Abstract
The purpose of this paper is to introduce the class of multivalued operators by the technique of interpolation of operators. Our results extend and generalize several results from the existing literature. Moreover, we also study the data dependence problem of the fixed point set and Ulam–Hyers stability of the fixed point problem for the operators introduced herein. Moreover, as an application, we obtain a homotopy result.
Introduction and preliminaries
Let X and Y be two normed spaces equipped with norms \(\left\ \cdot \right\ _{X}\) and \(\left\ \cdot \right\ _{Y},\) respectively. Suppose that \(X\subseteq Y.\) If the inclusion map
is continuous, that is, there exists a constant \(c\ge 0\) such that
for every \(x\in X,\) then X is said to be continuously embedded in Y. We use \(\hookrightarrow \) to denote a continuous embedding \(X\hookrightarrow Y\), where \(\left\ \cdot \right\ _{A}\) means that a normed space A is equipped with a norm \(\left\ \cdot \right\ _{A}.\)
A pair \((X_{0},X_{1})\) of normed spaces \(X_{0}\) and \(X_{1}\) is called a compatible couple if there is some Hausdorff topological vector space \( {\mathcal {C}}\) (say) such that each of \(X_{0}\) and \(X_{1}\) are continuously embedded in \({\mathcal {C}}\). Then, we can find the sum \(X_{0}+X_{1}\) and intersection \(X_{0}\cap X_{1}\) as follows:
The sum \(X_{0}+X_{1}\) consists of all \(x\in {\mathcal {C}}\) such that we can write \(x=x_{0}+x_{1}\) for some \(x_{0}\in X_{0}\) and \(x_{1}\in X_{1}\) and the intersection \(X_{0}\cap X_{1}\) consists of all \(x\in {\mathcal {C}}\) such that \( x\in X_{0}\) and \(x\in X_{1}.\)
Suppose \((X_{0},X_{1})\) is a compatible couple. Then, \(X_{0}\cap X_{1}\) is normed space with a norm defined by
Moreover, \(X_{0}+X_{1}\) is also a normed space with norm
Let X and Y be two normed spaces. An operator \(T:X\rightarrow Y\) is called a bounded linear operator if \(T(\alpha x+\beta y) =\alpha T(x)+\beta T(y)\) for all scalars \(\alpha ,\beta \) and \(x\in X,\) \(y\in Y\) and
The space of all bounded linear operators from X to Y is denoted by BL(X, Y) and is a normed space with norm \(\left\ T\right\ _{BL(X,Y)}.\)
If \((X_{0},X_{1})\) is a compatible couple, then a normed space X is said to be an intermediate space between \(X_{0}\) and \(X_{1}\) if
Let \(T:X_{0}+X_{1}\rightarrow Y_{0}+Y_{1}\) be a bounded linear operator. Then, T is said to be an admissible operator with respect to the couples \( (X_{0},X_{1})\) and \((Y_{0},Y_{1})\) if,
are bounded linear operators where for each \(i=0,1,\) \(T_{_{X_{i}}}\) denotes the restriction of T to \(X_{i}.\)
The class of all admissible operators with respect to the couples \( (X_{0},X_{1})\) and \((Y_{0},Y_{1})\) is denoted by \({\mathcal {A}} (X_{0},X_{1};Y_{0},Y_{1}).\) The norm of an admissible operator T is defined by
Let X and Y be intermediate spaces for the compatible couples \( (X_{0},X_{1})\) and \((Y_{0},Y_{1}),\) respectively. The pair (X, Y) is said to have an interpolation property if every \(T\in {\mathcal {A}} (X_{0},X_{1};Y_{0},Y_{1})\) maps X to Y and for \(0\le \alpha \le 1\), following inequality holds:
for some constant c.
For more properties of the interpolation of operator, we refer to [11, 33].
Motivated by the interpolation property, the interpolative Kannan contraction has been described by Karapinar [26] as follows: Given a metric space (X, d), the operator \(T:X\rightarrow X\) is said to be an interpolative Kannan type contraction operator if there exists \(a\in [0,1)\) such that for all \(x,y\in X\setminus \{z\in X:T(z)=z\},\)
where \(\alpha \in (0,1).\)
For more results in this direction, we refer to [3, 8, 9, 19,20,21, 27,28,29,30,31, 38] and references mentioned therein.
On the other hand, Banach [10] contributed a nice result known as the Banach contraction principle and initiated a fixed point theory in the framework of metric spaces. Owing to its applications in the various fields of nonlinear analysis and applied mathematical analysis, this principle has been generalized and extended in different ways.
One of the interesting and famous generalizations was given by Nadler [37] by applying the concept of the Pompeiu–Hausdorff metric (see [13]).
Throughout this paper, the standard notations and terminologies in the nonlinear analysis are used. For the convenience of the reader, we recall some of them.
Let (X, d) be a metric space. Let CB(X) be the class of all nonempty closed and bounded subsets of X and K(X) be the class of all compact subsets of X.
The symmetric functional \(H:CB(X)\times CB(X)\rightarrow [0,\infty )\) defined by
for all \(A,B\in CB(X)\) is a metric called the Pompeiu–Hausdorff metric, where
Let \(T:X\rightarrow CB(X)\) be a multivalued operator. It is said to be a multivalued contraction if for all \(x,y\in X,\) there exists a constant \( k\in (0,1)\) such that the following inequality
holds. The set \(\{x\in X:x\in Tx\}\) of all fixed points of T is denoted by \({\mathrm{Fix}}(T)\).
Nadler proved the following fixed point theorem for multivalued operators.
Theorem 1.1
[37] Let (X, d) be a complete metric space and \( T:X\rightarrow CB(X)\) a multivalued contraction operator. Then, \({\mathrm{Fix}}(T)\ne \emptyset .\)
Later, several interesting fixed point theorems for multivalued operators were obtained (see [4, 16,17,18, 24, 25, 36, 41] and especially the monographs of Rus [42,43,44]).
One of the interesting generalization of Theorem 1.1 in the setting of Banach space was given by Abbas et al. [1] in 2021, by extending the definition of multivalued contraction to the case of enriched multivalued contractions. The enrichment as done as follows:
Let \((X,\left\ \cdot \right\ )\) be a normed space. An operator \( T:X\rightarrow CB(X)\) is called multivalued enriched contraction if there exist constants \(b\in [0,\infty )\) and \(\theta \in [0,b+1)\) such that for all \(x,y\in X,\) the following holds:
It is clear that every multivalued contraction (1.4) is multivalued enriched contraction.
Using multivalued enriched contraction, Abbas et al. [1] proved the following result.
Theorem 1.2
[1] Let \((X,\left\ \cdot \right\ )\) be Banach space and \(T:X\rightarrow CB(X)\) a multivalued enriched contraction. Then, \({\mathrm{Fix}}(T)\ne \emptyset .\)
On the basis of multivalued enriched contraction Hacioğulu and Gürsoy introduced the concept of multivalued enriched Ćirić–Reich–Rus type contraction as follows. An operator \(T:X\rightarrow CB(X)\) is called a multivalued enriched Ćirić–Reich–Rus type contraction operator [22], if there exist constants \(a,b,c\in [0,\infty )\) satisfying \(a+2c<1\) such that for all \(x,y\in X\), we have
It was proved that every multivalued enriched Ćirić–Reich–Rus type contraction operator defined on Banach space has fixed point.
The main idea of Theorem 1.2 is based on the following lemma, which will be useful in this paper.
Lemma 1.3
[1] Let \((X,\left\ \cdot \right\ )\) be a normed space, \(T:X\rightarrow CB(X)\) and \(T_{\lambda }(x)=\{(1\lambda )x+\lambda y:\ y\in Tx\}.\) Then, for any \(\lambda \in [0,1]\)
In this way, the following classes of mappings were introduced and studied: enriched quasi contraction [2], modified Kannan enriched contraction pair [5], enriched cyclic contraction [6],enriched contractions and enriched \(\phi \)contractions [12], enriched Kannan contractions [14], enriched Chatterjea mappings [15] and enriched Ćirić–Reich–Rus contractions [39] etc.
Let \(T_{1},T_{2}:X\rightarrow P(X)\) be two multivalued operators such that \({\mathrm{Fix}}(T_{1})\) and \({\mathrm{Fix}}(T_{2})\) are non empty and there exists \(\delta >0\) such that \(H(T_{1}x,T_{2}x)\le \delta \) for all \(x\in X\) where P(X) is the power set of X. Under these conditions, an estimate of \( H({\mathrm{Fix}}(T_{1}),{\mathrm{Fix}}(T_{2}))\) is found which is the basis of wellknown data dependence problem in metric fixed point theory. Several partial answers to this problem are given in [34, 35, 40].
Similarly, the stability problem is also of great interest in metric fixed point theory.
Let (X, d) be a metric space and \(T:X\rightarrow CB(X)\). A problem of finding the solution of an inclusion \(x\in Tx\) is termed as a fixed point problem for \(T:X\rightarrow CB(X).\) Let \(\epsilon >0.\) An element \(w^{*}\in X\) is called an \(\epsilon \)solution of \(x\in Tx\), if
We now state the notion of Ulam–Hyers stability [23, 45, 46].
The fixed point problem for \(T:X\rightarrow CB(X)\) is called Ulam–Hyers stable if and only if there exists \(c>0\) such that for each \(\epsilon \)solution \(w^{*}\in X\) of the fixed point problem
there exists a solution \(x^{*}\) of \(x\in Tx\) in X such that
where \(\epsilon >0.\)
Motivated by the work of Karapinar [26], we propose a new class of multivalued enriched interpolative Ćirić–Riech–Rus Type contraction operators and prove a fixed point result. Moreover, we study Data Dependence and Ulam–Hyers stability for the operators introduced herein. We obtain a homotopy result as an application of the result presented in this paper.
Main results
The notations \({\mathbb {N}}\) and \({\mathbb {R}}\) will denote the set of all natural numbers and real numbers, respectively.
We first introduce the following concepts.
Definition 2.1
Let \((X,\left\ \cdot \right\ )\) be a normed space. A multivalued operator \(T:X\rightarrow K(X)\) is called:

(1)
enriched interpolative Ćirić–Reich–Rus type contraction operator if there exist constants \(b\in [0,\infty )\), \(a\in [0,b+1)\) and \(\alpha ,\beta \in (0,1)\) such that for all \(x,y\in X\setminus {\mathrm{Fix}}(T)\), we have
$$\begin{aligned} H(bx+Tx,by+Ty)\le a(b+1)^{\alpha }(d(x,y))^{\alpha }(D(x,Tx))^{\beta }(D(y,Ty))^{1\alpha \beta }. \end{aligned}$$(2.1) 
(2)
enriched Gaba interpolative Ćirić–Reich–Rus type contraction operator if there exist constants \(b\in [0,\infty )\), \(a\in [0,b+1)\) and \(\alpha ,\beta ,\gamma \in (0,1)\) such that for all \(x,y\in \setminus {\mathrm{Fix}}(T)\), we have
$$\begin{aligned} H(bx+Tx,by+Ty)\le a(b+1)^{\beta +\gamma 1}(d(x,y))^{\alpha }(D(x,Tx))^{\beta }(D(y,Ty))^{\gamma }. \end{aligned}$$(2.2)
To highlight the involvement of constants in (2.1) and (2.2), we also call T a multivalued \((b,a,\alpha ,\beta )\)enriched interpolative Ćirić–Reich–Rus type contraction and multivalued \( (b,a,\alpha ,\beta ,\gamma )\)enriched Gaba interpolative Ćirić–Reich–Rus type contraction, respectively.
We need the following lemmas in the sequel.
Lemma 2.2
[37] Let F and G be nonempty closed and bounded subsets of a metric space. For any \(f\in F\), \(D(f,G)\le H(F,G).\)
Lemma 2.3
[7] Let (X, d) be a metric space and F a compact subset of X. Then for \(x\in X,\) there exists \(f\in F\) such that \( d(x,f)=D(x,F).\)
Lemma 2.4
[37] Let (X, d) be a metric space, \(A,B \subset X\) and \(q>1.\) Then, for each \(a \in A,\) there exists \(b \in B\) such that
We start with the following result.
Theorem 2.5
Let \((X,\left\ \cdot \right\ )\) be Banach space and \( T:X\rightarrow K(X)\) a multivalued \((b,a,\alpha ,\beta )\)enriched interpolative Ćirić–Reich–Rus type contraction operator. Then \({\mathrm{Fix}}(T)\ne \emptyset .\)
Proof
Take \(\lambda =\frac{1}{b+1}.\) Clearly, \(0<\lambda <1.\) For this value of \( \lambda \), (2.1) becomes,
and hence,
On simplifications, we write (2.5) in an equivalent form as below:
In view of the Krasnoselskij iteration defined in [32] is exactly the Picard’s iteration associated with \(T_{\lambda },\) i.e.,
By taking \(x=x_{n}\) and \(y=x_{n1}\) in (2.6) and using the Lemma 2.2 that for \(x_{n+1}\in T_{\lambda }x_{n}\), we have
From (2.6), we have
As \(T_{\lambda }x_{n}\) and \(T_{\lambda }x_{n1}\) are compact subsets of X, by Lemma 2.3, there exits \(x_{n+1}\in T_{\lambda }x_{n}\) and \( x_{n}\in T_{\lambda }x_{n1}\) such that, \(d(x_{n},x_{n+1})=D(x_{n},T_{ \lambda }x_{n})\) and \(d(x_{n1},x_{n})=D(x_{n1},T_{\lambda }x_{n1})\). So
If \(d(x_{n},x_{n1})\le d(x_{n+1},x_{n})\), then from (2.7), we have
a contradiction to the fact that \(a\in [0,1)\). Hence \(d(x_{n+1},x_{n})\le d(x_{n1},x_{n})\). Using this fact in (2.7), we obtain that
for all \(n\in {\mathbb {N}}.\) Let \(m,n\in {\mathbb {N}}\) with \(n\le m,\) then
which shows that \(\{x_{n}\}\) is Cauchy sequence. Since X is a Banach space, there exists \(p\in X\) such that \(x_{n}\rightarrow p\) as \(n\rightarrow \infty .\) Consider,
On taking limit as \(n\rightarrow \infty ,\) we have, \(D(p,T_{\lambda }p)=0\) and hence
So p is the fixed point of \(T_{\lambda }\) and hence of T. \(\square \)
Theorem 2.6
Let \((X,\left\ \cdot \right\ )\) be Banach space and \( T:X\rightarrow CB(X)\) be a multivalued \((b,a,\alpha ,\beta )\)enriched interpolative Ćirić–Reich–Rus type contraction operator. Then \( {\mathrm{Fix}}(T)\ne \emptyset .\)
Proof
Following arguments given in the proof of Theorem 2.5, we obtained Krasnoselskij iteration, which is exactly the Picard’s iteration associated with \(T_{\lambda },\)
Using Lemma 2.4, there exists \(q>1\) such that for \(x_{n}\in T_{\lambda }x_{n1}\) and \(x_{n+1}\in T_{\lambda }x_{n},\) we have
Consider,
where \(c=qa\), we choose \(q>1\) such that \(c<1.\) The result follows using the similar arguments given in the proof of Theorem 2.5. \(\square \)
For \(b=0\), we get a particular case of Theorem 2.11 of [7].
Corollary 2.7
[7] Let \((X,\left\ \cdot \right\ )\) be Banach space and \(T:X\rightarrow CB(X)\) a multivalued \((0,a,\alpha ,\beta )\)enriched interpolative Ćirić–Reich–Rus type contraction operator. Then T has fixed point.
As a corollary of our result, we can obtain Theorem 6 of [22].
Corollary 2.8
[22] Let \((X,\left\ \cdot \right\ )\) be Banach space and \( T:X\rightarrow CB(X)\) a multivalued enriched Ćirić–Reich–Rus type contraction operator. Then T has fixed point.
Single valued case of the above result yields Theorem 2.3 of [39].
Corollary 2.9
[39] Let \((X,\left\ \cdot \right\ )\) be Banach space and \( T:X\rightarrow X\) a enriched Ćirić–Reich–Rus type contraction operator. Then T has a unique fixed point.
We now prove the fixed point result for the multivalued \((b,a,\alpha ,\beta ,\gamma )\)enriched Gaba interpolative Ćirić–Reich–Rus type contraction operators.
Theorem 2.10
Let \((X,\left\ \cdot \right\ )\) be Banach space and \( T:X\rightarrow CB(X)\) a multivalued \((b,a,\alpha ,\beta ,\gamma )\)enriched Gaba interpolative Ćirić–Reich–Rus type contraction operator with the condition that for \(x\ne y,\) \(d(x,y)\ge 1\). Then \({\mathrm{Fix}}(T)\ne \emptyset .\)
Proof
Following the steps of proof in Theorem 2.5, we get a sequence \( x_{n+1}=T_{\lambda }x_{n},\) for \(n\ge 0.\) By taking \(x=x_{n}\) and \(y=x_{n1} \) in (2.2), we get
If \(d(x_{n},x_{n1})\le d(x_{n+1},x_{n}),\) then from (2.8) we have,
As \(\alpha +\beta +\gamma <1\), so
is a contradiction to the fact that \(a\in [0,1)\). Hence \( d(x_{n+1},x_{n})\le d(x_{n1},x_{n})\). Following arguments similar to those in the proof of Theorem 2.5, we obtain \({\mathrm{Fix}}(T)\ne \emptyset .\) \(\square \)
Data dependence and Ulam–Hyers stability
Data dependence
We present the following data dependence result for multivalued \( (b,a,\alpha ,\beta )\)enriched interpolative Ćirić–Reich–Rus type contraction operators.
Theorem 3.1
Let \((X,\left\ \cdot \right\ )\) be Banach space and \( S,T:X\rightarrow CB(X)\) be multivalued operators. Assume that

(1)
S is a \((b_{1},a_{1},\alpha ,\beta )\)enriched interpolative Ćirić–Reich–Rus type contraction.

(2)
T is a \((b_{2},a_{2},\alpha ,\beta )\)enriched interpolative Ćirić–Reich–Rus type contraction.

(3)
There exists \(\gamma >0\) such that \(H(Tx,Sx)\le \gamma ,\ \ \forall x \in X\).
Then, \({\mathrm{Fix}}(S),{\mathrm{Fix}}(T) \in CB(X).\)
Moreover,
where \(h= aq,\) for \(q>1.\)
Proof
It follows from Lemma 1.3, that \({\mathrm{Fix}}(T)={\mathrm{Fix}}(T_{\lambda })\) and \({\mathrm{Fix}}(S)={\mathrm{Fix}}(S_{\lambda })\). As S and T are \((b_{1},a_{1},\alpha ,\beta )\) and \((b_{2},a_{2},\alpha ,\beta )\)enriched interpolative Ćirić–Reich–Rus type contractions, respectively, by Theorem 2.5, \({\mathrm{Fix}}(T)\ne \emptyset \) and \({\mathrm{Fix}}(S)\ne \emptyset \).
Let \(z_{n}\in {\mathrm{Fix}}(T_{\lambda })\) such that \(z_{n}\rightarrow z\) as \( n\rightarrow \infty \). Using (2.6), we have
On taking limit as \(n\rightarrow \infty \), we obtain \(D(z,T_{\lambda }z)=0\). Hence \({\mathrm{Fix}}(T)\) is closed. Similarly, \({\mathrm{Fix}}(S)\) is also closed. To prove the remaining part of the theorem, let \(q>1.\) Using Lemma 2.4 and condition (3) for an arbitrary \(x_{0}\in {\mathrm{Fix}}(S_{\lambda }),\) one finds \( x_{1}\in T_{\lambda }x_{0}\) such that
As \(x_{1}\in T_{\lambda }x_{0}\), there exists \(x_{2}\in T_{\lambda }x_{1}\) such that
Thus, we have
Choose \(q>1\) such that \(h=aq<1.\) Following the arguments similar to those given in the proof of Theorem 2.5, we get
where \(n,m\in {\mathbb {N}}\). On taking limit as \(n\rightarrow \infty \), we obtain that \(\{x_{n}\}\) is a Cauchy sequence in X. Since X is complete, we have \(x_{n}\rightarrow x,\) for some \(x\in X.\) In addition, \(x\in T_{\lambda }x.\) By (3.2), we have
In particular, we get
which implies that,
For \(q \rightarrow 1\), we have
Similarly, for an arbitrary \(c_{0}\in {\mathrm{Fix}}(T_{\lambda })\), we can find \(c \in {\mathrm{Fix}}(S_{\lambda })\) such that
From (3.3) and (3.4), we obtain
and hence,
\(\square \)
Ulam–Hyers stability
Now we will prove Ulam–Hyers Stability result for multivalued \( (b,a,\alpha ,\beta )\)enriched interpolative Ćirić–Reich–Rus type contraction operator.
Theorem 3.2
Let \((X,\left\ \cdot \right\ )\) be a Banach space. Suppose that T is an operator on X as given in the Theorem 2.5. Then the fixed point problem (1.6) is Ulam–Hyers stable.
Proof
By Lemma 1.3, it follows that the fixed point problem (1.6) is Ulam–Hyers stable if and only if the fixed point problem
is Ulam–Hyers stable. Let \(w^{*}\) be \(\epsilon \)solution of the fixed point equation (3.7), that is,
Consider
This implies that
\(\square \)
Application
In this section, as application of Theorem 2.6, we derive a homotopy result.
Theorem 4.1
Let \((X,\left\ \cdot \right\ )\) be a Banach space and A any open subset of X such that \({\mathcal {A}}\subseteq C\), where C is a closed subset of X. Let \(F:C\times [0,1]\rightarrow CB(X)\) be an operator such that the following conditions are satisfied,

(1)
\(x \notin F(x,t) \) for each \(x \in C\setminus A\) and \(t \in [0,1]\).

(2)
There exist \(b \in [0,\infty ), a \in [0,b+1)\) and \(\alpha \in (0,1)\) such that, for all \(x, y \in C\) and \(t \in [0,1]\), we have
$$\begin{aligned} H\big (bx+F(x,t),by+F(y,t)\big )&\le \frac{a}{(b+1)^{\alpha 1}} [(d(x,y))^{\alpha }(D(x,F(x,t)))^{\beta } \nonumber \\&\quad (D(y,F(y,t)))^{1 \alpha  \beta }]. \end{aligned}$$(4.1) 
(3)
There exist a continuous function \(\eta : [0,1] \rightarrow {\mathbb {R}}\), such that
$$\begin{aligned} H\big (bx+F(x,t),bx+F(x,t_{0})\big )\le a\eta (t) \eta (t_{0}) \end{aligned}$$(4.2)for all \(x\in C\) and \(t,t_{0} \in [0,1]\).

(4)
If \(x \in F(x,t)\), then \(F(x,t) = \{x\}.\)
Then \(F(\cdot ,1)\) has a fixed point in A if \(F(\cdot ,0)\) has a fixed point in A.
Proof
For each fix \(t\in [0,1],\) we denote the operator \(F(\cdot ,t):C\rightarrow CB(X)\) by \(T^{(t)}(\cdot )=F(\cdot ,t)\) for all \(x\in C.\) As argued in the proof of Theorem 2.6, for \(\lambda =\frac{1}{b+1},\) the contractive conditions (4.1) and (4.2) become
respectively, where \(\nu =a\lambda .\) As \(a\in [0,b+1),\) so \(\nu \in [0,1).\) Consider the set
Suppose that \(F(\cdot ,0)=T^{(0)}(\cdot )\) has a fixed point, which implies that \( T_{\lambda }^{(0)}(\cdot )\) also posses fixed point. Hence \(0\in Q\), which means \(Q\ne \emptyset .\) Now, we show that Q is both closed and open in [0, 1], and by the connectedness of [0, 1], the result will follow.
First, we show that Q is open. Let \(t_{0}\in Q\) and \(x_{0}\in A\) with \( x_{0}\in T_{\lambda }^{(t_{0})}x_{0}\). Since A is open in X, there exists \(r>0\) such that \(B(x_{0},r)\subseteq A\). Consider \(\epsilon =(1\nu )r>0. \) Since \(\eta \) is continuous at \(t_{0}\in [0,1]\), there exists a nonnegative real number \(\gamma (\epsilon )\) depending upon \(\epsilon \) such that for \(t\in (t_{0}\gamma (\epsilon ),t_{0}+\gamma (\epsilon ))\), we have
Take \(t\in (t_{0}\gamma (\epsilon ),t_{0}+\gamma (\epsilon ))\) and \(x\in \overline{B(x_{0},r)}\) and consider,
This implies that,
Thus, for each \(t\in (t_{0}\gamma (\epsilon ),t_{0}+\gamma (\epsilon ))\), \( T_{\lambda }^{(t)}(\cdot ):\overline{B(x_{0},r)}\rightarrow CB(X)\) and satisfies all the hypothesis of Theorem 2.6, hence \(T_{\lambda }^{(t)}(\cdot )\) has fixed point in \(\overline{B(x_{0},r)}\subseteq C\). But by condition (1), this fixed point must be in A and hence \((t_{0}\gamma (\epsilon ),t_{0}+\gamma (\epsilon ))\subseteq Q\), which means that Q is open.
To prove that Q is closed; let \(\{t_{n}\}\) be a sequence in Q such that for \(n\rightarrow \infty \), \(t_{n}\rightarrow t\) for some \(t\in [0,1]\). We now show that \(t\in Q\). Since \(t_{n}\in Q\), there exist \(x_{n}\in A\) such that \(x_{n}\in T_{\lambda }^{(t_{n})}x_{n},\) for all \(n\in {\mathbb {N}}\). Thus for \(m,n\in {\mathbb {N}}\), we have
On taking limit as \(m,n\rightarrow \infty \) in the above inequality, we get \( \lim _{m,n\rightarrow }d(x_{m},x_{n})=0\). This proves that \(\{x_{n}\}\) is a Cauchy sequence in X. Since X is Banach space and C is a closed subset of X, there exists \(x\in C\) such that \(\lim _{n\rightarrow \infty }x_{n}=x\) . Consider,
Hence, \(\lim _{n\rightarrow \infty }D(x_{n},T_{\lambda }^{(t)}x)=0\). This implies that \(D(x,T_{\lambda }^{(t)}x)=0\) and \(x\in T_{\lambda }^{(t)}x\). Since (1) holds, thus \(t\in Q\) which means that Q is closed, and we have \(Q=[0,1]\). As a result of this argument, \(T_{\lambda }^{(1)}(\cdot )\) has a fixed point and hence of \(T^{(1)}(\cdot ).\) \(\square \)
Conclusions

(1)
We presented a large class of contractive operators, called multivalued enriched interpolative Ćirić–Reich–Rus type contraction operator, which includes usual multivalued interpolative Ćirić–Reich–Rus type contraction operator and multivalued enriched Ćirić–Reich–Rus type contraction operator.

(2)
We studied the set of fixed point (Theorems 2.5 and 2.6) and constructed an algorithm of Krasnoselskij type in order to approximate fixed point of operators involved therein and proved a convergence theorem.

(3)
We obtained Theorems 3.1 and 3.2 for data dependence and Ulam–Hyers stability problem of the fixed point problem for multivalued enriched interpolative Ćirić–Reich–Rus type contraction operator.

(4)
Moreover, we gave a homotopy result (Theorem 4.1) as an application of our main result.
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Abbas, M., Anjum, R. & Riasat, S. A new type of fixed point theorem via interpolation of operators with application in homotopy theory. Arab. J. Math. (2022). https://doi.org/10.1007/s4006502200402z
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DOI: https://doi.org/10.1007/s4006502200402z
Mathematics Subject Classification
 46B70
 47H10
 47H04
 47H09