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On interpolative contractions that involve rational forms
Advances in Difference Equations volume 2021, Article number: 448 (2021)
Abstract
The aim of this paper is to investigate the interpolative contractions involving rational forms in the framework of bmetric spaces. We prove the existence of a fixed point of such a mapping with different combinations of the rational forms. A certain example is considered to indicate the validity of the observed result.
1 Introduction and preliminaries
It is worth noting that Caccioppoli [1] is the first author who extended the results of Banach [2] from normed space to metric space. After that, a number of authors have studied different abstract spaces to advance the Banach and Caccioppoli results. One of the successive generalizations was given Bakhtin [3] (and independently by Czerwik [4]) from metric space to bmetric space. Following this success, many authors have continued to work on this trend and reported several improvements, advances in the setting of bmetric spaces, see e.g. [5–12].
Let be a nonempty set and be a metric on . The notion of bmetric (reported in several papers, e.g., Bakhtin [3], Czerwik [4]) as an extension of a metric notion is obtained by replacing the triangle inequality of the metric with a general one
 \((B)\):

for every ,
for fixed \(s \geq 1\). The triplet is said to be a bmetric space. (It is worth pointing out that in case \(s=1\) the space coincides with a corresponding standard metric space.)
One of the basic examples for bmetric is the following.
Example
([5])
Let be a metric space. Then the function defined as with \(p>1\) forms a bmetric (here \(s=2^{p1}\)).
For more examples, see e.g. [5–12].
Like metric spaces, bmetric spaces admit a nice topology. On the other hand, alike metric, bmetric does not need to be continuous. For the sake of the integrity of the article, we recollect the basic topological notions here.
We say that a sequence in a bmetric space is

(1)
convergent to if . The limit of a convergent sequence is unique;

(2)
Cauchy if as \(n,m\rightarrow \infty \).
Each convergent sequence in a bmetric space is Cauchy and, as usual, if each Cauchy sequence is convergent, then the bmetric space is said to be complete.
Definition 1.1
Let be a bmetric space and be a mapping. For , the orbit of at is the set
The mapping is said to be orbitally continuous at a point if
Additionally, if every Cauchy sequence is convergent in , then the bmetric space is said to be orbitally complete.
Definition 1.2
([13])
Let be a bmetric space. We say that the mapping is mcontinuous, where \(m=1,2,\ldots\) , if , whenever the sequence in is such that .
Remark 1.3
We note that every continuous mapping is orbitally continuous in and also every complete bmetric space is orbitally complete for any , but the converse is not necessarily true.
On the other hand, it is clear that 1continuity (which coincides with usual continuity) implies 2continuity implies 3continuity and so on, but the converse does not hold. Indeed, for example, considering the mapping , where , defined by
we can easily see that is not continuous (in ), but it is 2continuous because .
Let us consider the following class of functions (named the set of bcomparison functions):
here \(\phi ^{n}\) represents the nth iterate of ϕ. It can be shown that every function \(\phi \in \Theta \) fulfills the following properties:
 \((\phi 1)\):

\(\phi (\theta )<\theta \) for any \(\theta >0\);
 \((\phi 2)\):

\(\phi (0)=0\).
Let be a nonempty set and \(\alpha :\mathcal{X}\times \mathcal{X}\rightarrow [0,\infty )\) be a function. We say that the mapping is αorbital admissible if
for all .
Moreover, we say that the bmetric space is αregular if for any sequence \(\{ \eta _{m} \} \) in such that \(\lim_{m\rightarrow \infty }\eta _{m}=\eta \) and \(\alpha (\eta _{m}, \eta _{m+1})\geq 1\) we have \(\alpha (\eta _{m},\eta )\geq 1\).
(For more details and examples, see [14].)
Very recently, the notion of the interpolative contraction was introduced in [15]. The goal of this paper is to revisit the wellknown Kannan type contraction in the setting of interpolation. After that, several famous contractions (Ćirić [16], Reich [17], Rus [18], Hardy– Rogers [19], Kannan [20], Bianchini [21]) are revisited in this new setting, see e.g. [15, 22–26]
In this paper, we combine all these notions and trends to get more general results on the topic in the literature. We observe some interpolative contractions involving distinct rational forms that provide a fixed point in the framework of bmetric spaces.
2 Main results
Definition 2.1
Let be a bmetric space. A selfmapping is called admissible interpolative contraction (\(l=1,2\)) if there exist \(\phi \in \Theta \) and such that
where , \(i=1, 2, 3, 4, 5\), are such that and
and
for any . (.)
The first main results of this paper is given in the following theorem.
Theorem 2.2
Let be a complete bmetric space and be an admissible interpolative contraction such that
 \((i)\):

is αorbital admissible;
 \((\mathit{ii})\):

there exists such that ;
 \((\mathit{iii}_{1})\):

is mcontinuous for \(m\geq 1\), or
 \((\mathit{iii}_{2})\):

is orbitally continuous.
Then possesses a fixed point and the sequence converges to this point ϖ.
Proof
Let in be an arbitrary point and the sequence \(\{ \eta _{n} \} \) be defined as , for all \(n\in \mathbb{N}\). If we can find some \(q\in \mathbb{N}\) such that , then it follows that \(\eta _{q}\) is a fixed point of and the proof is closed. For this reason, we can assume from now on that \(\eta _{n}\neq \eta _{n1}\) for any \(n\in \mathbb{N}\). Using assumption \((i)\), is αorbital admissible, we have
On the other hand, we have that
Now, taking into account the main assumption that is an admissible interpolative contraction, if we substitute with \(\eta _{n1}\) and ω with \(\eta _{n}\) in (2.1), we get
But by \((B)\), together with the monotony of the function ϕ, it follows
moreover, by \((\phi 1)\) we have
If there exists \(m_{0}\in \mathbb{N}\) such that , then the above inequality becomes
which is a contradiction since (keeping in mind that ) it is equivalent with
Therefore, for any \(n\in \mathbb{N}\),
Furthermore, returning to inequality (2.5), we have
Let \(q\in \mathbb{N}\). Then, by \((B)\), together with (2.6), we obtain
It follows that \(\{ \eta _{n} \} \) is a Cauchy sequence in a orbitally complete bmetric space. Therefore, we can find such that .
We claim that ϖ is a fixed point of the mapping under of any hypothesis, \((\mathit{iii})_{1}\) or \((\mathit{iii})_{2}\).
Indeed,
Moreover,
If is mcontinuous, then , and by (2.7) it follows that .
If is assumed to be orbitally continuous on , then
Therefore, . □
Example
Let and be the bmetric defined as for all . Let the mapping be defined by
and a function , where
Let also the comparison function \(\phi :[0,\infty )\rightarrow [0,\infty )\), \(\phi (t)=t/3\), and we choose , , . Thus, we can easily observe that assumptions (i) and (ii) are satisfied, and since is continuous, assumption (iv) is also verified.
Case \((i.)\) For , we have , so inequality (2.1) holds.
Case \((\mathit{ii}.)\) For and \(\omega =2\), we have and . Thus, (2.1) holds.
Case \((\mathit{iii}.) \) For and \(\omega =3\), we have ⇒
Case \((\mathit{iv}.) \) For and \(\omega =9\), we have ⇒
All other cases are of no interest because and (2.1) is satisfied.
Therefore, the mapping is an admissible interpolative contraction. On the other hand, since is continuous and is αorbital continuous, by Theorem 2.2 we get that there exists a fixed point of the mapping ; that is, .
Theorem 2.3
Let be a complete bmetric space and be an admissible interpolative contraction such that
 \((i)\):

is αorbital admissible;
 \((\mathit{ii})\):

there exists such that ;
 \((\mathit{iii}_{1})\):

is mcontinuous for \(m\geq 1\), or
 \((\mathit{iii}_{2})\):

is orbitally continuous.
Then possesses a fixed point .
Proof
As in the previous proof, for , we build the sequence \(\{ \eta _{n} \} \), where and for any \(n\in \mathbb{N}\). Since \(\eta _{n1}\neq \eta _{n}\) for any \(n\in \mathbb{N}\cup {0}\), taking into account that the mapping is supposed to be admissible interpolative contraction, we have
where
Therefore, since by assumption \((i)\) it follows that \(\alpha (\eta _{n1},\eta _{n})\geq 1\) for all \(n\in \mathbb{N}\), we have
(Here, we used the property \((\phi 1)\) of the function ϕ.)
Thus,
and then for any \(n\in \mathbb{N}\). Furthermore, by (2.8) and keeping in mind \((\phi 2)\), we obtain
and following the same steps as in the proof of Theorem 2.2, we can easily find that the sequence \(\{ \eta _{n} \} \) is Cauchy. Moreover, since is supposed to be orbitally complete, we can find a point such that . Assuming that is mcontinuous, we have
and assuming that is orbitally continuous, we get
that is, ϖ is a fixed point of . □
In case we replace the continuity condition of the mapping with the continuity of the bmetric , we get the following results.
Theorem 2.4
Let be a complete, αregular bmetric space, where the bmetric is continuous, and is such that
where \(\phi \in \Theta \) and , for \(l=1,2\) are given by (2.2) and (2.3). If
 \((i)\):

is αorbital admissible;
 \((\mathit{ii})\):

there exists such that .
Then possesses a fixed point , and the sequence converges to this point ϖ.
Proof
From the proof of Theorem 2.2 we know that the sequence \(\{ \eta _{n} \} \), where converges to a point , and we claim that ϖ is a fixed point of the mapping . For this purpose, we claim that
or
Indeed, supposing the contrary
we get that
This is a contradiction, and then (2.10) or (2.11) holds. Under the regularity assumption of the space , we have that \(\alpha (\eta _{n},\varpi )\geq 1\) for any \(n\in \mathbb{N}\).
Case 1. (\(l=1\))
We can distinguish the following two situations:

(i)
.
Letting \(n\rightarrow \infty \) in (2.12) respectively (2.13), we obtain . Thus, .

(ii)
.
In this case, when \(n\rightarrow \infty \), from (2.12), (2.13) and keeping in mind the continuity of bmetric , we get
which is a contradiction.
Consequently, , that is, ϖ is a fixed point of the mapping .
Case 2. (\(l=2\))
We can distinguish the following two situations:

(i)
.
Letting \(n\rightarrow \infty \) in (2.14), respectively (2.15), we obtain . Thus, .

(ii)
.
In this case, when \(n\rightarrow \infty \), from (2.14) and (2.15), we get
which is a contradiction.
Consequently, , that is, ϖ is a fixed point of the mapping . □
Example
Let and be a bmetric space (\(s=2\)), defined by
Let be a selfmapping on , with and . Taking , for all , \(\phi (t)=t/2\) and the constants for \(i\in \{ 1,2,3,4,5 \} \), we have
Thus, by Theorem 2.4, the mapping has (at least) a fixed point.
3 Consequences
Corollary 3.1
Let be a complete bmetric space and be a mapping such that
for any , where , \(l=1,2\), are defined by (2.2) and (2.3) and \(\phi \in \Theta \). Then possesses a fixed point provided that
 \((i)\):

is αorbital admissible;
 \((\mathit{ii})\):

there exists such that ;
 \((\mathit{iii}_{1})\):

is mcontinuous for \(m\geq 1\), or
 \((\mathit{iii}_{2})\):

is orbitally continuous.
Corollary 3.2
Let be a complete bmetric space and be a mapping such that
for any , where , \(l=1,2\), are defined by (2.2) and (2.3). Then possesses a fixed point , provided that either is mcontinuous for \(m\geq 1\) or is orbitally continuous.
Proof
Put in Theorem 2.2, respectively 2.3. □
Corollary 3.3
Let be a complete bmetric space and be a mapping such that there exists \(\kappa \in [0,1)\) such that
for any , where , \(l=1,2\), are defined by (2.2) and (2.3). Then possesses a fixed point , provided that either is mcontinuous for \(m\geq 1\), or is orbitally continuous.
Proof
Put \(\phi (t)=\kappa \cdot t\) in Corollary 3.2. □
Corollary 3.4
Let be a complete bmetric space such that is continuous. A mapping has a fixed point in provided that
where \(\phi \in \Theta \) and , for \(l=1,2\) are given by (2.2) and (2.3).
Proof
Put in Theorem 2.4. □
Corollary 3.5
Let be a complete bmetric space such that is continuous. A mapping has a fixed point in provided that there exists \(\kappa \in [0,1)\) such that
where for \(l=1,2\) are given by (2.2) and (2.3).
Proof
Put \(\phi (t)=\kappa \cdot t\) in Corollary 3.4. □
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Fulga, A. On interpolative contractions that involve rational forms. Adv Differ Equ 2021, 448 (2021). https://doi.org/10.1186/s13662021036054
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DOI: https://doi.org/10.1186/s13662021036054