PII. S0161171202110350 http://ijmms.hindawi.com © Hindawi Publishing Corp.
SOME RESULTS ON FIXED POINT THEOREMS FOR MULTIVALUED
MAPPINGS IN COMPLETE METRIC SPACES
JEONG SHEOK UME, BYUNG SOO LEE, and SUNG JIN CHO
Received 17 October 2001
Using the concept ofw-distance, we improve some well-known fixed point theorems.
2000 Mathematics Subject Classification: 47H10.
1. Introduction. Recently, Ume [3] improved the fixed point theorems in a com-plete metric space using the concept ofw-distance, introduced by Kada, Suzuki, and Takahashi [2], and more general contractive mappings than quasi-contractive map-pings.
In this paper, using the concept ofw-distance, we first prove common fixed point theorems for multivalued mappings in complete metric spaces, then these theorems are used to improve ´Ciri´c’s fixed point theorem [1], Kada-Suzuki-Takahashi’s fixed point theorem [2], and Ume’s fixed point theorem [3].
2. Preliminaries. Throughout, we denote byNthe set of all positive integers and byRthe set of all real numbers.
Definition2.1(see [2]). Let(X,d)be a metric space, 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 allx,y,z∈X;
(2) for anyx∈X,p(x,·):X→[0,∞)is lower semicontinuous;
(3) for any >0, there existsδ >0 such that p(z,x)≤δand p(z,y)≤δimply
d(x,y)≤.
Definition2.2. Let(X,d)be a metric space with aw-distancep, then
(1) for any x ∈ X and A⊆ X, d(x,A) := inf{d(x,y): y ∈ A} and d(A,x) :=
inf{d(y,x):y∈A};
(2) for any x ∈X and A ⊆X, p(x,A):= inf{p(x,y) : y ∈ A} and p(A,x):=
inf{p(y,x):y∈A};
(3) for anyA,B⊆X,p(A,B):=inf{p(x,y):x∈A, y∈B};
(4) CBp(X)= {A|Ais nonempty closed subset ofXand supx,y∈Ap(x,y) <∞}.
The following lemmas are fundamental.
(1) ifp(xn,y)≤αnandp(xn,z)≤βnfor anyn∈N, theny=z. In particular, if p(x,y)=0andp(x,z)=0, theny=z;
(2) ifp(xn,yn)≤αnandp(xn,z)≤βnfor anyn∈N, then{yn}converges toz; (3) ifp(xn,xm)≤αnfor anyn,m∈Nwithm > n, then{xn}is a Cauchy sequence; (4) ifp(y,xn)≤αnfor anyn∈N, then{xn}is a Cauchy sequence.
Lemma2.4(see [3]). LetXbe a metric space with a metricd, letpbe aw-distance onX, and letT be a mapping ofXinto itself satisfying
p(T x,T y)≤q·maxp(x,y),p(x,T x),p(y,T y),p(x,T y),p(y,T x) (2.1)
for allx,y∈Xand someq∈[0,1). Then
(1) for eachx∈X,n∈N, andi,j∈Nwithi,j≤n,
pTix,Tjx≤q·δO(x,n); (2.2)
(2) for eachx∈Xandn∈N, there existk,l∈Nwithk,l≤nsuch that
δO(x,n)=maxp(x,x),px,Tkx,pTlx,x; (2.3)
(3) for eachx∈X,
δO(x,∞)≤ 1
1−q
p(x,x)+p(x,T x)+p(T x,x); (2.4)
(4) for eachx∈X,{Tnx}∞
n=1is a Cauchy sequence.
3. Main results
Theorem 3.1. LetX be a complete metric space with a metricdand let p be a
w-distance onX. Suppose thatS andT are two mappings ofXintoCBp(X)andϕ: X×X→[0,∞)is a mapping such that
maxpu1,u2
,pv1,v2
≤q·ϕ(x,y) (3.1)
for all nonempty subsetsA,B of X, u1∈SA,u2∈S2A, v1∈T B, v2∈T2B,x∈A,
y∈B, and someq∈[0,1),
sup
sup
ϕ(x,y)
minp(x,SA),p(y,T B):x∈A, y∈B :A,B⊆X
< 1
q, (3.2)
infp(y,u)+p(x,Sx)+p(y,T y):x,y∈X>0, (3.3)
for everyu∈Xwithu∉Suoru∉T u, whereSAmeansa∈ASa. ThenSandT have a common fixed point inX.
Proof. Let
β=sup
sup
ϕ(x,y)
minp(x,SA),p(y,T B):x∈A, y∈B :A, B⊆X
and k=βq. Define xn+1∈Sxn and yn+1∈T yn for all n∈N. Thenxn ∈Sxn−1, xn+1∈S2xn−1,yn∈T yn−1, andyn+1∈T2yn−1. From (3.1) and (3.2), we have
pxn,xn+1
≤kpxn−1,xn
≤ ··· ≤kn−1px 1,x2
, (3.5)
pyn,yn+1
≤kpyn−1,yn
≤ ··· ≤kn−1py1,y2
, (3.6)
for alln∈Nand some k∈[0,1). Let nand mbe any positive integers such that
n < m. Then, from (3.6), we obtain
pyn,ym
≤pyn,yn+1
+···+pym−1,ym
=
m−n−1
i=0
pyn+i,yn+i+1
≤
m−n−1
i=0
kn+i−1py1,y2
≤ kn−1
(1−k)p
y1,y2
.
(3.7)
ByLemma 2.3,{yn}is a Cauchy sequence. Since X is complete,{yn}converges to u∈X. Then, sincep(yn,·)is lower semicontinuous, from (3.7) we have
pyn,u
≤ lim m→∞infp
yn,ym
≤ kn−1
(1−k)p
y1,y2
. (3.8)
Suppose thatu∉Suoru∉T u. Then, by (3.3), (3.5), (3.6), and (3.8), we have
0<infp(y,u)+p(x,Sx)+p(y,T y):x,y∈X
≤infpyn,u
+pxn,xn+1
+pyn,yn+1
:n∈N
≤inf
kn−1
(1−k)p
y1,y2
+kn−1px 1,x2
+kn−1py 1,y2
:n∈N
=
2− k (1−k)p
y1,y2
+px1,x2
infkn−1:n∈N
=0.
(3.9)
This is a contradiction. Therefore we haveu∈Suandu∈T u.
Theorem 3.2. LetX be a complete metric space with a metricdand let p be a
w-distance onX. Suppose thatS andT are two mappings ofXintoCBp(X)andϕ: X×X→[0,∞)is a mapping such that
maxpu1,u2
,pv1,v2
≤q·ϕ(x,y) (3.10)
for allx,y∈X,u1∈Sx,u2∈S2x,v1∈T y,v2∈T2y, and someq∈[0,1),
sup
sup
ϕ(x,y)
minp(x,Sx),p(y,T y):x∈A, y∈B :A,B⊆X
< 1
q, (3.11)
Proof. By a method similar to that in the proof ofTheorem 3.1, the result follows.
Theorem 3.3. LetX be a complete metric space with a metricdand let p be a
w-distance onX. Suppose thatT is a mapping ofXintoCBp(X)andψ:X→[0,∞)is a mapping such that
pu1,u2
≤q·ψ(x) (3.12)
for allx∈X,u1∈T x,u2∈T2xand someq∈[0,1),
sup
ψ(x)
p(x,T x):x∈X
<1 q,
infp(x,u)+p(x,T x):x∈X>0,
(3.13)
for everyu∈Xwithu∉T u. ThenThas a fixed point inX.
Proof. By a method similar to that in the proof ofTheorem 3.1, the result follows.
Theorem 3.4. LetX be a complete metric space with a metricdand let p be a
w-distance onX. Suppose thatS andT are self-mapping ofXandϕ:X×X→[0,∞)
is a mapping such that
maxpSx,S2x,pT y,T2y≤q·ϕ(x,y) (3.14)
for allx,y∈Xand someq∈[0,1),
sup
ϕ(x,y)
minp(x,Sx),p(y,T y):x,y∈X
<1 q,
infp(y,u)+p(x,Sx)+p(y,T y):x,y∈X>0,
(3.15)
for everyu∈Xwithu≠Suoru≠T u. ThenSandT have a common fixed point inX.
Proof. By a method similar to that in the proof ofTheorem 3.1, the result follows.
FromTheorem 3.1, we have the following corollary.
Corollary 3.5. LetX be a complete metric space with a metricdand let p be aw-distance onX. Suppose thatS andT are two mappings of X intoCBp(X)and ϕ:X×X→[0,∞)is a mapping such that
maxsuppu1,u2
:u1∈Sx, u2∈S2x
,
suppv1,v2
:v1∈T x, v2∈T2x
≤q·ϕ(x,y)
(3.16)
for allx,y∈Xand someq∈[0,1), and that (3.3) and (3.11) are satisfied. ThenS and
FromTheorem 3.3, we have the following corollaries.
Corollary3.6. LetXbe a complete metric space with a metricdand letp be a
w-distance onX. Suppose thatT is a mapping ofXintoCBp(X)andψ:X→[0,∞)is a mapping such that
suppu1,u2:u1∈T x, u2∈T2x≤q·ψ(x) (3.17)
for allx∈Xand someq∈[0,1), and that (3.13) is satisfied. ThenT has a fixed point inX.
Corollary3.7. LetXbe a complete metric space with a metricdand letp be a
w-distance onX. Suppose thatTis a self-mapping ofXandψ:X→[0,∞)is a mapping such that
pT x,T2x≤q·ψ(x) (3.18)
for allx∈Xand someq∈[0,1),
sup
ψ(x)
p(x,T x):x∈X
< 1 q,
infp(x,u)+p(x,T x):x∈X>0,
(3.19)
for everyu∈Xwithu≠T u. ThenT has a fixed point inX.
FromCorollary 3.7, we have the following corollaries.
Corollary3.8(see [3]). LetXbe a complete metric space with a metricdand let
pbe aw-distance onX. Suppose thatT is a self-mapping ofXsuch that
p(T x,T y)≤q·maxp(x,y),p(x,T x),p(y,T y),p(x,T y),p(y,T x) (3.20)
for allx,y∈Xand someq∈[0,1), and that
infp(x,u)+p(x,T x):x∈X>0 (3.21)
for everyu∈Xwithu≠T u. ThenT has a unique fixed point inX.
Proof. By (3.20) andLemma 2.4(3), we have
suppTix,Tjx|i, j∈N∪{0}<∞ (3.22)
for everyx∈X. Thus we may define a functionr:X×X→[0,∞)by
r (x,y)=maxsuppTix,Tjx|i, j∈N∪{0},p(x,y) (3.23)
for everyx,y∈X. Clearly,ris aw-distance onX. Letxbe a given element ofX, then, by usingLemma 2.4(1), (3.20), and (3.23), we have
rT x,T2x=suppTix,Tjx|i, j∈N
≤q·suppTix,Tjx|i, j∈N∪{0} =q·r (x,T x).
By (3.21) and (3.23), we obtain
infr (x,u)+r (x,T x):x∈X>0 (3.25)
for everyu∈Xwithu≠T u. From (3.24), (3.25), andCorollary 3.7,Thas a fixed point inX. By (3.20) andLemma 2.4, it is clear that the fixed point ofT is unique.
Corollary3.9(see [2]). LetXbe a complete metric space, letpbe aw-distance on
X, and letT be a mapping fromXinto itself. Suppose that there existsq∈[0,1)such that
pT x,T2x≤q·p(x,T x) (3.26)
for everyx∈Xand that
infp(x,y)+p(x,T x):x∈X>0 (3.27)
for everyy∈Xwithy≠T y. ThenT has a fixed point inX.
Proof. Defineψ:X→[0,∞)by
ψ(x)=p(x,T x) (3.28)
for allx∈X. Thus the conditions ofCorollary 3.7are satisfied. HenceT has a fixed point inX.
FromCorollary 3.8, we have the following corollary.
Corollary3.10(see [1]). LetXbe a complete metric space with a metricdand let
T be a mapping fromXinto itself. Suppose thatT is a quasicontraction, that is, there existsq∈[0,1)such that
d(T x,T y)≤q·maxd(x,y),d(x,T x),d(y,T y),d(x,T y),d(y,T x) (3.29)
for everyx,y∈X. ThenT has a unique fixed point inX.
Proof. It is clear that the metricdis aw-distance and
infd(x,y)+d(x,T x):x∈X>0 (3.30)
for everyy∈Xwithy≠T y. Thus, by Corollary3.8or3.9,Thas a unique fixed point inX.
Acknowledgment. This work was supported by grant No. 2001-1-10100-005-2
from the Basic Research Program of the Korea Science & Engineering Foundation.
References
[1] L. B. ´Ciri´c,A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc.45 (1974), 267–273.
[3] J. S. Ume,Fixed point theorems related to ´Ciri´c’s contraction principle, J. Math. Anal. Appl. 225(1998), no. 2, 630–640.
Jeong Sheok Ume: Department of Applied Mathematics, Changwon National
Univer-sity, Changwon641-773, Korea
E-mail address:jsume@sarim.changwon.ac.kr
Byung Soo Lee: Department of Mathematics, Kyungsung University, Pusan608-736,
Korea
E-mail address:bslee@star.kyungsung.ac.kr
Sung Jin Cho: Department of Applied Mathematics, Pukyong National University, Pusan608-737, Korea
