-Algebra-Valued -Metric Spaces and Related Fixed-Point Theorems

Abstract

We introduce the notion of the -algebra-valued -metric space. The existence and uniqueness of some fixed-point theorems for self-mappings with contractive or expansive conditions on complete -algebra-valued -metric spaces are proved. As an application, we prove the existence and uniqueness of the solution of a type of differential equations.

1. Introduction

As is known to all, the proverbial fixed-point theorem of Banach has been widely used in many branches of mathematics and physics. There are a large number of generalizations for such a theorem. In general, the theorem has been extended in two directions. On the one hand, the usual contractive condition is replaced with weakly contractive conditions [1–7]. On the other hand, the action spaces are replaced with different types of metric spaces. Particularly, in 2006, Mustafa and Sims [8] introduced the concept of generalized metric spaces (-metric spaces). Since then, many scholars studied fixed-point theory in -metric spaces and many meaningful results are obtained.

Let us recall the basic definitions and conclusions on -metric spaces. Details can be seen in [8–20].

Definition 1 (see [8]). Let be a nonempty set. Suppose that is a mapping satisfying(1) if ;(2) for all with ;(3) for all with ;(4) (symmetry in all three variables);(5) for all (rectangle inequality). Then is called a generalized metric, or a -metric on . The pair is called a -metric space.

Definition 2 (see [8]). Let be a -metric space, If there exists , such that , we say that the sequence is called -convergent to . If for any , there is an , such that, for all , is called a -Cauchy sequence.

Notice that, in a -metric space , the following statements are equivalent:(1) is -convergent to ;(2) as ;(3) as .

As we have known that -algebra, which was first proposed for its use in quantum mechanics to model algebras of physical observable, is an important research field of modern mathematics [21–29]. In 1947, I. Segal [28] introduced the term “-algebra” to describe a “uniformly closed, self-adjoint algebra of bounded operators on a Hilbert space”.

Throughout this paper, will denote a unital -algebra with a unit ; namely, is a unital Banach algebra with an involution such that (). Let be a Hilbert space and the set of all bounded linear operators on , then is a -algebra with the operator norm. Let be the set of all self-adjoint elements in , and define the spectrum of to be the set . An element is positive (denoted by ) if and , set , then [27]. Using the positive element, one can define a partial ordering “” on as follows: if and only if . It is clear that if and , then , and that if are invertible, then

Using the partial ordering “” on , Ma introduced the notion of -algebra-valued metric spaces and gave the fixed-point theory for contractive or expansion mapping on such a space [30, 31]. Let us recall the definition first.

Definition 3 (see [30]). Let be a nonempty set. If is a mapping satisfying(1);(2) for all ;(3) for all , then is called a -algebra-valued metric on . is called a -algebra-valued metric space.

Definition 4 (see [30]). Let be a -algebra-valued metric space, If for any , there is an , such that for all , , then we say this converges to . We denote it by
If for any , there is an , such that, for all , then we say is a Cauchy sequence with respect to We say that is a complete -algebra-valued metric space if every Cauchy sequence with respect to is convergent.

Definition 5 (see [30]). Let be a -algebra-valued metric space. We call a mapping is a contractive mapping on if there exists with such that ,

Theorem 6 (see [30]). If is a complete -algebra-valued metric space and is a contractive mapping, there exists a unique fixed point in .

In this paper, we will define -algebra-valued -metric spaces and prove some fixed point theorems on such spaces. We also provide an application of the theory for a type of differential equations.

2. Main Results

In this section, we first give the definition of a -algebra-valued -metric spaces.

Definition 7. Let be a nonempty set and be the permutation group on . If is a mapping satisfying(1);(2) for all ;(3) for all with ;(4) for all ,   is called a -algebra-valued -metric on , and is called a -algebra-valued -metric space.

Example 8. Let Notice that of -matrices with entries in is identified with . This is a -algebra. Let and extend to all of by symmetry in the variables. Then is a -algebra-valued -metric on , and is a -algebra-valued -metric space.

Definition 9. Let be a -algebra-valued -metric space, . If for any , there is an , such that for all , then we say is -convergent to , and denote by , .

Proposition 10. Let be a -algebra-valued -metric space, . The following statements are equivalent:(1).(2), as .(3), as .

Proof. If , that is, for all , , such that, for all, , especially, Hence , as .
If , , such that, for all , then when , that is, , as .
If , as , then, for any , , such that for all ; , such that, for all . Let , for , that is, .

Definition 11. Let be a -algebra-valued -metric space, If for any , there is an such that for all , then the sequence is called a -Cauchy sequence. If any -Cauchy sequence in is -convergent, then is called a complete -algebra-valued -metric space.

Proposition 12. Let be a -algebra-valued -metric space, . Then is a -Cauchy sequence if and only if for any , there is an , such that, for all .

Proof. It suffices to show the necessity. For any , , such that for all ; , such that for all . So for the above , let , when , that is, is a -Cauchy sequence.

Example 13. Set . Let ; set where , then is a -algebra-valued -metric on , and is a complete -algebra-valued -metric space.
It is easy to see that is a -algebra-valued -metric space. We only need to prove the completeness. Let be a -Cauchy sequence. Then for any , there is an , such that for all , So . Since is complete, there exists , such that Hence there is such that, for any , . It follows thatTherefore, , and is complete.

Example 14. Suppose is a compact Hausdorff space and is a positive regular Borel measure on . Let , the set of all essentially bounded complex-valued measurable functions on , then is a Banach space with the essential supremum norm . Set ; is a Hilbert space with the inner-product For , defineThen is bounded and moreover . Let by Then is a -algebra-valued -metric and is a complete -algebra-valued -metric space. We omit its proof and leave it to readers.

Next, we define the contractive mapping on -algebra-valued -metric space and prove the fixed point theorem for contractive mappings.

Definition 15. Let be a -algebra-valued -metric space and is a mapping. If there exists with such that then is called a contractive mapping on

Theorem 16. Let be a complete -algebra-valued -metric space. If is a contractive mapping on , then there is a unique fixed point of on

Proof. Let .
We first show that is a -algebra-valued metric on . It suffices to show that That is, Sincewe have
Next, we show is a complete -algebra-valued metric space. Let be a Cauchy sequence with respect to Then for any , there is an such that for all , , that is, Since , So is a -Cauchy sequence. By the completeness of , there exists an , such that That is, for any , there is an such that, for all ; there is an such that for all . Let , then for all , we have Therefore,Hence , and is complete.
Moreover, is a contractive mapping on In fact,It follows from Theorem 6 that there is an such that .
Finally, we show the uniqueness of this fixed point. Let is another fixed point of If , then Since , This is a contradiction. So .

Remark 17. (1) In the theorem above, the completeness of is essential. For example, let and satisfy , then is a -algebra-valued -metric space, but is not complete. Considering the mapping is a contractive mapping, but has no fixed point.
(2) In the definition of contractive mapping, the element does not depend on the choice of If depends on , then may not have a fixed point.
Indeed, let and be defined by . Then is a complete -algebra-valued -metric space. Let . If , thenwhere , , , and depends on , but has no fixed point.
(3) When , may not have a unique fixed point.
Consider and , for , and let . Define by Then is a complete -algebra-valued -metric space. Let Then But for each , is a fixed point, which means the fixed point is not unique.

What follows is the definition of the expansion mapping on -algebra-valued -metric space and the fixed-point theorem for expansion mappings.

Definition 18. Let be a -algebra-valued -metric space. If satisfies the condition where is invertible and , we call an expansion mapping on

Theorem 19. Let be a complete -algebra-valued -metric space and a expansion mapping on . If is surjective, then there is a unique fixed point for .

Proof. First we show is injective. Indeed, if , then Since , , and since is invertible, Therefore
Next we show that has a unique fixed point in In fact, is bijective and so is invertible. Since , Replace by , , , respectively, we get Hence Thus By Theorem 16, there is a unique , such that , and therefore there is a unique , such that

The following lemma is necessary for another fixed-point theorem, for detail, see [27].

Lemma 20. Set .
(1) If and , then
(2) If , with and is invertible, then .

Theorem 21. Let be a complete -algebra-valued -metric space, . If there exists an , such that for any , orthen has a unique fixed point in

Proof. Without loss of generality, we can assume
(1) Suppose that satisfies Since , For , set , and ThenThat is, Since with , we have and furthermore with . Therefore, Let , for ,This implies that is a -Cauchy sequence in . Since is complete, there exists an such that , i.e., SinceThat isThenand hence and is the fixed point of on
Next, we show the uniqueness of If there exists another fixed point , theni.e.,Since , Hence , and
(2) The case when can be proved similarly and we omitted it.

Definition 22. Let be a -algebra-valued -metric space. We say is symmetric if for all

It is easy to show that, in Example 8, is not symmetric and, in Example 13, is symmetric.

Theorem 23. Let be a complete -algebra-valued -metric space and symmetric. If is a mapping satisfying that for or where and , then has a unique fixed point in

Proof. Without loss of generality, one can assume
We only consider the case when satisfies Since
For , set , and If is symmetric,That is, Since with , then is invertible and and Therefore Just like the proof of Theorem 21, we can prove that, for , This implies that is a -Cauchy sequence in ; thus there exists an such that i.e., Similarly, we can show that , is the fixed point of on
The uniqueness of the fixed point can be proved similarly.

3. Applications

For fixed-point theorems, there are a number of applications in differential equations and integral equations.

Consider the second-order differential equation: where and are continuous. Then the equation has a unique solution if and only if has a unique solution, where

Let , then is a complete -algebra-valued -metric space. Let be a mapping defined bythen has a unique solution if and only if has a unique fixed point.

Theorem 24. If , equation has a unique solution.

Proof. For a fixed , Therefore, and has a unique fixed point.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The project is supported by the Scientific Research Foundation of Education Bureau of Hebei Province (QN2016191) and Natural Science Foundation of China (11371222).