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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:n∈w},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:n∈w} ⊆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]). LetA⊆XandAn⊆Xfor alln∈N. 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 Aεis 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=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 existsϕ∈Φandp, q, m, n∈wwithp+q, m+n∈Nsuch that
dfpgqx, fmgny≤ϕ 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∈Xwithfpgqx≠fmgny.
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, fky≤r 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 x≠gy. 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
(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 anyi∈wandx, y∈X, 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
=ϕ 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)
Suppose thatf and g have another common fixed pointv ∈X. It follows from (2.4) that
d(u, v)≤δfigiX≤ϕ[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 alli∈w. Thus, for anyi > cwe have
figiX⊆B(u, ε)= x∈X: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)
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∈Xwithfpx≠gmy. 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∈Xwithfpx≠fmy. Thenf has a unique fixed pointu∈Xandhu=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:X→Xbyg=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) .
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