*for all*
.

*Suppose that either*
- (a)

- (b)
*if for any two sequences*
,

*with*
,

*for all*
, *then*
*for all*
.

*If there exists*
*such that*
*and*
*M*
*is an*
*F*-*invariant set which satisfies the transitive property*, *then there exists*
*such that*
*and*
, *that is*, *F*
*has a coupled fixed point*.

*Proof* From

, we can construct two sequences

and

in

*X* such that

for all

. If there exists

such that

and

, then

Thus,
is a coupled fixed point of *F*. This finishes the proof. Therefore, we may assume that
or
for all
.

Since

and

*M* is an

*F*-invariant set, we get

Again, using the fact that

*M* is an

*F*-invariant set, we have

By repeating this argument, we get

for all
. Denote
for all
.

for all

. Since

for all

, from (3.1), it follows that

Since

*M* is an

*F*-invariant set and

for all

, we get

for all

. From (3.1) and

for all

, we get

Adding (3.3) and (3.4), we get

for all

. From (3.5) and

for all

, we have

for all
, that is,
is a monotone decreasing sequence. Therefore,
for some
.

Now, we show that

. Suppose that

. Taking

of both sides of (3.5), from

for all

, it follows that

which is a contradiction. Thus,

and

Next, we prove that

and

are Cauchy sequences. Suppose that at least one,

or

, is not a Cauchy sequence. Then there exists

and two subsequences of integers

and

with

such that

for all

. Further, corresponding to

, we can choose

in such a way that it is the smallest integer with

satisfying (3.7). Then we have

Using (3.7), (3.8) and the triangle inequality, we have

Letting
and using (3.6), we have
.

Since

and

*M* satisfies the transitive property, we get

From (3.1) and (3.10), we get

Adding (3.11) and (3.12), we get

for all

. Taking

of both sides of (3.13), from

for all

, it follows that

which is a contradiction. Therefore,

and

are Cauchy sequences. Since

*X* is complete, there exists

such that

Finally, we show that

and

. If the assumption (a) holds, then we have

Therefore,
and
, that is, *F* has a coupled fixed point.

Suppose that (b) holds. We obtain that a sequence

converges to

*x* and a sequence

converges to

*y* for some

. By the assumption, we have

for all

. Since

for all

, by the triangle inequality and (3.1), we get

Taking
, we have
, and so
. Similarly, we can conclude that
. Therefore, *F* has a coupled fixed point. This completes the proof. □

Now, we give an example to validate Theorem 3.1.

**Example 3.2** Let

endowed with the usual metric

for all

and endowed with the usual partial order defined by

. Define a continuous mapping

by

for all
. Let
and
. Then we have
, but
, and so the mapping *F* does not satisfy the mixed monotone property.

Now, let

be a function defined by

for all

. Then we obtain

and

for any

. By simple calculation, we see that for all

,

Therefore, if we apply Theorem 3.1 with
, we know that *F* has a unique coupled fixed point, that is, a point
is a unique coupled fixed point.

**Remark 3.3** Although the mixed monotone property is an essential tool in the partially ordered metric spaces to show the existence of coupled fixed points, the mappings do not have the mixed monotone property in a general case as in the above example. Therefore, Theorem 3.1 is interesting, as a new auxiliary tool, in showing the existence of a coupled fixed point.

If we take the mapping
for some
in Theorem 3.1, then we get the following:

*for all*
.

*Suppose that either*
- (a)

- (b)
*for any two sequences*
,

*with*
,

*if*

*for all*
, *then*
*for all*
.

*If there exists*
*such that*
*and*
*M*
*is an*
*F*-*invariant set which satisfies the transitive property*, *then there exists*
*such that*
*and*
, *that is*, *F*
*has a coupled fixed point*.

Now, from Theorem 3.1, we have the following question:

**(Q1)** Is it possible to guarantee the uniqueness of the coupled fixed point of *F*?

Now, we give positive answers to this question.

**Theorem 3.5**
*In addition to the hypotheses of Theorem *3.1, *suppose that for all*
, *there exists*
*such that*
*and*
. *Then*
*F*
*has a unique coupled fixed point*.

*Proof* From Theorem 3.1, we know that *F* has a coupled fixed point. Suppose that
and
are coupled fixed points of *F*, that is,
,
,
and
.

Now, we show that

and

. By the hypothesis, there exists

such that

and

. We put

and

and construct two sequences

and

by

for all
.

Since

*M* is

*F*-invariant and

, we have

From

, if we use again the property of

*F*-invariant, then it follows that

By repeating this process, we get

for all

. From (3.1) and (3.19), we have

Since

*M* is

*F*-invariant and

for all

, we have

for all

. From (3.1) and (3.21), we get

Thus, from (3.20) and (3.22), we have

for all

. By repeating this process, we get

for all

. From

and

, it follows that

for each

. Therefore, from (3.24), we have

Similarly, we can prove that

By the triangle inequality, for all

, we have

Taking
in (3.27) and using (3.25) and (3.26), we have
, and so
and
. Therefore, *F* has a unique coupled fixed point. This completes the proof. □

Next, we give a simple application of our results to coupled fixed point theorems in partially ordered metric spaces.

*for all*
*for which*
*and*
.

*Suppose that either*
- (a)

- (b)
*X*
*has the following property*:

- (i)

- (ii)

*If there exists*
*such that*
*then there exists*
*such that*
*and*
, *that is*, *F*
*has a coupled fixed point*.

*Proof* First, we define a subset

by

From Example 2.10, we can conclude that

*M* is an

*F*-invariant set which satisfies the transitive property. By (3.28), we have

for all

with

. Since

such that

For the assumption (b), for any two sequences

,

such that

is a non-decreasing sequence in

*X* with

and

is a non-increasing sequence in

*X* with

, we have

for all
. Therefore, we have
for all
, and so the assumption (b) of Theorem 3.1 holds.

Now, since all the hypotheses of Theorem 3.1 hold, *F* has a coupled fixed point. This completes the proof. □

**Corollary 3.7**
*In addition to the hypotheses of Corollary *3.6, *suppose that for all*
, *there exists*
*such that*
,
*and*
,
. *Then*
*F*
*has a unique coupled fixed point*.

*Proof* First, we define a subset

by

From Example 2.10, we can conclude that *M* is an *F*-invariant set which satisfies the transitive property. Thus, the proof of the existence of a coupled fixed point is straightforward by following the same lines as in the proof of Corollary 3.6.

Next, we show the uniqueness of a coupled fixed point of *F*. Since for all
, there exists
such that
,
and
,
, we can conclude that
and
. Therefore, since all the hypotheses of Theorem 3.5 hold, *F* has a unique coupled fixed point. This completes the proof. □

**Corollary 3.8** (Bhaskar and Lakshmikantham [21])

*then there exists*
*such that*
*and*
.

*Proof* Taking
for some
in Corollary 3.6(a), we can get the conclusion. □

**Corollary 3.9** (Bhaskar and Lakshmikantham [21])

*Let*
*be a partially ordered set and suppose that there is a metric*
*d*
*on*
*X*
*such that*
*is a complete metric space*.

*Suppose that*
*X*
*has the following property*:

- (i)

- (ii)

*Let*
*be a continuous mapping having the mixed monotone property on*
*X*.

*Assume that there exists*
*with*
*then there exists*
*such that*
*and*
.

*Proof* Taking
for some
in Corollary 3.6(b), we can get the conclusion. □