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Journal of Function Spaces
Volume 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

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; moc.621@207naynah

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 [24] 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 [917]. 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, 917]. 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 thatMaking full use of (6), we obtain which implies thatThus, we can find the sum of (8) and (10) as follows: Further, we set . Then, we seeSince , 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 thatSince 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 thatSince , 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 thatSimilarly by using (38), we obtain which implies thatThus, (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