Some Fixed Point Theorems of Contractive Mappings in -Cone Metric Spaces over Banach Algebras

Abstract

In this paper, we introduce the concept of a -cone metric space over Banach algebras and prove some fixed point results under various contractive mappings in such a space. Some examples are given to elucidate the results. Our results extend and generalize many existing results in the literature.

1. Introduction

In 2017, George et al. [1] introduced the concept of a rectangular cone -metric space over Banach algebras as a generalization of metric space and many of its generalizations. They proved some fixed point results in such a space. Very recently, Fernandez et al. [2] introduced -cone metric space over Banach algebras as a generalization of partial metric space and many of its generalizations. Motivated and inspired by these papers [1, 2], we introduce the concept of a -cone metric space over Banach algebras which generalized both rectangular cone -metric space over Banach algebras and -cone metric space over Banach algebras. Furthermore, we prove some fixed point results under various contractive mappings in such a space. Examples are also given to elucidate our results. Our results extend and generalize many results in [1, 3–10].

2. Preliminaries

We start with definitions and some basic facts needed in the sequel.

Definition 1. (see [11]). Let be a real Banach algebra, i.e., is a real Banach space in which an operation of multiplication is defined, for all and , and the following are satisfied:(i)(ii) and (iii)(iv)A Banach algebra is called unital if there exists a unit such that , for any .

Definition 2. (see [5]). A subset of Banach algebra is called a cone if(i) is nonempty, closed, and , where is the zero of (ii) for all nonnegative real numbers (iii)(iv)For a given cone , we define a partial ordering with respect to by if and only if . The notation will stand for , where denotes the interior of . is called a solid cone if .

Definition 3. (see [6]). Let be a solid cone in a Banach algebra . A sequence is said to be a -sequence if, for every , there exists such that for all .

Lemma 1 (see [8]). Let be a solid cone in a Banach algebra :(1)If , , and , then (2)If , where and , then (3)If and , then for any fixed

Lemma 2 (see [11, 12]). Let be a unital Banach algebra and ; then, exists and the spectral radius satisfiesIf , then is invertible in . Moreover,where is a complex constant.

Lemma 3 (see [11]). Let be a unital Banach algebra and such that commutes with . Then,

Lemma 4 (see [13]). Let be a solid cone in a Banach algebra , and be two -sequences in . If are two given vectors, then is also a -sequence in .

Lemma 5 (see [13]). Let be a unital Banach algebra. Let and . Then, is a -sequence in .

Lemma 6 (see [6]). Let be a solid cone in a Banach algebra :(1)If and , then (2)If and for each , then (3) is a -sequence provided that as

Definition 4. (see [2]). Let be a nonempty set and a Banach algebra. Suppose that, for all , a mapping satisfies(1)(2)(3)(4)Then, is called a partial cone -metric space over with coefficient .

Definition 5. (see [1]). Let be a nonempty set and a solid cone in a Banach algebra . Suppose that, for all and all distinct points , a mapping satisfies(1) and .(2).(3)There exists with such thatThen, is called a rectangular cone -metric on , and is called a rectangular cone -metric space over with coefficient .

3. Main Results

In this section, we introduce the concept of a partial rectangular cone -metric space (-cone metric space) over Banach algebras and give some of its topological property. Furthermore, the notions of convergent sequence, -Cauchy sequence, and -completeness in the setting of this new space are defined. Moreover, some fixed point theorems under various contractive mappings are proved in such a space.

Definition 6. Let be a nonempty set and be a solid cone in a unital Banach algebra . Suppose that, for all and all distinct points , a mapping satisfies (P1) . (P2) . (P3) . (P4) There exists with such thatThen, is called a partial rectangular cone -metric on , and is called a partial rectangular cone -metric space over with coefficient (in short PRCbMS-BA).

Remark 1. In any PRCbMS-BA , if for all , then , but the converse may not be true. Also, every rectangular cone -metric space over is a -cone metric space over with zero self distance, but there are -cone metric spaces over which are not a rectangular cone -metric space over .

Example 1. Let with the normDefine multiplication pointwisely on . Then, is a Banach algebra with unit . Let . Then, is a solid cone in . Let and, for all , define a mapping byThen, is a PRCbMS-BA with coefficient which is not a rectangular cone -metric space over because and .

Definition 7. Let be a PRCbMS-BA and be a solid cone in . For each and each , letThen,is a topology on , is a -ball in , is a subbase for the topology on , and is a base generated by the subbase .

Definition 8. Let be a PRCbMS-BA, be a solid cone in , , and be a sequence in . If, for every , there exists such that for all , then is said to be convergent in and converges to . This fact is denoted by as or .

Definition 9. Let be a PCbMS-BA, be a solid cone in , and be a sequence in . Then, is called a -Cauchy sequence if is a -sequence in . That is, if, for every , there exists such that for all .

Definition 10. Let be a PRCbMS-BA, be a solid cone in , , and be a sequence in . Then, is called -complete if every -Cauchy sequence in converges to a point . That is,

Lemma 7. Let be a PRCbMS-BA and be a sequence in . If converges to , then(1) is a -sequence.(2)For any , is a -sequence.

Proof. It follows from Definitions 3, 6, and 8.
Firstly, we present a variant of the Banach contraction principle on -cone metric space over Banach algebra .

Theorem 1. Let be a -complete PRCbMS-BA with such that . Suppose is a function satisfyingwhere such that commutes with and . Then, has a unique fixed point.

Proof. Let be a point in . We define a sequence in byIf for some , then is a fixed point of , and the result is proved. Hence, we assume that . We will show that and . Suppose that ; then, and . Then, (11) implies thatUsing Lemma 1, we obtain that , that is, , which is a contradiction. Therefore, . Hence, from (11) and (12), we have thatSimilarly, for all , we obtain thatObserve that exists because of Lemma 2, and since , there exists such that holds. Since commutes with , by Lemmas 2 and 3, we have thatHence, by condition (P4), for all , we haveThis, using (15) and (16), implies thatUsing Lemmas 4 and 5, we deduce that is a -sequence in . Therefore, is a -Cauchy sequence in . From the hypothesis, is -complete; hence, there exists a point such that converges to . That is,Next, we will show that is the unique fixed point of :By Lemmas 6 and 7, we have as and as . Hence, we deduce that . That is, . So, is a fixed point of . For uniqueness, we let be another fixed point of . Then, it follows from (11) thatBy Lemma 1, we get that , and hence, .
Kindly, observe that Theorem 1 extends and generalizes Theorem 3.5 in [1], Theorem 3.1 in [3], Theorem 2.1 in [4], Theorem 2.1 in [5], and Theorem 3.1 in [6].

Example 2. Let with the normDefine multiplication pointwisely on . Then, is a Banach algebra with unit . Let . Then, is a solid cone in . Let and, for all , define a mapping byThen, is a -complete PRCbMS-BA with coefficient . Define a mapping as follows:Hence, the mapping satisfies all the conditions of Theorem 1 and is the unique fixed point of .
Secondly, we present a variant of the Reich contraction principle on -cone metric space over Banach algebra .

Theorem 2. Let be a -complete PRCbMS-BA with such that . Suppose is a function satisfyingfor all , where commutes, , and . Then, has a unique fixed point.

Proof. Let be a point in . We define a sequence in byFrom (25) and (26), we haveOn the contrary, we haveAdding up (27) and (28), we havewhere . Now, we observe thatThis implies that ; then, by Lemma 2, it follows that is invertible and . From (29), we obtainwhere . Hence,We claim that . Indeed, since commutes with , it follows thatTherefore, commutes with . Then, by Lemmas 2 and 3, we obtainIf for some , then is a fixed point of , and the result is proved. Hence, we assume that . We will show that and . Suppose that ; then, and . Then, (31) implies thatUsing Lemma 1, we obtain that , that is, , which is a contradiction. Therefore, . Next, from (25), (26), and (32), we havewhere such that . Hence, from (36), we also obtainfor all . Observe that exists because of Lemma 2, and since , there exists such that holds. Furthermore, since commutes with , by Lemmas 2 and 3, we have thatTherefore, from (37), we further obtainHence, from (P4), (38), (39), (40) and (41), we haveUsing Lemmas 4 and 5, we deduce that is a -sequence in . Therefore, is a -Cauchy sequence in . From the hypothesis, is -complete; hence, there exists a point such that converges to . That is,Next, we will show that is the unique fixed point of :On the contrary, we haveBy Lemmas 6 and 7, we have as and as . Hence, from (44) and (45), we deduce that and . Since , by Lemma 1, we have so that . That is, is a fixed point of . For uniqueness, we let be another fixed point of . Then, it follows from (25) thatBy Lemmas 1 and 3, we have that , and hence, , i.e., the fixed point of is unique.
Note that Theorem 2 extends and generalizes Theorem 3.1 in [7].
Thirdly, we present a variant of the quasi-contraction principle on -cone metric space over Banach algebra .

Corollary 1. Let be a -complete PRCbMS-BA with such that . Let be a function satisfyingwhere , commutes with , and . Then, has a unique fixed point.

Proof. Since (47) implies (25), the proof follows from Theorem 2.
Note that Corollary 1 generalizes Theorem 3.1 in [8–10].
Finally, we present a variant of the Kannan contraction principle on -cone metric space over Banach algebra .

Corollary 2. Let be a -complete PRCbMS-BA with such that . Suppose is a function satisfyingwhere such that and . Then, has a unique fixed point.

Proof. Put and in Theorem 2, and the result follows.
Note that Corollary 2 generalizes Theorem 2.4 in [4], Theorem 2.3 in [5], and Theorem 3.3 in [6].

4. Conclusion

The concept of a partial rectangular cone -metric space over Banach algebras was introduced, and some fixed point results under various contractive mappings were proved in such a space. Some examples were also given to elucidate the results. Our results extend and generalized many existing results in [1, 3–10].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors have contributed equally and significantly in writing this article.