#### 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.  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.  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, 310].

#### 2. Preliminaries

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

Definition 1. (see ). 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 ). 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 ). 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 ). 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 ). Let be a unital Banach algebra and such that commutes with . Then,

Lemma 4 (see ). 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 ). Let be a unital Banach algebra. Let and . Then, is a -sequence in .

Lemma 6 (see ). 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 ). 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 ). 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 , Theorem 3.1 in , Theorem 2.1 in , Theorem 2.1 in , and Theorem 3.1 in .

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 .
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 .
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 , Theorem 2.3 in , and Theorem 3.3 in .

#### 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, 310].

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.