Abstract

In the present paper, we define -cone metric spaces over a Banach algebra which is a generalization of -metric space (-MS) and cone metric space (CMS) over a Banach algebra. We give new fixed-point theorems assuring generalized contractive and expansive maps without continuity. Examples and an application are given at the end to support the usability of our results.

1. Introduction

The notion of a generalized partial -metric (for short, a -metric) space was introduced by Hussain et al. [1] in 2014 by generalizing the notions of a -metric space (-MS) and a partial -metric space. They studied the related topological properties and provided fixed point theorems for some contractive maps.

Huang and Zhang [2] generalized the notion of metric spaces to a CMS. Later, the interesting concept of a cone metric space over a Banach algebra (for short, CMS over a BA) was proposed by Liu and Xu [3], by replacing a CMS with a CMS over a BA. Motivated by these ideas, many authors further considered a CMS over a BA (see [4, 5]). Also, in [6], the authors introduced the concept of cone -metric space over Banach algebras which generalizes the notions of -metric space and cone metric spaces over Banach algebra. There are some very recent references (such as [7, 8]), where it is shown that the fixed point theory continues to provide useful tools for studying problems of Mathematical Physics.

The present paper is organized as follows. In Section 2, we recall some definitions. In Section 3, we generalize the concepts of a -metric space and a CMS over a Banach algebra by introducing a -CMS over a Banach algebra (for short, -CMS over a BA) with some examples. Section 4 is devoted to define generalized contractive and expansive maps. Finally, in Section 5 and Section 6, we prove some fixed point theorems for such certain contractive and expansive maps in the framework of a -CMS over a BA. Our work generalizes and extends some interesting results of [9]. Two examples and an application are given to verify the strength of our main results.

2. Preliminaries

We start with some known concepts. Denote by a real Banach algebra (for short, BA). From now on, assume that there is a unit element . An element is called invertible if there is so that . We denote by the inverse of . For more related details, one may check [10].

Proposition 1 [10]. Assume that the spectral radius of an element is less than 1, that is, then is invertible, where is unit. In addition,

Remark 2 ([11]). If , then as .

A set is said to be a cone if (1), closed and (2) for all (3)(4),where is the null of . The partial ordering on is given as iff . We write to indicate , but , while stands for (here, is the interior of ).

Definition 3 ([2, 3]). Let be a nonempty set. If verifies (1) for all and iff (2) for all (3) for all then is named as a cone metric on , and is said to be a CMS over the BA .

Definition 4 ([1]). Let be a nonempty set and . Assume that the function is so that:
  if  
for all with
, where is any permutation of and (symmetry in all three variables)
for all (rectangle inequality).

Then, is named as a -metric and is called a -metric space.

Using ( 4) and , we have

The -metric space is called symmetric if holds for all . Otherwise, is an asymmetric metric.

Example 5 [1]. Let and let be given by where .

Obviously, is a symmetric -metric space, which is not a -metric space. In fact, if , then . It is easy to see that are satisfied.

3. A -CMS over a Banach Algebra

Here, we introduce the notion of a -CMS over the BA , as a generalization of a generalized partial -CMS and a CMS over the BA .

Definition 6. Let be a nonempty set and . Suppose that so that:
(J₁)
(J₂) for all with
(J₃) , where is any permutation of and (symmetry in all three variables)
(J₄) for all (rectangle inequality).

Then, is called a -cone metric and is called a -CMS over the BA .

Since , from , we have

The -CMS is said to be symmetric if holds for all . Otherwise, is asymmetric.

We now present some examples.

Example 7. Let which is endowed with the norm . Under the pointwise multiplication, is a real Banach algebra with unit . Let . Moreover, is not normal (see [12]).

Let be given by where . Let be given by

Obviously, is not a -metric space. But is a -CMS over the BA . Indeed, if , then . Hence, are satisfied. Now, we show that holds. For all , we have

(by Example 12 in [1]). Therefore, for all (rectangle inequality).

The following examples show that a -CMS over a BA need not be a -CMS ([13]).

Example 8. Let and be the set of all real-valued continuously differentiable functions on with the norm . Define the multiplication in the usual way. Let . Clearly, is a nonnormal cone and is a BA with a unit . Consider as for all . Thus, is a -CMS over the BA , but it is not a -CMS since .

Example 9. Let and consider a norm on as for all . Let the multiplication on be the pointwise multiplication. Then, is a real unit BA with unit . Set which is a cone in . Let and be any constant. Consider as for all . This is a -CMS over the BA , but it is not a -CMS since .

Example 10. Let be the set of all continuous functions on with the norm . Taking the usual multiplication, then is a BA with a unit 1. Set and . Consider by for all . Then, is a -CMS over the BA . But it is not a -CMS, because .

In the following, is assumed to be a -CMS over the BA .

Lemma 11. (a) If , then .
(b) If , then .

Definition 12. For an and , the -ball with center and radius is

The -Cauchyness and convergence are given as follows.

Definition 13. A sequence in converges to , whenever for each , there is an integer so that for all .

Definition 14. A sequence in is called a -Cauchy sequence in if for every there is such that for all .

Definition 15. is said to be -complete if every -Cauchy sequence in is convergent to so that .

Definition 16. Let and be two -CMS over the BA . Then, a function is called continuous at iff for each convergent sequence to , we have is convergent to .

4. Generalized Contractive and Expansive Maps

In this section, we consider contractive and expansive maps in a -CMS over a BA. Some examples are also presented.

Definiton 17. Let be a -CMS over the BA and be a cone in . A map is named as a generalized contractive mapping if there is (with ) so that for all ,

Example 18. Let be a BA and be a cone (as in Example 8) and let . Define a mapping by for all . Then, is a -CMS over the BA . Take by . In view of for each , we have for all , where . Obviously, is a generalized contractive map on .

Definition 19. Let be a -CMS over the BA and be a cone in . A map is said to be an expansive mapping if for all , where and are called generalized contractive constants with .

Example 20. Let be a BA and be a cone (as given in Example 9) and let . As in Example 18, is a -CMS over the BA . Define for all . where . Clearly, is an expansive map in .