Abstract
Based on the concept and properties of -algebras, the paper introduces a concept of -algebra-valued metric spaces and gives some fixed point theorems for self-maps with contractive or expansive conditions on such spaces. As applications, existence and uniqueness results for a type of integral equation and operator equation are given.
MSC:47H10, 46L07.
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1 Introduction
We begin with the concept of -algebras.
Suppose that is a unital algebra with the unit I. An involution on is a conjugate-linear map on such that and for all . The pair is called a ∗-algebra. A Banach ∗-algebra is a ∗-algebra together with a complete submultiplicative norm such that (). A -algebra is a Banach ∗-algebra such that [1, 2].
Notice that the seeming mild requirement on a -algebra above is in fact very strong. Moreover, the existence of the involution -algebra theory can be thought of as infinite-dimensional real analysis. Clearly that under the norm topology, , the set of all bounded linear operators on a Hilbert space H, is a -algebra.
As we have known, the Banach contraction principle is a very useful, simple and classical tool in modern analysis. Also it is an important tool for solving existence problems in many branches of mathematics and physics. In general, the theorem has been generalized in two directions. On the one side, the usual contractive (expansive) condition is replaced by weakly contractive (expansive) condition. On the other side, the action spaces are replaced by metric spaces endowed with an ordered or partially ordered structure. In recent years, O’Regan and Petrusel [3] started the investigations concerning a fixed point theory in ordered metric spaces. Later, many authors followed this research by introducing and investigating the different types of contractive mappings. For example in [4] Caballero et al. considered contractive like mapping in ordered metric spaces and applied their results in ordinary differential equations. In 2007, Huang and Zhang [5] introduced the concept of cone metric space, replacing the set of real numbers by an ordered Banach space. Later, many authors generalized their fixed point theorems on different type of metric spaces [6–13]. In [14], the authors studied the operator-valued metric spaces and gave some fixed point theorems on the spaces. In this paper, we introduce a new type of metric spaces which generalize the concepts of metric spaces and operator-valued metric spaces, and give some related fixed point theorems for self-maps with contractive or expansive conditions on such spaces.
The paper is organized as follows: Based on the concept and properties of -algebras, we first introduce a concept of -algebra-valued metric spaces. Moreover, some fixed point theorems for mappings satisfying the contractive or expansive conditions on -algebra-valued metric spaces are established. Finally, as applications, existence and uniqueness results for a type of integral equation and operator equation are given.
2 Main results
To begin with, let us start from some basic definitions, which will be used later.
Throughout this paper, will denote an unital -algebra with a unit I. Set . We call an element a positive element, denote it by , if and , where is the spectrum of x. Using positive elements, one can define a partial ordering ⪯ on as follows: if and only if , where θ means the zero element in . From now on, by we denote the set and .
Remark 2.1 When is a unital -algebra, then for any we have [1, 2].
With the help of the positive element in , one can give the definition of a -algebra-valued metric space.
Definition 2.1 Let X be a nonempty set. Suppose the mapping satisfies:
-
(1)
for all and ;
-
(2)
for all ;
-
(3)
for all .
Then d is called a -algebra-valued metric on X and is called a -algebra-valued metric space.
It is obvious that -algebra-valued metric spaces generalize the concept of metric spaces, replacing the set of real numbers by .
Definition 2.2 Let be a -algebra-valued metric space. Suppose that and . If for any there is N such that for all , , then is said to be convergent with respect to and converges to x and x is the limit of . We denote it by .
If for any there is N such that for all , , then is called a Cauchy sequence with respect to .
We say is a complete -algebra-valued metric space if every Cauchy sequence with respect to is convergent.
It is obvious that if X is a Banach space, then is a complete -algebra-valued metric space if we set
The following are nontrivial examples of complete -algebra-valued metric space.
Example 2.1 Let and , where E is a Lebesgue measurable set. By we denote the set of bounded linear operators on Hilbert space H. Clearly is a -algebra with the usual operator norm.
Define by
where is the multiplication operator defined by
for . Then d is a -algebra-valued metric and is a complete -algebra-valued metric space.
Indeed, it suffices to verity the completeness. Let in X be a Cauchy sequence with respect to . Then for a given , there is a natural number such that for all ,
then is a Cauchy sequence in the space X. Thus, there is a function and a natural number such that if .
It follows that
Therefore, the sequence converges to the function f in X with respect to , that is, is complete with respect to .
Example 2.2 Let and . Define
where and is a constant. It is easy to verify d is a -algebra-valued metric and is a complete -algebra-valued metric space by the completeness of ℝ.
Now we give the definition of a -algebra-valued contractive mapping on X.
Definition 2.3 Suppose that is a -algebra-valued metric space. We call a mapping is a -algebra-valued contractive mapping on X, if there exists an with such that
Theorem 2.1 If is a complete -algebra-valued metric space and T is a contractive mapping, there exists a unique fixed point in X.
Proof It is clear that if , T maps the X into a single point. Thus without loss of generality, one can suppose that .
Choose and set , . For convenience, by B we denote the element in .
Notice that in a -algebra, if and , then for any both and are positive elements and [1]. Thus
So for ,
Therefore is a Cauchy sequence with respect to . By the completeness of , there exists an such that .
Since
hence, , i.e., x is a fixed point of T.
Now suppose that y (≠x) is another fixed point of T, since
we have
It is impossible. So and , which implies that the fixed point is unique. □
Similar to the concept of contractive mapping, we have the concept of an expansive mapping and furthermore have the related fixed point theorem.
Definition 2.4 Let X be a nonempty set. We call a mapping T is a -algebra-valued expansion mapping on X, if satisfies:
-
(1)
;
-
(2)
, ,
where is an invertible element and .
Theorem 2.2 Let be a complete -algebra-valued metric space. Then for the expansion mapping T, there exists a unique fixed point in X.
Proof Firstly, T is injective. Indeed, for any with , if , we have
Since , . Also A is invertible, , which is impossible. Thus T is injective.
Next, we will show T has a unique fixed point in X. In fact, since T is invertible and for any ,
In the above formula, substitute x, y with , , respectively, and we get
This means
and thus
Using Theorem 2.1, there exists a unique x such that , which means there has a unique fixed point such that . □
Before introducing another fixed point theorem, we give a lemma first. Such a result can be found in [1, 15].
Lemma 2.1 Suppose that is a unital -algebra with a unit I.
-
(1)
If with , then is invertible and ;
-
(2)
suppose that with and , then ;
-
(3)
by we denote the set . Let , if with and is a invertible operator, then
Notice that in a -algebra, if , one cannot conclude that . Indeed, consider the -algebra and set , , then . Clearly , while ab is not.
Theorem 2.3 Let be a complete -valued metric space. Suppose the mapping satisfies for all
where and . Then there exists a unique fixed point in X.
Proof Without loss of generality, one can suppose that . Notice that , is also a positive element.
Choose , set , , by B we denote the element in . Then
Thus,
Since with , one have and furthermore with by Lemma 2.1. Therefore,
where .
For ,
This implies that is a Cauchy sequence with respect to . By the completeness of , there exists such that , i.e. . Since
This is equivalent to
Then
This implies that i.e., x is a fixed point of T.
Now if y (≠x) is another fixed point of T, then
i.e.,
Since ,
This means that
Therefore the fixed point is unique and the proof is complete. □
3 Applications
As applications of contractive mapping theorem on complete -algebra-valued metric spaces, existence and uniqueness results for a type of integral equation and operator equation are given.
Example 3.1 Consider the integral equation
where E is a Lebesgue measurable set.
Suppose that
-
(1)
and ;
-
(2)
there exists a continuous function and such that
for and
-
(3)
.
Then the integral equation has a unique solution in .
Proof Let and . Set d as Example 2.1, then d is a -algebra-valued metric and is a complete -algebra-valued metric space with respect to .
Let be
Set , then and . For any ,
Since , the integral equation has a unique solution in . □
Example 3.2 Suppose that H is a Hilbert space, is the set of linear bounded operators on H. Let , which satisfy and , . Then the operator equation
has a unique solution in .
Proof Set . Clear that if , then the (), and the equation has a unique solution in . Without loss of generality, one can suppose that .
Choose a positive operator . For , set
It is easy to verify that is a -algebra-valued metric and is complete since is a Banach space.
Consider the map defined by
Then
Using Theorem 2.1, there exists a unique fixed point X in . Furthermore, since is a positive operator, the solution is a Hermitian operator. □
As a special case of Example 3.2, one can consider the following matrix equation, which can also be found in [16]:
where Q is a positive definite matrix and are arbitrary matrices. Using Example 3.2, there exists a unique Hermitian matrix solution.
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Acknowledgements
This work is supported financially by the NSFC (11371222) and by the project for the Construction of the Graduate Teaching Team of Beijing Institute of Technology (YJXTD-2014-A08).
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Ma, Z., Jiang, L. & Sun, H. -algebra-valued metric spaces and related fixed point theorems. Fixed Point Theory Appl 2014, 206 (2014). https://doi.org/10.1186/1687-1812-2014-206
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DOI: https://doi.org/10.1186/1687-1812-2014-206