Unified multivalued interpolative Reich–Rus–Ćirić-type contractions

Abstract

This article examines new multivalued interpolative Reich–Rus–Ćirić-type 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–Ćirić-type contraction conditions.

1 Introduction and preliminaries

A fixed point to a self-mapping L defined on a non-void 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 self-map 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], b-metric spaces [3, 4], and extended b-metric 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

$$ d_{B}(Lb,Lz)\leq \Omega \bigl[d_{B}(b,Lb)+ d_{B}(z,Lz)\bigr] $$

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

$$ d_{B}(Lb,Lz)\leq \Omega \bigl[d_{B}(b,Lz)+ d_{B}(z,Lb)\bigr] $$

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

$$ d_{B}(Lb,Lz)\leq \Omega \bigl[d_{B}(b,Lb) \bigr]^{\tau _{1}} \bigl[d_{B}(z,Lz)\bigr]^{1- \tau _{1}} $$

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

$$ d_{B}(Lb,Lz)\leq \Omega \bigl[d_{B}(b,Lb) \bigr]^{\tau _{1}} \bigl[d_{B}(z,Lz)\bigr]^{1- \tau _{1}} $$

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

$$ d_{B}(Lb,Lz)\leq \Omega \bigl[d_{B}(b,Lb) \bigr]^{\tau _{1}} \bigl[d_{B}(z,Lz)\bigr]^{ \tau _{2}} $$

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–Ćirić-type contraction, if

$$\begin{aligned} d_{B}(Lb,Lz) \leq & \Omega \bigl[\bigl[d_{B}(b,z) \bigr]^{\tau _{1}}\bigl[d_{B}(b,Lb)\bigr]^{ \tau _{2}} \bigl[d_{B}(z,Lz)\bigr]^{1-\tau _{1}- \tau _{2}} \bigr] \end{aligned}$$

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 set-valued/multivalued interpolative Reich–Rus–Ćirić-type contraction map was introduced by Debnath and Sen [16] in a b-metric space.

This article examines new multivalued interpolative Reich–Rus–Ćirić-type contraction maps and fixed point results for such maps. The new multivalued interpolative Reich–Rus–Ćirić-type 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–Ćirić-type contraction map in b-metric space, by assuming that all bounded and closed subsets of the b-metric 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 b-metric spaces, presented by Bakhtin [3] and Czerwik [4], with a few essential concepts.

Definition 1.1

([3, 4])

A function $d_{B} : B\times B \to [0,\infty )$ is called a b-metric 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 b-metric space along coefficient $\lambda \geq 1$.

The concept of b-metric 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 b-metric 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 b-metric space along coefficient $\lambda \geq 1$ and $CB(B)$ represents the collection of all non-empty bounded and closed subsets of B. The functional $H_{B}: CB(B)\times CB(B)\rightarrow [0,\infty )$ defined by

$$ H_{B}(D,E)=\max \bigl\{ \sup \bigl\{ d_{B}(\omega , E): \omega \in D\bigr\} , \sup \bigl\{ d_{B}( \eta , D): \eta \in E\bigr\} \bigr\} $$

is the Pompeiu–Hausdorff b-metric 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 b-metric space. Let $D,E \in CB(B)$ and $\omega \in D$. Then, for each $\Omega > 1$, there is $\eta \in E$ with

$$ d_{B}(\omega ,\eta )\leq \Omega H_{B}(D,E). $$

2 Main results

This section begins with the following definition.

Definition 2.1

Assume a b-metric 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 γ-interpolative Reich–Rus–Ćirić-I-contraction, if

$$\begin{aligned} \bigl[H_{B}(Lb,Lz)\bigr]^{\gamma (b,z)} \leq & \Omega \bigl[\bigl[d_{B}(b,z)\bigr]^{\tau _{1}} \bigl[d_{B}(b,Lb)\bigr]^{ \tau _{2}} \bigl[d_{B}(z,Lz) \bigr]^{\tau _{3}} \bigr] \end{aligned}$$

(2.1)

for each $b,z\in B$ with

$$ \min \bigl\{ d_{B}(b,z),d_{B}(b,Lb),d_{B}(z,Lz) \bigr\} >0, $$

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 b-metric space $(B,d_{B},\lambda )$ and γ-interpolative Reich–Rus–Ćirić-I-contraction 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

$$ \min \bigl\{ d_{B}(b_{0},b_{1}),d_{B}(b_{0},Lb_{0}),d_{B}(b_{1},Lb_{1}) \bigr\} =0 $$

then fixed point of L possesses in B. Suppose that

$$ \min \bigl\{ d_{B}(b_{0},b_{1}),d_{B}(b_{0},Lb_{0}),d_{B}(b_{1},Lb_{1}) \bigr\} >0. $$

By (2.1), we obtain

$$\begin{aligned} H_{B}(Lb_{0},Lb_{1}) =& \bigl[H_{B}(Lb_{0},Lb_{1})\bigr]^{\gamma (b_{0},b_{1})} \\ \leq & \Omega \bigl[\bigl[d_{B}(b_{0},b_{1}) \bigr]^{\tau _{1}}\bigl[d_{B}(b_{0},Lb_{0}) \bigr]^{ \tau _{2}} \bigl[d_{B}(b_{1},Lb_{1}) \bigr]^{\tau _{3}} \bigr]. \end{aligned}$$

(2.2)

From (2.2), we obtain

$$\begin{aligned} \frac{1}{\sqrt{\Omega }}d_{B}(b_{1},Lb_{1}) \leq & \frac{1}{\sqrt{\Omega }}H_{B}(Lb_{0},Lb_{1}) \\ \leq & \sqrt{\Omega } \bigl[\bigl[d_{B}(b_{0},b_{1}) \bigr]^{\tau _{1}}\bigl[d_{B}(b_{0},Lb_{0}) \bigr]^{ \tau _{2}} \bigl[d_{B}(b_{1},Lb_{1}) \bigr]^{\tau _{3}} \bigr]. \end{aligned}$$

(2.3)

As $\frac{1}{\sqrt{\Omega }}>1$, from Theorem 1.3, there should be $b_{2}\in Lb_{1}$ satisfying

$$ d_{B}(b_{1},b_{2})\leq \frac{1}{\sqrt{\Omega }}d_{B}(b_{1},Lb_{1}). $$

By (2.3) and the above fact, we conclude

$$\begin{aligned} d_{B}(b_{1},b_{2}) \leq \sqrt{\Omega } \bigl[\bigl[d_{B}(b_{0},b_{1}) \bigr]^{\tau _{1}}\bigl[d_{B}(b_{0},b_{1}) \bigr]^{ \tau _{2}} \bigl[d_{B}(b_{1},b_{2}) \bigr]^{\tau _{3}} \bigr]. \end{aligned}$$

(2.4)

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

$$ d_{B}(b_{1},b_{2})\leq (\sqrt{\Omega })^{\frac{1}{1-\tau _{3}}}d_{B}(b_{0},b_{1})< \sqrt{ \Omega }d_{B}(b_{0},b_{1}). $$

Hence, we arrive at

$$\begin{aligned} d_{B}(b_{1},b_{2}) \leq \sqrt{\Omega }d_{B}(b_{0},b_{1}). \end{aligned}$$

(2.6)

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

$$ \min \bigl\{ d_{B}(b_{1},b_{2}),d_{B}(b_{1},Lb_{1}),d_{B}(b_{2},Lb_{2}) \bigr\} >0 $$

then by (2.1) we get

$$\begin{aligned} \frac{1}{\sqrt{\Omega }}d_{B}(b_{2},Lb_{2}) \leq & \frac{1}{\sqrt{\Omega }}H_{B}(Lb_{1},Lb_{2}) \\ =&\frac{1}{\sqrt{\Omega }}\bigl[H_{B}(Lb_{1},Lb_{2}) \bigr]^{\gamma (b_{1},b_{2})} \\ \leq & \sqrt{\Omega } \bigl[\bigl[d_{B}(b_{1},b_{2}) \bigr]^{\tau _{1}}\bigl[d_{B}( b_{1},Lb_{1}) \bigr]^{ \tau _{2}} \bigl[d_{B}(b_{2},Lb_{2}) \bigr]^{\tau _{3}} \bigr]. \end{aligned}$$

(2.7)

As $\frac{1}{\sqrt{\Omega }}>1$, there should be $b_{3}\in Lb_{2}$ satisfying

$$ d_{B}(b_{2},b_{3})\leq \frac{1}{\sqrt{\Omega }}d_{B}(b_{2},Lb_{2}). $$

Thus, by (2.7) and the above inequality, we get

$$\begin{aligned} d_{B}(b_{2},b_{3}) \leq \sqrt{\Omega } \bigl[\bigl[d_{B}(b_{1},b_{2}) \bigr]^{\tau _{1}}\bigl[d_{B}(b_{1},b_{2}) \bigr]^{ \tau _{2}} \bigl[d_{B}(b_{2},b_{3}) \bigr]^{\tau _{3}} \bigr]. \end{aligned}$$

(2.8)

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

$$\begin{aligned} d_{B}(b_{2},b_{3}) \leq \sqrt{\Omega }d_{B}(b_{1},b_{2}). \end{aligned}$$

(2.10)

By (2.10) and (2.6) we obtain

$$\begin{aligned} d_{B}(b_{2},b_{3}) \leq (\sqrt{\Omega })^{2}d_{B}(b_{0},b_{1}). \end{aligned}$$

Induction yields a sequence $\{b_{m}\}$ in B with $b_{m}\in Lb_{m-1}$, $\gamma (b_{m},b_{m+1})=1$ $\forall m \in \mathbb{N}$ and

$$\begin{aligned} d_{B}(b_{m},b_{m+1}) \leq (\sqrt{\Omega })^{m} d_{B}(b_{0},b_{1}) \quad \forall m \in \mathbb{N}. \end{aligned}$$

Also, we get

$$ \min \bigl\{ d_{B}(b_{m},b_{m+1}),d_{B}(b_{m},Lb_{m}),d_{B}(b_{m+1},Lb_{m+1}) \bigr\} >0\quad \forall m\in \mathbb{N}. $$

By the triangle inequality, for $n>m$, we get

$$\begin{aligned} d_{B}(b_{n},b_{m}) \leq & \sum_{j=m}^{n-1} \lambda ^{j} d_{B}(b_{j},b_{j+1}) \\ \leq & \sum_{j=m}^{n-1} \lambda ^{j}(\sqrt{\Omega })^{j} d_{B}(b_{0},b_{1}). \end{aligned}$$

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

$$ \min \bigl\{ d_{B}(b_{m},b_{\ast }),d_{B}(b_{m},Lb_{m}),d_{B}(b_{\ast },Lb_{\ast }) \bigr\} >0\quad \forall m\geq n_{0} $$

for some natural number $n_{0}$. By (2.1) we get

$$\begin{aligned} d_{B}(b_{m+1},Lb_{\ast }) \leq & H_{B}(Lb_{m},Lb_{\ast }) \\ =&\bigl[H_{B}(Lb_{m},Lb_{\ast }) \bigr]^{\gamma (b_{m},b_{\ast })} \\ \leq & \Omega \bigl[\bigl[d_{B}(b_{m},b_{\ast }) \bigr]^{\tau _{1}}\bigl[d_{B}(b_{m},Lb_{m}) \bigr]^{ \tau _{2}} \bigl[d_{B}(b_{\ast },Lb_{\ast }) \bigr]^{\tau _{3}} \bigr] \\ \leq & \Omega \bigl[\bigl[d_{B}(b_{m},b_{\ast }) \bigr]^{\tau _{1}}\bigl[d_{B}(b_{m},b_{m+1}) \bigr]^{ \tau _{2}} \bigl[d_{B}(b_{\ast },Lb_{\ast }) \bigr]^{\tau _{3}} \bigr]\quad \forall m\geq n_{0}. \end{aligned}$$

(2.11)

By the triangle inequality and (2.11), we get

$$\begin{aligned} d_{B}(b_{\ast },Lb_{\ast }) \leq & \lambda \bigl[d_{B}(b_{\ast },b_{m+1})+d_{B}(b_{m+1},Lb_{ \ast }) \bigr] \\ \leq & \lambda d_{B}(b_{\ast },b_{m+1}) \\ &{} + \lambda \Omega \bigl[\bigl[d_{B}(b_{m},b_{\ast }) \bigr]^{\tau _{1}}\bigl[d_{B}(b_{m},b_{m+1}) \bigr]^{\tau _{2}} \bigl[d_{B}(b_{\ast },Lb_{\ast }) \bigr]^{\tau _{3}} \bigr]\quad \forall m\geq n_{0}. \end{aligned}$$

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 $d_{B}(b,b')=|b-b'|^{2}$ ∀b, $b'\in B$. Define $L:B\to CB(B)$ by

$$ L(b)= \textstyle\begin{cases} \{0,1\}, & b\in \{0, 1,2,3, \ldots \}, \\ \{b,-(b-2)^{2}\}, & b\in \{-1,-2,-3, \ldots \}, \end{cases} $$

and $\gamma :B\times B\to \mathbb{R}-\{0\}$ by

$$ \gamma \bigl(b,b'\bigr)= \textstyle\begin{cases} 1, \quad b,b'\in \{0, 1,2,3, \ldots \}, \\ -[ \vert b \vert + \vert b' \vert +8], \quad \text{otherwise}. \end{cases} $$

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 $H_{B}(Lb,Lb')^{\gamma (b,b')}= \frac{1}{[|-(b-2)^{2}+(b'-2)^{2}|^{2}]^{|b|+|b'|+8}}$.

• If $b<0$ and $b'\geq 2$, we obtain $H_{B}(Lb,Lb')^{\gamma (b,b')}= \frac{1}{[|-(b-2)^{2}|^{2}]^{|b|+|b'|+8}}$.

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 b-metric space $(B,d_{B},\lambda )$ and maps $L:B\to CB(B)$, $\gamma :B\times B\to \mathbb{R}-\{0\}$ such that

$$\begin{aligned} \bigl[H_{B}(Lb,Lz)\bigr]^{\gamma (b,z)} \leq & \Omega d_{B}(b,z) \end{aligned}$$

(2.12)

for each $b,z\in B$ with

$$ \min \bigl\{ d_{B}(b,z),d_{B}(b,Lb),d_{B}(z,Lz) \bigr\} >0, $$

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 b-metric space $(B,d_{B},\lambda )$ and a map $L:B\to CB(B)$ satisfying the following inequality:

$$\begin{aligned} H_{B}(Lb,Lz) \leq & \Omega d_{B}(b,z) \end{aligned}$$

(2.13)

for each $b,z\in B$ with

$$ \min \bigl\{ d_{B}(b,z),d_{B}(b,Lb),d_{B}(z,Lz) \bigr\} >0, $$

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 γ-interpolative Reich–Rus–Ćirić-I-contraction can be particularized to Nadler’s type contraction.

The right side of (2.14) is more analogous to interpolative Reich–Rus–Ćirić-contraction.

Definition 2.7

Assume a b-metric 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 γ-interpolative Reich–Rus–Ćirić-I-contraction, if

$$\begin{aligned} \bigl[H_{B}(Lb,Lz)\bigr]^{\gamma (b,z)} \leq & \Omega \bigl[\bigl[d_{B}(b,z)\bigr]^{\tau _{1}} \bigl[d_{B}(b,Lb)\bigr]^{ \tau _{2}} \bigl[d_{B}(z,Lz) \bigr]^{1-\tau _{1}-\tau _{2}} \bigr] \end{aligned}$$

(2.14)

for each $b,z\in B$ with

$$ \min \bigl\{ d_{B}(b,z),d_{B}(b,Lb),d_{B}(z,Lz) \bigr\} >0, $$

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 b-metric space $(B,d_{B},\lambda )$ and reduced γ-interpolative Reich–Rus–Ćirić-I-contraction 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 b-metric space $(B,d_{B},\lambda )$ and maps $L:B\to CB(B)$, $\gamma :B\times B\to \mathbb{R}-\{0\}$ such that

$$\begin{aligned} \bigl[H_{B}(Lb,Lz)\bigr]^{\gamma (b,z)} \leq & \Omega \bigl[ \bigl[d_{B}(b,Lb)\bigr]^{ \tau } \bigl[d_{B}(z,Lz)\bigr]^{1-\tau } \bigr] \end{aligned}$$

(2.15)

for each $b,z\in B$ with

$$ \min \bigl\{ d_{B}(b,z),d_{B}(b,Lb),d_{B}(z,Lz) \bigr\} >0, $$

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–Ćirić-contraction maps.

Definition 2.12

Assume a b-metric 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 γ-interpolative Reich–Rus–Ćirić-II-contraction, if

$$\begin{aligned} \gamma (b,z) H_{B}(Lb,Lz) \leq \Omega \bigl[ \bigl[d_{B}(b,z)\bigr]^{\tau _{1}}\bigl[d_{B}(b,Lb) \bigr]^{ \tau _{2}} \bigl[d_{B}(z,Lz)\bigr]^{\tau _{3}} \bigr] \end{aligned}$$

(2.16)

for each $b,z\in B$ with

$$ \min \bigl\{ d_{B}(b,z),d_{B}(b,Lb),d_{B}(z,Lz) \bigr\} >0, $$

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 b-metric space $(B,d_{B},\lambda )$ and γ-interpolative Reich–Rus–Ćirić-II-contraction 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

$$ \min \bigl\{ d_{B}(b_{0},b_{1}),d_{B}(b_{0},Lb_{0}),d_{B}(b_{1},Lb_{1}) \bigr\} >0; $$

otherwise a fixed point of L occurs in B. Then, by (2.16), we arrive at

$$\begin{aligned} H_{B}(Lb_{0},Lb_{1}) \leq &\gamma (b_{0},b_{1})H_{B}(Lb_{0},Lb_{1}) \\ \leq & \Omega \bigl[\bigl[d_{B}(b_{0},b_{1}) \bigr]^{\tau _{1}}\bigl[d_{B}(b_{0},Lb_{0}) \bigr]^{ \tau _{2}} \bigl[d_{B}(b_{1},Lb_{1}) \bigr]^{\tau _{3}} \bigr]. \end{aligned}$$

(2.17)

From (2.17), we obtain

$$\begin{aligned} \frac{1}{\sqrt{\Omega }}d_{B}(b_{1},Lb_{1}) \leq & \frac{1}{\sqrt{\Omega }}H_{B}(Lb_{0},Lb_{1}) \\ \leq & \sqrt{\Omega } \bigl[\bigl[d_{B}(b_{0},b_{1}) \bigr]^{\tau _{1}}\bigl[d_{B}(b_{0},Lb_{0}) \bigr]^{ \tau _{2}} \bigl[d_{B}(b_{1},Lb_{1}) \bigr]^{\tau _{3}} \bigr]. \end{aligned}$$

(2.18)

As $\frac{1}{\sqrt{\Omega }}>1$, there should be $b_{2}\in Lb_{1}$ satisfying

$$ d_{B}(b_{1},b_{2})\leq \frac{1}{\sqrt{\Omega }}d_{B}(b_{1},Lb_{1}). $$

By (2.18) and the above inequality, we get

$$\begin{aligned} d_{B}(b_{1},b_{2}) \leq \sqrt{\Omega } \bigl[d_{B}(b_{0},b_{1}) \bigr]^{\tau _{1}}\bigl[d_{B}(b_{0},b_{1}) \bigr]^{ \tau _{2}} \bigl[d_{B}(b_{1},b_{2}) \bigr]^{\tau _{3}} ]. \end{aligned}$$

(2.19)

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

$$ d_{B}(b_{1},b_{2})\leq (\sqrt{\Omega })^{\frac{1}{1-\tau _{3}}}d_{B}(b_{0},b_{1}). $$

Hence, we arrive at

$$\begin{aligned} d_{B}(b_{1},b_{2}) \leq \sqrt{\Omega }d_{B}(b_{0},b_{1}). \end{aligned}$$

(2.21)

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 $b_{m}\in Lb_{m-1}$, $\gamma (b_{m},b_{m+1})\geq 1 \forall m \in \mathbb{N}$ and

$$\begin{aligned} d_{B}(b_{m},b_{m+1}) \leq (\sqrt{\Omega })^{m} d_{B}(b_{0},b_{1}) \quad \forall m \in \mathbb{N}. \end{aligned}$$

Also,

$$ \min \bigl\{ d_{B}(b_{m},b_{m+1}),d_{B}(b_{m},Lb_{m}),d_{B}(b_{m+1},Lb_{m+1}) \bigr\} >0\quad \forall m\in \mathbb{N}. $$

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

$$\begin{aligned} d_{B}(b_{m+1},Lb_{\ast }) \leq & H_{B}(Lb_{m},Lb_{\ast }) \\ \leq &\gamma (b_{m},b_{\ast })H_{B}(Lb_{m},Lb_{\ast }) \\ \leq & \Omega \bigl[\bigl[d_{B}(b_{m},b_{\ast }) \bigr]^{\tau _{1}}\bigl[d_{B}(b_{m},Lb_{m}) \bigr]^{ \tau _{2}} \bigl[d_{B}(b_{\ast },Lb_{\ast }) \bigr]^{\tau _{3}} \bigr] \\ \leq & \Omega \bigl[\bigl[d_{B}(b_{m},b_{\ast }) \bigr]^{\tau _{1}}\bigl[d_{B}(b_{m},b_{m+1}) \bigr]^{ \tau _{2}} \bigl[d_{B}(b_{\ast },Lb_{\ast }) \bigr]^{\tau _{3}} \bigr]\quad \forall m\geq n_{0}. \end{aligned}$$

(2.22)

By (2.22) and the triangle inequality, we arrive at

$$\begin{aligned} d_{B}(b_{\ast },Lb_{\ast }) \leq & \lambda \bigl[d_{B}(b_{\ast },b_{m+1})+d_{B}(b_{m+1},Lb_{ \ast }) \bigr] \\ \leq & \lambda d_{B}(b_{\ast },b_{m+1}) \\ &{} + \lambda \Omega \bigl[\bigl[d_{B}(b_{m},b_{\ast }) \bigr]^{\tau _{1}}\bigl[d_{B}(b_{m},b_{m+1}) \bigr]^{\tau _{2}} \bigl[d_{B}(b_{\ast },Lb_{\ast }) \bigr]^{\tau _{3}} \bigr]\quad \forall m\geq n_{0}. \end{aligned}$$

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 γ-interpolative Reich–Rus–Ćirić-II-contraction map and related fixed point result.

Definition 2.14

Assume a b-metric 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 γ-interpolative Reich–Rus–Ćirić-II-contraction, if

$$\begin{aligned} \gamma (b,z) H_{B}(Lb,Lz) \leq \Omega \bigl[ \bigl[d_{B}(b,z)\bigr]^{\tau _{1}}\bigl[d_{B}(b,Lb) \bigr]^{ \tau _{2}} \bigl[d_{B}(z,Lz)\bigr]^{1-\tau _{1}-\tau _{2}} \bigr] \end{aligned}$$

(2.23)

for each $b,z\in B$ with

$$ \min \bigl\{ d_{B}(b,z),d_{B}(b,Lb),d_{B}(z,Lz) \bigr\} >0, $$

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 b-metric space $(B,d_{B},\lambda )$ and reduced γ-interpolative Reich–Rus–Ćirić-II-contraction 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 $d_{B}(b,b')=|b-b'|$ for all $b,b'\in B$. Define $L:B\to CB(B)$ by

$$ L(b)= \textstyle\begin{cases} [0,\frac{b}{8}], \quad b\geq 0, \\ \{0,2b\},\quad b\leq 0, \end{cases} $$

and $\xi :B\times B\to [0,\infty )$ by

$$ \xi \bigl(b,b'\bigr)= \textstyle\begin{cases} 1, \quad b,b'\geq 0, \\ 0,\quad \text{otherwise.} \end{cases} $$

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

$$ \bigl[d_{B}\bigl(b,b'\bigr)\bigr]^{\tau _{1}} \bigl[d_{B}(b,Lb)\bigr]^{\tau _{2}} \bigl[d_{B} \bigl(b',Lb'\bigr)\bigr]^{1- \tau _{1}-\tau _{2}}< 3 \quad \forall \tau _{1},\tau _{2}\in (0,1). $$

Thus, it can be seen that the set-valued versions based on the structure of b-metric 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–Ćirić-type contraction conditions and fixed point results for multivalued maps which fulfil these conditions in a complete b-metric 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–Ćirić-type contraction conditions. A few examples are given to support the findings of this article.