11.some fixed point theorems in generalised dislocated metric spaces

Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.1, No.4, 2011

Some Fixed Point Theorems in Generalized Dislocated Metric
Spaces
P Sumati Kumari

Department of Mathematics, FED – I, K L University.

Green fields, Vaddeswaram, A.P, 522502, India.

*Email: mumy143143143@gmail.com

Abstract: The notion of a generalized dislocated metric space is introduced, its topological properties are
established and an analogue of seghals fixed point theorem is obtained from which existence and
uniqueness of fixed points for self maps that satisfy the metric analogues of contractive conditions
mentioned and it can be derived.
Keywords: Generalized dislocated metric, Kuratowski’s axioms ,coincidence point, Contractive
conditions,
 -property.
Subject classification: 54H25 ,47H10.
1. Introduction: Pascal Hitzler 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 generalized dislocated
metric and show that this topology is metrizable , by actually showing a metric that induces the topology.
Rhoades 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 .We prove that these implications hold good for
self maps on a generalized dislocated metric space and prove the generalized dislocated metric version of
Seghal’s result then by deriving the generalized dislocated analogue’s of fixed point theorems of Banach,
Kannan ,Bianchini, Reich and others.
In what follows, R  the set of all positive real numbers.
  
1.1:Let binary operation ◊ : R  R  R satisfies the following conditions:
(I) ◊ is Associative and Commutative,
(II) ◊ is continuous w.r.t to the usual metric R 
A few typical examples are a ◊ b = max{ a , b }, a ◊ b = a + b , a ◊ b = a b , a ◊b = a b +
ab 
a + b and a ◊ b = for each a , b ∈ R
max a, b,1
In what follows we fix a binary operation ◊ that satisfies (l) and (ll)


Definition 1.2:A binary operation ◊ on R is said to satisfy  -property if
(lll)there exists a positive real number  such that a ◊ b ≤  max{ a , b } for
a , b ∈ R .
every
*
Definition 1.3: Let X be a non empty set. A generalized dislocated (simply gd) metric (or d
metric) on X is a function d : X 2  R that satisfies the following conditions:
* 

(1) d x , y   0 ,
*

(2) d x , y   0 Implies x  y
*

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(3) d * x , y  = d *  y , x 
(4) d * x , z  ≤ d * x , y  ◊ d *  y, z  for each? y, z  X.
x
* *
The pair (X, d ) is called a generalized dislocated (or simply d ) metric space.
Fix a d * metric space ( X , d * ). For r > 0 write Bd * ( x, r ) = { y ∈ X : d * x , y  < r }.
Definition 1.4: We say that a net { x /    } in X converges to x in (X, d * ) and write
lim{ x /   }  x if lim d * ( x , x)  0 i.e. for each  0 there exists  o   such that for all


   0  d * ( x, x ) .
Remark: If ◊ satisfies  -property with  > 0.then the limit of a net in ( X , d * ) is unique. Now on
we assume that( X , d ) has  -property with 0   ≤1
*

Notation: For A ⊂ X we write D ( A )={ x  X / x is a limit of a net in ( X , d )}
*

Proposition 1.5:Let A, B  X. Then
I. D ( A ) =  if A = 
II. D ( A )  D ( B ) if A  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,
Let x  D ( A  B ) and x = lim ( x ) where ( x / α   ) is a net in A  B . If 
x

 such that x  A for α   and α ≥  then ( x / α ≥  , α   ) is a cofinal subnet of
( x / α   ) and is in A and lim d * ( x , x ) = lim d * ( x , x ) =0 so that x  D( A ).
   
If no such  exists in  then for every α   , choose β(α)   such that β(α) ≥  and
x    B then {x   /   } is a cofinal subset in B of ( x / α   ) and lim d ( x    , x )
*
 
*
= lim d ( x , x ) =0 so that
 
x  D ( B ). It now follows that D ( A  B )  D ( A )  D ( B ) and hence (iii) holds. To prove
(iv) let x  D ( D ( A ) ) , x = lim x , x  D ( A ) for α   , and  α   , let ( x   / β 
 
 (α)) be a net in A 
1
x = lim x  .For each positive integer i  α i   such that d * ( x i , x )< .and β i   ( α i ) 
   x  i
1
d*( x  i , write α i =  i  i , then {  1, 2,.... } is directed set with  i <  j
x  i )<
i i i

If i < j , and d * ( x  i , x ) ≤ d * ( x  i , x  i ) ◊ d * ( x  i , x )
≤  max{ d * ( x i , x i ), d*( x i , x )}
* *
< d ( x i , x i )+ d (x i ,x)
2
< . This implies that x  D ( A ).
i

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Vol.1, No.4, 2011

As a corollary, we have the following
Theorem 1.6: If for A  X and A = A  D ( A ), then the operation A  A on P(X) satisfies
Kuratowski’s Closure axioms:
i.  =
ii. AA
iii. A = A and
iv. A B = A  B .
Consequently we have the following
Theorem 1.7 : Let  be the set of family of all subsets A of X for which A = A and  be the
complements of members of  . Then the  is a topology for X and the  -closure of a subset A of X
is A.
Definition 1.8:The topology  obtained in Theorem 1.7 is called the topology induced by d * and
simply referred to as the d-topology ofX and is denoted by ( X , d * ,  ).
Proposition 1.9: Let A  X . Then x  D ( A ) iff for every  >0 , B  x   A  
Proof: if x  D(A),there exist a net ( x / α   ) in A such that lim x = x , if  >0      such
that
d * ( x , x )<  .If α   and α    . Hence x  B x   A for α    .
Conversely if for every  >0, B  x   A   ,we choose one xn in B 1 ( x )  A for every integer
n

1
n  1 in (0,1). Clearly ( xn / n  1) is a net in A and let  >0 and N 0  if n > N 0 ,

1 1
d * ( xn , x )<< 
n N0
Hence x  D ( A ).
Corollary 1.1O: x  A  x  A or B  x   A    >0.
Corollary 1.11: A is open in ( X , d * ,  ) if and only if for every x  A   >0 such that
{ x } B x   A
Proposition 1.12: If x  X and  >0 then { x }  B  ( x ) is an open set in( X , d * ,  ).
Proof: Let A ={ x }  B  ( x ) , y  B  ( x ) and 0< r <  - d * ( x , y )
Then Br  y   B x   A
Since z  Br  y 
 d * ( y , z )< r <  - d * ( x , y )
 d * ( x , y ) + d * ( y , z )< 
Now d*( x , z )  d*( x , y ) ◊ d*( y , z )
  max{ d * ( x , y ) , d * ( y , z )}
 d*( x , y ) + d*( y , z ) < 
 z  B x  .

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Vol.1, No.4, 2011

Proposition 1.13: Ifx  X and V r ( x )= B r ( x )  { x } for r >0 then the collection {V r ( x ) /
x  X } is an open base at x in ( X , d * ,  ).
Proof: The first condition for a basis is trivial, before checking the second condition for a basis we show
 
that if ‘ y ’ is a basis element of V r ( x ).Then there is a basis element V y that is contained in V r ( x ).
Define  = r - d ( x ,y) then V  y  
*
V r ( x ).
For if z  V  y  then d ( y , z )<  = r - d * ( x , y )
*

 d * ( x , y )+ d * ( y , z ) < r
Now d*( x , z )  d*( x , y ) ◊ d*( y , z )
  max{ d * ( x , y ) , d * ( y , z )}
 d*( x , y ) + d*( y , z )
< r
Let V1 and V2 be two basis elements and let y  V1  V2 then there exist 1 ,  2 so that V1  y   V1
and V 2  y   V2 .
Let  =min{  1 ,  2 } ,then V  y   V1  V2 .
d ,  ) is a Hausdorff space and first countable.
Proposition 1.14: ( X ,
*

Proof: If x , y  X , and d ( x , y )>0 then V  ( x )  V  ( y ) =  .
*

2 2

Hence ( X , d ) is a Hausdorff space. If x  X the collection { B 1 ( x) } is base at X . Hence ( X , d * )
*

n
is first countable.
Remark: Proposition 1.14 enables us to deal with sequence instead of nets.
Definition 1.15: A sequence { x n } is called a Cauchy sequence in ( X , d ), if for each  >0, there exists
*

n0  N such that d * xn , xm   for each n, m  n0 . ( X , d * ) is said to be complete if every Cauchy
sequence is convergent.
Definition 1.16: Let ( X , d ) be a
*
d * metric space. If there is a number 0<   1 such that
d *  f x  , f  y    d * x , y   x, y  X then f is called a contraction.
* *
Let ( X , d ) be a d -metric space and f : X  X be a mapping write
V x   d x , f x  and z f    x / V x   0
*

Clearly every point of Z( f ) is a fixed point of f but the converse is not necessarily true. We
call points of Z(f) as coincidence points of f .The set Z( f ) is a closed subset of X .Mathew’s theorem
states that a contraction on a complete dislocated metric space has a unique fixed point . The same theorem
has been justified by an alternate proof by Pascal Hitzler.We present the generalized dislocated metric
version of this theorem for coincidence points.
2 MAIN RESULTS:
*
Theorem 2.1: If ( X , d ) is a complete and f : X  X is a contraction,then there is a unique
coincidence point for f.
Proof: For any x  X the sequence of iterates satisfies
d * ( f n x  , f n1 x  )   n d * x, f x  where  is any contractive constant.
Consequently if n  m ,

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Vol.1, No.4, 2011

d * ( f n x  , f m x  )  d * ( f n x  , f n1 x  )◊ d * ( f n1 x  , f n2 x  )◊….. d * ( f m1 x  ,
f m x )
  max { d * ( f n x  , f n1 x  ) , d * ( f n1 x  , f n2 x  )….. d * ( f m1 x 
, f x ) }
m

 d * ( f n x  , f n1 x  )+ d * ( f n1 x  , f n2 x  )+….. d * ( f m1 x  ,
f m x )
  n 1     2  .....   mn1 d * x, f x 
n
d * x, f x 

1 
Hence { f x  } is Cauchy sequence in X .
n

if  = lim f x 
n
n

then f (  )= lim f n1 x  so
n

d (  , f (  ) ) = lim d * ( f n x , f n1 x ) . Since d * ( f n x , f n1 x ) <  n d * x, f ( x) 
*
n

Since 0<   1 ; lim  d * x, f ( x) =0 Hence d * (  , f (  ) )=0
n

Uniqueness:If d (  , f (  ) )= d (  , f (  ))=0, then f (  ) =  and f (  )=  so that
* *

d * (  ,  )  d * (  , f (  ))◊ d * ( f (  ) , f (  ))◊ d * ( f ( ) ,  )
  max{ d * (  , f (  )), d * ( f (  ) , f (  )), d * ( f ( ) ,  )}
 d * (  , f (  ))+ d * ( f (  ) , f ( ))+ d * ( f ( ) ,  )
  d * (  ,  ) so that
d * (  ,  )=0, hence  =  .
Theorem 2.2: let ( X , d ) be any d -metric space and f : X  X be continuous .Assume that
* *

d*( f (x) , f ( y )) < max
*
{d (x , f ( x ))
* *
, d ( y , f ( y )), d ( x , y )}
whenever d ( x , y )  0.Then f has a unique coincidence point whenever cl O( x ) is nonempty for
*

some x  X .
Proof: Write V( x )= d ( x , f ( x )) , z ={ x / V( x )=0} ; O( x )= { f n ( x ) / n  0 }
*

Since f is continuous, V is continuous. If x  Z , then
* 2 * * 2 *
V ( f ( x ))= d ( f ( x ) , f ( x)) < max { d ( x , f ( x )) , d ( f ( x ) , f ( 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 =  then V( f
k 1
( x )) < V ( f ( x )) k
k

Hence V( f x  ) is convergent.
n
------------------------------- (2)
n
 be a cluster point of O( x ).  ( ni )   
let =lim f i ( x )
 f k   =lim f n k x 
i

 O(  )  cl 0( x ) , since V is continuous

20

Vol.1, No.4, 2011

V( f k   ) = lim V( f ni k x  )
Since O( x )  z =  by (2) { V ( f n x  ) } is convergent.
n
Let  = lim V( f i ( x ) ) = V ( )
Also  = lim V( f n 1 x  ) = V ( f (  )) ; k
i

 V ( f (  )) = V (  ) ---------------------------------- (3)
From (1) and (3) it follows that V (  ) =0
Uniqueness:If V(  ) =V(  )= 0 then
 = f (  ),  = f ( ) if d * (  ,  )  0
d * (  ,  ) = d * ( f (  ) , f ( )) < max {V(  ) , V( ) , d * (  ,  ) }
= d ( ,  )
*
which is a contradiction.
Hence d (  ,  )=0.
*

B.E Rhodes presented a list of definitions of contractive type conditions for a self map on a metric space
( X , d ) and established implications and nonimplications among them ,there by facilitating to check the
implication of any new contractive condition through any one of the condition mentioned so as to derive a
fixed point theorem. Among the conditions in seghal’s condition is significant as a good number of
contractive conditions imply seghal’s condition .These implications also hold good in the present context as
* *
well. In fact the usual metric on R is a d metric, if we write a ◊ b = a + b .We state the d metric
version of some of the contractive conditions mentioned in and derive various implications and non
implications and deduce fixed point theorems for d * metrics from 2.2
* *
Let (X, d ) be a d metric space with a ◊ b = a + b for each a , b  R  and f : X  X be a
mapping and x , y be any elements of X.
1. (Banach) : there exists a number  , 0    1 such that for each x , y  X
d * ( f ( x ) , f ( y ))   d * ( x , y )
2. (Rakotch) : there exists a monotone decreasing function  : (0, )  [0,1) such that
d * ( x , y )   d * ( x , y ) whenever d * ( x , y )  0
(Edelstein) : d ( f ( x ) , f ( y )) < d ( x , y ) whenever d ( x , y )  0
* * *
3.
1
4. (Kannan) :there exists a number  , 0  such that
2
d * ( f ( x ) , f ( y )) <  [ d * ( x , f ( x )) ◊ d * ( y , f ( y )) ]
5. (Bianchini): there exists a number h ,0  h  1 such that
d * ( 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 )) whenever d ( x , y )  0
* * * *

7. (Reich) : there exist nonnegative numbers a, b, c satisfying a + b + c < 1 such that
d * ( f ( x ) , f ( y ))  a d * ( x , f ( x )) ◊ b d * ( y , d * ( y )) ◊ c d * ( 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 ,

21

Vol.1, No.4, 2011

d * ( f ( x ) , f ( y ) ) < a(s) d * ( x , f ( x )) ◊ b(s) d * ( y , f ( y )) ◊ c(s)s where t= d * ( x ,
y) 0
9. there exist nonnegative functions a, b, c satisfying sup a( x , y )+b ( x , y )+c ( x , y ) < 1
x , yX
such 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 ) } if d * ( x
, y ) 0
*
Theorem 2.3: if f is a self map on a generalized dislocated metric space ( X , d ) and f satisfies any
of the conditions (1) through (9) then f has a unique coincidence point provided cl O( x ) is nonempty for
some x  X .
Proof: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)
These implications hold good in a d * - metric space as well since x  y  d * ( x , y )  0 in a
d * - metric space .It now follows from theorem 2.2 that f has a fixed point which is unique when O( x ) has
a cluster point for some x .
Remark: Every coincidence point in a generalized dislocated metric space is a fixed point but the converse
is not true.
Acknowledgement: The author is grateful to Dr. I.Ramabhadra Sarma for his valuable comments and
suggestions

Reference:
Pascal Hitzler: Generalized metrics and topology in logic programming semantics, Ph. D
Thesis,2001.
S.Sedghi: fixed point theorems for four mappings in d*-metric spaces, Thai journal of mathematics,
vol 7(2009) November 1:9-19
S.G. Mathews: Metric domains for completeness, Technical report 76 , Department of computer
science , University of Warwick, U.K, Ph.D Thesis 1985.
B.E.Rhoades : A comparison of various definitions of contractive mappings,Trans of the
Amer.Math.Society vol 226(1977) 257-290.
V.M.Sehgal : On fixed and periodic points for a class of mappings , journal of the London
mathematical society (2), 5, (1972) 571-576.
J. L. Kelley. General topology. D. Van Nostrand Company, Inc., 1960.

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11.some fixed point theorems in generalised dislocated metric spaces

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11.some fixed point theorems in generalised dislocated metric spaces