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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 b-metric 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 b-metric space. Following this success, many authors have continued to work on this trend and reported several improvements, advances in the setting of b-metric spaces, see e.g. [5–12].
Let be a nonempty set and
be a metric on
. The notion of b-metric (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 b-metric 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 b-metric is the following.
Example
([5])
Let be a metric space. Then the function
defined as
with \(p>1\) forms a b-metric (here \(s=2^{p-1}\)).
For more examples, see e.g. [5–12].
Like metric spaces, b-metric spaces admit a nice topology. On the other hand, alike metric, b-metric 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 b-metric 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 b-metric space is Cauchy and, as usual, if each Cauchy sequence is convergent, then the b-metric space is said to be complete.
Definition 1.1
Let be a b-metric space and
be a mapping. For
, the orbit of
at
is the set
![](http://media.springernature.com/lw191/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equa_HTML.png)
The mapping is said to be orbitally continuous at a point
if
![](http://media.springernature.com/lw288/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equb_HTML.png)
Additionally, if every Cauchy sequence is convergent in
, then the b-metric space
is said to be
-orbitally complete.
Definition 1.2
([13])
Let be a b-metric space. We say that the mapping
is m-continuous, 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 b-metric space is
-orbitally complete for any
, but the converse is not necessarily true.
On the other hand, it is clear that 1-continuity (which coincides with usual continuity) implies 2-continuity implies 3-continuity and so on, but the converse does not hold. Indeed, for example, considering the mapping , where
, defined by
![](http://media.springernature.com/lw156/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equc_HTML.png)
we can easily see that is not continuous (in
), but it is 2-continuous because
.
Let us consider the following class of functions (named the set of b-comparison 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 α-orbital admissible if
![](http://media.springernature.com/lw260/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equ1_HTML.png)
for all .
Moreover, we say that the b-metric space is α-regular 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 well-known 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 b-metric spaces.
2 Main results
Definition 2.1
Let be a b-metric space. A self-mapping
is called
-admissible interpolative contraction (\(l=1,2\)) if there exist \(\phi \in \Theta \) and
such that
![](http://media.springernature.com/lw395/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equ2_HTML.png)
where , \(i=1, 2, 3, 4, 5\), are such that
and
![](http://media.springernature.com/lw522/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equ3_HTML.png)
and
![](http://media.springernature.com/lw471/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equ4_HTML.png)
for any . (
.)
The first main results of this paper is given in the following theorem.
Theorem 2.2
Let be a complete b-metric space and
be an
-admissible interpolative contraction such that
- \((i)\):
-
is α-orbital admissible;
- \((\mathit{ii})\):
-
there exists
such that
;
- \((\mathit{iii}_{1})\):
-
is m-continuous 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 \(\eta _{n}\neq \eta _{n-1}\) for any \(n\in \mathbb{N}\). Using assumption \((i)\),
is α-orbital admissible, we have
![](http://media.springernature.com/lw466/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Eque_HTML.png)
On the other hand, we have that
![](http://media.springernature.com/lw289/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equf_HTML.png)
Now, taking into account the main assumption that is an
-admissible interpolative contraction, if we substitute
with \(\eta _{n-1}\) and ω with \(\eta _{n}\) in (2.1), we get
![](http://media.springernature.com/lw420/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equ5_HTML.png)
But by \((B)\), together with the monotony of the function ϕ, it follows
![](http://media.springernature.com/lw494/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equ6_HTML.png)
moreover, by \((\phi 1)\) we have
![](http://media.springernature.com/lw511/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equg_HTML.png)
If there exists \(m_{0}\in \mathbb{N}\) such that , then the above inequality becomes
![](http://media.springernature.com/lw385/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equh_HTML.png)
which is a contradiction since (keeping in mind that ) it is equivalent with
![](http://media.springernature.com/lw190/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equi_HTML.png)
Therefore, for any \(n\in \mathbb{N}\),
![](http://media.springernature.com/lw154/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equj_HTML.png)
Furthermore, returning to inequality (2.5), we have
![](http://media.springernature.com/lw322/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equ7_HTML.png)
Let \(q\in \mathbb{N}\). Then, by \((B)\), together with (2.6), we obtain
![](http://media.springernature.com/lw474/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equk_HTML.png)
It follows that \(\{ \eta _{n} \} \) is a Cauchy sequence in a -orbitally complete b-metric 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,
![](http://media.springernature.com/lw181/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equl_HTML.png)
Moreover,
![](http://media.springernature.com/lw225/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equ8_HTML.png)
If is m-continuous, then
, and by (2.7) it follows that
.
If is assumed to be orbitally continuous on
, then
![](http://media.springernature.com/lw344/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equm_HTML.png)
Therefore, . □
Example
Let and
be the b-metric defined as
for all
. Let the mapping
be defined by
![](http://media.springernature.com/lw296/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equn_HTML.png)
and a function , where
![](http://media.springernature.com/lw301/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equo_HTML.png)
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
⇒
![](http://media.springernature.com/lw439/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equp_HTML.png)
Case \((\mathit{iv}.) \) For and \(\omega =9\), we have
⇒
![](http://media.springernature.com/lw439/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equq_HTML.png)
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 α-orbital 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 b-metric space and
be an
-admissible interpolative contraction such that
- \((i)\):
-
is α-orbital admissible;
- \((\mathit{ii})\):
-
there exists
such that
;
- \((\mathit{iii}_{1})\):
-
is m-continuous 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 \(\eta _{n-1}\neq \eta _{n}\) for any \(n\in \mathbb{N}\cup {0}\), taking into account that the mapping
is supposed to be
-admissible interpolative contraction, we have
![](http://media.springernature.com/lw324/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equr_HTML.png)
where
![](http://media.springernature.com/lw488/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equs_HTML.png)
Therefore, since by assumption \((i)\) it follows that \(\alpha (\eta _{n-1},\eta _{n})\geq 1\) for all \(n\in \mathbb{N}\), we have
![](http://media.springernature.com/lw368/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equ9_HTML.png)
(Here, we used the property \((\phi 1)\) of the function ϕ.)
Thus,
![](http://media.springernature.com/lw386/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equt_HTML.png)
and then for any \(n\in \mathbb{N}\). Furthermore, by (2.8) and keeping in mind \((\phi 2)\), we obtain
![](http://media.springernature.com/lw438/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equu_HTML.png)
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 m-continuous, we have
![](http://media.springernature.com/lw228/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equv_HTML.png)
and assuming that is orbitally continuous, we get
![](http://media.springernature.com/lw319/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equw_HTML.png)
that is, ϖ is a fixed point of . □
In case we replace the continuity condition of the mapping with the continuity of the b-metric , we get the following results.
Theorem 2.4
Let be a complete, α-regular b-metric space, where the b-metric
is continuous, and
is such that
![](http://media.springernature.com/lw400/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equ10_HTML.png)
where \(\phi \in \Theta \) and , for \(l=1,2\) are given by (2.2) and (2.3). If
- \((i)\):
-
is α-orbital 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
![](http://media.springernature.com/lw156/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equ11_HTML.png)
or
![](http://media.springernature.com/lw206/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equ12_HTML.png)
Indeed, supposing the contrary
![](http://media.springernature.com/lw416/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equx_HTML.png)
we get that
![](http://media.springernature.com/lw363/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equy_HTML.png)
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 b-metric
, 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 b-metric space (\(s=2\)), defined by
![](http://media.springernature.com/lw197/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equab_HTML.png)
Let be a self-mapping on
, with
and
. Taking
,
for all
, \(\phi (t)=t/2\) and the constants
for \(i\in \{ 1,2,3,4,5 \} \), we have
![](http://media.springernature.com/lw312/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equac_HTML.png)
Thus, by Theorem 2.4, the mapping has (at least) a fixed point.
3 Consequences
Corollary 3.1
Let be a complete b-metric space and
be a mapping such that
![](http://media.springernature.com/lw208/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equad_HTML.png)
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 α-orbital admissible;
- \((\mathit{ii})\):
-
there exists
such that
;
- \((\mathit{iii}_{1})\):
-
is m-continuous for \(m\geq 1\), or
- \((\mathit{iii}_{2})\):
-
is orbitally continuous.
Corollary 3.2
Let be a complete b-metric space and
be a mapping such that
![](http://media.springernature.com/lw342/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equae_HTML.png)
for any , where
, \(l=1,2\), are defined by (2.2) and (2.3). Then
possesses a fixed point
, provided that either
is m-continuous for \(m\geq 1\) or
is orbitally continuous.
Proof
Put in Theorem 2.2, respectively 2.3. □
Corollary 3.3
Let be a complete b-metric space and
be a mapping such that there exists \(\kappa \in [0,1)\) such that
![](http://media.springernature.com/lw340/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equaf_HTML.png)
for any , where
, \(l=1,2\), are defined by (2.2) and (2.3). Then
possesses a fixed point
, provided that either
is m-continuous 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 b-metric space such that
is continuous. A mapping
has a fixed point in
provided that
![](http://media.springernature.com/lw353/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equag_HTML.png)
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 b-metric space such that
is continuous. A mapping
has a fixed point in
provided that there exists \(\kappa \in [0,1)\) such that
![](http://media.springernature.com/lw339/springer-static/image/art%3A10.1186%2Fs13662-021-03605-4/MediaObjects/13662_2021_3605_Equah_HTML.png)
where for \(l=1,2\) are given by (2.2) and (2.3).
Proof
Put \(\phi (t)=\kappa \cdot t\) in Corollary 3.4. □
Availability of data and materials
No data was generated or used during the study; this paper was not about data.
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Acknowledgements
The author thanks the anonymous referees for their outstanding comments, suggestions, and ideas that helped improve this work.
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Fulga, A. On interpolative contractions that involve rational forms. Adv Differ Equ 2021, 448 (2021). https://doi.org/10.1186/s13662-021-03605-4
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DOI: https://doi.org/10.1186/s13662-021-03605-4