Common Fixed Point Theorems for Mappings of ...

Nepal Journal of Science and Technology Vol. 16, No.1 (2015) 79-86

Common Fixed Point Theorems for Mappings of Compatible Type(A) in Dislocated Metric Space Dinesh Panthi1 and P. Sumati Kumari2 1

Department of Mathematics, Valmeeki Campus, Nepal Sanskrit University, Kathmandu, Nepal. 2 Department of Mathematics, K L University, Vaddeswaram, A. P., India. e-mail: [email protected]

Abstract In this article we establish a common fixed point result for two pairs of mappings of compatible type(A) in dislocated metric space which generalizes and extends the similar results in the literature.

Key words: d-metric space, common fixed point, compatible of type(A), cauchy sequence.

Introduction

Preliminaries

In 1922, S. Banach established a fixed point theorem for contraction mapping in metric space. Since then a number of fixed point theorems have been proved by many authors and various generalizations of this theorem have been established. G. Jungck (1976) initiated the concept of commuting maps and generalized it with the concept of compatible maps (Jungck 1986, 1988) and established some important fixed point results. G. Jungck, P. P. Murthy and Y. J. Cho (1993) initiated the concept of compatible mappings of type (A) in metric space.

We start with the following definitions, lemmas and theorems. Definition 1 Let

X be a non empty set and let d : X × X → [0, ∞ ) be a function satisfying the

following conditions: (i)

d ( x, y ) = d ( y , x )

d ( x, y ) = d ( y , x ) = 0 implies x = y. (iii) d(x, y) ≤ d(x, z) + d(z, y) for all x, y, z ∈ X. (ii)

The study of common fixed point of mappings satisfying contractive type conditions has been a very active field of research activity. S. G. Matthews (1986)introduced the concept of dislocated metric space under the name of metric domains in domain theory. P. Hitzler and A. K. Seda (2000) generalized the famous Banach Contraction Principle in dislocated metric space. The study of dislocated metric plays very important role in topology, logic programming and in electronics engineering.

Then d is called dislocated metric(or simply d-metric) on X. Definition 2 A sequence

{xn } in a d-metric space

( X , d ) is called a Cauchy sequence if for given

ε > 0, there corresponds n0 ∈ N such that for all m, n ≥ n0 , we have d ( xm , xn ) < ε . Definition 3 A sequence in d-metric space converges with respect to d (or in d) if there exists

The purpose of this article is to establish a common fixed point theorem for two pairs of mappings of compatible type (A) in dislocated metric spaces which generalize and improve similar results of fixed point in the literature.

d ( x n , x ) → 0 as n → ∞. In this case, x is called limit of {xn } (in d)and we write x n → x.

x∈X

79

such that


Definition 4 A d-metric space

Sxn → x ⇒ SSx n → Sx

( X , d ) is called

complete if every Cauchy sequence in it is convergent with respect to

and

d.

Tx n → x ⇒ STx n → Sx.

Therefore, Definition 5 Let

d (TSxn , Sx) ≤ d (TSxn , SSx n ) + d ( SSx n , STx n ) + d ( STxn , Sx )

( X , d ) be a d-metric space. A map

T : X → X is called contraction if there exists a number λ with 0 ≤ λ < 1 such that d (Tx, Ty ) ≤ λ d ( x, y ).

Now taking limit as n → ∞ and using above relations we get

We state the following lemmas without proof.

lim d (TSxn , Sx) = 0 ⇒ lim TSx n = Sx.

Lemma 1 Limits in a d-metric space are unique.

n→∞

n →∞

Theorem 1 Let

Now we establish the following result

and let T

Main Results Theorem 2 Let (X, d) be a complete d-metric space.

T

( X , d ) be a complete d-metric space : X → X be a contraction mapping, then

has a unique fixed point.

Let

A and S be two self mappings on X A and S are said to be commuting if ASx = SAx ∀x ∈ X .

A, B, S , T : X → X be mappings satisfying

Definition 6 Let

the condition

a set

T ( X ) ⊂ A( X ) & S ( X ) ⊂ B( X )

. Mappings

The A and S be two self mappings on a set X . If Ax = Sx for some x ∈ X , then x is called coincidence point of A and S . Definition 7 Let

pairs (T , B) & ( S , A) are

compatible of type( A)

(2)

 d ( Ay , Sy )d ( Bx , Ay)  d (Tx, Sy ) ≤ α    d ( Ax, Tx ) + d ( Sy , Ax)   d (Tx, Ax) d (Ty, By )  + β   d ( Ax, Tx) + d ( Sy, Ax) 

Definition 8 Two mappings S and T from a metric space (X, d) into itself are called compatible of type(A) if

lim d ( STxn , TTxn ) = 0 n→∞

(1)

(3)

and

lim d (TSxn , SSxn ) = 0 n→∞ whenever {xn } is a sequence in X such that lim Sxn = lim Txn = x for some x ∈ X n→∞

 d( Ax, Sx)d(Sy, Ay)   d (Bx, Ay)d (Tx, Sy)  +γ  + κ    d(Ax, Tx) + d (Sy, Ax) d ( Bx,Tx) + d (Bx, Sy)  d(Bx, Tx)d(Ay, Sy)   d (Ax, Sx)d (By,Ty)  +δ  + µ    d( Ax, Ty) + d(Bx,Ty) d (Bx, Ay) + d ( Ax, Ty)

n→ ∞

Proposition 1 Let S and T be mappings of compatible type(A) from a metric space (X, d) into itself. Suppose

for all

that

lim Sxn = lim Txn = x n→∞

n→ ∞

If S is continuous then

x, y ∈ X and α , β , γ , δ , κ , µ ≥ 0 such that

0 ≤ α + β + γ + δ + κ + µ < 1 . If any one of A, B, S,

for some x ∈ X

T is continuous then A, B, S and T have a unique common fixed point in X.

lim TSx n = Sx . n→∞

Proof. Let us define a sequence

Proof. Since S is continuous, so

80

{ yn }∈ X such that

Dinesh Panthi and P. Sumati Kumari/Common Fixed Point Theorems for.............

 d ( y2 n , y2 n +1 ) d ( y2 n +1 , y2 n + 2 )  +µ   d ( y2 n , y2 n +1 ) + d ( y2 n , y2 n + 2 ) 

Tx 2n +1 = y 2 n+ 2 , Ax2 n = y 2n , Sx2 n +1 = y 2n + 2 , (4) Bx2n = y 2 n for n = 1,2,3,... Let us consider

therefore

d (Tx 2 n , Sx2 n +1 )

d ( y 2 n +1 , y 2 n +2 )

 d ( Ax2 n +1 , Sx2 n +1 )d ( Bx2 n , Ax 2 n +1 )  ≤α   d ( Ax2 n , Tx2 n ) + d ( Sx2 n +1 , Ax 2 n ) 

≤ (α + β + γ + δ + κ + µ ) d ( y 2 n , y 2 n +1 ) Let

 d (Tx2 n , Ax2 n )d (Tx2 n +1 , Bx2 n +1 )  + β   d ( Ax2 n ,Tx2 n ) + d ( Sx2 n +1 , Ax2 n ) 

h = (α + β + γ + δ + κ + µ ) < 1

d ( y2 n +1 , y2 n +2 ) ≤ hd ( y2 n , y2 n +1 )

Then

Similarly we can obtain

 d ( Ax2 n , Sx2 n ) d ( Sx2 n +1 , Ax2 n +1 )  +γ   d ( Ax2 n , Tx2 n ) + d ( Sx2 n +1 , Ax2 n ) 

d ( y 2 n , y2 n +1 ) ≤ hd ( y2 n −1 , y2 n ) Hence

d ( y 2 n +1 , y 2 n +2 ) ≤ hd ( y2 n , y 2 n +1 )

≤ h 2 d ( y 2 n −1 , y 2 n ) ≤ ... ≤ h n d ( y0 , y1 )

 d ( Bx2 n , Tx2 n )d ( Ax2 n +1 , Sx2 n +1 )  +δ   d ( Ax2 n , Ax2 n +1 ) + d ( Bx2 n , Tx2 n +1 ) 

This shows that

d ( y n +1 , y n ) ≤ hd ( yn , yn −1 ) ≤ ... ≤ h n d ( y1 , y0 )

 d ( Bx2n , Ax2n+1) d (Tx2n , Sx2n+1)  + κ   d ( Bx2n , Tx2n ) + d ( Bx2n , Sx2n+1 ) 

For every integer q > 0 we have

d ( y n +q , yn ) ≤ d ( y n +q , y n +q −1 ) + ... + d ( yn + 2 , y n +1 ) + d ( y n +1 , yn )

 d ( Ax2n , Sx2n )d ( Bx2n+1, Tx2n+1)  + µ   d ( Bx2n , Ax2 n+1 ) + d ( Ax2n , Tx2n+1) 

≤ ( hq −1 + ... + h2 + h + 1) d ( yn +1 , yn ) ≤ ( hq −1 + ... + h2 + h + 1) hn d ( y1 , y0 )

Now,

h < 1 , so h n → 0 as n → ∞ .

d ( y 2 n +1 , y 2 n + 2 ) ≤

Since

 d ( y 2 n +1 , y 2 n + 2 ) d ( y 2 n , y 2 n +1 )  α   d ( y2 n , y 2 n +1 ) + d ( y 2 n + 2 , y 2 n )   d ( y 2n+1, y2n ) d ( y2n+ 2, y2n+1 )  +β   d ( y2n , y2n+1 ) + d ( y2n+ 2, y2n ) 

Therefore d ( yn +q , yn ) → 0 as This implies that

n → ∞.

{ yn } is a cauchy sequence.

Since X is complete, so there exists a po int z ∈ X such thart { yn } → z . Consequently subsequences

 d ( y2 n , y2 n +1 )d ( y2 n + 2 , y2 n +1 )  +γ   d ( y 2 n , y 2 n +1 ) + d ( y 2 n + 2 , y 2 n )   d ( y2 n , y 2n+1 )d ( y 2n+1, y 2n+ 2 )  +δ   d ( y 2 n , y 2 n +1 ) + d ( y 2 n , y 2 n + 2 ) 

{ Ax 2 n } = { Bx2 n } → z & {Sx2 n +11 } = {Tx 2 n +1 } → z.

(5)

Since the pair (T, B) is compatible of type(A) and let T is continuous, then by proposition (1) we have

 d ( y2 n , y2 n +1 )d ( y2 n +1 , y2 n + 2 )  +κ    d ( y2 n , y2 n +1 ) + d ( y2 n , y2 n +2 ) 

TTx 2 n → Tz

TTx as

81

and

→ Tz and

n → ∞.

BTx2 n → Tz

BTx

→ Tz

(6)


Now, we show that z is the fixe d point of T i.e Tz = z. For this,

We show that z is the fixed point of B i.e

d (TTx2 n , Sx2 n+1 ) ≤

For this put (3), we have

 d ( Ax 2n+1 , Sx2 n+1 )d ( BTx2 n , Ax2 n+1 )  α   d ( ATx 2n , TTx2 n ) + d (Sx 2n +1 , ATx2 n ) 

 d ( Ax2 n +1 , Sx2 n +1 )d ( BBx2 n , Ax2n +1 )  α   d ( ABx2 n , TBx2 n ) + d (Sx2 n +1 , ABx2 n ) 

 d (TBx2 n , ABx2 n ) d (Tx2 n +1 , Bx2 n +1 )  +β   d ( ABx2 n , TBx2 n ) + d (Sx2 n +1 , ABx2 n ) 

 d ( ATx2n , STx 2 n )d (Sx2 n+1 , Ax2 n+1 )  +γ   d ( ATx2n , TTx2 n ) + d (Sx2 n+1 , ATx2n ) 

 d ( ABx2 n , SBx 2 n )d ( Sx2 n +1 , Ax 2 n +1 )  +γ    d ( ABx2 n , TBx 2 n ) + d ( Sx2 n +1 , ABx2 n ) 

 d (BTx2 n , TTx 2 n )d ( Ax2n +1 , Sx2n +1 )  +δ    d ( ATx2 n , Ax2n ) + d (BTx 2 n , Tx2n +1 ) 

 d ( BBx 2 n ,TBx2 n ) d ( Ax2 n +1 , Sx2 n +1 )  +δ    d ( ABx2 n , Ax2 n ) + d ( BBx2 n , Tx2 n +1 ) 

 d ( BTx2n , Ax2n +1 )d (TTx2 n , Sx2 n+1 )  +κ    d (BTx2 n , TTx 2 n ) + d ( BTx2n , Sx2n +1) 

 d ( BBx 2 n , Ax2 n +1 )d (Tbx2 n , Sx2 n +1 )  +κ    d ( BBx 2 n , TBx2 n ) + d (BBx2 n , Sx2 n +1 ) 

 d ( ATx2n , STx 2 n )d ( Bx2n +1 , Tx2 n+1 )  + µ   d (BTx2n , Ax2 n+1 ) + d ( ATx2n , Tx 2n+ 1) 

 d ( ABx2 n , SBx2 n )d ( Bx2 n +1 , Tx 2 n +1 )  +µ   d ( BBx2n , Ax2 n +1 ) + d ( ABx2 n , Tx 2 n+1 ) 

n → ∞ and using relations (5) and

 d (Tz, z ) d (Tz, z )  d (Tz , z) ≤ κ    d (Tz , Tz ) + d (Tz , z )  κ ≤ d (Tz , z) 3

Now taking limit as (7) we have

d (Bz, z ) ≤ κ

n → ∞ and using relations (5) and

d (Bz, z )d (bz, z ) d (Bz, Bz ) + d (Bz , z )

1 ≤ κ d (Bz , z ) 3

which is a contradiction, hence

d (Tz , z ) = 0 ⇒ Tz = z .

which is a contradiction, hence

d ( Bz, z) = 0 ⇒ Bz = z

Again, since the pair (T , B ) is compatible of type(A) and B is continuous then

BBx2 n → Bz & TBx2 n → Bz

x = Bx2 n & y = x2 n+1 in the relation

d (TBx2 n , Sx2 n +1 ) ≤

 d (TTx 2n , ATx 2n )d (Tx2 n+1 , Bx2n +1 )  + β   d ( ATx2n , TTx 2 n ) + d (Sx2 n+1 , ATx2n ) 

Now taking limit as (6) we have

Bz = z.

Again, Since the pair (S , A) is compatible of type(A) (7)

and if

82

S is continuous, then by proposition (1) we have


SSx 2 n → Sz If we consider

and

ASx 2 n → Sz

κ  2α 4( β + γ ) ≤ + + 2(δ + µ ) + d ( z, w) 3 3  3

x = x2 n +1 and y = Sx2 n in

which is a contradiction. Therefore d ( z , w) = 0 ⇒ z = w. Hence A, B, S and T have a unique common fixed point. Now we have following corollaries. If we put B = A in theorem (2) , then it is reduced to following corollary

relation (3) and proceed as in above cases, then we obtain

Sz = z Similarly one can show that if the pair ( S , A) is compatible of type(A) and A is continuous then

Az = z Therefore from above relations, we have

Az = Bz = Sz = Tz = z Thus z is the common fixed point of the mappings A, B, S & T .

Corrollary 1 Let (X, d) be a complete d-metric space. Let A, S , T : X → X be mappings satisfying the condition

Uniqueness : Let z and w are two common fixed points of the mappings A, B, S and T . Then

compatible of

T ( X ) & S ( X ) ⊂ A( X ) The pairs (T , A) & ( S , A) are type( A)

and

 d ( Ay, Sy)d ( Ax, Ay)  d (Tx , Sy) ≤ α    d ( Ax,Tx ) + d ( Sy, Ax)   d (Tx , Ax)d (Ty , Ay)  + β   d ( Ax, Tx ) + d ( Sy, Ax) 

d ( z , w) = d (Tz, Sw) ≤  d ( Aw, Sw)d ( Bz, Aw)  α   d ( Az, Tz ) + d ( Sw, Az )   d (Tz, Az )d (Tw, Bw)  +β   d ( Az, Tz ) + d ( Sw, Az ) 

 d ( Ax, Sx)d ( Sy, Ay)  +γ   d ( Ax,Tx ) + d ( Sy, Ax) 

 d ( Az, Sz )d (Sw, Aw)  +γ    d ( Az, Tz ) + d (Sw, Az)   d ( Bz, Tz )d ( Aw, Sw)  +δ    d ( Az, Aw) + d ( Bz, Tw) 

 d ( Ax, Tx ) d ( Ay, Sy)  +δ   d ( Ax, Ty ) + d ( Ax, Ty )   d ( Ax, Ay) d (Tx , Sy)  + κ   d ( Ax,Tx) + d ( Ax, Sy) 

 d ( Bz, Aw) d (Tz , Sw)  +κ   d (Bz, Tz) + d ( Bz , Sw)   d ( Az, Sz) d ( Bw, Tw)  + µ   d (Bz , Aw) + d ( Az , Tw) 

 d ( Ax, Sx)d ( Ay,Ty )  + µ   d ( Ax, Ay) + d ( Ax,Ty )  for all x, y ∈ X and α , β , γ ,δ , κ , µ ≥ 0 such that 0 ≤ α + β + γ + δ + κ + µ < 1 . If any one of A, S, T is continuous then A, S and T have a unique common fixed point in X. If we put T = S and B = A then the above theorem (2) is reduced to the following corollary Corrollary 2 Let (X, d) be a complete d-metric space. Let A, S : X → X be mappings satisfying the condition

 d ( w, w) d ( z, w )   d ( z, z ) d ( w, w )  = α   +β     d ( z , z ) + d ( w, z )   d ( z , z ) + d (w , z ) 

 d ( z , z ) d ( w, w)   d ( z , z ) d ( w, w)  +γ  +δ     d ( z , z ) + d ( w, z )   d ( z , w) + d ( z , w)   d ( z , w) d ( z , w)   d ( z , z )d ( w, w)  +κ  + µ    d ( z, z ) + d ( z , w)   d ( z , w) + d ( z , w)  83


S ( X ) ⊂ A( X )

 d ( x, Sx ) d ( y, Sy)  +δ   d ( Ax, Sy ) + d ( Ax, Sy )   d ( x, y) d ( Sx, Sy )  + κ   d ( x, Sx ) + d ( x, Sy ) 

The pair (S , A) is compatible of type( A) and

 d ( Ay, Sy)d ( Ax, Ay)  d ( Sx, Sy) ≤ α    d ( Ax, Sx) + d (Sy, Ax)   d ( Sx, Ax)d ( Sy, Ay)  + β   d ( Ax, Sx) + d ( Sy, Ax)   d ( Ax, Sx ) d (Sy , Ay)  +γ   d ( Ax, Sx) + d (Sy, Ax) 

 d ( x, Sx )d ( y , Sy )  + µ   d ( x, y) + d ( x , Sy )  for all x, y ∈ X and α , β , γ , δ , κ , µ ≥ 0 such that 0 ≤ α + β + γ + δ + κ + µ < 1 . If S is continuous then S have a unique common fixed point in X. If we put A = B = I in the above theorem (2) then we obtain the following corollary

 d ( Ax, Sx )d ( Ay, Sy )  +δ   d ( Ax, Sy ) + d ( Ax, Sy )   d ( Ax, Ay)d ( Sx, Sy)  + κ   d ( Ax, Sx) + d ( Ax, Sy) 

Corrollary 4 Let (X, d) be a complete d-metric space. Let

T(X ) & S(X ) ⊂ I (X )

 d ( Ax, Sx)d ( Ay, Sy)  + µ   d ( Ax, Ay) + d ( Ax, Sy)  for all x, y ∈ X and α , β , γ , δ , κ , µ ≥ 0 such that 0 ≤ α + β + γ + δ + κ + µ < 1.

The pairs (T , I ) & ( S , I ) compatible of type( A)

are

and  d ( y, Sy) d ( x, Ay)  d (Tx, Sy) ≤ α    d ( x, Tx) + d ( Sy, x ) 

If any one of A, S is continuous then A and S have a unique common fixed point in X.

 d (Tx, x) d (Ty , y)  + β   d ( x , Tx) + d ( Sy, x) 

If we put A = B = I and T = S in the above theorem (2) then we obtain Corrollary 3 Let (X, d) be a complete d-metric space. Let S , I : X → X be mappings satisfying the condition

S( X ) ⊂ X The pair

S , T , I : X → X be mappings satisfying the

condition

 d ( x, Sx )d (Sy , y )   d ( x, Tx )d ( y , Sy )  +γ  +δ     d ( x, Tx ) + d ( Sy , x )   d ( x, Ty ) + d (x ,Ty ) 

( S , I ) is compatible of

 d (x , y )d (Tx , Sy )   d ( x , Sx) d ( y , Ty )  +κ + µ    d ( x, Tx ) + d ( x , Sy )   d ( x , y ) + d ( x, Ty ) 

type( A) and

 d ( y, Sy) d ( x , y )  d ( Sx, Sy) ≤ α    d ( x, Sx) + d ( Sy, x)   d ( Sx, x) d ( Sy, y )  + β   d ( x , Sx) + d ( Sy, x ) 

for all

x, y ∈ X α , β ,γ ,δ ,κ, µ ≥ 0 x, y ∈ X and α , β , γ , δ , κ , µ ≥ 0 such that

0 ≤ α + β + γ +δ + κ + µ < 1. If any one of S and T is continuous then S and T have a unique common fixed point in X.

 d ( x, Sx) d ( Sy, y )  +γ   d ( x, Sx) + d ( Sy, x ) 

If we put T = S in the above theorem (2) then we obtain Corrollary 5 Let (X, d) be a complete d metric space.

84


Let

A, B, S : X → X be mappings satisfying the

 d (T s x, A p x) d (T s y, B q y)  + β  p s r p  d ( A x , T x ) + d (S y , A x ) 

condition

S ( X ) ⊂ A( X ) & S ( X ) ⊂ B( X ) The pairs ( S , B ) & ( S , A ) compatible of type ( A)

 d ( A p x, S r x )d ( S r y, A p y)  +γ  p s r p  d ( A x, T x ) + d ( S y, A x)   d ( B q x, T s x )d ( A p y, S r y)  +δ  p s q s  d ( A x, T y) + d ( B x, T y ) 

are

and

 d ( Ay, Sy) d ( Bx, Ay )  d ( Sx , Sy ) ≤ α    d ( Ax, Sx ) + d ( Sy , Ax)   d ( Sx, Ax )d ( Sy , By )  + β   d ( Ax, Sx) + d ( Sy, Ax )   d ( Ax, Sx) d ( Sy, Ay )  +γ   d ( Ax, Sx) + d ( Sy , Ax) 

 d ( B q x, A p y )d (T s x, S r y )  + κ  q s q r  d ( B x, T x ) + d ( B x , S y )   d ( A p x, S r x) d ( B q y ,T s y )  + µ  q p p s  d ( B x, A y) + d ( A x, T y ) 

 d ( Bx, Sx) d ( Ay , Sy )  +δ   d ( Ax, Sy) + d ( Bx, Sy)   d ( Bx, Ay ) d ( Sx , Sy )  + κ   d ( Bx, Sx ) + d ( Bx , Sy ) 

for all

0 ≤ α + β + γ + δ + κ + µ < 1 then A, B, S and T have a unique common fixed point in X. Proof. The proof of this theorem is similar to the above theorem (2).

 d ( Ax , Sx) d ( By, Sy )  + µ   d ( Bx, Ay ) + d ( Ax, Sy)  for all

References Banach, S. 1922. Sur les operations dans les ensembles abstraits et leur applications aux equations integrales, fundamental Mathematicae 3(7): 133-181. Hitzler, P. and A. K. Seda. 2000. Dislocated Topologies, J. Electr. Engg. 51 (12): 3-7. Jungck, G. 1976. Commuting mappings and fixed points, Amer. Math. Monthl. 83: 261-263. Jungck, G. 1986. Compatible mappings and common fixed points, Int. J. Math. Sci. 9: 771-779. Jungck, G.1988. Compatible mappings and common fixed points(2), Int. J. Math. Sci. 11: 285-288. Jungck, G., P.P.Murthy and Y. J. Cho. 1993. Compatible mappings of type (A) and common fixed points, Math. Japon. 38(2): 381 - 390 . Matthews, S. G. 1986.Metric domains for completeness. Technical report 76, Department of Computer Science, University of Warwick, Coventry, UK. Panthi D., K. Jha, P.K. Jha and P. S. Kumari. 2015. A common fixed point theorem for twopairs of mappings in dislocated metric space, American J. Computational Mathematics 5: 106-112.


0 ≤ α + β + γ + δ + κ + µ < 1 . If any one of A, B, S is continuous then A, B and S have a unique common fixed point in X. Now we have the following theorem Theorem 3 Let (X, d) be a complete d-metric space. Let

A, B, S , T : X → X . Suppose that any one of

A, B, S, T is continuous and for some positive integers p, q, r, s which satisfy the following conditions

T s ( X ) ⊂ Ap ( X ) & S r ( X ) ⊂ Bq( X ) The are

pairs (T , B ) & ( S , A ) compatible of type ( A )


and

 d ( A p y , S r y )d ( B q x , A p y )  d (T s x, S r y) ≤ α   p s r p  d ( A x, T x) + d ( S y, A x) 

85


86