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In this paper, we establish the existence and uniqueness of fixed points of operator , when n is an arbitrary positive integer and X is a partially ordered complete metric space. We have shown examples to verify our work. Our results generalize the recent fixed point theorems cited in [1]-[4] etc. and include several recent developments.

The metric fixed point theory plays a vital role to solve the problems related to variational inequalities, optimization, approximation theory, etc. Many authors (for detail, see [

Bhaskar and Lakshmikantham [

Jungck [

Definition 2.1 [

For

Definition 2.2 [

Definition 2.3 [

Definition 2.4 [

Definition 2.5 [

Definition 2.6 [

Imdad et al. [

Definition 2.7 Let

monotone property if F is non-decreasing in its odd position arguments and non-increasing in its even positions arguments, that is, if,

1) For all

2) For all

3) For all

For all

For all

Definition 2.8 Let

Then F is said to have the mixed g-monotone property if F is g-non-decreasing in its odd position arguments and g-non-increasing in its even positions arguments, that is, if,

1) For all

2) For all

3) For all

For all

For all

Definition 2.9 [

Example 1. Let (R, d) be a partial ordered metric space under natural setting and let

then

Definition 2.10 [

Example 2. Let (R, d) be a partial ordered metric space under natural setting and let

for any

Definition 2.11 [

Now, we define the concept of compatible maps for r-tupled maps.

Definition 2.12 Let

whenever,

For some

Imdad et al. [

Theorem 3.1 Let

(i)

(ii) g is continuous and monotonically increasing,

(iii) the pair (g, F) is commuting,

(iv)

a) F is continuous or

b) X has the following properties:

(i) If a non-decreasing sequence

(ii) If a non-increasing sequence

If there exist

Then F and g have a r-tupled coincidence point, i.e. there exist

Now, we prove our main result as follows:

Theorem 3.2 Let

(3.2) g is continuous and monotonically increasing,

(3.3) the pair (g, F) is compatible,

(3.4)

For all

a) F is continuous or

b) X has the following properties:

(i) If a non-decreasing sequence

(ii) If a non-increasing sequence

If there exist

Then F and g have a r-tupled coincidence point, i.e. there exist

Proof. Starting with

Now, we prove that for all n ≥ 0,

So (3.8) holds for n = 0. Suppose that (3.8) holds for some n > 0. Consider

and

Thus by induction (3.8) holds for all

Similarly, we can inductively write

Therefore, by putting

We have,

Since

We shall show that

this contradiction gives

Next we show that all the sequences

and

Now,

Similarly,

Thus,

Again, the triangular inequality and (3.17) gives

and

i.e., we have

Also,

Using (3.17), (3.19) and (3.22), we have

Letting

Finally, letting

which is a contradiction. Therefore,

As g is continuous, so from (2.26), we have

By the compatibility of g and F, we have

Now, we show that F and g have an r-tupled coincidence point. To accomplish this, suppose (a) holds. i.e. F is continuous, then using (3.28) and (3.8), we see that

which gives

Hence

If (b) holds, since

Now, using triangle inequality together with (3.8), we get

Therefore,

Thus the theorem follows.

Corollary 3.1 Under the hypothesis of theorem 3.2 and satisfying contractive condition as (3.31)

Then F and g have a r-tupled coincidence point.

Proof: If we put

Uniqueness of r-tupled fixed point

For all

We say that

Theorem 3.3 In addition to the hypothesis of theorem 3.1, suppose that for every

Then exist

And

Then F and g have a unique r-coincidence point, which is a fixed point of

Proof. By theorem 3.2, the set of r-coincidence points is non-empty. Now, suppose that

We will show that

By assumption, there exists

is comparable to

and

Let

In addition, let

And

are comparable, then

We have

Then

Summing, we get

It follows that

For all

Similarly, one can prove that

Using (3.34), (3.35) and triangle inequality we get

As

Since

Denote

Hence

It follows from (3.32)

This means that

Now, from (3.37), we have

Hence,

To prove the uniqueness of the fixed point, assume that

Thus,

Authors are highly thankful for the financial support of this paper to Deanship of Scientific Research, Jazan University, K.S.A.

Authors declare that they have no conflict of interest.

Ibtisam Masmali,Sumitra Dalal, (2016) Study of Fixed Point Theorems for Higher Dimension in Partially Ordered Metric Spaces. Applied Mathematics,07,399-412. doi: 10.4236/am.2016.75037