SELF-MAPS OF HAUSDORFF TOPOLOGICAL SPACES
GERALD F. JUNGCK
Received 13 July 2004 and in revised form 3 February 2005
The concept ofproper orbitsof a mapg is introduced and results of the following type are obtained. If a continuous self-mapgof a Hausdorfftopological spaceXhas relatively compact proper orbits, thenghas a fixed point. In fact,ghas a common fixed point with every continuous self-mapf ofXwhich is nontrivially compatible withg. A collection of metric and semimetric space fixed point theorems follows as a consequence. Specifically, a theorem by Kirk regarding diminishing orbital diameters is generalized, and a fixed point theorem for maps with no recurrent points is proved.
1. Introduction
Letg be a mapping of a topological spaceXinto itself. LetN denote the set of positive integers andω=N∪ {0}. Forx∈X,ᏻ(x) is called theorbit of g at x and defined by ᏻ(x)= {gk(x) :k∈ω}, wherego(x)=x. Thus, ifn∈ω, the orbit ofg atgn(x) is the set ᏻ(gn(x))= {gk(x) :k∈ωandk≥n}. (Clearly,ᏻ(gn(x))⊂ᏻ(x) forn∈N.) And ifXhas a metric or semimetricd, we will designate thediameterof a setM⊂X byδ(M) which of course is definedδ(M)=sup{d(x,y) :x,y∈M}.
The purpose of this paper is to introduce the concept of proper orbitsand to dem- onstrate its role in obtaining fixed points. (We use cl(A) to denote the closure of the setA.)
Definition 1.1. Letgbe a self-map of a topological spaceXand letx∈X. The orbitᏻ(x) of gatxisproperif and only ifᏻ(x)= {x}or there existsn=nx∈Nsuch that cl(ᏻ(gn(x))) is a proper subset of cl(ᏻ(x)). Ifᏻ(x) is proper for eachx∈M⊂X, we will say thatghas proper orbits onM. IfM=X, we sayghasproper orbits.
The concept of proper orbits generalizes the concept of diminishing orbital diameters, which was introduced by Belluce and Kirk [1] in 1969. They introduced the concept of mappings with diminishing orbital diameters to obtain fixed point theorems for non- expansive self-maps of metric spaces. A self-mapg of a metric spaceXhasdiminishing orbital diametersif for eachx∈X,δ(ᏻ(x))<∞, and wheneverδ(ᏻ(x))>0, there exists n=nx∈Nsuch thatδ(ᏻ(x))> δ(ᏻ(gn(x))). (If the given property holds for a specificx,
Copyright©2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:3 (2005) 355–363 DOI:10.1155/FPTA.2005.355
we will say thatᏻ(x) has diminishing diameters.) Subsequently Kirk [13] (1969) extended the concept to more general mappings. In particular, he proved the following interesting result for a metric spaceM.
Theorem1.2 (Kirk [13]). SupposeMis compact andg:M→Mis continuous with dimin- ishing orbital diameters. Then for eachx∈M, some subsequence{gnk(x)}of the sequence {gn(x)}of iterates ofxhas a limit which is a fixed point ofg.
One purpose of this paper is to extend Theorem 1.2, and in the following manner.
The underlying space will be a Hausdorfftopological space. Instead of requiring that the spaceM be compact, we will require that the orbits be relatively compact. And we will replace the requirement that orbits have diminishing diameters by the demand that they be proper. Moreover, it will be shown thatghas a common fixed point with everyf ∈Kg, a set we now define.
Definition 1.3. Ifg is a continuous self-map of a topological spaceX,Kgis the set of all continuous maps f :X→X such thatM= {x∈X:f x=gx} = ∅and f gx=g f x for x∈M. (Kg= ∅, sinceg∈Kg.)
We will expand further on these concepts, but first note that the motivation behind our approach and our interest in the setKgis the following theorem.
Theorem1.4 [10]. A continuous self-mapg ofI=[0, 1]has a common fixed point with every function inKgif and only if every periodic point ofgis a fixed point ofg(i.e.,gk(x)= x⇒g(x)=x).
To appreciate the significance ofTheorem 1.4in the present context, suppose there existx∈Iandk∈Nsuch thatk >1,gk(x)=xandgi(x)=xfor 0< i < k. Then it is clear thatᏻ(x)=ᏻ(gn(x))= {x}for alln∈N. Thus,g does not have proper orbits. So, if the functiongofTheorem 1.4has proper orbits, thenghas a common fixed point with each
f ∈Kg. We now show that this implication extends to very general settings.
2. Preliminaries
We first review background regarding semimetric spaces and compatible maps. Semimet- ric spaces give us a vehicle for extending metric space results. Moreover, since semimetric spaces are first countable and orbits are virtually sequences, semimetric spaces provide a natural and relatively unrestricted arena for examples and results involving orbits.
A semimetric on a setXis a functiond:X×X→[0,∞) such thatd(x,y)=0 if and only ifx=y, andd(x,y)=d(y,x) forx,y∈X. Forp∈Xand>0 we letS(p,)= {x∈ X:d(x,p)<}. A semimetric space is a pair (X;d) in whichXis a Hausdorfftopological space anddis a semimetric onX. The topology onXis the familyt(d)= {U⊂X:p∈ U⇒S(p,)⊂Ufor some>0}. We require that the topological interior ofS(p,) be nonempty and containp; that is, there existsU∈t(d) such thatp∈U⊂S(p,). Con- sequently, a sequence{xn}inXconverges int(d) to p∈X, denotedxn→p, if and only ifd(xn,p)→0. And a function (map) g:X→X is continuous if and only if gxn→gx wheneverxn→x. See [5,11] for further details on semimetric spaces.
Compatible maps were introduced by Jungck [8] in 1986 for metric spaces as a gener- alization of commuting maps. The concept proved useful in obtaining generalizations of established fixed point theorems. The definition for a semimetric space (X;d) is formally the same and is as follows. Maps f,g:X→Xare compatible if and only ifd(g f xn,f gxn)
→0 whenever{xn}is a sequence inXsuch that f xn,gxn→pfor somep∈X. If f and g are compatible and do have a coincidence point, f andg arenontriviallycompatible.
An immediate consequence of the definition is that compatible maps commute at coinci- dence points, a property we highlight. So if fandg are continuous nontrivially compat- ible self-maps ofX, then f ∈Kgandg∈Kf. In fact, if f andgare continuous andX is compact, then f andgare compatible if and only iff x=gximplies thatf gx=g f x. This result is proved in [9] for metric spaces but also holds for semimetric spaces (the triangle inequality is not used). (See [8,9] for properties of compatible maps.)
Before continuing, we pay for the freedom granted by semimetric spaces. LetAbe a subset of a semimetric space. Since we have no triangle inequality, it is not necessarily true—as the next example shows—thatδ(A)=δ(cl(A)). (Kirk’s proof of Theorem 1.2 appealed to this “metric space” equality.)
Example 2.1. LetX=[0, 1] and defined(x,y)= |x−y|ifx,y∈(0, 1] andd(b, 0)=d(0, b)=2b if b∈[0, 1]. Then δ((0, 1/2])=1/2, whereas δ([0, 1/2])=δ( cl((0, 1/2]))=1.
Note that the topological space (X,t(d)) is metrizable butdis not a metric.
WithA=(0, 1/2] andB=[0, 1/2],Example 2.1also shows that in the context of semi- metric spaces, even thoughA⊂Bandδ(A)< δ(B), cl(A) may not be a proper subset of cl(B).
The above observations prompt the following definition.
Definition 2.2. Let (X;d) be a semimetric space. A mappingg:X→X has orbits with diminishing closure diametersif and only if for eachx∈X,δ(cl(ᏻ(x)))<∞, and whenever δ( cl(ᏻ(x)))>0, there existsn∈Nsuch thatδ(cl(ᏻ(x)))> δ(cl(ᏻ(gn(x)))).
Of course, if (X;d) is a semimetric space andg :X→X has orbits with diminish- ing closure diameters (d.c.d.), theng has proper orbits. For supposeᏻ(x)= {x}. Then δ(cl(ᏻ(x)))>0. Sinceghas d.c.d., there is ann∈Nsuch thatδ(cl(ᏻ(gn(x))))< δ(cl(ᏻ(x))).
But this implies that cl(ᏻ(gn(x))) is a proper subset of cl(ᏻ(x)), as desired.
Observe also that in metric spaces (X,d)Definition 2.2reduces to that of diminishing orbital diameters (d.o.d.). Thus, ifghas d.o.d. on (X,d), thenghas proper orbits.
3. Results and examples
We now state and prove our results. The statements and proofs of the initial theorems and corollaries are completely topological in nature. The proof of our first theorem repeats an argument used by Schwartz in the proof of Proposition 1 on page 353 of [18].
Theorem3.1. LetXbe a Hausdorfftopological space and letg:X→Xbe continuous. If g has relatively compact proper orbits, then any nonemptyg-invariant closed subset ofX contains a fixed point ofg. Specifically, the closure of each orbitᏻ(x)has a fixed point ofg.
Proof. LetMbe a nonemptyg-invariant closed subset ofX, and letx∈M. By hypothesis, gn(x)∈Mfor eachn∈Nso thatᏻ(x)⊂M. Then cl(ᏻ(x))⊂MsinceMis closed. Clearly, g(ᏻ(x))⊂ᏻ(x); therefore the continuity of g implies that g(cl(ᏻ(x)))⊂cl(g(ᏻ(x)))⊂ cl(ᏻ(x)) (⊂M). But cl(ᏻ(x)) is compact by hypothesis. Consequently, cl(ᏻ(x)) is a non- emptyg-invariant compact subset ofM. A Zorn’s lemma argument produces a minimal nonemptyg-invariant compact subsetAofM.
Leta∈A. Since A isg-invariant and closed, cl(ᏻ(a))⊂A. As above,g(cl(ᏻ(a)))⊂ cl(O(a)) (⊂A). But then cl(ᏻ(a)) is ag-invariant nonempty compact subset ofAand thus ofM. By the minimality ofA, cl(ᏻ(a))=A. Consequently,A=cl(ᏻ(gn(a))) forn∈ω, sinceawas an arbitrary element ofAandgn(a)∈A. Thus, cl(ᏻ(a))=cl(ᏻ(gn(a))) for all n∈N, and thereforeᏻ(a)= {a}sinceᏻ(a) isproper. We conclude,g(a)=a, andais the desired fixed point ofM.
Moreover, the above argument shows that for eachx∈X, cl(ᏻ(x)) is a nonvoidg- invariant compact (and therefore, closed) subset ofX. As such, cl(ᏻ(x)) contains a fixed
point ofgby the preceding result.
Corollary3.2. A continuous self-mapg of a Hausdorfftopological spaceX has a fixed point if and only if there existx∈Xsuch thatcl(ᏻ(x))is compact andg has proper orbits oncl(ᏻ(x)).
Proof. Ifg(x)=x, thenᏻ(x)= {x} =cl(ᏻ(x)). Thus cl(ᏻ(x)) is compact, and the only orbit ofgon cl(ᏻ(x)), namely{x}, is proper. So the condition is necessary.
To see that the condition is sufficient, letx∈Xsuch that cl(ᏻ(x)) is compact and g has proper orbits on cl(ᏻ(x)). Sinceg is continuous, cl(ᏻ(x)) isg-invariant. Therefore, we applyTheorem 3.1withX=M=cl(ᏻ(x)). Sinceg has relatively compact orbits on cl(ᏻ(x)) (cl(ᏻ(x)) is compact) and g has proper orbits on cl(ᏻ(x)) by hypothesis, the
conclusion follows.
Why would it not be sufficient in the above result to merely require that there exist a proper orbitᏻ(x) ofghaving compact closure? Consider the following.
Example 3.3. Let 0< a <1. LetXbe the planar annulus defined in polar coordinates by X= {(r,θ) : 0< a≤r≤1}.
Defineg:X→Xbyg((r,θ))=(r/2 + 1/2,θ+π). Then forr <1 andn∈N,gn((r,θ))= (r/2n+nk=1(1/2k),θ+nπ).
Clearly,g2n((r,θ))→(1,θ) and g2n+1((r,θ))→(1,θ+π), and consequently, cl(ᏻ((r, θ)))=ᏻ((r,θ))∪ {(1,θ), (1,θ+π)}. Thus for allr <1,ᏻ((r,θ)) is proper and has com- pact closure; butgdoes not have proper orbits on cl(ᏻ((r,θ))) sinceᏻ((1,θ))= {(1,θ), (1, θ+π)}(i.e.,gis periodic of period 2 onr=1). And of course,ghas no fixed points. Note also that the diameter of every orbit inXis 2, and therefore no orbit inXhas d.o.d.
The following example satisfies the hypothesis ofCorollary 3.2and thus the mapping gof this example has a fixed point.
Example 3.4. LetX=[−1, 1] with the usual metric and letg(x)= −x1/3forx∈X. It is easy to verify that every point in (−1, 1) has a proper orbit. However,x=0, the only fixed point ofg, is the only point inXwith proper orbits on the closure of its orbit{0}.
We now applyTheorem 3.1to obtain information regarding the setKg, ofDefinition 1.3.
Corollary3.5. Letgbe a continuous self-map of a Hausdorfftopological spaceX. Ifghas relatively compact proper orbits, thenghas a common fixed point with each f ∈Kg. Proof. Let f ∈Kg, and letM= {x∈X: f x=gx}. ThenM= ∅by definition ofKg. If x∈M, then f x=gx. Since f ∈Kg, f2x=f gx=g f x=g2x. But f2x=g f ximplies that
f x∈M, and f gx=g2ximplies thatgx∈M. Thus f(M),g(M)⊂M.
It is well known thatMis closed. Thus,Mis a nonemptyg-invariant closed subset of X. ByTheorem 3.1,Mhas a fixed pointpofg. By definition ofM,f(p)=g(p)=p, and
pis the promised common fixed point.
For purposes of reference we now combineTheorem 3.1andCorollary 3.5.
Theorem3.6. Letg be a continuous self-map of a Hausdorfftopological spaceX. Ifg has relatively compact proper orbits, thenghas a common fixed point with each functionf ∈Kg. Moreover, each nonemptyg-invariant closed subset of X has a fixed point ofg. Specifically, the closure of each orbitᏻ(x)contains a fixed point ofg.
If a Hausdorfftopological spaceXis compact, then all orbits of a self-map ofXare relatively compact. Consequently,Theorem 3.6yields the following result which stresses the role of proper orbits.
Corollary3.7. Any continuous self-mapgof a compact Hausdorfftopological space with proper orbits has a fixed point. In fact, the closure of each orbit contains a fixed point ofg.
Moreover,ghas a common fixed point with each f ∈Kg.
Ifg is a self-map of a topological spaceX, a pointx∈Xis called arecurrent pointif and only ifxis a limit point (accumulation point) ofᏻ(x). Andxis anontrivial periodic pointif and only ifgkx=xfor somek∈Nbutgx=x.
Theorem3.8. Letg be a continuous self-map of a Hausdorfftopological spaceX. Ifg has no recurrent or nontrivial periodic points, thenghas proper orbits.
Proof. We showᏻ(x) is proper. Ifx=gx, then we are done. So letx=gx. Sincexis not a recurrent point, there exists a neighborhoodN(x) ofxsuch thatN(x)∩(ᏻ(x)\{x})= ∅. By hypothesisxis not a periodic point, soᏻ(x)\{x} =ᏻ(gx). ThusN(x)∩ᏻ(gx)= ∅, that is,x /∈cl(ᏻ(gx)). Hence cl(ᏻ(gx))=cl(ᏻ(x)) which meansᏻ(x) is proper.
Corollary 3.7andTheorem 3.8yield the following.
Corollary3.9. Any continuous self-map of a compact Hausdorfftopological space which has no recurrent or nontrivial periodic points has a fixed point.
Of course, if inCorollary 3.9 g has nontrivial periodic points,g may have no fixed points. Defineg byg(x)=1−xforx∈X=[0, 1/4]∪[3/4, 1]. Theng is a continuous self-map of a compact spaceXof period 2 which has neither recurrent points nor fixed points.
As an application ofCorollary 3.9, consider the following.
Example 3.10. LetC= {(r,θ) :θ∈R}be a circle in polar coordinates of radiusr >0 and letgbe a rational rotation ofC. Thus,g((r,θ))=(r,θ+φ) for some rationalφ∈(0, 2π).
Then every point ofCis a recurrent point ofg. To see this, first note thatgn((r,θ))= (r,θ+nφ) for n∈N and that (r,θ+nφ)=(r,θ+ 2kπ) for non,k∈N since φis ra- tional. Thus,g has no fixed or periodic points so thatg has a recurrent point inCby Corollary 3.9. We leave it to the reader to show that therefore every point ofCis a re- current point ofg. Consequently, ifAis a rational rotation of the plane about the origin, then every point ofR2− {(0, 0)}is a recurrent point ofA.
A self-mapgof a spaceXis compact [4] if and only ifXhas a compact subsetY and g(X)⊂Y.
Corollary3.11. Letgbe continuous self-map of a Hausdorfftopological space with proper orbits. Ifgis compact, thenghas a fixed point; indeed,ghas a common fixed point with each
f ∈Kg.
Proof. Sinceg is compact there exists a compact setY⊂X such thatg(X)⊂Y. Then g(Y)⊂g(X)⊂Y, and thereforeYisginvariant and compact. Sinceghas proper orbits, the restriction ofgtoYsatisfies the hypothesis ofCorollary 3.7.
Corollary3.12. Letgbe a continuous self-map of a first countable Hausdorfftopological spaceX. Ifghas relatively compact proper orbits, thenghas a common fixed point with each f ∈Kg. Moreover, for eachx∈X, there is a subsequence of{gn(x)}which converges to a fixed point ofg.
Proof. Corollary 3.12is an immediate consequence ofTheorem 3.6, except for possibly the concluding statement. This last statement follows by noting that byTheorem 3.6, for eachx∈X, cl(ᏻ(x)) contains a fixed point p ofg. SinceX is first countable, there is a sequence inᏻ(x) which converges to p. This sequence can be chosen so as to be a
legitimate subsequence of{gn(x)}.
Since any semimetric space is Hausdorffand first countable we have:
Corollary3.13. Let(X;d)be a semimetric space and letg:X→X be continuous. Ifg has relatively compact orbits with proper orbits or diminishing closure diameters, thenghas a common fixed point with each f ∈Kg. Moreover, for eachx∈X, some subsequence of {gn(x)}converges to a fixed point ofg.
As noted above, in metric spaces diminishing orbital diameters (d.o.d.) yield orbits with diminishing closure diameters (d.c.d.) and therefore, proper orbits. Thus,Corollary 3.13is valid for metric spaces (X,d) with d.c.d. replaced by d.o.d.
The next example shows that common fixed points ofg and the members ofKg in Corollary 3.13need not be unique, even ifX=[0, 1] with the usual metric. It also suggests thatKgcan be large.
Example 3.14. LetX=[0, 1] andd(x,y)= |x−y|. Define f(x)=2x−x2andg(x)=x2. The common fixed points of f andgare 0 and 1 and are their only coincidence points. In general f g=g f, but since f andgclearly commute at 0 and 1, f andg are nontrivially compatible. Ifx∈[0, 1),gn(x)↓0, andgn(1)=1 for allnsoghas proper orbits. Note also
that any continuous function f :X→Xsuch that f(x)> x2or f(x)< x2forx∈(0, 1) is a member ofKg.
The following example shows us thatCorollary 3.13does generalize Kirk’sTheorem 1.2since (X;d) is a noncompact semimetric space andg has proper orbits but no d.o.d.
except atx=0.
Example 3.15. LetX=[0, 1] and ifx,y=0, definedbyd(x,y)=0 ifx=yand 1 ifx=y.
Andd(0,x)=d(x, 0)=xforx∈X. Defineg:X→Xbyg(x)=x/2. Thengn(x)=x/2n↓ 0 forx∈X. Sinceᏻ(x)= {x/2k:k∈ω},ᏻ(x) is clearly proper for allx. In fact, cl(ᏻ(g(x))) is a proper subset of cl(ᏻ(x)) for allx=0. Butδ(ᏻ(gn(x)))=δ(cl(ᏻ(gn(x))))=1 for all n∈ωandx=0; that is,ᏻ(x), has diminishing closure diameters or d.o.d. only at 0.
Corollary3.16. Any continuous self-mapg of a compact metric space with d.o.d. has a common fixed point with each f ∈Kg. Moreover, for eachx∈Xsome subsequence{gkn(x)} of the sequence{gn(x)}has a limit which is a fixed point ofg.
The need for compatibility of functions to ensure the existence of a common fixed point is demonstrated in the next simple example.
Example 3.17. LetX= {1, 2, 3, 4}with the usual metric. Letf(1)=2, f(2)=2, f(3)=1, and f(4)=3. Also letg(1)=2,g(2)=4,g(3)=3,g(4)=3. Then each of f andg has d.o.d., each is continuous, and f and g have a coincidence point. In fact, f(1)=2= g(1). Butg f(1)=4 whereas f g(1)=2; that is, f andgare not compatible and have no common fixed point.
Of course, proper orbits also play an essential role.
Example 3.18. LetX=[0, 1] with the usual metric and define f,g:X→Xbyg(x)=1−x and f(x)=1−x2forx∈X. Then f ∈Kgsince f andg coincide only at 0 and 1, and commute at 0 and at 1. But neither f norg has proper orbits since both have periodic points of period 2 at 0 and at 1. And f andg have no common fixed point.
Since any closed and bounded subset ofRnis compact, and since the orbits of a map with d.o.d. are bounded by definition, we have the following somewhat amazing conse- quence ofCorollary 3.13.
Corollary3.19. Ifg is a continuous self-map ofRnwith diminishing orbital diameters, thenghas a fixed point. In fact,g has a common fixed point with each f ∈Kg. Moreover, each orbitᏻ(x)has a sequence which converges to a fixed point ofg.
4. Retrospect
Compatible maps were introduced in [8] as a generalization of commuting maps, which had served as a vehicle for various generalizations of the Banach Contraction Theorem (see [7], and, e.g., [3]). Compatible maps, in turn, proved productive in producing com- mon fixed points and further generalizations of Banach’s theorem: [2,6,8,9,10,11,12, 14,15,16,18,19,20]. On the other hand, the concept of proper orbits now provides new
results, and a means of extending a bevy of known results. We conclude with such an example.
Supposeg is any continuous self-map of complete metric spaceX such that{gn(x)} converges for eachx∈X. Then the orbitᏻ(x) is proper and relatively compact for each x, and consequently,ghas a common fixed point with each f ∈Kg. With this in mind, consider a result presented at the Third World Congress of Nonlinear Analysis.
Theorem4.1 (Rhoades [17]). Let(X,d)be a complete metric space,ga weakly contractive map. Thenghas a unique fixed pointpinX.
The mapg:X→Xis weakly contractive if and only ifd(gx,g y)< d(x,y)−ψ(d(x,y)) forx,y∈X andψ: [0,∞)→[0,∞) is a continuous, nondecreasing function such that ψ(0)=0,ψis positive on (0,∞) and limt→∞ψ(t)= ∞. Sinceψis nonnegative,gis non- expansive, hence continuous. Moreover, in the proof it is shown thatgn(x)→pfor each x∈X. By the above comments we can therefore augmentTheorem 4.1with the state- ment,pis also the common fixed point ofgand each f ∈Kg.
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Gerald F. Jungck: Department of Mathematics, Bradley University, Peoria, IL 61625, USA E-mail address:[email protected]
