Involutions with fixed points in 2 Banach spaces

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VOL. 16 NO. 3 (1993) 429-434

429

INVOLUTIONS WITH

FIXED POINTS IN

2-BANACH

SPACES

M.S.KHAN

Department

ofMathematics andComputing Collegeof Science, SultanQaboosUniversity

P.O.Box 32486, Alkhod,

Muscat,

Sultanate of

Oman

and

M.D.KHAN

Department

of Mathematics, Faculty of Science Aligarh Muslim University,Aligarh- 202002,India

(Received January

28, 1992 andin revisedform April 2,

1992)

ABSTRACT.

Some results on fixed points of involution maps in 2-Banach spaces have been obtained. These are extensionsof thoseproved earlier by Goebel-Zlotkiewicz, Sharma-Sharma, Assad-Sessa andIeki.

KEY WORDS

AND

PHRASES.

Involutions, 2-Banachspaces, coincidencepoints,fixedpoints. 1991

AMS

SUBJECT

CLASSIFICATION

CODE. 54H25.

1.

INTRODUCTION.

Ghler

([1]-[3])

initiated theconceptsof2-metric and 2-Banach spaces ina series of papers. These new spaces have subsequently been studied by several mathematicians in recent years. Like other spaces, the fixed point theory has also been developed in the frame work of these spaces. Itwas Ieki

([4],

[5])

who for the first time, obtained basicresults onfixed points in 2-metric and 2-Banach spaces. Since then quite a number of authors have extended and generalized fixed point theorems of Ieki and various other results involving contraction type mappings. Foranextensivebibliographyone isreferred to Ieki

([6]).*

In

thisnote, somefixedpoint theorems forcertain involutions in2-Banach spaceshavebeen obtained whichcan beviewed as a2-Banachspace extensionofaresult duetoAssad and Sessa

[7],

which in turngeneralizes afixed point theorem of Goebel and Zlotkiewiez

[8]

concerning an

involution ofa closed convex subset of a Banach space. The work of Assad and

Sessa

[7]

was

inspired by the contractive condition introduced by Delbosco

[9].

Our result also generalizes a

theorem of Sharma and Sharma

[10].

Itis important tonote thatin our proofcontinuity of the map under considerationis not essentiallyneeded, and hence thesameisunnecessarily stringent inSharmaand Sharma

[10]

andIkeki

[5].

2.

PRELIMINARIES.

We assumethe familiarity withthebasic theory of 2-Banach spacesas given inWhite

[11].

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The followingnotions areessentially duetoGhler

[1].

DEFINITION 2.1. Let

x

bealinear space, and

.,-II

beareal-valuedfunction definedon

x.

Thenthe pair (x

II-,.

II)iscalledalinear2-normedspaceif, fora,b,cCX,

(i)

a,b -0ifandonlyifaand arelinearlydependent,

(ii)

a, b b,a

II,

(iii)

a,b

I1

a,b

II,

being real,

(iv)

Ila, b+cll_< Ila, bll+lla,

cll.

Here

.,-II

iscalleda2-normandisanon-negative function.

DEFINITION 2.2.

A

sequence {Zn} in a linear 2-normed space X is called a convergent aequence if there is an element zeX such that the

I,/

xn-

z, 0 for all

converges to-z, we writezn---,z and call zthe limit of

{xn}.

Ofcourse, here dimX_>2 otherwise everysequenceof pointsinsuchaspacewould convergeto every pointof thespace.

DEFINITION 2.3.

A

sequence

{zn}

in a linear 2-normed space X is called a Cauchy sequence if

m,limn--,o

_tm in, 0forevery X.

DEFINITION

2.4.

A

linear2-normed spacein whicheveryCauchy sequenceis aconvergent sequence is calleda2-Baaachapace.

Wealso need the followingnotionfrom Assad andSessa

[7].

Let be the family ofcontinuousfunctions

:ta+--.+,

(where

t+

stands for thesetof

non-negative

reals)

satisfyingthefollowingconditions:

(i)

@(1,1,1) k<2,

(ii)

fors_>0,t_>0, the inequalitys<_ (t,2t, s) implies thats_<:t.

3.

RESULTS.

Throughout this section, X stands for a 2-Banach space with dimX>_2, and I denotes the identitymapon

x.

THEOREM3.1. LetTbeaself-mappingofXand such that

(A)

T2=

I,

(B)

Tz- Ty,a <@( z- y,a[[, z Tz,a[[, y-Ty,a ), for all z,y,ain X. Then Thas at leastonefixedpoint.

PROOF. Let z bean arbitrary point in X. Put y

Tz

+

z), z Ty and u 2y-z. Itis easytoobserve that

Nowwehave

2 Tz I1,a Tz,a 2

,,

II.

and also

z,.

T2:

T!l,a

<( Tz-//,a

II,

Tz

T2z,

a

II,

V-Tll,a[[) (

=

Y, a

II,

2

=

Y, a

II,

Y-Ty,a

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2-BANACH

Ontheotherhand,wehave

Hence

By

the hypothesis

(ii),

weobtain

LetusputGr

(Tr

+

t), for yzin X. Then by the foregoing inequMity,weget

Y T,a

forM1

,.

in X.

Now,

for bitry

int

_%iX, le

ao

a_

1,n 1,2, I] m n 1, then

=m-

=.,a

Gmzo-Gnzo,

a

IIGmzo-Gm-Zo,

all

+...

+

G"+

;zo-G%o,

all

5 _/2

Go

.o,a

II.

From

this,it follows that

{}

isa Cauchysequencewhich convergesin

x.

Put z*=

Iz,.

ow

consider

z* -Gz*,a

II

<-

z*

.

+

,a

+

II

Gzn-Gz*,a

_<

IIx*-xn+

1,all

+1/211xn-*,all +1/211Tn-Tx*,all

<

I1*-%+

1,11

+ll=.-t*,all +(ll.-t*,all,

II.Tzn,

all,

II*-T=*,all)

I1=*-=+

,11

+11=-*,11

+(ll==-=*,ll,211(G=-=),ll,211(=*-G=*),ll)-ttingn,weget

2

*-G=*,all

5 (0,0,2

II

=*-G=*,

II

).

So

agn

bycondition

(ii),

weget

-G*,a

0, for fllainX.

Hence, (z*-Gz*)

d a e linely dependent for M1 a in X. Since dimX 2, the only way

(r*-Gz*)

c be linely dependent with M1 a in X, is that z*-Gz*=0. Hence r*=

Tz*

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KHAN AND M.D. KHAN

COROLLARY

3.1

(Sharma

and Sharma

[10]).

LetT:X--,Xbe such that T2 Iand

I[Tz-Ty,a[ <_a[[z-y,a[I +/(l[z-T,all

+

Ily-Ty,al[),

(3.1)

for all z,y,ain X, wherea_>0,/>0anda+4/ <2. ThenThas at leastonefixed point. PROOF. Thecondition

(3.1)

implies that

_<

+

(

+

2)

2

,- y,

II,

II

-

T,II, Y-Ty, }. Nowifwesumethat

then, by Theorem 3.1,That letonefixedpoint. Thiscompletes theprof.

REMARKS.

(a)

A

critical observation of the prfof the

mMn

threm in Sharma d Shma

[10],

reveMs

that theyhave used the continuity of theinvolutionmap but fMled tomentionthese.

However,

inoutprfthisadditioaMcontion isnot required.

(b)

When

x

is the

usuM

Bhspace, Corolly 3.1 reducedtoathrem ofIeki

[5].

In

a private communication Profesr Ieki agrd that the continuity of the involution map is essentiMly nded for hisprftohold.

COROLLARY

3.2. Under the hypothesis of Threm3.1, suppose, inaddition, thatat let

oneof thefollowingstrictinequMity holds:

*

T,a <

*

,

a

+

T,a

II,

(3.2)

*

,a <

*

T,a

+

Tz ,a

for allaand {

*)

in

x.

Then

*

isthe uniquefixedpoint ofT.

PROOF.

By

Theorem 3.1, T* *.

Suppose

also that Ty* y* for somey* X. A .ume

that

*

y*. Then using

(3.2},

wehave

II*-y*,=ll

II*-Ty*,=II

< II*-y*,=ll

+

IlY*-Ty*,all

which is impossible.

Therefore,

*=y*. Similarly, other condition in

(3.2)

also implie., that x* y*.

Now,

weapplyTheorem 3.1 toobtain a coincidencetheorem.

THEOREM

3.2. Let T and 5’be the self-mappings of

x,

and such that thefollowing hold:

(i)

T2 I,S2 I, TS ST,

(ii)

11T

Ty,a <( S- Sy,

II,

S-T,a

II,

Sy-Ty,a _{), for all} t,y,a in X. Then thereexistsatleastonepointt_oin

x

such thatTt_o St_o.

PROOF.

Itfollows from Theorem 3.1 that TS hasatleast onefixedpoint_%. Then clearly

T%

S%.

Thiscompletestheproof.

REMARK. In

case,oneassumessomeadditional conditionsonTS,asinCorollary 3.2, then

t_o inTheorem 3.2 becomes the unique fixed point ofTS. Then, commutativity of T,S and the uniqueness oft_o can be used to show that Zo is actually a common fixed point of S and T.

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FIXED POINTS IN 2-BANACH SPACES 433

ACKNOWLEDGEMENT. The first author is grateful to Professor S. Sessa for supplying the preprintof

[7],

which motivatedthepresent study.

REFERENCES

1. G/i,

HLER, S.,

Lineare 2-normierteRiume, Math. Nachr. 28

(1965),

1-43.

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G,HLER,

S.,

2-metrische R/iume undihre topologische Strucktur,Math. Nachr. 26

(1963).

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3.

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S.,

0bet

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(1965),

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K.,

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Kobe Univ. 3

(1975),

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K.,

Fixed point theorems in Banach spaces, Maths Seminar

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Kobe Univ. 2

(1976),

11-13.

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K.,

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(2)

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N.A.

&

SESSA, S.,

Involution maps and fixed points in Banaeh spaces, Math. J.

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Univ. 14

(1991),

141-146.

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K.

z

ZLOTKIEWICZ, E., Some

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(1971),

103-106.

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DELBOSCO, D., A

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&

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B.K.,

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