Abstract
In this article, we obtain fixed point results under some rational contractive conditions for mappings in the setup of ordered ρ-metric spaces. Also, three examples and an application to existence of local solutions for first-order periodic problems are provided to illustrate the results presented herein.
PUBLIC INTEREST STATEMENT
Metric fixed point theory is a powerful tool for solving several problems in various parts of mathematics and its applications. In this article, we propose a new generalized approach to fixed point problems using so-called p-metric (or extended b-metric) and more general contractive conditions. We show how the obtained results can be applied to prove the existence of solutions of some involved boundary valued problems for differential equations.
1. Introduction
The field of extension of metric space is a growing field these years. The famous extensions of the concept of metric spaces have been done by (Czerwik, Citation1993) and (Matthews, Citation1994) where they introduced and studied the concepts of -metric spaces and partial metric spaces, respectively. In this article, we introduce some type of Geraghty contractive mappings and comparison contractive mappings in extended
-metric spaces and we establish some fixed point results in complete partially ordered extended
-metric spaces. Our results generalize several comparable results in the literature.
2. Preliminaries
Recall (see (Czerwik, Citation1993)) that a -metric
on a set
is a generalization of standard metric, where the triangular inequality is replaced by
for some fixed . For more details on fixed point results and their applications in
-metric spaces we refer the reader to (Ali, Kamran, & Postolache, Citation2017; Kamran, Postolache, Ali, & Kiran, Citation2016; Shatanawi, Pitea, & Lazovic, Citation2014). Motivated by the concept of
-metric spaces, the following further generalization has been recently presented by Parvaneh and Ghoncheh.
Definition 2.1. (CitationParvaneh & Hosseini Ghoncheh) Let be a (nonempty) set. A function
is an extended
-metric (
-metric, for short) if there exists a strictly increasing continuous function
with
for all
and
such that for all
, the following conditions hold:
(P1) iff
,
(P2) ,
(P3)
In this case, the pair is called a
-metric space, or an extended
-metric space.
It should be noted that each -metric is a
-metric, with
for some
, while each metric is a
-metric, with
. More general examples of
-metrics can be constructed using the following easy proposition.
Proposition 2.2. Let be a
-metric space with coefficient
and let
where
is a strictly increasing continuous function with
for
and
. Then,
is a
-metric with
.
Taking various functions in the previous proposition, we can obtain a lot of examples of
-metrics. We state just a few of them which will be used later in the text.
Example 2.3. (1) If , we get
and
. Note that
.
(2) If , we get
and
. Note that
.
3) If , then
and
. Note that in this case
, for
, where
is the Lambert
-function (see, e.g. (Dence, Citation2013)).
4) If , then
and
, for
.
Note that such functions and
generate
-metric spaces which are usually not
-metric spaces. For instance, in the case (2) of the previous example (if
), it was shown in (CitationParvaneh, Dinmohammadi, & Kadelburg) that there is no
such that
is a
-metric with parameter
.
Definition 2.4. (CitationParvaneh & Hosseini Ghoncheh) Let be a
-metric space. Then a sequence
in
is called:
(a) -convergent if there exists
such that
, as
. In this case, we write
;
(b) -Cauchy if
as
.
(c) The -metric space
is
-complete if every
-Cauchy sequence in
-converges.
We will need the following simple lemma about the -convergent sequences.
Lemma 2.5. (CitationParvaneh & Hosseini Ghoncheh) Let be a
-metric space with the function
, and suppose that and
-converge to, respectively. Then, we have
In particular, if , then
. Moreover, for each
we have
Some generalizations of Banach Contraction Principle were obtained, starting from (Jaggi, Citation1977), by using rational contractive conditions in various spaces. On the other hand, Ran and Reurings initiated the studying of fixed point results on partially ordered sets in (Ran & Reurings, Citation2004), which was widely used in a lot of subsequent papers. Geraghty-type contractive conditions (Geraghty, Citation1973) were also the matter of investigation of several researchers (see, e.g. (Duki, Kadelburg, & Radenovi, Citation2011)).
In this article, we introduce the notions of rational -Geraghty contractive mappings (in three versions) and rational
-
contractive mappings in
-metric spaces and we establish some fixed point results in complete partially ordered
-metric spaces. Our results generalize several comparable results in the literature. Also, three examples and an application to existence of local solutions for first-order periodic problems are provided to illustrate the results presented herein.
3. Main results
In the rest of the article, will always be a
-metric space with function
.
3.1. Fixed point results via generalized Geraghty functions
Let denote the class of all functions
satisfying the following condition:
Definition 3.1. Let be a partially ordered
-metric space. A mapping
is called a rational Geraghty contraction of type I if there exists
such that
holds for all with
, where
A partially ordered -metric space
is said to have the sequential limit comparison property (s.l.c. property) if for every non-decreasing sequence
in
, the convergence of
to some
yields that
for all
.
Theorem 3.2. Let be a partially ordered
-complete
-metric space. Let
be a nondecreasing mapping with respect to
such that there exists an element
with
. Suppose that
is a rational Geraghty contraction of type
. If
(I) is continuous, or
(II) has the s.l.c. property,
then has a fixed point. Moreover, the set of fixed points of
is well ordered if and only if
has a unique fixed point.
Proof. Let for all
. Since
and
is a nondecreasing mapping, we obtain by induction that
If for some
, there is nothing to prove. So, we will assume that
for all
. We will realize the proof in the following steps.
Step I: We will show that . Since
for each
, then by (3.1) we have
where
If , then from (3.2) we would have
which is a contradiction. Hence, . So, from (3.2),
Since is a decreasing sequence, there exists
such that
. We will prove that
. Suppose on contrary that
. Then, letting
in (3.3) we have
which implies that . Now, as
we conclude that
a contradiction. That is,
Step II: Now, we prove that the sequence is a
-Cauchy sequence. Suppose the contrary. Then there exists
for which we can find two subsequences
and
of
such that
is the smallest index for which
This means that
From (3.4) and using the triangular inequality, we get
By taking the upper limit as , we get
From the definition of and the above limits,
Now, from (3.1) and the above inequalities, we have
which implies that . Now, as
we conclude that
. Consequently,
a contradiction to (3.5). Therefore, is a
-Cauchy sequence.
-completeness of
yields that
-converges to a point
.
Step III: is a fixed point of
.
First, let be continuous. Then, we have
Now, let (II) hold. Using the assumption on we have
. By Lemma 2.5
where
Therefore, from the above relations, we deduce that , so,
.
Finally, suppose that the set of fixed point of is well ordered. Assume on contrary, that
and
are two fixed points of
such that
. Then, for example
, and by (3.1), we have
Since
we get , a contradiction. Hence,
, and the fixed point of
is unique.
Definition 3.3. Let be a partially ordered
-metric space. A mapping
is called a rational Geraghty contraction of type II if there exists
such that
for all with
, where
Theorem 3.4. Let be a partially ordered
-complete
-metric space. Let
be a nondecreasing mapping with respect to
such that there exists an element
with
. Suppose that
is a rational Geraghty contractive mapping of type
. If
(I) is continuous, or
(II) has the s.l.c. property.
Then has a fixed point. Moreover, the set of fixed points of
is well ordered if and only if
has a unique fixed point.
Proof. The method of proof is similar to the proof of Theorem 6 of (Zabihi & Razani, Citation2014) and the previous theorem.
Definition 3.5. Let be a partially ordered
-metric space. A mapping
is called a rational Geraghty contraction of type III if there exists
such that
for all with
, where
Theorem 3.6. Let be a partially ordered
-complete
-metric space. Let
be a nondecreasing mapping with respect to
such that there exists an element
with
. Suppose that
is a rational Geraghty contractive mapping of type
. If
(I) is continuous, or
(II) has the s.l.c. property,
then has a fixed point. Moreover, the set of fixed points of
is well ordered if and only if
has a unique fixed point.
Proof. Put .
Step I: We will show that . Since
for each
, then by (3.6) we have
because
Therefore, is decreasing. Similar to what have been done in Theorem 3.2, we have
Step II: Now, we prove that the sequence is a
-Cauchy sequence. Suppose the contrary. Then there exists
for which we can find two subsequences
and
of
such that
is the smallest index for which
This means that
From (3.8) and using the triangular inequality, we get
By taking the upper limit as , we get
Using the triangular inequality, we have
Taking the upper limit as in the above inequality and using (3.7) and (3.9) we get
Again, using the triangular inequality, we have
Taking the upper limit as in the above inequality and using (3.9) we get
From the definition of and the above limits,
Now, from (3.6) and the above inequalities, we have
which implies that . Now, as
we conclude that
is a p-Cauchy sequence. p-completeness of
yields that
p-converges to a point
.
The rest of the proof is done in a similar manner as in Theorem 3.2.
Choosing different -metrics, as well as functions
, we can obtain various corollaries from the above theorems. For example, using the
-metric space mentioned in Example 2.3.(3) and an appropriate constant function
, we obtain from Theorems 3.2 and 3.4 the following two corollaries.
Corollary 3.7. Let be a partially ordered
-complete
-metric space with parameter
, and let
be an increasing mapping with respect to
such that there exists an element
with
. Suppose that
for all with
, where
,
, and
or
If is continuous, or
has the s.l.c. property, then
has a fixed point.
Corollary 3.8. Let be a partially ordered
-complete
-metric space with parameter
, and let
be an increasing mapping with respect to
such that there exists an element
with
. Suppose that
or
for all with
, where
and
,
. If
is continuous, or
has the s.l.c. property, then
has a fixed point.
Consider now the -metric defined as in Example 2.3.(1) (for simplicity, take
) and the function
given as
where and
(recall that
,
and
). Then we obtain the following corollary of Theorems 3.2 and 3.4.
Corollary 3.9. Let be a partially ordered complete metric space, and let
be an increasing mapping with respect to
such that there exists an element
with
. Suppose that
for some and all
with
, where
or
If is continuous, or
has the s.l.c. property, then
has a fixed point.
3.2. Fixed point results via comparison functions
Let be the family of all nondecreasing functions
such that
for all .
Lemma 3.10. If , then the following are satisfied:
(a) for all
;
(b) .
Theorem 3.11. Let be a partially ordered
-complete
-metric space, and let
be an increasing mapping with respect to
such that there exists an element
with
. Suppose that
for some and for all
with
, where
If is continuous, or
has the s.l.c. property, then
has a fixed point. Moreover, the set of fixed points of
is well ordered if and only if
has one and only one fixed point.
Proof. Repeating the beginning of the proof of Theorem 3.2, we obtain the increasing sequence ,
such that
and, this time,
By induction, we get that
As , we conclude that
Now, we prove that the sequence is a
-Cauchy sequence. Suppose the contrary, i.e. Then, again as in Theorem 3.2, there exists
for which we can find two subsequences
and
of
such that
is the smallest index for which
This means that
Obviously,
From the definition of and the above limits we can conclude that
Now, from (3.13) and the above inequalities, we have
which is a contradiction. Consequently, is a
-Cauchy sequence.
-completeness of
yields that
-converges to a point
. If
is continuous, we have
and is a fixed point of
.
Let have the s.l.c. property. Then we have
. By (3.13) we have
where
Letting , we get
Again, taking the upper limit as in (3.14) and using Lemma 2.5 and (3.15) we get
So, we get , i.e.
. □
Remark 3.12. In Theorem 3.11, we can replace by the expressions used in Theorems 3.4 or 3.6.
Similarly as for Theorems 3.2, 3.4 and 3.6, we can get several consequences of Theorem 3.11, using various -metrics and/or various functions
. We state just the following, based on Example 2.3.(4).
Corollary 3.13. Let be a partially ordered
-complete
-metric space with parameter
, and let
be an increasing mapping with respect to
such that there exists an element
with
. Suppose that
where
for some and all
with
. If
is continuous, or
has the s.l.c. property, then
has a fixed point. Moreover, the set of fixed points of
is well ordered if and only if
has one and only one fixed point.
Remark 3.14. As any -metric is a
-metric with
for all
, so our results modify the obtained results in (Shahkoohi & Razani, Citation2014; Zabihi & Razani, Citation2014) and several other articles.
4. Examples
Example 4.1. Let be equipped with the
-metric
for all
, where
, with
(Example 2.3.(1)).
Define a relation on
by
iff
, a mapping
by
and a function by
. For all
with
, we have:
Therefore,
So, from Theorem 3.2, has a fixed point.
Example 4.2. Let be equipped with the
-metric generated from the standard metric
as in Corollary 3.9, i.e. take
. Define a partial order on
by
and consider the mapping given by
Then, is a partially ordered
-complete
-metric space and
is a nondecreasing mapping. Take the function
given by
where (i.e. take
in Corollary 3.9). We will check the condition (3.11) of this corollary. Considering elements
with
, the following cases are nontrivial.
(1) ,
. Then we have
,
and
(note that is an increasing function for
).
(2) ,
. Then we have
,
and
(3) ,
. Then we have
,
and
Hence, all the conditions of Corollary 3.9 are fulfilled and the mapping has a (unique) fixed point (which is
).
Note that the same conclusion could not be obtained if the space without partial order were used. Indeed, in this case we should have also the following case to consider.
(4) ,
. Then
and
and it is trivial to check that the condition (3.11) does not hold. In fact, none of the known fixed point results can be used to obtain the conclusion in this case.
Example 4.3. Let be also equipped with the
-metric
for all
, where
. Define a relation
on
by
iff
, a mapping
by
and a function by
. It is easy to see that
and
for all
.
For all with
, by Mean Value Theorem, we have
So, from Theorem 3.11, has a fixed point.
5. Application to existence of local solutions for first-order periodic problems
In this section, we present an application to existence of a solution for a periodic problem which is a consequence of Theorem 3.11. This kind of application first appeared in (Nieto & Rodríguez-López, Citation2007).
Let be the set of all real continuous functions on
where
. Obviously, this space with the
-metric given by
for all is a
-complete
-metric space with
. Secondly,
can also be equipped with a partial order given by
Moreover, as in (Nieto & Rodríguez-López, Citation2007), it can be proved that enjoys the s.l.c. property.
Consider the following first-order periodic boundary value problem
where and
is a given continuous function. A lower solution for (5.1) is a function
such that
where .
Assume that there exists such that, for all
and
, we have
Problem (5.1) can be rewritten as
where . It is well known that this problem is equivalent to the integral equation
where is the Green’s function given as
Now define an operator by
The mapping is nondecreasing (Harjani & Sadarangani, Citation2009). Note that if
is a fixed point of
then
is a solution of (5.1).
Let . Then we have
where from , where
Finally, let be a lower solution for (5.1). In (Harjani & Sadarangani, Citation2009), it was shown that
.
Hence, the hypotheses of Theorem 3.11 are satisfied with . Therefore, there exists a fixed point
such that
. This
is then a solution of problem (5.1).
6. Conclusion
Taking , our obtained results coincide with the results in usual
-metric spaces and taking
, the obtained results coincide with the results in usual metric spaces.
Additional information
Funding
Notes on contributors
Vahid Parvaneh
Vahid Parvaneh The authors of this article have been working on problems in metric fixed point theory for several years, together or with some other coauthors. Various results were obtained in different setups, generalizing some of the well-known theorems of this theory. In particular, V. Parvaneh, together with Ghoncheh, introduced a new environment, called p-metric spaces, which was further developed to obtain various new fixed point results. It is the area of research in which several new results are expected.
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