Results on common fixed points

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RESULTS ON COMMON FIXED POINTS

ZEQING LIU and JEONG SHEOK UME

(Received 9 October 2000)

Abstract.We establish common fixed point theorems related with families of self-mappings on metric spaces. Our results extend, improve, and unify the results due to Fisher (1977, 1978, 1979, 1981, 1984), Jungck (1988), and Ohta and Nikaido (1994).

2000 Mathematics Subject Classification. 47H10, 54H25.

1. Introduction. LetwandNdenote the sets of nonnegative integers and positive integers, respectively. Fort∈[0,∞),[t]denotes the largest integer not exceedingt. Letf andgbe mappings from a metric space(X, d)into itself andCf = {h:h:X→ Xandhf=f h}. Forx, y∈XandA⊆X, defineOf(x)= {fnx:nw},Of(x, y)= Of(x)∪Of(y), Of ,g(x)= {figjx :i, j w}, Of ,g(x, y)=Of ,g(x)Of ,g(y), and δ(A)=sup{d(x, y):x, y∈A}. LetiXdenote the identity mapping onX. It is easy to see that{fn:nw} ⊆Cf. LetΦ= {ϕ:ϕ:[0,∞)[0,∞)is upper semicontinuous and nondecreasing andϕ(t) < tfort >0}. The following definition and lemmas were introduced by Fisher [8], Singh and Meade [13], and Jungck [10], respectively.

Definition1.1(see [8]). LetAXandAnXfor allnN. The sequence{An}n∈N

is said to converge toAif

(i) each pointa∈A is the limit of some convergent sequence{an}n∈N, where

an∈Anfor alln∈N;

(ii) for arbitraryε >0, there exists an integerksuch thatAn⊆Aεforn > k, where is the union of all open spheres with centers inAand radiusε.

Lemma1.2(see [13]). Letϕ∈Φ, then(a) limn→∞ϕn(t)=0for allt >0and(b)t=0 provided thatt≤ϕ(t)for somet≥0.

Lemma1.3(see [10]). Letfandgbe commuting self-mappings of a compact metric

space(X, d)such thatgf is continuous. IfA=n∈N(gf )nX, then

(i) hA⊆Afor allh∈Cgf;

(ii) A=f A=gAφ;

(iii) Ais compact.

In recent years, a number of generalizations of a well-known contraction mapping principles due to Banach have appeared in the literature (cf. [1,2,3,4,5,6,7,8,9,10,

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First we list the following general conditions.

(1) There existsϕ∈Φandp, q, m, n∈wwithp+q, m+n∈Nsuch that

dfpgqx, fmgnyϕ  δ

h∈Cg∪Cf

hOg,f(x, y)   

 (1.1)

for allx, y∈X.

(2) There existp, q, m, n∈wwithp+q, m+n∈Nsuch that

dfpgqx, fmgny< δ

h∈Cgf

hOgf(x, y)

 (1.2)

for allx, y∈Xwithfpgqxfmgny.

In the literature of fixed point theory there exist conditions which are special cases of (1) or (2). Now we list below some of the contractive mappings for which various fixed point theorems have been established.

(3) (See [6].) There existsr∈[0,1)andp, q, m, n∈wwithp+q, m+n∈Nsuch that dfpgqx, fmgny

≤r·max dfrgrx, fsgsy,

dfrgrx, ftgtx, dfsgsx, figiy:

0≤r , t≤p, 0≤r, t≤q, 0≤s, i≤m,0≤s, i≤n (1.3)

for allx, y∈X.

(4) (See [6].) There existp, q, m, n∈wwithp+q, m+n∈Nsuch that dfpgqx, fmgny

<max dfrgrx, fsgsy,

dfrgrx, ftgtx, dfsgsx, figiy:

0≤r , t≤p, 0≤r, t≤q,0≤s, i≤m,0≤s, i≤n

(1.4)

for allx, y∈Xfor which the right-hand side of the inequality is positive. (5) (See [11].) There existk∈Nandr∈[0,1)such that

dfkx, fkyr δOf(x, y) (1.5)

for allx, y∈X.

(6) (See [10].) For allx, y∈X, d(f x, gy) < δ(ht:t∈ {x, y}, h∈Cgf)withf xgy. In this note, we prove common fixed point theorems forCf∩CgandCgfin bounded complete metric spaces and compact metric spaces. Our results extend, improve and unify the corresponding results in [1,2,3,4,5,6,7,9,10,11].

2. Common fixed point theorems. Our main results are as follows.

Theorem2.1. Letfandgbe commuting mappings fromabounded complete metric

space(X, d)into itself. Iff andgare continuous and satisfy (1), then

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(ii) d(fi+agi+bx, u)≤ ϕ[i/k](δ(X)) for all a, b ∈ {0,1} and i N, where k= max{p, q, m, n};

(iii) limi→∞fi+agi+bx=ufor alla, b∈ {0,1}; (iv) {figiX}i∈

Nconverges to{u}.

Proof. For anyiwandx, yX, it follows from (1) that

dfi+kgi+kx, fi+kgi+ky

≤ϕ  δ

h∈Cf∩Cg

hOf ,gfi+k−pgi+k−qx, fi+k−mgi+k−ny    

≤ϕ  δ

h∈Cf∩Cg

hOf ,gfigix, figiy    

≤ϕ  δ

h∈Cf∩Cg

hfigiX    

 δ

h∈Cf∩Cg

figihX    

=ϕδfigiX

(2.1)

which implies that

δfi+kgi+kX= sup x,y∈Xd

fi+kgi+kx, fi+kgi+ky≤ϕδfigiX (2.2)

for alli∈w. We can writei=jk+tuniquely for somej, k∈wwitht < k. Thus

δfigiX≤ϕδf(j−1)k+tg(j−1)k+tX≤ϕjδftgtX≤ϕjδ(X). (2.3)

It follows from the boundedness ofX,Lemma 1.2and (2.3) that

lim i→∞δ

figiX=lim j→∞ϕ

jδ(X). (2.4)

Since d(figix, fi+tgi+tx)δ(figiX)for all i, t w and x X, {figix}i∈

N is a

Cauchy sequence and converges to someu∈Xby completeness ofX. For anyi, c∈w, a, b∈ {0,1}andx∈X, by (2.3) and (2.4) we have

dfi+agi+bx, fi+cgi+cx≤δfigiX≤ϕ[i/k]δ(X). (2.5)

Lettingctend to infinity, we get

dfi+agi+bx, u≤ϕ[i/k]δ(X) (2.6)

forx∈X. The continuity offandg, and (2.4) and (2.6) ensure that

u=lim i→∞f

igix=lim i→∞f

agbfigix=fagbu (2.7)

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Suppose thatf and g have another common fixed pointv ∈X. It follows from (2.4) that

d(u, v)≤δfigiXϕ[i/k]δ(X) 0 asi→ ∞. (2.8) That is,u=v. Thereforefandghave a unique common fixed point. Note thatf hu= hf u=hu=hgu=ghufor allh∈Cf∩Cg. By the uniqueness of common fixed point off andg, we havehu=ufor allh∈Cf∩Cg. Sincef , g∈Cf∩Cg, it follows thatu is a unique common fixed point ofCf∩Cg.

Letε >0. In view of (2.4), there existsc∈Nsuch that

δfigiX≤ϕ[i/k]δ(X)<1

2ε. (2.9)

for alli > c. Note thatu∈figiXfor alliw. Thus, for anyi > cwe have

figiXB(u, ε)= xX:d(u, x) < ε. (2.10)

Therefore{figiX}i

Nconverges to{u}. This completes the proof.

Remark2.2. [6, Theorem 2] is a special case ofTheorem 2.1.

As an immediate consequence ofTheorem 2.1, we have the following corollary.

Corollary2.3. Letf andg be commuting mappings fromabounded complete metric space(X, d)into itself. Iffandgare continuous and satisfy the inequality

dfpgqx, fmgny≤r δ

h∈Cf∩Cg

hOf ,g(x, y)

 (2.11)

for allx, y∈X, wherep, q, m, n∈w,p+q, m+n∈Nandr∈[0,1). Then (i), (iii), and (iv) ofTheorem 2.1and the following (v) hold:

(v) d(fi+agi+bx, u) r[i/k]δ(X) for all a, b ∈ {0,1} and i N, where k = max{p, q, m, n}.

Remark2.4. In casep=m,g=iX,Corollary 2.3reduces to a result which

gener-alizes [11, Theorem 3].

Theorem2.5. Letf andgbe commuting mappings fromacompact metric space

(X, d)into itself such that gf is continuous. If (2) is satisfied, thenf and ghave a unique common fixed pointu∈X. Moreover,u=hufor allh∈Cgf.

Proof. LetA=n∈N(gf )nX. It follows fromLemma 1.3thatA=f A=gA

and thatAis compact. We claim thatA={u}for someu∈X. Otherwiseδ(A) >0. By the compactness ofAthere exists distinctu, v∈Asuch thatδ(A)=d(u, v). Clearly, we can findx, y∈Asuch thatfpgqx=uandfmgny=v. Using (2) andLemma 1.3we have

δ(A)=dfpgqx, fmgny< δ

h∈Cgf

hOgf(x, y)  δ

h∈Cgf

hA

δ(A) (2.12)

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Remark2.6. Theorem 2.5includes [6, Theorem 5] as a special case.

As an immediate consequence ofTheorem 2.5, we have the following corollary.

Corollary2.7. Letfandgbe commuting mappings fromacompact metric space

(X, d)into itself. Ifgf is continuous and there existsp, m∈Nsuch that

dfpx, gmy< δ

h∈Cgf

hOgf(x, y)

 (2.13)

for allx, y∈Xwithfpxgmy. Then the conclusion ofTheorem 2.5holds.

Corollary2.8. Letfbe a continuous mapping from a compact metric space(X, d)

into itself. Assume that there existsp, m∈Nsuch that

dfpx, fmy< δ

h∈Cf

hOf(x, y)

 (2.14)

for allx, y∈Xwithfpxfmy. Thenf has a unique fixed pointuXandhu=u

for allh∈Cf.

Remark2.9. Corollary 2.7extends, improve and unifies [10, Theorem 4.2], [9,

Theo-rem 2], and [7, Theorem 5].

Remark2.10. [1, Theorem 4], [2, Theorem 4], [4, Theorem 9], [3, Theorem 2], and

[5, Theorem 4] are special cases ofCorollary 2.8.

Remark2.11. The following examples reveal that the condition thatf andgare

continuous is necessary in Theorems2.1,2.5, and Corollaries2.7,2.8.

Example2.12. LetX=[0,1]with the usual metricd. Definef , g:XXbyg=iX,

f0=1/4 andf x=(1/3)x for allX∈(0,1]. Takep=m=2, q=n=1, r=1/2, ϕ(t)=(1/2)tfor allt≥0. Then(X, d)is a bounded metric space,gis continuous, andf is discontinuous.

Forx, y∈(0,1], we have

dfpgqx, fmgny=19|x−y|

1

2|x−y| ≤ 1 2δ

Of ,g(x, y)

≤ϕ  δ

h∈Cf∩Cg

hOf ,g(x, y)    .

(2.15)

Forx=0,y∈(0,1]orx∈(0,1],y=0, we have

dfpgqx, fmgny≤max 1 12

1 9y

, 1

12 1 9x

<1

8

=1 2δ

Of ,g(0)≤ϕ  δ

h∈Cf∩Cg

hOf ,g(x, y)    .

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Forx=y=0, we have

dfpgqx, fmgny=0<1 8=

1 2δ

Of ,g(0,0)≤ϕ  δ

h∈Cf∩Cg

hOf ,g(x, y)    . (2.17)

That is,f andgsatisfy (1) and (2.11). Butf andghave no common fixed point inX.

Example2.13. Let (X, d), f , g, p, q, m, and nbe as inExample 2.12. It is easy to

check that the conditions ofTheorem 2.5andCorollary 2.8 are satisfied except for the continuity assumption. Howeverf has no fixed point inX.

Acknowledgement. The second author was financially supported by Changwon

National University in 2000.

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Zeqing Liu: Department of Mathematics, Liaoning Normal University, Dalian,

Liaoning116029, China

Jeong Sheok Ume: Department of Applied Mathematics, Changwon National

Univer-sity, Changwon641-773, Korea

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