WORDS AND

Internat. J. Math. & Math. Sci.

VOL. 21 NO. 3 (1998) 467-h70

467

AN APPLICATION OF FIXED POINT THEOREMS IN BEST APPROXIMATION THEORY

H.K.PATHAK

Department

of Mathematics

KalyanMahavidyalaya Bhilai

Nagar (M.P.)

^490006,

^INDIA

Y.J.CliO

Department

of Mathematics

Gyeongsang

NationalUniversity

Chinju 660-701,

KOREA

S.M. KANG

Department

of Mathematics

Gyeongsang

NationalUniversity

Chinju660-701, KOREA

(Received

February7,1996 andinrevised formJune 18,

1996)

ABSTRACT. In

thispaper, we give anapplication ofJungck’sfixed point theoremto best ap- proximationtheory, whichextends theresultsof Singh and Sahabetal.

KEY WORDS AND PHRASES:

Contractive operator, best approximant, compatiblemap- pings,fixedpoint.

1991 AMS SUBJECT

CLASSIFICATION

CODES: 54H25,47H10.

Let X

beanormedlinear space.

A

mapping

T X X

issaidtobe contractweon

X (resp.,

onasubset Cof

X)

^if

IITx- ^{Tyll <_} IIx Yll

^{for allx,yin}

^{X (resp., C).}

^Thesetoffixed points of

T

on

X

isdenotedby

F(T).

If is apoint of

X,

thenfor 0

<

^a

_<

1, wedefine theset

Da

^{of best}

(C,

a)-approximantsto consistsof thepoints y in

C

such that

Let

D

denote thesetofbest C-approximantsto

. ^For

^{a 1,ourdefinitionreducestotheset}

^D

^of

best C-approximantsto

. ^A

^subset

^C

^of

^X

^{issaidtobe starshapedwith,respectto apoint q E}

^C

if, forall xinCandall

A

5

[0,1],

Az

+ (1 A)q

C. The point piscalled the star-centre of

C. A

convexset isstarshapedwithrespect to each ofitspoints, butnotconversely. Foranexample, the setC

{0} [0,1]

^LI

[1, 0] {0}

^{isstarshapedwith}respectto

(0, 0) e ^C

^asthestar-centreof

C,

but it is not convex.

In

thispaper, we give anapplication ofJungck’sfixed point theorem to best approximation theory, which extends the results ofSahabetal.

[9]

andSingh

[10].

By

relaxing the linearity of the operator

T

and the convexity of

D

inthe originalstatementof Brosowski

[1],

Singh

[10]

proved the following:

Theorem 1. Let C be^a T-invariant subset ofanormedlinear space

X. Let T C C

bea contractiveoperatoron

C

and let

F(T).

^If

D c_ X

isnonempty, compact and starshaped, then

D

f

F(T) 0.

In

thesubsequent paper

[11],

Singh observed that only thenonexpansivenessof

T

on

D’ DU{}

isnecessary. Further,Hicks and Humphries

[4]

have shown that the assumption

T" C C

canbe weakenedtothe condition

T" OC C

if y

C,

i.e., y E

D

isnotnecessarilyintheinteriorof

C,

where

OC

denotestheboundary of

C.

Recently,Sahab,Khanand

Sessa [9]

generalized Theorem as inthe following:

468 H. K. PATHAK, Y. J. CHO AND S. M. KANG

Theorem 2. Let

X

beaBanach space.

Let T, I X X

beoperatorsand Cbe asubset of

X

such that

T" OC

Cand5:

F(T)f3 F(I).

Further,suppose that

T

and

I

satisfy

(1)

for allx,yin

D’, I

islinear, continuous on

D

and

ITx TIx

for allx in

D.

If

D

isnonempty, compact and starshaped with respectto apoint q

F(I)

and

I(D) D,

then

Df3F(T)f3F(I) .

Recallthattwoself-maps

I

and

T

ofa metricspace

(X, d)

^with

d(x, y) IIx- Yll

^forallx,y

^X

aresaidtobe compatibleon

X

if

.h_m ^{d(ITx., TIx. )(=} .h_rn ^{IlITz. Tlz.ll)}

⁰

wheneverthereisasequence

{x. }

ⁱⁿ

^X

such that

Tx., Ix.

^{t,as n} oo, forsome tin

X ([6]-[8]).

We

shalluse

N

todenote thesetof positiveintegersand

CI(S)

^todenote theclosure ofa set

S.

For

our maintheorem,weneed the following:

Proposition 3.

[8] Let T

and

I

be compatible self-maps ofametric space

(X, d)

^with

I

being continuous.

Suppose

that thereexistreal numbersr

>

0 anda

(0,1)

such thatfor allx,y

X,

d(Tx, Ty) <_ rd(Ix, Iy) +

^a

max(d(Tx, Ix), d(Ty, Iy)}.

,Then

Tw Iw

forsomew

X

if andonly if

A f3{Cl(T(Ko))

n

N} #

$, where for each

go {

^x

X" d(Tx, Ix) <_

- ^}.

On

theotherhand,usingthis proposition,Jungck

[8]

proved thefollowing:

Theorem4.

Let I

and

T

be compatibleself-mapsofaclosedconvexsubsetC ofaBanach space X.

Suppose

that

I

is continuousand linearwith

T(C) c_ I(C).

Ifthereexistsana

e (0,1)

^such

thatfor all x, y

C,

IITx ^{T,II _<} alllx lull + ^{(1 a) max{llTx} lxll, IITy- IulI}, (2)

then

I

and

T

haveauniquecommonfixed pointin

C.

By

using thistheorem,^weextend Theorem2asinthe following:

Theorem 5. Let

X

beaBanach space.

Let T, I X X

beoperators and

C

beasubsetof

X

such that

T"

c9C

C

and^5:

F(T)N F(1).

^Further,suppose that Tand

I

satisfy

(2)

for all x,y in

D’ ^Do

^t3

^{5:}

^U

^E,

^where

^{E {q X Ix=,Tx.}

^q,

^{xo}

^C

^Do},

⁰

^<

^a

^<

^1,

^I

^{is linear,}

continuous on

Do

^and

^{T, I}

^{arecompatible in}

D.

If

D

^is nonempty, cbmpact and convex, and

I(D) Do,

then

D

^f

^F(T)

^f

F(I) .

Proof.

Let

y

D

^andhence

^Iy

^isin

D

^since

I(D) D.

^{lurther, if y}

OC,

then

Ty

is in

C

since

T(OC) ^{c C. From} (2),

^itfollows that

[ITy- 5:[] ][Ty-

< allly I5:11 + (1 a) max{llTy I11, IITS:

which implies

a]]Ty 5:1] <IIIy- 11

^andso

T

^{is in}

D.

^Thus

T

maps

Do

^intoitself.

By

hypothesis,wehave5:

TS:

I5:. ThenProposition 3 implies that

A {CI(T(K.))’n N} # 0.

Suppose

thatw 6

A.

Then foreachn

N,

thereexists y.

T(Ko)

such that

d(w, I/o) < 1/n.

Consequently,for such n,we canand dochoose

x. K.

^{such that}

d(w, Tx.) < 1In

^andso

^Tx.

^w.

But

since

x.

⁶

K., d(Tx., Ix.) < 1/n

^andtherefore

Ixo

^w. Thuswehave

lira

Iz,

lira

Tx,

w.

(3)

FIXED POINT THEOREMS 69

Therefore,forasequence

{z }

ⁱⁿ

D=

^{theexistenceof}

(3)

^isguaranteed whenever

D=

^C

K,. Moreover,

w 6

E.

Since

I

and

T

are compatible and

I

is continuous, we have lirn_am

TIz,, Iw

and lim._am

I:x,., Iw. By (2),

wehave

IITIx. Y:ll ^{[]TIx. Vl] < al]Ix. I]1} + (1 a)max{I]Tlx.

whichimplies,asn

IIIw ll -< alllw- ll.

Hence Iw . ^{By (2)}

^{again, wehave}

w

gi

IIT- 11-< ^{(- )IIT- 11,}

^and

^T .

Next,

weconsider

IlTw T.II ^< allIw Ix-II + (1 -a)max{llVw -/toll, IITx.

which gives

II wll ^< all wll

^{asn oo, andso w, i.e., w}

^{Iw Tw. By}

^{Theorem4, w}

mustbe unique.

Hence E (w}.

Then

D; D

^U

(w} D’

Let {k. }

beamonotonically non-decreasing sequence of real numbers such that 0

_< k. <

1 and

li".am k.

^1.

Let {x,}

^beasequencein

D’

^satisfying

^{(3). For}

^eachn

^{e N,}

^{defineamapping}

T,’D’,, D’,,

by

T,,x, k.Tx, + (1- k.)p. (4)

Itispossibletodefine suchamapping

T,

for eachn

N

since

D’

^{isstarshapedwithrespectto}

p6F(I).

Since

I

islinear,wehave

T.Ix, k.TIx, + (1 k.)p, IT,x, k.ITx, + (1 k,.,)p.

By

compatibility of

I

and

T,

wehavefor eachn q

N,

0

<_

lim

liT. Ix, IT. x,

< k.

^lira

IIrlz, ITxjI +

^lira

(1 -/)llp-

=0 andso

lira lIT. ^{Iz, IT. z,} o

whenever

lim,._.am

Ixj

lim3_am T.x

^{w since wehave}

lira

T,.,x,

^{k, lira}

Tx, + (1 k,.,)w

Thus,

I

andT,arecompatibleon

D’

^{for eachnand}

^T,,(D’)

^C

^{D’,, I(D).}

On

the otherhand,by

(2),

for all x, y6

D’,

^{wehave,for allj}

^>

^nandnfixed,

liT: T.II

<_ alllz- I11 + (1 -a)max{llT- I11, IIT- ^I11}

<_

(X k.)llT- Pll

^/

IIT.- I11}-

470 H.K. PATHAK, Y. J. CHO AND S. M. KANG

Hencefor allj

_

^n,wehave

liT,a:- T.Yll ^< alllx- ^Iyll +

+ IIT,z ^{lall, (1 k,)llTy} Pll + ^{IIT.,y IylI}}

Thus,since

lim3_00 k

^{1, from}

^(5),

^foreverynE

^N,

^wehave

<

3--*00

+

which implies

()

liT,a: T.,yll- allIx -/Yll

^/

(- a) max{llT.,x Xxll, IIT. ^I11}

for allx,y E

D.

Therefore, by Theorem 4, forevery n

N, T,

and

I

haveaunique common fixed pointx,in

D’,

^{i.e., forevery n}

N,

^wehave

F(T,)

^N

F(I) {x, }.

Now,

the compactness of

D=

^ensuresthat

{x }

^hasaconvergentsubsequence

{x,,, }

which converges to apointzin

D=.

Since

x,,, T,, x, k,,Tx,,, + (1 k,,)q (6)

and

T

is continuous, wehave,as xin

(6),

^z

Tz,

i.e., z

D=

^f3

F(T).

Further,the continuityof

I

implies that

Iz I(,li_m ^z,.,,)= ,lim ^Ix,.,, ,li.m_ ^z,.,,

^z,

i.e.,z

F(I).

Therefore,^wehave z

D ^{a F(T)}

^N

F(I)

andso

D F(T) F(I) # .

Thiscompletes the proof.

ACKNOWLEDGEMENT.

The first authorwassupportedinpartby

U.G.C., New

Delhi, India, and thesecond and third authorsweresupportedinpartby the Basic ResearchInstitute

Program,

Ministry of Education,

Korea,

1996, Project

No.

BSRI-96-1405.

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