Abstract
In this paper, we improve the Proinov theorem by adding certain rational expressions to the definition of the corresponding contractions. After that, we prove fixed point theorems for these modified Proinov contractions in the framework of dislocated bmetric spaces. We show some illustrative examples to indicate the validity of the main results.
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1 Introduction and preliminaries
In the nature of mathematics, there is the purpose of generalizing, expanding, and obtaining the most general forms of existing concepts and results. The concept of metric, which is the most fundamental and solid basis of the analysis study, has been constantly expanded and generalized with this motivation. Examples of the new metrics that have been put forward for this purpose can be counted as quasimetric, bmetric, partialmetric, symmetric, Dmetric, modular metric, fuzzy metric, softmetric, Gmetric, and so on. On the other hand, it was understood that not all of these newly defined metrics provide a new and original structure. For instance, Gmetric can be reduced to semimetric or cone metric to a standard metric. More examples can be given, but here we stop to focus on the main motivation. Two of the new and original generalizations of metric notions are bmetrics [1–16] and dislocated metrics [17–21]. Very recently, these two notions have emerged under the name of dislocated bmetric [22, 23].
Metric fixed point theory is a field of study that needs an abstract metric framework (see, for instance, [24–27]). Very recently Proinov [28] proved a fixed point theorem that not only unifies but also generalizes a number of wellknown results in the framework of a standard metric space. In particular, he proved that Wardowski [29] and Jleli and Samet [30] results are not only equivalent to each other, but also they are a special case of one of the main results of [28].
In this paper, we improve the Proinov type contractions by involving certain rational expression to the corresponding contraction thought by Proinov [28]. After then, we prove fixed point theorems for these modified Proinov contractions in the framework of dislocated bmetrics. We bring forward illustrative examples to show the validity of the main results.
Let S be a nonempty set and \(\mathbb{N}=\{1,2,3,\ldots \}\). Some examples of rational contractivity conditions are shown in the following results (see also [31]).
Theorem 1
([32])
Let be a complete metric space and be a mapping such that there exist with such that
for all \(\mathsf {v},\mathsf {w}\in \mathsf {S}\). Then has a unique fixed point \(\mathsf {x}\in \mathsf {S}\), and the sequence converges to the fixed point x for all \(\mathsf {v}\in \mathsf {S}\).
Theorem 2
([33])
Let be a complete metric space and be a continuous mapping. If there exist with such that
for all distinct \(\mathsf {v},\mathsf {w}\in \mathsf {S}\), then possesses a unique fixed point in S.
Theorem 3
([28])
Let be a metric space and be a mapping such that
for all \(\mathsf {v}, \mathsf {w}\in \mathsf {S}\) with , where the functions \(\Psi, \Phi:(0,\infty )\rightarrow \mathbb{R}\) are such that the following conditions are satisfied:

1.
Ψ is nondecreasing;

2.
\(\Phi (\theta )<\Psi (\theta )\) for any \(\theta >0\);

3.
\(\limsup_{\theta \rightarrow \theta _{0}+}\Phi ( \theta )<\Psi (\theta _{0}+)\) for any \(\theta _{0}>0\).
Then admits a unique fixed point.
Definition 4
([34])
A function \(\mathsf{d}_{l}:\mathsf {S}\times \mathsf{S}\rightarrow [ 0,\infty )\) is a dislocatedmetric on S if it satisfies the conditions:
 .:

\(\mathsf{d}_{l}(\mathsf {v},\mathsf {w})=0 \Rightarrow \mathsf {v}= \mathsf {w}\);
 .:

symmetry: \(\mathsf{d}_{l} (\mathsf {w},\mathsf {v})=\mathsf{d}_{l}(\mathsf {v},\mathsf {w})\);
 .:

the triangle inequality
$$\begin{aligned} \mathsf{d}_{l}(\mathsf{u},\mathsf{w} )\leq \mathsf{d}_{l} ( \mathsf{u}, \mathsf{v})+\mathsf{d}_{l}(\mathsf{v},\mathsf{w} ) \end{aligned}$$
for all \(\mathsf{u},\mathsf{v},\mathsf{w} \in \mathsf {S}\). In this case, the pair is a dislocatedmetric space (shortly MS).
Definition 5
([35])
Let \(\mathsf{s}\in [1,\infty )\) be a real number. A function \(\mathsf{b}:\mathsf {S}\times \mathsf{S}\rightarrow [ 0,\infty )\) is a bmetric on S if it satisfies the conditions:
 \(\mathsf {b}_{1}\).:

\(\mathsf{b}(\mathsf {v},\mathsf {w})=0 \Leftrightarrow \mathsf {v}=\mathsf {w}\),
 \(\mathsf {b}_{2}\).:

symmetry: \(\mathsf{b}(\mathsf {w},\mathsf {v})=\mathsf{b}(\mathsf {v},\mathsf {w})\)
 \(\mathsf {b}_{3}\).:

the generalized version of the triangle inequality involving the number s
$$\begin{aligned} \mathsf{b}(\mathsf{u},\mathsf{w} )\leq \mathsf{s } \bigl[ \mathsf{ b}( \mathsf{u},\mathsf{v})+\mathsf{b}(\mathsf{v},\mathsf{w} )\mathsf{ } \bigr] \quad\text{for all }\mathsf{u},\mathsf{v},\mathsf{w} \in \mathsf {S}. \end{aligned}$$
In this case, the tripled \((\mathsf {S}, \mathsf {b}, \mathsf {s})\) forms a bmetric space (shortly bMS).
Obviously, for \(\mathsf{s}=1\), we find the notion of metric space.
Definition 6
([36])
Let \(\mathsf{s}\in [1,\infty )\) be a real number(given). A function is a dislocated bmetric on S if it satisfies the conditions:
 .:

;
 .:

;
 .:

for all \(\mathsf{u},\mathsf{v},\mathsf{w} \in \mathsf {S}\).
In this case, is a dislocated bmetric space (shortly MS).
We mention that, when \(\mathsf {s}=1\), a MS becomes a MS.
Definition 7
([36])
A sequence \(\{ \mathsf {v}_{n} \} \) on a MS is said to be:

convergent to a point \(\mathsf {v}\in \mathsf {S}\) ⇔ ;

Cauchy if and only if exists and tends to be finite.
Proposition 8
([36])
In a MS the limit of a convergent sequence is unique.
Proposition 9
([36])
In a MS every convergent sequence is Cauchy.
In case every Cauchy sequence is convergent, we say that the space is a complete MS. The next lemma will be useful in the sequel.
Lemma 10
Let a MS , a mapping , and \(\mathsf {v}_{0}\) be arbitrary, but fixed point in S. If there exists \(\mathcal{C}\in [ 0,1 ) \) such that
for every \(n\in \mathbb{N}\), then the sequence is a Cauchy sequence.
Proof
Let \(\mathsf {v}_{0}\) be an arbitrary point in S and the sequence \(\{ \mathsf {v}_{n} \} \) with
for \(n\in \mathbb{N}\cup \{ 0 \} \). Thus, by (3), we have
We split the proof in two cases, namely \(\mathsf {s}=1\) and \(\mathsf {s}>1\).

1.
For \(\mathsf {s}=1\), becomes a dislocated metric and by ., for \(n< p\), we have
Therefore, , that is, the sequence is Cauchy.

2.
For \(\mathsf {s}>1\), we distinguish two subcases:

(a)
If \(\mathcal{C}\in [0,\frac{1}{\mathsf {s}})\), by and taking into account (4), we get
that is, is Cauchy.

(b)
If \(\mathcal{C}\in [\frac{1}{\mathsf {s}},1)\), then \(\mathcal{C}^{n}\rightarrow 0\), and we can find \(l\in \mathbb{N}\) such that \(\mathcal{C}^{n}<\frac{1}{\mathsf {s}}\). Therefore, by (a), the sequence is Cauchy. But we have
$$\begin{aligned} \{ \mathsf {v}_{n} \} = \{ \mathsf {v}_{0}, \mathsf {v}_{1},\ldots, \mathsf {v}_{l1} \} \cup \{ \mathsf {v}_{l}, \mathsf {v}_{l+1},\ldots, \mathsf {v}_{l+n},\ldots \}, \end{aligned}$$and then the sequence is Cauchy.

(a)
□
2 Main results
Henceforth, we use the following notations:
and, respectively,
Let the functions \(R_{1}, R_{2}:\mathsf {S}\times \mathsf {S}\rightarrow [0,\infty )\) be defined by
where are nonnegative real numbers.
Theorem 11
Let be a complete ms, \(\Psi, \Phi \in \Theta \), a number \(\alpha \in [1,\infty )\), and two continuous mappings such that, for every distinct \(\mathsf {v}, \mathsf {w}\in \mathsf {S}\) with , the following inequality
holds. Assume that:
 \((\beta _{1})\):

and ;
 \((\beta _{2})\):

Ψ is nondecreasing.
Then . Moreover, if , then the set has exactly one element.
Proof
For an arbitrary (but fixed) point \(\mathsf {v}_{0}\in \mathsf {S}\), let \(\{ \mathsf {v}_{n} \} \) be the sequence defined as follows:
for all \(n\in \mathbb{N}_{0}\). First of all, we claim that \(\mathsf {v}_{n}\neq \mathsf {v}_{n+1}\) for any \(n\in \mathbb{N}_{0}\). Indeed, if we can find \(l_{0}\in \mathbb{N}\) such that \(\mathsf {v}_{l_{0}}=\mathsf {v}_{l_{0}+1}=\mathsf {v}_{l_{0}+2}=\mathsf {x}\), then .
Under this assumption, and letting \(\mathsf {v}=\mathsf {v}_{2n}\) and \(\mathsf {w}=\mathsf {v}_{2n+1}\) in (5), because the functions \(\Psi, \Phi \) belong to Θ, we have
Taking (\(\beta _{1}\)) into account, we get
or
where , holds due to the first assumption in \((\beta _{1})\).
In the same way, replacing in (5) v with \(\mathsf {v}_{2n1}\) and w with \(\mathsf {v}_{2n}\), and keeping in mind , we have
which leads us to
Consequently, (7) and (9) show us that
for any \(n\in \mathbb{N}\), where . By Lemma 10 it follows that \(\{ \mathsf {v}_{n} \} \) is a Cauchy sequence. Thus, exists and is finite. Moreover, since the ms is complete, we get that there exists \(\mathsf {x}\in \mathsf {S}\) such that \(\lim_{n\rightarrow \infty }\mathsf {v}_{n}=\mathsf {x}\) and
Since the mappings and are supposed to be continuous, we have
that is, . If we suppose that there exist such that \(\mathsf {x}\neq \mathsf {y}\), by (5) and since \(\Psi, \Phi \in \Theta \), we have
where
However, applying and taking into account ,
Moreover, by \((\beta _{2})\) we get
which is a contradiction. Therefore, and from it follows that \(\mathsf {x}=\mathsf {y}\), that is, the set has exactly one element. □
Corollary 12
Let be a complete ms, \(\Psi, \Phi \in \Theta \), a number \(\alpha \in [1,\infty )\), and a continuous mapping such that, for every distinct \(\mathsf {v}, \mathsf {w}\in \mathsf {S}\) with , the following inequality
holds, where for nonnegative real numbers,
for all \(\mathsf {v},\mathsf {w}\in \mathsf {S}, \mathsf {v}\neq \mathsf {w}\). Assume that:
 \((\beta _{1})\):

and ;
 \((\beta _{2})\):

Ψ is nondecreasing.
Then . Moreover, if , then the set has exactly one element.
Proof
Let in Theorem 11. □
Theorem 13
Let be a complete ms, \(\Psi, \Phi \in \Theta \), a number \(\alpha \in [1,\infty )\), and two mappings such that, for every distinct \(\mathsf {v}, \mathsf {w}\in \mathsf {S}\) with , the following inequality
holds. Assume that:
 \((\beta _{1})\):

, , , ;
 \((\beta _{2})\):

Ψ is nondecreasing.
Then . Moreover, if , then the set has exactly one element.
Proof
Let \(\mathsf {v}_{0}\in \mathsf {S}\) be a chosen point and \(\{ \mathsf {v}_{n} \} \) be the sequence defined by (6) in the proof of Theorem 11. Thus, following the same arguments, we can assume that and from (13) we get
Since by \((\beta _{2})\) Ψ is nondecreasing, we deduce that
which is equivalent to
where by \((\beta _{1})\). Similarly, taking \(\mathsf {v}=\mathsf {v}_{2n}\) and \(\mathsf {w}=\mathsf {v}_{2n1}\) in (5) and keeping in mind , we get
However, from relations (14), (15), together with Lemma 10, we find that \(\{ \mathsf {v}_{n} \} \) is a Cauchy sequence in a complete ms. Therefore, there exists \(\mathsf {x}\in \mathsf {S}\) such that
Without loss of generality, we can suppose that \(\mathsf {x}\neq \mathsf {v}_{n}\) for any \(n\in \mathbb{N}\). Supposing that , by (5), we have
or, taking \((\beta _{2})\) into account,
However, since
we obtain
On the other hand,
and then
which contradicts our assumption . Thus, we get , that is, . Moreover, if we suppose that , since ,
or, keeping in mind \((\beta _{2})\)
which is a contradiction. Therefore, which implies by that . That is, .
As a last step, we claim that x is the unique fixed point of the mappings and . Indeed, if we suppose that there exists another point such that \(\mathsf {x}\neq \upsilon \), by (13) we have
Since the function Ψ is supposed to be nondecreasing, it follows that
which is a contradiction. Therefore, the set has exactly one element. □
Corollary 14
Let be a complete ms, \(\Psi, \Phi \in \Theta \), a number \(\alpha \in [1,\infty )\), and a mapping such that, for every \(\mathsf {v}, \mathsf {w}\in \mathsf {S}\) with , the following inequality
holds, where for nonnegative real numbers,
Assume that:
 \((\beta _{1})\):

, , and ;
 \((\beta _{2})\):

Ψ is nondecreasing.
Then . Moreover, if , then the set has exactly one element.
Proof
Let in Theorem 13. □
Example 15
Let the set and the function be defined by
 m  n  p  q 

m  0  2  5  7 
n  2  6  8  5 
p  5  8  0  1 
q  7  5  1  0 
Obviously, is a metric, with \(\mathsf {s}=2\). Let be two mappings, where and . We have, in this case,
v  m  n  p  q 

 p  q  p  p 
 q  q  p  p 
 5  5  0  1 
 7  5  0  1 
v  m  n  p  q  


 p  q  p  p  
w 
 
m  q  1  0  1  1  
n  q  1  0  1  1  
p  p  0  1  0  0  
q  p  0  1  0  0 
Letting the functions \(\Psi, \Phi \in \Theta \), \(\Psi (\theta )=\theta \), \(\Phi (\theta )=\frac{3}{4}\theta \) and the numbers \(\alpha =2\), , , , we can easily see that assumptions \((\beta _{1})\) and \((\beta _{2})\) in Theorem 13 are satisfied. We show that (13) is satisfied for any pair , where
(the other cases are excluded by the hypotheses of Theorem 13).

\((\mathsf {v},\mathsf {w})=(m,n)\)

\((\mathsf {v},\mathsf {w})=(n,p)\)
The other cases are discussed similarly.
Thus .
Theorem 16
Let be a complete ms, the functions \(\Psi, \Phi \in \Theta \), a number \(\alpha \in [1,\infty )\), , and two mappings such that, for every distinct \(\mathsf {v}, \mathsf {w}\in \mathsf {S}\) with , the following inequality
holds, where
Assume that:
 \((\beta _{1})\):

;
 \((\beta _{2})\):

Ψ is nondecreasing.
Then the set has exactly one element.
Proof
Let us take in (20), \(\mathsf {v}=\mathsf {v}_{2n}\) and \(\mathsf {w}=\mathsf {v}_{2n+1}\), where the sequence \(\{ \mathsf {v}_{n} \} \) is defined as in Theorem 11. We have
with
Furthermore, taking \((\beta _{2})\) and the above relation into account, we get
which implies
Similarly, taking \(\mathsf {v}=\mathsf {v}_{2n}\), respectively \(\mathsf {w}=\mathsf {v}_{2n1}\), we obtain
Now, choosing (by assumption \((\beta _{1})\)), we have for any \(n\in \mathbb{N}\). Therefore, Lemma 10 leads us to the conclusion that \(\{ \mathsf {v}_{n} \} \) is a Cauchy sequence. Thus, since the space is complete, there exists \(\mathsf {x}\in \mathsf {S}\) such that
Supposing that , we have
Moreover, without loss of generality, we can assume that for any \(n\in \mathbb{N}\), and then from (20) we get
or, by \((\beta _{2})\),
Returning in (25), we have
Letting \(n\rightarrow \infty \) and keeping in mind (24), we get
which is a contradiction. Thus, and from we have .
Analogously, we have
or, by \((\beta _{2})\),
On the other hand, supposing that , we have
Combining the above inequalities and taking limit as \(n\rightarrow \infty \), we obtain , which is a contradiction. Therefore, , and then . Thus, x is a common fixed point for and , that is, and it remains to show that the set is in fact reduced to a single point. On the contrary, let with \(\upsilon \neq \mathsf {x}\). Replaced in (20), we have
and, due to \((\beta _{2})\),
which is a contradiction. Therefore, it follows that \(\mathsf {x}=\upsilon \) and the set has exactly one element. □
Example 17
Let \(\mathsf {S}= \{ 2,4,5,7 \} \) and two selfmappings be defined on S by
v  2  4  5  7 

 5  5  5  4 
 4  5  5  5 
Let be the metric on S (with \(\mathsf {s}=2\)) given by
Considering the functions \(\Psi, \Phi \in \Theta \) as in Example (15) and letting \(\alpha =2\), , , we have the following cases:

\((\mathsf {v}, \mathsf {w})=(4,2)\)

\((\mathsf {v}, \mathsf {w})=(5,2)\)

\((\mathsf {v}, \mathsf {w})=(7,2)\)

\((\mathsf {v}, \mathsf {w})=(7,4)\)

\((\mathsf {v}, \mathsf {w})=(7,5)\)
The other cases are excluded by the hypothesis of Theorem 16. Therefore, .
Corollary 18
Let be a complete ms, \(\Psi, \Phi \in \Theta \), a number \(\alpha \in [1,\infty )\), , and a mapping such that, for every distinct \(\mathsf {v}, \mathsf {w}\in \mathsf {S}\) with , the following inequality
holds, where
Assume that:
 \((\beta _{1})\):

;
 \((\beta _{2})\):

Ψ is nondecreasing.
Then the set has exactly one element.
Proof
Let in Theorem 16. □
3 Consequences
Taking particular functions Ψ and Φ, we obtain as consequences some known results. For example, let \(\Phi (\theta )=\beta (\theta )\Psi (\theta )\) for all \(\theta >0\) and \(\beta:[0,\infty )\rightarrow [0,\frac{1}{\mathsf {s}})\).
Corollary 19
Let be a complete ms, a number \(\alpha \in [1,\infty )\), and two continuous mappings such that, for every distinct \(\mathsf {v}, \mathsf {w}\in \mathsf {S}\) with , the following inequality
holds. Assume that:
 \((\beta _{1})\):

and ;
 \((\beta _{3})\):

\(\Psi:(0,\infty )\rightarrow (0,\infty )\) is nondecreasing;
 \((\beta _{4})\):

\(\beta:(0,\infty )\rightarrow (0,\frac{1}{\mathsf {s}})\) satisfies \(\limsup_{\theta \rightarrow \theta _{0}}\beta ( \theta )<\frac{1}{\mathsf {s}}\) for any \(\theta _{0}>0\).
Then . Moreover, if , then the set has exactly one element.
Corollary 20
Let be a complete ms, a number \(\alpha \in [1,\infty )\), and two mappings such that, for every distinct \(\mathsf {v}, \mathsf {w}\in \mathsf {S}\) with , the following inequality
holds. Assume that:
 \((\beta _{1})\):

, , , and ;
 \((\beta _{3})\):

\(\Psi:(0,\infty )\rightarrow (0,\infty )\) is nondecreasing;
 \((\beta _{4})\):

\(\beta:(0,\infty )\rightarrow (0,\frac{1}{\mathsf {s}})\) satisfies \(\limsup_{\theta \rightarrow \theta _{0}}\beta ( \theta )<\frac{1}{\mathsf {s}}\) for any \(\theta _{0}>0\).
Then . Moreover, if , then the set has exactly one element.
Corollary 21
Let be a complete ms, a number \(\alpha \in [1,\infty )\), and two mappings such that, for every distinct \(\mathsf {v}, \mathsf {w}\in \mathsf {S}\) with , the following inequality
holds. Assume that:
 \((\beta _{1})\):

, ;
 \((\beta _{3})\):

\(\Psi:(0,\infty )\rightarrow (0,\infty )\) is nondecreasing;
 \((\beta _{4})\):

\(\beta:(0,\infty )\rightarrow (0,\frac{1}{\mathsf {s}})\) satisfies \(\limsup_{\theta \rightarrow \theta _{0}}\beta ( \theta )<\frac{1}{\mathsf {s}}\) for any \(\theta _{0}>0\).
Then the set has exactly one element.
Considering \(\Phi (\theta )=\kappa \Psi (\theta )\) or \(\Phi (\theta )=\kappa \cdot \theta \) for all \(\theta >0\) in Theorems 11, 13 or 16, other consequences can be listed. On the other hand, many other corollaries can be deduced considering or letting \(\mathsf {s}=1\).
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Acknowledgements
The authors thank their universities. The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group no. RG1437037. A.F. Roldán López de Hierro is grateful to Project TIN201789517P of Ministerio de Economía, Industria y Competitividad and also to Junta de Andalucía by project FQM365 of the Andalusian CICYE.
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Alqahtani, B., Alzaid, S.S., Fulga, A. et al. Proinov type contractions on dislocated bmetric spaces. Adv Differ Equ 2021, 164 (2021). https://doi.org/10.1186/s13662021033295
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Accepted:
Published:
DOI: https://doi.org/10.1186/s13662021033295