem, the petal theorem and Ekeland’s variationalprinciple , Nonlinear Anal. (1986), 813–822[22] Prieß-Crampe, S.: Der Banachsche Fixpunktsatz f¨ur ultrametrische R¨aume ,Results in Mathematics (1990), 178–186[23] Prieß-Crampe, S. – Ribenboim, P.: Fixed Points, Combs and GeneralizedPower Series , Abh. Math. Sem. Hamburg (1993), 227–244[24] Prieß-Crampe, S. – Ribenboim, P.: The Common Point Theorem for Ultra-metric Spaces , Geom.Ded. (1998), 105–110[25] Prieß-Crampe, S. – Ribenboim, P.: Fixed Point and Attractor Theorems forUltrametric Spaces , Forum Math. (2000), 53–64[26] Prieß-Crampe, S. – Ribenboim, P.: Ultrametric spaces and logic programming ,J. Logic Programming (2000), 59–70[27] Prieß-Crampe, S. – Ribenboim, P.: Ultrametric dynamics , Illinois J. Math. (2011), 287–303[28] Shelah, S.: Quite Complete Real Closed Fields , Israel J. Math. (2004),261–272[29] Stepr¯ans, J. – Watson, S. – Just, W.:
A topological Banach fixed point theoremfor compact Hausdorff spaces , Canad. Bull. Math. (4) (1994), 552–555[30] Tarski, A.: A lattice-theoretical fixpoint theorem and its applications , Pacific J.Math. (1955), 285-309 Institute of Mathematics, ul. Wielkopolska 15, 70-451 Szczecin, Poland
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