The following is the first result.

**Theorem 3.1**.

*Let* (

*X*, ≤)

*be a partially ordered set and* (

*X*,

*G*)

*be a complete G-metric space. Let F* :

*X* ×

*X* →

*X and g* :

*X* →

*X be continuous mappings such that F has the mixed g-monotone property and g commutes with F. Assume that there are altering distance functions ψ and ϕ such that**for all x*, *y*, *u*, *v*, *w*, *z* ∈ *X with gw* ≤ *gu* ≤ *gx and gy* ≤ *gv* ≤ *gz*. *Also, suppose that F*(*X* × *X*) ⊆ *g*(*X*). *If there exist x*_{0}, *y*_{0} ∈ *X such that gx*_{0} ≤ *F*(*x*_{0}, *y*_{0}) *and F*(*y*_{0}, *x*_{0}) ≤ *gy*_{0}, *then F and g have a coupled coincidence point*.

**Proof**. Let

*x*_{0},

*y*_{0} ∈

*X* such that

*gx*_{0} ≤

*F*(

*x*_{0},

*y*_{0}) and

*F*(

*y*_{0},

*x*_{0}) ≤

*gy*_{0}. Since we have

*F*(

*X* ×

*X*) ⊆

*g*(

*X*), we can choose

*x*_{1},

*y*_{1} ∈

*X* such that

*gx*_{1} =

*F*(

*x*_{0},

*y*_{0}) and

*gy*_{1} =

*F*(

*y*_{0},

*x*_{0}). Again, since

*F*(

*X* ×

*X*) ⊆

*g*(

*X*), we can choose

*x*_{2},

*y*_{2} ∈

*X* such that

*gx*_{
2
}=

*F*(

*x*_{1},

*y*_{1}) and

*gy*_{2} =

*F*(

*y*_{1},

*x*_{1}). Since

*F* has the mixed

*g*-monotone property, we have

*gx*_{0} ≤

*gx*_{1} ≤

*gx*_{2} and

*gy*_{2} ≤

*gy*_{1} ≤

*gy*_{0}. Continuing this process, we can construct two sequences (

*x*_{
n
}) and (

*y*_{
n
}) in

*X* such that

If, for some integer *n*, we have (*gx*
_{n+1}, *gy*
_{n+1}) = (*gx*
_{
n
}, *gy*
_{
n
}), then *F*(*x*
_{
n
}, *y*
_{
n
}) = *gx*
_{
n
}and *F*(*y*
_{
n
}, *x*
_{
n
}) = *gy*
_{
n
}, that is, (*x*
_{
n
}, *y*
_{
n
}) is a coincidence point of *F* and *g*. So, from now on, we assume that (*gx*
_{n+1}, *gy*
_{n+1}) ≠ (*gx*
_{
n
}, *gy*
_{
n
}) for all *n* ∈ ℕ, that is, we assume that either *gx*
_{n+1 }≠ *gx*
_{
n
}or *gy*
_{n+1 }≠ *gy*
_{
n
}.

We complete the proof with the following steps.

For each

*n* ∈ ℕ, using the inequality (1), we obtain

Since

*ψ* is a non-decreasing function, we get

On the other hand, we have

Since

*ψ* is a non-decreasing function, we get

Thus, by (4) and (6), we have

Thus (max{

*G*(

*gx*
_{n-1},

*gx*
_{
n
},

*gx*
_{
n
}),

*G*(

*gy*
_{n-1},

*gy*
_{
n
},

*gy*
_{
n
})}) is a non-negative decreasing sequence. Hence, there exists

*r* ≥ 0 such that

Now, we show that

*r* = 0. Since

*ϕ* : [0, + ∞) → [0, + ∞) is a non-decreasing function, then, for any

*a*,

*b* ∈ [0, + ∞), we have

*ψ*(max{

*a*,

*b*}) = max{

*ψ*(

*a*)

*, ψ*(

*b*)}. Thus, by (3)) and (5), we have

Letting

*n* → +∞ in the above inequality and using the continuity of

*ψ*, we get

Hence *ϕ*(*r*) = 0. Thus *r* = 0 and (2) holds.

**Step 2**: We show that (

*gx*_{
n
}) and (

*gy*_{
n
}) are

*G*-Cauchy sequences. Assume that (

*x*_{
n
}) or (

*y*_{
n
}) is not a

*G*-Cauchy sequence, that is,

This means that there exists ϵ > 0 for which we can find subsequences of integers (

*m*(

*k*)) and (

*n*(

*k*)) with

*n*(

*k*) >

*m*(

*k*) >

*k* such that

Further, corresponding to

*m*(

*k*) we can choose

*n*(

*k*) in such a way that it is the smallest integer with

*n*(

*k*) >

*m*(

*k*) and satisfying (7). Then we have

Thus, by (

*G*
_{5}) and (8), we have

Thus, by (9) and (10), we have

Now, using the inequality (1), we obtain

Thus, by (12) and (13), we get

Letting

*k* → +∞ in the above inequality and using (11) and the fact that

*ψ* and

*ϕ* are continuous, we get

Hence *ϕ*(ϵ) = 0 and so ϵ = 0, which is a contradiction. Therefore, (*gx*
_{
n
}) and (*gy*
_{
n
}) are *G*-Cauchy sequences.

**Step 3**: The existence of a coupled coincidence point. Since (

*gx*_{
n
}) and (

*gy*_{
n
}) are

*G*-Cauchy sequences in a complete

*G*-metric space (

*X*,

*G*), there exist

*x, y* ∈

*X* such that (

*gx*_{
n
}) and (

*gyn*) are

*G*-convergent to points

*x* and

*y*, respectively, that is,

Then, by (14), (15) and the continuity of

*g*, we have

Therefore, (

*g*(

*gx*
_{
n
})) is

*G*-convergent to

*gx* and (

*g*(

*gy*
_{
n
})) is

*G*-convergent to

*gy*. Since

*F* and

*g* commute, we get

Using the continuity of *F* and letting *n* → +∞ in (18) and (19), we get *gx* = *F*(*x*, *y*) and *gy* = *F*(*y*, *x*). This implies that (*x*, *y*) is a coupled coincidence point of *F* and *g*. This completes the proof.

Tacking *g* = *I*
_{
X
}(: the identity mapping) in Theorem 3.1., we obtain the following coupled fixed point result.

**Corollary 3.1**.

*Let* (

*X*, ≤)

*be a partially ordered set and* (

*X*,

*G*)

*be a complete G-metric space. Let F* :

*X* ×

*X* →

*X be a continuous mapping satisfying the mixed monotone property. Assume that there exist the altering distance functions ψ and ϕ such that**for all x*, *y*, *u*, *v*, *w*, *z* ∈ *X with w* ≤ *u* ≤ *x and y* ≤*v* ≤ *z. If there exist x*_{0}, *y*_{0} ∈ *X such that x*_{0} ≤ *F*(*x*_{0}, *y*_{0}) *and F*(*y*_{0}, *x*_{0}) ≤ *y*_{0}, *then F has a coupled fixed point.*

Now, we derive coupled coincidence point results without the continuity hypothesis of the mappings *F, g* and the commutativity hypothesis of *F, g*. However, we consider the additional assumption on the partially ordered set (*X*, ≤).

We need the following definition.

**Definition 3.1**. Let (

*X*, ≤) be a partially ordered set and

*G* be a

*G*-metric on

*X*. We say that (

*X*,

*G*, ≤) is

*regular* if the following conditions hold:

- (1)
if a non-decreasing sequence (*x*
_{
n
}) is such that *x*
_{
n
}→ *x*, then *x*
_{
n
}≤ *x* for all *n* ∈ ℕ,

- (2)
if a non-increasing sequence (*y*
_{
n
}) is such that *y*
_{
n
}→ *y*, then *y* ≤ *y*
_{
n
}for all *n* ∈ ℕ.

The following is the second result.

**Theorem 3.2**.

*Let* (

*X*, ≤)

*be a partially ordered set and G be a G-metric on × such that* (

*X*,

*G*, ≤)

*is regular. Assume that there exist the altering distance functions ψ, ϕ and mappings F* :

*X* ×

*X* →

*X and g*:

*X* →

*X such that**for all x*, *y*, *u*, *v*, *w*, *z* ∈ *X with gw* ≤ *gu* ≤ *gx and gy* ≤ *gv* ≤ *gz. Suppose also that* (*g*(*X*), *G*) *is G-complete, F has the mixed g-monotone property and F*(*X* × *X*) ⊆ *g*(*X*). *If there exist x*_{0}, *y*_{0} ∈ *X such that gx*_{0} ≤ *F*(*x*_{0}, *y*_{0}) *and F*(*y*_{0}, *x*_{0}) ≤ *gy*_{0}, *then F and g have a coupled coincidence point*.

**Proof**. Following Steps 1 and 2 in the proof of Theorem 3.1., we know that (

*gx*_{
n
}) and (

*gy*_{
n
}) are

*G-*Cauchy sequences in

*g*(

*X*) with

*gx*_{
n
}≤

*gx*_{n+1 }and

*gy*_{
n
}≥

*gy*_{n+1 }for all

*n* ∈ ℕ. Since (

*g*(

*X*),

*G*) is

*G*-complete, there exist

*x, y* ∈

*X* such that

*gx*_{
n
}→

*gx* and

*gy*_{
n
}→

*gy*. Since (

*X*,

*G*, ≤) is regular, we have

*gx*_{
n
}≤

*gx* and

*gy* ≤

*gy*_{
n
}for all

*n* ∈ ℕ. Thus we have

Letting *n* → +∞ in the above inequality and using the continuity of *ψ* and *ϕ*, we obtain *ψ*(*G*(*F*(*x*, *y*),*gx*, *gx*)) = 0, which implies that *G*(*F*(*x*, *y*), *gx*, *gx*) = 0. Therefore, *F*(*x*, *y*) = *gx*.

Similarly, one can show that *F*(*y*, *x*) = *gy*. Thus (*x*, *y*) is a coupled coincidence point of *F* and *g*, this completes the proof.

Tacking *g* = *I*
_{
X
}in Theorem 3.2., we obtain the following result.

**Corollary 3.2**. *Let* (*X*, ≤) *be a partially ordered set and G be a G-metric on X such that* (*X*, *G*, ≤) *is regular and* (*X*, *G*) *is G-complete. Assume that there exist the altering distance functions ψ*, *ϕ and a mapping*

*F* :

*X* ×

*X* →

*X having the mixed monotone property such that**for all x, y, u, v, w, z* ∈ *X with w* ≤ *u* ≤ *x and y* ≤ *v* ≤ *z*. *If there exist x*_{0}, *y*_{0} ∈ *X such that x*_{0} ≤ *F*(*x*_{0}, *y*_{0}) *and F*(*y*_{0}, *x*_{0}) ≤ *y*_{0}, *then F has a coupled fixed point*.

Now, we prove the existence and uniqueness theorem of a coupled common fixed point. If (

*X*, ≤) is a partially ordered set, we endow the product set

*X* ×

*X* with the partial order defined by

**Theorem 3.3**. *In addition to the hypotheses of Theorem* 3.1.*, suppose that, for any* (*x*, *y*), (*x**, *y**) ∈ *X* × *X*, *there exists* (*u*, *v)* ∈ *X* × *X such that* (*F*(*u*, *v*), *F*(*v*, *u*)) *is comparable with* (*F*(*x*, *y*), *F*(*y*, *x*)) *and* (*F*(*x**, *y**), *F*(*y**, *x**)). *Then F and g have a unique coupled common fixed point, that is, there exists a unique* (*x*, *y*) ∈ *X* × *X such that x* = *gx* = *F*(*x*, *y*) *and y* = *gy* = *F*(*y*, *x*).

**Proof**. From Theorem 3.1., the set of coupled coincidence points is non-empty. We shall show that if (

*x*,

*y*) and (

*x**,

*y**) are coupled coincidence points, then

By the assumption, there exists (

*u*,

*v*) ∈

*X* ×

*X* such that (

*F*(

*u*,

*v*),

*F*(

*v*,

*u*)) is comparable to (

*F*(

*x*,

*y*),

*F*(

*y*,

*x*)) and (

*F*(

*x**,

*y**),

*F*(

*y**,

*x**)). Without the restriction to the generality, we can assume that (

*F*(

*x*,

*y*),

*F*(

*y*,

*x*)) ≤ (

*F*(

*u*,

*v*),

*F*(

*v*,

*u*)) and (

*F*(

*x**,

*y**),

*F*(

*y**,

*x**)) ≤ (

*F*(

*u*,

*v*),

*F*(

*v*,

*u*)). Put

*u*
_{0} =

*u*,

*v*
_{0} =

*v* and choose

*u*
_{1},

*v*
_{1} ∈

*X* so that

*gu*
_{1} =

*F*(

*u*
_{0},

*v*
_{0}) and

*gv*
_{1} =

*F*(

*v*
_{0},

*u*
_{0}). As in the proof of Theorem 3.1., we can inductively define the sequences (

*u*
_{
n
}) and (

*v*
_{
n
}) such that

Further, set

and, by the same way, define the sequences (

*x*
_{
n
}), (

*y*
_{
n
}) and

. Since (

*gx, gy*) = (

*F*(

*x*,

*y*),

*F*(

*y*,

*x*)) = (

*gx*
_{1},

*gy*
_{1}) and (

*F*(

*u*,

*v*),

*F*(

*v*,

*u*)) = (

*gu*
_{1},

*gv*
_{1}) are comparable,

*gx* ≤

*gu*
_{1} and

*gv*
_{1} ≤

*gy*. One can show, by induction, that

for all

*n* ∈ ℕ. From (1), we have

Since *ψ* is non-decreasing, it follows that (*max*{*G*(*gx*, *gx*, *gu*
_{
n
}),*G*(*gy*, *gy*, *gv*
_{
n
})}) is a decreasing sequence.

Hence there exists a non-negative real number

*r* such that

Using (21) and letting

*n* → +∞ in the above inequality, we get

Therefore,

*ϕ*(

*r*) = 0 and hence

*r* = 0. Thus

Similarly, we can show that

Thus, by (

*G*
_{5}), (22), and (23), we have, as

*n* → +∞,

Hence *gx* = *gx** and *gy* = *gy**. Thus we proved (20).

On the other hand, since

*gx* =

*F*(

*x*,

*y*) and

*gy* =

*F*(

*y*,

*x*), by commutativity of

*F* and

*g*, we have

Denote

*gx* =

*z* and

*gy* =

*w*. Then, from (24), it follows that

Thus (

*z*,

*w*) is a coupled coincidence point. Then, from (20) with

*x** =

*z* and

*y** =

*w*, it follows that

*gz* =

*gx* and

*gw* =

*gy*, that is,

Thus, from (25) and (26), we have *z* = *gz* = *F*(*z*, *w*) and *w* = *gw* = *F*(*w*, *z*). Therefore, (*z*, *w*) is a coupled common fixed point of *F* and *g*.

To prove the uniqueness of the point (

*z*,

*w*), assume that (

*s*,

*t*) is another coupled common fixed point of

*F* and

*g*. Then we have

Since the pair (*s*, *t*) is a coupled coincidence point of *F* and *g*, we have *gs* = *gx* = *z* and *gt* = *gy* = *w*. Thus *s* = *gs* = *gz* = *z* and *t* = *gt* = *gw* = *w*. Hence, the coupled fixed point is unique. this completes the proof.

Now, we present coupled coincidence and coupled common fixed point results for mappings satisfying contractions of integral type. Denote by Λ the set of functions *α* : [0, +∞) → [0, + ∞) satisfying the following hypotheses:

(h1) *α* is a Lebesgue integrable mapping on each compact subset of [0, + ∞),

(h2) for any *ε* > 0, we have
.

Finally, we give the following results.

**Theorem 3.4**.

*Let* (

*X*, ≤)

*be a partially ordered set and* (

*X*,

*G*)

*be a complete G-metric space. Let F* :

*X* ×

*X* →

*X and g* :

*X* →

*X be continuous mappings such that F has the mixed g-monotone property and g commutes with F. Assume that there exist α*,

*β* ∈ Λ

*such that**for all x*, *y*, *u*, *v*, *w*, *z* ∈ *X with gw* ≤ *gu* ≤ *gx and gy* ≤ *gv* ≤ *gz. Also, suppose that F*(*X* × *X*) ⊆ *g*(*X*). *If there exist x*_{0}, *y*_{0} ∈ *X such that gx*_{0} ≤ *F*(*x*_{0}, *y*_{0}) *and F*(*y*_{0}, *x*_{0}) ≤ *gy*_{0}, *then F and g have a coupled coincidence point.*

**Proof**. We consider the functions

*ψ, ϕ* : [0, +∞) → [0, +∞) defined by

for all *t* ≥ 0. It is clear that *ψ* and *ϕ* are altering distance functions. Then the results follow immediately from Theorem 3.1.. This completes the proof.

**Corollary 3.3**.

*Let* (

*X*, ≤)

*be a partially ordered set and* (

*X*,

*G*)

*be a complete G-metric space. Let F* :

*X* ×

*X* →

*X be a continuous mappings satisfying the mixed monotone property. Assume that there exist α*,

*β* ∈ Λ

*such that**for all x*, *y*, *u*, *v*, *w*, *z* ∈ *X with w* ≤ *u* ≤ *x and y* ≤ *v* ≤ *z. If there exist x*_{0}, *y*_{0} ∈ *X such that x*_{0} ≤ *F*(*x*_{0}, *y*_{0}) *and F*(*y*_{0}, *x*_{0}) ≤ *y*_{0}, *then F has a coupled fixed point.*

**Proof**. Tacking *g* = *I*_{
X
}in Theorem 3.3., we obtain Corollary 3.3..

Putting *β*(*s*) = (1 - *k*)*α*(*s*) with *k* ∈ [0,1) in Theorem 3.3., we obtain the following result.

**Corollary 3.4**.

*Let* (

*X*, ≤)

*be a partially ordered set and* (

*X*,

*G*)

*be a complete G-metric space. Let F* :

*X* ×

*X* →

*X and g*:

*X* →

*X be continuous mappings such that F has the mixed g-monotone property and g commutes with F. Assume that there exist α* ∈ Λ

*and k* ∈ [0, 1)

*such that**for all x*, *y*, *u*, *v*, *w*, *z* ∈ *X with gw* ≤ *gu* ≤ *gx and gy* ≤ *gv* ≤ *gz. Also, suppose that F*(*X* × *X)* ⊆ *g*(*X*).

*If there exist x*_{0}, *y*_{0} ∈ *X such that gx*_{0} ≤ *F*(*x*_{0}, *y*_{0}) *and F*(*y*_{0}, *x*_{0}) ≤ *gy*_{0}, *then F and g have a coupled coincidence point.*

Tacking *α*(*s*) = 1 in Corollary 3.4., we obtain the following result.

**Corollary 3.5**.

*Let* (

*X*, ≤)

*be a partially ordered set and* (

*X*,

*G*)

*be a complete G-metric space. Let F* :

*X* ×

*X* →

*X and g* :

*X* →

*X be continuous mappings such that F has the mixed g-monotone property and g commutes with F. Assume that there exists k* ∈ [0, 1)

*such that**for all x*, *y*, *u*, *v*, *w*, *z* ∈ *X with gw* ≤ *gu* ≤ *gx and gy* ≤ *gv* ≤ *gz. Also, suppose that F*(*X* × *X*) ⊆ *g*(*X*). *If there exist x*_{0}, *y*_{0} ∈ *X such that gx*_{0} ≤ *F*(*x*_{0}, *y*_{0}) *and F*(*y*_{0}, *x*_{0}) ≤ *gy*_{0}, *then F and g have a coupled coincidence point.*

**Corollary 3.6**.

*Let* (

*X*, ≤)

*be a partially ordered set and* (

*X*,

*G*)

*be a complete G-metric space. Let F* :

*X* ×

*X* →

*X and g* :

*X* →

*X be continuous mappings such that F has the mixed g-monotone property and g commutes with F. Assume that there exist non-negative real numbers a, b with a* +

*b* ∈ [0,1)

*such that**for all x*, *y*, *u*, *v*, *w*, *z* ∈ *X with gw* ≤ *gu* ≤ *gx and gy* ≤ *gv* ≤ *gz. Also, suppose that F*(*X* × *X*) ⊆ *g*(*X*). *If there exist x*_{0}, *y*_{0} ∈ *X such that gx*_{0} ≤ *F*(*x*_{0}, *y*_{0}) *and F*(*y*_{0}, *x*_{0}) ≤ *gy*_{0}, *then F and g have a coupled coincidence point*.

for all *x*, *y*, *u*, *v*, *w*, *z* ∈ *X* with *gw* ≤ *gu* ≤ *gx* and *gy* ≤ *gv* ≤ *gz*. Then Corollary 3.6. follows from Corollary 3.5..

**Remark 3.1**. Note that similar results can be deduced from Theorems 3.2. and 3.3..

**Remark 3.2**. (1) Theorem 3.1 in [36] is a special case of Theorem 3.1..

(2) Theorem 3.2 in [36] is a special case of Theorem 3.2..

**Example 3.1**. Let

*X* = 0,1, 2, 3,... and

*G* :

*X* ×

*X* ×

*X* →

*R*^{+} be defined as follows:

Then (

*X*,

*G*) is a complete

*G*-metric space [

36]. Let a partial order ≼ on

*X* be defined as follows: For

*x*,

*y* ∈

*X*,

*x* ≼

*y* holds if

*x* >

*y* and 3 divides (

*x* -

*y*) and 3 ≼ 1 and 0 ≼ 1 hold. Let

*F* :

*X* ×

*X* →

*X be* defined as follows:

*Let w* ≼ *u* ≼ *x* ≼ *y* ≼ *v* ≼ *z* hold, then equivalently, we have *w* ≥ *u* ≥ *x* ≥ *y* ≥ *v* ≥ *z*. Then *F*(*x*, *y*) = *F*(*u*, *v*) = *F*(*w*, *z*) = 1. Let
for *t* ≥ 0 and *k* ∈ [0,1) and let *g*(*x*) = *x* for *x* ∈ *X*. Thus left-hand side of (1) is *G*(1, 1,1) = 0 and hence (1) is satisfied. Then with *x*_{0} = 81 and *y*_{0} = 0 the Theorem 3.2. is applicable to this example. It may be observed that in this example the coupled fixed point is not unique. Hence, (0,0) and (1,0) are two coupled fixed point of *F*.