Common fixed point theorems in complex valued generalized metric spaces

J. Math. Comput. Sci. 7 (2017), No. 4, 739-754 ISSN: 1927-5307

COMMON FIXED POINT THEOREMS IN COMPLEX VALUED GENERALIZED METRIC SPACES

ZHONGNA SUN∗, MEIMEI SONG

Department of Mathematics, Tianjin University of Technology, Tianjin 300384, P.R. China

Copyright c2017 Sun and Song. This is an open access article distributed under the Creative Commons Attribution License, which permits

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract. In this paper, using(CLR)property and the Common(E.A)property common fixed point results for weakly compatible mappings, satisfying integral type contractive condition in complex valued generalized metric spaces are investigated.

Keywords:complex valued generalized metric spaces; weakly compatible mapping;(E.A)property;(CLR) prop-erty.

2010 AMS Subject Classification:47H10, 54H25.

1. Introduction

In 2011, Azam et al. [1] introduced the notion of complex valued metric space which is a generalization of the classical metric space and proved a unique common fixed point theorems for two self-mappings satisfying a rational type inequality. Though complex valued metric spaces form a special class of cone metric space, yet this idea is intended to define rational expressions which are not meaningful in cone metric spaces and thus many results of analysis

∗_{Corresponding author}

E-mail address: [email protected] Received November 7, 2016

cannot be generalized to cone metric spaces. However, in complex valued metric spaces, we can study improvements of a host of results of analysis involving division. Bhatt et al. [7] initiated the concept of weakly compatible maps to study common fixed point theorem for weakly compatible maps in complex valued metric spaces.

In 2013, Abbas et al. [2] introduced the notion of complex valued generalized metric space. They replaced the triangular inequality in the complex valued metric by the rectangular in-equality involving four points and extended the concept of complex valued metric spaces. They proved fixed point theorems involving the rational type contractive conditions for weakly con-tractive mappings in these spaces. The fixed point theorems in complex valued generalized metric space were obtained by many mathematicians (e.g. [2,3,8]).

In 2002, Branciari [9] proved fixed point theorems for two self-maps under contractive con-dition of integral type in metric space. In 2013, Manro et al. [10] extended and generalized the theorem of Branciari [9] for a pair of weakly compatible mappings satisfying a general contractive condition of integral type in complex valued metric space.

The aim of this paper is to prove common fixed point theorems for integral type contractive condition in complex valued generalized metric spaces.

2. Preliminaries

The following definitions and lemmas will be used in the sequel. Let_Cbe the set of complex numbers andz₁,z₂∈_C. We define a partial order_-on_Cas follows:z_1-z₂, iffRe(z₁)≤Re(z₂)

andIm(z₁)≤Im(z₂).

As a result, we can say thatz_1-z₂if one of the following conditions is satisfied: (I)Re(z₁) =Re(z₂),Im(z₁)<Im(z₂);

(II)Re(z₁)<Re(z₂),Im(z₁) =Im(z₂); (III)Re(z1)<Re(z2),Im(z1)<Im(z2); (IV)Re(z₁) =Re(z₂),Im(z₁) =Im(z₂).

In particular, we can writez1z2 ifz16=z2and one of (I)-(III) is satisfied. Also we will write

(1)z_1-z₂and 0≤r∈R=⇒rz_1-z₂; (2) 0_-z₁z₂=⇒ |z₁|<|z₂|;

(3)z1-z2andz2≺z3=⇒z1≺z3.

Definition 2.1.[1]LetXbe a nonempty set. If the mappingd:X×X→_Csatisfies the following axioms:

(i) 0_-d(x,y), for allx,y∈X andd(x,y) =0 if and only ifx=y; (ii)d(x,y) =d(y,x), for allx,y∈X;

(iii)d(x,y)_-d(x,z) +d(z,y). for allx,y,z∈X.

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

Example 2.1. LetX =_C, define the mappingd:X×X →_Cby

d(x,y) =2i|x−y|,

wherex,y∈X. Then(X,d)is a complex valued metric space.

Definition 2.2.[2]LetXbe a nonempty set. If the mappingd:X×X→_Csatisfies the following axioms:

(i) 0_-d(x,y), for allx,y∈X andd(x,y) =0 if and only ifx=y; (ii)d(x,y) =d(y,x), for allx,y∈X;

(iii) d(x,y)_-d(x,u) +d(u,v) +d(v,y), for all x,y∈X and all distinct u,v∈X, each one is different fromxandy.

Thendis called a complex valued generalized metric onX and(X,d)is called a complex valued generalized metric space.

Example 2.2. Let X ={−2,−1,1,2} and define the complex valued generalized metric on

d:X×X →_Cas

d(-2,2)= d(2,-2)= d(-1,2)= d(2,-1)= d(1,2)= d(2,1)=0.4i d(-2,-1)= d(-1,-2)= d(-1,1)= d(1,-1)=0.2i

d(-2,1)= d(1,-2)=0.5i

We can prove that (X,d) is a complex valued generalized metric space but it is not a com-plex valued metric space since it does not satisfy triangular inequality, d(−2,1) = 0.5i>

d(−2,−1) +d(−1,1) =0.2i+0.2i=0.4i.

LetX be a complex valued generalized metric space andA⊆X. A pointx∈X is called an interior point of a setAwhenever there exists 0≺r∈_Csuch thatB(x,r) ={y∈X :d(x,y)≺

r} ⊆A. A subsetAinX is called open whenever each point ofAis an interior point ofA. The familyF ={B(x,r):x∈X,0≺r}is a sub-basis for a Hausdorff topologyτ onX.

A pointx∈X is called a limit point ofAwhenever for every 0≺r∈_C,B(x,r)T

(A\x)6= /0. A subsetB⊆X is called closed whenever each limit point ofBbelongs toB.

Definition 2.3. [2]Let{x_n}be a sequence in complex valued generalized metric space(X,d)

andx∈X. Then

(1)xis called the limit of{x_n}if for everyω∈_Cwith 0≺ω there isn0∈Nsuch thatd(xn,x)≺

ω for alln>n0and we write limn→∞xn=x.

(2){x_n}is called a Cauchy sequence if for everyω ∈_C with 0≺ω there isn0∈N such that

d(x_n,x_m)≺ω for alln,m>n0.

(3) (X,d) is said to be a complete complex valued generalized metric space if every Cauchy sequence is convergent in(X,d).

Lemma 2.1. [2]Let (X,d) be a complex valued generalized metric space. Then a sequence

{x_n}inX converges toxif and only if|d(x_n,x)| →0 asn→∞.

Lemma 2.2. [2]Let (X,d) be a complex valued generalized metric space. Then a sequence

{x_n}inX is a Cauchy sequence if and only if|d(x_n,x_m)| →0 asn,m→∞.

Lemma 2.3. [11]d(xn,y)→d(x,y)and d(x,xn)→d(x,a)whenever {xn} is a sequence in X

withx_n→x.

Definition 2.4. [3]Let f andgbe two self-mapping on a nonempty setX. Then (I)x∈X is called to be fixed point of f if f x=x.

(II)x∈X is called to be a coincidence point of f andgif f x=gx.

Definition 2.5. [3] Let (X,d) be a complex valued generalized metric space. Then two self-mapping f andgare said to be weakly compatible if they commute at their coincidence points,i.e.,

x∈X with f x=gximplies that f gx=g f x.

Definition 2.6. [4]Let (X,d) be a complex valued generalized metric space. Then two self-mapping f andgare said to satisfy the common limit in the range ofgproperty(CLRg)if there

exists a sequence{x_n}inX,x_n6=x_mifn6=msuch that

limn→∞f xn=limn→∞gxn=gx

for somex∈X.

Definition 2.7. [5]Let(X,d)be a complex valued generalized metric space. Then two pairs of self maps (S,f) and (T,g) are said to satisfy the Common (E,A) property if there exists two sequence{x_n}and{y_n}inX x_n6=x_mandy_n6=y_mifn6=msuch that

limn→∞Sxn=limn→∞f xn=limn→∞Tyn=limn→∞gyn=z

for somez∈X.

3. Main results

In this section, we will prove the existence of fixed points for mappings f andgdefined on a complex valued generalized metric space(X,d)satisfying a contractive condition of integral type.

LetΦ={φ :φ:[0,∞)→[0,∞)is a Lebesgue-integrable mapping which is summable on each

compact subset of[0,∞), non-negative, non-decreasing and such that for eachε0,R₀εφ(t)dt

0}.

For anyz₁,z₂∈_C+, define

[z₁,z₂] ={r(s)∈_C:r(s) =z₁+s(z₂−z₁) f or some s∈[0,1]}.

(z₁,z₂] ={r(s)∈_C:r(s) =z₁+s(z₂−z₁) f or some s∈(0,1]}. Defineζ :[z1,z2]→Cas follow:

ζ(x+iy) =φ1(x) +iφ2(y), wherex+iy∈[z₁,z₂]andφ1,φ2∈Φ.

Lemma 3.1.[6]Supposeζ∈Γ1([z1,z2],C)and{zn}be a sequence inC+, then limn→∞R₀znζ(s)ds=

(0,0)if and only ifz_n→(0,0), asn→∞.

Definition 3.1. [6] A complex valued function ϕ :Rn →_C is measurable if both its real and

imaginary parts are measurable.

In 2016, Sarwar Muhammad defined the lebesgue integral of f to be R

E f =

R

ERe(f) +i

R

EIm(f) = (

R

ERe(f),

R

EIm(f)),

provided thatRe(f)andIm(f)are Lebesgue integrables and defined all complex valued lebesgue integrable functions byΓ1(E,C), whereE⊂Rnbe a measurable set.

We define Ψ ={ϕ :Rn →_C as a complex valued Lebesgue-integrable mapping(i.e.,ϕ ∈

Γ1(E,C)), which is summable and non-vanishing on each measurable subset of Rn, such that for eachε0,R₀εϕ(t)dt 0}

Theorem 3.1. Let f and g be two self-mapping of a complex valued generalized metric space

(X,d)which satisfy the following conditions:

(I) the pairs(f,g)is weakly compatible,

(II) the pairs(f,g)satisfy(CLR_g)property,

(III)∀x,y∈X , there exists

u(x,y)∈M(x,y)

such that

Z d(f x,f y)

0

ϕ(t)dt ≺α

Z u(x,y)

0

ϕ(t)dt (3.1)

whereα nonnegative real such thatα<1andϕ∈Ψ,

M(x,y) ={d(gx,gy),d(gy,f y),d(gx,f y),d(gy,f x)}, then f and g have a unique common fixed

point in X .

Proof.Since the mappings f andgsatisfy the(CLR_g)property, then there exists sequence{x_n}

such that

lim

n→∞f xn=nlim→∞gxn=gx,

for somex∈X. We now show that f x=gx. Suppose not, i.e., f x6=gx. From (3.1)

Z d(f x_n,f x)

0

ϕ(t)dt ≺α

Z u(x_n,x)

0

where

u(x_n,x)∈M(x_n,x)

M(x_n,x) ={d(gx_n,gx),d(gx_n,f x_n),d(gx,f x),d(gx_n,f x),d(gx,f x_n)}

Two cases arises: 1) If

u(xn,x) =d(gxn,gx),d(gxn,f xn)or d(gx,f xn)

Taking limit asn→∞, then (3.2) and lem (3.8) implies

Z d(gx,f x)

0

ϕ(t)dt ≺α

Z d(gx,gx)

0

ϕ(t)dt =0

Which is a contradiction. 2) If

u(xn,x) =d(gx,f x)or d(gxn,f x)

Taking limit asn→∞, then (3.2) and lem (3.8) implies

Z d(gx,f x)

0

ϕ(t)dt ≺α

Z d(gx,f x)

0

ϕ(t)dt

Sinceα <1, which is a contradiction.

Hence, from all cases, f x=gx. Now let z= f x=gx. Since (f,g) is weakly compatible,

f gx=g f x, i.e. f z=gz.

Next we show that f z=z. Suppose it is not, then Z d(f z,z)

0

ϕ(t)dt=

Z d(f z,f x)

0

ϕ(t)dt ≺α

Z u(z,x)

0

ϕ(t)dt (3.3)

where

u(z,x)∈M(z,x)

M(z,x) ={d(gz,gx),d(gz,f z),d(gx,f x),d(gz,f x),d(gx,f z)}

={d(f z,z),0,0,d(f z,z),d(z,f z)}

Two cases arises:

1) Ifu(z,x) =d(f z,z), then by (3.3), we have Z d(f z,z)

0

ϕ(t)dt ≺α

Z d(f z,z)

0

ϕ(t)dt

Sinceα <1, which is a contradiction.

2) Ifu(z,x) =0, then by (3.3), we have Z d(f z,z)

0

ϕ(t)dt ≺0

which is a contradiction. Thus, f z=gz=z.

Hence,zis a common fixed point of f andg.

To show that the fixed point is unique, supposez0 is another fixed point of f andg, i.e. f z0= gz0=z0.

From (3.3), we have

Z d(z,z0)

0

ϕ(t)dt=

Z d(f z,f z0)

0

ϕ(t)dt ≺α

Z u(z,z0)

0

ϕ(t)dt

where

u(z,z0)∈M(z,z0)

M(z,z0) ={d(gz,gz0),d(gz,f z),d(gz0,f z0),d(gz,f z0),d(gz0,f z)}

={d(z,z0),0,0,d(z,z0),d(z0,z)}

={d(z,z0),0}

Two possible cases arises:

1) Ifu(z,z0) =d(z,z0), then by (3.3), we have Z d(z,z0)

0

ϕ(t)dt ≺α

Z d(z,z0)

0

ϕ(t)dt

Sinceα <1, which is a contradiction.

2) Ifu(z,z0) =0, then by (3.46), we have Z d(z,z0)

0

which is a contradiction.

Hence,z=z0i.e., f andghave a unique common fixed point inX. This completes the proof. Theorem 3.2.Let S, T , f and g be four self-mapping of a complete complex valued generalized

metric space(X,d)which satisfy the following conditions:

(I) S(X)⊆g(X)and T(X)⊆ f(X),

(II) the pairs(S,f)and(T,g)are weakly compatible,

(III) the subspace f(X)or g(X)is closed,

(IV)∀x,y∈X ,

Z d(Sx,Ty)

0

ϕ(t)dt-(a+c)

Z d(f x,gy)

0

ϕ(t)dt+b

Z d(f x,Sx)

0

ϕ(t)dt+ (b+c)

Z d(gy,Ty)

0

ϕ(t)dt (3.4)

where a, b and c are nonnegative reals such that a+2b+2c<1andϕ ∈Ψ, then S, T , f and g

have a unique common fixed point in X.

Proof.We construct a sequence{y_n}inX such that,

y_2n=Sx_2n=gx_2n+1

and

y_2n+1=T x_2n+1= f x_2n+2,n≥0 (3.5)

where{x_2n}is another sequence inX. Using (3.4), we have, Z d(y_2n,y_2n+1)

0

ϕ(t)dt=

Z d(Sx_2n,T x_2n+1)

0

ϕ(t)dt -(a+c)

Z d(f x_2n,gx_2n+1)

0

ϕ(t)dt

+b

Z d(f x_2n,Sx₂n)

0

ϕ(t)dt+ (b+c)

Z d(gx_2n+1,T x_2n+1)

0

ϕ(t)dt

= (a+c)

Z d(y_2n−1,y_2n)

0

ϕ(t)dt+b

Z d(y_2n−1,y_2n)

0

ϕ(t)dt

+ (b+c)

Z d(y_2n,y_2n+1)

0

ϕ(t)dt

(3.6)

Hence,

Z d(y_2n,y_2n+1)

0

ϕ(t)dt -a+b+c

1−b−c

Z d(y_2n−1,y_2n)

0

ϕ(t)dt

Therefore,

Z d(y_2n,y_2n+1)

0

ϕ(t)dt -δ

Z d(y_2n−1,y_2n)

0

whereδ = a₁₋+b_b−+_cc <1, sincea+2b+2c<1. Proceeding in a similar way we have,

Z d(y_2n,y_2n+1)

0

ϕ(t)dt -δ

Z d(y_2n−1,y_2n)

0

ϕ(t)dt

-δ2

Z d(y_2n−2,y_2n−1)

0

ϕ(t)dt

-· · · ≤δ2n

Z d(y₀,y₁)

0

ϕ(t)dt

(3.7)

Finally we conclude that

Z d(y_n,y_n+1)

0

ϕ(t)dt -δn

Z d(y₀,y₁)

0

ϕ(t)dt

Taking limit asn→∞, we get

lim

n→∞

Z d(y_n,y_n+1)

0

ϕ(t)dt=0 (3.8)

Hence from lem(3.3), we getd(y_n,y_n+1)→0, asn→∞.

We now show that {yn} is Cauchy. Suppose that it is not. Then there exists an ε >0 and

subsequences{2m}and{2n}such that 2n>2m>n₀with

d(y_2m,y_2n)_%ε,d(y2m,y2n−1)≺ε (3.9)

Now from rectangular inequality,

d(y_2m−1,y_2n−1)_-d(y_2m−1,y_2m) +d(y_2m,y_2n−2) +d(y_2n−2,y_2n−1) (3.10)

Therefore, from (3.7), (3.9) and (3.10), we have, Z ε

0

ϕ(t)dt

-Z d(y_2m,y₂n)

0

ϕ(t)dt -δ

Z d(y_2m−1,y_2n−1)

0

ϕ(t)dt

-δ

Z d(y_2m−1,y_2m)

0

ϕ(t)dt+δ

Z d(y_2m,y_2n−2)

0

ϕ(t)dt

+δ

Z d(y_2n−2,y_2n−1)

0

ϕ(t)dt≺δ

Z ε

0

ϕ(t)dt

+δ

Z d(y_2m−1,y_2m)

0

ϕ(t)dt+δ

Z d(y_2n−2,y_2n−1)

0

ϕ(t)dt

Taking the limit asn,m→∞, from (3.8) we have,

Z ε

0

ϕ(t)dt ≺δ

Z ε

0

which is a contraction. Therefore {y_n} is Cauchy sequence inX. SinceX is complete, there exists pointzinX such that,

lim

n→∞Sx2n=nlim→∞gx2n+1=nlim→∞T x2n+1=nlim→∞f x2n+2=z

Assuming f(X) is closed, z∈ f(X)and z= f ufor some u∈X. We claim thatSu= f u=z. Using the rectangular inequality[Definition 2.2(iii)] we get,

d(Su,z)_-d(Su,T x_2n+1) +d(T x_2n+1,gx_2n+1) +d(gx_2n+1,z) (3.11)

Therefore from (3.11), Z d(Su,z)

0

ϕ(t)dt

-Z d(Su,T x_2n+1)

0

ϕ(t)dt+

Z d(T x_2n+1,gx_2n+1)

0

ϕ(t)dt+

Z d(gx_2n+1,z)

0

ϕ(t)dt

-(a+c)

Z d(f u,gx_2n+1)

0

ϕ(t)dt+ (b+c)

Z d(gx_2n+1,T x_2n+1)

0

ϕ(t)dt

+b

Z d(f u,Su)

0

ϕ(t)dt+

Z d(T x_2n+1,gx_2n+1)

0

ϕ(t)dt+

Z d(gx_2n+1,z)

0

ϕ(t)dt

Asn→∞, we get,

Z d(Su,z)

0

ϕ(t)dt -(a+c)

Z d(z,z)

0

ϕ(t)dt+ (b+c)

Z d(z,z)

0

ϕ(t)dt

+b

Z d(z,Su)

0

ϕ(t)dt+

Z d(z,z)

0

ϕ(t)dt+

Z d(z,z)

0

ϕ(t)dt

=b

Z d(z,Su)

0

ϕ(t)dt

Asb<1, Rd(z,Su)

0 ϕ(t)dt =0 implies thatSu=zi.e. f u=Su=zanduis a coincidence point of f andS. SinceS(X)⊆g(X), Su=gvfor somev∈X. Hence f u=Su=gv=z. We claim thatT v=z. By the inequality (3.4),

Z d(z,T v)

0

ϕ(t)dt =

Z d(Su,T v)

0

ϕ(t)dt-(a+c)

Z d(f u,gv)

0

ϕ(t)dt+b

Z d(f u,Su)

0

ϕ(t)dt

+ (b+c)

Z d(gv,T v)

0

ϕ(t)dt= (a+b+c)

Z d(z,z)

0

ϕ(t)dt

+ (b+c)

Z d(z,T v)

0

ϕ(t)dt = (b+c)

Z d(z,T v)

0

ϕ(t)dt

Sinceb+c<1,Rd(z,T v)

Sz6=z, then by (3.4), Z d(Sz,z)

0

ϕ(t)dt =

Z d(Sz,T v)

0

ϕ(t)dt -(a+c)

Z d(f z,gv)

0

ϕ(t)dt+b

Z d(f z,Sz)

0

ϕ(t)dt

+ (b+c)

Z d(gv,T v)

0

ϕ(t)dt= (a+2c)

Z d(Sz,z)

0

ϕ(t)dt+b

Z d(f z,Sz)

0

ϕ(t)dt

+ (b+c)

Z d(z,z)

0

ϕ(t)dt = (a+2c)

Z d(Sz,z)

0

ϕ(t)dt

Sincea+2c<1,Rd(Sz,z)

0 ϕ(t)dt implies thatSz=zi.e.Sz= f z=z. Thuszis a fixed point ofS and f. Also sinceT andgare weakly compatible,T gv=gT vi.e. T z=gz. We now prove that

T z=z, suppose not,T z6=z, then by (3.4) again, Z d(z,T z)

0

ϕ(t)dt =

Z d(Sz,T z)

0

ϕ(t)dt -(a+c)

Z d(f z,gz)

0

ϕ(t)dt+b

Z d(f z,Sz)

0

ϕ(t)dt

+ (b+c)

Z d(gz,T z)

0

ϕ(t)dt = (a+c)

Z d(T z,z)

0

ϕ(t)dt+b

Z d(z,z)

0

ϕ(t)dt

+b

Z d(gz,T z)

0

ϕ(t)dt = (a+c)

Z d(T z,z)

0

ϕ(t)dt

Sincea+c<1,Rd(T z,z)

0 ϕ(t)dt implies thatT z=zandSz= f z=T z=gz=z,zis a common fixed point ofS,T, f andg. To show that the fixed point is unique, suppose that there is another pointw∈X such thatSw=Tw= f w=gw=w.

From (3.4), we have, Z d(w,z)

0

ϕ(t)dt=

Z d(Sw,T z)

0

ϕ(t)dt -(a+c)

Z d(f w,gz)

0

ϕ(t)dt+b

Z d(f w,Sw)

0

ϕ(t)dt

+ (b+c)

Z d(gz,T z)

0

ϕ(t)dt= (a+c)

Z d(w,z)

0

ϕ(t)dt+b

Z d(w,w)

0

ϕ(t)dt

+ (b+c)

Z d(z,z)

0

ϕ(t)dt = (a+c)

Z d(w,z)

0

ϕ(t)dt

ThereforeRd(w,z)

0 ϕ(t)dt =0 as a+c<1 and w=zwhich proves the uniqueness of the fixed point. Similar argument holds if g(X) is assumed to be closed. HenceS, T, f and g have a unique common fixed point inX. This completes the proof.

By puttingc=0, in Theorem 3.2 above, we get the following corollary.

Corollary 3.3. Let S, T , f and g be four self-mapping of a complete complex valued generalized

metric space(X,d)which satisfy the following conditions:

(II) the pairs(S,f)and(T,g)are weakly compatible,

(III) the subspace f(X)or g(X)is closed,

(IV)∀x,y∈X ,

Z d(Sx,Ty)

0

ϕ(t)dt-a

Z d(f x,gy)

0

ϕ(t)dt+b(

Z d(f x,Sx)

0

ϕ(t)dt+

Z d(gy,Ty)

0

ϕ(t)dt)

where a and b are nonnegative reals such that a+2b<1andϕ∈Ψ, then S, T , f and g have a

unique common fixed point in X .

If we putT =Sandg= f,b=c=0 in Theorem 3.2 above, we get the following corollary. Corollary 3.4. Let S and f be two mappings of a complete complex valued generalized metric

space(X,d)which satisfy the following conditions:

(I) S(X)⊆ f(X),

(II) the pair(S,f)is weakly compatible,

(III) the subspace f(X)is closed,

(IV)∀x,y∈X ,

Z d(Sx,Ty)

0

ϕ(t)dt -a

Z d(f x,gy)

0

ϕ(t)dt

where a is nonnegative real such that a<1and ϕ∈Ψ, then S and f have a unique common

fixed point in X .

Theorem 3.5. Let S,T , f and g be four self-mapping of a complex valued generalized metric

space(X,d)which satisfy the following condition:

(I) the pairs(S,f)and(T,g)are weakly compatible;

(II) the pairs(S,f)and(T,g)satisfy the Common property(E,A);

(III) the subspaces f(X)and g(X)are closed;

(IV)∀x,y∈X ,

Z d(Sx,Ty)

0

ϕ(t)dt_-a

Z d(f x,gy)

0

ϕ(t)dt+b(

Z d(f x,Sx)

0

ϕ(t)dt+

Z d(gy,Ty)

0

ϕ(t)dt)

+c(

Z d(f x,Ty)

0

ϕ(t)dt+

Z d(gy,Sx)

0

ϕ(t)dt)

(3.12)

where a, b and c are nonnegative reals such that a+2b+2c<1andϕ ∈Ψ, then S, T , f and g

Proof. Since the pairs (S,f) and(T,g) satisfy the common property (E,A), there exists two sequences{x_n}and{y_n}inX such that

lim

n→∞Sxn=nlim→∞f xn=nlim→∞Tyn=nlim→∞gyn=z

for some z∈X. Since f(X) is closed andz∈ f(X), there exists u∈X such that z= f u. We claim thatSu= f u=z. From (3.12) we have,

Z d(Su,Ty_n)

0

ϕ(t)dt-a

Z d(f u,gy_n)

0

ϕ(t)dt+b(

Z d(f u,Su)

0

ϕ(t)dt+

Z d(gy_n,Ty_n)

0

ϕ(t)dt)

+c(

Z d(f u,Tyn)

0

ϕ(t)dt+

Z d(gy_n,Su)

0

ϕ(t)dt)

Asn→∞,

Z d(Su,z)

0

ϕ(t)dt-b

Z d(f u,Su)

0

ϕ(t)dt+c

Z d(z,Su)

0

ϕ(t)dt

Asb+c<1,d(Su,z) =0 andSu=zi.e. Su= f u=z. SinceSand f are weakly compatible,

S f u= f SuandSz= f zi.e. zis a coincidence point ofSand f. Also sinceg(X)is closed and

z∈g(X),z=gvfor somev∈X. HenceSu= f u=gv=z. We prove thatT v=gv=z. Consider from (3.12),

Z d(z,T v)

0

ϕ(t)dt =

Z d(Su,T v)

0

ϕ(t)dt-a

Z d(f u,gv)

0

ϕ(t)dt

+b(

Z d(f u,Su)

0

ϕ(t)dt+

Z d(gv,T v)

0

ϕ(t)dt)

+c(

Z d(f u,T v)

0

ϕ(t)dt+

Z d(gv,Su)

0

ϕ(t)dt

= (b+c)

Z d(z,T v)

0

ϕ(t)dt

Henced(z,T v) =0 asb+c<1. ThereforeT v=gv=z. SinceT andgare weakly compatible, we haveT gv=gT v i.e. T z=gzandz is a coincidence point ofT andg. To show that zis a fixed point, we claim thatSz=z. By (3.12),

Z d(Sz,z)

0

ϕ(t)dt=

Z d(Sz,T v)

0

ϕ(t)dt -a

Z d(f z,gv)

0

ϕ(t)dt

+b(

Z d(f z,Sz)

0

ϕ(t)dt+

Z d(gv,T v)

0

ϕ(t)dt)

+c(

Z d(f z,T v)

0

ϕ(t)dt+

Z d(gv,Sz)

0

ϕ(t)dt

= (a+2c)

Z d(z,Sz)

0

Since a+2c<1, d(Sz,z) =0 and Sz=z. Hence Sz= f z=z. Similarly we can show that

T z=gz=z. Thus we haveSz= f z=T z=gz=zandzis a common fixed point ofS,T, f and

ginX.

Uniqueness. letz0be another fixed point ofS,T, f andg, i.e. Sz0=T z0= f z0=gz0=z0. From (3.12), we have

Z d(z,z0)

0

ϕ(t)dt=

Z d(Sz,T z0)

0

ϕ(t)dt -a

Z d(f z,gz0)

0

ϕ(t)dt

+b(

Z d(f z,Sz)

0

ϕ(t)dt+

Z d(gz0,T z0)

0

ϕ(t)dt)

+c(

Z d(f z,T z0)

0

ϕ(t)dt+

Z d(gz0,Sz)

0

ϕ(t)dt

= (a+2c)

Z d(z,z0)

0

ϕ(t)dt

thus

(1−a−2c)

Z d(z,z0)

0

ϕ(t)dt -0

Since 1−a−2c>0, which is a contradiction, unlessz=z0. Hencezis a unique common fixed point ofS,T, f andginX. This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests.

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