On interpolative contractions that involve rational forms

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

The mapping is said to be orbitally continuous at a point if

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

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):

$$ \Theta = \biggl\{ \phi :[0,\infty )\rightarrow [0,\infty )\Bigm|\phi \text{ is nondecreasing and } \sum_{n\geq 1}s^{n}\phi ^{n}( \theta )< \infty \text{ for each } \theta >0 \biggr\} , $$

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

(1.1)

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

(2.1)

where , $i=1, 2, 3, 4, 5$, are such that and

(2.2)

and

(2.3)

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

On the other hand, we have that

Now, taking into account the main assumption that is an -admissible interpolative contraction, if we substitute with $\eta _{n-1}$ and ω with $\eta _{n}$ in (2.1), we get

(2.4)

But by $(B)$, together with the monotony of the function ϕ, it follows

(2.5)

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

(2.6)

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 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,

Moreover,

(2.7)

If is m-continuous, then , and by (2.7) it follows that .

If is assumed to be orbitally continuous on , then

Therefore, . □

Example

Let and be the b-metric 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 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

where

Therefore, since by assumption $(i)$ it follows that $\alpha (\eta _{n-1},\eta _{n})\geq 1$ for all $n\in \mathbb{N}$, we have

(2.8)

(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 m-continuous, 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 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

(2.9)

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

(2.10)

or

(2.11)

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. ($l=1$)

$(1.a)$:

If (2.10) holds, we get

(2.12)

$(1.b)$:

If (2.11) holds,

(2.13)

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. ($l=2$)

$(2.a)$:

If (2.10) holds, we get

(2.14)

$(2.b)$:

If (2.11) holds,

(2.15)

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

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

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

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

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

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

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

where for $l=1,2$ are given by (2.2) and (2.3).

Proof

Put $\phi (t)=\kappa \cdot t$ in Corollary 3.4. □