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@ Appl. Gen. Topol. 19, no. 2 (2018), 189-201doi:10.4995/agt.2018.7409
c© AGT, UPV, 2018

Fixed points and coupled fixed points in

partially ordered ν-generalized metric spaces

Mortaza Abtahi a, Zoran Kadelburg b and Stojan Radenović c

a
School of Mathematics and Computer Sciences, Damghan University, P.O.B. 36715-364,

Damghan, Iran (abtahi@du.ac.ir, mortaza.abtahi@gmail.com)
b

University of Belgrade, Faculty of Mathematics, Studentski trg 16, 11000 Beograd, Ser-

bia (kadelbur@matf.bg.ac.rs)
c
University of Belgrade, Faculty of Mechanical Engineering, Kraljice Marije 16, 11120 Beograd,

Serbia (radens@beotel.rs)

Communicated by M. Abbas

Abstract

New fixed point and coupled fixed point theorems in partially ordered

ν-generalized metric spaces are presented. Since the product of two

ν-generalized metric spaces is not in general a ν-generalized metric

space, a different approach is needed than in the case of standard metric

spaces.

2010 MSC: 47H10; 54H25.

Keywords: Meir-Keeler contractions; Ćirić-Matkowski contractions;
Proinov-type contractions; ν-generalized metric space; coupled

fixed point theorems.

1. Introduction and Preliminaries

Throughout the paper, the sets of integers, nonnegative integers, and posi-
tive integers are denoted, respectively, by Z, Z+, and N; the sets of real numbers
and nonnegative real numbers are denoted, respectively, by R and R+.

Received 15 March 2017 – Accepted 13 April 2018

http://dx.doi.org/10.4995/agt.2018.7409


M. Abtahi, Z. Kadelburg and S. Radenović

1.1. ν-generalized metric spaces. There are lots of works done on fixed
point theory by weakening the requirements of the Banach contraction princi-
ple. One direction of these generalizations was introduced by Branciari in [6],
where the triangle inequality was replaced by a so-called polygonal inequality.
In what follows, we briefly recall concepts of ν-generalized metric spaces. See
also [3, 8, 11, 19, 20].

Definition 1.1 (Branciari [6]). Let E be a nonempty set and ν ∈ N. A
mapping dν : E × E → R

+ is called a ν-generalized metric and the pair (E, dν)
is called a ν-generalized metric space if the following hold:

(1) dν(x, y) = 0 if and only if x = y;
(2) dν(x, y) = dν(y, x), for all x, y ∈ E;
(3) dν(x, y) ≤ dν(x, z1) + dν(z1, z2) + · · · + dν(zν, y), for each set

{x, z1, . . . , zν, y} of ν + 2 distinct elements of E.

Clearly, (E, dν) is a metric space if ν = 1, i.e., it is a 1-generalized metric
space. It is shown in [19], that the topology of a ν-generalized metric space
may be non-compatible.

Definition 1.2 ([3]). Let (E, dν) be a ν-generalized metric space. Given k ∈ N,
a sequence {xn} in E is said to be k-Cauchy if

lim
n→∞

sup{dν(xn, xn+1+mk) : m ∈ Z
+} = 0.

The sequence {xn} is called Cauchy if it is 1-Cauchy.

Cauchy sequences in ν-generalized metric spaces were investigated in [3, 6,
20].

Proposition 1.3 ([3, 20]). Let {xn} be a ν-Cauchy sequence with distinct
terms in (E, dν). If ν is odd, or if ν is even and dν(xn, xn+2) → 0 as n → ∞,
then {xn} is Cauchy.

As shown in [16] and [18], a sequence in a 2-generalized metric space may
converge to more than one point and a convergent sequence may not be a
Cauchy sequence. It is said [3, 20] that a sequence {xn} in E converges to x in
the strong sense if {xn} is Cauchy and {xn} converges to x. [18, Example 1.1]
shows that there exist convergent sequences that do not converge in the strong
sense. The completeness of ν-generalized metric spaces is investigated in [3].

Proposition 1.4 ([20]). Let {xn} and {yn} be sequences in (E, dν) that con-
verge to x and y in the strong sense, respectively. Then
dν(x, y) = limn→∞ dν(xn, yn).

Branciari proved in [6] a generalization of the Banach contraction principle.
Since, as was already said, a ν-generalized metric space does not necessarily
have a compatible topology, his proof needed some corrections, see [9, 16, 18,

19, 21]. Proofs of Kannan’s [10] and Ćirić’s [7] fixed point theorems in ν-
generalized metric spaces appear in [20]. The analogue of Proinov’s result from
[15], as an ultimate generalization of the Banach contraction principle in the
setting of ν-generalized metric spaces, was proved in [2].

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Coupled fixed points in ν-generalized metric spaces

1.2. Partially ordered spaces and coupled fixed points. Nieto and Ro-
dŕıguez-López initiated in [14] the use of another enrichment of metric space
structure by using additional partial order. A lot of researchers obtained sev-
eral results in such structures. Among them, Bhaskar and Lakshmikantham
started in [5] investigation of so-called coupled fixed points. They proved the
existence of coupled fixed points for contractive mappings in partially ordered
complete metric spaces. These and similar results were later obtained by dif-
ferent methods; see, e.g. [4, 12, 13, 17].

Assume that (E, �) is a partially ordered set and that F : E × E → E is
a mapping. The notions of a coupled fixed point of F and the (strict) mixed
monotone property has become standard, so we omit them here. Given a
pair (x, y) of elements in E, the Picard iterates {F n(x, y)} and {F n(y, x)} are
defined, inductively, as follows. Let F 0(x, y) = x, F 0(y, x) = y, and then, for
n ∈ Z+,

F n+1(x, y) = F
(

F n(x, y), F n(y, x)
)

,

F n+1(y, x) = F
(

F n(y, x), F n(x, y)
)

.
(1.1)

In 2012, Berinde and Păcurar [4] presented more general coupled fixed point
theorems in partially ordered metric spaces (E, �, d).

Theorem 1.5 ([4]). Let (E, �, d) be a complete partially ordered metric space,
and F : E ×E → E be a generalized symmetric Meir-Keeler type mapping, i.e.,
for every ǫ > 0, there exists δ > 0 such that, for x � u and y � v,

ǫ ≤ d(x, u) + d(y, v) <ǫ + δ =⇒

d(F(x, y), F(u, v)) + d(F(y, x), F(v, u)) < ǫ.

Suppose that

(i) the mapping F is continuous and has the mixed strict monotone prop-
erty,

(ii) there exist x0, y0 ∈ E such that
(1.2)

x0 � F(x0, y0), y0 � F(y0, x0), or x0 � F(x0, y0), y0 � F(y0, x0).

Then F has a coupled fixed point.

1.3. Fixed points of monotone contractions. Fixed point theorems of
Ćirić-Matkowski and Proinov types for monotone contractions in partially or-
dered ν-generalized metric spaces can be deduced from a sequence of lemmas
and propositions, similarly as it has been done in the setting of (ν-generalized)
metric spaces in [1] and [2]. Hence, we just state the respective formulations,
omitting the proofs.

Theorem 1.6. Let (E, �, dν) be a complete partially ordered ν-generalized

metric space and T : E → E be a monotone contraction of Ćirić-Matkowski
type, i.e.,

(1) the mapping T is nondecreasing,
(2) dν(T x, T y) < dν(x, y), for x ≺ y (that is x � y and x 6= y),

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M. Abtahi, Z. Kadelburg and S. Radenović

(3) for every ǫ > 0, there exists δ > 0 such that
(

x � y, ǫ < dν(x, y) < ǫ + δ
)

=⇒ dν(T x, T y) ≤ ǫ.

Then T has a fixed point provided there exists x0 ∈ E such that x0 � T x0.
Moreover, for any x ∈ E with x � T x, the sequence {T nx} converges to a fixed
point of T in the strong sense.

We also have a Proinov type fixed point theorem.

Theorem 1.7. Suppose that (E, �, dν) is a complete partially ordered ν-ge-
neralized metric space, and that T : E → E is sequentially continuous and
asymptotically regular, i.e.,

lim
n→∞

(

dν(T
nx, T n+1x) + dν(T

nx, T n+2x)
)

= 0, x ∈ E.

For γ > 0, define m(x, y) = dν(x, y) + γ
(

dν(x, T x) + dν(y, T y)
)

. Suppose that

dν(T x, T y) < m(x, y), x, y ∈ E, x ≺ y,

and that, for any ǫ > 0, there exist δ > 0 and N ∈ Z+ such that, for all
x, y ∈ E,

(1.3) x � y, m(T Nx, T Ny) < δ + ǫ =⇒ dν(T
N+1x, T N+1y) ≤ ǫ.

Then T has a fixed point provided there exists x0 ∈ E such that x0 � T x0.
Moreover, for any x ∈ E with x � T x, the sequence {T nx} converges to a fixed
point of T in the strong sense.

Remark 1.8. Similarly as in various other known fixed point results in ordered
spaces, the presented conditions are not sufficient to conclude that the fixed
point is unique. An additional assumption is needed, and this can be either
that arbitrary two elements of the fixed point set are comparable, or that there
exists a third element, comparable with both of them. We do not go into details
here, leaving them for the case of coupled fixed points in the next section.

1.4. Outline. The next (main) section is devoted to coupled fixed points, and
is divided into three parts. In Subsection 2.1, we let E be a partially ordered
metric space, and investigate the existence of coupled fixed points for different
types of symmetric contractions on E. Our technique in this section involves
considering induced metric and order on the set E = E × E and reducing a
symmetric contraction F : E×E → E to a monotone contraction T : E → E and
then applying results obtained in Section 1.3 to T . This technique appears in
several papers. However, we will show that this method is not applicable in the
case of partially ordered ν-generalized metric spaces (see further Example 2.2).
Hence, in Subsection 2.2, we shall take a different approach to obtain coupled
fixed point results in such spaces. Finally, in Subsection 2.3, we present a
brief discussion of the uniqueness of coupled fixed points. We conclude by an
illustrative example in the last subsection.

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Coupled fixed points in ν-generalized metric spaces

2. Coupled fixed points of symmetric contractions

In this section, we present coupled fixed point theorems for symmetric con-
tractions on (ν-generalized) metric spaces. We start with the following defini-

tion of symmetric contractions of Ćirić-Matkowski type.

Definition 2.1. Let (E, �, dν) be a partially ordered ν-generalized metric

space. A mapping F : E × E → E is called a symmetric contraction of Ćirić-
Matkowski type if

(1) for x � u and y � v, with (x, y) 6= (u, v),

(2.1) dν(F(x, y), F(u, v)) + dν(F(y, x), F(v, u)) < dν(x, u) + dν(y, v),

(2) for every ǫ > 0, there exists δ > 0 such that, for x � u and y � v,

ǫ < dν(x, u) + dν(y, v) < ǫ + δ =⇒

dν(F(x, y), F(u, v)) + dν(F(y, x), F(v, u)) ≤ ǫ.
(2.2)

To avoid repetitive writings and simplify calculations, the following conven-
tion seems to be convenient.

Convention. Let (E, �, dν) be a partially ordered (ν-generalized) metric space.
Set E = E × E and, for all elements x = (x, y) and u = (u, v) of E, define

x ⊑ u if and only if x � u and v � y.

Clearly, (E, ⊑) is a partially ordered set. Define ρν : E × E → R
+ by

(2.3) ρν(x, u) = dν(x, u) + dν(y, v), x = (x, y), u = (u, v).

Obviously, if (E, �, dν) is a (complete) metric space then (E, ⊑, ρν) is a (com-
plete) metric space. In general, however, as the following example shows, if E is
a ν-generalized metric space (ν ≥ 2) then (E, ρν) may fail to be a ν-generalized
metric space.

Example 2.2 ([8, Example 4.2]). Let E = {a, b, c} and define dν : E×E → R
+

by dν(a, b) = 4, dν(a, c) = dν(b, c) = 1, and dν(x, x) = 0, dν(x, y) = dν(y, x)
for all x, y ∈ E. Since four distinct points in E do not exist, the rectangular
inequality is trivially satisfied. Hence, (E, dν) is a 2-generalized metric space,
which is obviously not a metric space.

Now, consider the mappings ρ+, ρmax : E × E → R
+ defined by

ρ+(x, u) = dν(x, u) + dν(y, v),

ρmax(x, u) = max{dν(x, u), dν(y, v)},

where x = (x, y) and u = (u, v). Then, for the quadrilateral
{(a, b), (b, c), (a, c), (c, c)} in E, we have

ρ+
(

(a, b), (b, c)
)

= 5 > 1 + 1 + 1

= ρ+
(

(a, b), (a, c)
)

+ ρ+
(

(a, c), (c, c)
)

+ ρ+
(

(c, c), (b, c)
)

,

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M. Abtahi, Z. Kadelburg and S. Radenović

and

ρmax
(

(a, b), (b, c)
)

= 4 > 1 + 1 + 1

= ρmax
(

(a, b), (a, c)
)

+ ρmax
(

(a, c), (c, c)
)

+ ρmax
(

(c, c), (b, c)
)

.

Hence, in both cases, rectangular inequality is not satisfied, and so (E, ρ+) and
(E, ρmax) are not 2-generalized metric spaces.

The following notion of regularity for mappings F : E × E → E is needed
in this section.

Definition 2.3. Given x, y ∈ E, the mapping F : E × E → E is called
asymptotically regular at x = (x, y) if the Picard iterates xn = F

n(x, y) and
yn = F

n(y, x), defined by (1.1), satisfy the following condition

(2.4) ρν(xn, xn+1) + ρν(xn, xn+2) → 0, xn = (xn, yn).

Note that, if (E, dν) is a metric space, the summand ρν(xn, xn+2) in (2.4) is
redundant.

In the case of metric spaces, coupled fixed point results are usually obtained
by considering the induced space (E, ⊑, ρν) and reducing a symmetric contrac-
tion F : E × E → E to a monotone contraction T : E → E. This strategy, as
Example 2.2 shows, does not work in the case of ν-generalized metric spaces.
Hence, we shall take a different approach in this case.

2.1. Coupled fixed points in partially ordered metric spaces. In this
section, we assume that (E, �, d) is a partially ordered metric space. Given a
mapping F : E × E → E, define T : E → E by

(2.5) T x = (F(x, y), F(y, x)), x = (x, y).

The following properties are straightforward.

(i) If F is continuous then T is continuous.
(ii) If F is asymptotically regular in the sense of Definition 2.3, then T is

asymptotically regular in the sense of Theorem 1.7.
(iii) If F has the mixed monotone property, then T is nondecreasing on

(E, ⊑).

(iv) If F is a symmetric contraction of Ćirić-Matkowski type, then T is a

monotone contraction of Ćirić-Matkowski type, in the sense of Theorem
1.6.

(v) F has a (unique) coupled fixed point if and only if T has a (unique)
fixed point.

These properties along with the results in Section 1.3 yield the following
coupled fixed point results.

Theorem 2.4. Let (E, �, d) be a complete partially ordered metric space. Sup-
pose that F : E × E → E has the following properties.

(i) F is continuous and has the mixed strict monotone property,

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Coupled fixed points in ν-generalized metric spaces

(ii) F is a symmetric contraction of Ćirić-Matkowski type,
(iii) there exist x0, y0 ∈ E such that x0 � F(x0, y0) and y0 � F(y0, x0).

Then F has a coupled fixed point.

Proof. Since F is continuous, T is continuous. Since F is a symmetric contrac-
tion of Ćirić-Matkowski type and has the mixed strict monotone property, T
is a monotone contraction of Ćirić-Matkowski type. Since x0 � F(x0, y0) and
y0 � F(y0, x0), we get x0 ⊑ T x0 with x0 = (x0, y0). All conditions of Theorem
1.6 are satisfied. Hence T has a fixed point, which in turn implies that F has
a coupled fixed point. �

Finally, we have the following Proinov type coupled fixed point theorem.

Theorem 2.5. Suppose that (E, �, d) is a complete partially ordered metric
space, and that F : E ×E → E satisfies conditions (i)-(ii) of Theorem 1.5. For
γ > 0, define m : E × E → R+ by (here x = (x, y), u = (u, v))

m(x, u) = d(x, u) + d(y, v) + γ
(

d(x, F(x, y)) + d(y, F(y, x))
)

+ γ
(

d(u, F(u, v)) + d(v, F(v, u))
)

.
(2.6)

Suppose that

(2.7) d
(

F(x, y), F(u, v)
)

+ d
(

F(y, x), F(v, u)
)

< m(x, u), x ⊑ u, x 6= u,

and that, for any ǫ > 0, there exists δ > 0 such that, for x ⊑ u,

(2.8) m(x, u) < δ + ǫ =⇒ d
(

F(x, y), F(u, v)
)

+ d
(

F(y, x), F(v, u)
)

≤ ǫ.

If F is asymptotically regular, then F has a coupled fixed point.

Proof. It is easily seen that

m(x, u) = ρν(x, u) + γ(ρν(x, T x) + ρν(u, T u)).

The assumptions of the theorem imply that

ρν(T x, T u) < m(x, u), x ⊑ u, x 6= u.

and that, for any ǫ > 0, there exists δ > 0 such that, for x ⊑ u,

(2.9) m(x, u) < δ + ǫ =⇒ ρν(T x, T u) ≤ ǫ.

Hence T is a monotone contraction of Ćirić-Matkowski type on (E, ρν) that
satisfies all conditions of Theorem 1.7. Hence T has a fixed point which, in
turn, implies that that F has a coupled fixed point. �

2.2. Coupled fixed points in partially ordered ν-generalized metric
spaces. In this subsection, we assume that (E, �, dν) is a partially ordered ν-
generalized metric space. As Example 2.2 shows, the induced space (E, ⊑, ρν)
may not be a partially ordered ν-generalized metric space. Hence, we take a
different approach to get coupled fixed point theorems. When we call a mapping
F : E × E → E continuous (since, in general, we do not have a topological
structure in E), we mean that F(xn, yn) → F(x, y) in E whenever {xn} and
{yn} are sequences in E such that xn → x and yn → y.

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M. Abtahi, Z. Kadelburg and S. Radenović

Lemma 2.6. Let {xn} and {yn} be two sequences in a ν-generalized metric
space (E, dν) and let xn = (xn, yn). The following statements are equivalent.

(i) Both {xn} and {yn} are ν-Cauchy sequences.
(ii) lim

n→∞
sup{ρν(xn, xn+1+mν) : m ∈ Z

+} = 0.

Proof. This follows easily from Definition 1.2 and the following simple inequal-
ities.

max{sup{dν(xn, xn+1+mν) : m ∈ Z
+}, sup{dν(yn, yn+1+mν) : m ∈ Z

+}}

≤ sup{dν(xn, xn+1+mν) + dν(yn, yn+1+mν) : m ∈ Z
+}

= sup{ρν(xn, xn+1+mν)m ∈ Z
+}

≤ sup{dν(xn, xn+1+mν) : m ∈ Z
+} + sup{dν(yn, yn+1+mν) : m ∈ Z

+}.

�

Lemma 2.7. Suppose that {xn} and {yn} are sequences in a ν-generalized
metric space (E, dν) satisfying (2.4), each of which consists of mutually distinct
elements. Suppose that, for every ǫ > 0 and for any two subsequences {xpi}
and {xqi}, if lim sup

i→∞

ρν(xpi, xqi ) ≤ ǫ then, for some N,

ρν(xpi+1, xqi+1) ≤ ǫ (i ≥ N).

Then both {xn} and {yn} are Cauchy sequences.

Proof. First, we show that both {xn} and {yn} are ν-Cauchy. Towards a
contradiction, assume that, for example, {xn} is not ν-Cauchy. Then condition
(2.6) of Lemma 2.6 fails to hold. Therefore, for some ǫ > 0, we have

(2.10) ∀k ≥ 1, ∃n ≥ k, sup{ρν(xn, xn+1+mν) : m ∈ Z
+} > ǫ.

Condition ρν(xn, xn+1) → 0 implies the existence of a sequence {ki} of positive
integers such that ki−1 < ki and

(2.11) ρν(xn, xn+1) < ǫ/i (n ≥ ki).

For each ki, by (2.10), there exist ni ≥ ki + 1 and mi ≥ 0 such that
ρν(xni, xni+1+miν) > ǫ. By (2.11), ρν(xni , xni+1) < ǫ. Hence, we must have
mi ≥ 1. Let mi be the smallest number with this property so that
ρν(xni, xni+1+miν−ν) ≤ ǫ. Take pi = ni −1 and qi = ni +miν. Then, for every
i ≥ 1, we get qi > pi ≥ ki, and

ρν(xpi+1, xqi+1) > ǫ,(2.12)

ρν(xpi+1, xqi+1−ν) ≤ ǫ.(2.13)

Since both {xn} and {yn} consist of mutually different elements, property (3)
in Definition 1.1 shows that, for every i ≥ 1,

dν(xpi , xqi) ≤ dν(xpi , xpi+1) + dν(xpi+1, xqi+1−ν)

+ dν(xqi+1−ν, xqi+2−ν) + · · · + dν(xqi−1, xqi ),

dν(ypi, yqi) ≤ dν(ypi, ypi+1) + dν(ypi+1, yqi+1−ν)

+ dν(yqi+1−ν, yqi+2−ν) + · · · + dν(yqi−1, yqi).

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Coupled fixed points in ν-generalized metric spaces

The above two inequalities along with (2.11) and (2.13) imply that ρν(xpi, xqi) ≤
2νǫ/i + ǫ, for all i ≥ 1, from which we get lim sup

i→∞

ρν(xpi , xqi) ≤ ǫ. This along

with (2.12) violate our assumption. Hence both {xn} and {yn} are ν-Cauchy.
Finally, the assumption ρν(xn, xn+2) → 0 as n → ∞ implies that

dν(xn, xn+2) → 0 and dν(yn, yn+2) → 0 as n → ∞. Proposition 1.3 now shows
that both {xn} and {yn} are Cauchy sequences. �

The following is a Ćirić-Matkowski type coupled fixed point theorem in par-
tially ordered ν-generalized metric spaces.

Theorem 2.8. Let (E, �, dν) be a complete partially ordered ν-generalized

metric space. If F : E ×E → E is a symmetric contraction of Ćirić-Matkowski
type satisfying conditions (i)-(ii) of Theorem 1.5, then F has a coupled fixed
point.

Proof. Suppose that (1.2) holds for x0 = (x0, y0) and let xn = (xn, yn) be the
Picard iterates of F at x0 defined by (1.1). Note that xp ⊑ xq if p ≤ q. In fact,
xn = T

nx0, n ≥ 1, where T is defined by (2.5). An argument similar to that
of [2, Theorem 3.4] shows that ρν(xn, xn+1) + ρν(xn, xn+2) → 0.

Now, let ǫ > 0 and assume that lim sup
i→∞

ρν(xpi, xqi ) ≤ ǫ. Since F is a

symmetric contraction of Ćirić-Matkowski type, by (2.2), there exists δ > 0
such that

p ≤ q, ǫ < ρν(xp, xq) < δ + ǫ =⇒ ρν(xp+1, xq+1) ≤ ǫ.

By [1, Lemma 3.1], there is N ∈ N, such that

ρν(xpi+1, xqi+1) ≤ ǫ (i ≥ N).

Now, Lemma 2.7 shows that the sequences {xn} and {yn} are both Cauchy
sequences. Since E is complete, there exist x and y in E such that xn → x and
yn → y. Since F is continuous, we conclude that F(x, y) = x and F(y, x) = y,
so that (x, y) is a coupled fixed point. �

The following is a Proinov type coupled fixed point result in the setting of
partially ordered ν-generalized metric spaces.

Theorem 2.9. Let (E, �, dν) be a complete partially ordered ν-generalized
metric space. Given F : E × E → E, define m : E × E → R+ by (2.6), and
assume that (2.8) hold. If F is asymptotically regular and satisfies conditions
(i)-(ii) of Theorem 1.5, then F has a coupled fixed point.

Proof. Suppose that (1.2) holds for x0 = (x0, y0) and let xn = (xn, yn) be the
Picard iterates of F at x0 defined by (1.1). Note that xp ⊑ xq if p ≤ q. Since
F is asymptotically regular, we have

(2.14) ρν(xn, xn+1) + ρν(xn, xn+2) → 0.

Let {xpi} and {xqi} be two subsequences of {xn}. Then

m(xpi, xqi) = ρν(xpi , xqi) + γ
(

ρν(xpi, xpi+1) + ρν(xqi, xqi+1)
)

.

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M. Abtahi, Z. Kadelburg and S. Radenović

Since ρν(xn, xn+1) → 0, we get

(2.15) lim sup
i→∞

m(xpi, xqi) = lim sup
i→∞

ρν(xpi, xqi).

Note that by (2.14) we get dν(xn, xn+1) → 0 and dν(yn, yn+1) → 0. Also,
by the condition (1.2) and the strict mixed monotone property of F , we have
that the sequences {xn} and {yn} consist of mutually distinct terms.

Now, let ǫ > 0 and assume that lim sup
i→∞

ρν(xpi, xqi ) ≤ ǫ. The equality in

(2.15) implies that lim sup
i→∞

m(xpi, xqi ) ≤ ǫ. By (2.8), there exists δ > 0 such

that, for p ≤ q,

m(xp, xq) < δ + ǫ =⇒ ρν(xp+1, xq+1) ≤ ǫ.

By [1, Lemma 3.1], there is N ∈ N, such that

ρν(xpi+1, xqi+1) ≤ ǫ (i ≥ N).

All conditions of Lemma 2.7 are fulfilled and so the sequences {xn} and {yn}
are both Cauchy sequences. Since E is complete, there exist x and y in E such
that xn → x and yn → y. Since F is continuous, we conclude that F(x, y) = x
and F(y, x) = y, so that (x, y) is a coupled fixed point. �

2.3. Uniqueness. In order to obtain the uniqueness of coupled fixed point in
the previous results, we need some additional assumption. We formulate it just
in the case of Theorem 2.8.

Proposition 2.10. Let (E, �, dν) and F be as in Theorem 2.8 and let CFix(F)
be the set of coupled fixed points of F. Then any two comparable elements of
CFix(F) (in the sense of order ⊑) are equal. In particular, if all the elements
of CFix(F) are comparable, then this set reduces to a singleton.

Proof. Suppose, to the contrary, that there exist two distinct coupled fixed
points (x, y) and (u, v) of F which are comparable, e.g., (x, y) ⊑ (u, v) and
(x, y) 6= (u, v). Then by (2.1) we get that

dν(x, y) + dν(u, v) < dν(x, y) + dν(u, v),

a contradiction. �

2.4. Illustrative examples. The following is a very easy example illustrating
a possible use of Theorem 2.8.

Example 2.11. Let (E, dν) be the space defined in Example 2.2. Introduce
an order � on E by a � a, b � b, b � a and c � c. Consider a mapping
F : E × E → E given by F(x, x) = a, for all x ∈ E, F(a, b) = F(b, a) = a,
and F(x, y) = c otherwise. It is easy to see that all conditions of Theorem 2.8
are satisfied. In particular, the only nontrivial case when conditions (2.1) and
(2.2) have to be checked (i.e., when x � u, y � v and (x, y) 6= (u, v)) is when
(x, y) = (a, a) and (u, v) = (b, a). It is easily seen that both of them are then
satisfied.

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Coupled fixed points in ν-generalized metric spaces

Example 2.12. Consider the following 2-generalized metric space, which is a
slight modification of the space considered in [18, Example 1.1]. Let E = A∪B,
with A = {0, 2}, B = (0, 1], and define dν : E × E → [0, +∞) by

dν(x, y) =



















0, if x = y

1, if x 6= y and ({x, y} ⊆ A or {x, y} ⊆ B)

y, if x ∈ A, y ∈ B

x, if x ∈ B, y ∈ A.

Take the usual order ≤ on E. Then (E, ≤, dν) is a complete, partially ordered,
2-generalized metric space which is not a metric space. Note that if we define ρν
on E = E ×E by (2.3), then (E, ρν) is not a 2-generalized metric space. Indeed,
if we take the points (0, 0), (b1, 0), (b1, b2), (2, 2) from E (here, 0 < b1, b2 ≤ 1),
we have

ρν((0, 0), (2, 2)) = dν(0, 2) + dν(0, 2) = 1 + 1 = 2,

ρν((0, 0), (b1, 0)) = dν(0, b1) + dν(0, 0) = b1,

ρν((b1, 0), (b1, b2)) = dν(b1, b1) + dν(0, b2) = b2,

ρν((b1, b2), (2, 2)) = dν(b1, 2) + dν(b2, 2) = b1 + b2.

Hence, if 2b1 + 2b2 < 2, we have

ρν((0, 0), (2, 2)) > ρν((0, 0), (b1, 0)) + ρν((b1, 0), (b1, b2)) + ρν((b1, b2), (2, 2)).

Consider now the mapping F : E × E → E given by

F(x, y) =







x − y

2
, if x ≥ y

0, if x < y.

The conditions (i)-(ii) of Theorem 1.5 are easy to check (for example, the second
one is satisfied for x0 = 2 and y0 = 0). In order to check the condition (2.8),
consider the mapping m given by

m(x, u) = dν(x, u) + dν(y, v)

+ dν(x, F(x, y)) + dν(y, F(y, x)) + dν(u, F(u, v)) + dν(v, F(v, u)),

for x = (x, y), u = (u, v) (i.e., take γ = 1). For u ⊑ x (i.e., u ≤ x, y ≤ v) and,
for example, 1 ≥ x > u > v > y > 0 (other possible cases can be treated in a
similar way), we have

m(x, u) = dν(x, u) + dν(y, v) + dν

(

x,
x − y

2

)

+ dν(y, 0) + dν

(

u,
u − v

2

)

+ dν(v, 0)

= 1 + 1 + 1 + y + 1 + v = 4 + y + v,

hence m(x, u) < δ + ǫ implies that ǫ > 4 + y + v − δ > 1 (if δ < 3). On the
other hand

dν(F(x, y), F(u, v))+dν(F(y, x), F(v, u)) = dν

(x − y

2
,
u − v

2

)

+dν(0, 0) = 1 < ǫ,

and the condition (2.8) is satisfied.

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 199



M. Abtahi, Z. Kadelburg and S. Radenović

Thus, all the conditions of Theorem 2.9 are fulfilled and we conclude that F
has a (unique) coupled fixed point (which is (0, 0)).

Acknowledgements. The second author is thankful to the Ministry of Ed-

ucation, Science and Technological Development of Serbia, Grant No. 174002.

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