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Unified multivalued interpolative Reich–Rus–Ćirićtype contractions
Advances in Difference Equations volume 2021, Article number: 311 (2021)
Abstract
This article examines new multivalued interpolative Reich–Rus–Ćirićtype contraction conditions and fixed point results for multivalued maps that fulfill these conditions. Earlier defined interpolative contraction type conditions cannot be particularized to any contraction type condition. This slackness of the interpolative contraction type condition is addressed through new multivalued interpolative Reich–Rus–Ćirićtype contraction conditions.
1 Introduction and preliminaries
A fixed point to a selfmapping L defined on a nonvoid abstract set B is a solution to an equation \(Lb=b\). Banach’s fixed point result [1] is the initial result in the metric fixed point theory which deals with the existence of a solution to the aforementioned equation for a selfmap L of a metric space \((B,d_{B})\). This result requires the following two conditions to ensure the existence and uniqueness of a solution to an equation \(Lb=b\), equivalently, fixed point of L:

(1)
The metric space should be complete;

(2)
L should be contraction map, that is, \(d_{B}(Lb,Lz)\leq \Omega d_{B}(b,z)\) for each \(b,z\in B\), where \(\Omega \in [0,1)\).
Above conditions have a pivotal role in the development of the metric fixed point theory. Several generalizations have been concluded by modifying these conditions. For instance, some modified types of metric spaces are known as partial metric spaces [2], bmetric spaces [3, 4], and extended bmetric spaces [5]. Meanwhile, the classical and the earliest modifications in contraction map are provided by Kannan [6], and Chatterjea [7], as follows:
A map \(L:(B,d_{B})\to (B,d_{B})\) is called a Kannan contraction, if
for all \(b,z\in B\), where \(\Omega \in [0,1/2)\).
A map \(L:(B,d_{B})\to (B,d_{B})\) is called a Chatterjea contraction, if
for all \(b,z\in B\), where \(\Omega \in [0,1/2)\).
An interpolative Kannan contraction seems like a modified form of Kannan contraction. This notion is derived by Karapınar [8] and further improved by Karapınar, Agarwal and Aydi [9]. Since the introduction of an interpolative Kannan contraction by Karapınar [8] many of the existing contraction type conditions have been modified utilizing the pattern of interpolative Kannan contraction. Details can be found in [10–18]. A few existing interpolative contraction type conditions are as follows:
A map \(L:(B,d_{B})\to (B,d_{B})\) is an interpolative Kannan contraction, if
for all \(b,z\in B\) with \(b \neq Lb\), where \(\Omega \in [0,1)\) and \(\tau _{1}\in (0,1)\).
A map \(L:(B,d_{B})\to (B,d_{B})\) is an improved interpolative Kannan contraction, if
for all \(b,z\in B{ \setminus }\operatorname{Fix}(L)\), where \(\Omega \in [0,1)\), \(\tau _{1}\in (0,1)\) and \(\operatorname{Fix}(L)=\{b\in B: Lb=b\}\).
A map \(L:(B,d_{B})\to (B,d_{B})\) is an \((\Omega ,\tau _{1},\tau _{2})\)interpolative Kannan contraction, if
for all \(b,z\in B{ \setminus }\operatorname{Fix}(L)\), where \(\Omega \in [0,1)\), \(\tau _{1},\tau _{2}\in (0,1)\) with \(\tau _{1}+\tau _{2}<1\).
A map \(L:(B,d_{B})\to (B,d_{B})\) is an interpolative Reich–Rus–Ćirićtype contraction, if
for each \(b,z\in B\setminus \operatorname{Fix}(L)\), where \(\Omega \in [0,1)\) and \(\tau _{1}, \tau _{2} \in (0,1)\) with \(\tau _{1} + \tau _{2} < 1\).
The setvalued/multivalued interpolative Reich–Rus–Ćirićtype contraction map was introduced by Debnath and Sen [16] in a bmetric space.
This article examines new multivalued interpolative Reich–Rus–Ćirićtype contraction maps and fixed point results for such maps. The new multivalued interpolative Reich–Rus–Ćirićtype contraction conditions which are being examined in this article cannot only be particularized to Nadler’s type contraction condition but also to some other types of interpolative contraction conditions. Debnath and Sen [16] discussed the existence of fixed points for multivalued interpolative Reich–Rus–Ćirićtype contraction map in bmetric space, by assuming that all bounded and closed subsets of the bmetric space are compact. Readers can see that the restriction of compactness is not required in the presented results of this article.
Before moving towards the main results, we discuss the notion of bmetric spaces, presented by Bakhtin [3] and Czerwik [4], with a few essential concepts.
Definition 1.1
A function \(d_{B} : B\times B \to [0,\infty )\) is called a bmetric on \(B\neq \emptyset \), if for all \(b,z,c\in B\) and for some \(\lambda \geq 1\), we get

(1)
\(d_{B}(b,z)=0\Leftrightarrow b=z\);

(2)
\(d_{B}(b,z) = d_{B}(z,b)\);

(3)
\(d_{B}(b,c)\leq \lambda [d_{B}(b,z) + d_{B}(z,c)]\).
Then \((B,d_{B},\lambda )\) denotes bmetric space along coefficient \(\lambda \geq 1\).
The concept of bmetric space is considered as the strongest generalization of metric space and it is reflected by the work of several researchers. The reader may refer to [19–27].
Definition 1.2
([4])
Let \((B,d_{B},\lambda )\) be a bmetric space along coefficient \(\lambda \geq 1\). Then:

a sequence \(\{b_{n}\}\) is Cauchy in B, if \(\lim_{n,m\rightarrow \infty }d_{B}(b_{n},b_{m})=0\);

a sequence \(\{b_{n}\}\) is convergent to \(b_{\ast }\) in B, if \(\lim_{n\rightarrow \infty }d_{B}(b_{n},b_{ \ast })=0\) and \(b_{\ast }\in B\);

\((B,d_{B},\lambda )\) is called complete if each Cauchy sequence \(\{b_{n}\} \) in B is convergent in B.
Now on, \((B,d_{B},\lambda )\) denotes the bmetric space along coefficient \(\lambda \geq 1\) and \(CB(B)\) represents the collection of all nonempty bounded and closed subsets of B. The functional \(H_{B}: CB(B)\times CB(B)\rightarrow [0,\infty )\) defined by
is the Pompeiu–Hausdorff bmetric on \(CB(B)\), where \(d_{B}(\omega , E)=\inf \{d_{B}(\omega ,\eta ), \eta \in E\}\).
The following theorem has an important role in the results presented by this article.
Theorem 1.3
([19])
Let \((B,d_{B},\lambda )\) be a bmetric space. Let \(D,E \in CB(B)\) and \(\omega \in D\). Then, for each \(\Omega > 1\), there is \(\eta \in E\) with
2 Main results
This section begins with the following definition.
Definition 2.1
Assume a bmetric space \((B,d_{B},\lambda )\) and maps \(L:B\to CB(B)\), \(\gamma :B\times B\to \mathbb{R}\{0\}\). The map L is called a γinterpolative Reich–Rus–ĆirićIcontraction, if
for each \(b,z\in B\) with
where \(\Omega \in (0,\frac{1}{\lambda ^{2}})\) and \(\tau _{1}, \tau _{2}, \tau _{3}\in [0,1]\) with \(\tau _{1} + \tau _{2}+\tau _{3} = 1\).
The existence of fixed points for the defined notion is discussed as follows.
Theorem 2.2
Assume a complete bmetric space \((B,d_{B},\lambda )\) and γinterpolative Reich–Rus–ĆirićIcontraction map L. Also, assume that:

(1)
there exist \(b_{0}\in B\) and \(b_{1}\in Lb_{0}\) with \(\gamma (b_{0},b_{1})=1\);

(2)
for each \(b,z\in B\) with \(\gamma (b,z)=1\), we have \(\gamma (c,d)=1 \forall c\in Lb\), \(d\in Lz\);

(3)
for each \(\{b_{m}\}\) in B with \(b_{m} \to b\) and \(\gamma (b_{m},b_{m+1})=1 \forall m\in \mathbb{N}\), we have \(\gamma (b_{m},b)=1 \forall m\in \mathbb{N}\).
Then L has a fixed point in B.
Proof
By (1), there exist \(b_{0}\in B\) and \(b_{1}\in Lb_{0}\) with \(\gamma (b_{0},b_{1})=1\). If
then fixed point of L possesses in B. Suppose that
By (2.1), we obtain
From (2.2), we obtain
As \(\frac{1}{\sqrt{\Omega }}>1\), from Theorem 1.3, there should be \(b_{2}\in Lb_{1}\) satisfying
By (2.3) and the above fact, we conclude
Now, we discuss the proof for the following three choices of \(\tau _{3}\):

If \(\tau _{3}=0\) in (2.4), then \(\tau _{1}+\tau _{2}=1\), thus \(d_{B}(b_{1},b_{2}) \leq \sqrt{\Omega }d_{B}(b_{0},b_{1})\).

If \(\tau _{3}=1\) in (2.4) then \(d_{B}(b_{1},b_{2})=0\), that is, \(b_{1}\) is a fixed point of L and it is not possible under the assumption.

If \(\tau _{3}\in (0,1)\) in (2.4) then we have the following:
$$\begin{aligned} \bigl[d_{B}(b_{1},b_{2}) \bigr]^{1\tau _{3}}\leq \sqrt{\Omega } \bigl[\bigl[d_{B}(b_{0},b_{1}) \bigr]^{ \tau _{1}+\tau _{2}} \bigr] \end{aligned}$$(2.5)
since \(1\tau _{3}=\tau _{1}+\tau _{2}\), thus, by the above inequality, we get
Hence, we arrive at
As \(b_{1}\in Lb_{0}\), \(b_{2}\in Lb_{1}\) and \(\gamma (b_{0},b_{1})=1\), then, by (2), we obtain \(\gamma (b_{1},b_{2})=1\). Again, we assume that
then by (2.1) we get
As \(\frac{1}{\sqrt{\Omega }}>1\), there should be \(b_{3}\in Lb_{2}\) satisfying
Thus, by (2.7) and the above inequality, we get
Again, we discuss the proof of the following three choices of \(\tau _{3}\):

If \(\tau _{3}=0\) in (2.8), then \(\tau _{1}+\tau _{2}=1\), thus \(d_{B}(b_{2},b_{3}) \leq \sqrt{\Omega }d_{B}(b_{1},b_{2})\).

If \(\tau _{3}=1\) in (2.8) then \(d_{B}(b_{2},b_{3})=0\), that is, \(b_{2}\) is a fixed point of L and it is not possible under the assumption.

If \(\tau _{3}\in (0,1)\) in (2.8) then we have the following:
$$\begin{aligned} \bigl[d_{B}(b_{2},b_{3}) \bigr]^{1\tau _{3}}\leq \sqrt{\Omega } \bigl[\bigl[d_{B}(b_{1},b_{2}) \bigr]^{ \tau _{1}+\tau _{2}} \bigr]. \end{aligned}$$(2.9)
Thus, we arrive at
Induction yields a sequence \(\{b_{m}\}\) in B with \(b_{m}\in Lb_{m1}\), \(\gamma (b_{m},b_{m+1})=1\) \(\forall m \in \mathbb{N}\) and
Also, we get
By the triangle inequality, for \(n>m\), we get
Since \(\sum_{j=1}^{\infty } \lambda ^{j}(\sqrt{\Omega })^{j}<\infty \), thus, \(\{b_{m}\}\) is a Cauchy in B. For \(\{b_{m}\}\) the completeness of B shall give \(b_{\ast }\) in B with \(b_{m} \to b_{\ast }\). By considering (3), we obtain \(\gamma (b_{m},b_{\ast })=1 \forall m \in \mathbb{N}\). Here, we claim that \(b_{\ast }\in Lb_{\ast }\). Let us suppose that if the claim is wrong then
for some natural number \(n_{0}\). By (2.1) we get
By the triangle inequality and (2.11), we get
Suppose that \(\tau _{3}\neq 1\) and \(m\to \infty \) in the above inequality, then we get \(d_{B}(b_{\ast },Lb_{\ast })=0\), that is, \(b_{\ast }\in Lb_{\ast }\). Suppose that \(\tau _{3}= 1\) and \(m\to \infty \) in the above inequality, then we get \(d_{B}(b_{\ast },Lb_{\ast })\leq \lambda \Omega d_{B}(b_{\ast },Lb_{\ast })\), which is not possible if \(d_{B}(b_{\ast },Lb_{\ast })\neq 0\). Hence, our claim is true, \(b_{\ast }\in Lb_{\ast }\). □
Example 2.3
Consider B as a set of all integers and define \(d_{B}(b,b')=bb'^{2}\) ∀b, \(b'\in B\). Define \(L:B\to CB(B)\) by
and \(\gamma :B\times B\to \mathbb{R}\{0\}\) by
Now, one can calculate the following cases.

If \(b,b'\in \{2,3,4, \ldots \}\) with \(b\neq b'\), we obtain \(H_{B}(Lb,Lb')^{\gamma (b,b')}=0\).

If \(b,b'<0\) with \(b\neq b'\), we obtain \(H_{B}(Lb,Lb')^{\gamma (b,b')}= \frac{1}{[(b2)^{2}+(b'2)^{2}^{2}]^{b+b'+8}}\).

If \(b<0\) and \(b'\geq 2\), we obtain \(H_{B}(Lb,Lb')^{\gamma (b,b')}= \frac{1}{[(b2)^{2}^{2}]^{b+b'+8}}\).
These calculations verify the validity of (2.1). The remaining axioms of Theorem 2.2 are also valid. Hence, L has a fixed point.
By assuming \(\tau _{1}=1\) and \(\tau _{2}=\tau _{3}=0\) in the above result, we arrive at the following results.
Corollary 2.4
Assume we have a complete bmetric space \((B,d_{B},\lambda )\) and maps \(L:B\to CB(B)\), \(\gamma :B\times B\to \mathbb{R}\{0\}\) such that
for each \(b,z\in B\) with
where \(\Omega \in [0,\frac{1}{\lambda ^{2}})\). Also, assume that:

(1)
there exist \(b_{0}\in B\) and \(b_{1}\in Lb_{0}\) with \(\gamma (b_{0},b_{1})=1\);

(2)
for each \(b,z\in B\) with \(\gamma (b,z)= 1\), we have \(\gamma (c,d)= 1\) \(\forall c\in Lb\), \(d\in Lz\);

(3)
for each \(\{b_{m}\}\) in B with \(b_{m} \to b\) and \(\gamma (b_{m},b_{m+1})= 1\) \(\forall m\in \mathbb{N}\), we have \(\gamma (b_{m},b)= 1\) \(\forall m\in \mathbb{N}\).
Then L has a fixed point in B.
By assuming \(\gamma (b,z)=1\) for all \(b,z\in B\) in the above corollary, we obtain the following result which can be considered as an extended form of Nadler’s fixed point theorem.
Corollary 2.5
Assume a complete bmetric space \((B,d_{B},\lambda )\) and a map \(L:B\to CB(B)\) satisfying the following inequality:
for each \(b,z\in B\) with
where \(\Omega \in [0,\frac{1}{\lambda ^{2}})\). Then L has a fixed point in B.
Remark 2.6
By considering (2.13) one can say that γinterpolative Reich–Rus–ĆirićIcontraction can be particularized to Nadler’s type contraction.
The right side of (2.14) is more analogous to interpolative Reich–Rus–Ćirićcontraction.
Definition 2.7
Assume a bmetric space \((B,d_{B},\lambda )\) and maps \(L:B\to CB(B)\), \(\gamma :B\times B\to \mathbb{R}\{0\}\). The map L is called a reduced γinterpolative Reich–Rus–ĆirićIcontraction, if
for each \(b,z\in B\) with
where \(\Omega \in (0,\frac{1}{\lambda ^{2}})\) and \(\tau _{1}, \tau _{2}\in [0,1)\) with \(0<\tau _{1} + \tau _{2} < 1\).
Remark 2.8
Consider \(\varsigma _{1},\varsigma _{2}\in [0,1)\) with \(0<\varsigma _{1}+\varsigma _{2}<1\). Define \(\tau _{1}=\varsigma _{1}\), \(\tau _{2}=\varsigma _{2}\) and \(\tau _{3}=1\varsigma _{1}\varsigma _{2}\), then \(\tau _{1}+\tau _{2}+\tau _{3}=\varsigma _{1}+\varsigma _{2}+(1 \varsigma _{1}\varsigma _{2})=1\). Thus, (2.1) of Definition 2.1 gives (2.14) of Definition 2.7.
Now one can easily understand that Theorem 2.9 is a simple consequence of Theorem 2.2.
Theorem 2.9
Assume a complete bmetric space \((B,d_{B},\lambda )\) and reduced γinterpolative Reich–Rus–ĆirićIcontraction map L. Also, assume that:

(1)
there exist \(b_{0}\in B\) and \(b_{1}\in Lb_{0}\) with \(\gamma (b_{0},b_{1})=1\);

(2)
for each \(b,z\in B\) with \(\gamma (b,z)=1\), we have \(\gamma (c,d)=1 \forall c\in Lb\), \(d\in Lz\);

(3)
for each \(\{b_{m}\}\) in B with \(b_{m} \to b\) and \(\gamma (b_{m},b_{m+1})=1 \forall m\in \mathbb{N}\), we have \(\gamma (b_{m},b)=1\) \(\forall m\in \mathbb{N}\).
Then L has a fixed point in B.
By assuming \(\tau _{1}=0\) and \(\tau _{2}=\tau \in (0,1)\) in the above result we reach the following result.
Corollary 2.10
Assume a complete bmetric space \((B,d_{B},\lambda )\) and maps \(L:B\to CB(B)\), \(\gamma :B\times B\to \mathbb{R}\{0\}\) such that
for each \(b,z\in B\) with
where \(\Omega \in [0,\frac{1}{\lambda ^{2}})\) and \(\tau \in (0,1)\). Also, assume that:

(1)
there exist \(b_{0}\in B\) and \(b_{1}\in Lb_{0}\) with \(\gamma (b_{0},b_{1})=1\);

(2)
for each \(b,z\in B\) with \(\gamma (b,z)= 1\), we have \(\gamma (c,d)= 1\) \(\forall c\in Lb\), \(d\in Lz\);

(3)
for each \(\{b_{m}\}\) in B with \(b_{m} \to b\) and \(\gamma (b_{m},b_{m+1})= 1\) \(\forall m\in \mathbb{N}\), we have \(\gamma (b_{m},b)= 1\) \(\forall m\in \mathbb{N}\).
Then L has a fixed point in B.
Remark 2.11
Inequality (2.15) is a generalized form of improved interpolative Kannan contraction.
The following definition provides another way to generalize interpolative Reich–Rus–Ćirićcontraction maps.
Definition 2.12
Assume a bmetric space \((B,d_{B},\lambda )\) and maps \(L:B\to CB(B)\), \(\gamma :B\times B\to [0,\infty )\). The map L is called a γinterpolative Reich–Rus–ĆirićIIcontraction, if
for each \(b,z\in B\) with
where \(\Omega \in (0,\frac{1}{\lambda ^{2}})\) and \(\tau _{1}, \tau _{2}, \tau _{3}\in [0,1]\) with \(\tau _{1} + \tau _{2}+\tau _{3} = 1\).
The existence of fixed points for the above defined notion are verified through the following result.
Theorem 2.13
Assume a complete bmetric space \((B,d_{B},\lambda )\) and γinterpolative Reich–Rus–ĆirićIIcontraction map L. Also, assume that:

(1)
there exist \(b_{0}\in B\) and \(b_{1}\in Lb_{0}\) with \(\gamma (b_{0},b_{1})\geq 1\);

(2)
for each \(b,z\in B\) with \(\gamma (b,z)\geq 1\), we have \(\gamma (c,d)\geq 1\) \(\forall c\in Lb\), \(d\in Lz\);

(3)
for each \(\{b_{m}\}\) in B with \(b_{m} \to b\) and \(\gamma (b_{m},b_{m+1})\geq 1\) \(\forall m\in \mathbb{N}\), we have \(\gamma (b_{m},b)\geq 1\) \(\forall m\in \mathbb{N}\).
Then L has a fixed point in B.
Proof
Axiom (1) says that there are elements \(b_{0}\in B\) and \(b_{1}\in Lb_{0}\) with \(\gamma (b_{0},b_{1})\geq 1\). Assume that
otherwise a fixed point of L occurs in B. Then, by (2.16), we arrive at
From (2.17), we obtain
As \(\frac{1}{\sqrt{\Omega }}>1\), there should be \(b_{2}\in Lb_{1}\) satisfying
By (2.18) and the above inequality, we get
Now, we discuss the proof for the following three choices of \(\tau _{3}\):

If \(\tau _{3}=0\) in (2.19), then \(\tau _{1}+\tau _{2}=1\), thus \(d_{B}(b_{1},b_{2}) \leq \sqrt{\Omega }d_{B}(b_{0},b_{1})\).

If \(\tau _{3}=1\) in (2.19) then \(d_{B}(b_{1},b_{2})=0\), that is, \(b_{1}\) is a fixed point of L and it is not possible under the assumption.

If \(\tau _{3}\in (0,1)\) in (2.19), then we get the following:
$$\begin{aligned} \bigl[d_{B}(b_{1},b_{2}) \bigr]^{1\tau _{3}}\leq \sqrt{\Omega } \bigl[\bigl[d_{B}(b_{0},b_{1}) \bigr]^{ \tau _{1}+\tau _{2}} \bigr] \end{aligned}$$(2.20)
since \(1\tau _{3}=\tau _{1}+\tau _{2}\), thus, by the above inequality, we get
Hence, we arrive at
As \(b_{1}\in Lb_{0}\), \(b_{2}\in Lb_{1}\) and \(\gamma (b_{0},b_{1})\geq 1\), then by axiom (2), we arrive at \(\gamma (b_{1},b_{2})\geq 1\). By the repetition of (2.16) and axiom (2), we arrive at a sequence \(\{b_{m}\}\) in B with \(b_{m}\in Lb_{m1}\), \(\gamma (b_{m},b_{m+1})\geq 1 \forall m \in \mathbb{N}\) and
Also,
From the proof of Theorem 2.2, we can see that \(\{b_{m}\}\) is a Cauchy in B and there should be \(b_{\ast }\) in B with \(b_{m} \to b_{\ast }\). Also, by (3), \(\gamma (b_{m},b_{\ast })\geq 1 \forall m \in \mathbb{N}\). Now we can claim that \(b_{\ast }\in Lb_{\ast }\). If our claim is wrong, then \(\min \{ d_{B}(b_{m},b_{\ast }),d_{B}(b_{m},Lb_{m}),d_{B}(b_{\ast },Lb_{\ast })\}>0\) for each \(m\geq n_{0}\) (for some natural number \(n_{0}\)). By (2.16), we arrive at
By (2.22) and the triangle inequality, we arrive at
Consider \(\tau _{3}\neq 1\) and \(m\to \infty \) in the above inequality, then we get \(d_{B}(b_{\ast },Lb_{\ast })=0\), that is, \(b_{\ast }\in Lb_{\ast }\). Consider \(\tau _{3}= 1\) and \(m\to \infty \) in the above inequality, then we get \(d_{B}(b_{\ast },Lb_{\ast })\leq \lambda \Omega d_{B}(b_{\ast },Lb_{\ast })\), which is not possible if \(d_{B}(b_{\ast },Lb_{\ast })\neq 0\). Hence, our claim is true, \(b_{\ast }\in Lb_{\ast }\). □
Now we shall discuss the notion of reduced γinterpolative Reich–Rus–ĆirićIIcontraction map and related fixed point result.
Definition 2.14
Assume a bmetric space \((B,d_{B},\lambda )\) and maps \(L:B\to CB(B)\), \(\gamma :B\times B\to [0,\infty )\). The map L is called a reduced γinterpolative Reich–Rus–ĆirićIIcontraction, if
for each \(b,z\in B\) with
where \(\Omega \in (0,\frac{1}{\lambda ^{2}})\) and \(\tau _{1}, \tau _{2}\in [0,1)\) with \(0<\tau _{1} + \tau _{2}< 1\).
The following result is a simple consequence of Theorem 2.13.
Theorem 2.15
Assume a complete bmetric space \((B,d_{B},\lambda )\) and reduced γinterpolative Reich–Rus–ĆirićIIcontraction map L. Also, assume that:

(1)
there exist \(b_{0}\in B\) and \(b_{1}\in Lb_{0}\) with \(\gamma (b_{0},b_{1})\geq 1\);

(2)
for each \(b,z\in B\) with \(\gamma (b,z)\geq 1\), we have \(\gamma (c,d)\geq 1\) \(\forall c\in Lb\), \(d\in Lz\);

(3)
for each \(\{b_{m}\}\) in B with \(b_{m} \to b\) and \(\gamma (b_{m},b_{m+1})\geq 1\) \(\forall m\in \mathbb{N}\), we have \(\gamma (b_{m},b)\geq 1\) \(\forall m\in \mathbb{N}\).
Then L has a fixed point in B.
Example 2.16
Consider B as a set of all real numbers and \(d_{B}(b,b')=bb'\) for all \(b,b'\in B\). Define \(L:B\to CB(B)\) by
and \(\xi :B\times B\to [0,\infty )\) by
For instance, take \(b=1\) and \(b'=3\), then \(H_{B}(Tb,Tb')=4\), \(d_{B}(b,b')=2\) \(d_{B}(b,Tb)=1\) and \(d_{B}(b',Tb')=3\). Also
Thus, it can be seen that the setvalued versions based on the structure of bmetric spaces for the interpolative contraction type conditions given in [8–12, 16], with many other existing interpolative contraction type conditions, are not applicable on the above defined function L with respect to the above \(d_{B}\). Meanwhile, all the axioms of Theorem 2.13 are valid on the above defined functions.
3 Conclusion
This article presents new multivalued interpolative Reich–Rus–Ćirićtype contraction conditions and fixed point results for multivalued maps which fulfil these conditions in a complete bmetric space. Earlier defined interpolative contraction type conditions cannot be particularized to any contraction type condition. This slackness of interpolative contraction type condition is addressed through the introduction of new multivalued interpolative Reich–Rus–Ćirićtype contraction conditions. A few examples are given to support the findings of this article.
Availability of data and materials
The data used to support the findings of this study are available from the corresponding author upon request.
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Acknowledgements
This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia, under grant number KEP513042. The authors, therefore, gratefully acknowledge the DSR for technical and financial support.
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The Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, Saudi Arabia, has funded this project under grant number KEP513042.
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Alansari, M., Ali, M.U. Unified multivalued interpolative Reich–Rus–Ćirićtype contractions. Adv Differ Equ 2021, 311 (2021). https://doi.org/10.1186/s13662021034621
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DOI: https://doi.org/10.1186/s13662021034621