A new type of fixed point theorem via interpolation of operators with application in homotopy theory

The purpose of this paper is to introduce the class of multi-valued 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.

The sum X 0 + X 1 consists of all x ∈ C such that we can write x = x 0 + x 1 for some x 0 ∈ X 0 and x 1 ∈ X 1 and the intersection X 0 ∩ X 1 consists of all x ∈ C such that x ∈ X 0 and x ∈ X 1 .
Suppose (X 0 , X 1 ) is a compatible couple. Then, X 0 ∩ 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 → Y is called a bounded linear operator if T (αx + βy) = αT (x) + βT (y) for all scalars α, β and x ∈ X, y ∈ Y and The space of all bounded linear operators from X to Y is denoted by B L(X, Y ) and is a normed space with norm T B L(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 → 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 , 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 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 ∈ A(X 0 , X 1 ; Y 0 , Y 1 ) maps X to Y and for 0 ≤ α ≤ 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 → X is said to be an interpolative Kannan type contraction operator if there exists a ∈ [0, 1) such that for all x, y ∈ X \ {z ∈ X : T (z) = z}, where α ∈ (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 C B(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 . Let T : X → C B(X ) be a multi-valued operator. It is said to be a multi-valued contraction if for all x, y ∈ X, there exists a constant k ∈ (0, 1) such that the following inequality holds. The set {x ∈ X : x ∈ T x} of all fixed points of T is denoted by Fix(T ). Nadler proved the following fixed point theorem for multi-valued operators.
Theorem 1.1 [37] Let (X, d) be a complete metric space and T : X → C B(X ) a multi-valued contraction operator. Then, Fix(T ) = ∅.
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, · ) be a normed space. An operator T : X → C B(X ) is called multi-valued enriched contraction if there exist constants b ∈ [0, ∞) and θ ∈ [0, b + 1) such that for all x, y ∈ X, the following holds: 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. On the basis of multi-valued enriched contraction Haciogulu and Gürsoy introduced the concept of multivalued enrichedĆirić-Reich-Rus type contraction as follows. An operator T : X → C B(X ) is called a multi-valued enrichedĆirić-Reich-Rus type contraction operator [22], if there exist constants a, b, c ∈ [0, ∞) satisfying a + 2c < 1 such that for all x, y ∈ X , we have (1.5) It was proved that every multi-valued enrichedĆirić-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.
Let T 1 , T 2 : X → P(X ) be two multi-valued operators such that Fix(T 1 ) and Fix(T 2 ) are non empty and there exists δ > 0 such that H (T 1 x, T 2 x) ≤ δ for all x ∈ X where P(X ) is the power set of X . Under these conditions, an estimate of H (Fix(T 1 ), 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 (X, d) be a metric space and T : X → C B(X ). A problem of finding the solution of an inclusion x ∈ T x is termed as a fixed point problem for T : We now state the notion of Ulam-Hyers stability [23,45].
The fixed point problem for T : X → C B(X ) is called Ulam-Hyers stable if and only if there exists c > 0 such that for each -solution w * ∈ X of the fixed point problem where > 0. Motivated by the work of Karapinar [26], we propose a new class of multi-valued enriched interpola-tiveĆirić-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 N and R will denote the set of all natural numbers and real numbers, respectively.
We first introduce the following concepts.
We need the following lemmas in the sequel.  H(A, B).

Lemma 2.2 [37] Let F and G be nonempty closed and bounded subsets of a metric space. For any f
We start with the following result. Proof Take λ = 1 b+1 . Clearly, 0 < λ < 1. For this value of λ, (2.1) becomes,

4)
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 λ , i.e., x n+1 ∈ T λ x n , n ≥ 0.
By taking x = x n and y = x n−1 in (2.6) and using the Lemma 2.2 that for x n+1 ∈ T λ x n , we have .
As T λ x n and T λ x n−1 are compact subsets of X, by Lemma 2.3, there exits x n+1 ∈ T λ x n and x n ∈ T λ x n−1 such that, d( a contradiction to the fact that a ∈ [0, 1). Hence d(x n+1 , x n ) ≤ d(x n−1 , x n ). Using this fact in (2.7), we obtain that d(x n+1 , x n ) ≤ ad(x n , x n−1 ) ≤ a n d(x 0 , x 1 ) for all n ∈ N. Let m, n ∈ N with n ≤ m, then which shows that {x n } is Cauchy sequence. Since X is a Banach space, there exists p ∈ X such that x n → p as n → ∞. Consider, On taking limit as n → ∞, we have, D( p, T λ p) = 0 and hence p ∈ T λ p.
So p is the fixed point of T λ and hence of T .

Theorem 2.6 Let (X, · ) be Banach space and T : X → C B(X ) be a multi-valued (b, a, α, β)-enriched interpolativeĆirić-Reich-Rus type contraction operator. Then Fix(T ) = ∅.
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 λ , x n+1 ∈ T λ x n , n ≥ 0.
Using Lemma 2.4, there exists q > 1 such that for x n ∈ T λ x n−1 and x n+1 ∈ T λ x n , we have 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.
For b = 0, we get a particular case of Theorem 2.11 of [7].

Corollary 2.8 [22] Let (X, · ) be Banach space and T : X → C B(X ) a multi-valued enrichedĆirić-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, · ) be Banach space and T : X → X a enrichedĆirić-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, α, β, γ )-enriched Gaba interpolativeĆirić-Reich-Rus type contraction operators. Proof Following the steps of proof in Theorem 2.5, we get a sequence x n+1 = T λ x n , for n ≥ 0. By taking x = x n and y = x n−1 in (2.2), we get As α + β + γ < 1, so is a contradiction to the fact that a ∈ [0, 1). Hence d(x n+1 , x n ) ≤ d(x n−1 , x n ). Following arguments similar to those in the proof of Theorem 2.5, we obtain Fix(T ) = ∅.

Data dependence
We present the following data dependence result for multi-valued (b, a, α, β)-enriched interpolativeĆirić-Reich-Rus type contraction operators.
To prove the remaining part of the theorem, let q > 1. Using Lemma 2.4 and condition (3) for an arbitrary x 0 ∈ Fix(S λ ), one finds x 1 ∈ T λ x 0 such that As 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 ∈ N. On taking limit as n → ∞, we obtain that {x n } is a Cauchy sequence in X . Since X is complete, we have x n → x, for some x ∈ X. In addition, x ∈ T λ x. By (3.2), we have In particular, we get which implies that, For q → 1, we have Similarly, for an arbitrary c 0 ∈ Fix(T λ ), we can find c ∈ Fix(S λ ) such that
is Ulam-Hyers stable. Let w * be -solution of the fixed point equation (3.7), that is, This implies that