On the -Contractive Mappings in -Algebra Valued b-Metric Spaces and Fixed Point Theorems

Abstract

In this paper, for a unital -algebra A, we introduce a version of --contractive mappings in -algebra valued b-metric spaces, and we prove some Banach fixed point theorems and give some examples to illustrate our results.

1. Introduction

Ma et al. [1] introduced the notion of -algebra valued metric spaces, where the set of real number was replaced by the positive cone of a unital -algebra. Later in [2], the class of -algebra valued b-metric spaces is considered. Many results are introduced in this direction (see [3–10]). The notion of --contractive mappings in metric spaces was introduced by Samet et al. [11]. Later in [12], Samet developed the notion of --contractive mappings in b-metric spaces. Several results have been introduced in some related studies of -admissible and -contractive mappings and related fixed point theorems [13–22]. In this present work, we introduced a version of -contractive mapping in a unital -algebra valued b-metric spaces and proved some basic Banach fixed point theorems.

Some nontrivial examples are given to support our results. Suppose that is a unital -algebra with a unit . Set . An element is a positive element, if and is the spectrum of . We define a partial ordering on as if , where means the zero element in , and we let denote the and .

Lemma 1. Suppose that is a unital -algebra with unit . The following holds:(1)If , with , then is invertible and (2)For any and , such that , we have and which are positive element and (3)If , then (4)If and , then (5)Let denote the set and let , if with and is an invertible element, then We refer [23] for more algebra details.

Definition 1. Let be a nonempty set and , , suppose the mapping satisfies the following:(1) for all and .(2) for all .(3) for all , where is zero element in and is the unit element in . Then, is called a -algebra valued b-metric on and is called -algebra valued b-metric space.

Definition 2. Let be a -algebra valued b-metric space, , and be a sequence in , then(i) is convergent to whenever, for every with , there is a natural number such thatfor all . We denote this by or as .(ii) is said to be a Cauchy sequence whenever, for every with , there is a natural number such thatfor all .

Lemma 2. (i) is a convergence sequence in if for any element there is such that for all , .(ii) is a Cauchy sequence in , if for any there is such that , for all . We say that is a complete -algebra valued b-metric space if every Cauchy sequence is convergent with respect to .

Example 1. Let and be the set of all matrices with entries in , and is a -algebra with the matrix norm. Definewhere and are two -matrices, , for all , .
One can define a partial ordering on as following if and only if .
And an element is positive in if and only if for all , we denote the set of all positive element in . Then, is -algebra valued b-metric space.

Definition 4. If is a linear mapping in -algebra, it is said to be positive if . In this case, , and the restriction map: is increasing.

Definition 5. Suppose that and are -algebras. A mapping is said to be -homomorphism if(a) for all and (b)(c)(d) maps the unit in A to the unit in B

Definition 6. Let be the set of positive functions satisfying the following conditions:(a) is continuous and nondecreasing(b) iff (c), for each , where is nth iterate of (d)The series for is increasing and continuous at 0

Corollary 1. Every -homomorphism is contractive and hence bounded.

Lemma 3. Every -homomorphism is positive.

2. Main Results

In [11] Samet et al. and in [12] Samet introduced the concept of --contractive mappings in metric space and --contractive mappings in b-metric space, respectively. Here, we will develop the definitions in case of unital -algebra and study some Banach fixed point theorems.

Definition 7. (see [11]). Let be self map and . Then, is called -admissible if for all with implies .

Definition 8. Let X be a nonempty set and be a function, we say that the self map is -admissible if , , where the unit of .

Definition 9. Let be a -algebra valued b-metric space and is mapping, we say that is an --contractive mapping if there exist two functions and such thatfor all .

Theorem 1 (Banach version fixed point). Let be a complete -algebra valued b-metric space and be an --contractive mapping satisfying the following conditions:(i) is -admissible(ii)There exists such that (iii) is continuousThen, has a fixed point in .

Proof. Let such that and define a sequence in such that for all . If for some , then is a fixed point for .
Suppose that for all , since is -admissible, we getBy induction, we have By inequalities (4) and (6), we getBy induction, we obtainFor and , it follows thatSince , using Definition 6, we obtainThus, is a Cauchy sequence in . Since is complete, there exists such that as , from continuity of if it follows that as . And by uniqueness of the limit, we get , that is, is a fixed point of . To prove the uniqueness of the fixed point, we will consider the following condition. : for all , there exists such that and .

Theorem 2. Adding condition to the hypothesis of Theorem 1, we obtain the uniqueness of the fixed point of .

Proof. Suppose that and are two fixed points of . From , there exists such thatSince is -admissible, we get Using (4) and (12), we obtainThus, . Similarly as . So, the uniqueness of the limit gives . This completes the proof.

Theorem 3 (Kannan version fixed point). .Let be a complete -algebra valued b-metric space and be a mapping satisfying, whereand the following conditions holds:(i) is -admissible(ii)There exists such that (iii) is continuousThen, has a fixed point in .

Proof. Following the proof of Theorem 1, we get for all .
By inequalities (14) and (16), we obtainfrom Lemma 1 and Definition 6, and let .
So, we get .
By induction, we obtain For and , it follows by similar calculation in Theorem 1 thatSince , using Definition 6, we obtainThus, is a Cauchy sequence in . Since is complete, there exists such that as , and from continuity of , it follows that as .
And by uniqueness of the limit, we get ; that is, is a fixed point of .
Now, if is another fixed point of , thenThis implies that . That is, , and this complete the proof.

Theorem 4 (Banach–Kannan version fixed point). Let be a complete -algebra valued b-metric space and be a mapping satisfying, wheresuch that , and the following conditions hold:(i) is -admissible(ii)There exists such that (iii) is continuousThen, has a fixed point in .

Proof. Following the proof of Theorem 1, we getBy using inequalities (22) and (24), we haveSince is additive, we getand putting , we getfor all ; for and and by similar calculation as in the proof of Theorem 1, we getas . This is a fixed point of .
To prove the uniqueness part of the fixed point of , if is another fixed point of , we haveThis is a contraction, so , and this gives . This completes the proof.

Example 2. Let and as given in Example 1, define , by , and and , so ; thus, T is admissible, where is the set of all positive elements in . Define , . This is clear that -contractive mapping and satisfies for all .

3. Applications

In this section, we shall apply Theorem 1 to prove the existence and uniqueness of solution an integral equation in -algebra.

Example 3. Let be a compact Hausdorff space, we denote by the algebra of all complex-valued continuous functions on with pointwise addition and multiplication. The algebra with the involution defined by for each , and with the norm is a commutative -algebra, with unit is the constant function. Let denote the positive Cone of , with partial order relation if and only if . Put as . It is clear that is a complete -algebra valued b-metric space.

Theorem 5 (Application). Consider the integral equationwhere is the compact topological Hausdorff space. Suppose(1).(2)There exists a continuous function and such that for all , .(3), then the integral equation (30) has a unique solution .

Proof. Let and as in the Example3, is a complete -algebra valued b-metric space, and let given by , . defined by . And defined by .
Now,Put since , this gives , and we getfor all .
Thus, T is an -contractive mapping and satisfies Theorem 1. So, T has a unique fixed point, and the integral equation (30) has a unique solution .

4. Conclusions

In this paper, we define a new version of --admissible in the case of self mappings . We prove the principal Banach fixed point theorem, Kannan fixed point theorem, and Banach–Kannan fixed point theorem in the -algebra valued b-metric space, which generalized the given results in [1, 2, 11, 12, 24].

Data Availability

No data were used to support the results.

Conflicts of Interest

The authors of this research declare that they have no conflicts of interest.

Acknowledgments

The first author thanks the South Valley University, Egypt, for partially supporting the study. The second author thanks the Jazan University, Saudi Arabia, for supporting the study.