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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

A

Let

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

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%.