Journal of Function Spaces
Volume 2018 (2018), Article ID 7463435, 10 pages
https://doi.org/10.1155/2018/7463435

Research Article

Some New Theorems on -Distance without Continuity in Cone Metric Spaces over Banach Algebras

Yan Han¹ and Shaoyuan Xu²

¹School of Mathematics and Statistics, Zhaotong University, Zhaotong 657000, China
²School of Mathematics and Statistics, Hanshan Normal University, Chaozhou 521041, China

Correspondence should be addressed to Yan Han; hanyan702@126.com

Received 25 February 2018; Accepted 8 April 2018; Published 15 May 2018

Academic Editor: Pasquale Vetro

Copyright © 2018 Yan Han and Shaoyuan Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The fixed point theorems for one mapping and the common fixed point theorems for two mappings satisfying generalized Lipschitz conditions are obtained, without appealing to continuity for mappings or normality for cone in the conditions. Furthermore, we not only get the existence of the fixed point but also get the uniqueness. These results greatly improve and generalize several well-known comparable results in the literature. Moreover, example is given to support our new results.

1. Introduction and Preliminaries

The cone metric space was initiated in 2007 by Huang and Zhang [1] as a generalization of metric space. Then, many fixed point results in cone metric spaces were introduced in [2–4] and references mentioned therein. However, it is not popular since some authors have appealed to the equivalence between some fixed point results in metric spaces and in (topological vector spaces valued) cone metric spaces. Afterwards, Liu and Xu [5] firstly defined cone metric space over Banach algebra and obtained some fixed point theorems in such spaces. Moreover, they gave an example to illustrate the nonequivalence of version of fixed point theorems between cone metric spaces over Banach algebras and metric spaces (in usual sense). In 2011, Cho et al. [6] and Wang and Guo [7] defined the concept of -distance which is a cone version of -distance of Kada et al. [8]. After that, lots of fixed point results on -distance in cone metric spaces and in tvs-cone metric spaces were introduced in [9–17]. However, the conditions relied strongly on the assumptions that the underlying cone is normal or the mappings are continuous.

In this paper, by exploiting the assumption of normality of the cone and the notion of continuity of the mappings at the same times, we establish some fixed point theorems on -distance in cone metric spaces over Banach algebras. Our main results improve and generalize some important known results in the literature [6, 7, 9–17]. In addition, we give an example to show that the main results are indeed real improvements and generalizations of the corresponding results in the literature.

First, we recall some basic terms and definitions about Banach algebras and cone metric spaces.

Let be a real Banach algebra; that is, is a real Banach space in which an operation of multiplication is defined, subject to the following properties: for all , ,

(1);(2) and ;(3);(4)

In this paper, we shall assume that the Banach algebra has a unit (i.e., a multiplicative identity) such that for all . An element is said to be invertible if there is an inverse element such that . The inverse of is denoted by . For more details, we refer to [18].

The following proposition is well-known (see [5, 18]).

Proposition 1. Let be a real Banach algebra with a unit and . If the spectral radius of is less than 1, that is,

then is invertible. Actually,

A subset of is called a cone if

(i) is nonempty, closed, and , where denotes the zero element of ;(ii) for all nonnegative real numbers ;(iii);(iv)

For a given cone , we can define a partial ordering with respect to by if and only if . We shall write if and , while will stand for , where denotes the interior of .

A cone is called normal if there is a number such that, for all ,

The least positive number satisfying the above inequality is called the normal constant of .

In the following, we always assume that is a cone in Banach algebra with and is the partial ordering with respect to .

Definition 2 (see [1, 5]). Let be a nonempty set. Suppose that the mapping satisfies

(d1) for all and if and only if ;(d2) for all ;(d3) for all .

Then is called a cone metric on and is called a cone metric space over a Banach algebra .

Definition 3 (see [1, 5]). Let be a cone metric space over a Banach algebra , and a sequence in . Then we say that is

(i) converges to if for every with , there is a natural number such that for all , ;(ii) is a Cauchy sequence if for every with , there is a natural number such that for all , ;(iii) is a complete cone metric space if every Cauchy sequence is convergent in .

Definition 4 (see [6, 7, 13]). Let be a cone metric space over a Banach algebra . Then the mapping is called a -distance on if the following are satisfied:

(q1) for all .(q2) for all .(q3)For each and , if for some , then whenever is a sequence in converging to a point .(q4)For all with , there exists such that , and imply .

Remark 5 (see [7]). The following facts are well-known:

(1) does not necessarily hold for all .(2) is not necessarily equivalent to for all .(3)The -distance is a great generalization of the -distance.(4)If , then is a -distance. That is, -distance is also a generalization of cone metric.

Lemma 6 (see [2]). The limit of a convergent sequence in cone metric space is unique.

Lemma 7 (see [6]). Let be a cone metric space and be a -distance on . Let and be sequences in and . Suppose that is a sequence in converging to . Then the following hold:

(1)If and , then .(2)If and , then converges to .(3)If for , then is a Cauchy sequence in .(4)If , then is a Cauchy sequence in .

Lemma 8 (see [12]). Let be a cone metric space and be a -distance on . Let be a sequence in . Suppose that and are two sequences in converging to . If and , then .

Lemma 9 (see [13]). Let be a cone metric space and be a -distance on . If such that and , then .

Lemma 10 (see [18]). Let be a Banach algebra with a unit . If , and commutes with , then

Lemma 11 (see [19]). Let be a Banach algebra with a unit and let be a vector in . If , then we have

Lemma 12 (see [20]). Let be a Banach algebra with a unit and be a solid cone in . Let hold and . If , then .

2. Main Results

In this section, without the assumption of normality of the cone and the notion of continuity of the mappings at the same times, we prove some fixed point theorems for -distance in the setting of cone metric spaces over Banach algebras. All conclusions are new. Moreover, we illustrate our results by an example.

Theorem 13. Let be a complete cone metric space over Banach algebra and the underlying solid cone . Let be a -distance on . Suppose the mapping satisfies generalized Lipschitz conditions:

for all , where are generalized Lipschitz constants with . If commutes with , then has a unique fixed point.

Proof. Suppose is an arbitrary point in and set , According to (5), we have

which means that

Making full use of (6), we obtain

which implies that

Thus, we can find the sum of (8) and (10) as follows:

Further, we set . Then, we see

Since , then, by Proposition 1, it is easy to see that is invertible. Furthermore,

Let . As commutes with , it follows that

that is to say commutes with . Then by Lemmas 10 and 11, we obtain

which means that and as .
By multiplying in both sides of (12) by , we get

Let . We infer

Owing to , it leads to . By Lemma 7, is a Cauchy sequence in .
Since is complete, there exists such that . By Definition 4, we obtain

Now, we show that . Substituting in (5), we get

which implies that Since , is invertible. So, it follows immediately from (16) and (18) that

Set and . As and , we know . Thus, by (18), (20), and Lemma 8, we get that .
In the following we shall show the fixed point is unique. Firstly, we have to prove . Making full use of (5), we know

In view of , , and Lemma 12, we have . Secondly, if there is another fixed point , then, from (5), we get

which establishes that

Since and , it follows immediately from Lemma 12 that . Actually, by (6), we also have that

Similar to the above proof, it is not difficult to obtain that . Thus, from Lemma 9, . The conclusion is true.

Corollary 14. Let be a complete cone metric space and the underlying solid cone . Let be a -distance on . Suppose the mapping satisfies generalized Lipschitz conditions:

for all , where and satisfies . Then, has a unique fixed point.

Proof. By taking for each in Theorem 13, we obtain the desired result.

Corollary 15. Let be a complete cone metric space over Banach algebra and the underlying solid cone . Let be a -distance on . Suppose the mapping satisfies generalized Lipschitz condition:

for all , where are generalized Lipschitz constants with . If commutes with , then has a unique fixed point.

Proof. Similar to Theorem 13, we get the sequence as follows: Then, we have

which means that

Since , then by Proposition 1 it is easy to see that is invertible. Furthermore,

Let . As commutes with , it follows that

that is to say commutes with . Then by Lemmas 10 and 11, we obtain

which means that and as .
Therefore, we get

Let . We infer

Owing to , it leads to . By Lemma 7, is a Cauchy sequence in .
Since is complete, there exists such that . By Definition 4, we obtain

Now, we show that . By the conditions in the theorem, we get

which implies that Since , is invertible. So,

Set and . As and , we know . Thus, by Lemma 8, we get that . Then, similar to the proof of Theorem 13, it is not difficult to obtain that is the unique fixed point of . So, the conclusion is true.

Now, we introduce some common fixed point theorems for two mappings which satisfy generalized Lipschitz conditions, without continuity of the mappings and normality of the cone.

Theorem 16. Let be a complete cone metric space over Banach algebra and the underlying solid cone . Let be a -distance on . Suppose the mappings satisfy generalized Lipschitz conditions:

Condition (37) holds for all and condition (38) holds for all , where are generalized Lipschitz constants with . If commutes with , then have a unique common fixed point.

Proof. Suppose is an arbitrary point in and set According to (37), we have

which means that

Similarly by using (38), we obtain

which implies that

Thus, (40) and (42) show that

Since , then by Proposition 1 it follows that is invertible. Furthermore,

Let . Since commutes with , we see

that is to say commutes with . Then by Lemmas 10 and 11, we obtain

which states that and as .
By multiplying in both sides of (43) by , we get

Let . It yields that

Note that ; then . By Lemma 7, is a Cauchy sequence in .
As is complete, there exists such that . By Definition 4, we obtain

which illustrates that

Now, we show that . Substituting , in (37), we get

which implies that Since , is invertible. So, it follows immediately from (47) and (49) that

Set and . As and , we know . Thus, by (50), (52), and Lemma 8, we get that . By repeating the similar method processes above, we suffice to prove that . As a result, , that is to say is a common fixed point of and .
Next, we shall show the common fixed point is unique. Firstly, we have to prove . Making full use of (37), we know

which means that . Since and is invertible, then, multiplying both sides with , we have . Secondly, if there is another common fixed point , then, from (37), we get

As and , it follows immediately from Lemma 12 that . Actually, by (38), we also have that

Similar to the above proof, it is not difficult to obtain that . Thus, from Lemma 9, . The conclusion is true.

Corollary 17. Let be a complete cone metric space and the underlying solid cone . Let be a -distance on . Suppose the mappings satisfy generalized Lipschitz conditions:

for all , where and satisfy . Then, have a unique common fixed point.

Proof. By taking for each in Theorem 16, the proof is clear.

Corollary 18. Let be a complete cone metric space over Banach algebra and the underlying solid cone . Let be a -distance on . Suppose the mappings satisfy generalized Lipschitz conditions:

for all , where are generalized Lipschitz constants and satisfies . Then, have a unique common fixed point.

Proof. It follows by taking in Theorem 16.

Corollary 19. Let be a complete cone metric space over Banach algebra and the underlying solid cone . Let be a -distance on . Suppose the mapping satisfies generalized Lipschitz condition:

for all , where are generalized Lipschitz constants and satisfies . Then, has a unique fixed point.

Proof. It follows by taking and in Theorem 16.

Corollary 20. Let be a complete cone metric space over Banach algebra and the underlying solid cone . Let be a -distance on . Suppose the mapping satisfies generalized Lipschitz condition:

for all , where is a generalized Lipschitz constant and satisfies . Then, has a unique fixed point.

Proof. Set and in Theorem 16, the proof is valid.

Remark 21. Taking in Corollary 20 the -distance defined by for all . Then, Corollary 20 corresponds to Theorem  3.1 in [19]. So, Corollary 20 improves and generalizes Theorem  3.1 in [19].

Remark 22. Comparing the main results in the literature [6, 7, 9–11, 13–17], we see that many conclusions in the literature relied strongly on either the assumption of the continuity of the mappings (Theorems 3.1 and 3.3 in [6]; Theorems 3.5 and 3.9 in [9]; Theorems 3.1 and 3.3 in [10]; Theorems 2–5 in [11]; Theorem  3.3 in [14]; Theorem 2.4 and Corollaries 2.5–2.7 in [16]; Theorems 3.1, 3.2, 3.5, and 3.6 in [17]); or the normality of the cone on -distance in cone metric space (Theorems 3.2 and 3.4 in [6]; Theorems 3.2 and 3.5 in [10]; Theorem 1 and Corollaries 1–3 in [13]; Theorems 3.1–3.3 and Corollaries 3.2 and 3.4 in [15]; Theorem 2.8 in [16]; Theorem 3.3, 3.4, 3.7, and 3.8 in [17]; Theorem 2.2 in [7]). They must have one of the two. In this paper, we get the fixed point theorems, which does not require the continuity of the mappings, nor does it require the normality of the cone. We delete the two conditions at the same times. Moreover, we not only obtain the existence of fixed points, but also get their uniqueness. So, the main results in this paper are indeed more different than the standard results of cone metric spaces presented in the literature. Based on these facts, our main results are very meaningful and may offer us a useful tool for the existence and uniqueness of fixed points.

Remark 23. -distance is great generalization of cone metric ([7]). Further, due to the nonequivalence of cone metric spaces with Banach algebra and (general) metric spaces, the conclusions in this paper are focused on fixed point theorems in cone metric spaces over Banach algebras instead of theorems only in usual metric spaces or cone metric spaces (see [1–4, 6–8, 10–12, 14–17, 21, 22]), which is more meaningful. If in these theorems, we obtain the corresponding (common) fixed point theorems in metric spaces and generalize many of the conclusions in [8].

As an application, we will give an example of the situation when Theorem 13 can be applied, while the results in the literature cannot.

Example 24. Let with the norm

Define the multiplication by

where , . Then is a Banach algebra with a unit . Let . It is clear that is a nonnormal cone. Let and define the cone metric by , . We make a conclusion that is a complete cone metric space over Banach algebra . Putting a mapping as , it is easy to show that is a -distance. Take , , , . We observe that

Furthermore, define the mapping by

where is equivalent to . Obviously, is not continuous. There are the following two cases.
  , ,

, ,

  , ,

, ,

Therefore, we can apply Theorem 13 and conclude that has a (unique) fixed point 0 in . Since is not continuous in and the cone is not normal, we can conclude that these known results in the literature cannot cope with Example 24, which shows that the main results in our paper are actual generalization of the known results.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The research was partially supported by Yunnan Applied Basic Research Projects, Yunnan Province, China (no. 2016FD082).

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