We introduce the notion of quasi-
Jleli and Samet [
For the sake of completeness, we collect some basic concepts and results from the literature. Let
We say that
There exists
In this case, the pair
If, in addition to the conditions in Definition
is satisfied for each
In what follows we shall define some basic topological notions for quasi-
Let
The
Let
Let
Let
Let
Then, clearly, Case 1: if Case 2: if
On the other hand, if
Let for every for every
Here, the pair
Any quasi-metric space is a quasi-
Proof is straightforward, so we omit it.
Let
In 2012, Shah and Hussain [
Let (QBM1) for every (QBM2) for every
Then,
The following proposition followed immediately from the previous definition.
Any quasi-
Let
In 2005, Zeyada et al. [
Let (QML1) (QML2)
In this case (QML3)
then it is called dislocated metric.
Any dislocated quasi-metric space is a quasi-
Let
In 1988, Kozlowski introduced the notion of modular spaces [
Let
for every
whenever
Let
The convergence in quasi-modular spaces is defined as follows.
Let
Let
(i) A quasi-modular space
(ii) A quasi-modular space
Let
whenever
whenever
Let
We have the following example of a quasi-
Let
Clearly,
Consequently, we have the following result.
Let a sequence a sequence
The Banach contraction principle was extended to a
Let
Let
Let
For each
The following theorem is an extension of Banach contraction principle in the context of a quasi-
Let
We shall prove that
Analogously, we have
Since any quasi-metric space and any quasi-
Let
Let
We can obtain a similar result in the context of complete dislocated quasi-metric spaces.
In this section, we consider the existence and uniqueness of fixed point for Ćirić type contraction in the setting of quasi-
Let
Suppose that
Let
Let
Note that as in (
Now, assume that
On the other hand, as
By continuing in the same manner, we deduce
Now, if
Let
Let
The pair
The pair
A function
Let
there exists
Then
Since
Thus, for every
Using the above inequality we have for every
Now, if
The weak continuity condition of
Let
there exists
Then
Following the steps of the previous proof we can prove that
Let
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.