#### Abstract

We discuss the extension of some fundamental results in nonlinear analysis to the setting of -complete metric-like spaces. Then, we show that these extensions can be obtained via the corresponding results in standard metric spaces.

#### 1. Introduction

It is well known that the Banach-Caccioppoli’s theorem [1, 2] is the starting point for the development of metric fixed point theory. Over the years, this theory has evolved by receiving the support and interest of many mathematicians. In fact, the fixed point theorems and constructive techniques have been successfully applied in pure and applied analysis, topology, and others. Consequently various generalizations and extensions of the Banach-Caccioppoli’s theorem have appeared in the literature; see for instance [3–10]. Precisely, these contributes have investigated the basic problems of the metric fixed point theory: existence and uniqueness under different contractive conditions, convergence of successive approximations, and well posedness of the fixed point problem.

In particular, Hitzler and Seda [11] presented the notion of dislocated metric space and proposed their generalization of the Banach-Caccioppoli’s theorem. Hitzler and Seda’s idea is to apply this theorem in order to obtain a unique supported model for acceptable logic programs. Also, many authors developed the fixed point theory in the setting of dislocated metric spaces; see for instance [12].

In 2012 Amini-Harandi rediscovered the notion of dislocated metric space in [13], as a generalization of a partial metric space [14]. These spaces were called metric-like spaces and used to introduce different notions of convergence and Cauchy sequence.

Inspired by the ideas in [11, 15], we characterize those metric-like spaces for which every Caristi’s mapping [16] has a fixed point in the sense of the Romaguera’s characterization of partial metric -completeness [17]. This will be done by means of the notion of a -complete metric-like space which is introduced in the sequel. Then, we present fixed point theorems in this setting, by using known and new classes of lower semicontinuous functions. Finally, as an application of our technique, we deduce Ekeland’s variational principle in a -complete metric-like space. The aim of this work is to underline the strong relation between standard metric spaces and their generalizations to better target the research on this topic. Applying the approach followed in this paper, for instance, the reader can obtain the extensions to a metric-like space of many recent results in fixed point theory.

#### 2. Metric-Like Spaces

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 [13].

##### 2.1. Preliminaries

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

*Definition 1 (see [13]). *A metric-like on a nonempty set is a function such that, for all , implies ;;. A metric-like space is a pair such that is a nonempty set and is a metric-like on .

Each metric-like on generates a topology on whose base is the family of open -balls , where

Then a sequence in the metric-like space converges to a point if and only if .

A sequence of elements of is called -Cauchy if the limit exists and is finite. The metric-like space is called complete if, for each -Cauchy sequence , there is some such that If , then is called a -Cauchy sequence. If every -Cauchy sequence in converges, with respect to , to a point such that , then is called -complete; see the paper of Romaguera [17] for a comparative discussion with partial metric spaces. Here we point out that every partial metric space is a metric-like space; see [13]. Also we give some examples of a metric-like space.

*Example 2. *Let and be defined by
Then is a metric-like space, which is not a metric space or a partial metric space.

*Example 3. *Let and be defined by
for all . Then is a complete metric-like space, which is not a metric space or a partial metric space.

*Example 4. *Let and be defined by
for all . Then is a -complete metric-like space, which is not a complete metric-like space.

##### 2.2. Metric Induced by a Metric-Like

We introduce useful tools for developing our theory.

Let be a -complete metric-like space and let be defined by

Lemma 5. *Let be a metric-like space and the function defined by (6). Then is a metric space. Moreover is a complete metric space if and only if is a -complete metric-like space.*

*Proof. *Clearly, and . Moreover, for all , we have
if or if and or . Also, if are distinct points, from
we get
and so the triangle inequality holds. Thus is a metric on and hence is a metric space.

Note that if is such that for all , then if and only if .

Now, suppose that is a -complete metric-like space and is a Cauchy sequence in . If for all , then the sequence converges to . Then we can assume that for all . In reason of the above observation, we get that is a -Cauchy sequence in . Using the fact that is a -complete metric-like space, then there exists such that ; that is, . Thus is a complete metric space.

Now, suppose that is a complete metric space and is a -Cauchy sequence in . Without loss of generality, assume that for all . Then
Hence, is a Cauchy sequence in . Since is complete, there exists such that . Thus, and so is -complete.

Let ; we have the following proposition.

Proposition 6. *Let be a metric-like space and the metric defined by (6). Let be a point of and let be such that . If for infinity values of , then . Moreover is a closed subset of .*

*Proof. *From for every such that it follows that there exists a subsequence of such that . This implies , since , as and so . Then for each sequence we have and so is a closed subset of .

*Definition 7. *Let be a metric-like space and a mapping. is called -continuous if, for all and with as , we have .

*Remark 8. *Let be a metric-like space and a mapping. If is -continuous, then . In fact, if and is a sequence such that as , then and so .

Proposition 9. *Let be a metric-like space, the metric defined in (6), and a mapping such that . Then is continuous in if and only if is -continuous in .*

*Proof. *First, we assume that is -continuous in and let be a sequence convergent to a point in . Clearly, if for all . Then, without loss of generality, we assume that for all . This implies that for all and hence . By Proposition 6, we get that . Then and is continuous in .

Now, we assume that is continuous in , is a given point in , and is a sequence convergent to . Without loss of generality, we assume that for all . From as , it follows that is -continuous in .

*Definition 10. *Let be a metric-like space. A mapping is a contraction if there exists such that

*Remark 11. *Let be a metric-like space. Every contraction is a -continuous mapping. In fact, for all and all sequences with as , we get .

Finally, we introduce the notion of -lower semicontinuous function.

*Definition 12. *Let be a metric-like space and the metric defined by (6). Assume that . A function is called -lower semicontinuous if, for all and every sequence with , we have

Lemma 13 13. *Let be a metric-like space with and the metric defined by (6). Then a function is lower semicontinuous in if and only if is -lower semicontinuous in .*

*Proof. *First, we assume that the function is -lower semicontinuous in and let be a sequence convergent to in . If for all with , then (12) holds since . Then we can assume that for all and also that for all . This implies as and hence (12) holds, since is -lower semicontinuous in and hence is lower semicontinuous in .

Now, we assume that the function is lower semicontinuous in and let be a sequence convergent to in . This implies as and hence (12) holds, since is lower semicontinuous in and so is -lower semicontinuous in .

#### 3. Fixed Point Theorems

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

##### 3.1. Caristi Type Fixed Point Theorems

The following theorem is an extension of the result of Caristi [16, Theorem ] in the setting of metric-like spaces. First, we say that a mapping satisfying the condition where is a -lower semicontinuous function, is a Caristi’s mapping on . Also, a point such that is called a fixed point of .

Theorem 14. *Let be a -complete metric-like space. Then any Caristi’s mapping on has a fixed point in with .*

*Proof. *Let be the metric defined by (6). Then, by Lemma 5, is a complete metric space. From for all and (13), we get
Since is lower semicontinuous in by Lemma 13, then by Caristi’s theorem has a fixed point . Finally, by (13), we get .

*Example 15. *Let and be defined by if and
Clearly, is a -complete metric-like space. Also, notice that . Consider the mapping defined by
Then, we get
It is easy to show that the function , defined by for all , is a -lower semicontinuous function. Also we get and so is a Caristi’s mapping. Thus Theorem 14 ensures that has a fixed point; here and are fixed points of .

The following results are some consequences of Theorem 14. In particular, the next theorem is the metric-like counterpart of Theorem 2.1 in [16].

Theorem 16. *Let be a -complete metric-like space and a -continuous mapping with . Suppose that is a mapping and there exists a negative real number such that
**
Then has a fixed point in with .*

*Proof. *Let be the metric defined by (6). Then, by Proposition 9, the mapping is continuous in . This implies that the function defined by
is lower semicontinuous in and hence is a -lower semicontinuous function in . From (18), we get
The existence of a fixed point follows by an application of Theorem 14.

Theorem 17. *Let be a -complete metric-like space and a mapping. Assume that there exists such that
**
If one of the following conditions holds, then has a fixed point in with :*(i)*the function defined by is -lower semicontinuous;*(ii)*the mapping is -continuous.*

*Proof. *Note that (ii) implies (i). In fact, let and such that as and assume that is -continuous. From
we get
This ensures that the function is -lower semicontinuous.

Now, we prove that has a fixed point in if (i) holds. By (21), we have
This implies
where is defined by , for all .

Now, by (i), the function is -lower semicontinuous. Thus, the existence of a fixed point follows by an application of Theorem 14.

##### 3.2. Banach-Caccioppoli, Ćirić, and Khamsi Type Results

First, we deduce the Banach-Caccioppoli’s theorem in the setting of a metric-like space by Theorem 17.

Theorem 18. *Let be a -complete metric-like space and let be a contraction. Then has a unique fixed point in with .*

*Proof. *Let such that (11) holds true. Then for all ; that is (21) holds true. Since, by Remark 11, the mapping is -continuous, then the existence of a fixed point follows by an application of Theorem 17. In view of the fact that is a contraction, the uniqueness of the fixed point , with , is an easy consequence of (11).

The proof of the following Ćirić type theorem (see [4]) proceeds on the same lines of the proof of Theorem 18 and so we omit it.

Theorem 19. *Let be a -complete metric-like space and let be a mapping. Assume that there exists such that
**
for all . Then has a unique fixed point in if one of the following conditions holds: *(i)*the function defined by is -lower semicontinuous;*(ii)*the mapping is -continuous.** If the mapping satisfies condition (26) and , then .*

In what follows we denote by the family of all functions nondecreasing, continuous at with , such that there exist and satisfying the condition , for each . Since is continuous at , then there exists such that .

We recall the following result due to Khamsi; see [18].

Theorem 20 (see [18, Theorem 2]). *Let be a complete metric space. Define the relation by
**
where and is a lower semicontinuous function. Then has a minimal element ; that is, if , then we must have .*

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

Theorem 21 (see [18, Theorem 3]). *Let be a complete metric space. Let be a mapping such that for all **
where the function and is a lower semicontinuous function. Then has a fixed point.*

Theorem 22 (see [18, Theorem 4]). *Let be a complete metric space. Let be a multivalued mapping such that is nonempty. Assume that for all there exists such that
**
where the function and is a lower semicontinuous function. Then has a fixed point; that is, there exists such that .*

Now, in the setting of a -complete metric-like space, we deduce the following results.

Theorem 23. *Let be a -complete metric-like space. Let be a mapping such that for all **
where the function and is a -lower semicontinuous function. Then has a fixed point.*

*Proof. *Let be the metric on defined by (6). By Lemma 13, is a complete metric space and, by Lemma 5, is a lower semicontinuous function in . Next, from for all , we get
for all . By an application of Theorem 21, we obtain that has a fixed point.

Theorem 24. *Let be a -complete metric-like space. Let be a multivalued mapping such that is nonempty. Assume that for all there exists such that
**
where the function and is a -lower semicontinuous function. Then has a fixed point.*

*Proof. *The multivalued mapping has a selection that satisfies the condition (30). Then by Theorem 23, the multivalued mapping has a fixed point.

*Example 25. *Let again and be defined by if and
so that is a -complete metric-like space and . Then, consider the multivalued mapping defined by
Clearly, for all , and there exists given by
such that
Thus Theorem 24 is applicable in this case with for all and for all .

#### 4. Ekeland’s Variational Principle

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 [19].

Theorem 26 (Ekeland’s variational principle). *Let be a -complete metric-like space with and consider a function that is -lower semicontinuous, bounded from below, and not identical to . Let be given and let be such that . Then there exists such that*(i)*;*(ii)*;*(iii)*for all in , .*

*Proof. *Let be the metric defined by (6); then, by Lemma 5, we deduce that is a complete metric space. Further, by Lemma 13 we deduce that the function is lower semicontinuous in . By the Ekeland’s variational principle in metric space there exists such that (j);(jj);(jjj) for all in , .

This implies that (i)–(iii) hold. In fact (i) reduces to (j). Next, if , then and so (ii) holds. Finally, (iii) holds since implies .

Building on Theorem 26, we present the second theorem of this section.

Theorem 27. *Let be a -complete metric-like space and a -lower semicontinuous function. Given , then there exists such that
*

*Proof. *To conclude, we recall that there exists at least a point such that . This implies that (37) follows from (i) and (iii) of Theorem 26, respectively.

*Remark 28. *By comparing Theorems 26 and 27, it is clear that the first is stronger than the second. In fact, the condition (ii) of Theorem 26, which gives the whereabouts of point in , does not have a counterpart in Theorem 27.

In view of Theorem 27, we can provide the following alternative proof of Theorem 14 described in this paper.

*Proof. *By an application of Theorem 27 with , we get that there exists some point such that
where we assume that the function satisfies (13). The above inequality also holds for ; therefore
Next, putting in (13), we obtain
Combining together the above inequalities, we get
This holds true unless and therefore we deduce that . Then, the existence of a fixed point is proved.

#### Conflict of Interests

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

#### Authors’ Contribution

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

#### Acknowledgment

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.