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We discuss the extension of some fundamental results in nonlinear analysis to the setting of

It is well known that the Banach-Caccioppoli’s theorem [

In particular, Hitzler and Seda [

In 2012 Amini-Harandi rediscovered the notion of dislocated metric space in [

Inspired by the ideas in [

In this section we collect first known notions and notations and then auxiliary concepts and tools to develop our theory. For a comprehensive discussion, we refer the reader to [

We start by recalling some basic definitions and properties of the setting which we will use.

A metric-like on a nonempty set

Each metric-like

Then a sequence

A sequence

Let

Let

Let

We introduce useful tools for developing our theory.

Let

Let

Clearly,

Note that if

Now, suppose that

Now, suppose that

Let

Let

From

Let

Let

Let

First, we assume that

Now, we assume that

Let

Let

Finally, we introduce the notion of

Let

Let

First, we assume that the function

Now, we assume that the function

The significance of the results given in the previous section will become clear as we proceed with the following applications of fixed points.

The following theorem is an extension of the result of Caristi [

Let

Let

Let

The following results are some consequences of Theorem

Let

Let

Let

the function

the mapping

Note that (ii) implies (i). In fact, let

Now, we prove that

Now, by (i), the function

First, we deduce the Banach-Caccioppoli’s theorem in the setting of a metric-like space by Theorem

Let

Let

The proof of the following Ćirić type theorem (see [

Let

the function

the mapping

In what follows we denote by

We recall the following result due to Khamsi; see [

Let

From this theorem, Khamsi deduced some generalizations of Caristi’s fixed point theorem.

Let

Let

Now, in the setting of a

Let

Let

Let

The multivalued mapping

Let again

As an application of our technique, we prove Ekeland’s variational principle in the setting of metric-like spaces. For a comparative study, see also [

Let

for all

Let

for all

This implies that (i)–(iii) hold. In fact (i) reduces to (j). Next, if

Building on Theorem

Let

To conclude, we recall that there exists at least a point

By comparing Theorems

In view of Theorem

By an application of Theorem

The authors declare that there is no conflict of interests regarding the publication of this paper.

All authors contributed equally and significantly to writing this paper. All authors read and approved the final paper.

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the International Research Group Project no. IRG14-04.