VOL. 16 NO. 2 (1993) 259-266

### FIXED

### POINTS FOR PAIRS OF

### MAPPINGS IN d-COMPLETE

### TOPOLOGICAL SPACES

TROYL.HICKS### Department

ofMathematics University of Missouri- RollaRolla, MO65401 U.S.A. and B.E.RHOADES

### Department

of Mathematics Indiana University Bloomington,IN47405 U.S.A.(Received October 7, 1991 and in revised form March 12, 1992)

ABSTRACT.Several importantlnetricspacefixedpoint theoremsareprovedforalnrgeclass

of non-metric spaces.

### In

some cases themetricspaceproofs need onlyminor changes. Thisissurprisingsince thedistance functionused need notbe symmetric and neednot sa.tisfy the

triangular inequality.

### KEY

WORDS AND PHRASES.Commuting,compatible, d-complete topologicalspaces,fixed points, pairsof mappings.1991 SUBJECT

### CLASSIFICATION

CODE. 47H10,54H25.1. INTRODUCTION.

Let

### (X,t)

be atopological spaceand d:X x### X

### [0, cx)

such that### d(x,y)

0 if and onlyifx y.### X

issaid tobed-completeif### ,,oo__

### d(x,,,x,,+l)

### <

cimpliesthatthesequence isconvergent in### (X, t). In

ametric space, such asequence isa### CauchSr

sequence.### T" X

### X

isw-continuousatx, if x,, x implies### Tx,,

Tx.Definition. Asymmetricon aset Xisareal-valuedfunction donX

### X

such that:### (1)

d(x,y)### >_

0andd(x,y) 0ifand onlyifx y;and### ()

### d(,u)

### d(u,).

Let dbe a symmetricon a set

### X

andfor any e### >

0 and any x### X,

let### S(x,e)

### {y

### X

d(x,y)

### <

### e}.

We definea topology t(d)on### X

byU### _

### t(d)

ifand only ifforeach x### U,

some### S(x,

e) C_ U.### A

symmetric d is asemi-metric if for each x### X

and each### >

### O,

### S(z,

e) is aneighborhood ofx in the topology t(d).

### A

topological space### X

is said to be symmetrizable### (semi-metrizable)

ifitstopology isitduced byasymmetric### (semi-metric)

onX.The d-complete synunctrizable

### (semi-metrizable)

spaces formanimportant class ofex-amples ofd-complete topological spaces. \Vegive other examples. Let

### X

be aninfinite set.260 T.L. HICKS AND B.E. RHOADES

,/fl)r Xsuch that

### <

tdandthe metrictopologytd isnon-discrete. Now### (X,t)

is d-conplete.iJc..

### d(a’,,x,,+l) <

cxz implies that### {x,}

is ad-Cauchy sequence. Thus, x, x in d andlmrefore in the topology t. See

### [4]

for the construction oftd. It should also be noted that,Jy COml,letequasi-metric space

### (X,d)

### (d(x,y)

### d(y,x))

is a d-completetopological space. Tlmc are several competing definitions for a Cauchy sequence, but### d(xn,xn+l)

### <

williq)ly that

### {x,

isaCauchysequence foranyreasonabledefinition. One reasonabledefinition is dcriw’d from requiring that the filter### generated

by### {x,}

be a Cauchy filter in thequasi-fiformitygeneratedby d. Thisgives

### {x,}

isaCauchysequence iffor each e_{>}

0thereexists
positive integer

### no

### ,(e)

andx### x(e)in X

suchthat### {x,’n

### >

### x0}

C_### {g

### X" d(z,y) <

### e}.

Tlw metric space definition alsoholdsif### d(x,,x,+a)

### <

c. Some fundamentaltheoremsford-c,,mplctctopologicalspacesaregivenin

### [4]

and### [5].

2. RESULTS. The resultsthroughCorollary2.2axegeneralizationsoftheorems due toJungck

### [7].

Even thoughonlyminorchangesareneededintheproofs, theyaregiven for completeness.### LEMMA

2.1. Let### (X,t)

be ad-complete topologicalspace._{If}

there exists c _{(0,1)}

such
thatd(y,,,y,+) ### <_

c_{d(y,_,y,)}

_{for}

atl n, then ### {Yn}

converges to apoint inX.On-1

eROOF.

### (u.,u.+) <

### .--(u,,u=)

gie### I.=

### a(U.,U.+l) _<

### (Ul,U=)E.=

The resultfollowssinceXis d-complete.THEOREM2.1. Let

### (X,t)

be a### Hausdorff

d-complete topologicalspace and_{f}

X ### X

a w-continuous_{function.}

Then_{f}

has a_{fixed}

pointin### X

### if

and only### if

there existsc_{(0,1)}

and
aw-continuous_{function}

g:X X whichcommutes with### f

and_{satisifes}

### (1)

### 9(X)

C_{f(X)}

andd(gx,gy)### <_

a### d(fx,fy)

### for

altz,y X. Indeed,### f

and_{9}have aunique common

_{fixed}

point_{if (1)}

holds.
PROOF. Suppose

### f(a)

aforsomea X. Put### g(x)

afor all x X. Then### g(fx)

a and f(gx)### f(a)

a. Also, g(x) a### f(a)

for allx### X

so that### g(X)

C_### f(X).

Forany### ,,

_{(O,),d(gz,gy)}

_{d(a,a)}

0### <

### ,

### d(fz,fy).

Since g isw-continuous,### (1)

holds.Supposethereexistsc and gsuchthat

### (1)

holds. Let x0### e

Xandlet x, besuch that### (2)

### .f(xn)

### 9(xn-).

Now### (1)

and### (2)

gives### d(fxn,fxn+x)

### d(gxn-l,gxn)

### d(fx,_,fxn).

Lemma 2.1 gives p in

### X

suchthat### f(x,)

p. 9 w-continuous impfies that lim9fx, 9P. Since9x,, p, thew-continuity of### f

impliesthat limfgx, fp. Thereforep isacoincidence point of### f

and9- Clearly, fgP### 9fP.

### f(fP)

### f(9P)

### 9(fP)

### 9(9P)

and### d(9P, 9(9P))

### d(fp, f(9p))

### d(9P, O(9P)).

### Hence,

### 9(P)

### 9(9P).

Thus_{9(P)}

### 9(9P)

### f(gP),

d_{9(P)}

isacooned point of### f

and_{9.}

Ifx

### f(x)

### 9(z)

and y### f(y)

### 9(Y),

then### (1)

gives### d(x,y)

### d(gx,9y)

### d(fx,fy)

d(x,y) orz y.

COROLLARY 2.1. Let

### (X,t)

be a_{Hausdorff}

d-complete topological space and _{f}

and g
commuting mappings_{from}

### X

toX. Suppose_{f}

andg arew-continuous and### g(S)

C_### f(X).

### If

there exists### (0,1)

and apositive integer k such thatthen

_{f}

and9 have aunique common_{fized}

point.
PROOF. Clearly,

### g

commutes with_{f}

and### g’(X)

C__{g(X)}

C_### f(X).

Applyingthetheorem### t,,gk

and_{f}

gives a uniquep ### X

withp--### g’(p)

### f(p).

Since_{f}

and g comnmtc 9(p)
,.l(.fp)

### f(g(p))

### .,:l’(gp)

or g(p) is a common fixed point of_{f}

and ### gk.

Uniqueness gives9(P) 1’

### f(P).

Letting

_{f}

i, the identity, gives a generalization of Banach’s theorem. If we also let
k= 1, weget Banach’stheoremin thisnew setting.
### COROLLARY

2.2. Let n be a positive integer and let k### >

1._{If}

g is a surjective
w-contm.,.o.us_{self}

map _{of}

a_{Haudorff}

_{d-complete topological}space

### X

such that### d(a".,

### anu) >_

### d(z,U)

### Io

### art

### ,

U### e

### X,

theng has a unique### fized

point.### PROOF. Let

_{f}

### g2,

and a## -

in Corollary 2.1. Now### d(x,y)

### <

o### d(g"x,g"y)

.gives d(g"z,g’y)### <_

o_{d(g2"x,g2’y)}

od(fz,fy), g surjectiveimpliesg(X)C_### f(X).

THEOREM 2.2 Let

### (X,t)

be a_{Hausdorff}

d-complete topologicalspace. ### Suppose

### f

andg are. commutingw-continuous selfmaps_{of}

### X

such that### g(X)

C_### f(X)

and### (a)

d(g"x, g"y)### <

### d,d(fx,

fy)### for

eachn and_{for}

allx,y in### X,

where Then:### ()

### f,f,..,

are_{fized}

_{?on}

_{of}

_{ff.}

_{]fp}

### ,

_{tenp}

_{{f,f,...,).}

_{If}

_{f}

_{f.}

then
### f

### f+

### for

### ec

1.### ()

### If

lim_{f}

_{"}

### (5)

### If

### I

inaddition,### f

is the identity, then p_{for}

eachx inX.
### PROOF.

Since limd,, 0, thereexistsa k with### d <

1. Thend(g

### ’x,

### gy)

### <_ dd(fx,

### fy).

### From

Corollary2.1,### f

andghaveauniquecommonfixed point p.Toprove

### (2),

fixxand sety gx in### (a).

Then### d(g’x,g"+z)

### <_ d,d(fx,

### fgx),

which gives### Z

### d(g’x’

### g"+’

### x)

### <_

### d(fx, fgx)

### Z

### d(fx,

gfx)### <

co.n=l n=l

### X

is d-complete, so there exists an such that x,, g"x Y.. Since g is w-continuous x,,+ g(x,,)### y..

Clearly262 T.L. HICKS AND B.E. RHOADES

To prove

### (3),

### g(fky)

### fk(g2)

### fk.

for each k### >

1. Ifp### .

then 0### < d(1,,.i’)

d(gp, g2)### <

dd(fp, f.i’)### dd(p,f),

and p### f.

Thus 0### <

d(p,]’) d(gp, gf.,’) d,d(fp,### f2)

dd(p,### fe)

andp_{f2.}

By induction p _{f"}

foranyn 1.
If

_{fh"}

_{f2"}

thcn 0 < ### d(f:r,.f2.)

### d(gf,gf2)

### dd(f2,f3)

and_{f2}

_{f.’.}

By
iMuction_{.f}

### fa+2-

foreach’t 1.To prove

### (4),

### 9f"

gy andgf"### f"9

### f"

y and_{9Y}Y.

### f"+;

Y and### .["+ii"

### ff"

fy,sofy y. From### (1)

y p.Toprove

### (5),

n+r--I n+r--I

### (".,+rx)

### e(.,+)

### (f.,.t)

k=n

T&ing thelimit asr yields

### (",)

### e(I,I)

### .

If

_{f}

is the identity,gha uniquefixed point from ### (1).

But### (2)

gives g2 2. Therefore.i:=p.

Altman

### [2]

called_{f-X}

### X

ageneralizedcontractionprovided### d(fz,fy)

### Q(d(x,y))for

allx,g### X

wherc### Q

satisfiesthe following:() 0

### < (t)

### <

fo### n

### e (0,t,],

(b) g(t)

### t/(t- Q(t))

is non-increasing,### (c)

### f2’

### g(t)dt <

### ,

and### (d) Q

isnon-decreasing.Heassumes

### (X,d)

is ametric space andprovesthat forany_{xo}

G ### X, xn

### f’xo

x### f(x).

Ve usehis definition in themore

### generM

setting to obtMn thefollowing.THEOREM 2.3. Let

### (X,t)

be ad-complete### Hausdorff

topologicalspace and_{f}

### X

X aw-continuous generalized contraction. LetXo E### X

and put x, --w### f’xo

### f(xn-1).

Thenlimz, z and x

### f(x).

PROOF. Put

### sa

### d(xo,xl)

ands,+l### Q(sn)

forn### _>

1. Using### (a), (b),

and### (c),

Altmanshows that theseries

### n__

s,generatedby### Q

isconvergent.### By

induction,### d(x,,xn+,)

### d(fxn_,,fx,)

### < O(d(x,,_,,x,)) < O(s,)

,5n+l.Thus,

### -:+,

### d(x,,,x+,

### <

cx and limx, x exists,_{f}

w-continuous gives x,+ ### f(xn)

### f(x).

Thus,### f(z)

x.Ifonereplaces topological bysymmetrizableinthetheorem,

_{f}

isforcedto bew-continuous.
### Even

though Altman asserted uniqueness of the fixed point, examples exist_{[11]}

that show
otherwise.

Ournext resultis thed-completeanalogueofTheorem 2.1 ofPark

### [9].

(i) there exists a.sequence

### {x,}

C X such thatu,+l := gu,,u2,+ := hu,+ and has a cluster point zn### X,

### (ii)

g andhg arew-continuous at### ,

(i,)

### G(x)

:=d(x,gz),H(x) :=_{d(z, hx)}

are orbitallycontinuous at### ,

and(i’v) g andh satisfy

### d(gx,hy) < d(x,y)

### for

each distinct z and y in### {h-,}

satisfying either x hy ory gx, then### (1)

g orh has a_{fixed}

potnt ,n_{{-7,}}

or
### (2)

is a common_{fixed}

point_{of}

g and h and u, as oo.
The proofis thesame asthat of Theorem2.1 of

### [9].

COROLLARY2.3. Letg andh be w-continuousselfinaps

### of

ad-completespace### X

satis-fyn.q

### d(hz,gy)

### <_

### max{d(z,y),

### d(z,hz),

### d(y,gy), d(y,hz)}

### for

allx,y in### X,

"where 0### <

k### <

and### d(x,gx)

and### d(x, hx)

are orbitallycontinuous. Theng orh has a_{fixed}

point org andh havea unique common### fixed

point.PROOF.

### Let

u,+ hu,,u.,+2 gu2,+. Thus,from thecontractivedefinition,### d(u2,+,u2n+2) <_ kmax{d(u2n, u2n+a),d(u2n, u2n+l),

### d(.+, .+), d(.+

### ,

### u.+ },

whichimplies that

### d(u,+,,u,,+) <

k### d(u2,,,u,+a).

### Therefore{u,}

is Cauchy, hencecon-vergesto apoint

### .

ThehypothesesofTheorem 4arenowsatisfied.The uniqueness ofacommonfixed point follows from thecontractivedefinition.

Wenowdemonstrate that several resultsin the literaturefollowasspecialcasesof Theorem

2.4. Whileourlist is notexhaustive, itindicates thegeneralityofthe theorem.

COROLLARY2.4.

### [3,

Theorem### 1].

Let### T,

_{T}

be twoselfmaps ### of

a_{Hausdorff}

F-complete
space ### X, F" X

### X

---,_{[0, oo)}

a continuou symmetric mapping such that ### F(x,y)

0_{for}

x y and,_{for}

each pair### of

distinctx,y in### X,

### F(Tx,Ty) <

### max{[F(x,y),F(x,Tx),F(y,Ty)]

U### min[F(y,Ty),F(y,Tx)]},

and_{for}

someXo inX the sequence ### {x,} defined

byx2n+ Tx2n,x2,,+2 Tx2n+, has subsequence converging to apoint inX._{If}

_{T}

and_{T2T}

or### T

and_{T}

_{T2}

arecontinuous at
then is a

_{fixed}

point _{of}

### T1

and_{T2 (or}

_{T}

or_{T2}

has a_{fixed}

point).
PROOF.
### F(Tx,TTlx)

### <

### max{[F(x, Tz),F(x,Tlx),F(Tlx,

### TTx)]

U### min[F(

### T

x,_{TT}

x### ),

### F(

### TI

x,### T

x### max{r(x,Tx),F(Tx,TTx)}

which implies that

### F(Tz,TTx) < F(z,

### TlZ),

andcondition### (iv)

ofTheorem 2.3issatisfied. With g### T,

h### T,

theremainingconditionsof Theorem2.4 aresatisfied.264 T.L. HICKS AND B.E. RHOADES

Ifwenow set g

### T,

h### T1

then, from the contractivedefinition,### F(T1T2x, T2x) <

### max{[F(T2x, x), F(T2x, T1T:x), F(x, Tx)]

U

### min[F(x,T2x),F(x,

### T1T:x)]}

### max{F(T2x, T,Tx),F(x,Tx)},

which implies that

### F(TITx,T:z)

### <

### F(x,Tx),

and again thehypothesesofTheorem 2.4 a.,’,’ satisfied.COROLLARY2.5.

### [S,

Theorem### 1].

Let_{f}

and g be w-continuousselfmaps### of

ad-complete L-space### X,

with### d(z,x)

0 and### d(x,y)

### d(y,x)

### for

each z,y inX._{If f}

and g satisfy
min

### d2(f

### x,gy), d(x,y)d(fx,gy),

d### (y,gy)

### -min{d(x,fx)d(y,gy), d(x,gy)d(y,f

### x)}

### <_

q### d(x,fx)d(y,gy)

### for

all x, in### X,

and 0### <

q### <

1, then_{f}

andg havea common_{fixed}

point, ### (or

### f

org has a_{fixed}

po,nt).
PROOF. Asin

### [S],

itcanbeshown that### {x,,}

converges,where### {x,,}

isdefinedbyX2n+l .fx2n,X2n+2 gx2n+lSettingy

_{fx}

in thecontractive definitionyields
o1"

### min{d2(f

x,gf### x),

### d(x,fx)d(f

### x,gyx),

### d2(f

x,gf### x)}

### i{d(,y)d(y,gy),d(,gy)d(y,Y)} <_

q### d(, fz)d(fz,gfz),

### rnin{d2(fx,gfx),d(x,

### fx)d(fx,gfx)} <

q### d(x,fx)d(fx,gfx).

If theminimum is### d2(fx,gfx),

thenwehave### d(fx,gfx)

### <

q### d(x,fx)d(fx,gfx),

whichimpliesthat either

_{fx}

is afixedpoint ofg,or### d(fx,gfx)

### <

q### dix,fx

### ).

If the minimumis

### d(x,fx)d(fx,gfx),

either x isafixedpoint of### f,

### fx

is afixedpoint ofg,or

### d(fx,gfx)

### <_

q### d(x,fx).

Therefore theconditionsof Theorem2.4 aresatisfied, except for### (iii).

### However,

forthis contractive definition### (iii)

is notneeded. Asin### [8], {x,,}

convergesto a point u inX. Since_{f}

isw-continuous andX2n+l ### fX2n,

it follows thatu### fu.

Similarly,### COROLLARY

2.6.### [1,

Theorem### 1]

### Let S,T

be orbitallyw-continuous selfmaps### of

ad-complete

### L-space

### X,

with d(x,y) 0_{iff}

x y. _{If}

### for

some 0### <

cr### <

1 and_{for}

each x,y in
### X,

### d(Sx,

### Ty)

### <_ a{a(x,

### sx)a(y,

### Ty)}

/:and

### d(x, Sx),d(x, Tx)

are orbitally continuous, then S and### T

have a unique common_{fixed}

point### (or

S orT has a_{fixed}

point).
PROOF. Nowthat thecontra(’tivedefinitionimplies that

### ,,max{d(x,

### Sz),d(y,

### Ty)},

and the result follows fromCorollary2.3.### COROLLARY

2.7._{[10,}

Theorem ### 1].

### Let

### X

be an F-complete### Hausdorff

space, Tl,### T.

continuousselfmaps

### of

X. Let### F" X

x### X

### IR+,

### F

continuous andsuch that### F(x,x)

0### for

allx in### X

and### F(Tx,Ty) <

a,E(x,y)### +

### a,F(x,Tx) +

### aaF(y,Ty)

### for

each distinct x,y in### X,

wherep andq arepositive integers, a,### >_

O,a_{+}

a2 -t"### aa <

1._{ff}

### for

some Xo### X,

the sequence### {x,}

C### X

_{defined}

byx2n+l ### Tz.,x2.+2

### Tz2.+,

has aconvergentsubsequeuce, then

### (T

or### T2

has a_{fixed}

point)or### T

and_{T2}

have aunique common
### .fixed

poini.PROOF. Sety

### Tx

toget### F(Tx,TTx)

### <

### a,F(x,Tx)

### +

### a:F(x,Tx)

### +

### aaF(Tx,TTx),

or

### F(T TIT)

### <

### +

### F(,

### T),

1 aaandthe conditions of Theorem 2.4aresatisfied with

_{f}

### T’,

g### T

### q.

The uniqueness followsas in

_{[10].}

### REMARK.

The wordsinparenthesesinCorollaries4-7 have been addedbythe presentauthors, inorder forthetheoremstobe correctly stated.266 T.L. HICKS AND B.E. RHOADES

REFERENCES

B. Ahmad and Fazal-ur-Retnnan, Fixedpoint theorems in

### L-spaces.

Math. Japonica36### (1991),

225-228.2. M. Altman,

### An

integraltestforseriesandgeneralizedcontractions. Amer. Math. Monthly82

### (1975),

827-829.3. H.Chatterjeeand BaradaK.

### Ray,

Onfurtherextensionofatheorem ofEdelstein. Indian J. Pure Appl. Math. 11### (1980),

256-257.### Troy

L. Hicksand Wilson### R.

Cfisler,### A

noteon### TI

topologies. Proc.### Amer.

Math. Soc.46,

### (1974),

94-96.### Troy L.

Hicks, Fixed point theorems for d-complete topological spaces I. Int.### J.

Math. and Math. Sci.,to appcar.### Troy

L. HicksandB.E. Rhoades, Fixedpointtheoremsford-completetopological spacesII. Math. Japonica,toappear.

Gerald

### Jungck,

Compatible mappings andcommonfixed points. Int. J. Math. and Math. Sci__.=9### (1986),

771-779.8.

### B.G.

Pachpatte, Commonfixedpoint theorems forcertainmappingsin### L-spaces.

J.

### M.A.C.T.

13### (1980),

55-58.9. SehiePark,Fixedpoints andperiodicpoints ofcontractivepairs of maps. Proc. Collegeof

Nat. Sci., SeoulNational Univ. 5

### (1980),

9-22.10.

11.

Barada

### K. Ray

and H. Chatterjee, Someresults on fixed points in metric and Banach spaces. Bull. L’Acad. Pol. des Sci. 25### (1977),

1243-1247.B.

### Watson,

B.A. Meade and C.W. Norris, Note on a theorem of Altman. IndianJ. Pure Appl. Math. 17### (1986)

1092-1093.**Mathematical Problems in Engineering**

**Special Issue on**

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**Call for Papers**

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*This proposed special edition of the Mathematical *

*Prob-lems in Engineering aims to provide a picture of the *

impor-tance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems. Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophis-ticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.

Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System athttp:// mts.hindawi.com/according to the following timetable:

Manuscript Due February 1, 2009 First Round of Reviews May 1, 2009 Publication Date August 1, 2009

**Guest Editors**

**José Roberto Castilho Piqueira, Telecommunication and**

Control Engineering Department, Polytechnic School, The University of São Paulo, 05508-970 São Paulo, Brazil; piqueira@lac.usp.br

**Elbert E. Neher Macau, Laboratório Associado de**

Matemática Aplicada e Computação (LAC), Instituto Nacional de Pesquisas Espaciais (INPE), São Josè dos Campos, 12227-010 São Paulo, Brazil ; elbert@lac.inpe.br

**Celso Grebogi, Department of Physics, King’s College,**

University of Aberdeen, Aberdeen AB24 3UE, UK; grebogi@abdn.ac.uk

*Hindawi Publishing Corporation*
*http://www.hindawi.com*