1. Introduction and Preliminaries
Banach contraction principle [1] is classical and powerful in fixed point theory. It has been widely generalized (see [2] [3] [4] and others). Recently, fixed point theory in partially ordered metric spaces has been presented by many scholars: Ran and Reurings [5] , Agarwal et al. [6] , Bhsakar and Lakshmikantham [7] , Samet [8] , Berinde and Borcut [9] , Amini-Harandi [10] , etc., considered some coupled and tripled fixed point theorems. For more fixed point theorems in partially ordered metric spaces, one can refer to [11] [12] [13] and others.
This paper focuses on the tripled and N-order fixed point theory. For convenience, we denote
. Let
denote a partially ordered set
endowed a metric d (i.e.,
is a metric space). Our work is carried out on the following two preliminaries: a result about fixed point in partially ordered metric space in [6] and a definition of generally Meir-Keeler type function for the case of coupled fixed points in [8] .
Lemma 1.1 ( [6] ). Let
be a partially ordered metric space and suppose the metric space
is complete. Assume there is a nondecreasing function
with
as
for each
. If
is a nondecreasing mapping with
Assume that either
1)
is continuous or,
2) If a nondecreasing sequence
, then
.
If
with
then
has a fixed point. If for each
, there exists
which is comparable to x and y, then the fixed point of
is unique.
Definition 1 ( [8] ) Let
be a partially ordered metric space and
be a mapping. F is called generalized Meir-Keeler type function if for all
there exists
such that
(1.1)
Let
be a partially ordered set with a metric d on X,
and
be a given mapping. Let
be the partial order on
:
. We employ the notion of tripled fixed point introduced by Samet and Vetro which is investigated by Amini-Harandi [10] .
Definition 2 ( [11] ) An element
is called a tripled fixed point of
if
In this paper, we first define N-order generalized Meir-Keeler type contraction by adding some parameters (see Definition 3 and Definition 5), which is an extension of Definition 1. Then we use a simple approach introduced by [10] to discuss N-order fixed point theorems. We start our discussions with the tripled case. Section 2 devotes to tripled fixed point theorems. Section 3 devotes to N-order fixed point theory. Section 4 gives two examples to illustrate the results obtained in Section 2.
2. Tripled Fixed Point Theory
Recalling that
is a partially ordered set with a metric d on X and
. Let
be the metric and
be the partially order on
. For each
, we define
and
Now, we define tripled generalized Meir-Keeler type contraction which is a useful tool for the following theorems in this section.
Definition 3 Let
be a partially ordered metric space and
be a mapping. F is called a tripled generalized Meir-Keeler type contraction if for all
there exists
such that
(2.1)
where
are constants with
.
Theorem 2.1 Let
be a partially ordered metric space. Let
be the given constants with
. If
is a tripled generalized Meir-Keeler contraction mapping, then
for all
.
Proof. Let
such that
. Then it follows that
Setting
we have
By
being a tripled generalized Meir-Keeler type contraction, then
. □
Let
be a mapping. We say F is nondecreasing in each of its variables if
and
By the monotone property of F, we can get
(2.2)
For all
,
, we define:
(2.3)
with
.
In order to investigate the tripled fixed point of F, we introduce a mapping
which is defined by
(2.4)
Obviously, by the definition of
, we have
(2.5)
Simultaneously, by (2.3) and (2.4), we have
with
, and we have
Theorem 2.2 Let
be a partially ordered metric space and
be the given constants with
. Let
be nondecreasing in each of its variables and be a tripled generalized Meir-Keeler type contraction. There exist
with
. Then, for
, we have
1)
;
2)
;
3)
.
Proof. We first prove 1). Since
, due to the monotone property of F and (2.2), we have
,
and
. By
and (2.4), 1) holds for
. Now we assume 1) holds for
, i.e.
Then, we obtain
which means
. Using the same strategy, we have
and
. Hence we have
, that is, 1) holds for
. Simultaneously, we can also obtain that
and
.
Now, we prove 2). We consider
It follows from Theorem 2.1 and 1) that
and
Thus,
Last, we prove 3). From 2), we know that
exists. If
, we suppose that
(2.6)
Then it follows that
By (2.6), we have
which implies that there exists
such that
(2.7)
Since F is a tripled generalized Meir-Keeler type contraction, we get
(2.8)
By (2.7), we also have
and
Then, we get
(2.9)
and
(2.10)
From (2.8)-(2.10), we get
This is a contradiction. The proof is completed.
From the definition of T, we observe that the fixed point of T is exactly the tripled fixed point of F, that is,
We will obtain the tripled fixed point theorems by investigating the fixed point of T.
Theorem 2.3 Let
be a partially ordered metric space and
is a complete metric space. Let
be the given constants with
. Let
be nondecreasing in each of its variables and be a tripled generalized Meir-Keeler contraction.
be a mapping defined as (2.4) satisfying that there exists
with
. Then, there exists
which is a tripled fixed point of F, if either
1) F is continuous or
2) a nondecreasing sequence
, then
.
Furthermore, if
3) for
, there exists
that is comparable to
and
, we get the uniqueness of tripled fixed point of F and
.
Proof. Since
is a complete metric space, it is obvious that the metric space
is complete. By Theorem 2.2, T is non-decreasing. Meanwhile, by Theorem 2.1 and (2.5), for each
with
, we have
By Lemma 1.1, we deduce that T has a unique fixed point denoted by
, then
is the unique tripled fixed point of F.
However, we can check that
is also a tripled fixed point of F. In fact, since
is the tripled fixed point of F, i.e.,
, we have
which implies that
is also a tripled fixed point of F. By the uniqueness, we get
. □
Corollary 1 Suppose that all the hypotheses of Theorem 2.3 are satisfied, then the tripled fixed point
can be deduced by
(2.11)
Proof. By examining the proof of Theorem 2.3,
is actually the fixed point of T on
. According to the proof of Lemma 1.1 in [6] , we have
By the definition of
, we can easily get (2.11). □
Theorem 2.4 In addition to the hypotheses of Theorem 2.3 except (3), we have
by adding the hypotheses (3*):
in X are comparable.
Proof. Without the restriction of the generality, we assume that
. Setting
and
, it’s easy to see that
. From Theorem 1.1, we have
as
, which implies that
i.e.,
(2.12)
By the similar strategy, setting
and
, we can get
(2.13)
It follows from the triangular inequality that
Taking the limit as
, by (2.11), (2.12) and (2.13), we get
.
Similarly, by setting
and
we can get two equalities,
(2.14)
and
(2.15)
respectively. Then it follows from (2.11), (2.14) and (2.15) that
We get
. Hence we have
. □
3. N-Order Fixed Point Theorems
Let
be a partially ordered set with a metric d on X. Let
,
be the metric on
and
be the partially order. For each
, we define
and
Definition 4 [11] Let X be a non-empty set and
be a given mapping. An element
is called a N-order fixed point of F if
We introduce generally N-order generalized Meir-Keeler type contraction.
Definition 5 Let
be a partially ordered metric space and
be a mapping. F is called a N-order generalized Meir-Keeler contraction if for all
there exists
such that for
(3.16)
where
are constants with
.
Substituting the tripled case with N-order case in the discussions of Section 3, by the similar strategy, we can obtain the same results with Theorem 2.1, Theorem 2.2, Theorem 2.3, Corollary 1 and Theorem 2.4.
4. The Examples
This section provides two examples to illustrate Theorem 2.3 and Theorem 2.4.
Example 1 This example is aroused by [13] . Let
,
and
, defined by
It is easy to check that F satisfies all the hypotheses of Theorem 2.3 with
and
is the unique tripled fixed point of F.
Example 2 Let
For
,
and
.
is defined by
(4.1)
It is easy to check that:
1) F is continues on
;
2) F is a tripled generally Meir-Keeler type contraction. In fact, we can deduce that
3) Setting
, then we have
. Clearly, we have
;
4) Setting
, there are no elements in
which are comparable to
and
.
The above 4) implies that F doesn’t satisfy all the hypotheses of Theorem 2.3. However, the above 1)-3) imply that F satisfies all the hypotheses of Theorem 2.4, then F has the unique tripled fixed point
with
.
5. Conclusion
In this paper, we extend the definition generalized Meir-Keeler type contraction to N-ordered case. And we use it to discuss N-order fixed point theorems. In future work, we will study N-ordered fixed point theory with invariant set.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant No. 11701390).