some common fixed point theorems for weakly compatible ... · introduced by azam et al. [1]....

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International Journal of Innovation in Science and Mathematics

Volume 7, Issue 1, ISSN (Online): 2347–9051

Some Common Fixed Point theorems for Weakly

Compatible Mappings in Complex Valued

Rectangular Metric space

Garima Gadkari1*, M.S. Rathore2 and Naval Singh3 1*Department of Mathematics, Mahakal Institute of Technology, Ujjain-456664, M.P., India.

2Department of Mathematics, CSA Govt. P.G. College, Sehore, M.P., India. 3Govt. Science and Commerce P.G. College, Benazeer, Bhopal, M.P., India.

Date of publication (dd/mm/yyyy): 23/01/2019

Abstract –In this paper we introduce some common fixed point theorems for weakly compatible mappings under

contractive type condition in complex valued rectangular metric space. Moreover, we give some illustrative examples

which are useful to obtain the results. The recent works presented by authors were more generalized and improve

many existing results of literature.

Keywords – Complex Valued Metric Space, Weakly Compatible Mapping, (E.A) Property, (CLR) Property.

I. INTRODUCTION

Fixed point theory is one of the most famous and traditional theories in the fields of nonlinear analysis. In this

theory, contraction is one of the main tools to prove the existence and uniqueness of a fixed point. The Banach

contraction principle which gives an answer to the existence and uniqueness of a solution of an operator equation

Tx = x, is the most generally used fixed point theorem in all of analysis. The existing material of fixed point

theory contains a large of amount of generalizations of Banach contraction principle by using various form of

contraction condition in different spaces. The idea of complex valued metric spaces was introduced and studied

by Azam et.al[1].They have been established some common fixed points results for mappings satisfying a rational

inequality. Abbas et al. [2] discussed the common fixed point of mappings satisfying rational inequalities in

ordered complex valued generalized metric space. They changed the triangular inequality in the complex valued

metric by the rectangular inequality containing four points and extended the idea of complex valued metric spaces

introduced by Azam et al. [1]. Chauhan et al. [3] proved fixed point theorems in fuzzy metric spaces satisfying

ф-contractive condition with common limit range property. In 2013 Chandoke and Kumar [4] studied the some

fixed point result for rational contraction mappings in complex valued metric spaces. More recently Singh et al.

[5], and Patil and Salunke [6] proved some common fixed point theorems for weakly compatible mappings in

complex valued rectangular (generalized) metric spaces.

In this paper, we prove a fixed point theorems for four weakly compatible self maps in complex valued

rectangular metric spaces and also prove the same if the mapping satisfying the (E.A) property, the common limit

in the range of f, (CLRf) property, common (E.A) property and common limit in the range of f and g, i.e., the

(CLRf g) property. Some illustrative examples are also furnished to support the usability of our results.

II. PRELIMINARIES

Let be the set of complex numbers and let 1 2,z z

Define a partial order on as follows:


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𝑧1 𝑧2 iff Re(𝑧1) Re(𝑧2)and Im(𝑧1) Im(𝑧2). If follows that 𝑧1 𝑧2 if one of the following conditions is

satisfied:

(i) Re(Z1) = Re(Z2) and Im(Z1) = Im(Z2),

(ii) Re(Z1) < Re(Z2) and Im(Z1) = Im(Z2),

(iii) Re(Z1) = Re(Z2) and Im(Z1) < Im(Z2),

(iv) Re(Z1) < Re(Z2) and Im(Z1) < Im(Z2),

In particular, we will write Z1

Z2if Z1≠Z2 and one of ( )ii , ( )iii and ( )iv is satisfied and we will write Z1

Z2if only ( )iv is satisfied. Note that the following conditions hold:

(i) 0 𝑧1

𝑧2 implies that | 𝑧1 | < | 𝑧2|,

(ii) 𝑧1 𝑧2 and 𝑧2 𝑧3 implies that 𝑧1 < 𝑧3,

(iii) 0 𝑧1 𝑧2implies that | 𝑧1 | ≤ | 𝑧2 | ,

(iv) a1, a2 and a1≤ a2 implies that a1z a2 z, for all z .

Definition 2.1 [1]:

Let X be a non-empty set. Suppose that the mapping d: X×X→ is called a complex valued metric on X if

the following conditions are holds:

(a) 0 d (x, y) for all x, y∈X and d (x, y) = 0 ⟺x = y ;

(b) d (x, y) = d (y, x) for all x, y∈X ;

(c) d (x, y) d (x, z) + d (z, y) for all x, y,𝑧 ∈X ;

Then d is called complex valued metric on X and (X, d) is called a complex valued metric space.

Definition 2.2[2]:

Let X be a non-empty set. Suppose that the mapping d: X×X→ satisfies the following conditions:

(a) 0 d (x, y) for all x, y ∈ X and d (x, y) = 0 iff x = y;

(b) d(x, y) = d(y, x) for all x, y∈X ;

(c) d(x, y) d (x, u) +d(u, v) +d(v,y)for all x, y, z∈ X;and all distinct u, v X each one is different from x and

y.

Then d is called complex valued rectangular (generalized) metric on Xand (X, d) is called a complex valued rec-


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-tangular (generalized) metric space.

Example 2.3:

Let X = {-1,1,-i,i}. Define :d X X ;as follows:

d(1,-1)= d(-1,1)= 3eiθ ,

d(-1,i ) = d(i,-1)= d(1,i)=d(i,1)=eiθ ,

d(1,-i ) = d(-i,1)= d(-1,-i)=d(-i,-1)=d(i, -i)=d(-i,i)=5eiθ ,

d(1,1 ) = d(-1,-1)= d(i, i)=d(-i ,-i)=0 ,

It is easy to verify that (X,d) is a complex valued rectangular metric space

when 0,2

.

Note that 3eiθ= d(1,-1)> d(1, i) + d(i,-1) = 2eiθ , so d is not a complex valued metric space.

Definition 2.4[2]:

Let (X, d) be a complex valued rectangular metric space and let𝑥𝑛 be a sequence in X andx ∈X.

(1) If for every c ∈Cwith0 c, there exists 𝑛0 ∈ Nsuch that d(𝑥𝑛, x) c for all 0n n ,then nx is said to be

converges to x and x is a limit point of nx .We denote this by n xx as n or lim nn

x x

.

(2) If for every c ∈Cwith 0 c, there exists0 n N such that for all 0n n , ( , )n n md x x c where m N ,then

nx is said to be Cauchy sequence in X.

(3) If for every Cauchy sequence is convergent in (X, d), then (X, d) is called a complete complex valued

rectangular metric space.

Lemma 2.5 [1]:

Let ( , )X d be a complex valued rectangular metric space and let nx be a sequence in X. Then nx

converges to x if and only if ( , ) 0nd x x as .n

Lemma 2.6 [1]:

Let ( , )X d be a complex valued rectangular metric space and let nx be a sequence in X. Then nx is a

Cauchy sequence if and only if ( , ) 0n n md x x as n where .m

Definition 2.7[7]:

Let f and g be two self maps defined on a non-empty set X . If fx = gx = y for some x X , then x is called the

coincidence point of f and g and y is called the point of coincidence of f and g.

Definition 2.8[7]:


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Two self-mapsf and g are said to be weakly compatible if they commute at their coincidence points, i.e. fx =

gx implies that fgx = gfx.

III. A FIXED POINT THEOREM FOR WEAKLY COMPATIBLE MAPS

In this section, we establish a common fixed point theorem for four weakly compatible mappings in a complex

valued rectangular metric space.

Theorem 3.1:

Let M, N, f and g be four self-mappings of a complete complex valued rectangular (generalized) metric space

(X,d) which satisfy the following:

d(𝑀𝑥, 𝑁𝑦) 𝜇1d(𝑓𝑥,gy) 2 3

1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , )

1 ( , )d x y

d Mx gy d gy Ny d fx Mx d gy Ny

d fx gy

4 , for all , , (3.1)( , ) ( , )

( , ) ( , ) ( , )x y X

d Mx gy d fx Ny

d Mx gy d Ny fx d fx gy

in case ( , ) ( , ) ( , ) 0, 0d Mx gy d Ny fx d fx gyi for i=1 to 4 and

1,4

1i i

or

( , ) 0 ( , ) ( , ) ( , ) 0,d Mx Ny if d Mx gy d Ny fx d fx gy if

(i) ( )M X ( )g X and ( )N X ( ),f X

(ii) The Pairs ( , )M f and ( , )N g are weakly compatible,

(iii) The subspace ( )f X or ( )g X is a closed, then the mappings , ,M N f and g have a unique common fixed

point.

Proof:

We construct a sequence { }ny in X such that

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

n n n n n ny Mx gx and y Nx fx n

Where { }n

x is another sequence in .X Using (3.1), we have

( , ) ( , )2 2 1 2 2 1

d y y d Mx Nxn n n n

1

( , )2 2 1

d fx gxn n

2

1 ( , )

[ ( , ) ( , )]2 2 1 2 1 2 1

2 2 1d x

d Mx gx d gx Nxn n n n

xn n

3 4

[1 ( , )] ( , ) ( , ) ( , )2 2 2 1 2 1 2 2 1 2 2 11 ( , ) ( , ) ( , ) ( , )

2 2 1 2 2 1 2 1 2 2 2 1

d fx Mx d gx Nx d Mx gx d fx Nxn n n n n n n n

d fx gx d Mx gx d Nx fx d fx gxn n n n n n n n

1 ( , )2 1 2

d y yn n

2 3

1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , )2 2 2 2 1 2 1 2 2 2 1

1 ( , )2 2 1 2 1 2

d y y

d y y d y y d y y d y yn n n n n n n n

d y yn n n n

4

( , ) ( , )2 2 2 1 2 1

( , ) ( , ) ( , )2 2 2 1 2 1 2 1 2

d y y d y yn n n n

d y y d y y d y yn n n n n n

2 31

( , ) ( , ) ( , )2 1 2 2 2 1 2 2 1



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Hence, ( , )2 2 1

d y yn n

1 ( , )2 1 2(1 )

2 3

d y yn n

Therefore, ( , )2 2 1

d y yn n

( , )2 1 2

d y yn n

Where, 1, Since 1.4

1(1 ) 12 3 i i

Proceeding in a similar way we have,

2( , ) ( , ) ( , )2 2 1 2 1 2 2 2 2 1


0 1..................... ( , )

2d y y

n

Finally, we can conclude that 1

( , )n nd y y

0 1( , )nd y y

For all , , , we havem n m n N

( , )d y yn m ( , ) ( , ) ............. ( , )1 1 2 1

d y y d y y d y yn mn n n m

21 1( , ) ( , ) ( , ) ........... ( , )0 1 0 1 0 1 0 1

n n n md y y d y y d y y d y y

0 11( , )

n

d y y

0 1Therefore as , .

1

Thus,{ } is a Cauchy sequence in

( , ) ( , ) 0n

n

m n

y X

d y y d y yn m

Since X is complete, there exists point z in X such that,

2 2 1 2 1 2 2lim lim lim lim .

n n n nn n n nMx gx Nx fx z

Assuming ( )isclosed, ( )and forsome .f X z f X z fu u X

We claim that Mu = fu = z.

Using the rectangular inequality [Definition2.2 (iii)] we get,

( , )d Mu z 2 1 2 1 2 1 2 1) ( , ) ( , )( , n n n nd N g d g zd Mu Nx x x x

2 31 1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , )2 1 2 1 2 1 2 1 2 1( , )

2 1 1 ( , )2 1 2 1

d u

d Mu gx d gx Nx d fu Mu d gx Nxn n n n nd fu gx

n x d fu gxn n

4 , ).( , ) ( , )

2 1 2 1 ( , ) (2 1 2 1 2 1( , ) ( , ) ( , )

2 1 2 1 2 1

zd Mu gx d fu Nx

n n d Nx gx d gxn n nd Mu gx d Nx fu d fu gx

n n n

As n→∞, we get

2 3 41 ( , ) ( , ).1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , ) ( , ) ( , )( , )

1 ( , ) ( , ) ( , ) ( , )d z z d z z

d z z

d Mu z d z z d z Mu d z z d Mu z d z zd z z

d z z d Mu z d z z d z z


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( , )d Mu z 2 ( , )d Mu z

As 𝜇2< 1, |d (Mu, z)| = 0 implies that Mu = z i.e, fu = Mu = z and u is a coincidence point of f and A.

Since ( )M X ( ) , for some .g X Mu gv v X

Hence fu = Mu = gv = z. We claim that Nv=z. By inequality (3.1)

( , ) ( , )d z Nv d Mu Nv21 1 ( , )

[ ( , ) ( , )]( , )

d u v

d Mu gv d gv Nvd fu gv

3 4

[1 ( , )] ( , ) ( , ) ( , )

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

d fu Mu d gv Nv d Mu gv d fu Nv

d fu gv d Mu gv d Nv fu d fu gv

2 3 41 1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , ) ( , ) ( , )( , )

1 ( , ) ( , ) ( , ) ( , )d z z

d z z d z Nv d z z d z Nv d z z d z Nvd z z

d z z d z z d Nv z d z z

2 3 2 3( ) 0 and ,hence .( , ) since 1, ( , ) Nv z Mu fu Nv gv zd z Nv d z Nv

As M and f are weakly compatible, Mfu=fMu i.e., Mz=fz. We now prove that Mz=z, suppose not, Mz ≠ z,

then by (3.1),

( , ) ( , )d Mz z d Mz Nv 2 31 1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , )( , )

1 ( , )d z v

d Mz gv d gv Nv d fz Mz d gv Nvd fz gv

d fz gv

4

( , ) ( , )

( , ) ( , ) ( , )

d Mz gv d fz Nv

d Mz gv d Nv fz d fz gv

2 31 1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , )( , )

1 ( , )d z z

d Mz z d z z d Mz Mz d z zd Mz z

d Mz z

4

( , ) ( , )

( , ) ( , ) ( , )

d Mz z d Mz z

d Mz z d z Mz d Mz z

4

21 )3

( ( , )d Mz z

1, 0 and . , .4

Henceas ( , )1

Mz z i e Mz fz zd Mz zi i

Thus z is a fixed point of M and f. Also since N and g are weakly compatible, Ngv=gNv i.e., Nz=gz. We now

prove that Nz = z, suppose not, Nz ≠ z, then by (3. 1), again,

( , ) ( , )d z Nz d Mz Nz1 2

[ ( , ) ( , )]( , )

1 ( , )

d Mz gz d gz Nzd fz gz

d z z

3 4

[1 ( , )] ( , ) ( , ) ( , )

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

d fz Mz d gz N z d Mz gz d fz Nz

d fz g z d Mz gz d Nz fz d fz gz

2 31 1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , )( , )

1 ( , )d z z

d z Nz d Nz Nz d z z d Nz N zd z Nz

d z N z

4

( , ) ( , )

( , ) ( , ) ( , )

d z Nz d z Nz

d z Nz d Nz z d z Nz

421 )

3( ( , )d z Nz

Hence │d (z, NZ)│= 0 i.e., NZ=Z and MZ= fZ= NZ = gZ= z, z is a common fixed point of M, N, f and g.


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To show that the fixed point is unique, suppose that there is another point w ∈ X such that Mw = Nw = fw =

gw = w. From (3.1), we have

( , ) ( , )d w z d Mw Nz21 1 ( , )

[ ( , ) ( , )]( , )

d w z

d Mw gz d gz Nzd fw gz

3 4

[1 ( , )] ( , ) ( , ) ( , )

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

d fw Mw d gz N z d Mw gz d fw Nz

d fw g z d Mw gz d Nz fw d fw gz

( , )d w z2 31 1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , )( , )

1 ( , )d w z

d w z d z z d w w d z zd w z

d w z

4

( , ) ( , )

( , ) ( , ) ( , )

d w z d w z

d w z d z w d w z

( , )d w z 4

21 )3

( ( , )d w z

0 1 and .4

Therefore ( , )1

as w zd w zi i

Which prove the uniqueness of the fixed point. Similar argument holds if g(X) is assumed to be closed. Hence

M, N, f and g have a unique common fixed point in X.

By putting 3 4, 0 , in Theorem 3.1 above, we get the following corollary.

Corollary 3.2:

Let M, N, f and g be four mappings of a complete complex valued rectangular metric space (X, d) which satisfy

the following:

( , )d Mx Ny 1 2

[ ( , ) ( , )]( , ) , (3.2)

1 ( , )

d fx gy d gy Nyd fx gy

d x y

where μ1 and μ2 are nonnegative reals such that μ1+ μ2< 1 and if,

(i) ( ) ( )and ( ) ( ),M X g X N X f X

(ii) The pairs (M, f) are (N, g) weakly compatible,

(iii) The subspace f(X) or g(X) is closed, then M, N,f and g have unique common fixed point.

If we put N=M and g = f, μ2, μ3, μ 4=0 in Theorem 3.1, we get the following corollary.

Corollary 3.3:

Let M and f be two mappings of a complete complex valued rectangular metric space (X, d) which satisfy the

following:

( , )d Mx Ny 1 ( , ) (3.3)d fx gy

where μ1 is nonnegative real’s such that μ1< 1 and if,

(i) ( ) ( ),M X f X

(ii) The pairs (M, f) is weakly compatible,


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(iii) The subspace f(X) is closed,

then M and f have unique common fixed point.

Example 3.4:

Let X = 1 1 1 1, where , , , , & [1,2].

2 3 4 5P Q P Q

Define the rectangular metric on :d X X as

1 1 1 1, , 0.3 ,

2 3 4 5d d i

1 1 1 1, , 0.2 ,

2 5 3 4d d i

1 1 1 1, , 0.6 ,

2 4 5 3d d i

1 1 1 1 1 1 1 1, , , , 0

2 2 3 3 4 4 5 5d d d d

and ( , ) , , , .d x y i x y if x y Qor x P y Q or x Q y P

Clearly, d does not satisfy the triangle inequality on P. Indeed,

1 1 1 1 1 10.6 , , ,

2 4 2 5 3 4

0.3 0.2

0.5

i d d d

i i

i

Notice that (RM3) holds, so d is a rectangular metric.

Let M, f:X→X be define as

1for all and

4

1[1,2],

5

1 1 1 1( ) , , ,

4 2 3 4

1 1.

3 5

Mx x X

if x

f x if x

if x

Here ( ) ( )M X f X , also M and f are weakly compatible , for all x X , it can be shown that ( , )d Mx My

1 1( , ), . , ( , ) , 0 ( , ),

4 4d fx fy i e d Mx My d d fx fy

where is any non-negative real such that < 1.

Therefore all the conditions of corollary 3.3 are satisfied and 1

4X is the unique commonfixed point of M and f.


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IV. FIXED POINT THEOREMS FOR MAPPINGS SATISFYING THE (E.A) PROPERTY AND (CLR)

PROPERTY

Many researchers such as Aamri and Moutawakil [8], Sintunavarat and Kumam [9], and Verma and Pathak

[10] have been studied on common fixe point theorems for mappings satisfying the (E.A) property and the limit

in the range of f (CLR f) property in complex valued metric spaces. In this paper we obtain such results in

complex valued rectangular metric spaces.

The definitions of (E.A) property and common limit in the range (CLR) in complex valued rectangular metric

spaces are as follows:

Definition 4.1[11]:

Let A,B : X → X be two self-mappings of a complex valued rectangular metric space (X, d).The pair (A,B) is

said to satisfy property (E.A), if there exists a sequence {xn}in X such that

lim ( , ) lim ( , ) for some .n n

d Ax u d Bx u u u Xn n

Example 4.2:

Let(X, d) be a complex valued rectangular metric space.Let us define mappings A,B:X X as

21 5

12 3

Ax x and2

3

xBx for all x X and sequence

2

11nx

n

.

Then

2

2

1 1 5 1lim lim 1 1 ,

2 3 3n

n nAx

n

2

2

11

1lim lim

3 3n

n n

nBx

Since1

3X , the mapping A and B satisfy the (E.A) property.

Definition 4.3[9]:

Let A,B: X → X be two self-mappings of a complex valued rectangular metric space (X,d), A and B is said to

satisfy the common limit in the range of B property if lim lim , forsome .Ax Bx Bx x Xn nn n

Example 4.4:

Let(X, d) be a complex valued rectangular metric space.Let us define mappings A,B:X X as

21 5

12 3

Ax x and2

for all3

xBx x X and sequence

2

11nx

n

.

Then

2

2

1 1 5 1lim lim 1 1 ,

2 3 3n

n nAx

n


10



2

2

11

1lim lim (1),

3 3n

n n

nBx B

where1 ,X hence A and B satisfy the common limit in the range of

B, i.e. (CLR B) property.

Now we prove a fixed point theorem for four weakly compatible self-maps in complex valued rectangular

metric space (X, d) satisfying the (E.A) property.

Theorem 4.5:

Let M, N, f and g be four self-mappings of a complex valued rectangular (generalized) metric space (X, d)

which satisfy the following:

( , )d Mx Ny2 3 41 ,

1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , ) ( , ) ( , )( , )

1 ( , ) ( , ) ( , ) ( , )d x y

d Mx gy d gy Ny d fx Mx d gy Ny d Mx gy d fx Nyd fx gy

d fx gy d Mx gy d Ny fx d fx gy

for all , , (4.1)x y X

in case ( , ) ( , ) ( , ) 0, 0d Mx gy d Ny fx d fx gy i for i=1 to 4 and 1,4

1i i

or


(i) One of the pairs (M, f) or (N, g) satisfies the property (E.A),

(ii) ( )M X ( )g X or ( )N X ( ),f X

(iii) The subspace ( )f X or ( )g X is closed,

(iv) The Pairs ( , )M f and ( , )N g are weakly compatible,

Then the mappings , ,M N f and g have a unique common fixed point.

Proof:

Suppose the pair (N, g) satisfies the property (E.A), then there exists a sequence {xn} in X such that

lim lim ,forsome .Nx gx t t Xn nn n

Since N(X) ⊆f(X), there exists a sequence {yn} in X such that 𝑁𝑥𝑛 = 𝑓𝑦𝑛, lim𝑛→∞

𝑓𝑦𝑛 = t.

weclaimthat lim My tnn

suppose that*lim ,My t tnn

then from (4.1) we get,

( , )n nM Nd y x 211 ( , )

[ ( , ) ( , )]( , ) n n n n

n n

n n

M N

d y x

d y gx d gx xd fy gx

3 4

[1 ( , )] ( , ) ( , ) ( , )

1 ( , ) ( , ) ( , ) ( , )n n n n n n n n

n n n n n n n n

d fy My d gx Nx d My gx d fy Nx

d fy gx d My gx d Nx fy d fy gx

As n → ∞


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*( , )d t t* * *

2 3 4 *

,1 1 ( , )

[ ( ) ( , )] [1 ( , )] ( , ) ( , ) ( , )( , )

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

t

d t t

d t d t t d t t d t t d t t d t td t t

d t t d t t d t t d t t

*( , )d tt * * *

2 2( , ) ,as 1, ( , ) 0and , .thus limd t d t tt t t My tnn

Hence lim lim lim limM gy Nx fy x tn n n nn n n n

Suppose isclosed. Then for some .

Therefore lim lim lim lim =

g(X) t= gu u X

Nx gx fy My t gun n n nn n n n

We claim that Nu = gu, from (4.1), we have

( , )nd My Nu1 2 3

,

1 ( , )

[ ( ) ( , )] [1 ( , )] ( , )( , )

1 ( , )n n n

n

n n

gu

d y u

d My d gu Nu d fy My d gu Nud fy gu

d fy gu

4

( , ) ( , )

( , ) ( , ) ( , )n n

n n n

d My gu d fy Nu

d My gu d Nu fy d fy gu

As n→∞, we get,

( , )d t Nu 1 2 3 4

,

1 ( , )

[ ( ) ( , )] [1 ( , )] ( , ) ( , ) ( , )( , )

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

t

d t u

d t d t Nu d t t d t Nu d t t d t Nud t t

d t t d t t d Nu t d t t

Thus, d(t, Nu) 2 3 2 3( ) ) 1, 0 and i.e, ( , ) as( ( , ) Nu t gu = t.d t Nu d t Nu Nu

Since N and g are weakly compatible, Ngu=gNu, i.e., Nt=gt and t is a coincidence point of N and

g. Since ( ) ( ), forsome .N X f X Nu fv v X

Hence Nu = gu =fv = t, we claim that Mv = fv = t. From (4.1), we have

( , )d Mv Nu1 2 3

1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , )( , )

1 ( , )d v u fv

d Mv gu d gu Nu d fv Mv d gu Nud fv gu

d gu

4

( , ) ( , )

( , ) ( , ) ( , )

d Mv gu d fv Nu

d Mv gu d Nu fv d fv gu

Hence d (Mv, t) 1 2 3 4

1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , ) ( , ) ( , )( , )

1 ( , ) ( , ) ( , ) ( , )d v t

d Mv t d t t d t Mv d t t d Mv t d t td t t

d t t d Mv t d t t d t t

. , ( , )Mi e d v t2 2

( , ) as ( , )1, 0 and .d Mv t d Mv t Mv t

Thus Mv =fv = t. Since M and f are weakly compatible, Mfv = fMv, i.e., Mt = ft.

We claim that Mt = t, consider from (4.1),

( , ) ( , )d Mt t d Mt Nu1 2 3

1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , )( , )

1 ( , )d t u

d Mt gu d gu Nu d ft Mt d gu Nud ft gu

d ft gu

4

( , ) ( , )

( , ) ( , ) ( , )

d Mt gu d ft Nu

d Mt gu d Nu ft d ft gu

1 2 3

1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , )( , )

1 ( , )d t t

d Mt t d t t d Mt Mt d t td Mt t

d Mt t


12



44 1 2As( ) 1, ( , ) 0and .

3

( , ) ( , )

( , ) ( , ) ( , )d Mt t Mt t

d Mt t d Mt t

d Mt t d t Mt d Mt t

Similarly we can show that Nt = t. Hence Mt = Nt = ft = gt = t and the mappings M, N, f and g have a

common fixed point. To prove uniqueness of the fixed point, assume that

isanother point such that . Then using(4.1)w X Mw Nw fw gw w

( , ) ( , )d t w d Mt Nw1 2 3 4

1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , ) ( , ) ( , )( , )

1 ( , ) ( , ) ( , ) ( , )d t w

d t w d w w d t t d w w d t w d t wd t w

d t w d t w d w t d t w

41 2( )

3( , )d t w

41 2As( ) 1, ( , ) 0and .

3d t w t w

Which proves the uniqueness of the fixed point, similar result can be obtained assuming that the pair (M, f)

satisfies the (E.A) property and f(X) is closed. Hence complete the proof.

Now we obtain fixed point theorem for mapping satisfying the common limit in the range of f(CLRf)

(or(CLRg)) property in complex valued rectangular (generalized) metric spaces. Here the results are obtained

without the closeness assumption of the subspaces.

Theorem 4.6:

Let M, N, f and g be four self mappings of a complex valued rectangular (generalized) metric space (X, d)


( , )d Mx Ny 1 2 3

[ ( , ) ( , )] [1 ( , )] ( , )( , )

1 ( , ) 1 ( , )

d Mx gy d gy Ny d fx Mx d gy Nyd fx gy

d x y d fx gy

4 , for all , , (4.2)( , ) ( , )

( , ) ( , ) ( , )x y X

d Mx gy d fx Ny


in case ( , ) ( , ) ( , ) 0, 0d Mx gy d Ny fx d fx gyi for i=1 to 4 and 1,

4

1i i

or


(i) ( )M X ( )g X (or ( )N X ( )),f X

(ii) The Pairs ( , )M f and ( , )N g are weakly compatible,

(iii) The Pair ( , )M f satisfies the ( )fCLR property ( , )or N g satisfies the ( )gCLR property,


Proof:

Suppose the mappings M and f satisfy the (CLR f) property, there exists sequence {xn} such that,

lim lim ,sayM n nn nx fx fx t


13



For some ,x X since ( )M X ( ),g X there exists a sequence {yn} in X such that .n ngyMx

Therefore lim nn

gy t

*we claim that lim ,Suppose that lim , then from(4.2),wegetn nn n

Ny t Ny t t

( , )n nd Mx Ny 1 2

,

1 ( , )

[ ( ) ( , )]( , ) n n n n

n n

n n

gy

d x y

d Mx d gy Nyd fx gy

3 4

,[1 ( , )] ( , ) ( , ) ( )

1 ( , ) ( , ) ( , ) ( , )n n n n n n n n

n n n n n n n n

Nd fx Mx d gy Ny d Mx gy d fx y


As n→∞, we have

*)( ,d t t* * *

1 2 3 4 *1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , ) ( , ) ( , )( , )

1 ( , ) ( , ) ( , ) ( , )d t t

d t t d t t d t t d t t d t t d t td t t

d t t d t t d t t d t t

*)( ,d t t * *

2 3( , ) ( , )d t t d t t *

2 3( ) ( , )d t t

* *.

2 3( ( , ) 0and .ThusSince ) 1, limd t t t t Ny tnn

Hence lim lim lim lim .M N gyx fx y t fxn n n nn n n n

We claim that Mx = fx, suppose not, then from (4.2), we have

( , )nd Mx Ny 1 21 ( , )

[ ( , ) ( , )]( , ) n n n

n

nd x y

d Mx gy d gy Nyd fx gy

3 4

[1 ( , )] ( , ) ( , ) ( , )

1 ( , ) ( , ) ( , ) ( , )n n n n

n n n n

d fx Mx d gy Ny d Mx gy d fx Ny


As n→∞, we have

( , )fxd Mx1 2 3 4

1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , ) ( , ) ( , )( , )

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

fx fx fx fx fx fxfx

d x x fx fx fx fx

d Mx fx d d fx Mx d d Mx fx dd fx

d fx d Mx d fx d fx

( , ))d Mx fx 2 2( , ) , 1, .since ( ) ( ), for some .d Mx fx as Mx fx t M X g X Mx gv v X

Hence Mx = fx = gv t. We claim that Nv = t, then from (4.2).

( , )d Mx Nv1 2

1 ( , )

[ ( , ) ( , )]( , )

d x v

d Mx gv d gv Nvd fx gv

3

[1 ( , )] ( , )

1 ( , )

d fx Mx d gv Nv

d fx gv

4 ,( , ) ( , )

( , ) ( , ) ( , )

fxd Mx gv d Nv

d Mx gv d Nv fx d fx gv

1 2 3 4Hence1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , ) ( , ) ( , )( , ) ( , )

1 ( , ) ( , ) ( , ) ( , )d t t

d t t d t Nv d t t d t Nv d t t d t Nvd t Nv d t t

d t t d t t d Nv t d t t

2 3 2 3( ) ( , ). As( ) 1,Wehave ( , ) 0and .( , ) d t Nv d t Nv Nv td t Nv

Hence .Mx fx gv Nv t


14



As M and f are weakly compatible, M fx =f Mx and Mt = ft, and also since N and g are weakly compatible

Ngv=gNv i.e., Nt = gt. we claim Mt = t using (4.2),consider,

( , ) ( , )d Mt t d Mt Nv1 2 3

1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , )( , )

1 ( , )d t v

d Mt gv d gv Nv d ft M t d gv Nvd ft gv

d ft gv

4

( , ) ( , )

( , ) ( , ) ( , )

d Mt gv d ft Nv

d Mt gv d Nv ft d ft gv

1 2 3

1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , )( , )

1 ( , )d t t

d Mt t d t t d Mt Mt d t td Mt t

d Mt t

4

( , ) ( , )

( , ) ( , ) ( , )

d Mt t d Mt t

d Mt t d t Mt d Mt t

4 4

1 2 1 2( ) ( ) 0and .3 3

( , ),as 1, ( , ) Mt td Mt t d Mt t

Similarly we can show that Nt = t i.e., Nt = gt = t, hence Mt = ft = Nt = gt =t and t is a common fixed point of

the mappings M, f, N and g. Uniqueness of the common fixed point follows easily from (4.2).Similarly existence

and uniqueness of the fixed point can be proved assuming that (N, g) satisfies the (CLR g) property.

Remark 4.7:

We can conclude that if mapping A and f satisfy the property (E.A) and f(X) is closed, then M and f satisfy the

Common limit in the range of f (CLRf) property.

V. FIXED POINT FOR MAPPINGS SATISFYING THE COMMON (E.A) PROPERTY

Liu et al. [11] introduced the concept of common (E.A) property.

Definition 5.1:

Two pairs of self maps (A,f) and (B,g) of a complex valued rectangular metric space (X, d) are said to satisfy

the common (E.A) property if there exist two sequence {xn} and {yn} in X such that

lim lim lim lim , for some .Ax fx By gy t t Xn n n nn n n n

Next we prove a fixed point theorem in complex valued rectangular metric space in which the two pairs of self

mappings of X satisfy the common property (E.A).

Theorem 5.2:

Let M, N, f and g be four self mappings of a complex valued rectangular (generalized) metric space (X, d)


( , )d Mx Ny2 31 1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , )( , )

1 ( , )d x y


d fx gy

4 , , , (5.1)( , ) ( , )

( , ) ( , ) ( , )for all x y X

d Mx gy d fx Ny


in case ( , ) ( , ) ( , ) 0, 0d Mx gy d Ny fx d fx gyi for i=1 to 4 and 1,

4

1i i

or


(i) The Pairs ( , )M f and ( , )N g are weakly compatible,


15



(ii) The Pairs ( , )M f and ( , )N g satisfy the common property (E.A),

(iii) The subspace ( )f X and ( )g X are closed,


Proof:

Since the pairs ( , )M f and ( , )N g satisfy the common property (E.A), there exist two sequences

and inn nx y X such that

lim lim lim lim ,M gyx fx Ny zn n n nn n n n

for some z X . Since ( )f X is closed and ( )t f X , there exists u X such that z fu .we claim that

Mu fu z . From (5.1) we have,

( , )nd Mu Ny 1 2 31 ( , )

[ ( , ) ( , )] [1 ( , )] ( , )( , )

1 ( , )n n n n n

n

n nd u y

d Mu gy d gy Ny d fu Mu d gy Nyd fu gy

d fu gy

4

( , ) ( , )

( , ) ( , ) ( , )n n

n n n

d Mu gy d fu Ny

d Mu gy d Ny fu d fu gy

As n→∞

( , )d Mu z1 2 3 4

1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , ) ( , ) ( , )( , )

1 ( , ) ( , ) ( , ) ( , )d z z

d Mu z d z z d z Mu d z z d Mu z d z zd z z

d z z d Mu z d z z d z z

2 2 0 and .i.e, .( , ),as 1, ( , ) Mu z Mu fu zd z Mu d Mu z

Since M and f are weakly compatible, Mfu fMu and Mz fz i.e., z is a coincidence point of M and f. Also

since ( )g X is closed and ( ),z g X z gv for some .v X Hence .Mu fu gv z we prove that .Nv gv z

consider from (5.1)

( , ) ( , )d z Nv d Mu Nv 1 21 ( , )

[ ( , ) ( , )]( , )

d u v

d Mu gv d gv Nvd fu gv

3 4

[1 ( , )] ( , ) ( , ) ( , )

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

d fu Mu d gv Nv d Mu gv d fu Nv

d fu gv d Mu gv d Nv fu d fu gv

1 2 3 41 ( , )

[ ( , ) ( , )] [1 ( , )] ( , ) ( , ) ( , )( , )

1 ( , ) ( , ) ( , ) ( , )d z z

d z z d z Nv d z z d z Nv d z z d z Nvd z z

d z z d z z d Nv z d z z

2 3 2 3( 0 as( . ) ( , ),Hence ( , ) ) 1d z Nv d Mu z

Therefore, .Nv gv z Since N and g are weakly compatible, we have Ngv gNv i.e., Nz gz and z is a

coincidence point of N and g. To show that z is a fixed point, we claim that Mz z , from (5.1)


16



( , ) ( , )d Mz z d Mz Nv 1 2 31 ( , )

[ ( , ) ( , )] [1 ( , )] ( , )( , )

1 ( , )d z v

d Mz gv d gv Nv d fz Mz d gv Nvd fz gv

d fz gv

4

( , ) ( , )

( , ) ( , ) ( , )

d Mz gv d fz Nv

d Mz gv d Nv fz d fz gv

Since fz = Mz,

( , )d Mz z

1 2 3 41 ( , )

[ ( , ) ( , )] [1 ( , )] ( , ) ( , ) ( , )( , )

1 ( , ) ( , ) ( , ) ( , )d z z

d Mz z d z z d Mz Mz d z z d Mz z d Mz zd Mz z

d Mz z d Mz z d z Mz d Mz z

4 41 2 1 2( ) ( ) 0and ,Hence .

3 3( , ),Since 1, ( , ) Mz z Mz fz zd Mz z d Mz z

Similarly we can show that .Nz gz z Thus we have Mz fz Nz gz z and z is a common fixed point

of M, N, f and g. Uniqueness of the fixed point follows easily using inequality (5.1).

Remark5.2:

It can be pointed that in proving the existence and uniqueness of the unique fixed point of M, N, f and g, when

one of the pairs ( , ) ( , )M f or N g satisfies the property (E.A), closeness of any one of ( ) ( )f X or g X and

containment of one pair of subspaces is required, whereas when the pairs ( , )and( , )M f N g satisfy the common

property (E.A), it is required that both ( )and ( )f X g X are closed. Here the assumption of containment of

subspaces is omitted.

VI. FIXED POINT THEOREM FOR MAPPINGS SATISFYING (CLR f g) PROPERTY

Imdad et al. [12] extended the (CLR f) or (CLR g) property to the (CLR fg) property. The (CLR fg) property does

not require closedness of the range subspaces nor their containment.

Definition 6.1:

Two pairs of self mappings (A, f) and (B, g) in a complex valued rectangular metric space (X, d) are said to

satisfy the (CLR f g) property with respect to maps f and g if there exist two sequence{xn} and {yn }in X such

that

lim lim lim lim , where ( ) ( ).Ax fx By gy t t f X g Xn n n nn n n n

Theorem 6.2:

Let M, N, f and g be four self-mappings of a complex valued rectangular (generalized) metric space (X, d)


( , )d Mx Ny2 31 1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , )( , )

1 ( , )d x y


d fx gy

4 , for all , , (6.1)( , ) ( , )

( , ) ( , ) ( , )x y X

d Mx gy d fx Ny



17



in case ( , ) ( , ) ( , ) 0, 0d Mx gy d Ny fx d fx gy i for i=1 to 4 and 41,

1 ii

or ( , ) 0d Mx Ny

( , ) ( , ) ( , ) 0,if d Mx gy d Ny fx d fx gy if

(i) The Pairs ( , )M f and ( , )N g are weakly compatible,

(ii) The Pairs ( , )M f and ( , )N g satisfy the (CLR f g) property,


Proof:

The pairs ( , )M f and ( , )N g satisfy the (CLR f g) property, so there exist sequences

and inn nx y X such that, lim lim lim lim ,Mx fx Ny gy tn n n nn n n n

where ( ) ( ).t f X g X Since ( )t f X , t fu for some .u X we prove that Mu fu t . From (6.1) we have,

( , )nd Mu Ny1 2 3

1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , )( , )

1 ( , )n n n n n

n

n nd u y

d Mu gy d gy Ny d fu Mu d gy Nyd fu gy

d fu gy

4

( , ) ( , )

( , ) ( , ) ( , )n n

n n n

d Mu gy d fu Ny

d Mu gy d Ny fu d fu gy

As n→∞

( , )d Mu t1 2 3 4

1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , ) ( , ) ( , )( , )

1 ( , ) ( , ) ( , ) ( , )d t t

d Mu t d t t d t Mu d t t d Mu t d t td t t

d t t d Mu t d t t d t t

2 2 0and .( , ),Since 1, ( , ) Mu td Mu t d Mu t

Thus Mu fu t and u is a coincidence point of M and f. since M and f are weakly compatible, Mfu fMu

and Mt ft . Also ( )t g X , there exists v X such that t gv . Hence Mu fu gv t . To prove that

Nv gv t , suppose Nv gv , consider using (6.1)

( , ) ( , )d t Nv d Mu Nv1 2 3

1 ( , )

[ ( , ) ( , )] [1 ( , )] ( , )( , )

1 ( , )d u v

d Mu gv d gv Nv d fu Mu d gv Nvd fu gv

d fu gv

4

( , ) ( , )

( , ) ( , ) ( , )

d Mu gv d fu Nv

d Mu gv d Nv fu d fu gv

1 2

1 ( , )

[ ( , ) ( , )]( , )

d t t

d t t d t Nvd t t

3 4

[1 ( , )] ( , ) ( , ) ( , )

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

d t t d t Nv d t t d t Nv

d t t d t t d Nv t d t t

2 3 2 3( ) 0and .( , ),Since ( ) 1, ( , ) Nv td t Nv d t Nv

i.e, .Nv gv t As N and g are weakly compatible, Ngv gNv i.e., Nt gt .. In a similar way as in Theorem

4.5 we can prove that Mt t i.e., ,Mt ft t and also Nt t , .Nt gt t It implies that Mt Nt ft gt t


18



and t is a common fixed point of M, N, f and g. The uniqueness of the common fixed point follows easily from

(6.1).

Remark 6.3:

We can conclude that when the pairs (M, f) and (N, g) satisfy the common (E.A) property and if f(X) and g(X),

both are closed, then the pairs satisfy the (CLR f g) property.

VII. CONCLUSIONS

In this article, we continued the study of complex valued metric space to common fixed point theorems for

weakly compatible mappings for rational contractions. Theorems approved in this article will be supportable for

researchers to work on rational contractions.

REFERENCES

[1] A. Azam, B. Fisher and M. Khan,“Common fixed point theorems in complex valued metric spaces”, Numerical functional Analysis and

optimization, Vol. 3, 2011, pp. 243-253

[2] M. Abbas,V. Raji, T. Nazir and S. Radenovi,“Common fixed point of mappings satisfying rational inequalities in ordered complex valued generalized metric spaces”, AfrikaMatematika, Vol. 26, 1-2,2013, pp. 17-30

[3] S. Chauhan, M.A. Khan, W. Sintunavarat, “Common fixed point theorems in fuzzy metric spaces satisfying ф-contractive condition with

common limit range property”, Abstract and Applied Analysis, Vol. 2013, 2013, Article ID 735217, 14 pages [4] S. Chandok and D. Kumar, “Some common fixed point results for rational type contraction mappings in complex valued metric spaces”,

Journal of operators, Vol. 2013, 2013, Article ID 813707,6 pages.

[5] D. Singh,O.P. Chauhan, N. Singh and V. Joshi, “Complex valued rectangular metric spaces and common fixed point theorems, Bulletin of Mathematical Analysis and Applications, Vol.7, 2, 2015, pp. 1-13

[6] S.R. Patil and J.N. Salunke, “Common fixed point theorems in complex valued rectangular metric spaces, South Asian Journal of

Mathematics, Vol. 6, 1, 2016, pp.10-23. [7] S. Bhatt, S. Chaukiyal and R. C. Dimri, “Common fixed point of mappings satisfying rational inequality in complex valued metric

space”, International Journal of Pure and applied Mathematics, Vol.73, 2011, pp. 159-164 [8] M. Aamri and D. El. Moutawakil, “Some new common fixed point theorems under strict contractive conditions”, Journal of

Mathematical Analysis and Applications, Vol. 270, 2002, pp. 181-188.

[9] W. Sintunavarat and P. Kumam, “Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces”, Journal of Applied Mathematics,Vol. 2011, 2011, Article ID 637958, 14 pages

[10] R.K. Verma and H.K. Pathak, “Common fixed point theorems using property (E.A) complex valued metric spaces”, Thai Journal of

Mathematics, Vol. 11, 2013, pp.347-355 [11] Y. Liu, J. Wu and Z. Li, “Common fixed points of single-valued and multivalued maps”, Int. J.Math. Math.Sci., Vol. 19, 2005, pp.3045-

3055.

[12] M. Imdad, B.D. Pant and S. Chauhan, “Fixed point theorems in Menger spaces using the (CLRST) property and applications”, Journal of Nonlinear Analysis and Optimization Theory and Applications, Vol. 3, 2012, pp. 225-237.

AUTHORS PROFILE’

Garima Gadkari having more than 10 years of teaching experience as an Assistant professor. Currently perusing Ph.D.

degree from Department of Mathematics, Barkatullah University, Bhopal, (M.P.), India.

M.S. Rathore is working as Professor in Department of Mathematics at Chandra Shekhar Azad Government P.G. College

Sehore, (M.P.), India. He completed his Ph.D. in 1983 from Vikram University, Ujjain, India. He has published more than

60 research papers in the field of fixed point theory.

Naval Singh is working as Assistant Professor in Department of Mathematics in Government Science and Commerce P.G.

College, Benazeer, Bhopal, (M.P.), India. He completed his Ph.D. in 2004 from Barkatullah University, Bhopal, India. He

has published many research papers in the area of fixed point theory.

some common fixed point theorems for weakly compatible ... · introduced by azam et al. [1]....

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