Rectangular b-Metric Spaces and Contraction Principle
Reny George∗,1 , S.Radenovic2 , K.P Reshma3 and S.Shukla4
1
Present Affiliation : Department of Mathematics College of Science, Salmanbin
Abdulaziz University Al-Kharj, Kingdom of Saudi Arabia
Permanent Affiliation : Department of Mathematics and Computer Science St.
Thomas College, Bhilai Chhattisgarh, India
2
Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120
Beograd , Serbia
3
Department of Mathematics, Government VYT PG Autonomous College, Durg,
Chhattisgarh, India
4
Department of Applied Mathematics S.V.I.T.S. Indore (M.P.), 453331 India
Abstract
The concept of rectangular b-metric space is introduced as a generalization of
metric space and rectangular metric space. An analogue of Banach contraction
principle and Kannan’s fixed point theorem is proved in this space. Our result
generalises many known results in fixed point theory.
Keywords : Fixed points, rectangular metric space, rectangular b-metric space
Corresponding Author: Reny George
E-mail: [email protected]
2000 Mathematics Subject Classification47H10
1
Introduction
Since the introduction of Banach contraction principle in 1922, because of its wide applications, the study of existence and uniqueness of fixed points and common fixed
points has become a subject of great interest. Many authors proved the Banach contraction Principle in various generalised metric spaces. In the sequel Branciari [5] introduced the concept of rectangular metric space (RMS) by replacing the sum of the right
hand side of the triangular inequality in metric space by a three-term expression and
2
proved an analog of the Banach Contraction Principle in such spaces. Since then many
fixed point theorems for various contractions on rectangular metric spaces appeared (see
[2, 3, 9, 14, 15, 16, 17, 18, 20]).
On the other hand, in [4] Bakhtin introduced b-metric spaces as a generalization of
metric spaces. He proved the contraction mapping principle in b-metric spaces that generalized the famous Banach contraction principle in metric spaces. Since then, several
papers have dealt with fixed point theory or the variational principle for single-valued
and multi-valued operators in b-metric spaces (see [6, 7, 8, 10, 11, 12, 13] and the references therein).
In this paper we have introduced the concept of rectangular b-metric space, which
is not necessarily Hausdorff and which generalizes the concepts of metric space, rectangular metric space and b-metric space. Note that spaces with non Hausdorff topology
plays an importnat role in Tarskian approach to programming language semantics used
in computer science (For some details see [21]). An analog of the Banach contraction
principle as well as the Kannan type fixed point theorem in rectangular b-metric spaces
are also proved. Some examples are included which show that our generalizations are
genuine.
2
Preliminaries
Definition 2.1 [4] Let X be a nonempty set and the mapping d : X × X → [0, ∞)
satisfies:
(bM1) d(x, y) = 0 if and only if x = y for all x, y ∈ X;
(bM2) d(x, y) = d(y, x) for all x, y ∈ X;
(bM3) there exist a real number s ≥ 1 such that d(x, y) ≤ s[d(x, z) + d(z, y) for all
x, y, z ∈ X.
Then d is called a b-metric on X and (X, d) is called a b-metric space (in short bMS)
with coefficient s.
Definition 2.2 [5] Let X be a nonempty set and the mapping d : X × X → [0, ∞)
satisfies:
(RM1) d(x, y) = 0 if and only if x = y for all x, y ∈ X;
(RM2) d(x, y) = d(y, x) for all x, y ∈ X;
3
(RM3) d(x, y) ≤ d(x, u) + d(u, v) + d(v, y) for all x, y ∈ X and all distinct points
u, v ∈ X \ {x, y}.
Then d is called a rectangular metric on X and (X, d) is called a rectangular metric
space (in short RMS).
Now we will define a rectangular b-metric space.
Definition 2.3 Let X be a nonempty set and the mapping d : X × X → [0, ∞) satisfies:
(RbM1) d(x, y) = 0 if and only if x = y;
(RbM2) d(x, y) = d(y, x) for all x, y ∈ X;
(RbM3) there exists a real number s ≥ 1 such that d(x, y) ≤ s[d(x, u) + d(u, v) + d(v, y)]
for all x, y ∈ X and all distinct points u, v ∈ X \ {x, y}.
Then d is called a rectangular b-metric on X and (X, d) is called a rectangular b-metric
space (in short RbMS) with coefficient s.
Note that every metric space is a rectangular metric space and every rectangular metric
space is a rectangular b-metric space (with coefficient s = 1). However the converse of
the above implication is not necessarily true.
Example 2.4 Let X = N, define d :

 0,
4α,
d(x, y) =

α,
X × X → X by
if x = y;
if x, y ∈ {1, 2} and x 6= y;
if x or y 6∈ {1, 2} and x 6= y,
where α > 0 is a constant. Then (X, d) is a rectangular b-metric space with coefficient
s = 43 > 1, but (X, d) is not a rectangular metric space, as d(1, 2) = 4α > 3α =
d(1, 3) + d(3, 4) + d(4, 2).
Example 2.5 Let X = N, define
x, y ∈ X and

0,




 10α,
α,
d(x, y) =


2α,



3α,
d : X × X → X such that d(x, y) = d(y, x) for all
if
if
if
if
if
x = y;
x = 1, y = 2;
x ∈ {1, 2} and y ∈ {3};
x ∈ {1, 2, 3} and y ∈ {4};
x or y 6∈ {1, 2, 3, 4} and x 6= y,
where α > 0 is a constant. Then (X, d) is a rectangular b-metric space with coefficient
s = 2 > 1, but (X, d) is not a rectangular metric space, as d(1, 2) = 10α > 5α =
d(1, 3) + d(3, 4) + d(4, 2).
4
Note that every b-metric space with coefficient s is a RbM S with coefficient s2 but
the converse is not necessarily true. (See Example 2.7 below).
For any x ∈ X we define the open ball with centre x and radius r > 0 by
Br (x) = {y ∈ X : d(x, y) < r}
The open balls in RbM S are not necessarily open(See Example 2.7 below). Let U be the
collection of all subsets A of X satisfying the condition that for each x ∈ A there exist
r > 0 such that Br (x) ⊆ A. Then U defines a topology for the RbM S (X, d), which is
not necessarily Hausdorff(See Example 2.7 below).
Now we define convergence and Cauchy sequence in rectangular b-metric space and
completeness of rectangular b-metric space.
Definition 2.6 Let (X, d) be a rectangular b-metric space, {xn } be a sequence in X and
x ∈ X. Then
(a) The sequence {xn } is said to be convergent in (X, d) and converges to x, if for every
ε > 0 there exists n0 ∈ N such that d(xn , x) < ε for all n > n0 and this fact is
represented by lim xn = x or xn → x as n → ∞.
n→∞
(b) The sequence {xn } is said to be Cauchy sequence in (X, d) if for every ε > 0 there
exists n0 ∈ N such that d(xn , xn+p ) < ε for all n > n0 , p > 0 or equivalently, if
lim d(xn , xn+p ) = 0 for all p > 0.
n→∞
(c) (X, d) is said to be a complete rectangular b-metric space if every Cauchy sequence
in X converges to some x ∈ X.
Note that, limit of a sequence in a RbM S is not necessarily unique and also every
RbM S-convergent sequence in a RbM S is not necessarily RbM S-Cauchy. The following
example illustrates this fact.
Example 2.7 Let X = A ∪ B, where A = { n1 : n ∈ N} and B is the set of all positive
integers. Define d : X × X → [0, ∞) such that d(x, y) = d(y, x) for all x, y ∈ X and

0, if x = y;



2α, if x, y ∈ A;
d(x, y) =
α
, if x ∈ A and y ∈ {2, 3};

2n


α, otherwise,
where α > 0 is a constant. Then (X, d) is a rectangular b-metric space with coefficient
s = 2 > 1. However we have the following :
5
1) (X, d) is not a rectangular metric space, as d( 12 , 31 ) = 2α > 17
= d( 12 , 4) + d(4, 3) +
12
1
d(3, 3 ) and hence not a metric space.
2) There does not exist s > 0 satisfying d(x, y) ≤ s[d(x, z) + d(z, y)] for all x, y, z ∈ X,
and so (X, d) is not a b-metric space.
3) B α2 ( 12 ) = {2, 3, 12 } and there does not exist any open ball with centre 2 and contained
in B α2 ( 21 ). So B α2 ( 12 ) is not an open set.
4) The sequence { n1 } converges to 2 and 3 in RbM S and so limit is not unique. Also
1
d( n1 , n+p
) = 2α 6→ 0 as n → ∞, therefore { n1 } is not a Cauchy sequence in RbM S.
T
5) There does not exist any r1 , r2 > 0 such that Br1 (2) Br2 (3) = φ and so (X, d) is not
Hausdorff.
3
Main Results
Following theorem is the analog to Banach contraction principle in rectangular b-metric
space.
Theorem 3.1 Let (X, d) be a complete rectangular b-metric space with coefficient s > 1
and T : X → X be a mapping satisfying:
d(T x, T y) ≤ λd(x, y)
(3.1)
for all x, y ∈ X, where λ ∈ [0, 1s ]. Then T has a unique fixed point.
Proof Let x0 ∈ X be arbitrary. We define a sequence {xn } by xn+1 = T xn for all n ≥ 0.
We shall show that {xn } is Cauchy sequence. If xn = xn+1 then xn is fixed point of T.
So, suppose that xn 6= xn+1 for all n ≥ 0. Setting d(xn , xn+1 ) = dn , it follows from (3.1)
that
d(xn , xn+1 ) = d(T xn−1 , T xn ) ≤ λd(xn−1 , xn )
dn ≤ λdn−1 .
Repeating this process we obtain
dn ≤ λn d0 .
(3.2)
Also, we can assume that x0 is not a periodic point of T. Indeed, if x0 = xn then using
(3.2), for any n ≥ 2, we have
d(x0 , T x0 )
d(x0 , x1 )
d0
d0
=
=
=
≤
d(xn , T xn )
d(xn , xn+1 )
dn
λn d0 ,
6
a contradiction. Therefore, we must have d0 = 0, i.e., x0 = x1 , and so x0 is a fixed
point of T. Thus we assume that xn 6= xm for all distinct n, m ∈ N. Again setting
d(xn , xn+2 ) = d∗n and using (3.1) for any n ∈ N, we obtain
d(xn , xn+2 ) = d(T xn−1 , T xn+1 ) ≤ λd(xn−1 , xn+1 )
d∗n ≤ λd∗n−1
Repeating this process we obtain
d(xn , xn+2 ) ≤ λn d∗0 .
(3.3)
For the sequence {xn } we consider d(xn , xn+p ) in two cases.
If p is odd say 2m + 1 then using (3.2) we obtain
d(xn , xn+2m+1 ) ≤ s[d(xn , xn+1 ) + d(xn+1 , xn+2 ) + d(xn+2 , xn+2m+1 )]
≤ s[dn + dn+1 ] + s2 [d(xn+2 , xn+3 ) + d(xn+3 , xn+4 )
+d(xn+4 , xn+2m+1 )]
≤ s[dn + dn+1 ] + s2 [dn+2 + dn+3 ] + s3 [dn+4 + dn+5 ]
+ · · · + sm dn+2m
≤ s[λn d0 + λn+1 d0 ] + s2 [λn+2 d0 + λn+3 d0 ] + s3 [λn+4 d0 + λn+5 d0 ]
+ · · · + sm λn+2m d0
≤ sλn [1 + sλ2 + s2 λ4 + · · · ]d0 + sλn+1 [1 + sλ2 + s2 λ4 + · · · ]d0
1+λ
=
sλn d0
(note that sλ2 < 1).
1 − sλ2
Therefore,
1+λ
d(xn , xn+2m+1 ) ≤
sλn d0 .
(3.4)
1 − sλ2
If p is even say 2m then using (3.2) and (3.3) we obtain
d(xn , xn+2m ) ≤ s[d(xn , xn+1 ) + d(xn+1 , xn+2 ) + d(xn+2 , xn+2m )]
≤ s[dn + dn+1 ] + s2 [d(xn+2 , xn+3 ) + d(xn+3 , xn+4 )
+d(xn+4 , xn+2m )]
≤ s[dn + dn+1 ] + s2 [dn+2 + dn+3 ] + s3 [dn+4 + dn+5 ]
+ · · · + sm−1 [d2m−4 + d2m−3 ] + sm−1 d(xn+2m−2 , xn+2m )
≤ s[λn d0 + λn+1 d0 ] + s2 [λn+2 d0 + λn+3 d0 ] + s3 [λn+4 d0 + λn+5 d0 ]
+ · · · + sm−1 [λ2m−4 d0 + λ2m−3 d0 ] + sm−1 λn+2m−2 d∗0
≤ sλn [1 + sλ2 + s2 λ4 + · · · ]d0 + sλn+1 [1 + sλ2 + s2 λ4 + · · · ]d0
+sm−1 λn+2m−2 d∗0 ,
7
i.e.
1+λ
sλn d0 + sm−1 λn+2m−2 d∗0
1 − sλ2
1+λ
sλn d0 + (sλ)2m λn−2 d∗0 (as 1 < s)
<
1 − sλ2
1+λ
sλn d0 + λn−2 d∗0 (as λ ≤ 1s ).
≤
2
1 − sλ
d(xn , xn+2m ) ≤
Therefore
d(xn , xn+2m ) ≤
1+λ
sλn d0 + βλn−2 d0 .
1 − sλ2
(3.5)
It follows from (3.4) and (3.5) that
lim d(xn , xn+p ) = 0 for all p > 0.
n→∞
(3.6)
Thus {xn } is a Cauchy sequence in X. By completeness of (X, d) there exists u ∈ X
such that
lim xn = u.
(3.7)
n→∞
We shall show that u is a fixed point of T. Again, for any n ∈ N we have
d(u, T u) ≤ s[d(u, xn ) + d(xn , xn+1 ) + d(xn+1 , T u)]
= s[d(u, xn ) + dn + d(T xn , T u)]
≤ s[d(u, xn ) + dn + λd(xn , u)].
Using (3.6) and (3.7) it follows from above inequality that d(u, T u) = 0, i.e., T u = u.
Thus u is a fixed point of T.
For uniqueness, let v be another fixed point of T. Then it follows from (3.1) that d(u, v) =
d(T u, T v) ≤ λd(u, v) < d(u, v), a contradiction. Therefore, we must have d(u, v) = 0,
i.e., u = v. Thus fixed point is unique.
Example 3.2 Let X = A ∪ B, where A = { n1 : n ∈ {2, 3, 4, 5}} and B = [1, 2]. Define
d : X × X → [0, ∞) such that d(x, y) = d(y, x) for all x, y ∈ X and
 1 1
d( 2 , 3 ) = d( 14 , 51 ) = 0.03


 1 1
d( 2 , 5 ) = d( 13 , 41 ) = 0.02
d( 1 , 1 ) = d( 15 , 31 ) = 0.6


 2 4
d(x, y) = |x − y|2
otherwise
Then (X, d) is a rectangular b-metric space with coefficient s = 4 > 1. But (X, d) is
neither a metric space nor a rectangular metric space. Let T : X → X be defined as :
1
if x ∈ A
4
Tx =
1
if x ∈ B
5
Then T satisfies the condition of Theorem 3.1 and has a unique fixed point x = 14 .
8
Remark 3.3 We say that T : X → X has property P if F (T ) = F (T n ) (see [19]) where
F (T ) = {x ∈ X : T x = x}. It is an easy exercise to see that under the assumptions of
Theorem 3.1, T has property P .
Following theorem is the fixed point result for Kannan type contraction in rectangular
b-metric space.
Theorem 3.4 Let (X, d) be a complete rectangular b-metric space with coefficient s > 1
and T : X → X be a mapping satisfying:
d(T x, T y) ≤ λ[d(x, T x) + d(y, T y)]
(3.8)
1
for all x, y ∈ X, where λ ∈ [0, s+1
]. Then T has a unique fixed point.
Proof 1 Let x0 ∈ X be arbitrary. We define a sequence {xn } by xn+1 = T xn for all
n ≥ 0. We shall show that {xn } is Cauchy sequence. If xn = xn+1 then xn is fixed point
of T. So, suppose that xn 6= xn+1 for all n ≥ 0. Setting d(xn , xn+1 ) = dn , it follows from
(3.8) that
d(xn , xn+1 ) = d(T xn−1 , T xn ) ≤ λ[d(xn−1 , T xn−1 ) + d(xn , T xn )]
d(xn , xn+1 ) = λ[d(xn−1 , xn ) + d(xn , xn+1 )]
dn = λ[dn−1 + dn ]
λ
dn ≤
dn−1 = βdn−1 ,
1−λ
where β =
λ
1−λ
<
1
s
(as, λ <
1
).
s+1
Repeating this process we obtain
dn ≤ β n d0 .
(3.9)
Also, we can assume that x0 is not a periodic point of T. Indeed, if x0 = xn then using
(3.9), for any n ≥ 2, we have
d(x0 , T x0 )
d(x0 , x1 )
d0
d0
=
=
=
≤
d(xn , T xn )
d(xn , xn+1 )
dn
β n d0 ,
a contradiction. Therefore, we must have d0 = 0, i.e., x0 = x1 , and so x0 is a fixed point
of T. Thus we assume that xn 6= xm for all distinct n, m ∈ N. Again using (3.8) and
9
(3.9) for any n ∈ N, we obtain
d(xn , xn+2 ) =
=
≤
=
=
d(T xn−1 , T xn+1 ) ≤ λ[d(xn−1 , T xn−1 ) + d(xn+1 , T xn+1 )]
λ[d(xn−1 , xn ) + d(xn+1 , xn+2 )] = λ[dn−1 + dn+1 ]
λ[β n−1 d0 + β n+1 d0 ]
λβ n−1 [1 + β 2 ]d0
γβ n−1 d0 .
Therefore,
d(xn , xn+2 ) ≤ γβ n−1 d0 ,
(3.10)
where γ = λ[1 + β 2 ] > 0.
For the sequence {xn } we consider d(xn , xn+p ) in two cases.
If p is odd say 2m + 1 then using (3.9) we obtain
d(xn , xn+2m+1 ) ≤ s[d(xn , xn+1 ) + d(xn+1 , xn+2 ) + d(xn+2 , xn+2m+1 )]
≤ s[dn + dn+1 ] + s2 [d(xn+2 , xn+3 ) + d(xn+3 , xn+4 )
+d(xn+4 , xn+2m+1 )]
≤ s[dn + dn+1 ] + s2 [dn+2 + dn+3 ] + s3 [dn+4 + dn+5 ]
+ · · · + sm dn+2m
≤ s[β n d0 + β n+1 d0 ] + s2 [β n+2 d0 + β n+3 d0 ] + s3 [β n+4 d0 + β n+5 d0 ]
+ · · · + sm β n+2m d0
≤ sβ n [1 + sβ 2 + s2 β 4 + · · · ]d0 + sβ n+1 [1 + sβ 2 + s2 β 4 + · · · ]d0
1+β
=
sβ n d0
(note that sβ 2 < 1).
2
1 − sβ
Therefore,
d(xn , xn+2m+1 ) ≤
1+β
sβ n d0 .
2
1 − sβ
(3.11)
10
If p is even say 2m then using (3.9) and (3.10) we obtain
d(xn , xn+2m ) ≤ s[d(xn , xn+1 ) + d(xn+1 , xn+2 ) + d(xn+2 , xn+2m )]
≤ s[dn + dn+1 ] + s2 [d(xn+2 , xn+3 ) + d(xn+3 , xn+4 )
+d(xn+4 , xn+2m )]
≤ s[dn + dn+1 ] + s2 [dn+2 + dn+3 ] + s3 [dn+4 + dn+5 ]
+ · · · + sm−1 [d2m−4 + d2m−3 ] + sm−1 d(xn+2m−2 , xn+2m )
≤ s[β n d0 + β n+1 d0 ] + s2 [β n+2 d0 + β n+3 d0 ] + s3 [β n+4 d0 + β n+5 d0 ]
+ · · · + sm−1 [β 2m−4 d0 + β 2m−3 d0 ] + sm−1 γβ n+2m−3 d0
≤ sβ n [1 + sβ 2 + s2 β 4 + · · · ]d0 + sβ n+1 [1 + sβ 2 + s2 β 4 + · · · ]d0
+sm−1 γβ n+2m−3 d0 ,
i.e.
1+β
sβ n d0 + sm−1 γβ n+2m−3 d0
2
1 − sβ
1+β
sβ n d0 + γ(sβ)2m β n−3 d0 (as 1 < s)
<
2
1 − sβ
1+β
sβ n d0 + γβ n−3 d0 (as β ≤ 1s ).
≤
1 − sβ 2
d(xn , xn+2m ) ≤
Therefore
d(xn , xn+2m ) ≤
1+β
sβ n d0 + γβ n−3 d0 .
1 − sβ 2
(3.12)
It follows from (3.11) and (3.12) that
lim d(xn , xn+p ) = 0 for all p > 0.
n→∞
(3.13)
Thus {xn } is a Cauchy sequence in X. By completeness of (X, d) there exists u ∈ X
such that
lim xn = u.
(3.14)
n→∞
We shall show that u is a fixed point of T. Again, for any n ∈ N we have
d(u, T u) ≤
=
≤
=
(1 − sλ)d(u, T u) ≤
s[d(u, xn ) + d(xn , xn+1 ) + d(xn+1 , T u)]
s[d(u, xn ) + dn + d(T xn , T u)]
s[d(u, xn ) + dn + λ{d(xn , T xn ) + d(u, T u)}]
s[d(u, xn ) + dn + λ{d(xn , xn+1 ) + d(u, T u)}]
s[d(u, xn ) + β n d0 + λd(xn , xn+1 )]
11
1
Using (3.13) and (3.14) and the fact that λ < s+1
, it follows from above inequality that
d(u, T u) = 0, i.e., T u = u. Thus u is a fixed point of T.
For uniqueness, let v be another fixed point of T. Then it follows from (3.8) that d(u, v) =
d(T u, T v) ≤ λ[d(u, T u) + d(v, T v)] = λ[d(u, u) + d(v, v)] = 0. Therefore, we have
d(u, v) = 0, i.e., u = v. Thus fixed point is unique
Remark 3.5 On the basis of discussion contained in this paper, we have the following:
1) The open ball defined in b-metric space, RMS and RbMS are not necessarily open set.
2) The collection of open balls in RbMS, RMS and b-metric space do not necessarily
form a basis for a topology.
3) RbMS, RMS and b-metric space are not necessarily Hausdorff.
OpenProblems :
1) In Theorem 3.1 can we extent the range of λ to the case
1
s
< λ < 1.
2) Prove analogue of Chatterjee contraction, Reich contraction, Ciric contraction and
Hardy-Rogers contraction in RbM S.
References
[1] Aydi H, Bota M.F, Karapinar E, Moradi S, A common fixed pointfor weak φcontractions on b-metric spaces, Fixed Point Theory, 13(2), 337-346,2012.
[2] Azam A, Arshad M: Kannan Fixed Point Theorems on generalised metric spaces,
J. Nonlinear Sci. Appl. 1, 45-48, 2008.
[3] Azam A, Arshad M, Beg I: Banach contraction principle on cone rectangular metric
spaces, Applicable Analysis and Discrete Math, 3, 236-241, 2009.
[4] Bakhtin I.A., The contraction mapping principle in quasimetric spaces, Funct.
Anal., Unianowsk Gos. Ped. Inst. 30 (1989), 2637.
[5] Branciari A :A fixed point theorem of Banach-Caccippoli type on a class of generalised metric spaces, Publ. Math. Debrecen, 57, 31-37, 2000.
[6] Boriceanu M, Strict fixed point theorems for multivalued operators in b-metric
spaces, International J. Modern Math., 4(3), 285-301, 2009.
[7] Boriceanu M., Bota M., Petrusel A., Mutivalued fractals in b-metric spaces, Cen.
Eur. J. Math. 8(2)(2010), 367-377.
12
[8] Bota M., Molnar A., Csaba V., On Ekeland’s variational principle in b-metric
spaces, Fixed Point Theory, 12(2011), 21-28.
[9] Chen C.M, Common fixed point theorem in complete generalised metric spaces, J.
Appl. Math., Article ID 945915, 2012.
[10] Czerwik S, Contraction mappings in b-metric spaces, Acta. Math. Inform. Univ.
Ostraviensis, 1, 5-11, 1993.
[11] Czerwik S., Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem.
Mat. Univ. Modena, 1998, 46, 263-276.
[12] Czerwik S, Dlutek K, Singh S.L, Round-off stability of iteration procedures for operators in b-metric spaces, J. Natur. Phys. Sci. , 11, 87-94, 1997.
[13] Czerwik S, Dlutek K, Singh S.L, Round-off stability of iteration procedures for set
valued operators in b-metric spaces, J. Natur. Phys. Sci. , 15, 1-8, 2001.
[14] Das P : A fixed point theorem on a class of generalised metric spaces, Korean j.
Math. Sci. 9, 29-33, 2002.
[15] Das P : A fixed point theorem in generalised metric spaces, Soochow J. Math, 33,
33-39, 2007.
[16] Das P, Lahri BK: Fixed point of a Ljubomir Ciric’s quasi-contraction mapping in a
generalised metric space, Publ. Math. Debrecen, 61, 589-594, 2002.
[17] Das P, Lahri BK: Fixed Point of contractive mappings in generalised metric space,
Math. Slovaca, 59, 499-504, 2009.
[18] Inci M Erhan, E Karapinar, Tanja Sekulic Fixed Points of (psi, phi) contractions on generalised metric spaces, Fixed Point Theory and Appl 2012, 2012:138,
doi:10.1186/1687-1812-2012-138.
[19] Jeong G.S, Rhoades B.E, Maps for which F (T ) = F (T n ), Fixed Point Theory Appl.,
6, 2005.
[20] Lakzian H, Samet B: Fixed Points for (ψ, φ)-weakly contractive mapping in generalised metric spaces, Appl Math Lett, 25, 902-906, 2112.
[21] Mathews S.G, Partial Metric Topology, Papers on general topology and applications, Eighth summer conference at Queens college, Annals of New York Academy
of Sciences, Vol 728, 183-197.