Quasi Controlled -Metric Spaces over -Algebras with an Application to Stochastic Integral Equations
1 Department of Mathematics, Laboratory of Partial Differential Equations, Algebra and Spectral Geometry, Faculty of Sciences, Ibn Tofail University, Kenitra, BP 133, Morocco
2 Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh, 11586, Saudi Arabia
3 Department of Medical Research, China Medical University, Taichung, 40402, Taiwan
* Corresponding Author: Thabet Abdeljawad. Email:
(This article belongs to this Special Issue: Computational Aspects of Nonlinear Operator and Fixed Point Theory with Applications)
Computer Modeling in Engineering & Sciences 2023, 135(3), 2649-2663. https://doi.org/10.32604/cmes.2023.023496
Received 28 April 2022; Accepted 16 August 2022; Issue published 23 November 2022
AbstractGenerally, the field of fixed point theory has attracted the attention of researchers in different fields of science and engineering due to its use in proving the existence and uniqueness of solutions of real-world dynamic models. C*-algebra is being continually used to explain a physical system in quantum field theory and statistical mechanics and has subsequently become an important area of research. The concept of a C*-algebra-valued metric space was introduced in 2014 to generalize the concept of metric space. In fact, It is a generalization by replacing the set of real numbers with a C*-algebra. After that, this line of research continued, where several fixed point results have been obtained in the framework of C*-algebra valued metric, as well as (more general) C*-algebra-valued b-metric spaces and C*-algebra-valued extended b-metric spaces. Very recently, based on the concept and properties of C*-algebras, we have studied the quasi-case of such spaces to give a more general notion of relaxing the triangular inequality in the asymmetric case. In this paper, we first introduce the concept of C*-algebra-valued quasi-controlled -metric spaces and prove some fixed point theorems that remain valid in this setting. To support our main results, we also furnish some examples which demonstrate the utility of our main result. Finally, as an application, we use our results to prove the existence and uniqueness of the solution to a nonlinear stochastic integral equation.
One of the most relevant theories marking the passage from classical to modern analysis is the fixed point theory which was implemented by Banach . Several mathematicians have created diverse generalizations of Banach fixed point theory. Wilson, on the other hand, introduced the quasi-metric space that is one of the abstractions of the metric spaces . This theory, however, does not include the commutative condition. Numerous mathematicians have adopted this concept to demonstrate some fixed point outcomes, see .
The b-metric spaces concept was first set up by Bakhtin  and Czerwik . Besides, numerous authors obtained a lot of fixed point results. For example, see [6–10]. The extended b-metric spaces idea was elaborated by Kamran et al.  and generalized by Abdeljawad et al.  by imposing the control or the double control of the s-relaxed inequality by one or two functions. Mudasir et al.  stated new results in the context of dislocated b-metric spaces and presented an application related to electrical engineering and extended the notion of Kannan maps in view of the F-contraction in this framework, see .
In [15,16], Ma et al. introduced -algebra valued b-metric spaces by considering metrics that take values in the set of positive elements of a unitary -algebra. Lately, Asim et al.  enlarged this class by defining -algebra-valued extended b-metric spaces. Very recently, Kabbaj et al.  have investigated the quasi case of such a metric and they give a more general notion of relaxing the triangular inequality in the asymmetric case . Recently, for some work on fixed point theory in the mentioned area, we refer to some published work as [20–34].
In this work, we introduce the notion of -algebra-valued quasi controlled -metric spaces. We give basic definitions and then employ them to demonstrate fixed point results in such spaces. Examples are also provided to verify the usefulness of our main results. Finally, as an application, we verify the existence of the solution for a nonlinear stochastic integral equation in this setting.
Throughout this paper, will be a unitary -algebra with and is the spectrum of . We set
Note that is a cone , which induces a partial order on by
To prove our main results, it will be useful to introduce the following lemma.
Lemma 2.1.  Suppose that is a unital -algebra with a unit .
1. if and , then for each ;
2. if , and then ;
3. for all , ;
Definition 2.1.  Let and . A -algebra-valued extended b-metric is a mapping : such that
1. if and only if ;
The triplet is called a -algebra valued extended b-metric space.
In this section, by omitting the symmetry condition, we introduce the notion of -algebra-valued quasi controlled -metric spaces, where is a control function.
Definition 3.1. A -algebra-valued quasi controlled -metric space is the triplet where is a non empty set, is a -control function and is a mapping that
1. if and only if ;
Remark 3.1. In particular, by taking , is a -algebra-valued quasi b-metric space .
Example 3.1. Let and . We know that is a -algebra where partial ordering on is given as
Define a -algebra-valued quasi controlled -metric by:
Given the -control function as
Then, is a -algebra-valued quasi controlled -metric space.
Example 3.2. Let and . Define a mapping as
Let the -control function be defined by
Example 3.3. Consider the space of all continuous functions where is compact. Let the usual unital -algebra with the sup norm and given for each as
Thus, is a -algebra-valued quasi controlled -metric space.
Next, we introduce some topological concepts on -algebra-valued quasi controlled -metric spaces.
Definition 3.2. Let be a -algebra-valued quasi controlled -metric space. The open ball of center and radius is given by
Example 3.4. Let us define a -algebra-valued quasi controlled -metric as
with the -controlled function given by
Then, it is evident that
The open ball B is given by
if , then
if , then
Remark 3.2. We can also define the closed ball by
Definition 3.3. Let be a -algebra-valued quasi controlled -metric space and let be a sequence in .
1. is called left-converges to with respect to , if and only if such that
2. is called right-converges to with respect to , if and only if such that
3. is called converges to with respect to , if and only if
Definition 3.4. Let be a -algebra-valued quasi controlled -metric space. Then
1. is called right-Cauchy with respect to , if for each there exists such that ,
2. is called left-Cauchy with respect to , if for each there exists such that ,
3. is called Cauchy sequence with respect to if and only if ,
4. If every Cauchy sequence in converges to some point in , then, the triplet is said to be a complete -algebra-valued quasi controlled -metric space.
Example 3.5. Take and
Let be the mapping defined by
Then, is a complete -algebra-valued quasi controlled -metric space.
Example 3.6. Let be a compact Hausdorff space and be the set of complex valued continuous functions on . Note that is a unitary commutative -algebra with the usual sup norm such that the involution is defined by . Setting where E is a Lebesgue mensurable set and let us define a -algebra-valued quasi controlled -metric by
Let us define the -control operator by
The condition (i) of Definition 3.1 is clearly satisfied by Now we check the condition . We take as arbitrary. Then
This prove that is a -algebra-valued quasi controlled -metric. Now we want to verify that is a complete -algebra-valued quasi controlled -metric space. Let be a Cauchy sequence in with respect to . Then
We deduce , so is a Cauchy sequence in the space . Since is complete, has a limit that is also in . Hence it follows that
We conclude that the sequence converges to the function in respecting
We will fix the notion of a continuous metric in the context presented in this paper since in the literature during the proof of the results in fixed point certain problems arise due to the possible discontinuity of the b-metric with respect to the topology it generates.
Definition 3.5. Let be a -algebra-valued quasi controlled -metric. is said to be continuous at if the sequence converges to and converges to then
Lemma 3.1. Let be a -algebra-valued quasi controlled -metric space. Such is continuous in each variable. If a sequence has a limit, then this limit is unique.
Proof. Fix By assumption, converges to so there exists such that for all We also assume that converges to so there exists such that for all . Then for all
As was arbitrary, we deduce that which implies .
Our main result runs as follows.
Theorem 3.1. Let be complete -algebra-valued quasi controlled -metric space such that is a continuous and satisfies the following:
where with and such that for an arbitrary . Then has a unique fixed point .
Proof. Let the sequence be defined by . From Eq. (1), we obtain by induction
Now we prove that is a right-Cauchy sequence. For any , we have
Since so that the series converges by ratio test for each . Let
Thus, the above inequality implies
Letting , we conclude that is a right-Cauchy sequence. Similarly, we prove that is a left-Cauchy sequence. The fact that is complete involves such that
Remains to see that is a fixed point of . Indeed for any , we have
Therefore, is a fixed point of . To prove uniqueness, we can assume and such that . Then by employing Eq. (1), we have
Then, we get a contradiction, as a result .
Dynamic programming is a powerful technique for solving some complex problems in computer sciences. We illustrate Theorem 3.2 by studying the existence and uniqueness of the solutions of the functional equation presented in the following example.
Example 3.7. Let X and Y be Banach spaces. is the state space and is the decision space. Let , and . Denote by B(S) the set of all real-valued bounded functions on S. Let the usual unital -algebra with the sup norm and given for each as
is a complete -algebra-valued quasi controlled -metric space. We consider the functional equation
such that and are bounded and
for all in , where and . We define a mapping by where
It is easy to get satisfies with
Therefore, the Eq. (1) possesses unique bounded solution on S.
Example 3.8. Let and . For any , we define its norm as . Define a mapping such that for all and ,
Let the -control function by:
We define a mapping by
It is easy to get
Definition 3.6. Let and for an arbitrary . A function is said to be -orbitally lower semi continuous at with respect to if the sequence is such that with respect to implies
Definition 3.7. Let be a -algebra valued quasi controlled -metric space. is a -left-contractive (respectively -right-contractive mapping) if there exists and an such that
with for every
Theorem 3.2. Let be a complete -algebra valued quasi controlled -metric space such that is continuous. Suppose that is -left-contractive for some , and exists for every such that . Then as . Besides is a fixed point of if and only if is -orbitally l.s.c at .
Proof. Similar to Theorem 3.1, we prove that is a Cauchy sequence. Since is complete then . Assume that is -orbitally l.s.c at , we obtain
We find . It follows that . Conversely, let and a sequence in with . Then
and this completes the proof.
By applying the previous results and involving the -algebra valued quasi controlled -metric space, we prove the existence and uniqueness of a solution of a nonlinear stochastic integral equation given by
1. is the support of a complete probability space;
2. , is the continuous stochastic free where ;
3. is the stochastic kernel where belongs to such that
4. is the unknown continuous real-valued stochastic process such that
Let be the space of all continuous functions from into the space such that , and is continuous from into for every .
We consider . Now, we define the integral operator on by
We now claim is bounded and continuous in mean-square. Indeed
where . This proves , that means is an operator from into .
Assume now the function is a bounded continuous function from into and the function is in the satisfying the condition
where and are constants with and is defined as
Define the operator from into by
Moreover, under the conditions , we get
Hence, so is self mapping on .
We prove the existence of solutions to problem 4 utilising our deduced fixed point theorems. Now, let and . We denote the set of all bounded linear operators on Hilbert space by . Note that is a unitary -algebra. We define a -algebra quasi controlled -metric by:
Similar to the Example 6, one can easily verify the completeness of . Then, we get by using our assumptions
Since , satisfies the inequality (1). Therefore, the integral Eq. (4) has a unique solution by Theorem 3.1.
Example 4.1. Let and . We consider
Note that for all , the function is continuous from into .
Assume that and we take Then, we can check that condition 5 is satisfied with .
We see that , so all the assumptions mentioned in the application section are well insured. Hence, there exists unique solution of the nonlinear integral equation given by
The results obtained are supported by non-trivial examples and complement and extend some of the most recent results from the literature. We have made a contribution by establishing some basic fixed-point problems considering a -algebra valued quasi controlled -metric. We have proved some existence results for maps satisfying a new class of contractive conditions. The fixed point theorems are essential notions in the theory of integral equations. We have proved that the solution of a nonlinear stochastic integral equation of the Hammerstein type of a more general context using a -algebra quasi controlled -metric spaces.
Future study is to investigate the sufficient conditions to guarantee the existence of a unique positive definite solution of the nonlinear matrix equations in the setting of -algebra-valued quasi controlled -metric spaces. The conditions of Theorem 3.1 will be verified numerically by giving various values for the given matrices, and the convergence analysis of nonlinear matrix equations will be shown through graphical representations.
Acknowledgement: The authors Thabet Abdeljawad and Aziz Khan would like to thank Prince Sultan University for the support through the TAS research lab.
Funding Statement: The article is financially supported by Prince Sultan University.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
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