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COMMON FIXED POINT THEOREMS FOR A WEAK DISTANCE
IN COMPLETE METRIC SPACES
JEONG SHEOK UME and SUCHEOL YI
Received 17 October 2001
Using the concept of aw-distance, we obtain common fixed point theorems on complete metric spaces. Our results generalize the corresponding theorems of Jungck, Fisher, Dien, and Liu.
2000 Mathematics Subject Classification: 47H10.
1. Introduction. In 1976, Caristi [1] proved a fixed point theorem in a complete metric space which generalizes the Banach contraction principle. This theorem is very useful and has many applications. Later, Dien [3] showed that a pair of mappings satisfying both the Banach contraction principle and Caristi’s condition in a complete metric space has a common fixed point. That is to say, let(X, d)be a complete metric space and letSandT be two orbitally continuous mappings ofXinto itself. Suppose that there exists a finite number of functions{ϕi}1≤i≤N0ofXintoR+such that
d(Sx, T y)≤q·d(x, y)+ N0
i=1
ϕi(x)−ϕi(Sx)+ϕi(y)−ϕi(T y) (1.1)
for allx, y∈Xand someq∈[0,1). ThenSandT have a unique common fixed point zinX. Further, ifx∈XthenSnx→zandTnx→zasn→ ∞. In particular, ifSis an identity mapping,q=0, andN0=1, then this means a Caristi’s fixed point theorem.
Recently, Liu [7] obtained necessary and sufficient conditions for the existence of fixed point of continuous self-mapping by using the ideas of Jungck [5] and Dien [3]: letf be a continuous self-mapping of a metric space(X, d), thenfhas a fixed point inXif and only if there existz∈X, a mappingg:X→X, and a functionΦfromX into[0,∞)such thatf andgare compatible,g(X)⊂f (X),gis continuous, and
d(gx, z)≤r d(f x, z)+Φ(f x)−Φ(gx) (1.2)
for allx∈Xand somer∈[0,1).
In 1996, Kada et al. [6] introduced the concept ofw-distance on a metric space as follows: letX be a metric space with metricd, then a functionp:X×X→[0,∞)is called aw-distance onXif the following are satisfied:
(1) p(x, z)≤p(x, y)+p(y, z)for anyx, y, z∈X;
(3) for any >0, there existsδ >0 such that p(z, x)≤δand p(z, y)≤δimply d(x, y)≤.
In this paper, using the concept of aw-distance, we obtain common fixed point the-orems on complete metric spaces. Our results generalize the corresponding thethe-orems of Jungck [5], Fisher [4], Dien [3], and Liu [7].
2. Definitions and preliminaries. Throughout, we denote byNthe set of positive integers and byR+the set of nonnegative real numbers, that is,R+:=[0,∞).
Definition2.1(see [3]). A mappingT of a spaceX into itself is said to be or-bitally continuous ifx0∈Xsuch thatx0=limi→∞Tnixfor somex∈X, thenT x0= limi→∞T (Tnix).
Definition2.2(see [2]). LetT be a mapping of a metric spaceX into itself. For eachx∈X, let
O(T , x, n)=x, T x, . . . , Tnx, n=1,2, . . . ,
O(T , x,∞)= {x, T x, . . .}. (2.1)
A spaceX is said to be T-orbitally complete if and only if every Cauchy sequence, which is contained inO(T , x,∞)for somex∈X, converges inX.
Definition2.3(see [6]). LetX be a metric space with metricd. Then a function
p:X×X→R+is called aw-distance onXif the following properties are satisfied:
(1) p(x, z)≤p(x, y)+p(y, z)for anyx, y, z∈X;
(2) for anyx∈X,p(x,·):X→R+is lower semicontinuous;
(3) for any >0, there existsδ >0 such that p(z, x)≤δand p(z, y)≤δimply d(x, y)≤.
The metricdis aw-distance onX. Other examples ofw-distance are stated in [6].
Definition2.4(see [5]). Let(X, d)be a metric space andf , g:X→X. The map-pingsf andgare called compatible if and only if for every sequence{xn}n∈N such that limn→∞f xn=limn→∞gxn=tfor somet∈X, it implies
lim n→∞d
f gxn, gf xn=0. (2.2)
Lemma2.5(see [6]). LetXbe a metric space with metricd, andpaw-distance onX.
Let{xn}and{yn}be sequences inX, let{αn}and{βn}be sequences inR+converging to0, and letx, y, z∈X. Then the following properties hold:
(i) ifp(xn, y)≤αnandp(xn, z)≤βnfor anyn∈N, theny=z. In particular, if p(x, y)=0andp(x, z)=0, theny=z;
3. Main results
Theorem3.1. Let(X, d)be a complete metric space with aw-distancep. Suppose that two mappingsf , g:X→X and a functionϕ fromX intoR+ are satisfying the following conditions:
(i) g(X)⊆f (X),
(ii) there existst∈X such thatp(t, gx)≤r·p(t, f x)+[ϕ(f x)−ϕ(gx)]for all x∈Xand somer∈[0,1),
(iii) for every sequence{xn}n∈N inXsatisfying
lim n→∞p
t, f xn=lim n→∞p
t, gxn=0, (3.1)
it implies that
lim n→∞max
pt, f xn, pt, gxn, pf gxn, gf xn=0, (3.2)
(iv) for eachu∈Xwithu≠f uoru≠gu,
infp(u, f x)+p(u, gx)+p(f gx, gf x):x∈X>0. (3.3)
Thenf andghave a unique common fixed point inX.
Proof. Letx0be a given point ofX. By (i), there existsxn∈Xsuch thatgxn−1= f xnforn≥1. FromTheorem 3.1(ii), we have
pt, f xj+1
=pt, gxj≤r·pt, f xj+ϕf xj−ϕgxj, (3.4)
which implies that
n−1
j=0
pt, f xj+1≤r· n−1
j=0
pt, f xj+ n−1
j=0
ϕf xj−ϕgxj, (3.5)
that is,
n
j=1
pt, f xj≤1−rrpt, f x0+1−1rϕf x0−ϕf xn,
≤ r
1−rp
t, f x0+ 1
1−rϕ
f x0,
(3.6)
which means that the series∞n=1p(t, f xn)is convergent, so
lim n→∞p
t, f xn=lim n→∞p
Suppose thatt≠f tort≠gt. Then, fromTheorem 3.1(iii) and (iv) we obtain that
0<infp(t, f x)+p(t, gx)+p(f gx, gf x):x∈X
≤infpt, f xn+pt, gxn+pf gxn, gf xn:n∈N
=0.
(3.8)
This is a contradiction. Hencetis a common fixed point off andg.
We prove thattis a unique common fixed point off andg. Letube a common fixed point offandg. Then, byTheorem 3.1(ii),
p(t, t)=p(t, gt)≤r·p(t, f t)+ϕ(f t)−ϕ(gt)=r·p(t, t),
p(t, u)=p(t, gu)≤r·p(t, f u)+ϕ(f u)−ϕ(gu)=r·p(t, u). (3.9)
Thusp(t, t)=p(t, u)=0. FromLemma 2.5, we obtaint=u. Thereforetis a unique common fixed point off andg.
Remark3.2. Theorem 3.1generalizes and improves Dien [3, Theorem 2.2] and Liu
[7, Theorem 3.2].
Theorem3.3. Letf be a continuous self-mapping of metric space(X, d). Assume thatf has a fixed point in X. Then there exists aw-distancep, t∈X, a continuous mappingg:X→X, and a functionϕfromXintoR+satisfyingTheorem 3.1(i), (ii), (iii), and (iv).
Proof. Letzbe a fixed point off,r=1/2,gx=t=z, andϕ(x)=1 for allx∈X.
Definep:X×X→R+by
p(x, y)=maxd(f x, x), d(f x, y), d(f x, f y) ∀x, y∈X. (3.10)
Suppose that
lim n→∞p
t, f xn=lim n→∞p
t, gxn=0. (3.11)
Then it is easy to verify that the results ofTheorem 3.3follow.
Theorem3.4. Letf andgbe a continuous compatible self-mappings of the metric space(X, d). There existst∈Xsatisfying
d(t, gx)≤r·d(t, f x)+ϕ(f x)−ϕ(gx) (3.12)
for allx∈Xand somer∈[0,1). Then
(i) for every sequence{xn}n∈N inXsuch that
lim n→∞d
t, f xn=lim n→∞d
t, gxn=0 (3.13)
for somet∈X, it implies that
lim n→∞max
(ii) for eachu∈Xwithu≠f uoru≠gu,
infd(u, f x)+d(u, gx)+d(f gx, gf x):x∈X>0. (3.15)
Proof. The results follow by elementary calculation.
Remark3.5. Since the metricdisw-distance, from Theorems3.1,3.3, and3.4, we obtain Liu [7, Theorem 3.1].
Theorem3.6. Let(X, d)be a complete metric space with aw-distancep, two map-pingsf , g:X→X, and two functionsϕ,ψfromXintoR+such thatTheorem 3.1(i), (iv) are satisfied,
(i) for every sequence{xn}n∈N inXsuch that
lim
n→∞f xn=nlim→∞gxn=t (3.16)
for somet∈X, it implies that
lim n→∞max
pt, f xn, pt, gxn, pf gxn, gf xn=0, (3.17)
(ii)
p(gx, gy)≤a1p(f x, f y)+a2p(f x, gx)+a3p(f y, gy)
+a4p(f x, gy)+a5p(gx, f y)d(f y, gx)1/2
+ϕ(f x)−ϕ(gx)+ψ(f y)−ψ(gy)
(3.18)
for allx, y∈X, wherea1,a2,a3,a4, anda5are in[0,1)witha1+a4+a5<1 anda1+a2+a3+2a4<1.
Thenf andghave a unique common fixed point inX.
Proof. Letx0be an arbitrary point ofX. ByTheorem 3.1(i), we obtain a sequence
{xn}inX such that gxn−1=f xn forn≥1. Letγn=p(f xn, f xn+1) forn≥0. It
follows fromTheorem 3.6(ii) that
γj+1=p
gxj, gxj+1
≤a1pf xj, f xj+1
+a2pf xj, gxj+a3pf xj+1, gxj+1
+a4pf xj, gxj+1
+a5pgxj, f xj+1
df xj+1, gxj1/2 +ϕf xj−ϕgxj+ψf xj+1
−ψgxj+1
≤a1+a2+a4γj+a3+a4γj+1
+ϕf xj−ϕf xj+1
+ψf xj+1
−ψf xj+2
,
(3.19)
which implies that
γj+1≤L1γj+L2
ϕf xj−ϕf xj+1
+ψf xj+1
−ψf xj+2
, (3.20)
where
L1=a1+a2+a4
1−a3−a4 , L2= 1
Thus
n
j=1
γj≤1−L1L1γ0+1−L2L1ϕf x0+ψf x1 (3.22)
for alln≥1. Hence, the series∞n=1γnis convergent. For anyn, r≥1, we have
pf xn, f xn+r≤ n+r−1
i=n
γi. (3.23)
By Lemma 2.5, this implies that {f xn}∞
n=1 is a Cauchy sequence in X. Since X is a complete metric space, there exists t ∈ X such that f xn →t as n→ ∞. From
Theorem 3.6(i), we have
lim n→∞max
pt, f xn, pt, gxn, pf gxn, gf xn=0. (3.24)
Suppose thatt≠f tort≠gt, then fromTheorem 3.1(iv) we obtain that
0<infp(t, f x)+p(t, gx)+p(f gx, gf x):x∈X
≤infpt, f xn+pt, gxn+pf gxn, gf xn:n∈N
=0.
(3.25)
which is a contradiction. Thereforetis a common fixed point off andg. It follows fromLemma 2.5andTheorem 3.6(ii) thattis a unique common fixed point offandg.
Theorem3.7. Letfbe a continuous self-mapping of a metric space(X, d). Assume that f has a fixed point inX. Then there exist aw-distancep, t∈X, a continuous mappingg:X→X, and functionsϕ,ψfromXintoR+satisfyingTheorem 3.1(i), (iv) andTheorem 3.6(i), (ii).
Proof. By a method similar to that in the proof ofTheorem 3.3, the results follow.
Remark3.8. Since the metricdisw-distance, from Theorems3.4,3.6, and3.7, we obtain Jungck [5, Theorem], Fisher [4, Theorem 2], and Liu [7, Theorem 3.3].
Acknowledgment. This work was supported by Korea Research Foundation
Grant (KRF-2001-015-DP0025).
References
[1] J. Caristi,Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc.215(1976), 241–251.
[2] L. B. ´Ciri´c,A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc.45
(1974), 267–273.
[3] N. H. Dien,Some remarks on common fixed point theorems, J. Math. Anal. Appl.187(1994), no. 1, 76–90.
[5] G. Jungck,Commuting mappings and fixed points, Amer. Math. Monthly83(1976), no. 4, 261–263.
[6] O. Kada, T. Suzuki, and W. Takahashi,Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japon.44(1996), no. 2, 381–391. [7] Z. Liu, Y. Xu, and Y. J. Cho,On characterizations of fixed and common fixed points, J. Math.
Anal. Appl.222(1998), no. 2, 494–504.
Jeong Sheok Ume: Department of Applied Mathematics, Changwon National
Univer-sity, Changwon641-773, Korea
E-mail address:jsume@sarim.changwon.ac.kr
Sucheol Yi: Department of Applied Mathematics, Changwon National University,
Changwon641-773, Korea
