We introduce and study the notion of common coupled fixed points for a pair of mappings in complex valued metric space and demonstrate the existence and uniqueness of the common coupled fixed points in a complete complex-valued metric space in view of diverse contractive conditions. In addition, our investigations
are well supported by nontrivial examples.
1. Introduction
Azam et al. [1] introduced the concept of complex-valued metric spaces and obtained sufficient conditions for the existence of common fixed points of a pair of contractive type mappings involving rational expressions. Subsequently, several authors have studied the existence and uniqueness of the fixed points and common fixed points of self-mappings in view of contrasting contractive conditions. Some of these investigations are noted in [2–26].
In [27], Bhaskar and Lakshmikantham introduced the concept of coupled fixed points for a given partially ordered set X. Recently Samet et al. [28, 29] proved that most of the coupled fixed point theorems (on ordered metric spaces) are in fact immediate consequences of well-known fixed point theorems in the literature. In this paper, we deal with the corresponding definition of coupled fixed point for mappings on a complex-valued metric space along with generalized contraction involving rational expressions. Our results extend and improve several fixed point theorems in the literature.
2. Preliminaries
Let ℂ be the set of complex numbers and z1,z2∈ℂ. Define a partial order ⪯ on ℂ as follows:
(1)z1⪯z2iffRe(z1)≤Re(z2),Im(z1)≤Im(z2).
Note that 0⪯z1,z2 and z1≠z2, z1⪯z2 implies |z1|<|z2|.
Definition 1.
Let X be a nonempty set. Suppose that the self-mapping d:X×X→ℂ satisfies the following:
0⪯d(x,y),for all x,y∈X and d(x,y)=0 if and only if x=y;
d(x,y)=d(y,x) for all x,y∈X;
d(x,y)⪯d(x,z)+d(z,y), for all x,y,z∈X.
Then d is called a complex valued metric on X, and (X,d) is known as a complex valued metric space. A point x∈X is called interior point of a set A⊆X whenever, there exists 0≺r∈ℂ such that
(2)B(x,r)={y∈X:d(x,y)≺r}⊆A.
A point x∈X is a limit point of A whenever, for every 0≺r∈ℂ,
(3)B(x,r)∩(A∖{x})≠∅.
A is called open whenever each element of A is an interior point of A. Moreover, a subset B⊆X is called closed whenever each limit point of B belongs to B. The family
(4)F={B(x,r):x∈X,0≺r∈ℂ}
is a subbasis for a Hausdorff topology τ on X.
Let {xn} be a sequence in X and x∈X. If for every c∈ℂ with 0≺c there is n0∈ℕ such that, for all n>n0,d(xn,x)≺c, then {xn} is said to be convergent, {xn} converges to x, and x is the limit point of {xn}. We denote this by limn→+∞xn=x, or xn→x, as n→+∞. If for every c∈ℂ with 0≺c there is n0∈ℕ such that, for all n>n0,d(xn,xn+m)≺c, then {xn} is called a Cauchy sequence in (X,d). If every Cauchy sequence is convergent in (X,d), then (X,d) is called a complete complex valued metric space. We require the following lemmas.
Lemma 2 (see [1]).
Let (X,d) be a complex valued metric space, and let {xn} be a sequence in X. Then {xn} converges to x if and only if |d(xn,x)|→0asn→+∞.
Lemma 3 (see [1]).
Let (X,d) be a complex valued metric space, and let {xn} be a sequence in X. Then {xn} is a Cauchy sequence if and only if|d(xn,xn+m)|→0asn→+∞.
Definition 4 (see [27]).
An element (x,y)∈X×X is called a coupled fixed point of T:X×X→X if
(5)x=T(x,y),y=T(y,x).
Definition 5.
An element (x,y)∈X×X is called a coupled coincidence point of S,T:X×X→X if
(6)S(x,y)=T(x,y),S(y,x)=T(y,x).
Example 6.
Let X=ℝ and S,T:X×X→X defined as S(x,y)=x2y2 and T(x,y)=(4/3)(x+y) for all x,y∈X. Then (0,0), (1,2), and (2,1) are coupled coincidence points of S and T.
Example 7.
Let X=ℝ and S,T:X×X→X defined as S(x,y)=x+y+sin(x+y) and T(x,y)=x+y+xy+cos(x+y) for all x,y∈X. Then (0,π/4) and (π/4,0) are coupled coincidence points of S and T.
Definition 8.
An element (x,y)∈X×X is called a common coupled fixed point of S,T:X×X→X if
(7)x=S(x,y)=T(x,y),y=S(y,x)=T(y,x).
Example 9.
Let X=ℝ and S,T:X×X→X defined as S(x,y)=x((x+(y-1)2)/2) and T(x,y)=x(x2+y2+4-2) for all x,y∈X. Then (0,0), (1,2), and (2,1) are common coupled fixed points of S and T.
In the following, we provide common coupled fixed point theorem for a pair of mappings satisfying a rational inequality in complex valued metric spaces.
Theorem 10.
Let (X,d) be a complete complex-valued metric space, and let the mappings S,T:X×X→Xsatisfy
(8)d(S(x,y),T(u,v))⪯α(d(x,u)+d(y,v))2+(βd(x,S(x,y))d(u,T(u,v))(x,u)+γd(u,S(x,y))d(x,T(u,v)))×(1+d(x,u)+d(y,v))-1
for all x,y,u,v∈X and α,β, and γ are nonnegative reals with α+β+γ<1. Then S and T have a unique common coupled fixed point.
Proof.
Let x0 and y0 be arbitrary points in X. Define x2k+1=S(x2k,y2k), y2k+1=S(y2k,x2k) and x2k+2=T(x2k+1,y2k+1), y2k+2=T(y2k+1,x2k+1), for k=0,1,…
Then,
(9)d(x2k+1,x2k+2)=d(S(x2k,y2k),T(x2k+1,y2k+1))⪯α(d(x2k,x2k+1)+d(y2k,y2k+1))2+(βd(x2k,S(x2k,y2k))(zS(y))×d(x2k+1,T(x2k+1,y2k+1)))×(1+d(x2k,x2k+1)+d(y2k,y2k+1))-1+(γd(x2k+1,S(x2k,y2k))(zS(y))×d(x2k,T(x2k+1,y2k+1)))×(1+d(x2k,x2k+1)+d(y2k,y2k+1))-1⪯α(d(x2k,x2k+1)+d(y2k,y2k+1))2+βd(x2k,x2k+1)d(x2k+1,x2k+2)1+d(x2k,x2k+1)+d(y2k,y2k+1)+γd(x2k+1,x2k+1)d(x2k,x2k+2)1+d(x2k,x2k+1)+d(y2k,y2k+1)⪯α(d(x2k,x2k+1)+d(y2k,y2k+1))2+βd(x2k,x2k+1)d(x2k+1,x2k+2)1+d(x2k,x2k+1)+d(y2k,y2k+1),
which implies that
(10)|d(x2k+1,x2k+2)|≤α|d(x2k,x2k+1)+d(y2k,y2k+1)|2+β|d(x2k,x2k+1)d(x2k+1,x2k+2)||1+d(x2k,x2k+1)+d(y2k,y2k+1)|.
Since |1+d(x2k,x2k+1)+d(y2k,y2k+1)|>|d(x2k,x2k+1)|, so we get
(11)|d(x2k+1,x2k+2)|≤α|d(x2k,x2k+1)|+α|d(y2k,y2k+1)|2+β|d(x2k+1,x2k+2)|,
and hence
(12)|d(x2k+1,x2k+2)|≤12(α1-β)|d(x2k,x2k+1)|+12(α1-β)|d(y2k,y2k+1)|.
Similarly, one can show that
(13)|d(y2k+1,y2k+2)|≤12(α1-β)|d(y2k,y2k+1)|+12(α1-β)|d(x2k,x2k+1)|.
Also,
(14)d(x2k+2,x2k+3)=d(T(x2k+1,y2k+1),S(x2k+2,y2k+2))⪯α(d(x2k+1,x2k+2)+d(y2k+1,y2k+2))2+(βd(x2k+1,T(x2k+1,y2k+1))(x2k+2,)×d(x2k+2,S(x2k+2,y2k+2)))×(1+d(x2k+1,x2k+2)+d(y2k+1,y2k+2))-1+(γd(x2k+2,T(x2k+1,y2k+1))(x2k+2,)×d(x2k+1,S(x2k+2,y2k+2)))×(1+d(x2k+1,x2k+2)+d(y2k+1,y2k+2))-1⪯α(d(x2k+1,x2k+2)+d(y2k+1,y2k+2))2+βd(x2k+1,x2k+2)d(x2k+2,x2k+3)1+d(x2k+1,x2k+2)+d(y2k+1,y2k+2)+γd(x2k+2,x2k+2)d(x2k+1,x2k+3)1+d(x2k+1,x2k+2)+d(y2k+1,y2k+2)⪯α(d(x2k+1,x2k+2)+d(y2k+1,y2k+2))2+βd(x2k+1,x2k+2)d(x2k+2,x2k+3)1+d(x2k+1,x2k+2)+d(y2k+1,y2k+2),
so that
(15)|d(x2k+2,x2k+3)|≤α|d(x2k+1,x2k+2)+d(y2k+1,y2k+2)|2+β|d(x2k+1,x2k+2)||d(x2k+2,x2k+3)||1+d(x2k+1,x2k+2)+d(y2k+1,y2k+2)|.
As |1+d(x2k+1,x2k+2)+d(y2k+1,y2k+2)|>|d(x2k+1,x2k+2)|, therefore
(16)|d(x2k+2,x2k+3)|≤12(α1-β)|d(x2k+1,x2k+2)|+12(α1-β)|d(y2k+1,y2k+2)|.
Similarly, one can show that
(17)|d(y2k+2,y2k+3)|≤α1-β|d(y2k+1,y2k+2)|+12(α1-β)|d(x2k+1,x2k+2)|.
Adding (12)–(17), we get
(18)|d(x2k+1,x2k+2)|+|d(y2k+1,y2k+2)|≤α1-β|d(x2k,x2k+1)|+α1-β|d(y2k,y2k+1)||d(x2k+2,x2k+3)|+|d(y2k+2,y2k+3)|≤α1-β|d(x2k+1,x2k+2)|+α1-β|d(y2k+1,y2k+2)|.
If h=α/(1-β)<1, then from (18), we get
(19)|d(xn,xn+1)|+|d(yn,yn+1)|≤h(|d(xn-1,xn)|+|d(yn-1,yn)|)≤⋯≤hn(|d(x0,x1)|+|d(y0,y1)|).
Now if |d(xn,xn+1)|+|d(yn,yn+1)|=δn, then
(20)δn≤hδn-1≤⋯≤hnδ0.
Without loss of generality, we take m>n. Since 0≤h<1, so we get
(21)|d(xn,xm)|+|d(yn,ym)|≤|d(xn,xn+1)|+|d(yn,yn+1)|+⋯+|d(xm-1,xm)|+|d(ym-1,ym)|≤[hnδ0+hn+1δ0+⋯+hm-1δ0]≤∑i=nm-1hiδ0→0,asm,n→+∞.
This implies that {xn} and {yn} are Cauchy sequences in X. Since X is complete, there exists x,y∈X such that xn→x and yn→y as n→+∞. We now show that x=S(x,y) and y=S(y,x). We suppose on the contrary that x≠S(x,y) and y≠S(y,x) so that 0≺d(x,S(x,y))=l1 and 0≺d(y,S(y,x))=l2; we would then have
(22)l1=d(x,S(x,y))⪯d(x,x2k+2)+d(x2k+2,S(x,y))⪯d(x,x2k+2)+d(T(x2k+1,y2k+1),S(x,y))⪯d(x,x2k+2)+α(d(x2k+1,x)+d(y2k+1,y))2+βd(x2k+1,T(x2k+1,y2k+1))d(x,S(x,y))1+d(x2k+1,x)+d(y2k+1,y)+γd(x,T(x2k+1,y2k+1))d(x2k+1,S(x,y))1+d(x2k+1,x)+d(y2k+1,y)=d(x,x2k+2)+α(d(x2k+1,x)+d(y2k+1,y))2+βd(x2k+1,x2k+2)d(x,S(x,y))1+d(x2k+1,x)+d(y2k+1,y)+γd(x,x2k+2)d(x2k+1,S(x,y))1+d(x2k+1,x)+d(y2k+1,y)
so that
(23)|l1|≤|d(x,x2k+2)|+α|d(x2k+1,x)+d(y2k+1,y)|2+β|d(x2k+1,x2k+2)||d(x,S(x,y))||1+d(x2k+1,x)+d(y2k+1,y)|+γ|d(x,x2k+2)||d(x2k+1,S(x,y))||1+d(x2k+1,x)+d(y2k+1,y)|.
By taking k→+∞, we get |d(x,S(x,y))|=0 which is a contradiction so that x=S(x,y). Similarly, one can prove that y=S(y,x). It follows similarly that x=T(x,y) and y=T(y,x). So we have proved that (x,y) is a common coupled fixed point of S and T. We now show that S and T have a unique common coupled fixed point. For this, assume that (x*,y*)∈X is a second common coupled fixed point of S and T. Then
(24)d(x,x*)=d(S(x,y),T(x*,y*))⪯α(d(x,x*)+d(y,y*))2+βd(x,S(x,y))d(x*,T(x*,y*))1+d(x,x*)+d(y,y*)+γd(x,T(x*,y*))d(x*,S(x,y))1+d(x,x*)+d(y,y*)⪯α(d(x,x*)+d(y,y*))2+βd(x,x)d(x*,x*)1+d(x,x*)+d(y,y*)+γd(x,x*)d(x*,x)1+d(x,x*)+d(y,y*)
so that
(25)|d(x,x*)|≤|α(d(x,x*)+d(y,y*))2|+γ|d(x,x*)||d(x*,x)||1+d(x,x*)+d(y,y*)|.
Since |1+d(x,x*)+d(y,y*)|>|d(x,x*)|, so we get
(26)|d(x,x*)|≤|α(d(x,x*)+d(y,y*))2|+γ|d(x,x*)|=(α2-α-2γ)|d(y,y*)|.
Similarly, one can easily prove that
(27)|d(y,y*)|≤(α2-α-2γ)|d(x,x*)|.
If we add (26) and (27), we get
(28)|d(x,x*)|+|d(y,y*)|≤(α2-α-2γ)(|d(x,x*)|+|d(y,y*)|),
which is a contradiction because α+β+γ<1. Thus, we get x*=x and y*=y, which proves the uniqueness of common coupled fixed point of S and T.
By setting S=T in Theorem 10, one deduces the following.
Corollary 11.
Let (X,d) be a complete complex-valued metric space, and let the mapping T:X×X→X satisfy
(29)d(T(x,y),T(u,v))⪯α(d(x,u)+d(y,v))2+(βd(x,T(x,y))d(u,T(u,v))T(x,y),+γd(u,T(x,y))d(x,T(u,v)))×(1+d(x,u)+d(y,v))-1
for all x,y,u,v∈X, where α,β, and γ are nonnegative reals with α+β+γ<1. Then T has a unique coupled fixed point.
Corollary 12.
Let (X,d) be a complete complex-valued metric space, and let the mapping T:X×X→X satisfy
(30)d(Tn(x,y),Tn(u,v))⪯α(d(x,u)+d(y,v))2+(βd(x,Tn(x,y))d(u,Tn(u,v))T(x,y),+γd(u,Tn(x,y))d(x,Tn(u,v)))×(1+d(x,u)+d(y,v))-1
for all x,y,u,v∈X, where α,β, and γ are nonnegative reals with α+β+γ<1. Then, T has a unique coupled fixed point.
Theorem 13.
Let (X,d) be a complete complex-valued metric space, and let the mappings S,T:X×X→X satisfy
(31)d(S(x,y),T(u,v))⪯{α(d(x,u)+d(y,v))2+βd(x,S(x,y))d(u,T(u,v))d(x,T(u,v))+d(u,S(x,y))+d(x,u)+d(y,v),d(x,T(u,v))+d(u,S(x,y))d(x,u)d(y,v)ifD≠00,(x,T(u,v))+d(u,S(x,y))d(xu)d(y,v)ifD=0
for all x,y,u,v∈X, where D=d(x,T(u,y))+d(u,S(x,y))+d(x,u)+d(y,v) and α,β are nonnegative reals with α+β<1. Then S and T have a unique common coupled fixed point.
Proof.
Let x0 and y0 be arbitrary points in X. Define x2k+1=S(x2k,y2k), y2k+1=S(y2k,x2k) and x2k+2=T(x2k+1,y2k+1), y2k+2=T(y2k+1,x2k+1), for k=0,1,….
Now, we assume that
(32)DS(x2k,y2k)=d(x2k,T(x2k+1,y2k+1))+d(x2k+1,S(x2k,y2k))+d(x2k,x2k+1)+d(y2k,y2k+1)=d(x2k,x2k+2)+d(x2k,x2k+1)+d(y2k,y2k+1)≠0,DS(y2k,x2k)=d(y2k,T(y2k+1,x2k+1))+d(y2k+1,S(y2k,x2k))+d(x2k,x2k+1)+d(y2k,y2k+1)=d(y2k,y2k+2)+d(x2k,x2k+1)+d(y2k,y2k+1)≠0.
Then,
(33)d(x2k+1,x2k+2)=d(S(x2k,y2k),T(x2k+1,y2k+1))⪯α(d(x2k,x2k+1)+d(y2k,y2k+1))2+βd(x2k,S(x2k,y2k))d(x2k+1,T(x2k+1,y2k+1))DS(x2k,y2k)=α(d(x2k,x2k+1)+d(y2k,y2k+1))2+(βd(x2k,x2k+1)d(x2k+1,x2k+2))×(d(x2k,x2k+2)+d(x2k,x2k+1)x2kx2k1+d(y2k,y2k+1))-1
which implies that
(34)|d(x2k+1,x2k+2)|≤α|d(x2k,x2k+1)+d(y2k,y2k+1)|2+(β|d(x2k,x2k+1)||d(x2k+1,x2k+2)|)×(|d(x2k,x2k+2)+d(x2k,x2k+1)x2k1x2k2+d(y2k,y2k+1)|)-1≤α|d(x2k,x2k+1)+d(y2k,y2k+1)|2+β|d(x2k,x2k+1)|
as
(35)|d(x2k+1,x2k+2)|≤|d(x2k+1,x2k)+d(x2k,x2k+2)+d(y2k,y2k+1)|.
Therefore,
(36)|d(x2k+1,x2k+2)|≤(α+2β)2|d(x2k,x2k+1)|+α2|d(y2k,y2k+1)|.
Similarly, one can easily prove that
(37)|d(y2k+1,y2k+2)|≤(α+2β)2|d(y2k,y2k+1)|+α2|d(x2k,x2k+1)|.
Now, if
(38)DT(x2k+1,y2k+1)=d(x2k+2,T(x2k+1,y2k+1))+d(x2k+1,S(x2k+2,y2k+2))+d(x2k+2,x2k+1)+d(y2k+2,y2k+1)=d(x2k+1,x2k+3)+d(x2k+2,x2k+1)+d(y2k+2,y2k+1)≠0,
we get
(39)d(x2k+2,x2k+3)=d(T(x2k+1,y2k+1),S(x2k+2,y2k+2))⪯α(d(x2k+2,x2k+1)+d(y2k+2,y2k+1))2+(βd(x2k+2,S(x2k+2,y2k+2))x2k+2x3×d(x2k+1,T(x2k+1,y2k+1)))×(DT(x2k+1,y2k+1))-1=α(d(x2k+2,x2k+1)+d(y2k+2,y2k+1))2+(βd(x2k+2,x2k+3)d(x2k+1,x2k+2))×(d(x2k+1,x2k+3)+d(x2k+2,x2k+1)x2k+2,x+d(y2k+2,y2k+1))-1,
which implies that
(40)|d(x2k+2,x2k+3)|≤α|α(d(x2k+2,x2k+1)+d(y2k+2,y2k+1))2|+(β|d(x2k+2,x2k+3)||d(x2k+1,x2k+2)|)×(|d(x2k+1,x2k+3)+d(x2k+2,x2k+1)x2k1x2k3+d(y2k+2,y2k+1)|)-1≤α|α(d(x2k+2,x2k+1)+d(y2k+2,y2k+1))2|+β|d(x2k+1,x2k+2)|
as
(41)|d(x2k+2,x2k+3)|≤|d(x2k+2,x2k+1)+d(x2k+1,x2k+3)x2k2x1+d(y2k+2,y2k+1)|.
Therefore,
(42)|d(x2k+2,x2k+3)|≤α|d(x2k+2,x2k+1)|2+α|d(y2k+2,y2k+1)|2+β|d(x2k+1,x2k+2)|=(α+2β)2|d(x2k+1,x2k+2)|+α2|d(y2k+1,y2k+2)|.
Similarly, if DT(y2k+1,x2k+1)≠0, one can easily prove that
(43)|d(y2k+2,y2k+3)|≤(α+2β)2|d(y2k+1,y2k+2)|+α2|d(x2k+1,x2k+2)|.
Adding the inequalities (36)–(43), we get
(44)|d(x2k+1,x2k+2)|+|d(y2k+1,y2k+2)|≤(α+β)(|d(x2k,x2k+1)|+|d(y2k,y2k+1)|).|d(x2k+2,x2k+3)|+|d(y2k+2,y2k+3)|≤(α+β)(|d(x2k+1,x2k+2)|+|d(y2k+1,y2k+2)|).
If h=(α+β)<1, then, from (44), we get
(45)|d(xn,xn+1)|+|d(yn,yn+1)|≤h(|d(xn-1,xn)|+|d(yn-1,yn)|)≤⋯≤hn(|d(x0,x1)|+|d(y0,y1)|).
Now if |d(xn,xn+1)|+|d(yn,yn+1)|=δn, then
(46)δn≤hδn-1≤⋯≤hnδ0.
Without loss of generality, we take m>n. Since 0≤h<1, so we get
(47)|d(xn,xm)|+|d(yn,ym)|≤|d(xn,xn+1)|+|d(yn,yn+1)|+⋯+|d(xm-1,xm)|+|d(ym-1,ym)|≤[hnδ0+hn+1δ0+⋯+hm-1δ0]≤∑i=nm-1hiδ0→0,asm,n→+∞.
This implies that {xn} and {yn} are Cauchy sequences in X. Since X is complete, so there exists x,y∈X such that xn→x and yn→y as n→+∞. We now show that x=S(x,y) and y=S(y,x). We suppose on the contrary that x≠S(x,y) and y≠S(y,x) so that 0≺d(x,S(x,y))=l1 and 0≺d(y,S(y,x))=l2; we would then have
(48)l1=d(x,S(x,y))⪯d(x,x2k+2)+d(x2k+2,S(x,y))⪯d(x,x2k+2)+d(T(x2k+1,y2k+1),S(x,y))⪯d(x,x2k+2)+α(d(x2k+1,x)+d(y2k+1,y))2+(βd(x2k+1,T(x2k+1,y2k+1))d(x,S(x,y)))×(d(x2k+1,S(x,y))+d(x,T(x2k+1,y2k+1))xk1yk+d(x2k+1,x)+d(y2k+1,y))-1⪯d(x,x2k+2)+α(d(x2k+1,x)+d(y2k+1,y))2+(βl1d(x2k+1,x2k+2))×(d(x2k+1,S(x,y))+d(x,x2k+2)xk1yk+d(x2k+1,x)+d(y2k+1,y)(d(x2k+1,S(x,y))+d(x,x2k+2)))-1,
so that
(49)|l1|≤|d(x,x2k+2)|+α2|d(x2k+1,x)+d(y2k+1,y)|+(β|l1||d(x2k+1,x2k+2)|)×(|d(x2k+1,S(x,y))+d(x,x2k+2)x,x2k2+d(x2k+1,x)+d(y2k+1,y)(|d(x2k+1,S(x,y))+d(x,x2k+2))|)-1.
By taking k→+∞, we get |d(x,S(x,y))|=0 which is a contradiction so that x=S(x,y). Now
(50)l2=d(y,S(y,x))⪯d(y,y2k+2)+d(y2k+2,S(y,x))⪯d(y,y2k+2)+d(T(y2k+1,x2k+1),S(y,x))⪯d(y,y2k+2)+α(d(y2k+1,y)+d(x2k+1,x))2+(βd(y2k+1,T(y2k+1,x2k+1))d(y,S(y,x)))×(d(y2k+1,S(y,x))+d(y,T(y2k+1,x2k+1))(y+1)+d(y2k+1,y)+d(x2k+1,x))-1⪯d(y,y2k+2)+α(d(y2k+1,y)+d(x2k+1,x))2+(βl2d(y2k+1,y2k+2))×(d(y2k+1,S(y,x))+d(y,y2k+2)d(y,)+d(y2k+1,y)+d(x2k+1,x)(d(y2k+1,S(y,x))+d(y,y2k+2)))-1,
which implies that
(51)|l2|≤|d(y,y2k+2)|+α2|d(y2k+1,y)d(x2k+1,x)|+(β|l2||d(y2k+1,y2k+2)|)×(|d(y2k+1,S(y,x))+d(y,y2k+2)(y2k)+d(y2k+1,y)+d(x2k+1,x)|d(y2k+1,S(y,x))+d(y,y2k+2))|)-1,
Which, on making k→+∞, gives us |d(y,S(y,x))|=0 which is a contradiction so that y=S(y,x). It follows similarly that x=T(x,y) and y=T(y,x). So we have proved that (x,y) is a common coupled fixed point of S and T. As in Theorem 10, the uniqueness of common coupled fixed point remains a consequence of contraction condition (31).
We have obtained the existence and uniqueness of a unique common coupled fixed point if
(52)DS(x2k,y2k),DS(y2k,x2k),DT(x2k+1,y2k+1),DT(y2k+1,x2k+1)≠0
for all k∈ℕ. Now, assume that DS(x2k,y2k)=0 for some k∈ℕ. From
(53)d(x2k,x2k+2)+d(x2k,x2k+1)+d(y2k,y2k+1)=0,
we obtain that x2k=x2k+1=x2k+2 and y2k=y2k+1. If DS(y2k,x2k)≠0, using (8), we deduce
(54)d(y2k+1,y2k+2)=d(S(y2k,x2k),T(y2k+1,x2k+1))=0.
That is, y2k+1=y2k+2 (this equality holds also if DS(y2k,x2k)=0). The equalities
(55)x2k=x2k+1=x2k+2,y2k=y2k+1=y2k+2,
ensure that (x2k+1,y2k+1) is a unique common coupled fixed point of S and T. The same holds if either DS(y2k,x2k)=0, DT(x2k+1,y2k+1)=0, or DT(y2k+1,x2k+1)=0.
From Theorem 13, if we assume α=0, we obtain the following corollary.
Corollary 14.
Let (X,d) be a complete complex-valued metric space, and let the self-mappings S,T:X×X→X satisfy
(56)d(S(x,y),T(u,v))⪯{βd(x,S(x,y))d(u,T(u,v))d(x,T(u,v))+d(u,S(x,y))+d(x,u)+d(y,v),d(x,T(u,v))+d(u,S(x,y))(x,u)(y,v)ifD≠00,d(x,T(u,v))+(u,S(x,y))(x,u)(y,v)ifD≠0
for all x,y,u,v∈X, where D=d(x,T(u,y))+d(u,S(x,y))+d(x,y)+d(y,v) and β is a nonnegative real such that 0<β<1. Then S and T have a unique common coupled fixed point.
Corollary 15.
Let (X,d) be a complete complex-valued metric space, and let the mapping T:X×X→X satisfy
(57)d(T(x,y),T(u,v))⪯{α(d(x,u)+d(y,v))2+βd(x,T(x,y))d(u,T(u,v))d(x,T(u,v))+d(u,T(x,y))+d(x,u)+d(y,v),d(x,T(u,v))+d(u,T(x,y))d(x,u)d(y,v)ifD≠00,d(x,T(u,v))(u,T(x,y))cd(x,u)d(y,v)ifD=0
for all x,y,u,v∈X, where D=d(x,T(u,y))+d(u,T(x,y))+d(x,u)+d(y,v) and α,β are nonnegative reals with α+β<1. Then T has a unique coupled fixed point.
Corollary 16.
Let (X,d) be a complete complex-valued metric space, and let the mapping T:X×X→X satisfy
(58)d(Tn(x,y),Tn(u,v))⪯{α(d(x,u)+d(y,v))2+βd(x,Tn(x,y))d(u,Tn(u,v))d(x,Tn(u,v))+d(u,Tn(x,y))+d(x,u)+d(y,v),d(x,Tn(u,v))+d(u,Tn(x,y))+(x,u)d(y,v)ifD≠00,d(x,Tn(u,v))(u,Tn(x,y))ds(x,u)d(y,v)ifD=0
for all x,y,u,v∈X, where D=d(x,Tn(u,y))+d(u,Tn(x,y))+d(x,u)+d(y,v) and α,β are nonnegative reals with α+β<1. Then T has a unique coupled fixed point.
Now, we furnished a nontrivial example to support our main result (Theorem 10).
Example 17.
Let
(59)X1={z∈ℂ:Re(z)≥0,Im(z)=0},X2={z∈ℂ:Im(z)≥0,Re(z)=0},
and let X=X1∪X2. Consider a complex valued metric d:X×X→ℂ as follows:
(60)d(z1,z2)={23|x1-x2|+i2|x1-x2|,23|x1-x2|ifz1,z2∈X112|y1-y2|+i3|y1-y2|,23|x1-x2|ifz1,z2∈X229(x1+y2)+i6(x1+y2),23|xx2|ifz1∈X1,z2∈X2i3(x2+y1)+2i9(x2+y1),23|xx2|ifz1∈X2,z2∈X1,
with z1=x1+iy1 and z2=x2+iy2. Then (X,d) is a complex valued metric space. Define S,T:X×X→X as follows:
(61)S(z1,z2)={0+x1x24iifz1,z2∈X1y1y25+0iifz1,z2∈X20+x1y28iifz1∈X1andz2∈X2y1x29+0iifz1∈X2andz2∈X1,T(z1,z2)={0+x1x26iifz1,z2∈X1y1y27+0iifz1,z2∈X20+x1y210iifz1∈X1andz2∈X2y1x211+0iifz1∈X2andz2∈X1.
By a routine calculation, one can easily verify that the maps S and T satisfy the contraction condition (8) with α=3/4, β=1/15, and γ=2/15. Notice that the point (0,0) remains fixed under S and T and is indeed unique common coupled fixed point.
Conflict of Interests
The authors declare that they have no competing interests.
Authors’ Contribution
All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.
Acknowledgments
The authors thank the editor and the referees for their valuable comments and suggestions which improved greatly the quality of this paper. Marwan Amin Kutbi gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research. Cristina Di Bari is supported by Università degli Studi di Palermo (Local University Project R.S. ex 60%).
AzamA.FisherB.KhanM.Common fixed point theorems in complex valued metric spaces201132324325310.1080/01630563.2011.533046MR2748327ZBL1245.54036AhmadJ.ArshadM.VetroC.On a theorem of Khan in a generalized metric space20132013685272710.1155/2013/852727ArshadM.ShoaibA.BegI.Fixed point of a pair of contractive dominated mappings on a closed ball in an ordered complete dislocated metric space20132013, article 11510.1186/1687-1812-2013-115ArshadM.AhmadJ.KarapinarE.Some common fixed point results in rectangular metric spaces20132013730723410.1155/2013/307234ArshadM.AhmadJ.On multivalued contractions in cone metric spaces without normalityThe Scientific World Journal. In pressArshadM.KarapinarE.AhmadJ.Some unique fixed point theorem for rational contractions in partially ordered metric spaces20132013, article 24810.1186/1029-242X-2013-248AydiH.KarapınarE.ShatanawiW.Tripled fixed point results in generalized metric spaces201220121031427910.1155/2012/314279MR2923373ZBL1244.54085AydiH.KarapınarE.VetroC.Meir-Keeler type contractions for tripled fixed points20123262119213010.1016/S0252-9602(12)60164-7MR2989401AydiH.SametB.VetroC.Coupled fixed point results in cone metric spaces for ω~-compatible mappings20112011, article 2715MR2831139AzamA.ArshadM.Common fixed points of generalized contractive maps in cone metric spaces2009352255264MR2642938ZBL1201.47052di BariC.VetroP.φ-pairs and common fixed points in cone metric spaces200857227928510.1007/s12215-008-0020-9MR2452671ZBL1164.54031di BariC.VetroP.Weakly φ-pairs and common fixed points in cone metric spaces200958112513210.1007/s12215-009-0012-4MR2504991BhattS.ChaukiyalS.DimriR. C.Common fixed point of mappings satisfying rational inequality in complex valued metric space2011732159164MR2933952ZBL1246.54036HussainN.KhamsiM. A.LatifA.Banach operator pairs and common fixed points in hyperconvex metric spaces201174175956596110.1016/j.na.2011.05.072MR2833366ZBL1235.54037KutbiM. A.AhmadJ.HussainN.ArshadM.Common fixed point results for mappings with rational expressionsAbstract and Applied Analysis. In press10.1016/j.camwa.2010.03.062KarapınarE.Some nonunique fixed point theorems of Ćirić type on cone metric spaces201020101412309410.1155/2010/123094MR2652392ZBL1194.54064KarapınarE.Couple fixed point theorems for nonlinear contractions in cone metric spaces201059123656366810.1016/j.camwa.2010.03.062MR2651841ZBL1198.65097MongkolkehaC.SintunavaratW.KumamP.Fixed point theorems for contraction mappings in modular metric spaces20112011, article 9310.1186/1687-1812-2011-93MR2891973RouzkardF.ImdadM.Some common fixed point theorems on complex valued metric spaces20126461866187410.1016/j.camwa.2012.02.063MR2960809SintunavaratW.KumamP.Generalized common fixed point theorems in complex valued metric spaces and applications20122012, article 8410.1186/1029-242X-2012-84MR2922731SintunavaratW.ChoY. J.KumamP.Urysohn integral equations approach by common fixed points in complex valued metric spaces20132013, article 4910.1186/1687-1847-2013-49SintunavaratW.KumamP.Weak condition for generalized multi-valued (f,α,β)-weak contraction mappings201124446046510.1016/j.aml.2010.10.042MR2749727SintunavaratW.ChoY. J.KumamP.Common fixed point theorems for c-distance in ordered cone metric spaces20116241969197810.1016/j.camwa.2011.06.040MR2834821SintunavaratW.KumamP.Common fixed point theorems for generalized JH-operator classes and invariant approximations20112011, article 6710.1186/1029-242X-2011-67MR2837921SintunavaratW.KumamP.Common fixed points of f-weak contractors in cone metric spaces2012382293303MR3005062TahatN.AydiH.KarapinarE.ShatanawiW.Common fixed points for single-valued and multi-valued maps satisfying a generalized contraction in G-metric spaces20122012, article 4810.1186/1687-1812-2012-48Gnana BhaskarT.LakshmikanthamV.Fixed point theorems in partially ordered metric spaces and applications20066571379139310.1016/j.na.2005.10.017MR2245511ZBL1106.47047SametB.KarapinarE.AydiH.CojbasicV.Discussion on some coupled fixed point theorems20132013, article 5010.1186/1687-1812-2013-50SametB.VetroC.Coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces201174124260426810.1016/j.na.2011.04.007MR2803028ZBL1216.54021