Common Fixed Points of Two Mappings regarding a Generalized -Distance over a Banach Algebra

Abstract

In this article, applying the concept of a generalized -distance in cone -metric spaces over Banach algebra with a nonnormal solid cone therein, we establish several common fixed point theorems for two noncontinuous mappings satisfying the Han-Xu-type contraction. Our results are interesting, since they are not equivalent to former well-known results regarding a -distance in -metric spaces while they contain recent results corresponding to a generalized -distance in cone -metric spaces.

1. Introduction and Preliminaries

In 2015, Bao et al. [1] suggested a generalized -distance, which extended many former definitions in [2–10]) and references therein. Moreover, with regard to a survey on fixed point theory corresponding to this distance, see [11–13]).

In 2013, Liu and Xu [14] offered a cone metric space over Banach algebra by replacing a Banach space with a Banach algebra . After this definition, some other researchers suggested new several various spaces over a Banach algebra and extended available results in [15–17] and their references. In 2015, Huang et al. [18] proposed a -distance in cone metric spaces over a Banach algebra . In 2018, Han and Xu [19] proved some common fixed point results by removing the assumption of continuity of the mappings and by deleting the hypothesis of the normality of the cone. Recently, Arabnia et al. [20] suggested a generalized -distance in cone -metric spaces over a Banach algebra.

Here, we review some basic definitions and preliminary lemmas which are needed to continue.

Let be a Banach algebra with a unit element , a zero element , a norm , and a cone therein. Define a partial order with respect to by iff . Also, if and , and iff ( is the same as the interior of ). If , then the cone is named solid. Further, if there is so that deduced that for every , then the cone is named a normal cone.

Definition 1 (see [17]). Let be a nonempty set, be a constant, and be a Banach algebra. For all , assume that satisfies the following items: 2
(d₁) and iff
(d₂)
(d₃)

Then, is named a cone -metric on , and is named a cone -metric space over the Banach algebra .

For definitions such as convergent and Cauchy sequences, -sequence, completeness, continuity, and examples, see [16, 17]. In the sequel, assume that is a cone -metric space with the coefficient over a Banach algebra and is a solid cone therein.

Lemma 2 (see [17, 21]). Let be a Banach algebra and . Then, the following items hold:
(l₁) if where is the spectral radius and is a complex constant, then is invertible in and (l₂)if commutes with , then and
(l₃)if with , , and , then
(l₄)if , then is a -sequence. Furthermore, is a -sequence for each arbitrarily vector

Definition 3 (see [20]). Let be a cone -metric space over a Banach algebra with a constant . A function is named a generalized -distance on if, for all , it satisfies in the following properties:
(ν₁)
(ν₂)
(ν₃) for and a sequence converges to in , if for some and all , then
(ν₄)for all with , there exists with so that and imply

Notice that a generalized -distance contains both -distance and -distance. Further, is not necessarily true and the dose not imply that for every .

Example 4. Take , with . Let multiplication in be just pointwise multiplication. Then, is a Banach algebra with a unit for all . Also, let be a solid cone. Now, define by for all , where . Then, is a cone -metric space over a Banach algebra . Consider a mapping by for all . Then, is a generalized -distance on .

Lemma 5 (see [20]). Consider a generalized -distance on with two sequences and in , , and and be two -sequences. Then, the following cases hold: (1)if and for , then . In particular, if and , then (2)if and for , then converges to (3)if for , then is a Cauchy sequence in (4)if for , then is a Cauchy sequence in

Lemma 6 (see [20]). Let be a generalized -distance on . If for , then .

In this work, we establish several common fixed point theorems regarding a generalized -distance over a Banach algebra by removing the normality of the cone and the continuity of the mappings.

2. Main Results

The following theorem is the principal result of this paper using Han-Xu-type contraction [19].

Theorem 7. Consider a generalized -distance on a complete cone -metric space over a Banach algebra . Assume that two mappings for every satisfy the following relations: where so that commutes with and Then, and have a unique common fixed point.

Proof. Assume that is an arbitrary point with . Consider the sequence by putting and for all . Applying relation (2) by and , we get for all , which induces that Similarly, applying relation (3) by and , we get for all , which induces that Now, the inequalities (6) and (8) show that Since (by relation (4)), it follows from Lemma 2, (l₁), that is invertible and . Let . Since commutes with , we obtain Now, set . Then, by Lemma 2, (l₁) and (l₂), we obtain which implies that . Moreover, by multiplying in relation (9), we get Consider with . Using relation (12) and (ν₂), we deduce by a simple computation that Since and , we have which means that is a -sequence by Lemma 2, (l₄). Using Lemma 5, (3), is a Cauchy sequence. Due to the completeness of the space , there is a so that as . Using relation (13) and (ν₃), we have which shows that Now, we establish that . In relation (2), set and . Then, we get which induces that . Note that is invertible. Thus, by the inequalities (12) and (16), we get By considering the inequalities (15) and (18), Lemma 2, (l₄), and Lemma 5, (1), we conclude that . Now, in relation (3), set and . Then, we get which induces that . Note that is invertible. Thus, by the inequalities (12) and (15), we get By considering the inequalities (16) and (20), Lemma 2, (l₄), and Lemma 5, (1), we conclude that . Consequently, ; that is, is a common fixed point of and . Also, by using the relation (2), we have which induces that . Now, notice that . Thus, by relation (4), is invertible. Hence, by Lemma 2, (l₃), we have . Next, we prove that the common fixed point of and is unique. Assume that is another common fixed point and . It follows from relation (2) that Since and by using relation (4), we have by Lemma 2, (l₃). Also, it follows from relation (3) that which implies by the above procedure that . Now, by Lemma 6, we obtain . Consequently, the common fixed point of and is unique. Here, the proof ends.

Corollary 8. Consider a generalized -distance on a complete cone -metric space over a Banach algebra . Assume that two mappings for every satisfy the following relations: where with . Then, and have a unique common fixed point.

Proof. It is sufficient to set in Theorem 7.

Corollary 9. Consider a generalized -distance on a complete cone -metric space over a Banach algebra . Assume that a mapping for every satisfies the following relation: where with . Then, has a unique fixed point.

Proof. It follows by taking in Corollary 8.

Corollary 10. Consider a generalized -distance on a complete cone -metric space over a Banach algebra . Assume that a mapping for every satisfies the following relation: where with . Then, has a unique fixed point.

Proof. It is sufficient to set in Corollary 9.

Example 11. Let , with the norm and multiplication in be just pointwise multiplication. Then, is a real Banach algebra with a unit for all . Take a solid cone and define the cone -metric by , where and . Consider a mapping by for all . Then, is a generalized -distance in cone -metric space over Banach algebra . Take and define the mapping by Clearly, is not continuous. Also, On the other hand, we have the following two cases: (i)for all and , we get(ii)for all and , we get

That is, all hypotheses of Corollary 10 are held. Thus, has a unique fixed point at .

Corollary 12. Consider a generalized -distance on a complete cone -metric space . Assume that two mappings for every satisfy the following relations: where so that . Then, and have a unique common fixed point.

Proof. In Theorem 7, put , , and . The proof is evident.

Remark 13. In Theorem 7 and its corollaries, we take . Then, we obtain the same Theorem 16 and its next corollaries from Han and Xu [19] regarding a -distance over a Banach algebra . Also, these results generalize some main theorems and its next corollaries in [1, 3, 12, 13, 15, 18, 20].

3. Conclusions

In this paper, we established several fixed point results for two mappings and regarding a generalized -distance over a Banach algebra . Notice that the class of these distances is bigger than the class of usual -distances over the same Banach algebra. Also, this class is not equivalent to the class of -distances in -metric spaces. Further, we removed the continuity condition of the mappings and in expressing our results.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.