Vol 2015

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ISSN 2307-7743 http://scienceasia.asia

SOME COMMON FIXED POINT THEOREMS FOR CONTRACTIVE MAPPINGS IN COMPLEX-VALUED

B-METRIC SPACES

A.K. DUBEY, RITA SHUKLA, RAVI PRAKASH DUBEY

Abstract. In this paper, we obtain some common fixed point theorems for two single-valued mappings satisfying rational ex-pressions in complex valued b-metric spaces. Our results improve several well-known conventional results. Also, an example is given to illustrate our obtained result.

1. Introduction

The fixed point theorem, generally known as the Banach contraction mapping principle, appeared in explicit form in Banach’s thesis in 1922 [9]. It has applications in different branches of Mathematical analysis and provides the solution of many problems in mathematical analy-sis. Later, a lot of articles have been dedicated to the improvement and generalization of the Banach’s contraction mapping principle in different spaces.

In 1989, Bakhtin [6] introduced the concept of b-metric space as a generalization of metric spaces. Since then many papers have been devoted in this direction [4,7]. A new space called the complex-valued metric space which is more general form than metric spaces has been introduced by Azam et. al [1] and established the existence of fixed point theorems for maps satisfying the contraction condition. In 2012, Rouzkard and Imdad [3] proved some common fixed point theorems satisfying certain rational expressions in complex valued metric spaces which is extended and improved form of the results of Azam et al. [1]. Sintunavarat and Kumam [12] established common fixed point results by replacing constant of contractive condition to control functions.

The concept of complex valued b-metric spaces was introduced in 2013 by Rao et al. [8], which was more general than the well known complex valued metric spaces[1]. In sequel, AA.Mukheimer [2], proved some

2010Mathematics Subject Classification. 47H09, 47H10.

Key words and phrases. fixed point, contractive mappings, b-metric spaces.

c

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common fixed point theorems of two self mappings satisfying a rational inequality on complex valued b-metric spaces.

In this paper, some common fixed point theorems for a pair of maps un-der contraction involving rational expressions in the setting of complex valued b-metric spaces are proved. An examples is given to support the usability of our results. The obtained results are generalization-s of recent regeneralization-sultgeneralization-s proved by Azam et al. [1], AA. Mukheimer [2], H.K.Nashine [5], Bhatt et al.[10], Datta et al. [11], Chakkrid Klin-eam and Cholatis Suanoom [13] and Dubey et al.[14].

2. Preliminaries

Let_Cbe the set of complex numbers andz1, z2 ∈C.Define a partial

or-der_-on_Cas follows: z1 -z2 if and only ifRe(z1)≤Re(z2), Im(z1)≤

Im(z2).

Consequently, one can infer that z1 -z2 if one of the following

condi-tions 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 write z1 z2 if z1 6=z2 and one of (i),(ii) and (iii) is

satisfied and we writez1 ≺z2 if only (iii) is satisfied. Notice that

(a): If 0 _-z1 z2, then |z1|<|z2|,

(b): If z1 -z2 and z2 ≺z3 then z1 ≺z3,

(c): If a, b∈_R and a≤b then az _-bz for all z ∈_C.

The: following definition is recently introduced by Rao et al. [8].

Definition 2.1. LetX be a nonempty set and lets≥1 be a given real number. A functiond:X×X →_Cis called a complex valued b-metric onX if for all x, y, z∈X the following conditions are satisfied:

(i): 0_-d(x, y) and d(x, y) = 0 if and only if x=y; (ii): d(x, y) = d(y, x);

(iii): d(x, y)_-s[d(x, z) +d(z, y)].

The: pair (X,d) is called a complex valued b-metric space.

Example 2.2[8]. Let X = [0,1].Define the mapping d:X×X →_C byd(x, y) = |x−y|2₊_i_|_x₋_y_|2_, _{for all} _{x, y} _∈_X.

Then (X, d) is a complex valued b-metric space with s= 2.

Definition 2.3[8]. Let (X, d) be a complex valued b-metric space.

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(ii) A pointx∈X is called a limit point of a set Awhenever for every 0≺r∈_C, B(x, r)∩(A− {x})6=φ.

(iii) A subset A ⊆X is called open whenever each element of A is an interior point of A.

(iv) A subset A ⊆ X is called closed whenever each element of A belongs toA.

(v) A sub-basis for a Hausdorff topology τ on X is a family F = {B(x, r) :x∈X and0≺r}.

Definition 2.4[8]. Let (X, d) be a complex valued b-metric space, {xn}be a sequence in X and x∈X.

(i) If for every c ∈ _C, with 0 ≺ r there is N ∈ _N such that for all n > N, d(xn, x)≺c, then{xn}is said to be convergent, {xn}converges

toxand xis the limit point of{xn}. We denote this bylimn→∞xn=x

or{xn} →x as n→ ∞.

(ii) If for every c ∈ _C, with 0 ≺ r there is N ∈ _N such that for all n > N, d(xn, xn+m)≺ c, where m ∈N, then {xn}is said to be Cauchy

sequence.

(iii) If every Cauchy sequence in X is convergent, then (X, d) is said to be a complete complex valued b-metric space.

Lemma 2.5[8]. Let (X, d) be a complex valued b-metric space and let {xn}be a sequence in X. Then {xn}converges to x if and only if

|d(xn, x)| →0 as n → ∞.

Lemma 2.6[8]. Let (X, d) be a complex valued b-metric space and let {xn}be a sequence in X. Then {xn} is a Cauchy sequence if and

only if |d(xn, xn+m)| →0 as n → ∞,where m∈N. 3. Main Results

In this section, we will prove some common fixed point theorems for the contractive mappings in complex valued b-metric space.

Theorem 3.1. Let (X, d) be a complete complex valued b-metric space with the coefficient s ≥ 1 and let S, T : X → X are mappings satisfying:

d(Sx, T y)_-Ad(x, y) + Bd(x,Sx₁₊_d₍)_x,yd(y,T y₎ ) +Cd(y,Sx₁₊_d₍)_x,yd(x,T y₎ )

+Dd(x,Sx₁₊_d₍)_x,yd(x,T y₎ )+ Ed(y,Sx₁₊_d₍)_x,yd(y,T y₎ )− − −(3.1)

for all x, y ∈ X, where A, B, C, D and E are nonnegative reals with A+B +C+ 2sD+ 2sE < 1. Then S and T have a unique common fixed in X.

Proof. For any arbitrary point x0 ∈ X. Define sequence {xn} in X

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x2n+1 =Sx2n, x2n+2 =T x2n+1, f or n= 0,1,2,3, ...− − −(3.2)

Now, we show that the sequence {xn} is Cauchy. Let x = x2n and

y=x2n+1 in (3.1), we haved(x2n+1, x2n+2) = d(Sx2n, T x2n+1)

-Ad(x2n, x2n+1) + Bd(x2n,Sx2n₁₊_d₍_x2n,x2n+1)d(x2n+1,T x2n+1₎ )

+Cd(x2n+1,Sx2n₁₊_d₍_x2n,x2n+1)d(x2n,T x2n+1₎ )

+Dd(x2n,Sx2n₁₊_d₍_x2n,x2n+1)d(x2n,T x2n+1₎ )

+Ed(x2n+1,Sx2n₁₊_d₍_x2n,x2n+1)d(x2n+1,T x2n+1₎ )

-Ad(x2n, x2n+1) + Bd(x2n,x2n+1₁₊_d₍_x2n,x2n+1)d(x2n+1,x2n+2₎ )

+Dd(x2n,x2n+1₁₊_d₍_x2n,x2n+1)d(x2n,x2n+2₎ )

which implies that

|d(x2n+1, x2n+2)| ≤A|d(x2n, x2n+1)|+ B|d(x2n,x2n+1_|₁₊_d₍_x2n,x2n+1)||d(x2n+1,x2n+2₎_| )|

+D|d(x2n,x2n+1_|₁₊_d₍_x2n,x2n+1)||d(x2n,x2n+2₎_| )|.

Since |1 +d(x2n, x2n+1)|>|d(x2n, x2n+1)|,so we get

|d(x2n+1, x2n+2)| ≤A|d(x2n, x2n+1)|+B|d(x2n+1, x2n+2)|+D|d(x2n, x2n+2)|

and hence

|d(x2n+1, x2n+2)| ≤ ₁_−B−sDA+sD |d(x2n, x2n+1)|.− − −(3.3)

Similarly, we obtain

|d(x2n+2, x2n+3)| ≤ ₁_−B−sEA+sE |d(x2n+1, x2n+2)|.− − −(3.4)

Since A+B+C + 2sD+ 2sE < 1and s ≥ 1 we get A+B+ 2sD < 1and A+B + 2sE <1.

Therefore with δ=max₁_−B−sDA+sD ,₁_−B−sEA+sE <1,and for all n ≥0 and consequently, we have

|d(x2n+1, x2n+2)| ≤δ|d(x2n, x2n+1)| ≤δ2|d(x2n−1, x2n)| − − − − − −−

− − −− ≤δ2n+1_|_d₍_x 0, x1)|.

|d(xn+1, xn+2)| ≤δ|d(xn, xn+1)| ≤δ2|d(xn−1, xn)| − − − − − −−

− − − − − − − ≤δn+1_|_d₍_x

0, x1)|.− −−(3.5)

≤s|d(xn, xn+1)|+s2|d(xn+1, xn+2)|+s2|d(xn+2, xm)|

≤s|d(xn, xn+1)|+s2|d(xn+1, xn+2)|+s3|d(xn+2, xn+3)|+s3|d(xn+3, xm)|

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − −− ≤s|d(xn, xn+1)|+s2|d(xn+1, xn+2)|+s3|d(xn+2, xn+3)|+− − − − − − −

−−−−+sm−n−2_|_d₍_x

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By using (3.5), we get

|d(xn, xm)| ≤sδn|d(x0, x1)|+s2δn+1|d(x0, x1)|+s3δn+2|d(x0, x1)|

+−−−−+sm−n−2_δm−3_|_d₍_x

0, x1)|+sm−n−1δm−2|d(x0, x1)|+sm−nδm−1|d(x0, x1)|

=

m−n

X

i=1

si_δi+n−1_|_d₍_x 0, x1)|.

Therefore,

|d(xn, xm)| ≤ m−n

X

i=1

si+n−1δi+n−1|d(x0, x1)|

=

m−1 X

t=n

stδt|d(x0, x1)| ≤

∞

X

t=n

(sδ)t|d(x0, x1)|

= (₁sδ_−sδ)n|d(x0, x1)|

and hence

|d(xn, xm)| ≤ (sδ) n

1−sδ|d(x0, x1)| →0as m, n → ∞.

Thus{xn}is a Cauchy sequence inX.SinceX is complete, there exists

someu∈X such thatxn →uasn→ ∞.Assume not, then there exists

z ∈X such that

|d(u, Su)|=|z|>0.− − −(3.6)

So by using the trianglular inequality and (3.1), we get

z =d(u, Su)_-sd(u, x2n+2)+sd(x2n+2, Su)= sd(u, x2n+2)+sd(T x2n+1, Su)

-sd(u, x2n+2) +sAd(u, x2n+1) + sBd(u,Su₁₊)_dd₍(_u,x2n+1x2n+1,T x2n+1₎ )

+sCd(x2n+1,Su₁₊_d₍_u,x2n+1)d(u,T x2n+1₎ )+ sDd(₁₊u,Su_d₍_u,x2n+1)d(u,T x2n+1₎ ) +sEd(x2n+1,Su₁₊_d₍)_u,x2n+1d(x2n+1,T x2n+1₎ )

which implies that

|z|=|d(u, Su)| ≤s|d(u, x2n+2)|+sA|d(u, x2n+1)|

+sB|d(u,Su_|₁₊)_d||d₍_u,x2n+1(x2n+1,x2n+2₎_| )|

+sC|d(x2n+1,Su_|₁₊_d₍_u,x2n+1)||d(u,x2n+2₎_| )|

+sD|d_|(₁₊u,Su_d₍_u,x2n+1)||d(u,x2n+2₎_| )|

+sE|d(x2n+1,Su_|₁₊_d₍_u,x2n+1)||d(x2n+1,x2n+2₎_| )|.− −−(3.7)

Taking the limit of (3.7) asn→ ∞,we obtain that|z|=|d(u, Su)| ≤0, a contradiction with (3.6). So |z|= 0.

HenceSu=u.It follows that similarlyT u=u.Therefore,uis common fixed point of S and T.

Finally, to prove the uniqueness of common fixed point, letu? _{is another}

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-Ad(u, u?_{) +} Bd(u,Su)d(u?_{,T u}?₎

1+d(u,u?₎ +

Cd(u?_,Su₎_d₍_{u,T u}?₎

1+d(u,u?₎

+Dd(u,Su₁₊_d₍)_u,ud(u,T u?₎ ?) +

Ed(u?_,Su₎_d₍_u?_{,T u}?₎

1+d(u,u?₎ .

Taking modulus of the above inequality, we get |d(u, u?)| ≤A|d(u, u?)|+ C|d(u_|?₁₊,Su_d₍)_u,u||d(?u,T u₎_| ?)|

=A|d(u, u?)|+C|d(u?, u)|_|₁₊|d(_du,u₍_u,u?)?|₎_|.

Since |1 +d(u, u?)|>|d(u, u?)|. Therefore

|d(u, u?_)|_{< A}_|_d₍_{u, u}?_)|₊_C_|_d₍_{u, u}?_{)|= (}_A₊_C_)|_d₍_{u, u}?_)|.

This is contradiction to A+C < 1. Hence u = u?_{which proves the}

uniqueness of common fixed point inX. This completes the proof.

Corollary 3.2. Let (X, d) be a complete complex valued b-metric space with the coefficient s ≥ 1 and let T : X → X be a mapping satisfying:

d(T x, T y)_-Ad(x, y) + Bd(x,T x₁₊_d₍)_x,yd(y,T y₎ )+ Cd(y,T x₁₊_d₍)_x,yd(x,T y₎ )

+Dd(x,T x₁₊_d₍)_x,yd(x,T y₎ )+ Ed(y,T x₁₊_d₍)_x,yd(y,T y₎ ) − − −(3.8)

for all x, y ∈ X, where A, B, C, D and E are nonnegative reals with A+B +C+ 2sD+ 2sE <1. Then T has a unique common fixed in X.

Proof. We can prove this result by applying Theorem 3.1 by setting S =T.

d(Tn_{x, T}n_y₎_-_Ad₍_{x, y}₎₊Bd(x,Tn_x₎_d₍_y,Tn_y₎

1+d(x,y) +

Cd(y,Tn_x₎_d₍_x,Tn_y₎

1+d(x,y) +

Dd(x,Tn_x₎_d₍_x,Tn_y₎

1+d(x,y)

+Ed(y,T₁₊n_dx₍)_x,yd(y,T₎ ny) − −−(3.9)

for all x, y ∈ X, where A, B, C, D and E are nonnegative reals with A+B+C+ 2sD+ 2sE <1. Then T has a unique fixed point inX. Proof. From Corollary 3.2, we obtain u∈X such that

Tn_u₌_u_.

The uniqueness follows from

d(T u, u) =d(T Tn_{u, T}n_u_{) =}_d₍_Tn_{T u, T}n_u₎

-Ad(T u, u) + Bd(T u,T₁₊n_dT u₍_{T u,u})d(u,T₎ nu)

+Cd(u,T₁₊nT u_d₍_{T u,u})d(T u,T₎ nu)+ Dd(T u,T₁₊n_dT u₍_{T u,u})d(T u,T₎ nu) +Ed(u,T₁₊n_dT u₍_{T u,u})d(u,T₎ nu)

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By taking modulus of (3.10), we get

|d(T u, u)| ≤A|d(T u, u)|+C|d(T u, u)|_|₁₊|d(_dT u,u₍_{T u,u})|₎_|.

Since |1 +d(T u, u)|>|d(T u, u)|.

Therefore,

|d(T u, u)|<(A+C)|d(T u, u)|,a contradiction. So T u=u.

Hence T u=Tnu=u.

Therefore, the fixed point of T is unique. This completes the proof.

Corollary 3.4. Let (X, d) be a complete complex valued b-metric space with the coefficient s ≥ 1 and let S, T : X → X are mappings satisfying:

d(Sx, T y)_-Ad(x, y) + Bd(x,Sx₁₊_d₍)_x,yd(y,T y₎ )

+Cd(y,Sx₁₊_d₍)_x,yd(x,T y₎ ) + Dd(x,Sx₁₊_d₍)_x,yd(x,T y₎ ) − − −(3.11)

for all x, y ∈ X, where A, B, C and D are nonnegative reals with A+ B+C+ 2sD <1. Then S and T have a unique common fixed in X.

Proof. We can prove this result by applying Theorem 3.1 by setting E = 0.

d(T x, T y)_-Ad(x, y) + Bd(x,T x₁₊_d₍)_x,yd(y,T y₎ )

+Cd(y,T x₁₊_d₍)_x,yd(x,T y₎ ) +Dd(x,T x₁₊_d₍)_x,yd(x,T y₎ )− − −(3.12)

for all x, y ∈ X, where A, B, C and D are nonnegative reals with A+ B+C+ 2sD <1. Then T has a unique fixed point.

Proof. We can prove this result by applying Corollary 3.4 by setting T =S and E = 0.

Corollary 3.6. Let (X, d) be a complete complex valued b-metric space with the coefficient s ≥ 1 and let S, T : X → X are mappings satisfying:

d(Sx, T y)_-Ad(x, y) + Bd(x,Sx₁₊_d₍)_x,yd(y,T y₎ ) +Cd(y,Sx₁₊_d₍)_x,yd(x,T y₎ )

+Ed(y,Sx₁₊_d₍)_x,yd(y,T y₎ ) − − −(3.13)

for all x, y ∈X, where A, B, C and E are nonnegative reals with A+ B+C+ 2sE <1. Then S and T have a unique common fixed in X.

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d(T x, T y)_-Ad(x, y) + Bd(x,T x₁₊_d₍)_x,yd(y,T y₎ )+ Cd(y,T x₁₊_d₍)_x,yd(x,T y₎ )

+Ed(y,T x₁₊_d₍)_x,yd(y,T y₎ ) − − −(3.14)

for all x, y ∈X, where A, B, C and E are nonnegative reals with A+ B+C+ 2sE <1. Then T has a unique fixed point.

Proof. We can prove this result by applying Corollary 3.6 by setting T =S and D= 0.

Example 3.8. Let X = _C. Define a function d : X ×X → _C such that d(z1, z2) = |x1−x2|2+i|y1−y2|2

where z1 =x1+iy1 and z2 =x2+iy2.

It is clear that (X, d) is a complex valued b-metric space with s= 2.

Now, define two self mappings S, T :X →X as follows:

T z =T(x+iy) =

        

0, x, y ∈Q 1 +i, x, y∈Qc 1, x∈Qc_{, y} _∈_Q

i, x∈Q, y ∈Qc

such that S = T,and z = x+iy. Let x = √1

2 and y = 0 : and since

A∈[0,1) we have

d(T x, T y) = dT(√1

2), T(0)

=d(1,0) = 2A.√2

=Ad(√1

2,0) +

Bd_√1 2,T(

1

√

2)

d(0,T(0)) 1+d(√1

2,0)

+Cd

0,T(_√1 2)

d_√1

2,T(0)

1+d(√1

2,0)

+Dd

1

√

2,T( 1

√

2)

,d_√1 2,T(0)

1+d(√1

2,0)

+ Ed

0,T(_√1 2)

d(0,T(0)) 1+d(√1

2,0)

.

However, notice that Tnz= 0 for n >1, so

d(Tn_{x, T}n_y_{) = 0}

-Ad(x, y) + Bd(x,T₁₊n_dx₍)_x,yd(y,T₎ ny)+Cd(y,T₁₊n_dx₍)_x,yd(x,T₎ ny) +Dd(x,T₁₊n_dx₍)_x,yd(x,T₎ ny) +Ed(y,T₁₊n_dx₍)_x,yd(y,T₎ ny)

for allx, y ∈X,whereA, B, C, D, E ≥0 withA+B+C+2sD+2sE < 1. So all conditions of Corollary 3.3 are satisfied to get a unique fixed 0of T.

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Theorem 3.9 Let (X, d) be a complete complex valued b-metric s-pace with the coefficient s ≥ 1 and let S, T : X → X are mappings satisfying:

d(Sx, T y)_-αd(x, y) + βd(y,T y₁₊_d₍)_x,yd(x,Sx₎ )+γ[d(x, Sx) +d(y, T y)] +δ[d(x, T y) +d(y, Sx)]− −−(3.15)

for allx, y ∈X,whereα, β, γ, δare nonnegative reals withα+β+ 2γ+ 2sδ <1. Then S and T have a unique common fixed point in X. Proof. Let x0 be an arbitrary point in X and define sequence {xn} in

X such that

x2n+1 =Sx2n, x2n+2 =T x2n+1forn = 0,1,2, ....− − −(3.16)

Now, we show that the sequence {xn} is Cauchy. Let x = x2n and

y=x2n+1 in (3.15) we have

d(x2n+1, x2n+2) =d(Sx2n, T x2n+1)

-αd(x2n, x2n+1) + βd(x2n+1,T x2n+1₁₊_d₍_x2n,x2n+1)d(x2n,Sx2n₎ )

+γ[d(x2n, Sx2n)+d(x2n+1, T x2n+1)]+δ[d(x2n, T x2n+1)+d(x2n+1, Sx2n)]

=αd(x2n, x2n+1) + βd(x2n+1,x2n+2₁₊_d₍_x2n,x2n+1)d(x2n,x2n+1₎ )

+γ[d(x2n, x2n+1) +d(x2n+1, x2n+2)]

+δ[d(x2n, x2n+2) +d(x2n+1, x2n+1)]

:

-αd(x2n, x2n+1) + βd(x2n+1,x2n+2₁₊_d₍_x2n,x2n+1)d(x2n,x2n+1₎ )

+γ[d(x2n, x2n+1) +d(x2n+1, x2n+2)] +sδ[d(x2n, x2n+1) +d(x2n+1, x2n+2)]

which implies that

|d(x2n+1, x2n+2)| ≤α|d(x2n, x2n+1)|+

β|d(x2n+1,x2n+2)||d(x2n,x2n+1)| |1+d(x2n,x2n+1)|

+γ[|d(x2n, x2n+1)|+|d(x2n+1, x2n+2)|]

+sδ[|d(x2n, x2n+1)|+|d(x2n+1, x2n+2)|].

Since |d(x2n, x2n+1)| ≤ |1 +d(x2n, x2n+1)|,

+γ[|d(x2n, x2n+1)|+|d(x2n+1, x2n+2)|]

+sδ[|d(x2n, x2n+1)|+|d(x2n+1, x2n+2)|]

and hence

|d(x2n+1, x2n+2)| ≤

α+γ+sδ

1−β−γ−sδ

|d(x2n, x2n+1)|.− − −(3.17)

|d(x2n+2, x2n+3)| ≤

α+γ+sδ

1−β−γ−sδ

|d(x2n+1, x2n+2)|.− − −(3.18)

Putµ= ₁_{−β−γ−sδ}α+γ+sδ <1, we have

|d(xn+1, xn+2)| ≤µ|d(xn, xn+1)| ≤ −−− ≤µn+1|d(x0, x1)|.−−−(3.19)

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≤s|d(xn, xn+1)|+s2|d(xn+1, xn+2)|+s2|d(xn+2, xm)|

− − − − − − − − − − − − − − − − − − − − − − − − − − − − − ≤s|d(xn, xn+1)|+s2|d(xn+1, xn+2)|+s3|d(xn+2, xn+3)|+− − −

+sm−n−1_|_d₍_x

m−2, xm−1)|+sm−n|d(xm−1, xm)|.:

By using (3.19) we get

|d(xn, xm)| ≤sµn|d(x0, x1)|+s2µn+1|d(x0, x1)|+s3µn+2|d(x0, x1)|

+− − − − −+sm−n−1_µm−2_|_d₍_x

0, x1)|+sm−nµm−1|d(x0, x1)|

=

m−n

X

i=1

si_µi+n−1_|_d₍_x 0, x1)|.

Therefore,

|d(xn, xm)| ≤ m−n

X

i=1

si+n−1_µi+n−1_|_d₍_x 0, x1)|

=

m−1 X

t=n

st_µt_|_d₍_x

0, x1)| ≤

∞

X

t=n

(sµ)t_|_d₍_x

0, x1)|

= (₁sµ_−sµ)n|d(x0, x1)|

and so,

|d(xn, xm)| ≤ (sµ) n

1−sµ|d(x0, x1)| →0 as m, n→ ∞.

This implies that{xn}is a Cauchy sequence inX. SinceX is complete,

there exists some u∈X such that xn→u asn → ∞.Let on contrary

u6=Su,then there exists z∈X such that |d(u, Su)|=|z|>0.− − −(3.20)

So by using the triangluar inequality and (3.15), we get z =d(u, Su)_-sd(u, x2n+2) +sd(x2n+2, Su)

=sd(u, x2n+2) +sd(T x2n+1, Su)

-sd(u, x2n+2) +sαd(u, x2n+1) +

sβd(x2n+1,T x2n+1)d(u,Su) 1+d(u,x2n+1)

+sγ[d(u, Su) +d(x2n+1, T x2n+1)]

+sδ[d(u, T x2n+1) +d(x2n+1, Su)]

=sd(u, x2n+2) +sαd(u, x2n+1) +

sβd(x2n+1,x2n+2)d(u,Su) 1+d(u,x2n+1)

+sγ[d(u, Su) +d(x2n+1, x2n+2)] +sδ[d(u, x2n+2) +d(x2n+1, Su)].

This implies that

|z|=|d(u, Su)| ≤s|d(u, x2n+2)|+sα|d(x2n+1, u)|

+sβ|z||d_|₁₊(_dx2n+1,x2n+2₍_u,x2n+1₎_| )|

+sγ[|z|+|d(x2n+1, x2n+2)|]

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Taking the limit of (3.21) asn → ∞,we obtain that|z|=|d(u, Su)| ≤ 0,a contradiction with (3.20). So |z|= 0⇒Su=u. Similarly, one can show that u=T u.

We now show thatS and T have unique common fixed point. For this, assume that u?_in_X _{is another common fixed point of} _{S and T.} _Then

d(u, u?_{) =}_d₍_{Su, T u}?₎

-αd(u, u?) + βd(u,Su₁₊_d)₍d_u,u(u??,T u₎ ?) +γ[d(u, Su) +d(u

?_{, T u}?_)]

+δ[d(u, T u?_{) +}_d₍_u?_Su_)]_.

So that|d(u, u?_{)| ≤}_α_|_d₍_{u, u}?_)|₊β|d(u,Su)||d(u?_{,T u}?₎_| |1+d(u,u?₎_|

+γ[|d(u, Su)|+|d(u?, T u?)|]+δ[|d(u, T u?)|+|d(u?, Su)|] = (α+ 2δ)|d(u, u?)|.

This implies thatu? =u,which proves the uniqueness of common fixed point in X. This completes the proof.

d(T x, T y)_-αd(x, y) + βd(y,T y₁₊_d₍)_x,yd(x,T x₎ ) +γ[d(x, T x) +d(y, T y)]

+δ[d(x, T y) +d(y, T x)]− − −(3.22)

for allx, y ∈X,whereα, β, γ, δare nonnegative reals withα+β+ 2γ+ 2sδ <1. Then T has a unique common fixed point in X.

Proof. We can prove this result by applying Theorem 3.9 with S =T.

Corollary 3.11 Let (X, d) be a complete complex valued b-metric space with the coefficient s ≥ 1 and let T : X → X be a mapping satisfying (for some fixed n):

d(Tn_{x, T}n_y₎_-_αd₍_{x, y}_{) +} βd(y,Tny)d(x,Tnx) 1+d(x,y)

+γ[d(x, Tn_x_{) +}_d₍_{y, T}n_y_)]

+δ[d(x, Tn_y_{) +}_d₍_{y, T}n_x_)]_{− −−(3.23)}

for allx, y ∈X,whereα, β, γ, δare nonnegative reals withα+β+ 2γ+ 2sδ <1. Then T has a unique fixed point in X.

Proof. From Corollary 3.10, we obtain u∈X such that Tn_u₌_u.

The uniqueness follows from

d(T u, u) =d(T Tnu, Tnu) =d(TnT u, Tnu)

-αd(T u, u) + βd(u,T₁₊nu_d)₍d_{T u,u}(T u,T₎nT u) +γ[d(T u, Tn_{T u}_{) +}_d₍_{u, T}n_u_)]

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-αd(T u, u) + βd(u,u₁₊)d_d₍(T u,T T_{T u,u}₎ nu) +γ[d(T u, T Tn_u_{) +}_d₍_{u, u}_)]

+δ[d(T u, u) +d(u, T Tn_u_)]

= (α+ 2δ)d(T u, u).− − −(3.24)

SoT u=u.Hence

T u=Tnu=u.

Therefore, the fixed point of T is unique. This completes the proof.

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spaces, Fixed Point Theory and Applications, (2013), 2013:159.

[8] K.P.R.Rao, P.R.Swamy, J.R.Prasad, A common fixed point theorem in com-plex valued b-metric spaces, Bulletin of Mathematics and Statistics Research, Vo. 1 Issue 1(2013).

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[13] Chakkrid Klin-eam, Cholatis Suanoom, Some common fixed-point theorem-s for generalized-contractive type mappingtheorem-s on complex-valued metric theorem- s-paces Abstract and Applied Analysis, Vol.2013, Article ID 604215, 6 pages, http://dx.doi.org/10.1155/2013/604215.

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A.K. Dubey, Department of Mathematics, Bhilai Institute of Tech-nology, Bhilai House, Durg, 491001, Chhattisgarh, INDIA

Rita Shukla, Department of Mathematics, Shri Shankracharya Col-lege of Engineering and Technology, Bhilai, 490020 Chhattisgarh, IN-DIA