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Generalized Meir-Keeler type n-tupled fixed point theorems in ordered partial metric spaces
Fixed Point Theory and Applications volume 2014, Article number: 114 (2014)
Abstract
In this paper, we prove n-tupled fixed point theorems (for even n) for mappings satisfying Meir-Keeler type contractive condition besides enjoying mixed monotone property in ordered partial metric spaces. As applications, some results of integral type are also derived. Our results generalize the corresponding results of Erduran and Imdad (J. Nonlinear Anal. Appl. 2012:jnaa-00169, 2012).
1 Introduction
Existence of a fixed point for contraction type mappings in partially ordered metric spaces with possible applications have been considered recently by many authors (e.g. [1–31]). Recently many researchers have obtained fixed and common fixed point results on partially ordered metric spaces (see [5, 10, 11, 18, 22, 32, 33]). In 2006, Bhaskar and Lakshmikantham [13] initiated the idea of coupled fixed point and proved some interesting coupled fixed point theorems for mappings satisfying a mixed monotone property. In this continuation, Lakshmikantham and Ćirić [17] generalized these results for nonlinear ϕ-contraction mappings by introducing two ideas namely: coupled coincidence point and mixed g-monotone property. Thereafter Samet and Vetro [34] extended the idea of coupled fixed point to higher dimensions by introducing the notion of fixed point of n-order (or n-tupled fixed point, where n is natural number greater than or equal to 2) and presented some n-tupled fixed point results in complete metric spaces, using a new concept of F-invariant set. On the other hand, Imdad et al. [35] generalized the idea of n-tupled fixed point by considering even-tupled coincidence point besides exploiting the idea of mixed g-monotone property on and proved an even-tupled coincidence point theorem for nonlinear ϕ-contraction mappings satisfying mixed g-monotone property.
The concept of partial metric space was introduced by Matthews [36] in 1994, which is a generalization of usual metric space. In such spaces, the distance of a point to itself may not be zero. The main motivation behind the idea of a partial metric space is to transfer mathematical techniques into computer science. Following this initial work, Matthews [36] generalized the Banach contraction principle in the context of complete partial metric spaces. For more details, we refer the reader to [4, 7–9, 20, 24–26, 37–48].
Samet [27] introduced the concept of generalized Meir-Keeler type contraction function and proved some coupled fixed point theorems in partially ordered metric spaces. In different years, many researchers studied and worked on this contraction condition. Recently, in [12], Berinde and Pǎcurar gave the concept of symmetric Meir-Keeler type condition and generalized several results in the literature. Very recently, Erduran and Imdad [49] generalized the coupled fixed point theorems in the context of partial metric spaces. In this paper, we established some n-tupled fixed point theorems for generalized Meir-Keeler type contraction condition in ordered partial metric spaces enjoying strict mixed monotone property. In this paper, we prove the existence and uniqueness of some Meir-Keeler type n-tupled fixed point theorems in the context of partially ordered partial metric spaces. The presented theorems extend and improve the recent coupled fixed point theorems due to Erduran and Imdad [49].
2 Preliminaries
In this section, we collect some definitions and properties of partial metric space which are relevant to our presentation.
Definition 2.1 A partial metric on a nonempty set X is a function such that for all ,
(p1) ,
(p2) ,
(p3) ,
(p4) .
A partial metric space is a pair such that X is a nonempty set and p is a partial metric on X.
Remark 2.1 It is clear that if , then from (p1), (p2) and (p3), . But if , may not be zero.
Each partial metric p on X generates a topology on X which has as a base the family of open p-balls , where for all and .
If p is a partial metric on X, then the function given by
is a metric on X.
Consider with . Then is a partial metric space. It is clear that p is not a (usual) metric. Note that in this case .
Example 2.2 [50]
Let and define . Then is a partial metric space.
Example 2.3 [50]
Let and define by
Then is a complete partial metric space.
Example 2.4 [51]
Let and be metric space and partial metric space, respectively. Then the mappings () defined by
induce partial metrics on X, where is an arbitrary function and .
Definition 2.2 Let be a partial metric space and be a sequence in X. Then
-
(i)
converges to a point if and only if ,
-
(ii)
is said to be a Cauchy sequence if exists (and is finite).
Definition 2.3 A partial metric space is said to be complete if every Cauchy sequence converges with respect to , to a point , such that .
Lemma 2.1 Let be a partial metric space. Then
-
(i)
is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space ,
-
(ii)
is complete if and only if the metric space is complete. Furthermore, if and only if
In [52], Meir-Keeler generalized the well-known Banach fixed point theorem by proving the following interesting fixed point theorem.
Theorem 2.1 [52]
Let be a complete metric space and be a mapping. Suppose that for every there exists such that for all
Then T has a unique fixed point and for all , the sequence converges to z.
In recent years, many authors generalized Meir-Keeler fixed point theorems in various ways in various spaces which include complete metric space as well as ordered metric space. In [27], Samet introduced the concept of generalized Meir-Keeler type contraction function and proved some coupled fixed point results. Samet [27] introduced the definition below to modify the Meir-Keeler contraction and extended its applications.
Definition 2.4 [27]
Let be a partially ordered metric space and be a given mapping. Then F is a generalized Meir-Keeler type function if for all there exists such that
Very recently Erduran and Imdad [49] generalized the results of Samet [27] for ordered partial metric spaces. For more details, see [12, 27, 53, 54].
Erduran and Imdad [49] proved the following result:
Theorem 2.2 [49]
Let be a partially ordered set and suppose there is a partial metric p on X such that is complete partial metric space. Let be mapping satisfying the following hypotheses:
-
(1)
F has the mixed strict monotone property,
-
(2)
F is a generalized Meir-Keeler type function,
-
(3)
F is continuous or X has the following properties:
-
(a)
if a nondecreasing sequence , then for all n,
-
(b)
if a nonincreasing sequence , then for all n.
If there exist such that and , then there exists such that and . Furthermore, .
Note Throughout the paper we consider n to be an even integer.
Let be a partial metric. We endow , n times () with the partial metric η defined for by
Let be a given mapping. Then for all and for all , , we denote
In this paper, we used the concept of n-tupled fixed point given by Samet and Vetro [34]. We recall some basic concepts.
Definition 2.5 [35]
Let be a partially ordered set and be a mapping. The mapping F is said to have the mixed monotone property if F is nondecreasing in its odd position arguments and nonincreasing in its even position arguments, that is, if,
Definition 2.6 [34]
An element is called an n-tupled fixed point of the mapping if
Example 2.5 Let be a partially ordered metric space under natural setting and let be a mapping defined by , for any . Then is an n-tupled fixed point of F.
Remark 2.2 Definition 2.6 with respectively yields the definition of coupled fixed point [13] and quadrupled fixed point [55].
3 Main results
We begin this section by defining the following definitions:
Definition 3.1 Let be a partially ordered set and be a mapping. The mapping F is said to have the mixed strict monotone property if F is nondecreasing in its odd position arguments and nonincreasing in its even position arguments, that is, if,
Definition 3.2 Let be a partially ordered partial metric space and be a given mapping. We say that F is a generalized Meir-Keeler type function if for all there exists such that for with
The aim of this work is to prove the following results:
Lemma 3.1 Let be a partially ordered set and suppose that there is a partial metric p on X such that is a complete partial metric space. Let be a given mapping. If F is a generalized Meir-Keeler type function, then for
with or .
Proof Let such that
Then . Since F is a generalized Meir-Keeler type function. Therefore for , there exists such that
Putting and , we obtain the desired result. □
Lemma 3.2 Let be a partially ordered set and suppose that there is a partial metric p on X such that is a complete partial metric space. Let be a given mapping. Assume that the following hypotheses hold:
-
(1)
F has the mixed strict monotone property,
-
(2)
F is a generalized Meir-Keeler type function,
-
(3)
there exist with .
Then
Proof We claim that:
with the notation . Then by the mixed strict monotone property of F,
Then we have . Also
Therefore .
And similarly
Therefore . Thus (3.3) is satisfied for . For , we use the same strategy. We have
Thus we get
Now,
Therefore we get
In the same way,
Thus we have
Thus (3.3) is satisfied for . Repeating the same argument for each m, we see that (3.3) holds. Now using Lemma 3.1 and (3.3), we get
Also we have
Similarly we have,
Combining (3.4), (3.5), and (3.6), we get
This implies that
is a decreasing convergent sequence. Thus there exists such that
Now we show that . Assume that . This implies that there exists such that
In this case we have
It follows from (3.3) and hypothesis (2) that
that is,
On the other hand, we have
which implies that
Similarly,
Combining (3.7), (3.8), and (3.9), we have
which is a contradiction. Therefore, we have necessarily . That is,
□
Remark 3.1 Lemma 3.2 also holds if we replace condition (3) by such that .
Theorem 3.1 Let be a partially ordered set and suppose that there is a partial metric p on X such that is a complete partial metric space. Let be a given mapping satisfying the following hypotheses:
-
(1)
F is continuous,
-
(2)
F has the mixed strict monotone property,
-
(3)
F is a generalized Meir-Keeler type function,
-
(4)
there exist such that
(3.10)
Then there exist such that .
Proof Let us define sequences in X by
Since F has mixed monotone property and from (3.3) we have
Applying Lemma 3.2 by taking and , then we get
that is,
Denote
From the definition of , it is clear that
Using (3.11), we get
Let . It follows from (3.12) that there exists such that
Without restriction of generality, we can suppose that . We introduce the set defined by
Now we will prove that ,
Let . We have
We consider the following two cases.
Case I: .
By Lemma 3.1, we have
Case II: .
We have
In this case, we get
Since , by (3) we get
Also we have
By (3), this implies that
In the same way we have
Hence combining (3.15)-(3.18), we obtain
On the other hand, using (2), we can check easily that
Hence, we deduce that (3.14) holds. By (3.13), we have . This implies with (3.14) that
Thus for all , we have . This implies that for all , we have
We deduce that is a Cauchy sequence in the metric space . Since is complete, from Lemma 2.1, is a complete metric space. Therefore is complete. Hence there exist such that
which shows that
Therefore from Lemma 2.1 and using (3.12), we have
We will show that . Since F is continuous on X, then F is continuous at . Hence for any , there exists such that if verifying
means that
because , then we have
Since
for , there exists such that for , , . Then for , , we have
so we get
Now, for any ,
On the other hand, since F is a generalized Meir-Keeler type function, then from Lemma 3.1, we have
In this case, for any , . This implies that . Similarly we can show that
Thus we have proved that F has an n-tupled fixed point. □
Remark 3.2 Theorem 3.1 still holds if we replace (3.10) by such that
Theorem 3.2 Let be a partially ordered set and suppose that there is a partial metric p on X such that is a complete partial metric space. Assume that X has the following properties:
-
(a)
if a nondecreasing sequence then for all ,
-
(b)
if a nonincreasing sequence then for all .
Let be a given mapping satisfying the following hypotheses:
-
(1)
F is continuous,
-
(2)
F has the mixed strict monotone property,
-
(3)
F is a generalized Meir-Keeler type function,
-
(4)
there exist such that (3.10) holds.
Then there exists such that . Furthermore, .
Proof Following the proof of Theorem 3.1, we only have to prove that
Let . Since . Then there exist such that for all ,
Taking and using
by (3.21) and Lemma 3.1, we get
This implies that . Similarly, we can show that
which implies that .
This completes the proof. □
Now we endow the product space with the following partial order: for ,
One can prove that n-tupled fixed point is in fact unique and the product space endow with this partial order has the following property:
-
(A)
, that is comparable to and .
Theorem 3.3 Adding (A) to the hypotheses of Theorem 3.1 (respectively, Theorem 3.2), we obtain the uniqueness of n-tupled fixed point of F.
Proof Suppose that is another n-tupled fixed point of F. We distinguish two cases:
Case I: is comparable to with respect to ordering in , where
Without restriction of generality, we can suppose that
We have
Case II: is not comparable to . Then there exists that is comparable to and . Without restriction of generality, we can assume that
From (3.22) and Lemma 3.2, we have
Similarly we have
On the other hand, using the triangular inequality, we get
By (3.22) and (3.23), we have , we get
This completes the proof. □
4 Applications
In this section, using the earlier results proved in the preceding section, we obtain some n-tupled fixed point theorem for mappings satisfying a general contractive condition of integral type in partially ordered complete partial metric spaces.
Theorem 4.1 Let be a partially ordered set and suppose that there is a partial metric p on X such that is a complete partial metric space. Let be a given mapping. Assume that there exists a function θ from into itself satisfying the following:
-
(1)
and for every ,
-
(2)
θ is nondecreasing and right continuous,
-
(3)
for every , there exists such that
for all .
Then F is a generalized Meir-Keeler type function.
Proof Fix . Since , there exists and such that
From the right continuity of θ, there exists such that . Fix , then
Since θ is a nondecreasing function, we get
By (4.1) we get
and hence
□
The following result is an immediate consequence of Theorems 3.1, 3.2 and 4.1.
Corollary 4.1 Let be a partially ordered set and suppose that there is a partial metric p on X such that is a complete partial metric space. Let be a given mapping satisfying the following hypotheses:
-
(1)
F is continuous,
-
(2)
F has the mixed strict monotone property,
-
(3)
for all , there exists such that
for all , where φ is a locally integrable function from into itself satisfying
-
(4)
such that (3.10) holds.
Then there exists such that
Moreover, if property (A) is satisfied, then the n-tupled fixed point of F remains unique.
Remark 4.1 The conclusions of the preceding corollary remain valid if we replace the continuity hypothesis of F by hypotheses (a) and (b) of Theorem 3.2.
Corollary 4.2 Let be a partially ordered set and suppose that there is a partial metric p on X such that is a complete partial metric space. Let be a given mapping satisfying the following hypotheses:
-
(1)
F is continuous,
-
(2)
F has the mixed strict monotone property,
-
(3)
for all ,
where and φ is a locally integrable function from into itself satisfying
-
(4)
such that (3.10) holds.
Then there exists such that
Moreover, if property (A) is satisfied, then the n-tupled fixed point of F remains unique.
Proof For all , take and apply Corollary 4.1. □
Remark 4.2 We replace the continuity hypothesis of F by hypotheses (a) and (b) of Theorem 3.2, then this result also remains true.
5 Example
We give the following example to illustrate our main result.
Example 5.1 Let . Then is a partially ordered set under the natural ordering of real numbers. Define by , . Then is a complete partial metric space.
Now for any fixed even integer , consider the product space , n times (in short we write ). Define by
Then F has the mixed strict monotone property. Also F is a generalized Meir-Keeler type function. The proof follows in two parts, that is, we prove the following:
For with ,
The first part is trivial. For second part, we have
Hence all the hypotheses of Theorem 3.1 are satisfied. Therefore, F has a unique n-tupled fixed point. Here is an n-tupled fixed point of F.
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Imdad, M., Sharma, A. & Erduran, A. Generalized Meir-Keeler type n-tupled fixed point theorems in ordered partial metric spaces. Fixed Point Theory Appl 2014, 114 (2014). https://doi.org/10.1186/1687-1812-2014-114
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DOI: https://doi.org/10.1186/1687-1812-2014-114