Fixed point theorems of enriched multivalued mappings via sequentially equivalent Hausdor ﬀ metric

: Recently, Abbas et al. [ Enriched multivalued contractions with applications to di ﬀ erential inclusions and dynamic programming , Symmetry 13 (8) (2021), 1350] obtained an interesting generalization of the Nadler ﬁ xed point theorem by introducing the concept of enriched multivalued contraction in the framework of Banach spaces. In this article, we de ﬁ ne a new class of metrics on the family of closed and bounded subsets of a given metric space. Furthermore, ﬁ xed point theorems were established for enriched multi-valued contractions by substituting the Hausdor ﬀ metric with metrics from a speci ﬁ c class that are either metrically or sequentially equivalent to the Hausdor ﬀ metric. Some examples are provided to illustrate the concepts and results presented herein. These results improve, unify, and generalize several comparable results in the literature.


Introduction
Let ( ) X d , be a metric space and → T X X : .An element ∈ x X is called a fixed point of T if it remains invariant under the action of T , i.e., = Tx x.A mapping T on a metric space ( ) X d , is said to be a Banach contraction if there exists ≤ < k 0 1 such that ( ) ( ) ≤ d Tx Ty kd x y , , holds for all ∈ x y X , .The Banach contraction principle [5] states that a Banach contraction mapping defined on a complete metric space has a unique fixed point.Several authors have extended and generalized the Banach contraction principle in different directions.One of the results known as Nadler's fixed point theorem extended the Banach contraction principle from single-valued maps to multivalued maps.
Let ( ) P X ( ( ) CB X ) be the family of all nonempty subsets (the family of all nonempty closed and bounded subsets) of X .Let ( ) → T X P X : .An element ∈ x Tx is called a fixed point of T .The set { ( )} ∈ ∈ x X x T x : of all fixed points of a multivalued T is denoted by , is a metric on ( ) CB X called the Pompeiu-Hausdorff metric (see [10]).
A mapping holds for all ∈ x y X , ,where k is a constant such that ( ) ∈ k 0, 1 .The study of fixed point theorems for multivalued mappings was initiated by Markin [13] and Nadler [14].The following result is due to Nadler [14].
Theorem 1.1.[14] Let ( ) X d , be a complete metric space.Then, a multivalued contraction mapping ( ) → T X CB X : has a fixed point.
One of the interesting generalizations was proved by Abbas et al. [1] by extending the concept of multivalued contraction to enriched multivalued contraction.
Following [1], let ( ∥∥) ⋅ X , be a normed space.A multivalued mapping Note that a ( ) θ 0, -enriched multivalued contraction is a multivalued contraction (1).It was proved that any ( ) b θ , -enriched multivalued contraction defined on a Banach space has at least one fixed point.For more details on the study of enriched contraction mappings, we refer to [3,4,6,8,9] and references therein.
On the other hand, Kirk and Shahzad [12] extended Nadler's fixed point theorem by replacing the Hausdorff metric with a sequentially equivalent metric and obtained fixed points of multivalued contraction mappings.
Before stating the main result of [12], let us recall the following definition.
Note that + H is sequentially equivalent to the Hausdorff metric H . Indeed, + H is a metric on ( ) CB X (see [15]).In addition, for any ( ) Thus, for any sequence { } ( ) Then, T has a fixed point in X.
Recall that two metrics defined on a metric space are called equivalent if they induce the same metric topology.
The aim of this article is to extend the fixed point theorem of enriched multivalued contraction (2) by replacing the Hausdorff metric with metrics on ( ) CB X , which are either metrically or sequentially equivalent to the Pompeiu-Hausdorff metric.

H r class of metrics
For any metric space ( ) X d , , we now introduce a new class of metrics on ( ) CB X , named as H r class of metrics.
Definition 2. For any positive real number r, define Now we prove that for each fixed r, H r is a metric on ( ) CB X .
Proposition 2.1.Let ( ) X d , be any metric space.Then, for any real number > r 0, H r is a metric on ( ) CB X .Moreover, H r is sequentially equivalent to the Hausdorff metric H.
Proof.First, we prove that, for any > r 0, H r satisfies all the properties of a metric.(1) It is obvious to note that, for any ( ) ) where Q ¯and P ¯stand for closure of Q and P, respectively.Then, we have = P Q.On the other hand, for any ( ) . Then, for any ∈ p P, ∈ q Q and ∈ r R, we have As equation ( 7) is true for each ∈ p P, we obtain that and hence, Interchanging the roles of P and R, we obtain Taking sum of equations ( 8) and ( 9) and then multiplying it by r, we obtain Enriched multivalued mappings via sequentially equivalent Hausdorff metric  3 Furthermore, it is easy to check that H r is metrically equivalent to the Hausdorff metric; indeed, for ( ) By (10), we have be a normed space.We denote by Ω 1 the class of metrics U on ( ) CB X that satisfy the following conditions: (1) U is sequentially equivalent to the Hausdorff metric H ; be a normed space.As, for any > r 0, the metric H r on ( ) CB X is sequentially equivalent to the Hausdorff metric H .Moreover, we can obtain that Remark 1.For each positive real number r and for any ( ) In particular, Proof.The proof is straightforward by the definitions of H r and + H . □ We now define another class Ω 2 of metrics U on ( ) CB X that satisfy the following conditions: Clearly, be a normed space.We know that ∈ + H Ω 1 .In addition, from equations ( 10) and (11), it is easy to see that for any ( ) Therefore, We know that ∈ H Ω 1 which implies that .

Fixed point theorem for a new class of enriched multivalued contraction
In this section, we first define a new class of multivalued contractions in normed spaces by using the metrics of class Ω 1 .Afterward, we prove the existence theorem for the fixed point of such multivalued contractions in a Banach space.We need the following remark in order to prove the main results of this article.

Remark 2. [1]
Let M be a convex subset of a normed space X and ( ) → T M CB M : .Then, for any In other words, for each x in M , ( ) is the translation of the set λTx by the vector ( ) − λ x 1 . Clearly, Fix . λ We now introduce the following class of multivalued mappings.
be a normed space and ( ) → T X CB X : . Then, T is said to be a ( 1 and a metric ∈ U Ω 1 such that (1) for all ∈ y z X , , (2) for ∈ y X and ∈ z T y λ , where We start with the following result.
1 , and a metric or equivalently, (2) for ∈ y X and ∈ z T y λ , ( ) where Now we present following result.
and hence, we have Equivalently, it can be written as follows: where and hence, λ Then, for ∈ y X and ∈ z T y, λ we have , .
Similarly, there exists ∈ z Tz λ 3 2 such that Enriched multivalued mappings via sequentially equivalent Hausdorff metric  7 In general, for each ∈ i , there exists Therefore, Hence, { } z n is a Cauchy sequence in X , and there exists . Now by using (22), we have , .
Continuing this way, we obtain , .
and (2) for ∈ y X and ∈ z Ty, where ∈ U Ω 2 .Then, T has a fixed point in X.
If we take = U H in Theorem 3.2, we obtain the following result.Then, T has a fixed point in X.
If we take = U H in the Corollary 3.2.1,we obtain Theorem 3 of [7] in the setting of Banach space with restricted values of constant L.
We know that the space X equipped with the norm where 0 is the zero measurable function on Y .Clearly, we have which gives that Similarly, we have ( ) Now, if we take ∈ * h X .Then, , and for ∈ * w T h , we have Enriched multivalued mappings via sequentially equivalent Hausdorff metric  9 Therefore, In addition, Hence, T satisfies all the conditions of Theorem 3.2.
We now define a new class of single-valued enriched mappings in the following.
We now obtain the following results.
Corollary 3.2.4.Let ( ‖ ‖) ⋅ X , be a Banach space.Suppose that u is any metric on X induced by some other norm on X, which is sequentially equivalent to the metric d induced by the norm ‖ ‖ ⋅ .Combining equations ( 28) and (29), we have Cauchy sequence in X , there exists ∈ * λ is closed, this gives ∈ * * z Tz λ .Hence, ∈ z Tz .□ If we take = U H in Theorem 3.1, we obtain the Corollary 1 of [1] in the setting of Banach spaces.
Definition 5. Let ( ‖ ‖) ⋅ X ,be a normed space.Suppose that u is any metric on X induced by some other norm on X , which is sequentially equivalent to the metric d induced by the norm ‖ ‖