# FIXED POINTS FOR PAIRS OF

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

TROYL.HICKS

### Department

ofMathematics University of Missouri- Rolla

### 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. This

issurprisingsince 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

such that

### d(x,y)

0 if and onlyifx y.

### X

issaid tobed-completeif

### <

cimpliesthatthesequence isconvergent in

### (X, t). In

ametric space, such asequence isa

sequence.

### X

isw-continuousatx, if x,, x implies

### Tx,,

Tx.

Definition. Asymmetricon aset Xisareal-valuedfunction donX

such that:

d(x,y)

### >_

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

### d(u,).

Let dbe a symmetricon a set

andfor any e

0 and any x

let

d(x,y)

### e}.

We definea topology t(d)on

byU

### t(d)

ifand only ifforeach x

some

e) C_ U.

### A

symmetric d is asemi-metric if for each x

and each

### S(z,

e) is a

neighborhood 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 of

ex-amples ofd-complete topological spaces. \Vegive other examples. Let

### X

be aninfinite set.

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

,/fl)r Xsuch that

### <

tdandthe metrictopologytd isnon-discrete. Now

is d-conplete

.iJc..

cxz implies that

### {x,}

is ad-Cauchy sequence. Thus, x, x in d and

lmrefore in the topology t. See

### [4]

for the construction oftd. It should also be noted that

,Jy COml,letequasi-metric space

### d(y,x))

is a d-completetopological space. Tlmc are several competing definitions for a Cauchy sequence, but

will

iq)ly that

### {x,

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

by

### {x,}

be a Cauchy filter in the

quasi-fiformitygeneratedby d. Thisgives

### {x,}

isaCauchysequence iffor each e

0thereexists

positive integer

andx

suchthat

C_

### e}.

Tlw metric space definition alsoholdsif

### <

c. Some fundamentaltheoremsfor

d-c,,mplctctopologicalspacesaregivenin

and

### [5].

2. RESULTS. The resultsthroughCorollary2.2axegeneralizationsoftheorems due toJungck

### [7].

Even thoughonlyminorchangesareneededintheproofs, theyaregiven for completeness.

2.1. Let

there exists c

### (0,1)

such thatd(y,,,y,+)

c

atl n, then

### {Yn}

converges to apoint inX.

On-1

eROOF.

gie

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

The resultfollowssinceXis d-complete.

THEOREM2.1. Let

be a

### Hausdorff

d-complete topologicalspace and

X

a w-continuous

Then

has a

pointin

and only

there existsc

### (0,1)

and aw-continuous

### function

g:X X whichcommutes with

and

C

andd(gx,gy)

a

### for

altz,y X. Indeed,

### f

and 9 have aunique common

point

holds.

PROOF. Suppose

aforsomea X. Put

### g(x)

afor all x X. Then

a and f(gx)

a. Also, g(x) a

for allx

so that

C_

Forany

0

### d(fz,fy).

Since g isw-continuous,

### (1)

holds.

Supposethereexistsc and gsuchthat

holds. Let x0

### e

Xandlet x, besuch that

Now

and

gives

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

Lemma 2.1 gives p in

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

and

Thus

d

### 9(P)

isacooned point of

and9.

Ifx

and y

then

gives

### d(fx,fy)

d(x,y) orz y.

COROLLARY 2.1. Let

be a

### Hausdorff

d-complete topological space and

### f

and g commuting mappings

toX. Suppose

### f

andg arew-continuous and

C_

there exists

### (0,1)

and apositive integer k such that

(3)

then

### f

and9 have aunique common

point.

PROOF. Clearly,

commutes with

and

C_

C_

### f(X).

Applyingthetheorem

and

gives a uniquep

withp--

Since

### f

and g comnmtc 9(p)

,.l(.fp)

### .,:l’(gp)

or g(p) is a common fixed point of

and

Uniqueness gives

9(P) 1’

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

map

a

### Haudorff

d-complete topologicalspace

such that

U

### X,

theng has a unique

point.

and a

## -

in Corollary 2.1. Now

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_

THEOREM 2.2 Let

be a

### Hausdorff

d-complete topologicalspace.

### f

andg are. commutingw-continuous selfmaps

such that

C_

and

d(g"x, g"y)

fy)

eachn and

allx,y in

where Then:

are

ff. ]fp

then

1.

lim

### f

is the identity, then p

eachx inX.

### PROOF.

Since limd,, 0, thereexistsa k with

1. Then

d(g

Corollary2.1,

### f

andghaveauniquecommonfixed point p.

Toprove

### (2),

fixxand sety gx in

Then

which gives

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..

Clearly

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

To prove

for each k

1. Ifp

then 0

d(gp, g2)

dd(fp, f.i’)

and p

Thus 0

### <

d(p,]’) d(gp, gf.,’) d,d(fp,

dd(p,

andp

By induction p

foranyn 1.

If

thcn 0 <

and

By iMuction

foreach’t 1.

To prove

gy andgf"

y and 9Y Y.

Y and

fy,sofy y. From

y p.

Toprove

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

### (f.,.t)

k=n

T&ing thelimit asr yields

If

### f

is the identity,gha uniquefixed point from

But

### (2)

gives g2 2. Therefore

.i:=p.

Altman

called

### X

ageneralizedcontractionprovided

allx,g

wherc

### Q

satisfiesthe following:

() 0

fo

(b) g(t)

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

is non-increasing,

and

### (d) Q

isnon-decreasing.

Heassumes

### (X,d)

is ametric space andprovesthat forany

G

x

### f(x).

Ve usehis definition in themore

### generM

setting to obtMn thefollowing.

THEOREM 2.3. Let

### Hausdorff

topologicalspace and

### X

X aw-continuous generalized contraction. LetXo E

and put x, --w

Then

limz, z and x

PROOF. Put

ands,+l

forn

1. Using

and

### (c),

Altman

shows that theseries

s,generatedby

isconvergent.

induction,

,5n+l.

Thus,

### <

cx and limx, x exists,

### f

w-continuous gives 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].

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(i) there exists a.sequence

### {x,}

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

### (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

### for

each distinct z and y in

### {h-,}

satisfying either x hy ory gx, then

g orh has a

potnt ,n

or

is a common

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

satis-fyn.q

allx,y in

"where 0

k

and

and

### d(x, hx)

are orbitallycontinuous. Theng orh has a

### fixed

point org andh havea unique common

point.

PROOF.

### Let

u,+ hu,,u.,+2 gu2,+. Thus,from thecontractivedefinition,

### u.+ },

whichimplies that

k

### Therefore{u,}

is Cauchy, hence

con-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.

Theorem

Let

be twoselfmaps

a

F-complete space

---,

### [0, oo)

a continuou symmetric mapping such that

0

x y and,

each pair

distinctx,y in

U

and

### for

someXo inX the sequence

### {x,} defined

byx2n+ Tx2n,x2,,+2 Tx2n+, has subsequence converging to apoint inX.

and

or

and

arecontinuous at

then is a

point

and

or

has a

point). PROOF.

U

x,

x

x,

x

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

which implies that

andcondition

### (iv)

ofTheorem 2.3issatisfied. With g

h

### T,

theremainingconditionsof Theorem2.4 aresatisfied.

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

Ifwenow set g

h

### T1

then, from the contractivedefinition,

U

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

which implies that

### F(x,Tx),

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

COROLLARY2.5.

Theorem

Let

### f

and g be w-continuousselfmaps

with

0 and

each z,y inX.

and g satisfy

min

d

q

all x, in

and 0

q

1, then

### f

andg havea common

point,

org has a

po,nt).

PROOF. Asin

### [S],

itcanbeshown that

converges,where

### {x,,}

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

Settingy

### fx

in thecontractive definitionyields

o1"

x,gf

x,gf

q

q

If theminimum is

thenwehave

q

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

whichimpliesthat either

### fx

is afixedpoint ofg,or

q

If the minimumis

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

either x isafixedpoint of

### fx

is afixedpoint of

g,or

q

### d(x,fx).

Therefore theconditionsof Theorem2.4 aresatisfied, except for

### However,

forthis contractive definition

### (iii)

is notneeded. Asin

### [8], {x,,}

convergesto a point u inX. Since

### f

isw-continuous andX2n+l

it follows thatu

Similarly,

2.6.

Theorem

### Let S,T

be orbitallyw-continuous selfmaps

a

d-complete

with d(x,y) 0

x y.

some 0

cr

1 and

each x,y in

/:

and

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

are orbitally continuous, then S and

### T

have a unique common

point

S orT has a

### fixed

point).

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PROOF. Nowthat thecontra(’tivedefinitionimplies that

### Ty)},

and the result follows fromCorollary2.3.

2.7.

Theorem

be an F-complete

space, Tl,

### T.

continuousselfmaps

X. Let

x

### F

continuous andsuch that

0

allx in

and

a,E(x,y)

### for

each distinct x,y in

### X,

wherep andq arepositive integers, a,

O,a

a2 -t"

1.

some Xo

the sequence

C

byx2n+l

### Tz2.+,

has a

convergentsubsequeuce, then

or

has a

point)or

and

### T2

have aunique common

poini.

PROOF. Sety

toget

or

### T),

1 aa

andthe conditions of Theorem 2.4aresatisfied with

g

### q.

The uniqueness followsas in

### REMARK.

The wordsinparenthesesinCorollaries4-7 have been addedbythe presentauthors, inorder forthetheoremstobe correctly stated.

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

REFERENCES

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

Math. Japonica36

225-228.

2. M. Altman,

### An

integraltestforseriesandgeneralizedcontractions. Amer. Math. Monthly

82

827-829.

### Ray,

Onfurtherextensionofatheorem ofEdelstein. Indian J. Pure Appl. Math. 11

256-257.

### Troy

L. Hicksand Wilson

Cfisler,

noteon

### TI

topologies. Proc.

Math. Soc.

46,

94-96.

### Troy L.

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

### J.

Math. and Math. Sci.,to appcar.

### Troy

II. Math. Japonica,toappear.

Gerald

### Jungck,

Compatible mappings andcommonfixed points. Int. J. Math. and Math. Sci__.=9

771-779.

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### B.G.

Pachpatte, Commonfixedpoint theorems forcertainmappingsin

J.

13

### (1980),

55-58.

9. SehiePark,Fixedpoints andperiodicpoints ofcontractivepairs of maps. Proc. Collegeof

Nat. Sci., SeoulNational Univ. 5

9-22.

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11.

### K. Ray

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

1243-1247.

B.

### Watson,

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

1092-1093.

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### Call for Papers

Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.

Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from “Qualitative Theory of Diﬀerential Equations,” allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.

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

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