Adv. Fixed Point Theory, 6 (2016), No. 4, 341-351 ISSN: 1927-6303
COUPLED FIXED POINT THEOREMS IN COMPLEX VALUED Gb-METRIC
SPACES
JITENDER KUMAR1,∗AND SACHIN VASHISTHA2
1Department of Mathematics, University of Delhi, Delhi 110007, India
2Department of Mathematics, Hindu College (University of Delhi), Delhi 110007, India
Copyright c2016 Kumar and Vashistha. 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, we introduce the notion of coupled fixed point for a mapping in complex valued Gb metric space and prove some coupled fixed point theorems in this space and provide an example in support of our main
theorem.
Keywords:complex valuedGbmetric space; coupled fixed point theorem; contractive type mapping.
2010 AMS Subject Classification:47H10, 54H25.
1. Introduction
The concept of a metric space was introduced by Fr´echet [16]. The first important result on fixed-point for contractive-type mappings was the well-known Banach fixed point theorem, published for the first time in 1922. After that many researchers proved the Banach fixed point theorem in a number of generalized metric spaces. Bakhtin [7] presented b-metric spaces as a generalization of metric spaces. In 2011, Azam et al. [4] introduced the notion of complex
∗Corresponding author
E-mail address: kumar.jmaths@gmail.com
Received March 2, 2016
valued metric space which is a generalization of the classical metric space. Rao et al. [12] introduced the concept of complex valuedb-metric space.
Mustafa and Sims [9] presented the notion ofg-metric spaces, many researchers [1, 2, 3, 10, 11] obtained common fixed point results forG-metric spaces. The concept ofGb-metric space was given in [6].
E. Ozgur [15] presented the notion of complex valuedGb-metric space. In 2006, Bhaskar et al. [5] introduced the notion of coupled fixed point and proved some fixed point results in this context. Similarly, we introduced the notion of coupled fixed point for a mapping in complex valuedGb-metric spaces.
2. Preliminaries
In this section will recall some properties ofGb-metric spaces.
Definition 2.1([6]). LetX be a nonempty set ands≥1 be a given real number. Suppose that a mappingG:X×X×X →R+satisfies:
(Gb1) G(x,y,z) =0 ifx=y=z;
(Gb2) 0<G(x,x,y)for allx,y∈X withx6=y;
(Gb3) G(x,x,y)≤G(x,y,z)for allx,y,z∈X withy6=z;
(Gb4) G(x,y,z) =G(ρ{x,y,z}), whereρ is a permutation ofx,y,z;
(Gb5) G(x,y,z)≤s(G(x,a,a) +G(a,y,z))for allx,y,z,a∈X (rectangle inequality).
Then,Gis called a generalizedb-metric space and(X,G)is called a generalizedb-metric or a Gb-metric space.
Note that eachGb-metric space is aG-metric space withs=1.
Proposition 2.2([6]). Let X be a Gb-metric space. Then for each x,y,z,a∈X , it follows that: (1) If G(x,y,z) =0then x=y=z;
(2) G(x,y,z)≤s(G(x,x,y) +G(x,x,z)), (3) G(x,y,y)≤2sG(y,x,x);
LetCbe the set of complex numbers andz1,z2∈C.
Define a partial order onCas follows:
z1-z2if and only if Re(z1)≤Re(z2)and Im(z1)≤Im(z2).
It follows thatz1-z2if one of the following condition is satisfied. (1) Re(z1) =Re(z2), Im(z1)<Im(z2),
(2) Re(z1)<Re(z2), Im(z1) =Im(z2), (3) Re(z1)<Re(z2), Im(z1)<Im(z2),
(4) Re(z1) =Re(z2), Im(z1) =Im(z2).
In particular, we will writez1z2 Ifz16=z2 and one of (i), (ii) and (iii) is satisfied and we will writez1≺z2iff (iii) is satisfied.
The followed statements hold:
(i) Ifa,b∈Rwitha≤b, thenaz≺bz, for allz∈C. (ii) If 0-z1z2, then|z1|<|z2|.
(iii) Ifz1-z2,z2≺z3, themz1≺z3.
Definition 2.3([15]). Let X be a nonempty set and s≥1be a given real number. Suppose that a mapping G:X×X×X→Csatisfies:
(CGb1) G(x,y,z) =0if x=y=z;
(CGb2) 0≺G(x,x,y)for all x,y∈X with x6=y;
(CGb3) G(x,x,y)-G(x,y,z)for all x,y,z∈X with y6=z;
(CGb4) G(x,y,z) =G(ρ(x,y,z)), whereρ is a permutation of x,y,z;
(CGb5) G(x,y,z)-s(G(x,a,a) +G(a,y,z))for all x,y,z,a∈X (rectangle inequality).
Then, G is called a complex valued Gb-metric and(X,G)is called a complex valued Gb-metric space.
Proposition 2.4([15]). Let(X,G)be a complex valued Gb-metric space. Then for any x,y,z∈X ,
• G(x,y,z)-s(G(x,x,y) +G(x,x,z)),
• G(x,y,y)-2sG(y,x,y).
(i) {xn}is complex valued Gb-convergent to x if for every a∈Cwith0≺a, there exists N∈N
such that G(x,xn,xm)≺a for all n,m≥N.
(ii) A sequence{xn}is called complex valued Gb-Cauchy if for every a∈Cwith0≺a, there exists N∈Nsuch that G(xn,xm,x`)≺a for all n,m, `≥N.
(iii) If every complex valued Gb-Cauchy sequence is complex valued Gb-convergent in(X,G), then(X,G)is said to be complex valued Gb-complete.
3. Main Results
Theorem 3.1. Let(X,G)be a complete complex valued Gb-metric space with coefficient s>1 and F :X×X →X be a mapping satisfying:
G(F(x,y),F(u,v),F(z,w))-λG(x,u,z) +µG(y,v,w) (3.1)
for all x,y,u,v,z,w∈X , where λ and µ are non-negative constants with sλ+µ <1. Then F
has a unique coupled fixed point.
Proof. Choosex0,y0∈X and set
x1=F(x0,y0), y1=F(y0,x0) ..
.
xn+1=F(xn,yn), yn+1=F(yn,xn)
From (3.1), we have
G(xn,xn+1,xn+1) =G(F(xn−1,yn−1),F(xn,yn),F(xn,yn)) -λG(xn−1,xn,xn) +µG(yn−1,yn,yn)
and similarly
Therefore, by lettingGn=G(xn,xn+1,xn+1) +G(yn,yn+1,yn+1), we have
Gn=G(xn,xn+1,xn+1) +G(yn,yn+1,yn+1)
-λG(xn−1,xn,xn) +µG(yn−1,yn,yn) +λG(yn−1,yn,yn) +µG(xn−1,xn,xn) = (λ+µ)[G(xn−1,xn,xn) +G(yn−1,yn,yn)]
= (λ+µ)Gn−1.
That isGn-PGn−1, whereP=λ+µ <1.
In general, we have forn=0,1,2, . . .
Gn-PGn−1-P2Gn−2-· · ·-PnG0.
Now, for allm>n
G(xn,xm,xm)-s[G(xn,xn+1,xn+1) +G(xn+1,xm,xm)]
-sG(xn,xn+1,xn+1) +s2[G(xn+1,xn+2,xn+2) +G(xn+2,xm,xm)]
-sG(xn,xn+1,xn+1) +s2G(xn+1,xn+2,xn+2) +· · ·+sm−nG(xm−1,xm,xm) and
G(yn,ym,ym)-sG(yn,yn+1,yn+1) +s2G(yn+1,yn+2,yn+2) +· · ·+sm−nG(ym−1,ym,ym) Therefore, have
G(xn,xm,xm) +G(yn,ym,ym)-sGn+s2Gn+1+· · ·+sm−nGm−1
-sPnG0+s2Pn+1G0+· · ·+sm−nPm−1G0
=sPn[1+sP+ (sP)2+· · ·+ (sP)m−n−1]G0
≺ sP
n 1−sPG0. Thus, we obtain
|G(xn,xm,xm) +G(yn,ym,ym)| ≤ sPn 1−sP|G0|. SinceP<1, taking limit asn→∞, then
sPn
This means that|G(xn,xm,xm))| →0 and(G(yn,ym,ym))→0. By Proposition 2.4, we get
G(xn,xm,x`) +G(yn,ym,y`)
-G(xn,xm,xm) +G(x`,xm,xm) +G(yn,ym,ym) +G(y`,ym,ym)for`,m,n∈N.
Thus
|G(xn,xm,x`) +G(yn,ym,y`)|
≤ |G(xn,xm,xm) +G(yn,ym,ym)|+|(G(x`,xm,xm) +G(y`,ym,ym)|. If we take limit asn,m, `→∞, we obtain
|G(xn,xm,x`)| →0 and |G(yn,ym,y`)| →0.
which implies that {xn} and {yn} are complex valued Gb-Cauchy sequences in X. By X is complete, there existsx0,y0∈X such that
lim n→∞xn=x
0
and lim n→∞yn=y
0.
Letc∈Cwith 0≺c. For everym∈N, there exitsN∈Nsuch that
G(xn,xm,x`)≺c and G(yn,ym,y`)≺c ∀n,m, ` >N.
Thus, we have
G(xn+1,F(x0,y0),F(x0,y0)) =G(F(xn,yn),F(x0,y0),F(x0,y0)) -λG(xn,x0,x0) +µG(yn,y0,y0) This implies that
|(G(xn+1,F(x0,y0),F(x0,y0))| ≤λ|(G(xn,x0,x0)|+µ|G(yn,y0,y0)| Taking limit asn→∞, we get
|G(x0,F(x0,y0),F(x0,y0)| →0 that isG(x0,F(x0,y0),F(x0,y0)) =0 and henceF(x0,y0) =x0.
Hence(x0,y0)is a coupled fixed point ofF.
Now, if(x00,y00)is another coupled fixed point ofF, then
G(x0,x00,x00) =G(F(x0,y0),F(x00,y00),F(x00,y00)) -λG(x0,x00,x00) +µG(y0,y00,y00)
and
G(y0,y00,y00) =G(F(y0,x0),F(y00,x00),F(y00,x00) -λG((y0,y00,y00) +µG(x0,x00,x00)
Thus we have
G(x0,x00,x00) +G(y0,y00,y00)-(λ+µ)[G(x0,x00,x00) +F(y0,y00,y00)]
which implies that
|G(x0,x00,x00) +G(y0,y00,y00)|-(λ+µ)|G(x0,x00,x00) +G(y0,y00,y00)|
Sincesλ+µ <1, we have|G(x0,x00,x00) +G(y0,y00,y00)|=0.
That isG(x0,x00,x00) +G(y0,y00,y00) =0. Thus we have(x0,y0) = (x00,y00).
ThereforeF has a unique coupled fixed point.
From Theorem 3.1 withµ =λ, we have the following corollary:
Corollary 3.2. Let(X,G)be a complete complex valued Gb-metric space with coefficient s≥1 and F :X×X →X be mapping satisfying:
G(F(x,y),F(u,v),F(z,w))-λ[G(x,u,z) +G(y,v,w)] (3.2)
for all x,y,z,u,v,w∈X , where λ is a non-negative constant with λ < 12 then F has a unique
coupled fixed point.
Example 3.3. LetX = [−1,1]andG:X×X×X →Cbe defined as follows:
for allx,y,z∈X. (X,G)is complex valuedG-metric space. Define G∗(x,y,z) =G(x,y,z)2.
G∗is a complex valuedGb-metric withs=2 (see [6]).
If we defineF :X×X →X withF(x,y) = x+y3 i. ThenF satisfied the contractive condition (3.2) for 19 ≤λ < 12 that is
G(F(x,y),F(u,v),F(z,w)≤λ[G(x,u,z) +G(y,v,w)].
Here(0,0)is the unique coupled fixed point ofF.
Theorem 3.4. Let(X,G)be a complete complex valued Gbmetric space. Suppose that the map-ping F :X×X →X satisfies
G(F(x,y),F(u,v),F(u,v)-λ[G(x,F(x,y),F(x,y)+G(u,F(u,v),F(u,v)]. (3.4)
whereλ ∈0,12. Then F has a unique coupled fixed point.
Proof. Choosex0,y0∈X and set
x1=F(x0,y0), y1=F(y0,x0)
.. .
xn+1=F(xn,yn), yn+1=F(yn,xn) From (3.4), we have
G(xn,xn+1,xn+1) =G(F(xn−1,yn−1),F(xn,yn),F(xn,yn))
-λ[G(xn−1,F(xn−1,yn−1),F(xn−1,yn−1) +G(xn,F(xn,yn),F(xn,yn))] -λ[G(xn−1,xn,xn) +G(xn,xn+1,xn+1)]
which implies
G(xn,xn+1,xn+1) -λ
1−λG(xn−1,xn,xn)
and similarly
-λ[G(yn−1,F(yn−1,xn−1),F(yn−1,xn−1)) +G(yn,F(yn,xn),F(yn,xn)) which implies that
G(yn,yn+1,yn+1) -λ
1−λG(yn−1,yn,yn)
Now, by settingGn=G(xn,xn+1,xn+1) +G(yn,yn+1,yn+1), we have
Gn=G(xn,xn+1,xn+1) +G(yn,yn+1,yn+1)
- λ
1−λ[G(xn−1,xn,xn) +G(yn−1,yn,yn)]
that is
Gn-PGn−1 whereP= λ
1−λ <1.
In general, we have forn=0,1,2,· · ·
Gn-PGn−1-P2Gn−2-· · ·-PnG0.
This implies that{xn}and{yn}are Cauchy sequence in(X,G). and therefore, by completeness ofX, there existsx0,y0∈X such that lim
n→∞xn=x
0and lim
n→∞yn=y 0.
Letc∈Cwith 0≺c. For everym∈N, there existsN∈Nsuch that G(xn,xm,x`)≺candG(yn,ym,y`)≺c ∀m,n, ` >N.
Thus, we have
G(xn+1,F(x0,y0),F(x0,y0)) =G(F(xn,yn),F(x0,y0),F(x0,y0)]
-λ[G(xn,F(xn,yn),F(xn,yn) +G(x0,F(x0,y0),F(x0,y0)] which implies that
|G(xn+1,F(x0,y0),F(x0,y0)| ≤λ|G(xn,xn+1,xn+1) +G(x0,F(x0,y0),F(x0,y0)|.
Taking limit asn→∞, we get
|G(x0,F(x0,y0),F(x0,y0)| ≤λ|G(x0,F(x0,y0),F(x0,y0))|.
Now if(x00,y00)is another coupled fixed point ofF, then
G(x0,x00,x00) =G(F((x0,y0),F(x00,y00),(F(x00,y00))
-λ[G(x0,F(x0,y0),F(x0,y0) +G(x00,F(x00,y00),F(x00,y00)]
Thus we havex0=x00similarly, we gety0=y00. ThereforeF has a unique coupled fixed point.
Conflict of Interests
The authors declare that there is no conflict of interests.
REFERENCES
[1] M. Abbas, T. Nazir and P. Vetro, Common fixed point results for three maps inG-metric spaces, Filomat, 25
(4) (2011), 1–17.
[2] R.P. Agarwal, Z. Kadelburg and S. Radenovic, On coupled fixed point results in asymmetricG-metric spaces,
J. Inequalities Appl., 2013 (2013), 528.
[3] H. Aydi, W. Shatanawi and C. Vetro, On generalized weakly G-contractin mapping in G-metric spaces,
Computt. Math. Appl., 62 (2011), 4222-4229.
[4] A. Azam, B. Fisher and M. Khan, Common fixed point theorems in complex valued metric spaces, Numer.
Funct. Anal. & Optim., 32 (2011), 243–253.
[5] T.G. Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered cone metric spaces and
applications, Nonlinear Anal., 65 (2006), 825–832.
[6] A. Aghajani, M. Abbas, J.R. Roshan, Common fixed point of generalized weak contractive mappings in
partially orderedGb-metric spaces, Filomat, 20 (6) (2014), 1087-1101.
[7] I.A. Bakhtin, The contraction mappings principle in quasimetric spaces, Funct. Anal Unianouesk Gos. Ped
Inst., 30 (1989), 25-37.
[8] S. Czerwik, Contraction mappings inb-matric spaces, Acta Math. Inform. Univ., Ostraviensis, 1 (1993), 5-11.
[9] Z. Mustafa and B. Sims, A new approach to a generalized metric spaces, J. Nonlinear Convex Anal. 7 (2006),
289-297.
[10] Z. Mustafa and B. Sims, Fixed point theorems for contractive mappings in completeG-metric spaces, Fixed
Point Theory Appl. 2009 (2009), 917175.
[11] Z. Mustafa and B. Sims, A new approach to a generalized metric spaces, Int J. Math. Math. Sci., 2009 (2009),
[12] K.P.R. Rao, P.R. Swamy and J.R. Prasad, A common fixed point theorem in complex valuedb-metric spaces,
Bulletin of Mathematics and Statistics Research, 1 (1) (2013), 1-8.
[13] S. Sedghi, N. Shobkolaei, J.R. Roshan and W. Shatanawi, Coupled fixed point theorems inGb-metric spaces,
Mat. Vesnik, 66 (2), 190-201 (2014).
[14] J. Kumar and S. Vashistha, Common fixed point theorem for generalized contractive type maps on complex
valuedb-metric spaces, Int. Journal of Math. Anal., 9 (47) (2015), 2327-2334.
[15] E. Ozgur, Complex valuedGb-metric spaces, J. Comp. Anal. and Appl., 21 (2) (2016), 363-368.
[16] M. Frechet, Sur quelques point ducalcul fonctionnel, Rendicontiodel Cirolo Matematico di Palermo, (1906)
