Rev. Acad. Canar. Cienc., XX (Núms. l-2), 25-31 (2008) (publicado en septiembre de 2009)
SEMI GROUP OF QUASI-PROXIMAL MAPPINGS
AND THEIR FIXED POINTS
Mantu Saha', R. Chikkala2 & A. P. Baisnab'
Abstract
The purpose of this paper is to obtain sorne results on fixed point of certain proxirnally
contractive semi-groups of mapping in a suppurated q -proxirnity space.
Introduction
On a quasi proximity space, concept of quasi-metrices and its associated topologies was
first initialed by J. C. Kelly [ 4], R. D. Holrnes [6] had studied serni-group of rnappings of
proximally rnappings on a metric space. Sorne systematic study of quasi-proximity spaces
and its associated topologies was rnade by Singa! and La! [2] and Jas and Banerjee [1].
Sorne allied results also appear in Jas and Baisnab [3]. Chikkala and Baisnab [8] had also
proved sorne fixed point theorerns in this connection. The aim of this paper is to obtain
sorne results of fixed points of certain proximally contractive serni-groups of mappings in
a separated quasi- proximity spaces. Also we demonstrate how the rnotion of gauges can
be fruitfully employed in quasi-proxirnity spaces with positive character to derive sorne
fixed point Theorerns.
Key Words: Semi group, quasi proximity space, proximally contractive rnapping,
fixed point
AMS Subject Classification: 47H10, 54H25.
1Corresponding Author: Tel: +913422657741; Fax: (091) 342 2530452
E-mail: mantusaha@yahoo.com (Mantu Saha)
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Before proving the main theorems we need sorne preliminaries.
Definition 1: A binary relation ;ron the power set P(X) of X is said to be a quasi-proximity
( q -proximity) on X if the following axioms are satisfied.
P.l. A;r(BUC)if and only if A;rBor A;rCand (AUB);rCif and only if A;rCor
B;rC
P.2. A;rB ==:> A :t. 9and B :t. 9
P.3. A I B:t <J> ==:> A;rB
P.4. A J( B ==:> there is a subset E of X such that A J( B and (X 1 E) J( B.
For A e X, where (X,;r) is a q-proximity space
g" -cl(A) = {xE X: {x};rA} describes a Kuratowski closure operator on X including a
topology g" on X.
A q -proximity ;r on X defines a q -proximity ;r' on X by A ;r' B iff B ;r A: ;r' 1s
called the conjugate of ;r.
Definition 2. A set X (:f. 9) on which there are defined two topologies r and v, is
called a bi-topological space denoted by (X, r , v).
Examplel. Let 7t be a quasi-proximity on X, and 1C* be its conjugate on X, then
(X,""' r,,..) is a bitopological space, with topologies being 3,,. and g ". being T., while
(X, 1{, 1C*) is termed as a bi-quasi-proximity space.
Definition 3 : A bitopological space (X, T, v) is called pairwise Hausdorff (T2) if for
distinct points x, y E X there exists a r-open set U and a v-open set V su ch that x E U
and y E V and U íl V = 9.
Definition 4. A subfamily g;J e rU v of a bitopological space (X, T, v) is called a
pairwise open cover of X if it covers X.
Definition 5. A q-proximity space (X, r) is said to be separated if and only if for
x,yE X {x};r{y} implies x =y.
Definition 6: A quasi pseudometric don X is called a gauge in (X,.n') if for given
A ;r B and E> O there exists a E A and bE B such that d(a,b) <E.
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Definition 7. A bitopological space (X, r, v) is said to be compact if each pairwise open
cover of it has a finite subcover.
Definition 8. Let (X, tr) and (Y, tr1) be two quasi-proximity spaces. A mapping
f:(X,tr)~(Y,tr1 ) iscalledquasi-proximaliff AtrB implies f(A)tr¡f(B).Notethatf:
(X, 1t) ~(Y, tri) is quasi-proximal iff f: (X ,tr*) ~ (Y,tr1• ) is quasi-proximal.
Definition 9. Tis called proximally contractive if for any gauge d and any e> O, there
is a member g E Tsuch that
d(x, y)::; t: implies d (g(x), g(y)) < t: ;x, y E X.
Definition 10. Two points and x and y of (X, tr) is said to be proximal if for any gauge
d and for any t:> O, there is a member g E r satisfying d(g(x), g(y)) <E.
Lemma l. For any gauge don a quasi-proximity space (X, ít) ande> O. Let
Bd,< ={(x,y)E XxX :d(x,y)<t:}. Then Bd,eis a r,m,. neighborhood base
for the diagonal set L1(x) in (X x X, r m<Jr) .
Lemma 2. If (X, tr) is a separated quasi-proximity space, then
r m' -cl(L1(X)) = L1(X).
Theorem l. Let (X, ít) be separated and (X,""' r". ) is compact and pairwise T2 . If for
any xE X and f E r , x andfix) are proximal, then f has a fixed point in X.
Proof: Let L1 denote the family of gauges to generate trin X. Since x andfix) are proximal
we find a member, say, g k E r satisfying d ( gk ( x), gk (! ( x))) <e.
Put N ={Bd,< :dEL1,t:>O}
where Bd,< ={(x,y)E XxX :d*(x,y)< t:}
is a neighbourhood of (X). Then as in Lemma 1 Bd.e is a neighbourhood base for (X).
N is directed by set inclusion relation c.. So we consider ( g k ( x), g k ( f ( x))) <e as a
net in (X x X, r m<Jr' ) , which is compact by virtue of (X, r", r" · ) being assumed to be
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compact. Let { g k; ( x), g k; ( f ( x)) <E} be a convergent subnet of
{ g k (X) , g k ( f (X))} in X X X.
Put lifU{(gk; (x), gk; (f (x))) = (z,u )E X x X}.
Since the net is frequently in N, it follows that
(z,u)E N T""". -ct(Bd,E ) i.e., (z,u)E T"""-Cl(~(x))
i.e., ( z, u) E ~ ( x), by Lemma 2
Hence z =u. From above we have lim gk; ( x) = z and lim gk, (f ( x)) =u.
1 1
.. .... .. (1)
By continuity of d : (X x X, r """. ) ~ (9\, r") and using the fact that con vergence of a
sequence of reals to a real number with respect to usual topology of reals implies its
convergence with respect to r"with limit unchanged.
For d E ~we have
d(z,u)=d(lifllgk, (x),Iif118k, (j(x)))
= d ( lifll g k; (X) ,f ( lifll g k, (X)) )
=d( z,f(z)).
As z =u by (1), we have d(u,f(u)) =O.
That means {u}Jr{f(u)}. As Jris ~eparated we have f(u) =u.
Theorem 2: If Jr is a quasi-proximity on X and (X, r", r"*) is pairwise T2 and compact
and I'is commutative semigroup of quasi-proximal mapping, then each pair of points in
(X ,Jr) is proximal.
Proof: Let LI denote the family of all gauges to generate Jron X .
Let x, y E X and assume that they are not proximal. Then for sorne d E O and r > O, we
have μ=inf{d(g(x) ,g(y)):gE G}. Then clearly, we have μ?.r. Take a member
g1E G suchthat μ:s;d(g1(x),g1(y))<2μ.
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1
1+- μ., ,2μ
2
. 1
Then we ha ve μ., ~ μ, < min 1 + - μ1, μ2 , continuing this process we produce a
2
sequence { Ín } E G and a sequence satisfying
(i) μ ~ μn ~ μn+I < 2μ
('") 1 1 lll μn+I < +-- μn.
n+l
Let us now consider sequen ces Un (x)} and Un (y)}. Sin ce (X, T,, , r,,. ) is compact,
(X, Td) is also compact where (X, Td) is the induced topology on X by d.
Hence there is a subsequence { n;} of positive integers such that
{!n, ( x)}, {!n; (y)} and {μn;} are convergent.
Put lim fn (x) =u, Iimfn (y) =v, and limμ" =a ..
1 1 t 1 / 1
Clearly, a~μ~ r.
Also we have
μn,l ~μn,-1 ~d(fn; (x), Ín; (y))
l+-
n¡
1
~ l+- μn
n. '
1
Taking limitas i ---7oc in (2) one get d (u, v) =oc.
If gis any member of r, we ha ve by continuity of g ,
!~~ d ( g Ín, (X) , g Ín; ( Y)) = d ( g (U) , g (V)) .
Since d(g Í n, (x), gfn;(Y))~μn; always,wehave,
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...... ... (2)
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d ( g (u), g ( v)) :;=::a, a contradiction that I' is proximally contractive. The proof is now
complete.
Theorem 3. Let 1í be separated quasi-proximity on X and the bitopological space
(X , T;r, rlt') is compact and pairwise ~ . Let I' denote a commutative semigroup of quasi
proximal mapping on (X , 7tJ such that I' is proximally contractive. Then I' has a unique
common fixed point.
Proof: Let f E I' and x E X. By Theorem 2, x andf(x) are proximal, and Theorem 1
applies. If I denotes the identity mapping belonging to I' .The subfamily (/, f) now
possesses a common fixed point. We show that every finite subfamily of I'has a common
fixed point.
Since z and fiz) are proximal, there is a net { g k,} , g k, E I' as can be seen in the proof of
Theorem 1, such that lim gk, ( z) =u ( say ).
1
For 1 -:5.j -:5. m, we have
= limgk, ( z)
1
=u.
Hence u is a common fixed point of { J;, f 2 ,. • ., f m} . N ow suppose f E I' and let
<l> (f) denote the set of fixed points of f in (X , n-). If x is an accumulation point of
<l> (f), let {xk} be a net in <l> (f) such that Iimxk = x.
k
By continuity of f, we have
lim f ( xk ) = f ( x)
i.e., lim xk = f ( x), since xk E <1> (f)
k
i.e., x = fix), since (X ,n,n' ) is pairwise T;_. Hence <l> (f) is T;r - closed.
Finally, let <l>={<l> (f): f En
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By argurnent above <I> has finite intersection property, by cornpactness of (X, r,,.), the set
Let x, y E <I> with x =f:. y. Since (X ,1l) is separated we have
d(x,y)=r>O
for sorne gauge d for (X ,1l). Since I'is proxirnally contractive we have
d(g(x), g(y)) < r for sorne rnernber g E I',
.. .. .... .. (3)
i.e., d(x, y) < r. This is the desired contradiction in view of (3). Hence <I> is a singleton
i.e. the family possesses a unique comrnon fixed point in X .
References:
l. Jas Manoranjan and Banerjee Chhanda, Quasi-proximity and associated
bitopological spaces, Indian J. Pure Appl. Math, 1988, 12(8), 945.
2. Singal M. K. and Lal Sundar, Biquasi-proximity spaces and cornpactification of
a pairwise proximity space, Kyunpook Math. Jr.,1973, 13 (1), 41.
3. Jas Manoranjan and Baisnab A. P., Positive definiteness in Quasi-proximity
spaces and Fixed point theorerns ,Bull. Cal. Math. Soc. 1988, 80, 153.
4. Kelly J. C., Bitopological spaces, Proc. Lond. Math. Soc.,1963, 13(3), 71.
5. Kelly J. L. ,General Topology, D. Van Nostrand Cornpany. Inc., 1955.
6. Holrnes R. D., On contractive sernigroups, Pacific Jr. ofMath.,1971 , 37 (3), 701.
7. Bhakta P.C. and Chakrabarti B., On contractive semigroups of rnappings on
uniforrn spaces, Bull of the institute of Mathernatics Academia Sinica, 1990, 18.
8. Chikkala Raghu and Baisnab A. P., Fixed point theorerns in quasi -proximity
spaces, Joumal of the Indian Math. Soc., 1997, 63(1-4), 235.
Address for comrnunication:
l. Departrnent ofMathernatics, The University ofBurdwan, Burdwan-713104, W.
B., India.
2. Departrnent of Mathernatics ,Burdwan Raj College , Burdwan - 713 104, West
Bengal, India.
3. Departrnent of Mathernatics, Bengal Engineering and Science University,
Shibpur, Howrah-711 103, West Bengal, India
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