ISSN: 2008-6822 (electronic)
http://dx.doi.org/10.22075/ijnaa.2017.521
Coupled coincidence point and common coupled
fixed point theorems in complex valued metric
spaces
Fayyaz Rouzkarda,∗, Mohammad Imdadb
aFarhangian University, Shariati Pardis, Sari, Mazandaran Iran
bDepartment of Mathematics, Aligarh Muslim University, Aligarh, 202002, India (Communicated by A. Ebadian)
Abstract
In this paper, we introduce the concept of a w-compatible mappings and utilize the same to discuss the ideas of coupled coincidence point and coupled point of coincidence for nonlinear contractive mappings in the context of complex valued metric spaces besides proving existence theorems which are following by corresponding unique coupled common fixed point theorems for such mappings. Some illustrative examples are also given to substantiate our newly proved results.
Keywords: Common fixed point; Contractive type mapping; coupled coincidence point; coupled point of coincidence; Complex valued metric space.
2000 MSC: Primary 47H10; Secondary 54H25.
1. Introduction and preliminaries
The axiomatic development of a metric space was essentially carried out by French mathematician Frechet in the year 1906. The utility of metric spaces in the natural growth of Functional Analysis is enormous. Inspired from the impact of this natural idea to mathematics in general and to Functional Analysis in particular, several researchers attempted various generalizations of this notion in the recent past such as: rectangular metric spaces, semi metric spaces, quasi metric spaces, quasi semi metric spaces, pseudo metric spaces, probabilistic metric spaces, 2-metric spaces, D-metric spaces,
∗Corresponding author
Email addresses: fayyazrouzkard@gmail.com, fayyaz_rouzkard@yahoo.com(Fayyaz Rouzkard),
mhimdad@yahoo.co.in(Mohammad Imdad)
G-metric spaces, , K-metric spaces, Cone metric spaces etc and by now there exists considerable literature on all these generalizations of metric spaces (see [8]-[5]).
Most recently, Azam et al. [1] and Fayyaz et al. [12] studied complex valued metric spaces wherein some fixed point theorems for mappings satisfying a rational inequality were established. Naturally, this new idea can be utilized to define complex valued normed spaces and complex valued inner product spaces which, in turn, offer a lot of scopes for further investigation. 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 cannot be generalized to cone metric spaces. Indeed the definition of a cone metric space banks on the underlying Banach space which is not a division Ring . However, in complex valued metric spaces, we can study improvements of a host of results of analysis involving divisions.
In this paper we prove common fixed point theorems involving two pairs of weakly compatible mappings satisfying certain rational expressions in complex valued metric space.
In what follows, we recall some notations and definitions that will be utilized in our subsequent discussion. Let C be the set of complex numbers and z1, z2 ∈ C. Define a partial order - on C as
follows:
z1 -z2 if and only if Re(z1)≤Re(z2), Im(z1)≤Im(z2).
Consequently, one can infer that z1 -z2 if 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 write z1 z2 if z1 6= z2 and one of (i), (ii), and (iii) is satisfied and we write
z1 ≺z2 if only (iii) is satisfied. Notice that 0-z1 z2 ⇒ |z1|<|z2|,andz1 -z2, z2 ≺z3 ⇒z1 ≺z3.
Definition 1.1. [1] Let X be a nonempty set whereas C be the set of complex numbers. Suppose that the mappingd:X×X →C, satisfies the following conditions:
(d1). 0-d(x, y), for all x, y ∈X and d(x, y) = 0 if and only if x=y;
(d2). d(x, y) =d(y, x) for all x, y ∈X;
(d3). 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 called a complex valued metric space.
Example 1.2. [12]. Let X =C be a set of complex number. Define d:C×C→C, by
d(z1, z2) =|x1−x2|+i|y1−y2|
where z1 =x1+iy1 and z2 =x2+iy2. Then (X, d) is a complex valued metric space.
Example 1.3. [14] Let X =C be a set of complex number. Define d:C×C→C,by
d(z1, z2) = eik|z1−z2|
where 0≤k ≤ π
2, z1 =x1+iy1 and z2 =x2+iy2. Then (X, d) is a complex valued metric space.
(i) b∈B is called an interior point of a set B whenever there is 0≺r∈C such that
N(b, r)⊆B
where N(b, r) ={y ∈X :d(b, y)≺r}.
(ii) A point x∈X is called a limit point of B whenever for every 0≺r∈C,
N(x, r)∩(B\X)6=∅.
(iii) A subset A⊆X is called open whenever each element of A is an interior point of A.
(iv) A subset B ⊆X is called closed whenever each limit point of B belongs to B.
(v)The family
F ={N(x, r) :x∈X,0≺r}
is a sub-basis for a topology on X. We denote this complex topology by τc. Indeed, the topology τc
is Hausdorff.
Definition 1.5. [1] Let (X, d) be a complex valued metric space and {xn}n≥1 be a sequence in X
and x∈X. We say that
(i) the sequence{xn}n≥1 converges to x if for every c ∈ C, with 0 ≺ c there is n0 ∈ N such that
for all n > n0, d(xn, x)≺c, . We denote this by limnxn=x,or xn→x, asn → ∞,
(ii) the sequence{xn}n≥1 is Cauchy sequence if for every c ∈C with 0 ≺ c there is n0 ∈ N such
that for all n > n0, d(xn, xn+m)≺c,
(iii) the metric space (X, d) is a complete complex valued metric space If every Cauchy sequence is convergent.
In [1], Azam et al. established the following two lemmas.
Lemma 1.6. 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)| →0 as n→ ∞.
Lemma 1.7. 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)| →0 as n→ ∞.
2. Main results
Bhashkar et al. [2] introduced the concept of coupled fixed point of a mapping F : X ×X → X
Definition 2.1. [2] An element (x, y) ∈ X ×X is called a coupled fixed point of mapping F :
X×X →X if x=F(x, y) and y=F(y, x).
Inspired with Definition 2.1 in the following we introduce the concept of a coupled fixed point of a mapping F :X×X→X.
Definition 2.2. An element (x, y)∈X×X is called
(g1) a coupled coincidence point of mappingF :X×X →X and g :X →X if g(x) =F(x, y) and
g(y) =F(y, x), and (gx, gy) is called coupled point of coincidence;
(g2) a common coupled fixed point of mappingsF :X×X →X andg :X →Xifx=g(x) = F(x, y) and y=g(y) =F(y, x).
Note that ifgis the identity mapping, then Definition 2.2 reduces to Definition 2.1. We introduce the following definition.
Definition 2.3. The mappings F : X ×X → X and g : X → X are called w−compatible if
g(F(x, y)) = F(gx, gy) whenever g(x) =F(x, y) and g(y) =F(y, x).
Definition 2.4. [10] Let (X,) be a partially ordered set and letF :X×X →X andg :X →X.
The mapping F is said to have a mixed g-monotone property if F is monotone g-nondecreasing in its first argument and monotone g-nonincreasing in its second argument, that is, for any x, y ∈X,
x1, x2 ∈X, gx1 gx2 ⇒F(x1, y)F(x2, y)
and, y1, y2 ∈X, gy1 gy2 ⇒F(x, y2)F(x, y1).
Now we prove our main result.
Theorem 2.5. Let (X, d) be a complex valued metric space, F : X ×X → X and g : X → X be mappings satisfying
d(F(x, y), F(u, v))-a1d(gx, gu) +a2d(gy, gv)
+a3d(F(x, y), gx) +a4d(F(x, y), gu) 1 +d(u, v)
+a5d(F(u, v), gu) +a6d(F(u, v), gx) 1 +d(x, y) ,
(2.1)
for all x, y, u, v ∈X, where ai, i= 1,2, . . . ,6 are nonnegative real numbers such that
P6
i=1ai <1. If
F(X×X)⊆g(X)and g(X)is complete subset of X,then F and g have a coupled coincidence point in X.
Proof . Letx0, y0 be any two arbitrary point in X.Setg(x1) = F(x0, y0) and g(y1) = F(y0, x0),this
can be done because F(X ×X)⊆g(X). Continuing this process we obtain two sequence {xn} and {yn} inX such that
From 2.1, we have
d(gxn, gxn+1) =d(F(xn−1, yn−1), F(xn, yn))
-a1d(gxn−1, gxn) +a2d(gyn−1, gyn)
+ a3d(F(xn−1, yn−1), gxn−1) +a4d(F(xn−1, yn−1), gxn) 1 +d(xn, yn)
+ a5d(F(xn, yn), gxn) +a6d(F(xn, yn), gxn−1) 1 +d(xn−1, yn−1)
=a1d(gxn−1, gxn) +a2d(gyn−1, gyn)
+ a3d(gxn, gxn−1) +a4d(gxn, gxn) 1 +d(xn, yn)
+ a5d(gxn+1, gxn) +a6d(gxn+1, gxn−1) 1 +d(xn−1, yn−1)
-a1d(gxn−1, gxn) +a2d(gyn−1, gyn) +
a3d(gxn, gxn−1)
1 +d(xn, yn)
+ a5d(gxn+1, gxn) +a6d(gxn+1, gxn) +a6d(gxn, gxn−1) 1 +d(xn−1, yn−1)
.
Therefore
d(gxn, gxn+1)-a1d(gxn−1, gxn) +a2d(gyn−1, gyn) +
a3d(gxn, gxn−1)
1 +d(xn, yn)
+a5d(gxn+1, gxn) +a6d(gxn+1, gxn) +a6d(gxn, gxn−1) 1 +d(xn−1, yn−1)
so that
|d(gxn, gxn+1)| ≤a1|d(gxn−1, gxn)|+a2|d(gyn−1, gyn)|+
a3|d(gxn, gxn−1)| |1 +d(xn, yn)|
+a5|d(gxn+1, gxn)|+a6|d(gxn+1, gxn)|+a6|d(gxn, gxn−1)|
|1 +d(xn−1, yn−1)|
since |1 +d(xn, yn)| ≥1 and |1 +d(xn−1, yn−1)| ≥1,therefore
|d(gxn, gxn+1)| ≤a1|d(gxn−1, gxn)|+a2|d(gyn−1, gyn)|+a3|d(gxn, gxn−1)|
+a5|d(gxn+1, gxn)|+a6|d(gxn+1, gxn)|+a6|d(gxn, gxn−1)|
= (a1+a3 +a6)|d(gxn, gxn−1)|+ (a5+a6)|d(gxn+1, gxn)|
+a2|d(gyn−1, gyn)|
from which it follows
(1−a5 −a6)|d(gxn, gxn+1)| ≤(a1+a3+a6)|d(gxn, gxn−1)|+a2|d(gyn−1, gyn)|. (2.2)
Similarly, one can prove that
Because of the symmetry in 2.1,
d(gxn+1, gxn) = d(F(xn, yn), F(xn−1, yn−1)) -a1d(gxn, gxn−1) +a2d(gyn, gyn−1)
+a3d(F(xn, yn), gxn) +a4d(F(xn, yn), gxn−1) 1 +d(xn−1, yn−1)
+a5d(F(xn−1, yn−1), gxn−1) +a6d(F(xn−1, yn−1), gxn) 1 +d(xn, yn)
=a1d(gxn, gxn−1) +a2d(gyn, gyn−1)
+a3d(gxn+1, gxn) +a4d(gxn+1, gxn−1) 1 +d(xn−1, yn−1)
+a5d(gxn, gxn−1) +a6d(gxn, gxn) 1 +d(xn, yn)
-a1d(gxn, gxn−1) +a2d(gyn, gyn−1)
+a3d(gxn+1, gxn) +a4d(gxn+1, gxn) +a4d(gxn, gxn−1) 1 +d(xn, yn)
+a5d(gxn, gxn−1) 1 +d(xn, yn)
.
So
|d(gxn+1, gxn)| ≤a1|d(gxn, gxn−1)|+a2|d(gyn, gyn−1)|+a3|d(gxn+1, gxn)|
+a4|d(gxn+1, gxn)|+a4|d(gxn, gxn−1)|+a5|d(gxn, gxn−1)|.
From which it follows
(1−a3 −a4)|d(gxn+1, gxn)| ≤(a1+a4+a5)|d(gxn−1, gxn)|+a2|d(gyn, gyn−1)|. (2.4)
Similarly, we can prove that
(1−a3 −a4)|d(gyn+1, gyn)| ≤(a1+a4+a5)|d(gyn−1, gyn)|+a2|d(gxn, gxn−1)|. (2.5)
Letδn=|d(gxn, gxn+1)|+|d(gyn, gyn+1)|.Now, from 2.2 and 2.3 respectively 2.4 and 2.5 we obtain:
(1−a5 −a6)δn ≤(a1+a2+a3+a6)δn−1. (2.6)
(1−a3 −a4)δn ≤(a1+a2+a4+a5)δn−1. (2.7)
Finally, from 2.6 and 2.7 we have
(2−a3 −a4−a5−a6)δn≤(2a1+ 2a2+a3+a4+a5+a6)δn−1,
that is,
δn ≤ηδn−1, η=
2a1+ 2a2+a3+a4 +a5+a6
2−a3−a4−a5−a6
<1. (2.8)
Consequently, we have
Ifδ0 = 0 then (x0, y0) is a coupled coincidence point of F and g.So let 0< δ0. If m > n, we have
d(gxn+m, gxn)-d(gxn+m, gxn+m−1) +d(gxn+m−1, gxn+m−2) +· · ·+d(gxn−1, gxn),
and
d(gyn+m, gyn)-d(gyn+m, gyn+m−1) +d(gyn+m−1, gyn+m−2) +· · ·+d(gyn−1, gyn),
so that
|d(gxn+m, gxn)| ≤ |d(gxn+m, gxn+m−1)|+|d(gxn+m−1, gxn+m−2)|+· · ·+|d(gxn−1, gxn)|,
and
|d(gyn+m, gyn)| ≤ |d(gyn+m, gyn+m−1)|+|d(gyn+m−1, gyn+m−2)|+· · ·+|d(gyn−1, gyn)|.
Therefore,
|d(gxn+m, gxn)|+|d(gyn+m, gyn)| ≤(|d(gxn+m, gxn+m−1)|+|d(gyn+m, gyn+m−1)|)
+ (|d(gxn+m−1, gxn+m−2)|+|d(gyn+m−1, gyn+m−2)|)
+· · ·+ (|d(gxn−1, gxn)|+|d(gyn−1, gyn)|),
that is,
|d(gxn+m, gxn)|+|d(gyn+m, gyn)| ≤δn+m−1+δn+m−2+· · ·+δn ≤(ηn+m−1+ηn+m−2+· · ·+ηn)δ0
= η
n
1−ηδ0 →0, as n→ ∞,
hence |d(gxn+m, gxn)| → 0 and |d(gyn+m, gyn)| → 0 as n → ∞. Thus by Lemma 1.7, {gxn} and {gyn} are Cauchy sequence in g(X). Since g(X) is complete subset of X, so there exists x and y in
X such that gxn →gx and gyn →gy. Now, we prove thatF(x, y) =gx and F(y, x) =gy. For that,
we have
d(F(x, y), gx)-d(F(x, y), gxn+1) +d(gxn+1, gx)
=d(F(x, y), F(xn, yn)) +d(gxn+1, gx)
-a1d(gx, gxn) +a2d(gy, gyn) +
a3d(F(x, y), gx) +a4d(F(x, y), gxn)
1 +d(xn, yn)
+a5d(F(xn, yn), gxn) +a6d(F(xn, yn), gx)
1 +d(x, y) +d(gxn+1, gx)
=a1d(gx, gxn) +a2d(gy, gyn) +
a3d(F(x, y), gx) +a4d(F(x, y), gxn)
1 +d(xn, yn)
+a5d(gxn+1, gxn) +a6d(gxn+1, gx)
1 +d(x, y) +d(gxn+1, gx)
-a1d(gx, gxn) +a2d(gy, gyn) +
a3d(F(x, y), gx) +a4d(F(x, y), gx) +a4d(gx, gxn)
1 +d(xn, yn)
+a5d(gxn+1, gx) +a5d(gx, gxn) +a6d(gxn+1, gx)
so that
|d(F(x, y), gx)| ≤a1|d(gx, gxn)|+a2|d(gy, gyn)|+a3|d(F(x, y), gx)|
+a4|d(F(x, y), gx)|+a4|d(gx, gxn)|+a5|d(gxn+1, gx)|+a5|d(gx, gxn)|
+a6|d(gxn+1, gx)|+|d(gxn+1, gx)|,
which further implies that,
|d(F(x, y), gx)| ≤ a1+a4+a5
1−a3−a4
|d(gxn, gx)|+
a5+a6
1−a3−a4
|d(gxn+1, gx)|
+ a2
1−a3−a4
|d(gyn, gy)|.
(2.10)
Sincegxn→gxandgyn→gyasn → ∞,then,|d(F(x, y), gx)|= 0,it follows thatd(F(x, y), gx) = 0,
and hence F(x, y) = gx. Similarly, we can prove F(y, x) = gy. Hence (x, y) is coupled coincidence point of the mappings F and g.
By setting a1 = a2 =α, a3 =a4 = β and a5 = a6 =γ in Theorem 2.5, we deduce the following
corollary.
Corollary 2.6. Let (X, d) be a complex valued metric space, F : X ×X → X and g : X → X be mappings satisfying
d(F(x, y), F(u, v))-α[d(gx, gu) +d(gy, gv)] +βd(F(x, y), gx) +d(F(x, y), gu)
1 +d(u, v)
+γd(F(u, v), gu) +d(F(u, v), gx)
1 +d(x, y) ,
(2.11)
for all x, y, u, v ∈ X, where α, β and γ are nonnegative real numbers such that α+β +γ < 12. If
F(X×X)⊆g(X)and g(X)is complete subset of X,then F and g have a coupled coincidence point in X.
Now, we present two examples showing that Theorem 2.5 is a proper extension of known results.
Example 2.7. Let X = [0,∞).The mapping d:X×X →C defined as:
d(x, y) =|x−y|i.
Clearly, (X, d) is a complete complex valued metric space. We define the functionsF :X×X →X
and g :X →X by
g(x) = 3x and F(x, y) = x 5 +
y
5.
It is easy to verify that F and g satisfy all the conditions of Theorem 2.5, takinga1 =a2 = 101, a3 =
a4 = 121 and a5 =a6 = 141. Moreover (0,0), is common coupled coincidence point ofF and g.
Example 2.8. LetX = [0,∞) andd:X×X →Cis a mapping defined by: d(x, y) =|x−y|+|x−y|i.
Clearly, (X, d) is a complete complex valued metric space. We define the functionsF :X×X →X
and g :X →X by
g(x) = 3x and F(x, y) =x+ |siny| 5 .
It is easy to verify that F and g satisfy all the conditions of Theorem 2.5, taking a1 = 23, a2 = 1
9 and a3 =a4 =a5 =a6 = 1
Theorem 2.9. Let (X, d)be a complex valued metric space, F :X×X →X andg :X →X be two mappings which satisfy all the conditions of Theorem 2.5. If F and g arew−compatible, then F and
g have unique common coupled fixed point. Moreover, common fixed point of F and g has the form
(u, u) for someu∈X.
Proof . First we claim that coupled point of coincidence is unique. suppose that (x, y),(x∗, y∗) ∈
X×X with g(x) =F(x, y), g(y) =F(y, x), and g(x∗) =F(x∗, y∗), g(y∗) =F(y∗, x∗).Using 2.1, we get
d(gx, gx∗) =d(F(x, y), F(x∗, y∗))
-a1d(gx, gx∗) +a2d(gy, gy∗)
+ a3d(F(x, y), gx) +a4d(F(x, y), gx
∗)
1 +d(x∗, y∗)
+ a5d(F(x
∗, y∗), gx∗) +a
6d(F(x∗, y∗), gx)
1 +d(x, y)
=a1d(gx, gx∗) +a2d(gy, gy∗) +
a4d(gx, gx∗)
1 +d(x∗, y∗) +
a6d(gx∗, gx)
1 +d(x, y) ,
so that
|d(gx, gx∗)| ≤a1|d(gx, gx∗)|+a2|d(gy, gy∗)|+|
a4d(gx, gx∗)
1 +d(x∗, y∗)|+|
a6d(gx∗, gx)
1 +d(x, y) |,
and
|d(gx, gx∗)| ≤a1|d(gx, gx∗)|+a2|d(gy, gy∗)|+a4|d(gx, gx∗)|+a6|d(gx∗, gx).
Therefore
|d(gx, gx∗)| ≤(a1+a4+a6)|d(gx, gx∗)|+a2|d(gy, gy∗)|. (2.12)
Similarly, one gets
|d(gy, gy∗)| ≤(a1+a4+a6)|d(gy, gy∗)|+a2|d(gx, gx∗)|. (2.13)
Thus
|d(gx, gx∗)|+|d(gy, gy∗)| ≤(a1+a2+a4+a6)(|d(gx, gx∗)|+|d(gy, gy∗)|).
Since a1 +a2 +a4 +a6 < 1, therefore we have |d(gx, gx∗)|+|d(gy, gy∗)| = 0, which implies that
gx = gx∗ and gy =gy∗. Similarly we can prove that gx =gy∗ and gy =gx∗. Therefore (gx, gy) is unique coupled point of coincidence ofF andg.Now, letg(x) = u.Then we haveu=g(x) =F(x, x).
Byw−compatibility of F and g, we have
g(u) = g(g(x)) = g(F(x, x)) =F(gx, gx) =F(u, u).
Then (gu, gu) is coupled point of coincidence ofF and g.Consequentlygu=gx.Thereforeu=gu=
Theorem 2.10. Let (X, d) be a complex valued metric space, F : X×X → X and g : X → X be
w−compatible mappings such that
d(F(x, y), F(u, v))kU +mV (2.14)
for all x, y, u, v ∈X, where,
U, V ∈Su,vx,y ={d(gx, gu), d(gy, gv),d(F(x, y), gx)
1 +d(u, v) ,
d(F(x, y), gu) 1 +d(u, v) ,
d(F(u, v), gu) 1 +d(x, y) },
andk, m are nonnegative real numbers such thatk+m <1. If F(X×X)⊆g(X)than F andg have unique common coupled fixed point having the form (u, u) for some u∈X.
Proof . Following similar arguments to those given in Theorem 2.5, we construct two sequences
{xn} and {yn} inX such that
g(xn+1) =F(xn, yn) and g(yn+1) =F(yn, xn).
Now, from 2.14, we have
d(gxn, gxn+1) =d(F(xn−1, yn−1), F(xn, yn))kU+mV, (2.15)
where U, V ∈Sxn−1,yn−1 xn,yn , and
d(gyn, gyn+1) =d(F(yn−1, xn−1), F(yn, xn))kU+mV, (2.16)
where U, V ∈Syn−1,xn−1 yn,xn .
We have the following 15 cases:
(i) :U =d(gxn−1, gxn) and V =d(gxn−1, gxn).
(ii) :U =d(gxn−1, gxn) and V =d(gyn−1, gyn).
(iii) :U =d(gxn−1, gxn) and V = d(F(x1+n−d1,y(gxnn−,gy1),gxn)n−1).
(iv) :U =d(gxn−1, gxn) and V =
d(F(xn−1,yn−1),gxn)
1+d(gxn,gyn) .
(v) :U =d(gxn−1, gxn) and V = 1+d(dF((gxxnn,y−n1,gy),gxn−n)1).
(vi) :U =d(gyn−1, gyn) and V =d(gyn−1, gyn).
(vii) :U =d(gyn−1, gyn) and V =
d(F(xn−1,yn−1),gxn−1) 1+d(gxn,gyn) .
(viii) :U =d(gyn−1, gyn) and V = d(F1+(xnd−(1,ygxnn,gy−1)n,gx) n).
(ix) :U =d(gyn−1, gyn) and V = 1+d(dF((gxxn,yn),gxn)
n−1,gyn−1).
(x) :U = d(F(xn−1,yn−1),gxn−1)
1+d(gxn,gyn) and V =
d(F(xn−1,yn−1),gxn−1) 1+d(gxn,gyn) .
(xi) :U = d(F(xn−1,yn−1),gxn−1)
1+d(gxn,gyn) and V =
d(F(xn−1,yn−1),gxn)
(xii) :U = d(F(xn−1,yn−1),gxn−1)
1+d(gxn,gyn) and V =
d(F(xn,yn),gxn)
1+d(gxn−1,gyn−1).
(xiii) :U = d(F(xn−1,yn−1),gxn)
1+d(gxn,gyn) and V =
d(F(xn−1,yn−1),gxn)
1+d(gxn,gyn) .
(xiv) :U = d(F(xn−1,yn−1),gxn)
1+d(gxn,gyn) and V =
d(F(xn,yn),gxn)
1+d(gxn−1,gyn−1).
(xv) :U = d(F(xn,yn),gxn)
1+d(gxn−1,gyn−1) and V =
d(F(xn,yn),gxn)
1+d(gxn−1,gyn−1).
In case (i),according to 2.15 and 2.16 we get
d(gxn, gxn+1) =d(F(xn−1, yn−1), F(xn, yn))(k+m)d(gxn−1, gxn),
and
d(gyn, gyn+1) =d(F(yn−1, xn−1), F(yn, xn))(k+m)d(gyn−1, gyn).
Therefore,
|d(gxn, gxn+1)| ≤(k+m)|d(gxn−1, gxn)|,
and
|d(gyn, gyn+1)| ≤(k+m)|d(gyn−1, gyn)|.
Hence,
(|d(gxn, gxn+1)|+|d(gyn, gyn+1)|)≤(k+m)(|d(gxn−1, gxn)|+|d(gyn−1, gyn)|),
and also, in the cases (ii),(iii),(vi),(vii),(x),(xiii) and (xv) according to 2.15 and 2.16 we obtain that
(|d(gxn, gxn+1)|+|d(gyn, gyn+1)|)≤(k+m)(|d(gxn−1, gxn)|+|d(gyn−1, gyn)|).
Similarly, in the cases (ix),(xii) and (xiv), according to 2.15 and 2.16 we again obtain that
(|d(gxn, gxn+1)|+|d(gyn, gyn+1)|)≤
k
1−m(|d(gxn−1, gxn)|+|d(gyn−1, gyn)|),
and, in the cases (iv),(v),(viii) and (xi) we have
(|d(gxn, gxn+1)|+|d(gyn, gyn+1)|)≤k(|d(gxn−1, gxn)|+|d(gyn−1, gyn)|).
Thus we conclude that
(|d(gxn, gxn+1)|+|d(gyn, gyn+1)|)≤β(|d(gxn−1, gxn)|+|d(gyn−1, gyn)|),
for some β∈ {k,(k+m),1−km}(β <1) and for alln ≥1.
Following similar arguments to those given in Theorem 2.5, (x, y) is common coupled coincidence point of F and g, where gxn →gx and gyn → gy. Now we prove that coupled point of coincidence
Suppose that (x, y),(x∗, y∗) ∈ X ×X such that g(x) = F(x, y), g(y) = F(y, x) and g(x∗) =
F(x∗, y∗), g(y∗) =F(y∗, x∗).
By using 2.14 we get
d(gx, gx∗) =d(F(x, y), F(x∗, y∗))kU+mV (2.17)
where U, V ∈Sxx,y∗,y∗, and
d(gy, gy∗) = d(F(y, x), F(y∗, x∗))kU +mV, (2.18)
where U, V ∈Syy,x∗,x∗.
Again we have the following 15 cases: (i) :U =d(gx, gx∗) and V =d(gx, gx∗).
(ii) :U =d(gx, gx∗) and V =d(gy, gy∗).
(iii) :U =d(gx, gx∗) and V = 1+d(Fd((gxx,y∗),gy,gx∗)).
(iv) :U =d(gx, gx∗) and V = d1+(Fd((x,ygx∗),gx,gy∗∗)).
(v) :U =d(gx, gx∗) and V = d(F1+(xd∗(,ygx,gy∗),gx)∗).
(vi) :U =d(gy, gy∗) and V =d(gy, gy∗).
(vii) :U =d(gy, gy∗) and V = 1+d(Fd((gxx,y∗),gy,gx∗)).
(viii) :U =d(gy, gy∗) and V = d1+(Fd((x,ygx∗),gx,gy∗∗)).
(ix) :U =d(gy, gy∗) and V = d(F1+(xd∗(,ygx,gy∗),gx)∗).
(x) :U = 1+d(Fd((gxx,y∗),gy,gx∗)) and V =
d(F(x,y),gx) 1+d(gx∗,gy∗).
(xi) :U = 1+d(Fd((gxx,y∗),gy,gx∗)) and V =
d(F(x,y),gx∗) 1+d(gx∗,gy∗).
(xii) :U = 1+d(Fd((gxx,y∗),gy,gx∗)) and V =
d(F(x∗,y∗),gx∗)
1+d(gx,gy) .
(xiii) :U = d1+(Fd((x,ygx∗),gx,gy∗∗)) and V =
d(F(x,y),gx∗)
1+d(gx∗,gy∗).
(xiv) :U = d1+(Fd((x,ygx∗),gx,gy∗∗)) and V =
d(F(x∗,y∗),gx∗) 1+d(gx,gy) .
(xv) :U = d(F1+(xd∗(,ygx,gy∗),gx)∗) and V = d(F1+(xd∗(,ygx,gy∗),gx)∗).
In the cases (i),(ii),(iv),(vi),(viii) and (xiii) according to 2.17 and 2.18, we obtain that
|d(gx, gx∗)|+|d(gy, gy∗)| ≤(k+m)(|d(gx, gx∗)|+|d(gy, gy∗)|). (2.19)
Similarly, from 2.17 and 2.18, in the cases (iii),(v),(vii),(ix),(x),(xi),(xii),(xiv) and (xv) we again obtain that
where α ∈ {0, k, m} (α < 1). Since k+m < 1 and α < 1, therefore from 2.19 and 2.20, we have
|d(gx, gx∗)|+|d(gy, gy∗)| = 0, i.e. gx = gx∗ and gy =gy∗. That is, (gx, gx) is the unique common coupled point of coincidence. SinceF and g are w−compatible maps, we have,
g(g(x)) = F(gx, gx). (2.21)
Letu =g(x). By 2.21, we have g(u) =F(u, u). Therefore (gu, gu) is a coupled point of coincidence of F and g. Consequently, u = g(u) = F(u, u). Hence (u, u) is unique common coupled fixed point of F and g.
Example 2.11. Let X = {(x,0) : x ∈ [0,∞)} ∪ {(0, x) : x ∈ [0,∞)}. Define d : X ×X → C by
d(x, y) = |x1 −y1|+|x2 −y2|i, where x = (x1, x2) and y = (y1, y2). Clearly, (X, d) is a complete
complex valued metric space. Consider mappingsF :X×X →X and g :X →X,given by
g(x) = (
(0,t); x= (t,0), t∈[0,∞)
(t,0); x= (0, t), t∈[0,∞) and F (x1, x2),(y1, y2)
= (x1 2 ,
x2
2).
Note that F and g satisfy all the conditions of Theorem 2.9, if we take a1 = a2 = a3 = a4 = a5 =
a6 = 121. Moreover (0,0) is the unique common coupled fixed point of F and g.
Next, we give a example which supports Theorems 2.5 and 2.9, while a lot of results in partially ordered metric spaces, cone metric spaces, ordered cone metric spaces and complex valued metric spaces for example the results of Nashine et al. [11] do not.
Example 2.12. LetX ={ix:x∈[0,1]}.Defined:X×X →Cbyd(x, y) = i|x−y|, where x, y ∈
X. Clearly, (X, d) is a complete complex valued metric space. Consider mappings F :X×X →X
and g :X →X,given by
F(x, y) =ix
4+y4
16 and g(x) =i
x4
4 .
Ify1 = 34iand y2 = 12i,then g(y2) =i 1 16
4 i 81 44
4 =g(y1), but for x=i,we get
F(x, y2) =F(i,
1 2i) = i
1 + 161 16 i
1 + 8144
16 =F(i, 3
4i) =F(x, y1).
So, the mappings F and g do not satisfy the mixed g-monotone property. Therefore, Theorems 3.1 and 3.2 of Nashine et al. [11] cannot be supported to reach this conclusion.
Now, we show that Theorems 2.5 and 2.9 can be used for this case.
d(F(x, y), F(u, v)) =i|ix
4+y4
16 −i
u4+v4
16 |= 1 4i|i
x4−u4
4 +i
y4−v4
4 |
1
4i|i
x4−u4
4 |+ 1 4i|i
y4−v4
4 |= 1 4i|
x4−u4
4 |+ 1 4i|
y4−v4
4 |
= 1
4d(g(x), g(u)) + 1
4d(g(y), g(v)),
where a1 =a2 = 14 and a3 =a4 =a5 =a6 = 0. Note thata1+a2+a3+a4+a5+a6 <1.
F(X×X) ={ix:x∈[0,1
and g(X) is a complete subset of X. Hence, the conditions of Theorem 2.5 are satisfied. Therefore,
F and g have a coupled coincidence point in X.
Since, g(0) =F(0,0) and
g(F(0,0)) =g(0) =F(0,0) =F(g(0), g(0)),
then F and g are W-compatible. Therefore, conditions of Theorem 2.9 are satisfied. Now, we can apply Theorem 2.9 to conclude the existence of a unique common coupled fixed point of F and g that is a point (0,0).
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