A Note on the Constructive Scheme for Tetrad Fixed Point Theorem in Hilbert Spaces

Subject Areas : Fixed Point Theory and its Applications

Samuel Aniki ¹

1 - Department of Mathematics, Faculty of Science, Confluence University of Science and Technology, Osara, Kogi State, Nigeria

Received: 2022-08-29 Accepted : 2022-10-04 Published : 2024-06-15

Keywords: Tetrad fixed point, Parallelogram law, mapping, Mann iterative scheme,

Abstract :

The idea of fixed point theory has primarily become a focus of research interest in the area of mathematical analysis, especially for its vast application. Tetrad fixed point is an extension of tripled fixed point theory. This research presents tetrad fixed point iteration for approximating tetrad fixed points in linear spaces within the context of a Hilbert space. Consequently, a tetrad Mann iterative scheme is established and applied to resolve the problem of common tetrad fixed points of certain mappings. Hence, this work is an extension of recent research available in the literature.

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Research Paper

A Note on the Constructive Scheme for Tetrad Fixed Point Theorem in Hilbert Spaces
First Author
aDepartment of …, ….,
Article Info Article history: Received 11 March 2020 Accepted 10 June 2020   Keywords: Mann iterative scheme,  Tetrad fixed point, Mapping, Parallelogram law		Abstract
		The idea of fixed point theory has primarily become a focus of research interest in the area of mathematical analysis, especially for its vast application. Tetrad fixed point is an extension of tripled fixed point theory. This research presents tetrad fixed point iteration for approximating tetrad fixed points in linear spaces within the context of a Hilbert space. Consequently, a tetrad Mann iterative scheme is established and applied to resolve the problem of common tetrad fixed points of certain mappings. Hence, this work is an extension of recent research available in the literature.

 

 

1 Introduction

The notion of fixed point theory has gained a lot of momentum recently in the area of mathematical analysis and its applications. The existence of a fixed point for a contractive-type mapping in partially ordered metric spaces has been recently worked on by several authors [1-4]. After the presentation of the earlier works in this sense, research interest in this subject matter has expanded significantly. To ensure the existence and uniqueness of a solution to periodic boundary value problems, Bhaskar and Lakshmikantham [2] proved the existence and uniqueness of a coupled fixed point in the setting of partially ordered metric spaces. Consequently, so much research has been done on tripled fixed points, for their existence and uniqueness, and also the analysis of fixed point properties via mixed monotone mappings in complete metric spaces [5-9].

The Mann iterative procedure is the earliest known iterative procedure examined in linear spaces except for the most widely used Picard iteration. Some of the most recent references on Mann iteration can be found in [10, 11]. Recently, in the work of Choudhury and Kundu [12], the authors initiated the study of coupled fixed point iteration by introducing a related Mann iterative scheme and applied the same to the context of Hilbert space of approximate coupled fixed points of certain mappings.

Fixed points research has become the focus of interest in recent times, particularly for their potential applications. Very recently, Kim [13] extensively worked on a constructive scheme for common coupled fixed point problems in Hilbert space and recently extended to tripled fixed point by Aniki [14]. Based on this, we aim to generalize this works to tetrad fixed point for Mann pair iterative scheme in the context of Hilbert space.

 

 

2 Preliminaries

In this section, we will consider some useful definitions in the course of demonstrating our findings.

Definition 2.1 [15] The Parallelogram law states that for any vector  and  in a Hilbert space , we have

 

Definition 2.2 [14] The Mann iteration is as follows: Let  be a closed convex subset of a Hilbert space  and  be a self-mapping. Then for ,

 

where  satisfying suitable control conditions.

Definition 2.3 [14] For a non-empty set  and mappings  is a common couple fixed point of  and , if .

Definition 2.4 [14] Let  be a Hilbert space and C a nonempty closed convex subset of . Let  be a mapping. Also, let  and  be sequences in C. Then, the coupled Mann pair iterative scheme is as follows:

,

, ,

where  and .

Definition 2.5 [14] For a nonempty set  and mappings , with  is a common tripled fixed point of ,  and , if .

Definition 2.6 [14] Let H be a Hilbert space and C a nonempty closed convex subset of . Then,  be any mappings which satisfies any of the following contractive inequality conditions

I.        satisfies contractive inequality condition I if ,

,

II.        satisfies contractive inequality condition II if ,

 

where  with .

Definition 2.7 [15] Let  be a Hilbert space and C a nonempty closed convex subset of . then, let  be a mapping. Also, let  and  be sequences in C. Then, the tripled Mann pair iterative scheme is as follows:

 

 

 

Where  and .

3 Main results

In this part of the research, we will firstly define some terms that will be useful in the course of demonstrating our main result. Also, as a way of notational simplification, we will take  and consequently, . Also, .

Definition 3.1 Let  be nonempty and  be mappings, with  being a common tetrad fixed point of the mappings, if  and .

Definition 3.2 Let  be a Hilbert space and  be nonempty closed convex subset of . Then,  be mappings which satisfies any of the following conditions

I.        satisfies condition I if  

 

II.        satisfies condition II if  

 

where  with .

Definition 3.3 Let  be a Hilbert space and  be nonempty closed convex subset of . Then, let  be mappings. Also, let  be sequences in  Then, the tetrad Mann pair iterative scheme is as follows:

 

where

 

and

 

Theorem 3.1 Let  be mappings defined on a closed nonempty convex subset  of a Hilbert space  such that  satisfies conditions I and II. Hence, the tetrad Mann pair iterative scheme which is constructed in (1)-(3), and if  is convergent, it satisfies  and converges to a common tetrad fixed point of .

Proof. Let  as .

1.                      On using the parallelogram law, we have

 

 . Since

 

Similarly,

 

 

 

Using condition I of Definition 3.3 and applying (5), (6), (7) and (8) in (4), we obtain

 

 

 

Since

 

 

 

and

 

From (9) we obtain

 

If  in (14), and going by (3), we have

 

Since  and  we obtain

.

Also, since  we obtain

 

By (15) and (16), we obtain

 

Then,

, ,  and

Therefore,

, , , .

2.                      Using the parallelogram law, we have

 

. Since

 

Similarly,

 

 

 

Therefore, by using condition II of Definition 3.3 and applying (18), (19), (20) and (21) in (17), we obtain

 

 

 

Since

 

 

 

and

 

On using (25), (26), (27) and (28) in (24), we have

 

Taking  in (29), and going by (3), we have

 

Since  and  we obtain

 

From (30) and (31), we have

 

Then,

, ,  and

Therefore,

, , , .

Then, by conditions 1 and 2,  is a common tetrad fixed point of , ,  and . Therefore, this completes the proof.

4 Conclusions

This work has shown that tetrad Mann iterative scheme can be applied to resolve the problem of common tetrad fixed points of certain mappings. Hence, the work can further be extended to fixed point theory via mixed monotone mappings.

Acknowledgements

The author acknowledges with thanks all those whose contributions had enriched this work.

References

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