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

$$\begin{aligned} i:X\rightarrow Y\ :\ x\mapsto x \end{aligned}$$

is continuous, that is, there exists a constant \(c\ge 0\) such that

$$\begin{aligned} \left\| x\right\| _{Y}\le c\left\| x\right\| _{X}, \end{aligned}$$

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

$$\begin{aligned} \left\| x\right\| _{X_{0}\cap X_{1}}=\max \{\left\| x\right\| _{X_{0}},\left\| x\right\| _{X_{1}}\}. \end{aligned}$$

Moreover, \(X_{0}+X_{1}\) is also a normed space with norm

$$\begin{aligned} \left\| x\right\| _{X_{0}+X_{1}}=\inf _{x=x_{0}+x_{1}}\{\left\| x_{0}\right\| _{X_{0}}+\left\| x_{1}\right\| _{X_{1}}\}. \end{aligned}$$

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

$$\begin{aligned} \left\| T\right\| _{BL(X,Y)}=\sup _{x\ne 0}\frac{\left\| Tx\right\| _{Y}}{\left\| x\right\| _{X}}<\infty . \end{aligned}$$

The space of all bounded linear operators from X to Y is denoted by BL(XY) 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

$$\begin{aligned} X_{0}\cap X_{1}\hookrightarrow X\hookrightarrow X_{0}+X_{1}. \end{aligned}$$

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,

$$\begin{aligned} T_{|_{X_{0}}}:X_{0}\rightarrow Y_{0},\quad T_{|_{X_{1}}}:X_{1}\rightarrow Y_{1}, \end{aligned}$$

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

$$\begin{aligned} \left\| T\right\| _{{\mathcal {A}}(X_{0},X_{1};Y_{0},Y_{1})}=\max \big \{ \left\| T\right\| _{BL(X_{0},Y_{0})},\left\| T\right\| _{BL(X_{1},Y_{1})}\big \}. \end{aligned}$$

Let X and Y be intermediate spaces for the compatible couples \( (X_{0},X_{1})\) and \((Y_{0},Y_{1}),\) respectively. The pair (XY) 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:

$$\begin{aligned} \left\| T\right\| _{BL(X,Y)}\le c\left\| T\right\| _{BL(X_{0},X_{1})}^{\alpha }\left\| T\right\| _{BL(Y_{0},Y_{1})}^{1-\alpha }, \end{aligned}$$
(1.1)

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 (Xd),  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 \(x,y\in X\setminus \{z\in X:T(z)=z\},\)

$$\begin{aligned} d(Tx,Ty)\le a[d(x,Tx)]^{\alpha }[d(y,Ty)]^{1-\alpha }, \end{aligned}$$
(1.2)

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 (Xd) 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

$$\begin{aligned} H(A,B)=\max \big \{D(A,B),D(B,A)\big \}, \end{aligned}$$
(1.3)

for all \(A,B\in CB(X)\) is a metric called the Pompeiu–Hausdorff metric, where

$$\begin{aligned} D(A,B)=\sup _{x\in A}\inf _{y\in B}d(x,y). \end{aligned}$$

Let \(T:X\rightarrow CB(X)\) be a multi-valued operator. It is said to be a multi-valued contraction if for all \(x,y\in X,\) there exists a constant \( k\in (0,1)\) such that the following inequality

$$\begin{aligned} H(Tx,Ty)\le kd(x,y) \end{aligned}$$
(1.4)

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 multi-valued operators.

Theorem 1.1

[37] Let (Xd) be a complete metric space and \( T:X\rightarrow CB(X)\) a multi-valued contraction operator. Then, \({\mathrm{Fix}}(T)\ne \emptyset .\)

Later, several interesting fixed point theorems for multi-valued 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 multi-valued contraction to the case of enriched multi-valued 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 multi-valued 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:

$$\begin{aligned} H(bx+Tx,by+Ty)\le \theta \left\| x-y\right\| . \end{aligned}$$

It is clear that every multi-valued contraction (1.4) is multi-valued enriched contraction.

Using multi-valued 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 multi-valued enriched contraction. Then, \({\mathrm{Fix}}(T)\ne \emptyset .\)

On the basis of multi-valued enriched contraction Hacioğulu and Gürsoy introduced the concept of multi-valued enriched Ćirić–Reich–Rus type contraction as follows. An operator \(T:X\rightarrow CB(X)\) is called a multi-valued 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

$$\begin{aligned} H(bx+Tx,by+Ty)\le a\left\| x-y\right\| +c[D(x,Tx)+D(y,Ty)]. \end{aligned}$$
(1.5)

It was proved that every multi-valued 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 \(\lambda \in [0,1]\)

$$\begin{aligned} {\mathrm{Fix}}(T)={\mathrm{Fix}}(T_{\lambda }). \end{aligned}$$

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 multi-valued 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 well-known 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 (Xd) 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

$$\begin{aligned} D(w^{*},Tw^{*})\le \epsilon . \end{aligned}$$

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

$$\begin{aligned} x\in Tx, \end{aligned}$$
(1.6)

there exists a solution \(x^{*}\) of \(x\in Tx\) in X such that

$$\begin{aligned} d(x^{*},w^{*})\le c\epsilon . \end{aligned}$$

where \(\epsilon >0.\)

Motivated by the work of Karapinar [26], we propose a new class of multi-valued 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 multi-valued operator \(T:X\rightarrow K(X)\) is called:

  1. (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. (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 multi-valued \((b,a,\alpha ,\beta )\)-enriched interpolative Ćirić–Reich–Rus type contraction and multi-valued \( (b,a,\alpha ,\beta ,\gamma )\)-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 (Xd) 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 (Xd) be a metric space,  \(A,B \subset X\) and \(q>1.\) Then,  for each \(a \in A,\) there exists \(b \in B\) such that

$$\begin{aligned} d(a, b) \le qH(A, B). \end{aligned}$$
(2.3)

We start with the following result.

Theorem 2.5

Let \((X,\left\| \cdot \right\| )\) be Banach space and \( T:X\rightarrow K(X)\) a multi-valued \((b,a,\alpha ,\beta )\)-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,

$$\begin{aligned} H\bigg (\bigg (\frac{1}{\lambda }-1\bigg )x+Tx,\bigg (\frac{1}{\lambda }-1\bigg ) y+Ty\bigg )\le a\lambda ^{-\alpha }(d(x,y))^{\alpha }(D(x,Tx))^{\beta }(D(y,Ty))^{1-\alpha -\beta }, \end{aligned}$$
(2.4)

and hence,

$$\begin{aligned} H((1-\lambda )x+\lambda Tx,(1-\lambda )y+\lambda Ty)\le a\lambda ^{1-\alpha }(d(x,y))^{\alpha }(D(x,Tx))^{\beta }(D(y,Ty))^{\gamma }. \end{aligned}$$
(2.5)

On simplifications, we write (2.5) in an equivalent form as below:

$$\begin{aligned} H(T_{\lambda }x,T_{\lambda }y)\le a(d(x,y))^{\alpha }(D(x,T_{\lambda }x))^{\beta }(D(y,T_{\lambda }y))^{1-\alpha -\beta }. \end{aligned}$$
(2.6)

In view of the Krasnoselskij iteration defined in [32] is exactly the Picard’s iteration associated with \(T_{\lambda },\) i.e.,

$$\begin{aligned} x_{n+1}\in T_{\lambda }x_{n},\quad n\ge 0. \end{aligned}$$

By taking \(x=x_{n}\) and \(y=x_{n-1}\) in (2.6) and using the Lemma 2.2 that for \(x_{n+1}\in T_{\lambda }x_{n}\), we have

$$\begin{aligned} D(x_{n+1},T_{\lambda }x_{n-1})\le H(T_{\lambda }x_{n},T_{\lambda }x_{n-1}). \end{aligned}$$

From (2.6), we have

$$\begin{aligned} D(x_{n+1},T_{\lambda }x_{n-1})&\le H(T_{\lambda }x_{n},T_{\lambda }x_{n-1})\\&\le a(d(x_{n},x_{n-1}))^{\alpha }(D(x_{n},T_{\lambda }x_{n}))^{\beta }(D(x_{n-1},T_{\lambda }x_{n-1}))^{1-\alpha -\beta }. \end{aligned}$$

As \(T_{\lambda }x_{n}\) and \(T_{\lambda }x_{n-1}\) are compact subsets of X,  by Lemma 2.3, there exits \(x_{n+1}\in T_{\lambda }x_{n}\) and \( x_{n}\in T_{\lambda }x_{n-1}\) such that, \(d(x_{n},x_{n+1})=D(x_{n},T_{ \lambda }x_{n})\) and \(d(x_{n-1},x_{n})=D(x_{n-1},T_{\lambda }x_{n-1})\). So

$$\begin{aligned} d(x_{n+1},x_{n})\le H(T_{\lambda }x_{n},T_{\lambda }x_{n-1})\le a(d(x_{n},x_{n-1}))^{\alpha }(d(x_{n},x_{n+1}))^{\beta }(d(x_{n-1},x_{n}))^{1-\alpha -\beta }. \end{aligned}$$
(2.7)

If \(d(x_{n},x_{n-1})\le d(x_{n+1},x_{n})\), then from (2.7), we have

$$\begin{aligned} d(x_{n+1},x_{n})\le ad(x_{n+1},x_{n}) \end{aligned}$$

a contradiction to the fact that \(a\in [0,1)\). Hence \(d(x_{n+1},x_{n})\le d(x_{n-1},x_{n})\). Using this fact in (2.7), we obtain that

$$\begin{aligned} d(x_{n+1},x_{n})\le ad(x_{n},x_{n-1})\le a^{n}d(x_{0},x_{1}) \end{aligned}$$

for all \(n\in {\mathbb {N}}.\) Let \(m,n\in {\mathbb {N}}\) with \(n\le m,\) then

$$\begin{aligned} d(x_{n},x_{m})&\le d(x_{n},x_{n+1})+\cdots +d(x_{m-1},x_{m}) \\&\le a^{n}d(x_{1},x_{o})+\cdots +a^{m-1}d(x_{1},x_{0}) \\&\le \frac{a^{n}}{1-a}d(x_{1},x_{0}) \end{aligned}$$

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,

$$\begin{aligned} D(p,T_{\lambda }p)&\le d(p,x_{n+1})+D(x_{n+1},T_{\lambda }p) \\&\le d(p,x_{n+1})+H(T_{\lambda }x_{n},T_{\lambda }p) \\&\le d(p,x_{n+1})+a(d(x_{n},p))^{\alpha }(D(x_{n},T_{\lambda }x_{n}))^{\beta }(D(p,T_{\lambda }p))^{1-\alpha -\beta }. \end{aligned}$$

On taking limit as \(n\rightarrow \infty ,\) we have, \(D(p,T_{\lambda }p)=0\) and hence

$$\begin{aligned} p\in T_{\lambda }p. \end{aligned}$$

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 multi-valued \((b,a,\alpha ,\beta )\)-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 \(T_{\lambda },\)

$$\begin{aligned} x_{n+1}\in T_{\lambda }x_{n},\quad n\ge 0. \end{aligned}$$

Using Lemma 2.4, there exists \(q>1\) such that for \(x_{n}\in T_{\lambda }x_{n-1}\) and \(x_{n+1}\in T_{\lambda }x_{n},\) we have

$$\begin{aligned} d(x_{n+1},x_{n})\le qH(T_{\lambda }x_{n},T_{\lambda }x_{n-1}). \end{aligned}$$

Consider,

$$\begin{aligned} d(x_{n+1},x_{n})&\le qH(T_{\lambda }x_{n},T_{\lambda }x_{n-1}) \\&\le qa(d(x_{n},x_{n-1}))^{\alpha }(D(x_{n},T_{\lambda }x_{n}))^{\beta }(D(x_{n-1},T_{\lambda }x_{n-1}))^{1-\alpha -\beta } \\&\le qa(d(x_{n},x_{n-1}))^{\alpha }(d(x_{n},x_{n+1}))^{\beta }(d(x_{n-1},x_{n}))^{1-\alpha -\beta } \\&\le qad(x_{n},x_{n-1}) \\&=cd(x_{n},x_{n-1}), \end{aligned}$$

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 multi-valued \((0,a,\alpha ,\beta )\)-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 multi-valued 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 multi-valued \((b,a,\alpha ,\beta ,\gamma )\)-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 multi-valued \((b,a,\alpha ,\beta ,\gamma )\)-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 \(y=x_{n-1} \) in (2.2), we get

$$\begin{aligned} d(x_{n+1},x_{n})\le H(T_{\lambda }x_{n},T_{\lambda }x_{n-1})\le a[d(x_{n},x_{n-1})]^{\alpha }[d(x_{n},x_{n+1})]^{\beta }[d(x_{n-1},x_{n})]^{\gamma }. \end{aligned}$$
(2.8)

If \(d(x_{n},x_{n-1})\le d(x_{n+1},x_{n}),\) then from (2.8) we have,

$$\begin{aligned} d(x_{n+1},x_{n})\le a(d(x_{n+1},x_{n}))^{\alpha +\beta +\gamma }. \end{aligned}$$

As \(\alpha +\beta +\gamma <1\), so

$$\begin{aligned} d(x_{n+1},x_{n})\le a(d(x_{n+1},x_{n}))^{\alpha +\beta +\gamma }\le ad(x_{n+1},x_{n}), \end{aligned}$$

is a contradiction to the fact that \(a\in [0,1)\). Hence \( d(x_{n+1},x_{n})\le d(x_{n-1},x_{n})\). 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 multi-valued \( (b,a,\alpha ,\beta )\)-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 multi-valued operators. Assume that

  1. (1)

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

  2. (2)

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

  3. (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, 

$$\begin{aligned} H({\mathrm{Fix}}(S),{\mathrm{Fix}}(T))\le \frac{1}{1-h}\gamma , \end{aligned}$$
(3.1)

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

$$\begin{aligned} D(z,T_{\lambda }z)&\le d(z,z_{n})+D(z_{n},T_{\lambda }z) \\&\le d(z,z_{n})+H(T_{\lambda }z_{n},T_{\lambda }z) \\&\le d(z,z_{n})+a(d(z_{n},z))^{\alpha }(D(z_{n},T_{\lambda }z_{n}))^{\beta }(D(z,T_{\lambda }z))^{1-\alpha -\beta }. \end{aligned}$$

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

$$\begin{aligned} d(x_{0},x_{1})\le qH(S_{\lambda }x_{0},T_{\lambda }x_{0})\le q\gamma . \end{aligned}$$

As \(x_{1}\in T_{\lambda }x_{0}\), there exists \(x_{2}\in T_{\lambda }x_{1}\) such that

$$\begin{aligned} d(x_{1},x_{2})\le qH(T_{\lambda }x_{0},T_{\lambda }x_{1}). \end{aligned}$$

Thus, we have

$$\begin{aligned} d(x_{1},x_{2})&\le qH(T_{\lambda }x_{0},T_{\lambda }x_{1}) \\&\le qa(d(x_{0},x_{1}))^{\alpha }(D(x_{0},T_{\lambda }x_{0}))^{\beta }(D(x_{1},T_{\lambda }x_{1}))^{1-\alpha -\beta } \\&\le qa(d(x_{0},x_{1}))^{\alpha }(d(x_{0},x_{1}))^{\beta }(d(x_{2},x_{1}))^{1-\alpha -\beta } \\&\le qad(x_{0},x_{1}) \\&=hd(x_{0},x_{1}). \end{aligned}$$

Choose \(q>1\) such that \(h=aq<1.\) Following the arguments similar to those given in the proof of Theorem 2.5, we get

$$\begin{aligned} d(x_{n},x_{n+m})\le \frac{h^{n}}{1-h}d(x_{0},x_{1}), \end{aligned}$$
(3.2)

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

$$\begin{aligned} d(x_{n},x)\le \frac{h^{n}}{1-h}d(x_{0},x_{1}). \end{aligned}$$

In particular, we get

$$\begin{aligned} d(x_{0},x)\le & {} \frac{1}{1-h}d(x_{0},x_{1}), \\ d(x_{0},x)\le & {} \frac{ 1}{1-h}q \gamma , \end{aligned}$$

which implies that,

$$\begin{aligned} \sup _{x_{0} \in {\mathrm{Fix}}(S_{\lambda })} D(x_{0},{\mathrm{Fix}}(T_{\lambda }) \le \frac{ 1}{1-h }q \gamma . \end{aligned}$$

For \(q \rightarrow 1\), we have

$$\begin{aligned} \sup _{x_{0} \in {\mathrm{Fix}}(S_{\lambda })} D(x_{0},{\mathrm{Fix}}(T_{\lambda })) \le \frac{ 1}{ 1-h}\gamma . \end{aligned}$$
(3.3)

Similarly, for an arbitrary \(c_{0}\in {\mathrm{Fix}}(T_{\lambda })\), we can find \(c \in {\mathrm{Fix}}(S_{\lambda })\) such that

$$\begin{aligned} \sup _{c_{0} \in {\mathrm{Fix}}(T_{\lambda })} D({\mathrm{Fix}}(S_{\lambda }),c_{0}) \le \frac{ 1}{ 1-h}\gamma . \end{aligned}$$
(3.4)

From (3.3) and (3.4), we obtain

$$\begin{aligned} H({\mathrm{Fix}}(S_{\lambda }),{\mathrm{Fix}}(T_{\lambda }))\le \frac{1}{1-h}\gamma , \end{aligned}$$
(3.5)

and hence,

$$\begin{aligned} H({\mathrm{Fix}}(S),{\mathrm{Fix}}(T))\le \frac{1}{1-h}\gamma . \end{aligned}$$
(3.6)

\(\square \)

Ulam–Hyers stability

Now we will prove Ulam–Hyers Stability result for multi-valued \( (b,a,\alpha ,\beta )\)-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

$$\begin{aligned} x \in T_{\lambda }x, \end{aligned}$$
(3.7)

is Ulam–Hyers stable. Let \(w^{*}\) be \(\epsilon \)-solution of the fixed point equation (3.7), that is,

$$\begin{aligned} D(w^{*},T_{\lambda }w^{*})\le \epsilon . \end{aligned}$$
(3.8)

Consider

$$\begin{aligned} d(x^{*},w^{*})&\le D(x^{*},T_{\lambda }w^{*})+D(w^{*}, T_{\lambda }w^{*}) \\&\le H(T_{\lambda }x^{*},T_{\lambda }w^{*}) + \epsilon \\&\le a(d(x^{*},w^{*}))^{\alpha }(D(x^{*},T_{\lambda }x^{*}))^{\beta }(D(w^{*},T_{\lambda }w^{*}))^{1-\alpha -\beta } + \epsilon . \end{aligned}$$

This implies that

$$\begin{aligned} d(x^{*},w^{*})\le \epsilon . \end{aligned}$$

\(\square \)

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. (1)

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

  2. (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. (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. (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

$$\begin{aligned} H(T_{\lambda }^{(t)}x,T_{\lambda }^{(t)}y)\le & {} \nu (d(x,y))^{\alpha }(D(x,T_{\lambda }^{(t)}x))^{\beta }(D(y,T_{\lambda }^{(t)}y))^{1-\alpha -\beta }, \end{aligned}$$
(4.3)
$$\begin{aligned} H(T_{\lambda }^{(t)}x,T_{\lambda }^{(t_{0})}x)\le & {} \nu |\eta (t)-\eta (t_{0})|, \end{aligned}$$
(4.4)

respectively, where \(\nu =a\lambda .\) As \(a\in [0,b+1),\) so \(\nu \in [0,1).\) Consider the set

$$\begin{aligned} Q:=\{t\in [0,1]:\ x\in T_{\lambda }^{(t)}x\ \ \text {for some}\ x\in A\}. \end{aligned}$$
(4.5)

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

$$\begin{aligned} |\eta (t)-\eta (t_{0})|\le \epsilon . \end{aligned}$$

Take \(t\in (t_{0}-\gamma (\epsilon ),t_{0}+\gamma (\epsilon ))\) and \(x\in \overline{B(x_{0},r)}\) and consider,

$$\begin{aligned} D(T_{\lambda }^{(t)}x,x_{0})&\le H(T_{\lambda }^{(t)}x,T_{\lambda }^{(t_{0})}x_{0})) \\&\le H(T_{\lambda }^{(t)}x,T_{\lambda }^{(t_{0})}x)+H(T_{\lambda }^{(t_{0})}x,T_{\lambda }^{(t_{0})}x_{0}) \\&\le \nu |\eta (t)-\eta (t_{0})|+\nu (d(x,x_{0}))^{\alpha }(D(x,T_{\lambda }^{(t_{0})}x))^{\beta } \\&\quad (D(x_{0},T_{\lambda }^{(t_{0})}x_{0}))^{1-\alpha -\beta }. \end{aligned}$$

This implies that,

$$\begin{aligned} D(T_{\lambda }^{(t)}x,x_{0})&\le \nu |\eta (t)-\eta (t_{0})| \\&\le \nu \epsilon <\epsilon \\&=r-\nu r\le r. \end{aligned}$$

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

$$\begin{aligned} d(x_{m},x_{n})&\le H(T_{\lambda }^{(t_{m})}x_{m},T_{\lambda }^{(t_{n})}x_{n}) \\&\le H(T_{\lambda }^{(t_{m})}x_{m},T_{\lambda }^{(t_{n})}x_{m})+H(T_{\lambda }^{(t_{n})}x_{m},T_{\lambda }^{(t_{n})}x_{n})\\&\le \nu |\eta (t_{m})-\eta (t_{n})|+\nu [(d(x_{m},x_{n}))^{\alpha }(D(x_{m},T_{\lambda }^{(t_{n})}x_{m}))^{\beta } \\&\quad (D(x_{n},T_{\lambda }^{(t_{n})}x_{n})))^{1-\alpha -\beta }]. \end{aligned}$$

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,

$$\begin{aligned} D(x_{n},T_{\lambda }^{(t)}x)&\le H(T_{\lambda }^{(t_{n})}x_{n},T_{\lambda }^{(t)}x) \\&\le H(T_{\lambda }^{(t_{n})}x_{n},T_{\lambda }^{(t)}x_{n})+H(T_{\lambda }^{(t)}x_{n},T_{\lambda }^{(t)}x) \\&\le \nu |\eta (t_{n})-\eta (t)| \\&\quad +\nu [(d(x_{n},x))^{\alpha }(D(x_{n},T_{\lambda }^{(t)}x_{n}))^{\beta }(D(x,T_{\lambda }^{(t)}x))^{1-\alpha -\beta }]. \end{aligned}$$

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. (1)

    We presented a large class of contractive operators, called multi-valued enriched interpolative Ćirić–Reich–Rus type contraction operator, which includes usual multi-valued interpolative Ćirić–Reich–Rus type contraction operator and multi-valued enriched Ćirić–Reich–Rus type contraction operator.

  2. (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. (3)

    We obtained Theorems 3.1 and 3.2 for data dependence and Ulam–Hyers stability problem of the fixed point problem for multi-valued enriched interpolative Ćirić–Reich–Rus type contraction operator.

  4. (4)

    Moreover, we gave a homotopy result (Theorem 4.1) as an application of our main result.