Hadar COMMON

Internat. J. Math. & Math. Sci.

VOL. 16 NO. 4 (1993) 733-736

733

COMMON STATIONARY POINTS FOR SET-VALUED MAPPINGS

M.S. KAHN

Departmentof Mathematics,Facultyof Science SultanQaboosUniversity, P.O.

Box

32486

A1-Khod,

Muscat,

Sultanate ofOman

K.R.

RAO

Department

of Mathematics,

D.A.R.

College Nuzvid-521201, KrishnaDistrict, AndhraPradesh, India

Y.J.CHO

Department

ofMathematics,

Gyeongsang

National University Jinju660-701, Korea

(Received January

30, 1992 andinrevisedform

May

25,

1992)

ABSTRACT.

Several theorems on stationary points for set-valued mappings have obtained.

Theseareimprovementsupon some earlierresults duetoFisher.

KEY

WORDS AND

PHRASES.

Generalized Hausdorff distance, nearly-densifying mappings, orbit,commonstationary points.

1991 AMS

SUBJECT CLASSIFICATION CODE.

54H25.

1.

INTRODUCTION AND

PRELIMINARIES.

In

this paper, we prove several common stationary point theorems for four set-valued mappings,whichareimprovements uponsomeearlier results obtainedbyFisher

[1], [2], [3].

Let (X,d) beametric space and CL(X) be the class of allnonemptyclosed subset of X. For

z

x

^andAc_X, let _D(z,A) iny{d(z,y):yeA}.

DEFINITION 1.1. For A,B CL(X), define

H(A’B)=fmaz{suPreAD(z’B)’I suPs’BD(A’Y)}’

^ifitexists,

too, otherwise.

Then//iscalled the getera//zed

Hadar

^d/.Cm:ejmct/an for the classCL(X) inducedby the metricd.

DEFINITION 1.2.

For

A,B^6_CL(X), define h:CL(X)^xCL(X}--.I + by

sup{d(x,y)

^{x A,y B},} if it exists, h(A,B)

too, otherwise.

DEFINITION

1.3.

A

set-valued mapping S:X-.CL(X) is said to be nettr/y-dem.j// if ct(q(A))<ct(A) for any bounded and $-invariant subset of X with a(A)>0, where a is the Kuratowski’smeasureofnon-compactness.

DEFINITION

1.4. Let F,G,S,T X-.CL(X) be set-valued mappings. For^somez X, define thearb/tO(z) of^rby

O(z) {y X:y zory=/’(z) forsome

"

^},

734 M.S. KHAN, K.P.R. RAO, AND Y.J. CHO

’Y being the subsemigroup generated by F,G,Sand Tin the semigroup of all self-mappings on X withcomposition operation.

DEFINITION 1.5.

A

point issaid tobea common

stionarll

pointof set-valued mappings Fand

2.

THE MAIN RESULTS.

Throughout this paper, for any set-valued mapping S:X-,CL(X), we assume that all the powers of^$ map X intoCL(X). First ofall,we provethefollowing ^{crucial result tobeused} in thesequel.

LEMMA

2.1. Let (X,d)beacompactmetricspace and S:X-CL(X) beaset-valued mapping such thatS’ is continuous with respecttothegeneralized Hausdorffdistance function Hforsome positiveinteger^i. If‘4

nF= 1Sk’(X),

then S(‘4) .4.

PROOF. Clearly,SIk+

)’(x)

c

Sk’(X)

^for/ 1,2,. Also, c

x

implies

S C_ A.

(1.1)

Let e

^A. Then e

$(

⁺

)’(X)

for k 1,2,

.,

and so thereexists

S’,(X)

such that y

S’x

for

:

1,2,.... Since

x

is compact, thereexistsaconvergent subsequence

{,}

^{of{rk} withthe}

limitz. Further, since{zj,zj+

,

^c_

SJ,(X)

^forj 1,2,

.,

wehavezeA. Also,wehave D(y,S’z)<D(y,

S’zt ^{+ H(S’zt,Sz}

^).

Letting I-0, we get ySiz. Hence there exist z,,zi_,.

.,zX

^{such that} ySt,, z,(Sz,_,..-,z3(SSz2, ^and

zSz. ^{By (1.1),}

^{sincezA, it} follows that SzC_Aand so

zA. ^A

repeatedapplication of

(1.1)

yields that z (A.

Therefore,

^wehave St forsomez A. Thus, Ac_S(A). Fromthisand (1.1),weconclude that S(A) A. Thiscompletestheproof.

Now,

wearein apositiontopresentourmainresults. Wedenote

M(z,y,FP,Gq,SS,T

t)

rnaz{h(SSz,Tty),h(SSz, FPz),h(Tty,Gqy), h(SSz,Gqy),h(Tty, Ft’z)}

and

m(z,y, Ft’,Gq,SS,T

t)

maz{h(SSz,Tty),h(SSz, Gqy),h(Tty, F’z)}, where,q,sand arepositivefixedintegers.

THEOREM 2.1. Let (X,d)beacompletemetric spaceand F,G,S,T:X-,CL(X)beset-valued mappings such that

(2.1)

F,G,S,T and (FG) are continuous with respect to the distance function H for some positive integer^i. Also,F,G,SandTarenearly-densifying,

(2.2)

^forsomet X, theorbitO(zo)isbounded,

(2.3)

H(FPz,Gqy)<M(z,y,FP,Gq,Ss,

Tt),

(2.4)

^FG GF,(FG)’S" S’(FG) ^and

(FG)iT Tt(FG) .

ThenF, G,SandThaveauniquecommonstationary point in X.

PROOF. PuttingA O(zo), ^wehaveclearlyI(A)= A for I {F,G,S,T}. Also, the continuity of set-valued mappings F,G,S and T yields that

I()c_

^{for I} {F,G,S,T}. Further, we have

A={ro}UF(A)UG(A)US(A)UT(A

). Thus,

a(A)=maz{a(zo),a(F(A)),a(G(A)),a(S(A)),a(T(A))

and also is compact.

Now,

define B

t3=(FG)’-().

^{Then B is} compact.

By Lemma

2.1, (FG)(B) Band the condition

(2.4)

^ensures that F(B) B G(B), S’(B)C_ Band

Tt(B) c_

B. Since B is compact, there exist z,z e^{B such that}

d(z,z)=

^{stp{d(z,y) z,y}

e

B}={B), say. Also, thereexist w,wa^q.Bsuch that

z

^e

FPw

^and

_ Gqw. Suppose

that (B) >0. Then,by

(2.3),

^we

COMMON STATIONARY POINTS FOR SET-VALUED MAPPINGS 735 have

<M(wl,

w:,

^F

’

^{Gv,S T}

⁾

_< (B),

which is a contradiction. Thus, (B)=0 and hence B {z}, say. Therefore, z is a common stationary point of F,G,S and T. The uniqueness of z follows from condition

(2.3).

^This completestheproof.

THEOREM 2.2.

Let

(X,d) beacompactmetric space andF,G,$,T:X-,CL(X) beset-valued mappingssuchthat

(2.5)

(FG)’is continuousforsomepositive integeri,

(2.6)

H(F’z, Gqy) < M(z,y,F,Gq, S’,T

t)

whenever theleft-handsideis positive,

(2.7)

^FG

GF,(FG)iS"

S’(FG) and

(FG)iT Tt(FG) i.

Then F,G,S and T have a unique common stationary point z in X. Further, z is the unique commonstationary point ofFandG.

PROOF. IfweputB

n= ^1(FG)n(X),

^{asin theproofof}Theorem 2.1,wehaveB {z) and

zis auniquecommon stationary point of F,G,S andT. Since anycommon stationary point ofF

ad

G is a point ofB {z}, ^it follows thatzis the uniquecommonstationary pointofF andG.

Thiscompletestheproof.

REMARK.

Theorem 2 of Fisher

[2]

and theorems in Fisher

[3]

^followas corollaries ofour Theorem 2.2.

In

fact,ourtheoremcanbe

regarded

asan improvementover the above theorems due to Fisher.

THEOREM

2.3.

Let

(X,d) beacompletemetricspace and F,G,S,T:X-,CL(X) be set-valued mappings such that

(2.8)

^{F,G,S,T,F and G are continuous with respect to the} distance function H for some positive integers andj. Also,F,G,SandT arenearly-densifying,

(2.9)

forsomez X, the orbitO(zo)^isbounded,

(2.10)

H(FPz, GqF) < m(z,F,FP,Gq,S’,T

t)

^{whenevertheleft-handsideis}positive,

(2.11)

^SF

FiS"

and

TtG

^{q GqT}

t.

ThenF, G,Sand Thaveaunique common stationary pointzin X.

PROOF. Let

A O(zo). ^Thenasinthe proof of Theorem 2.1, iscompact. Ifwedefine

lFin ^1Gin

B

t=

^(A)andK

t=

^(A),

by Lemma2.1, F(B)=Band G(K)=K.

Also,

^itfollows that BandK arecompactsubsets ofX.

By

the condition

(2.11),

^{alsowe have} $(B)_B and

T(K)C_

K.

Then,

^thereexistzl, B and

yl,y Ksuch that

d(Zl,Yl)

sup{d(z,y):z B,I/K) (B,K), say,

with

z FPwl

^{andy Gqw}T

Suppose

that (B,K) >0. Then, bythe condition

(2.10),

^wehave

5(B,K)

_

d(Zl,yl)H(FPwI,Gqw)

< m(Wl, w2,F^pG

q,

S

,

^T

<_,(,K),

736 M.S. KHAN, K.P.R. RAO AND Y.J. CHO

which is acontradiction. Therefore, ,S(B,K)=0and B K {z}. Thus zis a commonstationary point of F,G,S and T. The uniqueness of follows easily from the condition

(2.10).

^This

completestheproof.

THEOREM 2.4. Let (X,d) be acompactmetricspace and F,G,S,T:X--,CL(X) be set-valued mappings such that

(2.12)

F’ and GJ are continuous with respect to the distance function H for some positive integers and j,

(2.13)

H(Ft’x,Gqy) < m(x,y, Ft’,Gq,SS,T

t)

whenevertheleft-hand sideis positive,

(2.14)

^{F’S S’F’andGqT}

TtG

^q.

Then F,G,S and T have a unique common stationary point in X. Further, z is the unique common stationary point of the pairs F,S and G,T. Also, is the unique common stationary point ofFandG.

PROOF. Let

B

r,= 1F’"(X)

^andK

r.= IGJ"(X).

^{Then asinthe}proofofTheorem 2.3,

we get B K {z} and is a unique common stationary point of F,G,S and T. Since any stationary point ofF is a point ofB {z} and anystationarypoint ofG is a point ofK {z}, it follows that is the unique stationarypointofFaswellasofG. Thiscompletestheproof.

REMARK.

Theorem 5 of Fisher

[1]

followsasacorollaryofourTheorem 2.3.

ACKNOWLEDGEMENT. We

aregratefultoProfessorB. Fisherfor providingusreprints of his papers which motivated the present work. Thanks are also due to Professor

S. Sessa

and the referee for theirusefulcommentsontheearlier versionofourpaper.

REFERENGE$

FISHER, B., Four

mappingswithacommonfixedpoint,J. Univ. Kuwait

(Sci.),

⁸

(1981),

131-138.

2.

FISHER, B.,

Three mappingswithacommonfixed point, Math. Sem.

Notes,

Kobe Univ., 10

(1982),

298-302.

3.

FISHER, B., A

commonfixed point theorem for four mappings on acompactmetricspace, Inst. Math. Acad. Sinica, 12

(1984),

^249-252.