
We present
The Banach Contraction Principle (BCP) is the most famous elementary result in the metric fixed point theory. A huge amount of literature contains applications, generalizations and extensions of this principle carried out by several authors in different directions, e.g., by weakening the hypotheses, using different setups, considering various types of mappings and generalized form of metric spaces, see, e.g., [2, 6, 7, 10, 12, 14, 15, 18].
In this context, Matthews [12] introduced the notion of a partial metric space as a part of the study of denotational semantics of data-flow networks. He showed that BCP can be generalized to the partial metric context for applications in program verifications. Note that in partial metric spaces, self-distance of an arbitrary point need not be equal to zero.
Hitzler and Seda [6], resp. Amini-Harandi [2] made a further generalization under the name of dislocated, resp. metric-like space, also having the property of “non-zero self-distance”. Amini-Harandi defined
We recall some definitions and facts which will be used throughout the paper.
A mapping , for a nonempty set
, is called
,
(
(
(
In this case, the pair ( ,
Let and
be defined by
Then ( ,
Let . Then the mappings
, defined by
where .
Let ( ,
A sequence { converges to a point
(denoted as
,
such that
A sequence {,
,
converges to a point
such that
A mapping is
,
It is easy to show (e.g., [2]) that the limit of a sequence in a metric-like space might not be unique.
,
.
It is clear that if a metric-like space is
Let and
be defined by
for all . Then (
,
,
In the paper [18], Wardowski introduced a new type of contractions which he called
Denote by the family of all functions
(F1)
(F2) for each sequence {
(F3) There exists
Let
Let ( ,
is called an
and
for all with
Note that, taking defined by
i.e., such mapping is a (Banach-type) contraction. However, other functions may produce new concepts (see the respective examples in [18]). Hence, the notion of
Contraction-type mappings have been also generalized in other directions. In the series of generalizations, Samet et al. [13] introduced the concept of
With the above discussion in mind, we introduce in this paper the notion of
Recently, Sintunavarat [16] introduced the notion of weakly
For a nonempty set , let
and
be two mappings. Then
Let . Define mappings
and
by
It is easy to see that
In what follows, we use the following terminology from the paper [17]. For a nonempty set and a mapping
, we use
and
to denote the collection of all
and the collection of all weakly
, respectively. Obviously,
and, by the previous example, the inclusion can be strict.
We introduce now the notion of
Let ( ,
. A self-mapping
is called an
and
where
We denote by the collection of all
,
If we take , new conditions can be obtained (see further Remark 2.7).
We are equipped now to state our first main result.
,
(;
(
(,
(
Starting from the given satisfying
by
Using that and
Repeating this process, we obtain
It follows from that
If ϱ
Therefore we derive
that is,
From (
Now by the property (F3), there exists
By (
Passing to the limit as
From (
In order to show that {
By the convergence of the series ,
is a 0-
such that
Now, since
This proves that
We note that the previous result can still be valid for
,
,
(
(
Following the proof of Theorem 2.4, we obtain a 0-,
such that
We have to prove that , we have
Passing to the limit as
To ensure the uniqueness of the fixed point, we will consider the following hypothesis.
(H0): For all
Suppose that
a contradiction, which implies that
Taking various concrete functions in the condition (
Taking
Taking
Taking
Taking
The following examples can be used to illustrate the usage of Theorems 2.4–2.6.
(This example demonstrates the use of rational terms in the contractive condition.)
Let and
be given by
Then ( ,
and
by
Moreover, take defined by
needs to be checked. In this case it reduces to
for all with
Case 1:
Hence, (
Case 2:
Hence, (
Case 3:
Hence, (
Thus, all the conditions of Theorem 2.6 are satisfied and the mapping
(This example demonstrates the advantage of using a metric-like instead of a standard metric; it is inspired by [11, Example 6]. Also, it shows the advantage of using
Let be equipped with
given by
,
and
by
Moreover, take defined by
needs to be checked. In this case it reduces to
for all with
Case 1:
Hence, (
Case 2:
Hence, (
Case 3:
Hence, (
Case 4:
Hence, (
Thus, all the conditions of Theorem 2.6 are satisfied and the mapping
Consider now the same example, but using the standard metric . Then, e.g., for
whatever
(This example, inspired by [18, Example 2.5] and [1, Example 22], shows the reason for using various functions —it cannot be treated by the simplest example
Let , where
by
Then ( ,
and
given by
Take given by
for with
for
Case 1:
Case 2:
Now,
Hence,
Therefore, all the conditions of Theorem 2.6 are satisfied, and
The same conclusion cannot be obtained if the simplest function is used, because the Banach-type condition (
i.e.,
In order to complete the results, we first need the following notion.
For a nonempty set , let
and
be mappings. We say that (
, we have
We use to denote the collection of all generalized
If the operator
If
Now we introduce the notion of rational
Let ( ,
be a mapping. Two self-mappings
are said to form a
,
We denote by the collection of all rational
,
We are equipped now to state the first result of this section.
,
,
(
(
(
(
By the given condition ( such that
by
If there exists
Since and
Repeating this process, we obtain
Again from (
Repeating this process, we obtain
Therefore implies that
If ϱ2
Again using , we get
With similar arguments, we get
Combining (
Using the arguments of proof of Theorem 2.4, we reach to (,
is a 0-complete metric-like space, there exists an
such that lim
This implies that
Using (
Further we shall prove (
Using and
By the ,
From (
Similarly using
Thus we have reached (
Let be endowed with the
Consider the mappings given by
and by
Moreover, take defined by
where
In order to show , suppose that
Hence, (.
Next we prove . Let
be such that
and
Then, .
We divide in three parts the proof that .
1° For
2° For
3° For
Thus,
Moreover, the mappings ,
We note that the previous result can be sharpened when continuity of
(H1): If { such that if
Now we have the following result.
,
Following the proof of Theorem 3.4, we obtain a 0-,
such that
that is, , there exists a subsequence {
Applying the limit as
a contradiction. Therefore we have
Similarly, by taking
To ensure the uniqueness of the common fixed point, we will consider the following hypothesis as a generalized form of (
(H2): For all
Suppose that
a contradiction, which implies that
This section is devoted to the existence of solutions of an integral equation as an application of Theorem 2.5.
Let , where
We will consider the integral equation
where
define the respective operator .
(F1)
(F2)
(F3) ,
(F4)
(F5)
(F6)
.
Define a function by
where .
We are going to check that . For this, let
be such that
, we have
Now, using the assumption (F6), after routine calculations, we obtain
where is given in (
Now, by considering given by
for all with
. Thus
. Therefore, all the hypotheses of Theorem 2.5 are satisfied and we conclude that there is a fixed point
of the operator
. It is well known that in this case
This section is devoted to the existence of solutions of a nonlinear fractional differential equation as an application of Theorem 2.5. It is inspired by the paper [4].
Recall that the Caputo derivative of fractional order
where
We consider the nonlinear fractional differential equation of the form
with the integral boundary conditions
where
Let the corresponding operator be defined by
for ,
given by
.
Here we have to check the contraction condition ( (see Remark 2.7.(III)).
For each , we have
Applying now Cauchy-Schwartz inequality, we get
Now, using the assumption (F6′), after routine calculations, we obtain
This implies that
where is given in (
Now, by considering given by
for all with
. Thus
. Therefore, all the hypotheses of Theorem 2.5 are satisfied and we conclude that there is a fixed point
of the operator
. It is well known (see, e.g., [4, Theorem 3.1]) that in this case
Consider the following nonlinear fractional differential equation
with the three-point integral boundary value condition
Here of the