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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.** Let*w*andNdenote the sets of nonnegative integers and positive
integers, respectively. For*t∈[0,∞),[t]*denotes the largest integer not exceeding*t.*
Let*f* and*g*be mappings from a metric space*(X, d)*into itself and*Cf* *= {h*:*h*:*X→*
*X*and*hf=f h}. Forx, y∈X*and*A⊆X, deﬁneOf(x)= {fn _{x}*

_{:}

_{n}_{∈}_{w},}_{Of}_{(x, y)}_{=}*Of(x)∪Of(y),*

*Of ,g(x)= {fi*

_{g}j_{x}_{:}

_{i, j}

_{∈}_{w}_{}}_{,}

_{Of ,g(x, y)}_{=}_{Of ,g(x)}_{∪}_{Of ,g(y), and}*δ(A)=*sup

*{d(x, y)*:

*x, y∈A}*. Let

*iX*denote the identity mapping on

*X. It is easy to*see that

*{fn*

_{:}

_{n}_{∈}_{w} ⊆}_{Cf}_{. Let}

_{Φ}

_{= {ϕ}_{:}

_{ϕ}_{:}

_{[0,∞)}_{→}_{[0,∞)}_{is upper semicontinuous}and nondecreasing and

*ϕ(t) < t*for

*t >*0

*}*. The following deﬁnition and lemmas were introduced by Fisher [8], Singh and Meade [13], and Jungck [10], respectively.

**Deﬁnition _{1.1}**

_{(see [}

_{8}

_{])}

**.**

_{Let}

_{A}⊆_{X}_{and}

_{An}⊆_{X}_{for all}

*N*

_{n}∈_{. The sequence}

*{An}n∈*

_{N}

is said to converge to*A*if

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

*an∈An*for all*n∈*N;

(ii) for arbitrary*ε >*0, there exists an integer*k*such that*An⊆Aε*for*n > k, where*
*Aε*is the union of all open spheres with centers in*A*and radius*ε.*

**Lemma1.2**(see [13])**.** *Letϕ∈*Φ*, then*(a) limn* _{→∞}ϕn(t)=*0

*for allt >*0

*and*(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=gA*≠*φ;*

(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,

First we list the following general conditions.

(1) There exists*ϕ∈*Φand*p, q, m, n∈w*with*p+q, m+n∈*Nsuch that

*dfp _{g}q_{x, f}m_{g}n_{y}_{≤}_{ϕ}*

*δ*

*h∈Cg∪Cf*

*hOg,f(x, y)*

(1.1)

for all*x, y∈X.*

(2) There exist*p, q, m, n∈w*with*p+q, m+n∈*Nsuch that

*dfpgqx, fmgny< δ*

*h∈Cgf*

*hOgf(x, y)*

(1.2)

for all*x, y∈X*with*fp _{g}q_{x}*

_{≠}

_{f}m_{g}n_{y.}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 exists*r∈[0,1)*and*p, q, m, n∈w*with*p+q, m+n∈*Nsuch that
*dfp _{g}q_{x, f}m_{g}n_{y}*

*≤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 all*x, y∈X.*

(4) (See [6].) There exist*p, q, m, n∈w*with*p+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 all*x, y∈X*for which the right-hand side of the inequality is positive.
(5) (See [11].) There exist*k∈*Nand*r∈[0,1)*such that

*dfk _{x, f}k_{y}_{≤}_{r δ}_{Of}_{(x, y)}*

_{(1.5)}

for all*x, y∈X.*

(6) (See [10].) For all*x, y∈X, d(f x, gy) < δ(ht*:*t∈ {x, y}, h∈Cgf)*with*f x*≠*gy.*
In this note, we prove common ﬁxed point theorems for*Cf∩Cg*and*Cgf*in 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*

(ii) *d(fi+agi+bx, u)≤* *ϕ[i/k] _{(δ(X))}*

_{for all}

_{a, b}

_{∈ {}_{0,1}

_{}}

_{and}

_{i}_{∈}_{N}

_{, where}*max{p, q, m, n}*

_{k}_{=}*;*

(iii) limi→∞*fi+a _{g}i+b_{x}_{=}_{u}_{for all}_{a, b}_{∈ {}*

_{0,1}

*(iv)*

_{}}_{;}*{fi*

_{g}i_{X}_{}i∈}N*converges to{u}.*

**Proof.** _{For any}_{i}∈_{w}_{and}_{x, y}∈_{X, it follows from (1) that}

*dfi+kgi+kx, fi+kgi+ky*

*≤ϕ*
*δ*

*h∈C _{f}∩Cg*

*hOf ,gfi+k−pgi+k−qx, fi+k−mgi+k−ny*

*≤ϕ*
*δ*

*h∈C _{f}∩Cg*

*hOf ,gfi _{g}i_{x, f}i_{g}i_{y}*

*≤ϕ*
*δ*

*h∈Cf∩Cg*

*hfi _{g}i_{X}*

*=ϕ*
*δ*

*h∈Cf∩Cg*

*figihX*

*=ϕδfi _{g}i_{X}*

(2.1)

which implies that

*δfi+kgi+kX=* sup
*x,y∈Xd*

*fi+kgi+kx, fi+kgi+ky≤ϕδfigiX* (2.2)

for all*i∈w. We can writei=jk+t*uniquely for some*j, k∈w*with*t < k. Thus*

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

It follows from the boundedness of*X,*Lemma 1.2and (2.3) that

lim
*i→∞δ*

*figiX=*lim
*j→∞ϕ*

*j _{δ(X)}_{.}*

_{(2.4)}

Since *d(fi _{g}i_{x, f}i+t_{g}i+t_{x)}_{≤}_{δ(f}i_{g}i_{X)}*

_{for all}

_{i, t}

_{∈}

_{w}_{and}

_{x}

_{∈}_{X,}

_{{}_{f}i_{g}i_{x}_{}i∈}N is a

Cauchy sequence and converges to some*u∈X*by completeness of*X. For anyi, c∈w,*
*a, b∈ {*0,1*}*and*x∈X, by (*2.3) and (2.4) we have

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

Letting*c*tend to inﬁnity, we get

*dfi+agi+bx, u≤ϕ[i/k] _{δ(X)}*

_{(2.6)}

for*x∈X. The continuity off*and*g, and (*2.4) and (2.6) ensure that

*u=*lim
*i→∞f*

*i _{g}i_{x}_{=}*

_{lim}

*i→∞f*

*a _{g}b_{f}i_{g}i_{x}_{=}_{f}a_{g}b_{u}*

_{(2.7)}

Suppose that*f* and *g* have another common ﬁxed point*v* *∈X. It follows from*
(2.4) that

*d(u, v)≤δfi _{g}i_{X}_{≤}_{ϕ}[i/k]_{δ(X)}*

_{}

_{→}_{0}

_{as}

_{i}_{}

_{→ ∞}_{.}_{(2.8)}That is,

*u=v. Thereforef*and

*g*have a unique common ﬁxed point. Note that

*f hu=*

*hf u=hu=hgu=ghu*for all

*h∈Cf∩Cg*. By the uniqueness of common ﬁxed point of

*f*and

*g, we havehu=u*for all

*h∈Cf∩Cg. Sincef , g∈Cf∩Cg, it follows thatu*is a unique common ﬁxed point of

*Cf∩Cg.*

Let*ε >*0. In view of (2.4), there exists*c∈*Nsuch that

*δfigiX≤ϕ[i/k]δ(X)<*1

2*ε.* (2.9)

for all*i > c. Note thatu∈fi _{g}i_{X}*

_{for all}

_{i}_{∈}_{w. Thus, for any}_{i > c}_{we have}

*fi _{g}i_{X}_{⊆}_{B(u, ε)}_{=}*

_{x}_{∈}_{X}_{:}

_{d(u, x) < ε}_{.}_{(2.10)}

Therefore*{fi _{g}i_{X}i}*

*∈*Nconverges to*{u}. This completes the proof.*

**Remark _{2.2}.**

_{[}

_{6}

_{, Theorem 2] is a special case of}

_{Theorem 2.1}

_{.}

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

**Corollary _{2.3}.**

_{Let}_{f}

_{and}_{g}

_{be commuting mappings from}_{a}_{bounded 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∈*N*andr∈[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 case*p=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.** _{Let}_{A}=_{n∈}_{N}_{(gf )}n_{X. It follows from}_{Lemma 1.3}_{that}* _{A}=_{f A}=_{gA}*≠

*∅*

and that*A*is compact. We claim that*A={u}*for some*u∈X. Otherwiseδ(A) >0. By the*
compactness of*A*there exists distinct*u, v∈A*such that*δ(A)=d(u, v). Clearly, we can*
ﬁnd*x, y∈A*such that*fp _{g}q_{x}_{=}_{u}*

_{and}

_{f}m_{g}n_{y}_{=}_{v. Using (2) and}_{Lemma 1.3}

_{we have}

*δ(A)=dfpgqx, fmgny< δ*

*h∈Cgf*

*hOgf(x, y)*
_{≤}δ

*h∈Cgf*

*hA*

* _{≤}δ(A)* (2.12)

**Remark _{2.6}.**

_{Theorem 2.5}

_{includes [}

_{6}

_{, Theorem 5] as a special case.}

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

**Corollary _{2.7}.**

_{Let}_{f}_{and}_{g}_{be commuting mappings from}_{a}_{compact metric space}*(X, d)into itself. Ifgf* *is continuous and there existsp, m∈*N*such that*

*dfp _{x, g}m_{y}_{< δ}*

*h∈Cgf*

*hOgf(x, y)*

(2.13)

*for allx, y∈Xwithfp _{x}*

_{≠}

_{g}m_{y}_{. Then the conclusion of}_{Theorem 2.5}_{holds.}**Corollary _{2.8}.**

_{Let}_{f}_{be a continuous mapping from a compact metric space}_{(X, d)}*into itself. Assume that there existsp, m∈*N*such that*

*dfp _{x, f}m_{y}_{< δ}*

*h∈C _{f}*

*hOf(x, y)*

(2.14)

*for allx, y∈Xwithfp _{x}*

_{≠}

_{f}m_{y}_{. Then}_{f}

_{has a unique ﬁxed point}_{u}_{∈}_{X}_{and}_{hu}_{=}_{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.

**Remark _{2.11}.**

_{The following examples reveal that the condition that}

_{f}_{and}

_{g}_{are}

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

**Example _{2.12}.**

_{Let}

_{X}=_{[0,1]}_{with the usual metric}

_{d. Deﬁne}_{f , g}_{:}

_{X}→_{X}_{by}

_{g}=_{iX,}*f*0*=*1/4 and*f x=(1/3)x* for all*X∈(0,*1]. Take*p=m=*2, *q=n=*1, *r=*1/2,
*ϕ(t)=(1/2)t*for all*t≥*0. Then*(X, d)*is a bounded metric space,*g*is continuous,
and*f* is discontinuous.

For*x, y∈(0,1], we have*

*dfpgqx, fmgny=*1_{9}*|x−y|*

*≤*1

2*|x−y| ≤*
1
2*δ*

*Of ,g(x, y)*

*≤ϕ*
*δ*

*h∈Cf∩Cg*

*hOf ,g(x, y)*
*.*

(2.15)

For*x=*0,*y∈(0,1]*or*x∈(0,1],y=*0, we have

*dfpgqx, fmgny≤*max 1
12*−*

1
9*y*

*,* 1

12*−*
1
9*x*

*<*1

8

*=*1
2*δ*

*Of ,g(0)≤ϕ*
*δ*

*h∈Cf∩Cg*

*hOf ,g(x, y)*
*.*

For*x=y=*0, we have

*dfpgqx, fmgny=*0*<*1
8*=*

1
2*δ*

*Of ,g(0,0)≤ϕ*
*δ*

*h∈Cf∩Cg*

*hOf ,g(x, y)*
*.* (2.17)

That is,*f* and*g*satisfy (1) and (2.11). But*f* and*g*have no common ﬁxed point in*X.*

**Example _{2.13}.**

_{Let}

_{(X, d), f , g, p, q, m, and}

_{n}_{be as in}

_{Example 2.12}

_{. It is easy to}

check that the conditions ofTheorem 2.5andCorollary 2.8 are satisﬁed except for
the continuity assumption. However*f* has no ﬁxed point in*X.*

**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