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COMMON STATIONARY POINTS OF MULTIVALUED MAPPINGS ON BOUNDED METRIC SPACES
ZEQING LIU and SHIN MIN KANG
(Received 31 August 1999 and in revised form 17 March 2000)
Abstract.Necessary and sufficient conditions for the existence of common stationary points of two multivalued mappings and common stationary point theorems for multival- ued mappings on bounded metric spaces are given. Our results extend the theorems due to Fisher in 1979, 1980, and 1983 and Ohta and Nikaido in 1994.
Keywords and phrases. Common stationary point, multivalued mappings, complete bounded metric space.
2000 Mathematics Subject Classification. Primary 54H25.
1. Introduction. Let(X,d)be a metric space andB(X)denote the set of all nonempty bounded subsets ofX. ForA,B∈X, letδ(A,B)=sup{d(a,b):a∈A, b∈B}and δ(A)=δ(A,A). IfAconsists of a single pointa, we writeδ(A,B)=δ(a,B). IfBalso consists of a single pointb, we writeδ(A,B)=δ(a,b)=d(a,b). LetNandωdenote the sets of positive integers and nonnegative integers, respectively. LetΦ denote a family of mappings such that for eachφ∈Φ, φ:[0,∞)→[0,∞)is upper semicontin- uous, nondecreasing andφ(t) < tfort >0.
The following definitions and lemmas were introduced by Fisher [3] and Singh and Meade [6].
Definition1.1[3]. Let{An}be a sequence of sets in B(X) and A∈B(X). The sequence{An}is said toconvergeto the setAif
(i) each pointa∈Ais the limit of some convergent sequence{an}, wherean∈An
forn∈N;
(ii) for arbitrary >0,there existsk∈Nsuch thatAn⊆A, forn > k, whereA
is the union of all open spheres with centres inAand radius.
Definition1.2[3]. LetF be a multivalued mapping of(X,d)intoB(X).The map- pingF is calledcontinuousinXif whenever{xn}is a sequence of points inX con- verging tox∈X, the sequence{Fxn}inB(X)converges toFx∈B(X).
Lemma1.3[3]. If{An}and{Bn}are sequences of bounded subsets of a complete metric space(X,d)which converge to the bounded subsetsAandB, respectively, then the sequence{δ(An,Bn)}converges toδ(A,B).
Lemma1.4[6]. Letφ∈Φ.Thenφ(t) < tfor eacht >0if and only if
n→∞limφn(t)=0, (1.1)
whereφndenotes then-times composition ofφ.
Let F and G be mappings of (X,d) into B(X). A point x ∈ X is called a com- mon stationary pointof F and G if Fx =Gx= {x}. ForA⊆X, let FA= ∪a∈AFa and GFA=G(FA). The mappingsF and G are said tocommute ifFGx=GFxfor x∈X. Define CF = {T : Tis a mapping ofXintoB(X)andT andF commute} and CCF = {T :T is continuous andT ∈CF}. It follows that CF ⊇ {Fn:n∈ω}, where F0x= {x}forx∈X.
Throughout the rest of the paper, we assume that(X,d) is a complete bounded metric space.
In 1979, Fisher [1] proved a common fixed point theorem for commuting mappings fandgof(X,d)into itself satisfying
d(f x,gy)≤cmax
d(x,y),d(x,f x),d(y,gy),d(x,gy),d(y,f x)
(1.2) for allx,y∈X, where 0≤c <1.
In 1980, Fisher [2] generalized the result to multivalued mappingsF andGof(X,d) intoB(X)satisfying the condition
δ(Fx,Gy)≤cmax
δ(x,y),δ(x,Fx),δ(y,Gy),δ(x,Gy),δ(y,Fx)
(1.3) for allx,y∈X, where 0≤c <1.
In 1983, Fisher [4] established a common fixed point theorem for continuous, com- muting mappingsFandGof(X,d)intoB(X)satisfying
δ
FpGpx,FpGpy
≤cmax δ
FqGrx,FsGty ,δ
FqGrx,FsGtx , δ
FqGry,FsGty
: 0≤q,r ,s,t≤p (1.4) for allx,y∈X, where 0≤c <1 andpis a fixed positive integer.
In 1994, Ohta and Nikaido [5] obtained the existence of fixed point for a continuous self mappingfof(X,d)satisfying
d
fkx,fky
≤cδ
fit:t∈ {x,y}, i∈ω
(1.5) for allx,y∈X, where 0≤c <1 andkis a fixed positive integer.
The first purpose of the paper is to establish criteria for the existence of common stationary points of commuting mappingsF andGof(X,d)intoB(X).The second purpose of the paper is to prove common stationary point theorems for commuting mappingsF andGof(X,d)intoB(X)satisfying one of the following:
δ
FpGpx,FqGqy
≤φ δ
∪D∈CFGD{x,y}
(1.6) for allx,y∈X, whereφ∈Φandp,qare fixed positive integers;
δ
Fpx,Gqy
≤φ δ
∪D∈CF∩CGD{x,y}
(1.7) for allx,y∈X, whereφ∈Φandp,qare fixed positive integers;
δ(Fx,Gy)≤φ max
δ(x,Fx),δ(y,Gy),δ(x,Gy),δ(y,Fx),δ
∪D∈CCFGD{x,y}
(1.8) for allx,y∈X, whereφ∈Φ.
It is easy to see that (1.2) and (1.3) are special cases of (1.8), that (1.4) and (1.5) are special cases of (1.6), and that (1.2) and (1.5) are special cases of (1.7). Our results extend and unify the theorems of Fisher [1, 2, 4] and Ohta and Nikaido [5].
2. Common stationary points. Our main results are as follows.
Theorem2.1. LetF andGbe continuous and commuting mappings of(X,d)into B(X). Then the following statements are equivalent:
(i) F andGhave a common stationary point;
(ii) there existS,T∈CCF∩CCGwithS∈CT andφ∈Φsuch that δ(Sx,T y)≤φ
δ
∪D∈CCS∩CCTD{x,y}
∀x,y∈X; (2.1)
(iii) there existS,T∈CF∩CGwithS∈CTandφ∈Φsuch that δ(Sx,T y)≤φ
δ
∪D∈CS∩CTD{x,y}
∀x,y∈X; (2.2)
(iv) there exist mappingsS,T of(X,d)intoB(X)withS∈CT andφ∈Φsuch that F,G∈CSTand
δ(Sx,T y)≤φ δ
∪D∈CSTD{x,y}
∀x,y∈X. (2.3)
Proof. We shall verify the following implications: (i)⇒(ii)⇒(iii)⇒(iv)⇒(i). Suppose, first of all, thatF andGhave a common stationary pointz. Define mappingsS and T of(X,d)intoB(X)bySx=T x= {z}for allx∈X.It is easy to check thatS,T∈ CCF∩CCGand
δ(Sx,T y)=0≤φ δ
∪D∈CCS∩CCTD{x,y}
(2.4) for allx,y∈X,φ∈Φ, that is, (ii) holds.
Note thatCCF⊆CF andCS∩CT⊆CST. Therefore (ii)⇒(iii)⇒(iv) are clear.
We now assume that (iv) holds. Then for anyA,B∈B(X), we have δ(SA,T B)≤φ
δ
∪D∈CSTD(A∪B)
, (2.5)
by (iv). SinceXis bounded,M=δ(X) <∞.SetXn=SnTnXforn∈N.ThenXn⊆Xn−1
forn∈N.We now will prove by induction that δ
Xn
≤φn(M) forn∈N. (2.6)
Note thatSandT commute. From (2.5), we have δ
X1
=δ(ST X,T SX)≤φ δ
∪D∈CSTD(T X∪SX)
≤φ δ(X)
=φ(M), (2.7) that is, (2.6) holds forn=1. Assume now that (2.6) holds for somen∈N.It follows from (2.5) that
δ Xn+1
=δ
Sn+1Tn+1X,Tn+1Sn+1X
≤φ δ
∪D∈CSTD
SnTn+1X∪TnSn+1X
=φ δ
∪D∈CSTSnTn(DT X∪DSX)
≤φ δ
Xn
≤φn+1(M)
(2.8)
by our assumption. Hence (2.6) follows by induction. Choosexn∈Xnforn∈N.Then, by (2.6), we get
d xn,xm
≤δ Xn,Xm
≤δ Xn
≤φn(M) form > n. (2.9) Consequently,{xn}is a Cauchy sequence by Lemma 1.4. SinceXis complete, there exists a pointzinXsuch thatxn→zasn→ ∞. From (2.6), we have
δ z,Xn
≤d z,xm
+δ xm,Xn
≤d z,xm
+δ Xn
≤d z,xm
+φn(M) (2.10) form,n∈Nwithm > n. Lettingmtend to infinity, we obtain
δ z,Xn
≤φn(M) forn∈N. (2.11)
SinceF is continuous andxn→z, then{Fxn}converges to{Fz}. Note that
Fxn⊆FSnTnX=SnTnFX⊆Xn forn∈N. (2.12) Thenδ(z,Fxn)≤δ(z,Xn)forn∈N. Lettingntend to infinity, we haveδ(z,Fz)≤0 by (2.11) and Lemmas 1.3 and 1.4, that is,Fz= {z}. Similarly, we haveGz= {z}. This completes the proof.
Theorem2.2. LetF andGbe continuous and commuting mappings of(X,d)into B(X)satisfying (1.6) or (1.7). ThenF andGhave a unique common stationary pointz and the sequence{FnGnx}converges to{z}for allx∈X.
Proof. LetM=δ(X), k=p+q,Xn=FnGnXandxn∈Xnforn∈N. Note that everyn∈Ncan be written as
n=kj+i, (2.13)
wherej∈ωand 0≤i < k. Now we claim that δ
Xn
≤φj(M). (2.14)
If (1.6) is satisfied, then δ
Xn
=δ FpGp
Fq+iGq+iXk(j−1) ,FqGq
Fp+iGp+iXk(j−1)
≤φ δ
∪D∈CFGD
Fq+iGq+iXk(j−1)∪Fp+iGp+iXk(j−1)
=φ δ
∪D∈CFG
Fk(j−1)Gk(j−1)Fq+iGq+iDX∪Fk(j−1)Gk(j−1)Fp+iGp+iDX
≤φ δ
Xk(j−1)
(2.15) which implies that
δ Xkj
≤φ δ
Xk(j−1)
≤ ··· ≤φj−1 δ
Xk
≤φj(M). (2.16) Note thatXn⊆Xn−1.Thus (2.14) follows from (2.15) and (2.16). If (1.7) is satisfied, then
δ Xn
=δ Fp
Fq+iGk+iXk(j−1) ,Gq
Fk+iGp+iXk(j−1)
≤φ δ
∪D∈CF∩CGD
Fq+iGk+iXk(j−1)∪Fk+iGp+iXk(j−1)
≤φ δ
Xk(j−1) .
(2.17)
Similarly, (2.16) holds also. It follows from (2.16) and (2.17) that (2.14) holds.
Givenxn∈Xnfor alln∈N. For anym > n > k, by (2.13) and (2.14) we have d
xn,xm
≤δ Xn
≤φj(M). (2.18)
As in the proof of Theorem 2.1, we conclude thatFandGhave a common stationary point zand thatxn→z asn→ ∞.Suppose that F and G have a second common stationary pointw. Then{u} =FnGnu⊆Xn foru∈ {z,w}andn∈N. In view of (2.13) and (2.14), we infer that
d(z,w)≤δ Xn
≤φj(M). (2.19)
Lettingntend to infinity we haved(z,w)≤0 by Lemma 1.4, that is,z=w. Hence F and G have a unique common stationary point z. Forx ∈X and n∈N, choose yn∈FnGnx.Using (2.13) and (2.14), we have
d yn,z
≤δ
FnGnx,z
≤δ Xn,z
≤δ Xn
≤φj(M). (2.20) Lettingntend to infinity, by Lemma 1.4 and Definition 1.1 and the above inequalities, we conclude that{FnGnx}converges to{z}. This completes the proof.
As a consequence of Theorem 2.2, we have the following corollary.
Corollary2.3. LetFandGbe continuous and commuting mappings of(X,d)into B(X)satisfying one of the following:
δ
Fqx,Gqy
≤φ δ
∪i,j∈ωFiGj{x,y}
∀x,y∈X, (2.21)
whereφ∈Φandp,qare fixed positive integers;
δ
FpGpx,FqGqy
≤φ δ
∪i,j∈ωFiGj{x,y}
∀x,y∈X, (2.22) whereφ∈Φandp,qare fixed positive integers. ThenFandGhave a unique common stationary pointzand the sequence{FnGnx}converges to{z}for allx∈X.
From Corollary 2.3, we have the following.
Corollary2.4[4, Theorem 1]. LetF andG be continuous and commuting map- pings of(X,d)intoB(X)satisfying (1.3). ThenF andGhave a unique common station- ary pointzand the sequence{FnGnx}converges to{z}for allx∈X.
Corollary2.5[5, Theorem 3]. Letfbe a continuous mapping of(X,d)into itself satisfying (1.5). Thenf has a unique fixed point zand for each x∈X,fnx→zas n→ ∞.
Theorem2.6. LetF andG be commuting mappings of (X,d)into B(X) satisfy- ing (1.8). ThenF andGhave a unique common stationary pointzand the sequence {FnGnx}converges to{z}for allx∈X. Further,{z} =Dzfor allD∈CCFG.
Proof. LetM=δ(X), Xn=FnGnX, andxn∈Xn forn∈N. As in the proof of Theorem 2.1, we conclude thatδ(Xn)≤φn(M)forn∈Nand thatxn→z,δ(z,Xn)→
0 asn→ ∞.Consequently, the sequences{Xn}and{{z} ∪Xn}converge to{z}. For eachD∈CCFG,we haveδ(Dxn,z)→δ(Dz,z)asn→ ∞,by the continuity ofDand Lemma 1.3. Note that
δ Dxn,z
≤δ
Dxn,xn +d
xn,z
≤δ Xn
+d xn,z
≤φn(M)+d xn,z
→0 asn→ ∞, (2.23)
which implies thatδ(Dz,z)≤0, that is,Dz= {z}. Using (1.8), we have forn∈N, δ
FnGnX,Gz
≤φ max
δ
Fn−1GnX,FnGnX
,δ(z,Gz),δ
Fn−1GnX,Gz , δ
z,FnGnX ,δ
∪D∈CCFGD
Fn−1GnX∪{z}
≤φ max
δ
Xn−1,Xn
,δ(z,Gz),δ
Xn−1,Gz , δ
z,Xn ,δ
∪D∈CCFG
Fn−1Gn−1DGX∪Dz
≤φ max
δ Xn−1
,δ(z,Gz),δ
Xn−1,Gz , δ
z,Xn ,δ
Xn−1∪{z}
,
(2.24)
which implies that δ
xn,Gz
≤δ
FnGnX,Gz
≤φ max
δ(z,Gz),δ
Xn−1,Gz ,δ
z,Xn ,δ
Xn−1∪{z}
. (2.25) Lettingntend to infinity, we get
δ(z,Gz)≤φ max
δ(z,Gz),δ(z,Gz),0,0
=φ
δ(z,Gz)
. (2.26)
Suppose thatδ(z,Gz) >0. Then δ(z,Gz)≤φ
δ(z,Gz)
< δ(z,Gz), (2.27) which is a contradiction. Thereforeδ(z,Gz)=0, that is,Gz= {z}.Similarly we have Fz= {z}.The rest of the proof is exactly the same as that of Theorem 2.2. This com- pletes the proof.
From Theorem 2.6, we have the following corollary.
Corollary2.7[2, Theorem 2]. LetFandGbe commuting mappings of(X,d)into B(X)satisfying (1.3). ThenFandGhave a unique common stationary pointzand the sequence{FnGnx}converges to{z}for allx∈X.
Corollary2.8[1, Theorem 4]. Letfandgbe commuting mappings of(X,d)into itself satisfying (1.2). Thenfandghave a unique common fixed pointzand for each x∈X, fngn→zasn→ ∞.
The following example shows that Theorem 2.6 extends properly Corollaries 2.7 and 2.8.
Example2.9. LetX= {1,2,5,8}with the usual metric. Define self mappingsfand gof(X,d)by
f1=1, f2=f5=g1=g2=g5=5, f8=g8=2. (2.28) SetFx= {f x}andGx= {gx}forx∈X. Letφ(t)=(1/2)tfort≥0.It is easy to check thatFandGsatisfy the conditions of Theorem 2.6. But Corollaries 2.7 and 2.8 are not applicable since
d(f1,g1)=4=max
d(1,1),d(1,f1),d(1,g1),d(1,g1),d(1,f1)
, (2.29) that is,f andgdo not satisfy (1.2). SimilarlyF andGdo not satisfy (1.3).
Acknowledgement. The authors would like to thank the referee for his many helpful comments. The second author was supported by Korea Research Foundation Grant (KRF-99-005-D00003).
References
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[3] ,Common fixed points of mappings and set-valued mappings, Rostock. Math. Kolloq.
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[4] ,Common fixed points of set-valued mappings on bounded metric spaces, Math. Sem.
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Zeqing Liu: Department of Mathematics, Liaoning Normal University, Dalian, Liaoning116029, China
Shin Min Kang: Department of Mathematics, Gyeongsang National University, Chinju660-701, Korea
E-mail address:[email protected]
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