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In the paper, we consider some fixed point results of -contractions for triangular -admissible and triangular weak -admissible mappings in metric-like spaces. The results on -contraction type mappings in the context of metric-like spaces are generalized, improved, unified, and enriched. We prove the main result but using only the property () of the strictly increasing mapping . Our approach gives a proper generalization of several results given in current literature.
First, we recall some notions introduced recently in several papers.
In 2012, Samet et al.  introduced the concept of -admissible mappings as follows.
Let and . Then, is called α-admissible if for all with implies .
Furthermore, one says that is a triangular -admissible mapping if it is -admissible and if
For triangular -admissible mapping, the following result is known (, Lemma 7):
Let be a triangular α-admissible mapping. Assume that there exists such that Define sequence by Then,
In , the author presented the notion of weak -admissible mappings as follows:
Let be a nonempty set and let be a given mapping. A mapping is said to be a weak α-admissible one if the following condition holds:
It is customary to write and as the collection of all (triangular) α-admissible mappings on and the collection of all (triangular) weak α-admissible mappings on (see). One can verify that
Now, we recall some basic concepts, notations, and known results from partial metric and metric-like spaces. In 1994 Matthews () introduced notion of partial metric space as follows.
Let be a nonempty set. A mapping is said to be a partial metric on if for all the following four conditions hold:
if and only if ;
In this case, the pair is called a partial metric space. Obviously, every metric space is a partial metric space. The inverse is not true. Indeed, let and . Under these conditions is a partial metric space but is not a metric space because . For more details, see ([5,6,7,8,9,10,11]).
Let be a nonempty set. A mapping is said to be a metric-like on if for all the following three conditions hold:
The pair is called a metric-like space or dislocated metric space by some authors. A metric-like mapping on satisfies all the conditions of a metric except that may be positive for some . The following is a list of some metric-like spaces:
1. where for all
One can see that is a metric-like space, but it is not a metric space, due to the fact that On the other hand, is a partial metric space.
2. where for all
It is clear that is a metric-like space where for each Since , it follows that does not hold. Hence, is not a partial metric space.
3. where and ,
It is clear that is a metric-like (that is a dislocated metric) space with . This means that is not a standard metric space. However, is also not a partial metric space because
4. where is the set of real continuous functions on and for all
This is an example of metric-like space that is not a partial metric space. Indeed, for we obtain Putting for all we obtain that
Note that some of the metric-like spaces given in the list are not partial metric spaces. It is clear that a partial metric space is a metric-like space and the inverse is not true. Now, we give the definitions of convergence and Cauchyiness of the sequences in metric-like space (see ).
Let be a sequence in a metric-like space .
The sequence is said to be convergent to if
The sequence is said to be —Cauchy in if exists and is finite;
A metric-like space is —complete if for every Cauchy sequence in there exists an such that
In metric-like space (as in the partial metric space), the limit of a sequence need not be unique and a convergent sequence need not be a —Cauchy sequence (see examples in Remark 1.4 (1) and (2) in ). However, if the sequence is Cauchy such that in complete metric-like space , then the limit of such sequence is unique. Indeed, in such a case if as , we get that (by (iii) of Definition 5). Now, if and we obtain
By (1) from Definition 4, it follows that which is a contradiction.
Now, we give the definition of the continuity for self-mapping defined on a metric-like space as follows (see for example [10,11,34]):
Let be a metric-like space and be a self-mapping. We say that is continuous in point if , for each sequence such that . In other words, the mapping is continuous if the following holds true:
Let be a metric-like space. A sequence in it is called Cauchy sequence if . The space is said to be complete if every Cauchy sequence in converges to a point such that
It is obvious that every Cauchy sequence is a Cauchy sequence in and every complete metric-like space is a complete metric-like space. In addition, every complete partial metric space is a complete metric-like space. In the sequel, some results on metric-like spaces are given. Proofs to most of the results are self-evident.
Let be a metric-like space. Then, we have the following:
If the sequence converges to as and if then, for all , it follows that
If , then
If is a sequence such that , then
If , then
holds for all where
Let be a sequence such that If then there exists and sequences and such that and the following sequences tend to ε when
Notice that, if the condition (vi) is satisfied then the sequences and also converge to when where For more details on (i)–(vi), the reader can see in ([26,27,36]). The concept of -contraction was introduced by Wardowski in  (for more details, see also: [5,9,14,15,16,17,18,24,28,31,32,33]).
Let be a mapping satisfying the following:
is a strictly increasing, that is, for , implies
For each sequence , if and only if
There exists such that
Let be a metric space. A mapping is said to be an -contraction if there exist satisfying (), () and () and such that
In 2014, Piri and Kumam  investigated some fixed point results concerning contraction in complete metric spaces by replacing the condition () with the condition:
is continuous on
Recently, in 2018, Qawaqueh et al. () defined and proved the following:
Let be a metric-like space and . A mapping is said to be an -Geraghty contraction mapping if there exist and such that, for all with and
is strictly increasing function satisfying (), () and () and is a family of all functions which satisfy the condition: implies as
It is worth noticing that authors in  denote with the collection of all almost generalized -contractive mappings. However, it is not clear what “almost generalized -contractive mappings” mean.
Let be a metric-like space and . A mapping be an -Geraghty contraction mapping. Assume that the following conditions are satisfied:
There exists such that .
Then, has a unique fixed point with
2. Main Result
In this section, we improve the whole concept by introducing a new definition and new approaches. Firstly, we introduce the following:
Let be a metric-like space and . A mapping is said to be a triangular -contraction one if there exists such that, for all with and holds true,
is strictly increasing function.
Example 3 from , for instance, illustrates the validity of this definition but without the function . Definition 11 is an improvement of the definition given in  in several directions. Now, we prove the main result of our paper:
Let be a complete metric-like space and . Assume that a mapping is a triangular -contraction one. Suppose further that the following conditions are satisfied:
There exists such that
Then, has a unique fixed point with
First of all, we show the following two claims:
If is a fixed point of then
The uniqueness of a possible fixed point.
Firstly, we prove I. Indeed, if is a fixed point of and if then, putting in (8), we get
which is a contradiction. Hence, the assumption that is wrong. We proved claim I.
Now, we shall prove II. Suppose that has two distinct fixed point and in By (I), we get Since and , according to (8), we get:
In other words, taking into consideration,
is a contradiction. Hence, the uniqueness of fixed point is proved.
In the sequel, we prove the existence of the fixed point of .
Let be such that Furthermore, we define the sequence in with for all . If for some , then by the previous, is a unique fixed point of and the proof of the theorem is finished. Now, let us suppose that for all . Since and , we have
Using this process again, we get .
Because is a triangular -contraction mapping with , we have according to Lemma 1:
If , then a contradiction follows from
Thus, we conclude that for all . Therefore, since , we have
where from one can conclude that for all . This further means that there exists . If , we obtain a contradiction since by (), it follows:
where . We use the fact that strictly increasing function has a left and right limit in every point from . Hence, we obtain that . Now, we prove that the sequence is a Cauchy sequence by supposing the contrary. When we put in (8), we get
Since from the previous inequality, we get
We obtain the contradiction, which means that the sequence is a Cauchy. This means that there exists a unique (by Remark 2) point such that
Since the mapping is continuous, we get that , i.e., . According to Remark 2, it follows that , that is, is a fixed point of □
The following results are immediate corollaries of Theorem 2. Indeed, replacing in (8) with one of the following sets:
we get that Theorem 2 also holds true.
Immediate consequences of Theorem 2 are the following new contractive conditions that compliment the ones given in [23,35].
Let be a complete metric-like space and . Assume that a mapping is a triangular - contraction where is the strictly increasing mapping. Suppose further that the following conditions are satisfied:
There exists such that
In addition, suppose that there exist and, for all with and , the following inequalities hold true:
where is one of the following sets:
Then, in each of these cases, has a unique fixed point in .
If we put , and , , , , in Theorem 2, respectively, then every of the functions is strictly increasing on , and the result follows according to Theorem 2. □
Putting for all in the previous corollary, we get the following six new contractive conditions:
where is one of the following sets:
In every one of these cases, has a unique fixed point in The result can simply be obtained by putting and , in Theorem 2.
In , Ćirić introduced one of the most generalized contractive conditions (so-called quasicontraction) in the context of metric spaces as follows:
The self-mapping on metric space is called quasicontraction (in the sense of Ćirić) if there exists such that, for all holds true
Each quasicontraction on a complete metric space has a unique fixed point (say) η. Moreover, for all , the sequence converges to the fixed point η as .
Finally, we formulate the following notion and an open question:
Let be a metric-like space and . A mapping is said to be a triangular -contraction mapping of Ćirić type, if there exists such that, for all with and holds true:
is strictly increasing function satisfying only ().
An open question: Prove or disprove the following claim: each triangular-contraction mappingof Ćirić type defined on complete metric-like spacehas a unique fixed point.
Conceptualization, J.V. and S.R.; methodology, J.V., S.R., Z.D.M., and S.M.; formal analysis, Z.D.M. and S.M.; investigation, J.V., S.R., and Z.D.M.; data curation, S.R. and Z.D.M.; supervision, Z.D.M., J.V., S.R., and S.M.; project administration, S.R. and S.M. All authors have read and agreed to the published version of the manuscript.
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