/ / Article

Research Article | Open Access

Volume 2021 |Article ID 5511746 | https://doi.org/10.1155/2021/5511746

Vahid Parvaneh, Samira Hadi Bonab, Hasan Hosseinzadeh, Hassen Aydi, "A Tripled Fixed Point Theorem in -Algebra-Valued Metric Spaces and Application in Integral Equations", Advances in Mathematical Physics, vol. 2021, Article ID 5511746, 6 pages, 2021. https://doi.org/10.1155/2021/5511746

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

Revised27 Mar 2021
Accepted31 Mar 2021
Published13 Apr 2021

#### 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  on spaces equipped with vector-valued metrics. The result of Perov has been generalized in , 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 ). One can define a partial ordering on as . If and , then , and if are invertible, then .

Definition 1 (see ). 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  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 ). An element is named to be a coupled fixed point of if and .

Definition 3 (see ; see also ). 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 ). 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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