A Note on the Constructive Scheme for Tetrad Fixed Point Theorem in Hilbert Spaces
Subject Areas : Fixed Point Theory and its Applications
1 - Department of Mathematics, Faculty of Science, Confluence University of Science and Technology, Osara, Kogi State, Nigeria
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.
Research Paper
A Note on the Constructive Scheme for Tetrad Fixed Point Theorem in Hilbert Spaces
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First Author
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aDepartment of …, ….,
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Article Info Article history: Received 11 March 2020 Accepted 10 June 2020
Keywords: Mann iterative scheme, Tetrad fixed point, Mapping, Parallelogram law
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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
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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