In 2005, Mustafa and Sims (2006) introduced and studied a new class of generalized metric spaces, which are called
The literature of the last decades is rich of papers that focus on all matters related to the generalized metric spaces (i.e.,
In 2005, Mustafa and Sims introduced a new class of generalized metric spaces (see [
Here, we present the necessary definitions and results in
Let
Then
A
Let a a
A
Let
Let
An interesting observation is that any
It was observed that in the symmetric case (
In this section, we recall some well-known fixed point theorems on metric spaces.
The following Ćirić’s theorem for quasicontractive mappings [
Let
The next result is a consequence of Theorem 2.1 of [
Let
The following result is a consequence of Theorem 2.3 of [
Let
where
In
Let
For our purpose we need the following propositions which are key results.
Let
Let
If
Let
Let
If
Let
Recently, Mustafa and Obiedat [
Let
Now, we prove the following.
Theorem
Taking
In the same paper, the authors presented the following fixed point theorem.
Let (
Proceeding similarly as above we can prove the following.
Theorem
Taking
Now, we consider another theorem.
Let
In view of this result we give the following.
Theorem
Taking
Recently, Mustafa and Sims [
Let
We give the following.
Theorem
Taking
It follows from (
From the proof of Theorem
In this section, using our previous technique, we establish very easily some fixed point theorems on
We have the following result.
Let
Suppose that
Here, we give a fixed point result of Edelstein type in the setting of
Let
From (
To conclude this paper, we study the existence and uniqueness of common fixed points for two self-mappings satisfying a
Let
If one of
From condition (iii), we have
In this paper, we showed that some fixed point generalizations in fixed point theory are not real generalizations as they could easily be obtained from the corresponding fixed point theorems in metric spaces. Here, we reviewed only some theorems which appeared recently in the literature. It follows that researchers in fixed point theory should take care in obtaining real generalizations.
The authors are grateful to editors and anonymous reviewers for their constructive comments and suggestions. The second author is supported by Università degli Studi di Palermo, Local Project R.S. ex 60%.