FIXED POINTS FOR PAIRS OF

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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- Rolla

Rolla, 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. 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

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 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.

(2)

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 and

lmrefore 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)

<

will

iq)ly that

{x,

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

generated

by

{x,}

be a Cauchy filter in the

quasi-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 fundamentaltheoremsfor

d-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

and9.

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 that

(3)

then

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 gives

9(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 topologicalspace

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. Then

d(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..

Clearly

(4)

262 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).

Then

limz, z and x

f(x).

PROOF. Put

sa

d(xo,xl)

ands,+l

Q(sn)

forn

_>

1. Using

(a), (b),

and

(c),

Altman

shows 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].

(5)

(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, 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.

[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.

(6)

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+l

Settingy

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 of

g,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

a

d-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).

(7)

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 a

convergentsubsequeuce, 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 aa

andthe 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.

(8)

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

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

L-spaces.

Math. Japonica36

(1991),

225-228.

2. M. Altman,

An

integraltestforseriesandgeneralizedcontractions. Amer. Math. Monthly

82

(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

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TI

topologies. Proc.

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Troy L.

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Gerald

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B.G.

Pachpatte, Commonfixedpoint theorems forcertainmappingsin

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J.

M.A.C.T.

13

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11.

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K. Ray

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(9)

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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

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

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