The notion of metric space was introduced by Fréchet [1] in 1906. In almost all fields of quantitative sciences which require the use of analysis, metric spaces play a major role. Internet search engines, image classification, protein classification (see, *e.g.*, [2]) can be listed as examples in which metric spaces have been extensively used to solve major problems. It is conceivable that metric spaces will be needed to explore new problems that will arise in quantitative sciences in the future. Therefore, it is necessary to consider various generalizations of metrics and metric spaces to broaden the scope of applied sciences. In this respect, cone metric spaces, fuzzy metric spaces, partial metric spaces, quasi-metric spaces and *b*-metric spaces can be given as the main examples. Applications of these different approaches to metrics and metric spaces make it evident that fixed point theorems are important not only for the branches of mainstream mathematics, but also for many divisions of applied sciences.

Inspired by this motivation Mustafa and Sims [3] introduced the notion of a *G*-metric space in 2004 (see also [4–7]). In their introductory paper, the authors investigated versions of the celebrated theorems of the fixed point theory such as the Banach contraction mapping principle [8] from the point of view of *G*-metrics. Another fundamental aspect in the theory of existence and uniqueness of fixed points was considered by Ran and Reurings [9] in partially ordered metric spaces. After Ran and Reurings’ pioneering work, several authors have focused on the fixed points in ordered metric spaces and have used the obtained results to discuss the existence and uniqueness of solutions of differential equations, more precisely, of boundary value problems (see, *e.g.*, [10–20]). Upon the introduction of the notion of coupled fixed points by Guo and Laksmikantham [14], Gnana-Bhaskar and Lakshmikantham [15] obtained interesting results related to differential equations with periodic boundary conditions by developing the mixed monotone property in the context of partially ordered metric spaces. As a continuation of this trend, many authors conducted research on the coupled fixed point theory and many results in this direction were published (see, for example, [21–35]).

In this paper, we prove the theorem that amalgamates these three seminal approaches in the study of fixed point theory, the so called *G*-metrics, coupled fixed points and partially ordered spaces.

We shall start with some necessary definitions and a detailed overview of the fundamental results developed in the remarkable works mentioned above. Throughout this paper, ℕ and
denote the set of non-negative integers and the set of positive integers respectively.

**Definition 1** (See [3])

Let *X* be a non-empty set,
be a function satisfying the following properties:

(G1)
if
,

(G2)
for all
with
,

(G3)
for all
with
,

(G4)
(symmetry in all three variables),

(G5)
for all
(rectangle inequality).

Then the function *G* is called a generalized metric or, more specially, a *G*-metric on *X*, and the pair
is called a *G*-metric space.

It can be easily verified that every

*G*-metric on

*X* induces a metric

on

*X* given by

Trivial examples of *G*-metric are as follows.

**Example 2** Let

be a metric space. The function

, defined by

for all
, is a *G*-metric on *X*.

The concepts of convergence, continuity, completeness and Cauchy sequence have also been defined in [3].

**Definition 3** (See [3])

Let
be a *G*-metric space, and let
be a sequence of points of *X*. We say that
is *G*-convergent to
if
, that is, if for any
, there exists
such that
for all
. We call *x* the limit of the sequence and write
or
.

**Proposition 4** (See [3])

*Let*
*be a*
*G*-

*metric space*.

*The following statements are equivalent*:

- (1)
*is*
*G*-*convergent to*
*x*,

- (2)
*as*
,

- (3)
*as*
,

- (4)
*as*
.

**Definition 5** (See [3])

Let
be a *G*-metric space. A sequence
is called *G*-Cauchy sequence if for any
, there is
such that
for all
, that is,
as
.

**Proposition 6** (See [3])

*Let*
*be a*
*G*-

*metric space*.

*The following statements are equivalent*:

- (1)
*The sequence*
*is*
*G*-*Cauchy*.

- (2)

**Definition 7** (See [3])

A *G*-metric space
is called *G*-complete if every *G*-Cauchy sequence is *G*-convergent in
.

**Definition 8** Let
be a *G*-metric space. A mapping
is said to be continuous if for any three *G*-convergent sequences
,
and
converging to *x*, *y* and *z* respectively,
is *G*-convergent to
.

We define below *g*-ordered complete *G*-metric spaces.

**Definition 9** Let
be a partially ordered set,
be a *G*-metric space and
be a mapping. A partially ordered *G*-metric space,
, is called *g*-ordered complete if for each *G*-convergent sequence
, the following conditions hold:

(
) If
is a non-increasing sequence in *X* such that
, then
.

(
) If
is a non-decreasing sequence in *X* such that
, then
.

In particular, if *g* is the identity mapping in (
) and (
), the partially ordered *G*-metric space,
, is called ordered complete.

We next recall some basic notions from the coupled fixed point theory. In 1987 Guo and Lakshmikantham [14] defined the concept of a coupled fixed point. In 2006, in order to prove the existence and uniqueness of the coupled fixed point of an operator
on a partially ordered metric space, Gnana-Bhaskar and Lakshmikantham [15] reconsidered the notion of a coupled fixed point via the mixed monotone property.

**Definition 10** ([15])

Let

be a partially ordered set and

. The mapping

*F* is said to have the mixed monotone property if

is monotone non-decreasing in

*x* and is monotone non-increasing in

*y*, that is, for any

,

**Definition 11** ([15])

An element

is called a coupled fixed point of the mapping

if

The results in [15] were extended by Lakshmikantham and Ćirić in [16] by defining the mixed *g*-monotone property.

**Definition 12** Let

be a partially ordered set,

and

. The function

*F* is said to have mixed

*g*-monotone property if

is monotone

*g*-non-decreasing in

*x* and is monotone

*g*-non-increasing in

*y*, that is, for any

,

It is clear that Definition 12 reduces to Definition 10 when *g* is the identity mapping.

**Definition 13** An element

is called a coupled coincidence point of the mappings

and

if

and a common coupled fixed point of

*F* and

*g* if

**Definition 14** The mappings

and

are said to commute if

Throughout the rest of the paper, we shall use the notation *gx* instead of
, where
and
, for brevity. In [35], Nashine proved the following theorems.

**Theorem 15**
*Let*
*be a partially ordered*
*G*-

*metric space*.

*Let*
*and*
*be mappings such that*
*F*
*has the mixed*
*g*-

*monotone property*,

*and let there exist*
*such that*
*and*
.

*Suppose that there exists*
*such that for all*
*the following holds*:

*for all*
*and*
,

*where either*
*or*
.

*Assume the following hypotheses*:

- (i)
,

- (ii)
*is*
*G*-*complete*,

- (iii)
*g*
*is*
*G*-*continuous and commutes with*
*F*.

*Then*
*F*
*and*
*g*
*have a coupled coincidence point*, *that is*, *there exists*
*such that*
*and*
. *If*
*and*
, *then*
*F*
*and*
*g*
*have a common fixed point*, *that is*, *there exists*
*such that*
.

**Theorem 16**
*If in the above theorem*, *we replace the condition* (ii) *by the assumption that*
*X*
*is*
*g*-*ordered complete*, *then we have the conclusions of Theorem *15.

We next give the definition of *G*-compatible mappings inspired by the definition of compatible mappings in [13].

**Definition 17** Let

be a

*G*-metric space. The mappings

,

are said to be

*G*-compatible if

where
and
are sequences in *X* such that
and
for all
are satisfied.

In this paper, we aim to extend the results on coupled fixed points mentioned above. Our results improve, enrich and extend some existing theorems in the literature. We also give examples to illustrate our results. This paper can also be considered as a continuation of the works of Berinde [11, 12].