Applications to Solving Variational Inequality Problems via MR-Kannan Type Interpolative Contractions

: The aim of this paper is manifold. We ﬁrst deﬁne the new class of operators called MR-Kannan interpolative type contractions, which includes the Kannan, enriched Kannan, interpolative Kannan type, and enriched interpolative Kannan type operators. Secondly, we prove the existence of a unique ﬁxed point for this class of operators. Thirdly, we study Ulam-Hyers stability, well-posedness, and periodic point properties. Finally, an application of the main results to the variational inequality problem is given


Introduction
The Banach contraction principle, detailed in [1], stands as a landmark result with a profound impact on the development of metric fixed point theory.Banach's work is highly regarded, representing an adaptable and foundational contribution to fixed point theory.This principle not only initiated significant research in the field but also spurred exploration by numerous scholars from 1922 to the present.A notable extension of the BCP was presented by Kannan [2].We state the Kannan fixed point theorem in the context of Banach spaces.Theorem 1 ([2]).Let (Θ, • ) be a Banach space, and P : Θ → Θ be a Kannan contraction.This means that P satisfies the following condition: P g − P s ≤ a{ g − P g + s − P s }, ∀ g, s ∈ Θ, with 0 ≤ a < 1 2 .Then, P has a unique fixed point.
In the framework of Banach spaces, the primary finding of [3] can be summarized as follows: Theorem 2 ([3]).Let (Θ, • ) be a Banach space, and P : Θ → Θ be an interpolative Kannan type contraction.This means that P satisfies the following condition: P g − P s ≤ a( g − P g ) ξ ( s − P s ) 1−ξ , ∀ g, s ∈ Θ, when P g = g, where a ∈ [0, 1) and 0 < ξ < 1.As a consequence, it can be concluded that the operator P possesses a unique fixed point.
In 2020, Berinde and Pȃcurar [17] improved Theorem 1 by introducing the concept of enriched Kannan contraction.

Remark 1.
By substituting b = 0 into Theorem 3, we can derive Theorem 1.Therefore, Theorem 3 is a generalization of Theorem 1.

Remark 2.
By substituting b = 0 into Theorem 4, we can derive Theorem 2.Moreover, it follows from Corollary 2.8 of [4] that Theorem 4 is a generalization of Theorem 3.
On the other hand, in 2023, Anjum et al. [22] generalized Theorem 4 by introducing the concept of ( , a)-MR-Kannan type contraction.The principal outcome highlighted in [22] is presented as follows: Theorem 5 ([22]).Let (Θ, • ) be a Banach space and P : Θ → Θ be an ( , a)-MR-Kannan type contraction, that is an operator satisfying for all g, s ∈ Θ, where 0 ≤ a < 1 2 and Then, P has a unique fixed point.
Utilizing the ideas from Theorem 2 and Theorem 4, we now present the following.

Question
Under which condition can we attain an equivalent conclusion as stated in Theorem 5 by substituting the multiplication between the terms 1 1+ ( g) g − P g and 1 1+ (s) s − P s on the right-hand side of (5)?
This paper has multiple objectives.We first define the new class of operator called MR-Kannan interpolative type contraction, which includes the contractive conditions (1)- (5).Additionally, the existence of a unique fixed point for this class of operators is proven.
Furthermore, the study encompasses Ulam-Hyers stability, well-posedness, and periodic point properties.Finally, the main results are applied to a variational inequality problem.

Approximating Fixed Points of MR-Kannan Type Interpolative Contractions
We introduce the following definition.Definition 1.Let (Θ, • ) be a normed space.A operator P : Θ → Θ is said to be MRKI type contraction, if there exist ∈ η, 0 ≤ a < 1 and 0 < ξ < 1, such that for all g, s ∈ Θ with P g = g, g ( g) To emphasize the role of a, and ξ in ( 28), we shall also call P a (a, , ξ)-MRKI type contraction.
Before proceeding with the proof of the main theorem of our paper, the following findings are required from [22].
Recall that we denote the set { g ∈ Θ : P g = g} of fixed points of P by F(P).
Let P : Θ → Θ be an operator defined as where ∈ β is called a generalized averaged operator ( [22,23]).We would like to direct the reader's attention to the fact that the term generalized averaged operator refers to a specific type of admissible perturbations [23,24].It is worth noting that the class of generalized averaged operators includes the class of averaged operators (a term coined in [25]) .This is demonstrated by considering λ ∈ (0, 1) and defining ( g) = λ for all g ∈ Θ.
Then, P has a unique fixed point.
Let s * be another fixed point of P. Next, as shown by (11), we possess which, gives g * = s * .
We obtain Theorem 5 as a corollary of our main result.
We obtain Theorem 4 as a consequence of our main result.

Well-Posedness, Perodic Point Property and Ulam-Hyers Stability
We start this section with the following definition:

Well-Posedness
Recall that the goal of solving the fixed point problem of the operator P, represented by FPP(P), is to demonstrate the nonemptiness of F(P).
Theorem 7. Let P be an operator as defined in Theorem 6.Then, FPP(P) is well-posed.
Proof.Because F(P) = F(P ), we may derive that operator P is well-posed if and only if operator P is well-posed.

Perodic Point Result
Obviously, a fixed point g * of the operator P satisfies F(P ς ) = { g * } for all ς ∈ N; however, the reverse assertion does not hold.An operator P possesses a periodic point property ( [28]) if it satisfies the condition F(P) = F(P ς ) for every ς ∈ N.
Theorem 8. Let P be an operator defined in Theorem 6.Then, P possesses a preodic point property.
Proof.Because F(P) = F(P ), we may derive that P has preodic point property ⇔ P has preodic point property.

Ulam-Hyers Stability
Before presenting the definition, let's establish the following concept from [29].
Let (Θ, • ) be a normed space and P : Θ → Θ be an operator such that a point s * ∈ Θ as an ς-solution to the FPP(P), if it satisfies the inequality ), χ(0) = 0, χ is an increasing and continuous function}.
Let us begin with definition.
Theorem 9. Let P be an operator as in the Theorem 6.Then, FPP(P) possesses a Ulam-Hyers stability.
Proof.Because F(P) = F(P ), we may derive that P has Ulam-Hyers stability ⇔ the operator F(P ) has Ulam-Hyers stability.
Taking s * as an ς-solution to the Equation (3.3.2),we can infer the following: Utilizing ( 11) and ( 22), we obtain:
Let H be a Hilbert space with the inner product denoted by •, • , and consider a nonempty, closed, and convex subset C of H.This article is dedicated to exploring the classical variational inequality, seeking the presence of a point g * within C that satisfies where S : H → H represents an operator.We denote V IP(S, C) as the variational inequality problem associated with S and C. According to [33], it is well known that when Υ is a positive number, then g * ∈ C is a solution to V IP(S, C) if and only if g * satisfies the fixed-point problem: g = Pc(I − ΥG) g, (24) Here, the closest point projection onto C is indicated by Pc.
We choose an alternative approach by investigating V IP(S, C) with (a, , ξ)-MRKI contraction operators, which can exhibit discontinuity, unlike nonexpansive operators, that are inherently continuous.According to the next theorem, we expect that V IP(S, C) will have a unique solution in this situation.In addition, we anticipate substantial convergence of the algorithm outlined in (25) towards the V IP(S, C) solution.
Theorem 10.Let Υ be a positive value and P : C → C represent a (a, , ξ)-MRKI type operator satisfying Then, the iterative sequence { gς } ∞ ς=0 is given by where ∈ β, exhibits strong convergence towards the unique solution g * of the V IP(S, C), for any g0 ∈ C.
Proof.As C is a closed set, let Θ = C and employ the definition of P as given in (24).Subsequently, we apply Theorem 6.Consequently, there exists an element g * ∈ C such that Example 1.Let Θ = R 2 and the inner product for any g = ( g1 , g2 ) and s = (s 1 , s2 ) in Θ, is defined as follows: g, s = g1 s1 + g2 s2 .
With this definition, Θ becomes a Hilbert space.The associated norm is given by: Let's define the operator S : Θ → Θ as follows: where Υ is a fixed positive real number.Next, consider the operator P C : Θ → C defined by where C = { g ∈ Θ : g ≤ 1}, The operator P defined by ( 25) is (a, , ξ)-MRKI type operator.Certainly, when ( g) = 1 + g , ∀ g ∈ Θ, the left-hand side of (28) transforms to, Therefore, we obtain that It follows from (27) the condition in (28) satisfy for ( g) = 1 + g , ∀ g ∈ Θ.Hence, F(P) is a singleton set, which becomes a solution for VIP(S, C).

Conclusions
We provide a broad class of contractive operators called contractions of the MR-Kannan interpolative kind.Interpolative Kannan type, enriched interpolative Kannan type, Kannan, and enhanced Kannan are among the operators included in this class.A Krasnoselskii-type technique has been developed by us to estimate fixed points of MR-Kannan interpolative type operators.Our exploration involves the analysis of the set of fixed points (see Theorem 6).Furthermore, we have derived Theorems 7-9, which address well-posedness, periodic points, and Ulam-Hyers stability for the fixed-point problem of MR-Kannan interpolative type operators, respectively.Moreover, leveraging our primary findings (see Corollary 10), we have introduced Krasnoselskii projection-type algorithms to solve variational inequality problems within the class of MR-Kannan interpolative type operators.
Here, we now present an open problem.Open Problem: Following the approach proposed in [9] for the interpolation technique, we present a new problem.Suppose we have positive numbers a and b, where a + b < 1, and consider the following condition instead of (28): The open problem is whether the conclusion of Theorem 6 still holds under this new condition.