Vol 8, No 12 (2017)

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Available online throug

ISSN 2229 – 5046

International Journal of Mathematical Archive- 8(12), Dec. – 2017 48

A FIXED POINT THEOREM ON PRODUCT OF METRIC SPACES

U. C. GAIROLA, DEEPAK KHANTWAL*

Department of Mathematics,

H. N. B. Garhwal University, BGR Campus, Pauri Garhwal, Uttarakhand – 246001, India.

(Received On: 23-10-17; Revised & Accepted On: 18-11-17)

ABASTRACT

J. Matkowski [19], gave an important generalization of Banach contraction principle for a finite product of metric spaces. This result has been extended and generalized by several mathematicians. Recently Pant [22] gave an important concept of reciprocal continuity for a pair of maps. In this paper, we introduced the coordinatewise reciprocal continuity and proved a fixed point theorem which extend and unify the result of Jungck [13], Matkowski [op. cit.] and some of their generalization, for non continuous systems of maps.

Subject Classifications: 47H10, 54H25.

Key Words: Fixed point, coincidence point, Matkowski contraction, product space, reciprocal continuous maps,

coordinatewise commuting, weakly commuting, compatible and R-weakly commuting maps.

1. INTRODUCTION

In the galaxy of contraction principles, two important generalizations of well-known Banach contraction principle were obtained by Gerald Jungck [13] and Janusz Matkowski [18]-[19]. Jungck’s result being simple and elegant in nature has led to a massive growth of fixed point theorems for contractive type maps (see, [2], [3], [6]-[7], [12], [15], [24], [27]-[30], [34], [36]-[38]).Matkowski’s fixed point theorem (Matkowski contraction principle) being somewhat tedious in nature could draw the attention of only a few researchers in applicable mathematics (see, [4]-[5], [8]-[11], [16], [20], [25]-[26], [31]-[33]). Singh- Gairola [31], [32] extend and unify the result of Jungck [op. cit.] and Matkowski [op. cit.] and some of their generalizations by introducing a new class of maps- coordinatewise commuting and their weaker forms (see, also [8] and [11]).

With a view to generalizing fixed point theorems for non continuous maps Pant [22], introduce reciprocal continuity. If Sand Tare maps on a metric space

(

M d,

)

then the pair ( , )S T is said to be reciprocal continuous if and only if

lim _n

n

STx St

→∞ = and lim n n

TSx Tt

→∞ = whenever

{ }

xn be a sequence in M such that lim n lim n

n n

Sx Tx t

→∞ = →∞ = for some t inM . Motivated by the work of Singh-Gairola [31] and Pant [22], we extend and generalize the Matkowski contraction for non continuous maps. We do this by introducing a new class of maps- coordinatewise reciprocal continuous maps.

Throughout this paper we shall follow the following notations and definitions. Let

( )

a_ik be an n× square matrix with non-negative entries defined in Czerwik [4] and Matkowski [op.cit].

(0) _, _1,...,

1

a i k

ik

c i k n

ik _a _i _k

ik

≠ 

=_{ −} =

=

 (1.1) ( ) ( ) ( ) ( )

,

11 1, 1 1,1 1, 1

( 1)

( ) ( ) ( ) ( ) ,

11 1, 1 1,1 1, 1

t t t t

c c c_i c i k

i k k

t c

ik t t t t

c c c c i k

i

i k k

 ₊ _≠

+

+ + +

 + = 

 _{+ +} − ₊ ₊ =

 (1.2)

1,..., 1, , 1,..., .

t= n− i k= n t−

**Corresponding Author: Deepak Khantwal*, Department of Mathematics,**

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Let

(

X d_i, _i

)

,i=1,..., ,n be metric spaces,

1 2 ... n;

X =X ×X × ×X

(

1,..., n

)

;

x= x x

, : ; 1,...,

i i i

P T X →X i= nand

{ } (

{

1 ,...,

)

}

,

m m m

n

x = x x m∈(set of natural numbers) be a sequence in X.

Definition 1.1 [31]: Two systems of maps { ,...,P₁ Pn}and

{

T1,...,Tn

}

are coordinatewise commuting at a point x∈X if

and only if P T x_i

(

₁ ,...,T x_n

)

=T P x_i

(

₁ ,...,P x_n

)

for all i=1,.., .n Two systems of maps{ ,...,P₁ P_n}and

{

T₁,...,T_n

}

are coordinatewise commuting onX if and only if they are coordinatewise commuting at every point ofX.

Definition 1.2 [31]: Two systems of maps { ,...,P₁ Pn}and

{

T1,...,Tn

}

are coordinatewise weakly commuting at a point

x∈X if and only if d P T x_i

(

_i

(

₁ ,...,T x T P x_n

) (

, _i ₁ ,...,P x_n

)

≤d P x T x_i

(

_i , _i

)

for all i=1,..., .n Two systems of maps are coordinatewise weakly commuting on

X

if and only if they are coordinatewise weakly commuting at every point of

.

X

Remark 1.1: Evidently coordinatewise commuting systems of maps are coordinatewise weakly commuting. However, the weakly commuting systems of maps need not to be commuting (see, [31], [32]).

Definition 1.3 [11]: Two systems of maps { ,...,P₁ P_n}and

{

T₁,...,T_n

}

are coordinatewise asymptotically commuting or, following the terminology of Jungck [15], coordinatewise compatible, if and only if

(

) (

)

(

1 1

)

lim m,..., m , m,..., m 0

i i n i n

m→∞d P T x T x T P x P x =

whenever lim m lim m

i i i

m→∞P x =m→∞T x =u for some ui∈Xi, i=1,..., .n

Definition 1.4 [8]: Two systems of maps { ,...,P₁ Pn}and

{

T1,...,Tn

}

are coordinatewise R-weakly commuting at a point

x∈X if and only if d P T x_i

(

_i

(

₁ ,...,T x T P x_n

) (

, _i ₁ ,...,P x_n

)

≤ Rd P x T x_i

(

_i , _i

)

,for alli=1,...,n and for any positive real number R . Two systems of maps are coordinatewise R-weakly commuting on X if and only if they are coordinatewise R-weakly commuting at every point ofX.

Remark 1.2: Coordinatewise weakly commuting maps are coordinatewise R-weakly commuting. However, coordinatewise R-weakly commuting maps need not to be coordinatewise weakly commuting.

The following example shows the coordinatewise R-weak commutativity of two systems of maps and illustrates that the coordinatewise R-weak commutativity need not imply coordinatewise weak commutativity.

Example 1.1: Let X1= X2=

[ ]

0,1 be usual metric spaces andP Ti, i:X1×X2 →Xi, i=1, 2 such that

2

1 1 2

, 0,

2 1, 0.

P x x P x

T x x T x

= =

= − =

Since

(

) (

)

(

)

(

2

(

)

2

)

1 1 1 , 2 , 1 1 , 2 1 2 1 1, 2 1 1

d T P x P x P T x T x =d x − x − =2

(

x1−1

)

2 =2d1

(

2x1−1,x12

)

=2d T x P x1

(

1 , 1

)

and

(

) (

)

(

)

( )

(

)

2 2 1 , 2 , 2 1 , 2 2 0, 0 0 2 2 , 2 .

d T P x P x P T x T x =d = ≤d T x P x

Then two systems of maps { ,P P₁ ₂}and

{

T T₁, ₂

}

are coordinatewise R-weakly commuting. However, they are not coordinatewise weakly commuting for

(

) (

)

(

)

(

2

(

)

2

)

(

)

2

(

)

1 1 1 , 2 , 1 1 , 2 1 2 1 1, 2 1 1 2 1 1 1 1 , 1 ,

d T P x P x P T x T x =d x − x − = x − >d T x P x ∀ ∈x X .

Definition 1.5: Two systems of maps { ,...,P₁ P_n}and

{

T₁,...,T_n

}

are said to be coordinatewise reciprocal continuous if and only iflim

(

₁ m

,...,

m

)

m→∞Pi

T x

n

=

P z

i andlim

(

1

,...,

)

m m

m→∞

T P x

i

P x

n

=

T z

i , whenever there exist a sequence

{ }

m

x in X

such thatlim m lim m

m→∞Pix =m→∞T xi =zi for alli=1,..., .n

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© 2017, IJMA. All Rights Reserved 50 Remark 1.3: Notice that definitions above with n=1are standard ones for commuting, weakly commuting (see [15] and [28]), asymptotically commuting (see, [37]) (also called compatible [14]), R-weakly commuting (see [21]) and reciprocal continuous maps ([22] and see also [23]).

Remark 1.4: Asymptotically commuting (or compatible) class of maps includes commuting and weakly commuting maps. Commuting maps are necessarily weakly and asymptotically commuting both (see, for instance, [14], [28], [31], [37]).

Remark 1.5: The commutativity, weak commutativity and asymptotic commutativity (or compatibility) are equivalent at the point of coincidence of two (or two systems of) maps (see, [1], [14]).

The following example illustrates the coordinatewise reciprocal continuity of systems of maps and shows that coordinatewise reciprocal continuity of systems of maps does not imply continuity of any member of systems of maps.

Example 1.2: Let X₁=

[ ]

0, 1 ,X₂ =[0, 1]be metric spaces with usual metrics and mappingsP Ti, i:X₁×X₂ →Xi,for 1, 2

i= such that

1 2

1 1 2 2 1, 2

1 2

1 1 2 2 1 2

1 2

0 0 0 0

( , ) 1 ( ) 1

0 0

2 2

0 0 0 0

( , ) ( , ) .

1 0 1 0

if x if x

P x x P x x

if x if x

T x x T x x

if x if x

= =

 

 

= =

> >

 

 

= =

 

=_ =_

> >

 

Suppose

{ }

_xm _{be a sequence in}

1 2

X ×X such that m

i i

P x →z and m

i i

T x →zfor some z_i, i=1, 2,asm→ ∞.Then for (0, 0)

z= and

{ }

xm =

{

( )

0, 0

}

∈X for eachm,

(

₁ m

,

₂ m

)

0

i i

P

T x T x

→ =

P z

and

(

₁ m

,

₂ m

)

0

i i

T P x

P x

→ =

T z

asm→ ∞.

Hence systems of maps { ,P P₁ ₂}and

{

T T₁, ₂

}

are coordinatewise reciprocal continuous at z=(0, 0) but they are not

continuous atz=(0, 0). To see this, let

{ }

_xm 1 1, m m

 

= _ _

 be a sequence inX1×X2.Since

{ }

( )

0, 0

m

x → asm→ ∞but

then 1 (0, 0)

2

m

i i

P x → ≠P and m

1 (0, 0)

i i

T x

→ ≠

T

fori=1, 2,asm→ ∞.

The following Lemma is due to Matkowski [19] (see also [4], [33]).

Lemma 1.1: Let

c

_{i k}(0)_, ≥

0, ,

i k

=

1,...., ,

n n

≥

2,

then the system of inequalities

1

, 1,...,

n

ik k i k

a r r i n

=

< =

∑

(1.3) has a positive solution

r

₁

,...,

r

n if and only if the following inequalities hold:

( )t

_0,

_1,...,

_;

_0,1,...,

_1,

_2.

ii

c

>

i

=

n t

−

t

=

n

−

n

≥

(1.4) Moreover, there exists a positive number h<1 such that

1

, 1,..., ,

n

ik k i k

a r hr i n

=

≤ =

∑

(1.5)

for some positive number

r

₁

,..,

r

n.Indeed such an

h

may be found by

1

. max

n i ik k i

k

h r− a r

=

 

=  



∑



(1.6)

Now we will state our main results.

2. RESULTS

Theorem 2.1: Let(X d_i, _i),i=1,..., ,n be complete metric spaces and P Q S T_i, _i, _i, _i:X → X_i,for alli=1,..., ,n be such that

( ) ( ), ( ) ( ), 1, . . . ,.

i i i i

P X ⊂T X Q X ⊂S X i= n (2.1) The system ( ,...,P₁ Pn) coordinatewise R-weakly commutes with the system( ,...,S1 Sn)and the system (Q₁,...,Qn)

coordinatewise R-weakly commutes with the system( ,...,T₁ Tn). (2.2) The system of maps( ,...,P₁ P_n)coordinatewise reciprocal continuous with the system ( ,...,S₁ S_n)or the system of maps

1

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If there exist non- negative numbers b and

a

_ik, i k, =1,....,n such that (1.1), (1.2), (1.4) and the following hold:

0≤ < −b 1 h,where his defined in (1.6) and (2.4)

(

,

)

max

(

,

)

, max (₍ ,_, ),₎ ( ₍, _, ),₎ 1

2

d S x P x d T y Q y_i _i _i _i _i _i n

d_i P x Q y_i _i a d S x T y b _{d S x Q y} _{d T y P x}

ik k k k i i i i i i

i _k       ≤  ∑  +  =        (2.5)

for all x y, ∈X,then the system of equations

i

S

i i i i

P x=Q x=x = x=T x

(2.6) has a unique common solution ( ,...,x₁ x_n)such thatx_i∈X_i, i=1,...,n.

Proof: First we note that in view of the system of inequality (1.5),

h

defined in (1.6) exists and 0< <h 1(c.f. Czerwik [4]). From the Lemma 1.1 and (1.6) we may choose positive numbers

r

₁

,..,

r

nsuch that

1

, 1,..., .

n

ik k i k

a r hr i n

=

≤ =

∑

Pick 0 _, _{1,..., .}

i i

x ∈X i= n We in view of (2.1), construct sequences

{ }

x_im and

{ }

y_im in X_isuch that 2m 1 2m 2m 1

i i i

y + =P x =T x +

2 2 2 1 2 2

, 0,1, 2,...

m m m

i i i

y + =Q x + =S x + m= .

We may assume that 2 1

( , ) , 1,..., .

i i i i

d y y ≤r i= nFrom condition (2.5), we have

3 2 2 1

( , ) ( , )

i i i i i i

d y y =d P x Q x

2 2 1 1

2 1

2 1 1 2

1

( , ), ( , ),

max ( , ), max ₍ _, ₎ ₍ _, ₎

2

i i i i i i n

ik k k k _i _i _i _i _i _i

k

d S x P x d T x Q x a d S x T x b _{d S x Q x} _{d T x P x}

=       ≤   +   _ _  

∑

2 2 1 3

2 1 2 3 1 2

1

( , ) ( , )

max ( , ), max ( , ), ( , ),

2

n

i i i i i i ik k k k i i i i i i

k

d y y d y y

a d y y b d y y d y y

=   +     = _ _ _      

∑

 1 3

2 1 2 3 1 2

1

( , )

max ( , ), max ( , ), ( , ), ,

2

n

i i i ik k k k i i i i i i

k

d y y a d y y b d y y d y y

=       = _ _ _      

∑

 Since

1 3 1 2 2 3

( , ) ( , ) ( , )

2 2

d y_i _i y_i d y_i _i y_i +d y_i _i y_i

≤ implies that

{

}

3 2 2 1 2 3 1 2

( , ) max ( , ), max ( , ), ( , )

1 n

d y_i _i y_i a d y y b d y_i _i y_i d y_i _i y_i ik k k k

k

 

≤  ∑ 

=

 .

If 2 3 1 2

( , ) ( , ),

i i i i i i

d y y >d y y then

3 2 2 1

( , ) max ( , ),

1 n

d y_i _i y_i a d_{ik k} y_k y_k k  ≤  ∑ = 

}

2 3 ( , )

bd y_i _i y_i

{

}

2 3 2 3

max , ( , ) max , ( , )

1 n

a r_{ik k} bd y_i _i y_i hr bd y_i _i _i y_i hr_i k

 

≤  ∑ ≤ ≤

=

  .

Since otherwise, we get a contradiction.

If 2 3 1 2

( , ) ( , ),

i i i i i i

d y y <d y y then

3 2 2 1

( , ) max ( , ),

1

n

d y_i _i y_i a d y y ik k k k k  ≤  ∑ = 

}

1 2 ( , )

bd y_i _i y_i

1 2

max , ( , )

1

n

a r bd y_i _i y_i ik k k   ≤  ∑  =  ,

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4 3 2 3

( , ) ( , )

i i i i i i

d y y =d P x Q x

2 3 3 2

2 3 2 2 3 3

1

( , ) ( , )

max ( , ), max ( , ), ( , ),

2

n

i i i i i i

ik k k k i i i i i i

k

d S x Q x d T x P x a d S x T x b d S x P x d T x Q x

=   +     ≤         

∑



2 4 3 3

2 3 2 3 3 4

1

( , ) ( , )

max ( , ), max ( , ), ( , ),

2

n

i i i i i i ik k k k i i i i i i

k

d y y d y y

a d y y b d y y d y y

=   +     =         

∑

 2 4

2 3 2 3 3 4

1

( , )

max ( , ), max ( , ), ( , ),

2

n

k

=      =       

∑

. Since

2 4 2 3 3 4

( , ) ( , ) ( , )

2 2

i i i i i i i i i

d y y d y y +d y y

≤ implies that

{

}

4 3 2 3 2 3 3 4

1

( , ) max ( , ), max ( , ), ( , )

n

i i i ik k k k i i i i i i k

=   ≤   

∑



{

}

3 4 1

max , max ( , ),

n

ik k i i i i k

a cr b d y y cr

=

 

≤  



∑



and arguing same as before this implies

{

}

4 3

1

( , ) max , max ,

n

i i i ik k i i i

k

d y y a cr bcr hcr bcr

=

 

≤ _ _≤



∑



=crimax

{ }

h b, =c r2i, wherec=max{ , }.h b

Inductively

2 1

( m , m ) m , 0,1, 2,...

i i i i

d y + y + ≤c r m=

Since 0≤ <c 1,hence

{ }

1

m i _m

y ∞

= is a Cauchy sequence in Xifor alli=1,...,n. As Xiis a complete metric space, there

exists a point u_i (say) in X_i such that

{ }

y_im →u_i as m→ ∞ . Moreover 2m 1 2m 2m1

i i i i

y + =P x =T x + →u and

2m 2 2m 1 2m 2

i i i i

y + =Q x + =S x + →u asm→ ∞.

If systems of maps ( ,...,P₁ Pn)and ( ,...,S1 Sn)are coordinatewise reciprocal continuous then

2 2

1

mlim ( ,..., )

m m

i n i

P S x S x Pu

→∞ = and

2 2

1

mlim ( ,..., ) , 1,.., .

m m

i n i

S P x P x S u i n

→∞ = = (2.7)

Now, coordinatewise R- weak commutativity of systems of maps ( ,...,P₁ Pn)and ( ,...,S1 Sn) yields

(

2 2 2 2

)

(

2 2

)

1 1

( m,..., m), ( m,..., m) m, m .

i i n i n i i i

d P S x S x S P x P x ≤Rd P x S x

Taking limm→ ∞and using condition (2.7), we have

(

,

)

0, 1,...,

i i i

d Pu S u = i= n

and therefore

, 1,...,

i i

Pu=S u i= n. (2.8)

SinceP Xi( )⊂T Xi( ), so there exists a point w=

(

w1,...,wn

)

∈Xsuch that Pui =T wi fori=1,...,n.

Now from condition (2.5),

(

)

(

)

1

, .

( , ) ( )

, max , , max ( , ), ( , ),

2

n

i i i i i i

k

u w w u

P u w S u w u u w w d S Q d T P

d Q a d T b d S P d T Q

=

  + 

≤   

 



∑



Using condition (2.8), we get

(

)

(

)

1

,

( , ), ( , ),

, max , , max ( , ) ( )

2

i i i i i i n

k

u u u w

P u w P u w u w w u

d P P d P Q

d Q a d T b d P Q d T P

=       ≤   +   _ _  

∑

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So,

, 1,...,

i i

Pu

=

Q

w i= n.

By the above we then have

i

S ,

i i i

Pu= u=T w=Q w fori=1,...,n. (2.9)

Coordinatewise R-weak commutativity of systems of maps ( ,...,P₁ Pn)and ( ,...,S1 Sn) implies that there exists R>0

such that

(

( 1 ,..., ), ( 1 ,..., )

)

(

,

)

i i n i n i i i

d P S u S u S Pu P u ≤Rd Pu S u .

Using condition (2.9), we get

(

( 1 ,..., ), ( 1 ,..., )

)

0

i i n i n

d P S u S u S Pu P u =

and therefore

1 1

( ,..., ) ( ,..., ), 1,..., .

i n i n

P S u S u =S Pu P u i= n

Again using condition (2.9), we have

1 1 1 1

( ,..., ) ( ,..., ) ( ,..., ) ( ,..., ) , 1,..., .

i n i n i n i n

P S u S u =P Pu P u =S Pu P u =S S u S u i= n

Similarly, R-weak commutativity of systems of maps (Q₁,...,Q_n)and ( ,...,T₁ T_n)implies

1 1 1 1

( ,..., ) ( ,..., ) ( ,..., ) ( ,..., ) , 1,..., .

i n i n i n i n

Q T w T w =Q Q w Q w =T Q w Q w =T T w T w i= n

Now from condition (2.5) and assuming

(

( 1 ,..., ),

)

, 1,..., .

i i n i i

d P Pu P u Pu ≤r i= n

We have

(

( ₁ ,..., ),

) (

( ₁ ,..., ),

)

d_i P P u_i P u_n P u_i =d_i P P u_i P u Q w_n _i

(

)

(

)

(

) (

)

( ,..., ), ,

1 1

( ,..., ), ( ,..., ) , ( , ),

max 1 1

max

( ₁ ,..., ), , ( ₁ ,...,

2

n

a d S P u P u T w_n

ik k k k

k

d_i S P u_i P u P P u_n _i P u_n d T w Q w_{i i} _i

b

d_i S P u_i P u Q w_n _i d T w P P u_i _i _i P u_n

 _∑ 

 ₌ 

 

  

≤  __ __

 _ _

+

 _ _

 __ __

 

(

)

(

)

(

) (

)

( ,..., ), ,

1 1

( ,..., ), ( ,..., ) , ( , ),

max ₁ ₁

max ₍ _,..., _), _, ₍ _,...,

1 1

2

n

a d P P u P u P u_n

ik k k k

k

d_i P P u_i P u P P u_n _i P u_n d T w Q w_i _i _i

b _d _{P P u} _{P u P u} _d _{P u P P u} _{P u}

n n

i i i i i i

 _∑ 

 ₌ 

 

  

≤ _ _ __

 

 _ _

+

 _ _

 __ __

 

max ,

1 k

n

a r bdiik k



≤  ∑

=



(

P P ui( 1 ,...,P un ),P ui

)

}

≤max

{

hr bri, i

}

≤cri, where c=max{ , }.h b Inductively, we can prove that

(

( 1 ,..., ),

)

, 0,1, 2,...

m

i i n i i

d P Pu P u Pu ≤c r m=

and consequently

1

( ,..., ) , 1,...,

i n i

P Pu P u =Pu i= n.

By the above we then have

1 1

( ,..., ) ( ,..., )

i n i n i

P Pu P u =S Pu P u =Pu. (2.10)

Similarly, by using the condition (2.5), we can obtain

1 1

( ,..., ) ( ,..., )

i n i n i

Q Q w Q w =T Q w Q w =Q w. (2.11) Since

i i i

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i

S , 1,...,

i i i i

Pv=Q v= =v v=T v i= n.

This proves that the system of equation (2.6) has a common solution when systems of maps ( ,...,P₁ Pn)and ( ,...,S1 Sn)

are coordinatewise reciprocal continuous. In case, when systems of maps (Q₁,...,Qn)and ( ,...,T1 Tn)are coordinatewise

reciprocal continuous, the proof may be accomplished in an analogous manner.

Now, to prove the uniqueness of the solution, we assume that ,v vi i∈Xi such that vi ≠vi and

i i i i

P v=Q v=S v=T v=v. We can assume that

( )

, , 1,..., .

i i i i

d v v ≤r i= n

From condition (2.5), we get

( , ), ( , ),

( , ) ( , ) max ( , ), max ₍ _, ₎ ₍ _, ₎

1

2

d S v P v d T v Q v_i _i _i _i _i _i n

d v v_{i i i} d P v Q v_i _i _i a d_{ik k} S v T v b_k _k _{d S v Q v} _{d T v P v}

i i i i i i

k

  

  

 

= ≤  ∑  ₊ 

=

  

  

 

( , ), ( , ),

max ( , ), max _{( ,} ₎ _{( ,} ₎

1

2 d v v_{i i i} d v v_{i i i} n

a d_{ik k} v v_{i i} b _{d v v} _{d v v} i i i i i i k

  

  

 

≤  ∑  ₊ 

=

  

  

 

{

}

max , ( , ) max ,

1

n

a r bd v v_{i i i} hr br_i _i cr_i ik k

k

 

≤  ∑ = =

=

  , wherec=max{ , }.h b

Inductively

( , ) m , 0,1, 2,... d v v_{i i i} ≤c r_i m=

and consequently

, 1,..., v_i =v_i i= n.

This completes the proof.

Corollary 2.1: Let(X d_i, _i),i=1, 2...., ,n be complete metric spaces and P T_i, _i:X →X_i,i=1,..., ,n be such that ( ) ( ), 1,...,

i i

P X ⊂T X i= n; (2.13) Systems of maps ( ,...,P₁ Pn)and ( ,...,T1 Tn)are coordinatewise R- weakly commuting; (2.14)

Systems of maps ( ,...,P₁ P_n)and ( ,...,T₁ T_n)are coordinatewise reciprocal continuous; (2.15)

If there exist non- negative numbers band

a

_ik, ,i k=1, ....,n such that (1.1), (1.2), (1.4) and the following hold:

0≤ < −b 1 h, where his defined in (1.6) and (2.16)

(

)

(

)

1

( , ) ( , )

, max , , max ( , ), ( , ),

2

n

i i i i i i

i k

P d P x T y d P y T x

d x P y a d T x T y b d P x T x d P y T y

=

  + 

≤ _ _ _

 



∑

(2.17)

for all x y, ∈X,then the system of equations

i i i,

P x=T x=x (2.18) has a unique common solution ( ,...,x₁ xn)such thatxi∈Xi,i=1,..,n.

Proof: Proof may be completed by putting P_i =Q_iand Si =T ii, =1,... ,n in the proof of Theorem 2.1.

Example 2.1: Let X₁ =X₂ =

[

2, 20

]

be usual metric spaces andP Ti, i:X₁×X₂→Xi, i=1, 2,such that

(

)

(

)

(

)

(

)

1 2

1 2 1 1 2 2

1 2

1 2 1 1 2 2

1 2

1 1 2 2

2 2 2 2

, 6 2 6 , , 6 2 6 ,

2 6 2 6

2 2 2 2

, 12 2 6, , 12 2 6 ,

3 6 3 6

x x

P x x x P x x x

x x

T x x x T x x x

x x x x

= =

 

 

= < ≤ = < ≤

 _>  _>

 

= =

 

 

= < ≤ = < ≤

 ₋ _>  ₋ _>

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for all

(

x x₁, ₂

)

∈X₁×X₂.This example satisfy all the conditions of our Corollary 2.1 and have a unique common

solution at x=(2, 2).It is notable that two systems of maps { ,P P₁ ₂} and

{

T T₁, ₂

}

are coordinatewise reciprocal

continuous but none of P P T T₁, ₂, ,₁ ₂is continuous, not even at the pointx=(2, 2).

Remark 2.1: If we assume

(

M d,

) (

= X di, i

)

, P=P Qi, =Qi, S=Si,T=Ti, i=1, ..., ,n and n=1then (2.5) with a11=k may be written as:

, , , : , 0 1,

P Q S T M →M < <k

(

)

{

(

)

( , ) ( , )

}

, , ,

(*) , max , , ( , ), ( , ),

2 x y X

d Sx Qy d Ty Px

d Px Qy ≤β d Sx Ty d Sx Px d Ty Qy + ∈

where0<β =max

{ }

k b, <1.

Remark 2.2: A multitude of fixed point theorems generalizing the Jungck contraction principle have been establish under the condition (*) and its particular cases (see for [2], [3], [7], [12], [15], [17], [27], [29]-[30], [34]-[38]). It may also be mentioned that Kubiak main result proved exactly under the condition (*) uses the commutativity of Pwith S and that of QwithT.

ACKNOWLEDGEMENT

The second author is thankful to CSIR, New Delhi for financial Assistance vide file no. 09/386(0052)/2015- EMR-I.

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Source of support: CSIR, New Delhi, India. Conflict of interest: None Declared.