Mathematical Theory and Modeling                                                                www.iiste.orgISSN 2224-580...
Mathematical Theory and Modeling                                                                             www.iiste.org...
Mathematical Theory and Modeling                                                                                  www.iist...
Mathematical Theory and Modeling                                                             www.iiste.orgISSN 2224-5804 (...
Mathematical Theory and Modeling                                                                  www.iiste.orgISSN 2224-5...
Mathematical Theory and Modeling                                                                           www.iiste.orgIS...
Mathematical Theory and Modeling                                                                     www.iiste.orgISSN 222...
Mathematical Theory and Modeling                                                                                          ...
Mathematical Theory and Modeling                                                                                          ...
Mathematical Theory and Modeling                                                                           www.iiste.orgIS...
Mathematical Theory and Modeling                                                               www.iiste.orgISSN 2224-5804...
Mathematical Theory and Modeling                                                                            www.iiste.orgI...
Mathematical Theory and Modeling                                                               www.iiste.orgISSN 2224-5804...
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11.on generalized dislocated quasi metrics

  1. 1. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.2, 2012 On Generalized Dislocated Quasi Metrics P Sumati Kumari Department of Mathematics, FED – I, K L University, Green fields, Vaddeswaram, A.P, 522502, India. Email:mumy143143143@gmail.comAbstractThe notion of dislocated quasi metric is a generalization of metric that retains, an analogue of the illustriousBanach’s Contraction principle and has useful applications in the semantic analysis of logic programming.In this paper we introduce the concept of generalized dislocated quasi metric space.The purpose of this noteis to study topological properties of a gdq metric, its connection with generalized dislocated metric spaceand to derive some fixed point theorems.Keywords: Generalized dislocated metric, Generalized dislocated quasi metric, Contractive conditions,coincidence point,  -property.Introduction: Pascal Hitzler [1] presented variants of Banach’s Contraction principle for various modifiedforms of a metric space including dislocated metric space and applied them to semantic analysis of logicprograms. In this context Hitzler raised some related questions on the topological aspects of dislocatedmetrics. In this paper we establish the existence of a topology induced by a gdq metric, induce ageneralized dislocated metric (D metric) from a gdq metric and prove that fixed point theorems forgdq metric spaces can be derived from their analogues for D metric spaces . We then prove a D metricversion of Ciric’s fixed point theorem from which a good number of fixed point theorems can be deduced.   Definition 1.1:[2]Let binary operation ◊ : R  R  R satisfies the following conditions: (I) ◊ is Associative and Commutative, (II) ◊ is continuous. Five typical examples are a ◊b a , b }, a ◊ b = a + b , a ◊ b = a b , = max{ ab  a ◊ b = a b + a + b and a ◊ b = for each a , b ∈ R max a, b,1Definition 1.2: [2] The binary operation ◊ is said to satisfy  -property if there exists a positive realnumber  such that a ◊ b ≤  max{ a , b } for every a , b ∈ R . Definition 1.3: Let X be a nonempty set. A generalized dislocated quasi metric (or gdq metric) on X is a function gdq : X 2  R that satisfies the followingconditions for each x, y, z ∈ X .   (1) gdq x , y  0 ,   (2) gdq x , y  0 implies x  y    (3) gdqx , z  ≤ gdq x , y ◊ gdq y, z The pair ( X , gdq ) is called a generalized dislocated quasi metric (or gdq metric) space.If gdq satisfies gdq  x , y   gdq( y, x) also, then gdq is called generalized dislocated metric space.Unless specified otherwise in what follows ( X , gdq ) stands for a gdq metric space. 1
  2. 2. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.2, 2012Definition 1.4: If ( x /    )is a net in X and xX (x ) gdq converges to x we say thatand write gdq lim x = x if lim gdq( x x) = lim gdq( x x ) =0 .i.e For each  0 there exists   o   such that for all    0  gdq( x, x )  gdq( x , x) .We also call x, gdq limit of( x ). We introduce the followingDefinition 1.5: Let A  X . x  X is said to be a gdq limit point of A if there exists a net(x ) in X such that gdq lim x = x. Notation1.6:Let x  X , A  X and r >0,We writeD ( A )= {x / x  X and x is a gdq limit point of A }.Br ( x)  { y / y  X and min{ gdq( x, y), gdq( y, x)} r} and Vr ( x) {x}  Br ( x) .Remark 1.7: gdq lim x = x iff for every   0 there exists  0   such that  x B ( x) if    0Proposition 1.8: Let ( X , gdq ) be a gdq metric space such that ◊ satisfies  -property with   0.If a net ( x /    ) in X gdq converges to x then x is unique.Proof: Let ( x /    ) gdq converges to y and yxSince ( x /    ) gdq converges to x and y then for each  0 there exists 1 , 2   such that  for all   1  gdq( x, x )  gdq( x , x)  and    2  gdq( y, x )  gdq( x , y )   From triangular inequality we have, gdq  x, y   gdq( x, x )gdq( x , y)   max{gdq( x, x ), gdq( x , y)}     max{ , }   Which is a contradiction.Hence gdq ( x, y )=0 similarly gdq ( y, x )=0x  yProposition 1.9: x  X is a gdq limit point of A  X iff for every r>0 , A  B r (x)  Proof: Suppose x  D(A) . Then there exists a net ( x /    ) in A such that x = gdq lim x . If r  0 ,   0   such that Br (x)  A   for    0 .Conversely suppose that for every r >0 Br (x)  A   .Then for every positive integer n, there exists x n  B 1 ( x)  A so that ngdq ( xn , x )< 1 n , gdq ( x , xn )< 1 n and xn  A 2
  3. 3. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.2, 2012Hence gdq lim gdq ( xn x)  0 so that x  D(A) nTheorem 1.10: Let ( X , gdq ) be a gdq metric space such that ◊ satisfies  -property with   1 and A  X and B  X then i. D ( A )=  if A   ii. D ( A )  D ( B ) if A  B iii. D ( D ( A ))  D ( A ) iv. D ( A  B )= D ( A )  D ( B )Proof: (i) and (ii) are clear. That D ( A )  D ( B )  D ( A  B ) follows from (ii). To prove thereverse inclusion, let x  D ( A  B ) , x = gdq lim ( x ) where ( x / α   ) is a net in A  B .  If     such that x  A for α   and α ≥  then ( x /α ≥  ,α   ) is a cofinal subnet of( x /α   ) and lim gdq ( x , x ) = lim gdq ( x , x ) = lim gdq ( x , x )= lim gdq ( x , x )=0        so that x  D ( A ). If no such  exists in  then for every α   , choose β(α)   such that β(α)≥αand x  ( )  B. Then ( x ( ) / α   ) is a cofinal subnet in B of ( x / α   ) andlim gdq ( x ( ) , x )= lim gdq ( x , x )= lim gdq ( x , x ) = lim gdq ( 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 = gdq lim x , x  D ( A ) for α   , and  α   ,let ( x (  ) /β   (α))  be a net in A Such that x = gdq lim  x .For each positive integer i  α i   such that    1 1 gdq ( x i , x )= gdq ( x , x i )< , and β i   ( α i )  gdq ( x i , x i )= gdq ( x i , x i )< i . i i iIf we write α i =  i  i, then {  1, 2 ........ } is directed set with  i <  j  i if i  j, gdq x i , x   gdq( x i , xi )gdq( xi , x)   max{ gdq( x i , xi ), gdq( xi , x)}  gdq( x i , xi )  gdq( xi , x) 2 < i gdq x, x i   2Similarly, iHence x  D( A ).Corollory 1.11: If we write A  A  D(A) for A  X the operation A  A satisfies Kurotawski’s 3
  4. 4. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.2, 2012 _Closure axioms[6] so that the set   { A / A  X and AC  AC } is a topology on X . We call( X , gdq ,  ) topological space induced by gdq . We call A  X to be closed if A  A and open if A  .Corollory 1.12: A  X is open (i.e A   ) iff for every x  A there exists   0  V ( x)  AProposition 1.13: Let ( X , gdq ) be a gdq metric space such that ◊ satisfies  -property with   1.If x  X and   0 then V (x) is an open set in ( X , gdq ,  ).Proof: Let y  B (x) and 0  r  min{  d ( x, y) ,   d ( y , x)}.Then B ( y )  B ( x)  A , since z  Br ( y)  min{gdq( y, z ), gdq( z, y)}  r r   gdq( y, z)  r min{  gdq( x, y),   gdq( y , x)}Now gdq( x, z )  gdq( x, y )gdq ( y, z )   max{gdq( x, y ), gdq ( y, z )}  gdq( x, y )  gdq( y, z ) Similarly gdq( z, x)   therefore z  B ( x) Hence V (x) is open.Proposition 1.14: Let ( X , gdq ) be a gdq metric space such that ◊ satisfies  -property with Then( X , gdq ,  ) is a Hausodorff space and first countable.Proof: Suppose x  y we have to find  such that A  ( B ( x)  {x})  ( B ( y) { y})  Since x  y. One of gdq ( x , y ), gdq ( y , x ) is non zero. We may assume gdq ( y , x )>0 Choose   0 such that 2  gdq( y, x) .we show that A  .If z  A and z = y , z  x. z  B (x) .  gdq ( z , x )<  gdq( y, x)  gdq ( y , x )<  < which is a contradiction. 2Similarly if z  y and z  x , z  AIf y  z  x then gdq ( y , x )  gdq ( y , z )  gdq ( z , x )   max{gdq( y, z ), gdq ( z , x)}  gdq( y, z )  gdq( z, x)  2  gdq( y, x) which is a contradiction. Hence A   Hence ( X , gdq ,  ) is a Hausodorff space. 4
  5. 5. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.2, 2012If x  X then the collection V1 ( x) is base at x .Hence ( X , gdq ,  ) is first countable. nRemark 1.15: Proposition 1.14 enables us to deal with sequences instead of nets.Definition 1.16: A sequence {xn } in X is a gdq Cauchy sequence if for every  0 there correspondsa positive integerN 0  gdq( xn , xm )  or gdq( xm , xn ) whenever n  N 0 and m  N0ie min{gdq( xn , xm )  , gdq( xm , xn ) }. And ( X , gdq ) is said to be gdq complete if everygdq Cauchy sequence in X is gdq convergent. Result 1.17: Define D ( x , y )= gdq ( x , y )  gdq ( y , x ), where a  b = a + b, for a , b  R . 1. D is a generalized dislocated metric( D metric) on X . 2. For any {x /  }in X and x  X gdq lim( x )  x  D lim( x )  x 3. X is a gdq Complete  X is D complete.Proof:(i) and (ii) are clear.we prove (iii)Let X is a gdq CompleteLet {xn } be a Cauchy sequence in ( X , D ) and  0 then there exist a positive integern0  m, n  n0 . lim D( xn , xm )  lim[ gdq( xn xm )gdq( xm xn )]  lim[ gdq( xn xm )  gdq( xm xn )]  min{gdq( xn xm ), gdq( xm xn )}  {xn } is a gdq Cauchy sequence.Hence convergent. lim gdq( xn , x)  lim gdq( x, xn )  0 D( xn , x)  0Hencce X is D complete.Conversely suppose that,Let X is D complete. {xn } be a gdq Cauchy sequence in X and  0 there exist a positive integer n0 Let min{gdq( xn xm ), gdq( xm xn )}  2gdq( xn xm )  gdq( xm xn )} gdq( xn xm )gdq( xm xn )} D( xn xm ) Hence {xn } be a Cauchy sequence in ( X , D ) there exist x in X  D lim xn  x  lim D( xn x)  0 lim[ gdq( xn x )  gdq( x xn )]  0 lim gdq( xn x )  lim gdq( x xn )]  0Hence X is gdq D complete.Remark: As a consequence of 1.17 we can derive a fixed point theorem for gdq metric space if we canprove the same for D metric space and derive the contractive inequality for D from gdq . 5
  6. 6. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.2, 2012 TheD metric induced by a gdq metric on a set X is very useful in deriving fixed pointtheorems for self maps on ( X , gdq ) from their analogues for ( X , D ).If a self map f on a gdq metricspace ( X , gdq ) satisfies a contractive inequality gdq( f ( x), f ( y))  gdq ( x, y), Where  gdq is alinear function of { gdq(u, v) /{u, v}  {x, y, f ( x), f ( y)}} then f satisfies the contractive inequalityD( f ( x), f ( y))   D ( x, y) Where  D is obtained by replacing gdq in  gdq by D .B.E Rhodes[4] collected good number of contractive inequalities considered by various authors andestablished implications and nonimplications among them. We consider a few of them here. Let ( X , d ) be a metric space, x  X , y  Y , f a self map on X ,And a, b, c, h, a1 , a2 , a3 , a4 , a5 , ,  ,  nonnegative real numbers (=constants),a( x, y), b( x, y), c( x, y), p( x, y), q( x, y), r ( x, y)s( x, y) and t ( x, y) be nonnegative real valuedcontinuous function on X x X . 1. (Banach) : d ( f ( x ) , f ( y ))  a d ( x , y ) , 0  a  1 1 2. (Kannan) : d ( f ( x), f ( y))  a { d ( x, f ( x)) + d ( y, f ( y)} , 0  a  2 3. (Bianchini): d ( f ( x), f ( y))  h max { d ( x, f ( x)) , d ( y, f ( y)) } , 0  h  1 4. d ( f ( x ), f ( y ))  a d ( x , f ( x ))+ b d ( y , f ( y )) + c d ( x , y ) , a + b + c <1 5. d ( f ( x ) , f ( y ))  a( x, y) d ( x , f ( x ))+ b( x, y) d ( y , f ( y )) + c( x, y) d ( x , y ) , sup {a( x, y)  b( x, y)  c( x, y) / x  X , y  Y }  1 x , yX 1 6. (Chatterjea): d ( f ( x ) , f ( y ))  a [ d ( x , f ( y )) + d ( y , f ( x )) ] , a  2 7. d ( f ( x), f ( y))  h max [ d ( x, f ( y)) , d ( y, f ( x)) ] , 0  h  1 8. d ( f ( x ) , f ( y ))  a d ( x , f ( y )) + b d ( y , f ( x )) + c d ( x , y ) , a + b + c <1 9. d ( f ( x), f ( y))  a( x, y) d ( x , f ( y )) + b( x, y) d ( y , f ( x ))+ c( x, y) d ( x , y ) , sup {a( x, y)  b( x, y)  c( x, y) / x  X , y  Y }  1 x , yX 10. (Hardy and Rogers):d ( f ( x), f ( y))  a1d ( x, y)  a2 d ( x, f ( x))  a3 d ( y, f ( y))  a4 d ( x, f ( y))  a5 d ( y, f ( x)) , sup {  ai ( x, y) }  1 For every x  y x , yX i 11. (Zamfirescu):For each x, y  X at least one of the following is true: 6
  7. 7. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.2, 2012 I. d ( f ( x), f ( y))  d ( x, y) , 0    1 1 II. d ( f ( x), f ( y))   [ d ( x, f ( x)) + d ( y, f ( y)) ] , 0    2 1 III. d ( f ( x), f ( y))   [ d ( x, f ( y)) + d ( y, f ( x)) ] , 0  2 12. (Ciric): For each x, y  X d ( f ( x), f ( y))  q( x, y) d ( x , y )+ r ( x, y) d ( x , f ( x ))+ s( x, y) d ( y , f ( y )) + t ( x, y) [ d ( x, f ( y)) + d ( y, f ( x))] sup {q( x, y)  r ( x, y)  s( x, y)  2t ( x, y)    1 x , yX 13. (Ciric): For each x, y  X d ( f ( x), f ( y))  h max{ d ( x, y) d ( x, f ( x)) , d ( y, f ( y)) , d ( x, f ( y)) , d ( y, f ( x)) } , 0  h 1B.E Rhoades[4] established the following implications among the above inequalities: (2)  (3)  (5)  (12)  (13) (2)  (4)  (5)  (12)  (13) (6)  (7)  (9)  (13) (6)  (8)  (9)  (13) (6)  (10)  (11)  (12)  (13). If d is a D metric instead of a metric, it is possible that d ( x, x)  0. As such theseimplications hold good in a D metric space as well when “ x  y ” is replaced by “ D( x, y)  0 ”. Moreover all these implications end up with (13). Thus a fixed point theorem for f satisfying the D metricversion of Ciric’s Contraction principal (13) yields fixed point theorem for f satisfying the D metricversion of other inequalities. Moreover d ( x, f ( x)) =0  f (x) = x when d is a metric. However “ f (x) = x ”does notnecessarily imply d ( x, f ( x)) =0 where d is a D metric. We in fact prove the existence of x suchthat D( x, f ( x)) =0 which we call a coincidence point of f .We now prove the following analogue ofCiric’s Contraction principle.2 Main ResultsTheorem 2.1 : Let ( X , D ) be a complete D metric space such that ◊ satisfies  -property with   1 , f a self map on X and 0  h  1 .If for all x, y with D( x, y)  0. D ( f ( x ), f ( y ))  h max{ D ( x , y ), D( x , f ( x )), D ( x , f ( y )), D ( y , f ( x )), D ( y , f ( y ))} ----(*)Then f has a unique coincidence point.Proof: Assume that f satisfies (*). 7
  8. 8. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.2, 2012 x  X and any positive integer n write 0( x, m) ={ x , f ( x )…… f m ( x )} and For [ 0( x, m) ] =sup { D(u, v) /{u, v}  0( x, m) } .We first prove the followingLemma[7]: For every positive integer ‘ m ’ there exists a positive integer km such that [ 0( x, m) ]  D( x , f ( x) ) kProof: To prove this it is enough if we prove that  [ 0( x, m) ]   mWhere  m =max { D ( x , x )…….. D ( x , f m (x)) } ___________________(1)We prove this by using Induction,Assume that (1) is true for ‘ m ’ i.e  [ 0( x, m) ]   mNow we have to prove for m +1 i.e  [ 0( x, m  1) ]   m1 __________(2)We have  [ 0( f ( x), m) ]  max { D ( x , x ) ,…….. D ( x , f m (x)) , D ( x , f m1 ( x)) }Also D ( f i (x) , f m1 ( x))  max { D ( x , x ),…….. D ( x , f m (x)) , D ( f i 1 ( x) , f m1 ( x)) }______(3)  1i  mHence  [ 0( x, m  1) ] =Sup { D ( f ( x) , f ( x)) / 0  i  j  m  1} , i j =Sup { D ( x , x ),…….. D ( x , f m (x)) , D ( x , f m1 ( x)) }  Sup { D ( f i ( x) , f j ( x)) / 0  i  j  m  1} ,  max{ D ( x , x )…….. D (x, f m (x)) , D (x, f m1 ( x)) ,  ( 0( f ( x), m) ) }   m1 from (1) and (3)Hence  [ 0( x, m  1) ]   m1 . This proves the lemma.Proof of the Theorem:If 1  i  m , 1  j m i 1 j 1D ( f i(x), f j ( y )) = D(f(f ( x )), f(f ( y ))) h max { D ( f i 1 ( x ), f j 1 ( y )) , D ( f i 1 (x) , f i ( x )) , D ( f i 1 (x) j, f ( y )) , j 1 j 1 D( f (x) , f i ( x )) , D( f (x) j, f ( x ))} 8
  9. 9. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.2, 2012  h  [ 0( x, m) ] __________________(1)Also  [ 0( x, m) ]  max j 1 j m {D( f ( x ), f ( x )), D ( x , x ), D ( x , f ( x )),….. D ( x , f (x))}_____(2)If m , n are positive integers such that m  n then by (1) m n m n 1 n 1 n 1D( f (x), f ( x ))= D ( f ( f ( x )) , f ( f ( x ))) h (0( f n 1 ( x ) , m  n  1))  h D( f n 1 (x), f k1 n 1 ( x )) for some k1 ; 0  k1  m  n  1 (by above lemma) h2 (0( f n2 (x), m - n +2) ------------------------ ------------------------  h n  [ 0( x, m) ]By the lemma  k  0  k  m and  [ 0( x, m) ]  D ( x , f (x) ) kAssume k  1, D ( x , f (x) )  D ( x , f ( x ))  D ( f ( x ) , f (x) ) k k   max{ D ( x , f ( x )), D ( f ( x ) , f k ( x))}  D ( x , f ( x )) + D ( f ( x ) , f k (x) )  D ( x , f ( x )) + h  [ 0( x, m) ]  D ( x , f ( x )) + h D ( x , f k (x) ) 1  D ( x , f k (x) )  D ( x , f ( x )) 1 hIf k =0 ,  [ 0( x, m) ]  D ( x , x )  D ( x , f ( x ))  D ( f ( x ), x )   max{ D ( x , f ( x )) , D ( f ( x ), x ) }  D ( x , f ( x ))  D ( f ( x ), x )  D ( x , f ( x )) + h D ( x , x )  D  x, x   1 D  x, f  x   1 hHence D ( f ( x ), f ( x ))  h n  [ 0( x, m) ] m n hn  D  x, f  x   1 hThis is true for every m > n Since 0  h < 1. lim h n = 0.Hence { f m ( x )} is a Cauchy sequence in( X , D) .Since X is complete,  z  X so that lim f n ( x )= zWe prove that D( z, f ( z ))  0 9
  10. 10. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.2, 2012 0  D( z, f ( z ))  D( z, f n 1 ( z )) D( f n 1 ( z ), f ( z ))   max{D( z, f n 1 ( z )), D( f n 1 ( z ), f ( z ))}  D( z, f n 1 ( z ))  D( f n 1 ( z ), f ( z ))By continuity of f , lim D( f n1 ( x), f ( z))  0Hence D( z, f ( z))  0, hence z is a coincidence point of f .Suppose z1 , z 2 are coincidence point of f thenD ( z1 , z1 )= D ( z1 , f ( z1 )  0, similarly D ( z 2 , z 2 )=0If D ( z1 , z 2 )  0. Then by (*),D ( z1 , z 2 )= D ( f ( z1 ) , f ( z 2 ))  h max{D( z1 , z2 ), D( z1 , f ( z1 ), D( z1 , f ( z2 ) ), D( z2 , f ( z1 ) ), D(z2 ,f (z2 ))}  h D ( z1 , z 2 ) a contradictionHence D ( z1 , z 2 )=0.Hence z1  z2 .This completes the proof.We now prove a fixed point theorem for a self map on a D metric space satisfying the analogue of (12).Theorem2.2: Let ( X , D ) be a complete D metric space such that ◊ satisfies  -property with  1 and f : X  X be a continuous mapping such that there exist real numbers 1 1 1 1 1 , 0 ,   , 0    , 0  0  ,   min{ ,   ,  0} satisfying at least one of the 2 2 4 2 2following for each x, y  X i. D( f ( x), f ( y))   D( x, y) ii. D( f ( x), f ( y))   0 { D( x, f ( x))  D( y, f ( y)) } iii. D( f ( x), f ( y))   { D( x, f ( y))  D( y, f ( x)) }Then f has a unique coincidence point.Proof: Putting y= x in the above and  = max { 2 , 20 , 2 } we getD( f ( x), f ( x))   D( x, f ( x))Again putting y = f (x) in the above (i) (ii) (iii) yield D( f ( x), f 2 ( x))  2  D( x, f ( x)) 10
  11. 11. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.2, 2012 0 D( f ( x), f 2 ( x))  D( x, f ( x)) 1  0   D( f ( x), f 2 ( x))  D( x, f ( x)) 1    If h = max{2 , 0 , }then 0  h  1 and 1  0 1   D( f ( x), f 2 ( x))  h D( x, f ( x))If m , n are positive integers such that m > n then we can show that n D( f n ( x), f m ( x))  h D( x, f ( x)) 1h since 0  h  1 ; lim h  0 n nHence { f (x) } is a Cauchy sequence in ( X , D) .Since X is complete,  z in X  lim f (x) = z n n n1Since f is continuous, lim f ( x) = f ( z) in ( X , D).sin ce 0  D( z , f ( z ))  D( z , f n 1 ( x)) D( f n 1 ( x), f ( z ))   max{D( z , f n 1 ( x)), D( f n 1 ( x), f ( z ))}  D( z , f n 1 ( x))  D( f n 1 ( x), f ( z )) D( z , f ( z )) =0 , Hence z is a Coincidence point of f .It follows thatUniqueness : If z1 , z 2 are coincidence points of f then by hypothesis,Either D ( z1 , z 2 )   D ( z1 , z 2 ) or 0 or 2  D ( z1 , z 2 ) 1 1Since 0   and 0    we must have D ( z1 , z 2 )=0 2 4 Hence z1  z 2 . This completes the proof. The D metric version for the contractive inequality (10) in the modified form (**) given below yields the following X , D ) be a complete D metric space such that ◊ satisfies  -property with Theorem2.3 : Let (   1 and f : X  X be a continuous mapping. Assume that there exist non-negative constants a i satisfying a1  a2  a3  2a4  2a5 < 1 such that for each x , y  X with x  y D ( f ( x ), f ( y ))  a1 D ( x , y )  a2 D ( x , f ( x ))  a3 D ( y , f ( y ))  a4 D ( x , f ( y ))  a5 D ( y , f ( x )) -----(**). Then f has a unique coincidence point.Proof:Consider 11
  12. 12. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.2, 2012D( f ( x), f 2 ( x))  a1D( x, f ( x))a2 D( x, f ( x)) a3 D( f ( x), f 2 ( x)) a4 D( x, f 2 ( x)) a5 D ( f ( x), f ( x))   max{a1 D( x, f ( x)), a2 D( x, f ( x)), a3 D( f ( x), f 2 ( x)), a4 D( x, f 2 ( x )), a5 D ( f ( x ), f ( x))}  a1 D( x, f ( x))  a2 D( x, f ( x))  a3 D( f ( x), f 2 ( x))  a4 D( x, f 2 ( x))  a5 D( f ( x), f ( x))  (a1  a2 ) D( x, f ( x))  a3 D( f ( x), f 2 ( x))  a4 D( x, f 2 ( x))  2a5 D ( x, f ( x )) ( D( f ( x), f ( x))  D( x, f ( x)) + D( f ( x), x))  a  a 2  2a 5   a4   a4 D( f ( x), f 2 ( x))   1  D( x, f ( x)) +   D( x, f ( x)) +    1  a3  1  a3  1  a3 D( f ( x), f 2 ( x))  a1  a2  2a5  a4  D( f ( x), f ( x))   2  D( x, f ( x))  1  a3  a 4   a1  a2  2a5  a4  D( f ( x), f ( x))   D( x, f ( x)) 2 where    , 0<  < 1  1  a3  a4 If m > n thenD( 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 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))  ( n   n1  .......   m1 ) D( x, f ( x)) = (1     2 .......   mn1 ) D( x, f ( x)) n n < D( x, f ( x)) 1 Hence { f n (x) } is Cauchy sequence in ( X , D) , hence convergent.Let   lim ( f n ( x)) then f ( ) = lim ( f n1 ( x)) ( since f is continuous ) n nSo D( , f ( )) = lim D( f n ( x), f n1 ( x)) n 12
  13. 13. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.2, 2012 n  lim D( x, f ( x)) n 1 Since 0   < 1 , D( , f ( )) =0 .Hence f ( ) =  .Hence  is a coincidence point for f.Uniqueness: If D( , f ( ))  D( , f ( ))  0 f ( ) =  and f ( ) =ConsiderD( , )  D( f ( ), f ( ))  a1D( , ) a2 D( , f ( )) a3 D( , f ( )) a4 D( , f ( )) a5 D( , f ( ))   max{a1 D( , ), a2 D( , f ( )), a3 D( , f ( )), a4 D( , f ( )), a5 D( , f ( ))}  a1D( , )  a2 D( , f ( ))  a3 D( , f ( ))  a4 D( , f ( ))  a5 D( , f ( ))   D( , ) where   a1  a4  a5 1 D( , )  0 Hence   Acknowledgement: The author is grateful to Dr. I.Ramabhadra Sarma for his valuable comments andsuggestions.References: [1] Pascal Hitzler: Generalised metrices and topology in logic programming semantics, Ph. D Thesis,2001. [2] S.Sedghi: fixed point theorems for four mappings in d*-metric spaces, Thai journal of mathematics, vol 7(2009) November 1:9-19 [3] S.G. Mathews: Metric domains for completeness, Technical report 76 , Department of computer science , University of Warwick, U.K, Ph.D Thesis 1985. [4] B.E.Rhoades : A comparison of various definitions of contractive mappings,Trans of the Amer.Math.Society vol 226(1977) 257-290. [5] V.M.Sehgal : On fixed and periodic points for a class of mappings , journal of the London mathematical society (2), 5, (1972) 571-576. [6] J. L. Kelley. General topology. D. Van Nostrand Company, Inc., 1960. [7] Lj. B. Ciric : A Generalisation of Banach’s Contraction Principle, American Mathematical Society, volume 45, Number 2, August 1974. 13
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