Advances in Mathematical Physics
Volume 2021 (2021), Article ID 5511746, 6 pages
https://doi.org/10.1155/2021/5511746

Research Article

A Tripled Fixed Point Theorem in -Algebra-Valued Metric Spaces and Application in Integral Equations

Vahid Parvaneh,¹ Samira Hadi Bonab,² Hasan Hosseinzadeh,² and Hassen Aydi3,4,5

¹Department of Mathematics, Gilan-E-Gharb Branch, Islamic Azad University, Gilan-E-Gharb, Iran
²Department of Mathematics, Ardabil Branch, Islamic Azad University, Ardabil, Iran
³Institut Supérieur d’Informatique et des Techniques de Communication, Universite de Sousse, H. Sousse 4000, Tunisia
⁴Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa
⁵China Medical University Hospital, China Medical University, Taichung 40402, Taiwan

Correspondence should be addressed to Hasan Hosseinzadeh; hasan_hz2003@yahoo.com and Hassen Aydi; hassen.aydi@isima.rnu.tn

Received 20 February 2021; Revised 27 March 2021; Accepted 31 March 2021; Published 13 April 2021

Academic Editor: Sergey Shmarev

Copyright © 2021 Vahid Parvaneh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Our aim is to establish a tripled fixed and coincidence point result on generalized -algebra-valued metric spaces. We present an example on matrices. At the end, we give an application on integral equations.

1. Introduction

The Banach contraction principle (BCP) was considered by Perov [1] on spaces equipped with vector-valued metrics. The result of Perov has been generalized in [2], and its related fixed point property on generalized metric spaces was investigated.

Let be a unital algebra with the unit and be its zero element. An involution on is a conjugate linear map on so that for all . The pair is named as an -algebra. A Banach -algebra is an -algebra with the complete submultiplicative norm so that for all . A -algebra is a Banach -algebra such that for all . Let be a Hilbert space and be the family of all bounded linear operators on ; then, is a -algebra with the operator norm. Let be the family of all self-adjoint elements in , and define the spectrum of as . An element is positive (denoted by ) if and . Take , then (see [3]). One can define a partial ordering on as . If and , then , and if are invertible, then .

Definition 1 (see [4]). Let be a nonempty set. If the function is so that for all :

(i) and iff (ii)(iii)

then is named as a -algebra-valued metric space.

In this article, denote by the set of all matrices with coefficients in . Note that and .

Let , then is said to be convergent to zero, iff goes to as . See [5–8] for more details.

Denote by the family of all matrices so that . We provide the following examples.

Example 1.

are in . We have .

Example 2.

are in . Clearly,

Example 3. and are in . Then, for , one gets .

Definition 2 (see [9]). An element is named to be a coupled fixed point of if and .

Definition 3 (see [10]; see also [11]). Given and . An element so that and is named as a coupled coincidence point of and . is called a coupled point of coincidence.

Definition 4 (see [12]). An element is named to be a tripled fixed point of if and .

In this manuscript, we investigate a tripled common fixed point result for a sequence of mappings and in the class of complete -algebra-valued metric spaces. An example and an application are presented.

2. Main Results

Our main result is as follows.

Theorem 5. Let be a complete -algebra-valued metric space. Let and be a sequence of mappings from into so that

where , with . If and is complete in , then and have a tripled coincidence point. Further, if and are -compatible, then they have a unique tripled common fixed point in .

Proof. Take , and let

Continuing this technique, we get

By (3), we get

It follows that

Similarly,

Adding (7), (8), and (9), we have

Put . Then, for all ,

Using the triangle inequality, for all ,

We have

Now, taking the limit as , we conclude

This implies that

and , and are Cauchy sequences in , which is complete; there are so that

We have

Taking the limit as in the above relation, we obtain . Similarly, and . Therefore, is a tripled coincidence point of and .
Let and be tripled coincidence points, then

That is,

so

which further induces that

Therefore, , that is, . Similarly, we can prove that . So, . Therefore, and have a unique tripled coincidence point. . Now, set , then . By -compatibility of and ,

Then, is a tripled coincidence point of and . By the uniqueness, we know , which yields that . Hence, is a unique tripled common fixed point of and .

Letting in Theorem 5, we have the following.

Corollary 6. Let be a complete generalized -algebra-valued metric space. Suppose that is a sequence of mappings from into so that

where with , , . Then, has a unique tripled fixed point.

Example 4. Take . Given

Then, is a complete generalized -algebra-valued metric space.
Consider and as

Choose

By induction, (3) holds for all . Set and . Here, for , we have

Also,

So,

Clearly, and are -compatible. Therefore, all conditions in Theorem 5 hold, and is the unique tripled common fixed point of and .

3. Application

Consider the following sequence of the integral equations:

for all , where is a Lebesgue measurable set and .

Denote by the set of essentially bounded measurable functions on . We consider the following assumptions:

(i), , are integrable, and (ii)There is so that for all

for all with

(iii)

Theorem 7. Suppose that assumptions (i)–(iii) hold. Then, (30) has a unique solution in .

Proof. Let and be the set of bounded linear operators on the Hilbert space . We endow with the cone metric defined by , where is the multiplication operator on . It is clear that is a complete -algebra-valued metric space. Define the self-mapping by

for all and .
Now, we have

Using (31), we have

for all .
Therefore, for any , we have

Consequently,

Hence, all hypotheses of Corollary 6 hold. Hence, (30) possesses a unique solution in .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there is no competing interest regarding the publication of this manuscript.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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