Completeness Problem via Fixed Point Theory

The purpose of this paper is to present the notion of MR-Kannan type contractions using the generalized averaged operator. Several examples are provided to illustrate the concept presented herein. We provide a characterisation of normed spaces using MR-Kannan type contractions with a fixed point. We investigate the Ulam–Hyers stability and well-posedness result for the mappings presented here.

A metric space Xh is said to be complete if every Cauchy sequence in Xh is convergent in Xh .
The completeness problem is to determine the condition under which the underlying metric space is a complete space.
Let ( Xh , d) be a metric space and T : Xh → Xh .We denote the set { ˚ ∈ Xh : T ˚ = ˚ } of fixed points of T by Fi x(T ).A fixed point problem (FPP) of a mapping T has a solution if the set Fi x(T ) is nonempty.Let ϒ = {T : Xh → Xh : T satisfies a property P}.
We now state the relationship between the fixed point problem and the completeness problem for metric space ( Xh , d).
"( Xh , d) is a complete metric space if and only if every fixed point problem of mapping T ∈ ϒ has a solution." Let ( Xh , d) be metric space.A mapping T : Xh → Xh is called: 1.A Banach contraction ( [4]), if for ˚ , σ ∈ Xh there exists a ∈ 0, 1 such that 2. Kannan contraction ( [12]), if ∃ a ∈ 0, 1 2 such that For more details on Kannan contraction, please refer to [14].Let and Subrahmanyam [18] proved that Xh is a complete space if and only if the FPP for every T ∈ ϒ 1 has a solution.On the other hand, Connell [7] proved that a FPP for every T ∈ ϒ 2 has a solution but ( Xh , d) is not a complete metric space.
Throughout the paper, Xh , • denotes the nomred space over the field R ( the set of all real numbers).
The aim of this paper is to introduce the new class of mappings called MR-Kannan type contraction mappings in the setting of normed space Xh , • .We will also show that if the fixed point problem for every T ∈ ϒ = {T : Xh → Xh : T is a MR-Kannan type contraction mapping } has a solution if and only if then Xh , • is a complete normed space.
In this case, the condition (2.2) reduces to Before stating our main result, we first state the following lemma from [3].

Proof
The proof is obvious and hence omitted.
Let denotes the set of all functions ℵ : Xh → R satisfying the following property: We now present the following definition.
Definition 2.0.2A mapping T : Xh → Xh is said to be an MR-Kannan type contraction if there exist ℵ ∈ and a ∈ [0, 1  2 ) such that holds for all ˚ , σ ∈ Xh .
To highlight the involvement of ℵ and a in (2.6), we call T a (ℵ, a)-MR-Kannan type contraction.
which can be written in an equivalent form as: The contractive condition (2.7) given above is known as (b, a) -enriched Kannan contraction.
We start with the following result.
, for all ˚ ∈ Xh .Obviously, ℘ ∈ and the MR-Kannan-contraction condition (2.6) becomes which can be written in an equivalent form as where, T ℘ is generalized averaged operator defined in (2.2).As a ∈ [0, 1 2 ), by (2.11) T ℘ is a Kannan contraction with contraction constant a.
The generalized Krasnoselskii iteration process { ˚ n } ∞ n=0 , defined by (2.10) is exactly the Picard iteration associated with T ℘ (2.2), that is, (2.12) Take ˚ := ˚ n and σ := ˚ n−1 in (2.11) to get A simple argument shows that { ˚ n } ∞ n=0 is a Cauchy sequence and hence convergent in Xh .Let us denote (2.13) Note that On taking limit as n → ∞ on both sides of the above inequality, we get that ˚ * = T ℘ ˚ * .Next, we prove that ˚ * is the unique fixed point of T ℘ .Assume that ˚ * = σ * is another fixed point of T ℘ .Then, by (2.11) If, T : Xh → Xh is not an MR-Kannan type contraction mapping on the whole space Xh , but an MR-Kannan type contraction condition is satisfied on some neighbourhood of a given point in Xh , then we say that T is locally an MR-Kannan type contraction mapping.
In this case, we have the following result.
Then, T has a unique fixed point in B( υ, r ).
It follows from the inequality (2.14) that Note that the closed ball B( υ, r Indeed, for any ˚ ∈ B( υ, r ), we have and hence T ℘ : B( υ, r ) → B( υ, r ).Since B( υ, r ) is a complete, the result follows from Theorem 2.0.3.

Characterization of Normed Spaces
In this section, we will also show that if the FPP for every T ∈ ϒ = {T : Xh → Xh : T is a MR-Kannan type contraction mapping} has a solution if and only if then Xh , • is a complete normed space.
Theorem 3.0.1 Let ( Xh , • ) be a normed space and T : Xh → Xh has a unique fixed point if the following requirements are met.
, for all ˚ ∈ Xh .Obviously, ℘ ∈ .Then contractive condition (3.1) becomes: where More specifically, Using a similar logic for n ∈ N, there exists Define T : Xh → Xh by It is obvious that T has no fixed point.It still does, however, satisfy the requirements (1) and ( 2) stated in the Theorem's statement.Indeed, for Proof By similar arguments given in the proof of Theorem 2.0.3, (2.6) can be written in an equivalent form as, Since a ∈ [0, 1 2 ), by (3.6) T ℘ is a Kannan contraction with contraction constant a. Hence result follows from Corollary 5.5.6 of [17].

Ulam-Hyers Stability
In fact, the stability problem is a prominent field of study in metric fixed point theory.Let > 0. An element ẘ * ∈ Xh is called an -solution of ˚ = T ˚ , if We now state the notion of Ulam-Hyers stability [11,20]. The Hence the FPP (4.1) is Ulam-Hyers stable for c = 2.

Well-Posedness
Let us start with the following definition.Definition 5.0.1 [15] Let ( Xh , d) be a metric space and T : Xh → Xh .The FPP (4.1) is said to be well-posed if T has unique fixed point ˚ * (say) and for any sequence In this connection, see also pages 157-159 of the book [16].Since Fi x(T ) = Fi x(T ℘ ), we conclude that the FPP of T is well-posed if and only if the FPP of T ℘ is well-posed. (5.1) It follows from (5.1) that lim n→∞ ˚ n = ˚ * provided that lim n→∞ T ℘ ˚ n − ˚ n = 0.This completes the proof.

Conclusion
In this paper, we have introduced the notion of MR-Kannan type contraction using the generalized averaged operator.We have ensured a characterisation of normed spaces using such contractions with a fixed point.Several examples have been furnished to support of the effectiveness and usability of new theory.Also, we have investigated the Ulam-Hyers stability and well-posedness result for the mappings presented here.

. 8 )Example 2 . 2
We now present an example of an MR-Kannan type contraction which is neither an enriched Kannan nor a Kannan contraction.Let Xh = R be endowed with the usual norm.Define the mapping T : Xh → Xh by T ˚ = 5 ˚ .Then T is neither an enriched Kannan nor a Kannan contraction but T is an MR-Kannan contraction.

as a result, ( 1 )Corollary 3 . 0 . 2
is correct.This argument indicates that ( Xh , • ) is a Banach space.Let ( Xh , • ) be a normed space and T : Xh → Xh a (ℵ, a)-MR-Kannan type contraction mapping such that T has a unique fixed point, then ( Xh , • ) is a Banach space.
Now, we examine how well-posed a fixed point problem of maps in Theorem 2.0.3 is.Theorem 5.0.2Let ( Xh , • ) be a Banach space and T a selfmapping on Xh as in the Theorem 2.0.3.Then, fixed point problem (4.1) is well-posed.Proof It follows from Theorem 2.0.3 that ˚ * is the unique fixed point of T .Suppose that lim n→∞ T ℘ ˚ n − ˚ n = 0. Using (2.11), we have FPP for T : Xh → Xh is called Ulam-Hyers stable if and only if ∃ c > 0 such that for each -solution ẘ * ∈ Xh of the FPP Let ( Xh , • ) be a Banach space and T a mapping on Xh as given in the Theorem 2.0.3.Then, the FPP (4.1) is Ulam-Hyers stable.