ON DISLOCATED METRIC SPACES
Dr. I. Rambhadra Sarma and P. Sumati Kumari*
Department of Mathematics, FED – I, K L University, Green fields, Vaddeswaram, A.P, 522502, India
E-mail: [email protected]
(Received on: 23-12-11; Accepted on: 12-01-12)
________________________________________________________________________________________________
ABSTRACT
T
he notion of dislocated metric is one of the various generalizations of metric that retains a variant of the illustrious Banach’s Contraction principle and has useful applications in the semantic analysis of logic programming. The purpose of this note is to study topological aspects of a dislocated metric space and prove a dislocated metric version of Seghal’s fixed point theorem which ultimately implies existence(and uniqueness in some cases) of a fixed point for self maps that satisfy conditions analogous to those of Banach, Kannan, Bianchini, Reitch and Rakotch [4].Keywords: dislocated metric, kuratowski’s axioms, coincidence point, Contractive conditions.
Subject classification: 47H10.
________________________________________________________________________________________________
INTRODUCTION:
Pascal Hitzler [1] presented variants of Banach’s Contraction principle for various modified forms of a metric space including dislocated metric space and applied them to semantic analysis of logic programs. In this context Hitzler raised some related questions on the topological aspects of dislocated metrics.
In this paper we present results that establish existence of a topology induced by a dislocated metric and show that this topology is amortizable, by actually showing a metric that induces the topology.
Rhoades [4] collected a large number of variants of Banach’s Contractive conditions on self maps on a metric space and proved various implications or otherwise among them. We pick up a good number of these conditions which ultimately imply Seghal’s condition [4]. We prove that these implications hold good for self maps on a dislocated metric space and prove the dislocated metric version of Seghal’s result there by deriving the dislocated analogue’s of fixed point theorems of Banach, Kannan, Bianchini, Reich and others[4].
1. THE
d
-TOPOLOGY:Definition 1.1: Let
X
be a set andd
:X
×X
R
be a mapping satisfying the following conditions forx
,y
,z
in
X
.(i)
d
(x
,y
) o(ii)
d
(x
,y
) =d
(y
,x
)(iii)
d
(x
,y
) = 0 impliesx
=y
and(iv)
d
(x
,y
)d
(x
,z
) +d
(z
,y
)Then
d
is called dislocated (simplyd
-) metric onX
and the pair (X
,d
) is called a dislocated (d
-) metric space. In what follows (X
,d
) stands for ad
metric space.If
x
∈
X
and∈
>0 the set,B
∈(x
) = {y
/y
∈
X
andd
(x
,y
) <∈
} is called the open ball with centre atx
and radius∈
.Definition 1.2: We say that a net (
x
α /∈ ∆
) inX
converges tox
in (X
,d
) and writeα
lim
(x
α /∈ ∆
) =x
ifα
lim
d
(x
α,x
) = 0.Remark: The limit of a net in (
X
,d
) is unique.Notation: For
A
⊂
X
we writeD
(A
) = {x
∈
X
/x
is a limit of a net in (X
,d
)}Proposition 1.3: Let
A
⊆
X
andB
⊆
X
.Then (i)D
(A
) =φ
ifA
=φ
(ii)
D
(A
)⊆
D
(B
) ifA
⊆
B
(iii)
D
(A
∪
B
) =D
(A
)∪
D
(B
) and (iv)D
(D
(A
) )⊆
D
(A
)Proof: (i) and (ii) are clear. That
D
(A
)∪
D
(B
)⊆
D
(A
∪
B
) follows from (ii). To prove the reverse inclusion, Letx
∈
D
(A
∪
B
) andx
=∆ ∈
α
lim
(x
α) where (x
α /∈ ∆
) is a net inA
∪
B
. If∃
λ
∈ ∆
such thatx
α∈
A
for∈ ∆
andλ
then (x
α /λ
,∈ ∆
) is a cofinal subnet of (x
α /∈ ∆
) andis in
A
andλ
α≥
lim
d
(x
,x
α) =∆ ∈
α
lim
d
(x
,x
α) =0 so thatx
∈
D
(A
). If no suchλ
exists in∆
then for every∈ ∆
, choose ( )∈ ∆
such that ( ) andx
β( )α∈
B
.Then (x
β( )α /∈ ∆
) is a cofinal subnet inB
of(
x
α /∈ ∆
) and∆ ∈
α
lim
d
(x
β( )α ,x
) =∆ ∈
α
lim
d
(x
α,x
) =0 so thatx
∈
D
(B
).It now follows thatD
(
A
∪
B
)⊆
D
(A
)∪
D
(B
) and hence (iii) holds. To prove (iv) letx
∈
D
(D
(A
)),x
=∆ ∈
α
lim
x
α,x
α∈
D
(A
) for∈ ∆
, and∀
∈ ∆
, let (x
β
α /
∈
∆
( ))be a net inA
∋
x
α=βlim
∈∆( )αx
αβ .For eachpositive integer
i
∃
i∈ ∆
such thatd
(x
i
α ,
x
)<i
1
,and i
∈ ∆
( i)∋
d
(x
i iβ
α ,
x
αi)<i
1
. Write i iβ =
γ
i∀
i
, then {γ
1,γ
2,....} is directed set withγ
i <γ
j ifi
< j, andd
(x
i
γ ,
x
)d
(x
γi,x
αi) +d
(x
αi,x
) <i
2
.This implies that
x
∈
D
(A
).As a corollary, we have the following
Corollary 1.4: If for
A
⊂
X
andA
=A
∪
D
(A
), then the operationA
→
A
on P(X
) satisfies Kuratowski’s closure axioms [2 ]:(i)
φ
=φ
(ii)
A
⊂
A
(iii)
A
=A
and(iv)
A
∪
B
=A
∪
B
.Consequently we have the following
Theorem 1.5: Let
ℑ
be the family of all subsetsA
ofX
for whichA
=A
andτ
be the complements of members ofℑ
. Then theℑ
is a topology forX
and theℑ
-closure of a subsetA
ofX
isA
.Definition 1.6: The topology
ℑ
obtained in Theorem 1.5 is called the topology induced byd
and simply referred to as thed
-topology ofX
and is denoted by (X
,d
,ℑ
).1 / # # " 2 Proof: If
x
∈
D
(A
),there exist a net (x
α /∈ ∆
) inA
such that limx
α =x
, ifδ
>0∃
α
δ∈ ∆
such thatd
(x
α ,x
)<δ
.If∈ ∆
and≥
α
δ.Hencex
α∈
B
δ(x
)∩
A
for≥
α
δ. Conversely if for everyδ
>0,B
δ(x
)∩
A
≠
φ
. We choose onex
ninn
B
1(x
)∩
A
for every integer1
≥
n
.Clearly (x
n)is a net inA
andd
(x
n ,x
)<n
1
, so that n
lim
x
n =x
. Hencex
∈
A
.Corollary 1.8:
x
∈
A
⇔
x
∈
A
orB
δ(x
)∩
A
≠
φ
∀
δ
>0.Corollary 1.9:
A
is open in (X
,d
,ℑ
) if and only if for everyx
∈
A
,∃
δ
>0 such that {x
}∪
B
δ(x
)⊆
A
.Proposition 1.10: If
x
∈
X
andδ
>0 then {x
}∪
B
δ(x
) is an open set in(X
,d
,ℑ
).Proof: Let
A
= {x
}∪
B
δ(x
) ,y
∈
B
δ(x
) and 0<r
<δ
-d
(x
,y
) ThenB
r(y
)⊆
B
δ(x
)⊂
A
so thatB
r(y
)∩
(X
-A
) =φ
Hence (
X
-A
) is closed in (X
,d
,ℑ
).Corollary 1.11: If
x
∈
X
andV
r(x
)=B
r(x
)∪
{x
} forr
>0 then the collection {V
r (x
) /x
∈
X
} is an open base atx
in (X
,d
,ℑ
).Ifd
is a metric andV
=B
r(x
) ,ℑ
coincides with the metric topology.Proposition 1.12: (
X
,d
,ℑ
) is a Hausdorff space.Proof: If
x
,y
∈
X
, andd
(x
,y
)>0 thenV
2
δ(
x
)∩
V
2
δ(
y
) =φ
.Corollory 1.13: If
x
∈
X
, the collection {V
r(x
) /x
∈
X
} is an open base atx
for (X
,d
,ℑ
).Hence (X
,d
,ℑ
) is first countable.Remark: Corollory 1.13 enables us to deal with sequence instead of nets.
Proposition 1.14: Define
ρ
onX
xX
byρ
(x
,y
)==
≠
y
x
if
y
x
if
y
x
d
0
)
,
(
.Here
ρ
is a metric onX
.For
x
∈
X
and∈
>0 ,V
∈(x
) ={y
∈
X
/ρ
(x
,y
) <∈
}.Moreover a sequence (
x
n) inX
converges tox
in (X
,ρ
) if and only if(i)
x
n=x
except for finitely manyn
or(ii) lim
d
(x
k
n ,
x
) =0 for every subsequence(x
nk) of (x
n) withx
nk≠
x
. 2. CONTINUITY:Definition 2.1: Let (
X
,d
) and (y
,d
′
) bed
metric spaces.f
:X
→
y
is said to bed
continuous atx
∈
X
if for every∈
>0∃
δ
>0 such thatf
(B
δ(x
) )⊆
B
∈(f
(x
) ).f
isd
continuous iff
isd
continuous at everyx
inX
.Theorem 2.2:
f
: (X
,d
)→
(y
,d
′
) isd
continuous atx
if and only iff
: (X
,d
,ℑ
)→
(y
,d
′
,ℑ′
) is continuous atx
whereℑ
andℑ′
are the corresponding induced topologies.Proof: Assume that
f
isd
continuous atx
. Let V be a neighbourhood off
(x
) in (y
,d
′
,ℑ′
).Then∃
∈
>0∋
B
∈(f
(x
) )⊆
V
. By hypothesis,∃
δ
>0∋
f
(B
δ(x
) )⊆
B
∈(f
(x
) )Since {
x
}∪
B
δ(x
) is open in (X
,d
,ℑ
) into (y
,d
′ ℑ′
). It follows thatf
from (X
,d
,ℑ
) into (y
,d
′ ℑ′
) is continuous atx
. And∈
>0,B
∈(f
(x
)∪
f
(x
))∈
ℑ′
. So∃
δ
>0∋
f
({x
}∪
B
δ(x
)}⊆
B
∈(f
(x
)∪
f
(x
))Hence
f
(B
δ(x
))⊆
B
∈(f
(x
))f
:X
→
y
isd
continuous.Corollary 2.3: If (
X
,d
) , (y
,d
′
) ared
metric spaces andρ
,ρ
′
are the induced metrics corresponding tod
andd
′
respectively then,f
: (X
,d
)→
(y
,d
′
) isd
continuous atx
if and only iff
: (X
,ρ
)→
( Y,ρ
′
) is continuous.Let (
X
,d
) be ad
metric space andf
:X
→
X
be a mapping. WriteV
(x
) =d
(x
,f
(x
) ) and Z(f
)={x
/V
(x
)=0}.Clearly every point of Z(f
) is a fixed point off
but the converse is not necessarily true. We callpoints of Z (
f
) as coincidence points off
.The set Z (f
) is a closed subset ofX
. Mathew’s theorem [3] statesthat a contraction on a complete
d
metric space has a unique fixed point. The same theorem has been justified by an alternate proof by Pascal Hitzler [1].We present an extension of this theorem for coincidence points.3. MAIN RESULTS:
Theorem 2.4: Let (
X
,d
) be a completed
metric space andf
:
X
→
X
be a contraction. Then there is a unique coincidence point forf
.Proof: For any
x
∈
X
the sequence of iterates satisfiesd
(
f
n( )
x
,
f
n+1( )
x
)
≤
α
nd
(
x
,
f
( )
x
)
where
α
is any contractive constant. Consequently ifn
<m
d
(f
n( )
x
,f
m( )
x
)≤
(
α
n+
α
n+1+
...
+
α
m−1)
d
(
x
,
f
( )
x
)
=
(
)
α
α
α
−
−
−1
1
m nn
d
(
x
,
f
(
x
)
)
Hence {
f
n( )
x
} is Cauchy sequence inX
.If
ξ
=f
n( )
x
n
lim
thenf
(
ξ
)
=f
n( )
x
n 1
lim
+so
d
(ξ
,f
(ξ
) ) =lim
d
(
f
n( )
x
,
f
n1( )
x
)
n+
. Since
d
(
f
n( )
x
,
f
n+1( )
x
)
<α
nd
(
x
,
f
(
x
)
)
Since 0<
α
<
1
; limα
nd
(
x
,
f
(
x
)
)
= 0 Henced
(ξ
,f
(ξ
) )=0Uniqueness : If
d
(ξ
,f
(ξ
) )=d
(η
,f
(η
))=0, thenf
(ξ
) =ξ
and f(η
)=η
so thatd
(ξ
,η
)≤
d
(ξ
,f
(ξ
)) +d
(f
(ξ
) ,f
(η
)) +d
(f
(η
) ,η
)≤
α
d
(ξ
,η
)so that
d
(ξ
,η
)=0. Henceξ
=η
.Theorem 2.5: Let (
X
,d
) be ad
metric space andf
:X
→
X
be continuous .Assume thatd
(f
(x
) ,f
(
y
)) < max {d
(x
,f
(x
)) ,d
(y
,f
(y
)),d
(x
,y
)} wheneverd
(x
,y
)≠
0.Thenf
has a unique coincidence point wheneverclO
(
x
)
is nonempty for somex
∈
X
.1 / # # " 4 If
x
∉
Z then V(f
(x
))=d
(f
(x
) ,f
2(
x
))
< max {
d
(x
,f
(x
)) ,d
(f
(x
) ,f
2(
x
))
,
d
(x
,f
(x
))}= max {V (
x
) , V (f
(x
)) }V (
f
(x
)) < V (x
) whenever V (x
)≠
0
i.e.x
∉
Z (1)If
O
(
x
)
∩
Z=φ
then V (f
k+1(x
)) < V (f
k(x
))∀
k
Hence V (
f
n( )
x
) is convergent. (2)let
ξ
be a cluster point ofO
(
x
)
,∃
(n
i)↑
∋
ξ
=lim i nf
(x
)( )
ξ
k
f
=limf
ni+k( )
x
O
(ξ
)≤
clO
(
x
)
.Since V is continuous V (
f
k( )
ξ
) = lim V (f
ni+k( )
x
)Since
O
(x
)∩
Z =φ
by (2) { V (f
n( )
x
) } is convergent.Let
γ
= lim V( i nf
(x
) ) = V ( lim i nf
(x
) ) = V (ξ
) .Also
γ
= lim V (f
ni+1( )
x
) = V (
f
(ξ
)) ;∀
k
V (
f
(ξ
)) = V (ξ
) (4)From (1) and (3) it follows that V (
ξ
) =0If V (
ξ
) =V (η
) = 0 thenξ
=f
(ξ
),η
=f
(η
) ifd
(ξ
,η
)≠
0
d
(ξ
,η
) =d
(f
(ξ
) ,f
(η
)) < max {V(ξ
) , V(η
) ,d
(ξ
,η
)}=d
(ξ
,η
)which is a contradiction.
Hence
d
(ξ
,η
) = 0.B.E Rhodes [4] presented a list of definitions of contractive type conditions for a self map on a metric space (
X
,d
) and established implications and non implications among them ,there by facilitating to check the implication of any new contractive condition any one of the condition mentioned in [4] so as to derive a fixed point theorem. Among the conditions in [4], Seghal’s condition is significant as a good number of Contractive conditions imply Seghal’s condition. We now present the dislocated versions of these conditions.Let (
X
,d
) be a dislocated metric space andf
:X
→
X
be a mapping andx
,y
be any elements ofX
. Consider the following conditions1. (Banach): there exists a number
α
, 0≤
α
≤
1
such that for eachx
,y
∈
Xd
(f
(x
) ,f
(y
))≤
α
d
(x
,y
) .2. (Rakotch): there exists a monotone decreasing function
α
:
(
0
,
∞
)
→
[
0
,
1
)
such thatd
(f
(x
) ,f
(y
))≤
α
d
(x
,y
) wheneverd
(x
,y
)≠
0
.
3. (Edelstein):d
(f
(x
) ,f
(y
)) <d
(x
,y
) wheneverd
(x
,y
)≠
0
4. (Kannan): there exists a number
α
, 02
1
<
<
α
such thatd
(f
(x
),f
(y
))≤
h
max {d
(x
,f
(x
)) ,d
(y
,f
(y
)) }6.
d
(f
(x
),f
(y
)) < max {d
(x
,f
(x
)),d
(y
,f
(y
))} wheneverd
(x
,y
)≠
0
7. (Reich): there exist nonnegative numbers a, b, c satisfying a + b + c < 1 such thatd
(f
(x
),f
(y
) )≤
ad
(x
,f
(x
))+bd
(y
,f
(y
)) + cd
(x
,y
) 8. (Reich): there exist monotonically decreasing functions a, b, c from (0,∞
) to [0, 1) satisfying a(t) + b(t) + c(t) < 1 such that,d
(f
(x
),f
(y
))≤
ad
(x
,y
)d
(x
,f
(x
))+bd
(x
,y
)d
(y
,f
(y
))+cd
(x
,y
)d
(x
,y
)where
d
(x
,y
)≠
0
9. There exist nonnegative functions a, b, c satisfying ,
sup
{
,y X x ∈
a(
x
,y
)+b(x
,y
)+c(x
,y
) }< 1such that
d
(f
(x
),f
(y
))≤
a(t)d
(x
,f
(x
)) + b(t)d
(y
,f
(y
)) + c(t)t , where t=(x
,y
). 10. (Sehgal):d
(f
(x
),f
(y
)) < max{d
(x
,f
(x
)),d
(y
,f
(y
)) ,d
(x
,y
)} ifd
(x
,y
)≠
0
Theorem 2.6 If
f
is a self map on a dislocated metric space (X
,d
) and f satisfies any of the conditions (1) through(9) then
f
has a unique fixed point providedcl
O(x
) is nonempty for somex
∈
X
.Proof: In [4] B.E Rhodes proved that when
d
is a metric (1) (2) (3) (10)(4) (5) (6) (10) (4) (7) (8) (10) (5) (7) (9) (10)
With
d
(x
,y
)≠
0
is replaced byx
≠
y
.Consequently these implications hold good in ad
metric space as well sincex
≠
y
d
(x
,y
)≠
0
in ad
metric space .It now follows from 2.5 thatf
has a fixed point which is unique when O(x
) has a cluster point for somex
.Example 2.7: Define
d
(x
,y
) =|x
|+|y
| Forx
,y
inR
.d
is a dislocated metric onR
.If
∈
>0,B
0(∈
) = (-∈
,∈
)If
x
≠
0 ,∈
>0 ,B∈(x
) =∈<
∈>
−
∈
∈
−
|
|
|
|
|
|
,
|
|
x
if
x
if
x
x
φ
. Also 0≠
x
∈
B∈(x
) iff |x
|<2
∈
Result 2.6 Define
f
:R
→
R
byf
(x
) = |x
|.Every non negative real number is a fixed point off
, but 0 is the only coincidence point.d
(f
(x
),x
) = |x
|+|x
|=0⇔
x
=0 .Thus 0 is the only coincidence point while all non negative real numbers are fixed points.REFERENCE:
[1] P. Hitzler: Generalized metrics and topology in logic programming semantics, Ph. D Thesis, Schoolof Mathematics, Applied Mathematics and Statistics, National University Ireland, University college Cork, 2001.
[2] J. L. Kelle: General topology, D. Van Nostrand Company, Inc., 1960.
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