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.