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