Contribution of Fixed Point Theorem in Quasi Metric Spaces

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Contribution of Fixed Point Theorem in Quasi Metric
Spaces
Dr.Ayaz Ahmad,
Associate Professor, Department of Mathematics,
Millat College, Darbhanga, Bihar- 846004, India
ayazahmadmillatcollege@gmail.com
Manuscript History
Number: IJIRIS/RS/Vol.04/Issue07/JYIS10080
Received: 05, June 2017
Final Correction: 16, June 2017
Final Accepted: 25, June 2017
Published: July 2017
Citation: Ayaz.,Ahamad;(2017),'' Contribution of Fixed Point Theorem in Quasi Metric Spaces”, Associate
Professor, Department of Mathematics, Millat College, Darbhanga, Bihar, India.
Editor: Dr.A.Arul L.S, Chief Editor, IJIRIS, AM Publications, India
Copyright: ©2017 This is an open access article distributed under the terms of the Creative Commons Attribution License,
Which Permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are
credited
Abstract: In this paper we study on contribution of fixed point theorem in Metric spaces and Quasi Metric spaces.
Key words: Metric space, Contraction Mapping, Fixed point Theorem, Quasi Metric Space, p-Convergent, p-orbit
ally continuous.
Definition: 1 (Metric Space) Let X be a non-empty set-A function XxX →R (the set of reals) such that p:XxX→R is
called a metric or distance function (if ad only if) p satisfies the following conditions.
(i) p (x,y) ≥ 0 for all x, y x
(ii) p (x,y)= 0 if x=y
(iii) p (x,y) = p (y,x) for all x,y x
(iv) p (x,y) ≥ p (x,z)+ p (z ,y) for all x,y,z X
If p is a metric for X, then the pair (X, p) is called a metric space.
Definition: 2 (Cauchy sequence) Let (X,p) be a metric space .Then a sequence { xn} of points of X is said to be a
Cauchy sequence if for each >0,there exists a positive integer n0 such that m,nn0 implies p(xm,xn) < .
Definition: 3 (Completeness): A metric space(X,p) is said to be complete if every Cauchy sequence in X
converges to point of X.
Definition: 4 (Contraction mapping) Let (X, p) be a complete metric space. A mapping T:XX is said to be a
contraction mapping if there exists a real number  with 0 << 1 such that, p{T(x) ,T(y)} )  p (x,y) <p(x,y)  x, y,
 X
i.e In a contraction mapping, the distance between the images of any two points is less than the distance between
the points.

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Theorem: 5 .Let (X,p) be a complete metric space and let T be a contraction mapping defined on X. Then there
exists one and only one point x X such that T(x)=x, i.e, there exist a unique fixed point in X.
Proof: Let T is a contraction mapping on X there exists a real number  with 0 << 1,such that p{T(x) ,T(y)} ) 
p (x,y) where x,y X.
Now choose any point x0 X, Let us define a sequence { xn}by, x1=T(xo), x2=T(x1),
x3=T(x2),,………………………. xn1=T(xn),Then xn=Tn(x0),  n  N.
Thus the sequence { xn} can also be written as{x0,T(x0),T2(x0),…………Tn(x0)}
Now we shall show that the sequence { xn} is a Cauchy sequence.
For each positive integer n, we have P(xn,xn+1)=p{T(xn-1),T(xn)}
p(xn-1,xn)
p2(xn-2,xn-1)
p3(xn-3,xn-2)
……………………………
……………………………..
pn(x0,x1),
By triangle inequality, we have nm,
P(xm,xn) p (xm,xm+1) + p (xm+1,xm+2) + p (xm+2,xm+3) +……………..+ p (xn-1,xn)
mp (x0,x1) + m+1p(x0,x1) + m+2p (x0,x1),+………………..+ n-1p (x0,x1)
=m[1+ +2+………….+m-n-1]p(x0,x1)
= m (1-n-m-)p(x0,x1)/(1-) ( series are in G.P, whose c.r= )
< m p(x0,x1)/(1-)
 0 as m   since  < 1.
Hence { xn}is Cauchy sequence in X.
Let xnx, Since X is complete, so x X
Further ,since T is continuous ,therefore we have , T(x)=T(lim n  xn )= lim n  T( xn)= lim n  xn+1=x
Hence T(x)=x,This shows that x is a fixed point.
To prove uniqueness, Let T(y)=y for some y X is another fixed point. Then T(x)=x ,T(y)=y.
Now p(x.y)=p(Tx,Ty)p(x, y)
p(xn-1,xn)=(1-)p(x,y)0, Since 0 << 1,hence p(x,y) 0
But p is a metric and so p(x,y) 0
Hence p(x,y)=0 which shows that x=y
Thus there is a unique fixed point for T.
Quasi metric spaces
The mathematicians W.A. Wilson (6)J.C. Kelley(3) , Lane E.P(4) and Patty C.W. (5) have studied Quasi metric space
in this study of bi-topological spaces .
Definition:6 (Quasi-metric) Let X be a non-empty set and let p:XxX[0, ) be a function which satisfies
(i) p(x, y) = 0 iff x = y,  x, y  X
(ii) p(x, z)  p(x, y) + p(y, z);  x, y, z  X
Then p is called a Quasi-metric and the pair(X,p) is called a Quasi-metric space. For a quasi-metric p on
X there exists a quasi-metric q on X, called the conjugate of p given by q(x, y) = p(y, x),  x, y  X.
Definition: 7 (p -Convergent)
A sequence { xn} of points of X is said to be p-converge at a point x  X. If p(x, xn)  0 as n   and the sequence {
xn} is called p-convergent in X.

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Definition: 8 Let (x,p) be a quasi-metric space and {xn} be a sequence in X,we say that{xn}is left-cauchy if and
only if for every >0,,there exists a positive integer N=N()such that p(xn,xn) <  for all m  n >N.
Definition: 9 (Completeness)
A sequence { xn} of X is said to be p-Cauchy if and only if for >0, there exists a positive integer K such that p(xm, xn)
<  form m > n  K. If every p-cauchy sequence in quasi-metric space (X, p) is p-convergent in X then (X, p) is said
to be complete.
Definition:10 A self mapping T on a quasi-metric space (X, p) is said to be orbital continuous at x0 of X if for some
x  X2p (x, xni)  0 implies p(Tx, Txni)  0, where { xni} is a subsequence of sequence {xn} given by Txn = xn+1, n = 0,
1, 2, …..
In 1991,Chikkala R. and Baisnab A.P. [1] have proved the following fixed point theorem on complete quasi-metric
spaces.
Definition: 11 Let (x,p) be a complete quasi-metric space. Let T: X  X be a self-mapping which satisfies the
following conditions:
P(Tx, Ty)   [q(x, Tx) + q(y, Ty)] +  p(x, y)…………… (1)
Where ,   0 such that 2 +  < 1
T is p-orbitally continuous at some point x0 of x, …………… (2)
Then there is a unique fixed point of T in X. We generalize it for a complete quasi-metric space
Definition: 12
Let T be a self mapping of a complete quasi-metric space (x,p) satisfying the following conditions :
p(Tx, Ty)   [(x, Tx) + q(y, Ty)]
+ 
   
 yx
TyyTxxq
,
,,
+ (x, y) ………………. (3)
Where , ,   0 such that 2 +  +  < 1
T is d-orbitally continuous at some point x0 of X ……….. (4)
Then T has a unique fixed point in X.
Proof:
We define a sequence { xn} in X given by xn+1 = Txn , for n = 0, 1, 2, …. Let xn  xn+1.
Then on suitable application of (1) we obtain
p(x2, x1) = p(Txi, Tx0)
  [q(x1, x2) +q (x0, x1)]
+ 
   
 01
1021
,
,,
xxp
xxqxxq
+ p(x1, x0) =  [p(x2, x1) + q(x0, x1)]
+ 
   
 10
1012
,
,,
xxq
xxqxxp
+ q(x0, x1)
which implies p(x2, x1)  hq (x0, x1),
where
1
1






h
p(x3, x2) = p(Tx2, Tx1)
  [q(x2, x3) +q (x1, x2)]

_________________________________________________________________________________________________
+ 
   
 12
2132
,
,,
xxp
xxqxxq
+ p(x2, x1)
=  [p(x3, x2) +q (x1, x2)]
+ 
   
 21
2123
,
,,
xxq
xxqxxp
+ q d(x1, x2)
Which implies p(x3, x2)  hq (x1, x2) = hp(x2, x1)  hq
2 (x0, x1)
By induction, we get
p(xn+1, xn)  hq
n (x0, x1) ………………… (5)
Now for m > n, we get an inequality.
Now, p(xm, xn)  p(xm, xm-1) +p (xm-1, xm-2)+ …… +p (xn+1, xn)
 (hm-1 + hm-2 +……. + hn) q(x0, x1)
 h
hn
1 q(x0, x1)  0 as n  
This shows that { xn} is a Cauchy sequence in complete space X and therefore there exists a point z in X such that p
lim xn = z.
By the orbital continuity of self-mapping T we get
Tz = p lim Txn = p lim xn+1 = z, which shows that z is a fixed point of T.
We now prove its uniqueness for which let  be another fixed point of T. Then by applying the condition (1), we
get
p(z, ) = p(Tz, T)   [ (z, Tz) +q ( . )]
+ 
   
 

,
,,
zp
qzzq
+ p (z, )
p(z,  ) = p(Tz, T)  p (z, ) < (z, )
which is contradiction
Thus, T has a unique fixed point in X.
Definition: 13.
Let (X, p) be a complete quasi-metric space. Let T : XX be a self contradiction mapping which satisfies the
condition.
P(Tx, Ty)   max { p(x, y), ½ [ q(x, Tx) + q(y, Ty) ],
½ [ q(x, Ty) + q(y, Tx)] } ………………….. (6)
Where   [0, 1], then T has a unique fixed point.
Proof:
Let us define a sequence { xn} mix such that xn+1 = Txn, for n = 0, 1, 2, … and xn+1  xn.
By suitable application of (6), we get
p(x2, x1) = p(Tx1, Tx0)
  max { p(x1, x0), ½ [q (x1, x2) +q (x0, x1) ],
½ [q (x1, x1) + q(x0, x2)] }
Hence, we obtain an inequality
p(x2, x1)  p (x1, x0) = q (x0, x1),
 p(x2, x1)  

2 p(x0, x1),
or p(x2, x1)  2

q(x0, x2)  2

[p (x2, x1) +q (x0, x2)]

_________________________________________________________________________________________________
i.e. p(x2, x1)  

2 q(x0,).
In each case we have p(x2, x1)  hq(x0, x1), where 0  h <1. The remaining part is verifiable from theorem (6).
Theorem.14:
If in theorem x1 (5) condition (1) is replaced by
[p(Tx, Ty)]2  k max { q(x, Tx) q (y, Ty), p(x, Ty) q(y, Tx),
p(x, y) q(x, Tx), p(x, y) q(y, Ty), [p(x, y)]2} ……………………..(7)
where k  [0, 1], then T has a unique fixed point in X.
Proof: Sequence {xn} is defined in the same way as in theorem (6) On applying (7), we get
[p(x2, x1)]2 = [(Tx1, Tx0)]2
 k max { q(x1, x2) q (x0, x1), p(x1, x1) q (x0, x2),
p(x1, x0) q (x1, x2), p(x1, x0) q (x0, x1),
[p(x1, x0)]2 }
= k max { p(x2, x1) q (x0, x1), 0, q(x0, x1) p(x2, x1),
[q (x0, x1)]2, [q(x0, x1)]2 }
From the above it follows that
[p(x2, x1)]2  k p (x2, x1) q (x0, x1) implies,
p(x0, x1)  k1/2 q (x0, x1)
or [p(x2, x1)]2  k [q (x0, x1)]2 implies,
p(x2, x1)  k q (x0, x1)
As k  [0, 1] therefore 0  k < k1/2 < 1. Thus we have p(x2, x1)  h q(x0, x1) where h  [0, 1]. Rest proof is similar to
that of theorem (.6). Uniqueness of the fixed point follows easily.
REFERENCES
1. Chikkaler R. and Baisnab A.P.: An analogue caristic fixed point theorem in a quasi-metric spaces. Proc. Math.
Acad. Sci. India 61(A),11(1991),237-243.
2. Dhage B.C. and Dhobale V.V: Some fixed point theorem in metric spaces. Act Ciencia Inica 11(2) (1985), 138-
142.
3. Kelley J.C.: Bi-topological spaces. Proc.–London Math. Soc. 13(1963),71-89.
4. Lane E.P. : Bi-topological spaces and Quasi uniform spaces. Proc. London Math. Soc. 17(1967), 241-256.
5. Patty, C.W.: Biltopological spaces Duke Math Jr. 34(1967), 387-391.
6. Wilson W.A.: On Quasi Metric spaces. Amer. Math. Jour. 36(1931), 675-684.

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