1. Introduction
The Banach contraction principle [
1] is one of the most important tools of analysis and has many significant applications in various fields of science. It has been improved in many ways and generalized by many researchers. A map
, where
is a complete metric space, is said to be a contraction map if there exists
, such that for all
This result was introduced by Banach in 1922. Kannan [
2] in 1968 proved that, if
is a complete metric space and
is a map satisfying
where
for all
, then there is a unique fixed point on
T. Later, in 1972, Chatterjea [
3] proved that if
is a complete metric space and
is a mapping that exists
, such that
, the inequality
is satisfied; thus, T has a unique fixed point.
iri
[
4] in 1974 introduced an interesting general contraction condition. If there exists
, such that for all
, and
is a map satisfying
then
T has a unique fixed point.
On the other hand, Samet et al. [
5,
6] studied
-
-contractive mappings in metric spaces. Many researchers have established related studies to
-admissible and
-contractive mappings and related fixed-point theorems (see [
7,
8,
9,
10,
11,
12,
13,
14,
15]).
Recently, Ma et al. [
10] introduced the more generalized notion 0f a
-algebra-valued metric space by replacing real numbers with the positive cone of
-algebra. This line of research was continued in [
16,
17,
18,
19,
20,
21,
22], where several other fixed-point results were obtained in the framework of
-algebra-valued metric space.
Throughout this paper, we suppose that A is a unital -algebra with a unit . We mean that a unital -algebra is a complex Banach algebra A with an involution map , , such that , , and for , such that . Set . An element is a positive element if and , where is the spectrum of a. We define a partial ordering ⪯ on A as if , where means the zero element in A, and we let denote the and .
The results described in this article extend some fixed-point theorems in
-algebra-valued metric spaces.
-algebras are considered typical examples of quantum spaces and non-commutative spaces. They play an important role in the non-commutative geometry project introduced by Alain Connes [
23]. Thus, the theory of metric space-valued
-algebras should apply to many problems in quantum spaces, such as matrices and bounded linear operators on Hilbert spaces. Therefore,
-algebras and their metric provide a non-commutative version of ordinary metric spaces.
2. Preliminaries
In this section, we introduce some basic notions which will be used in the following work.
Lemma 1. Suppose that A is a unital -algebra with unit . The following holds.
(1) If , with , then is invertible and .
(2) If and , then .
(3) Let . If with and is an invertible element, then , where .
We refer to [24] for more algebra details. Definition 1. [10] Let Ω be a non-empty set. Suppose the mapping satisfies: (1) for all and .
(2) for all .
(3) for all .
Then, is called a -algebra-valued metric on Ω and is called -algebra-valued metric space.
Example 1. Let Ω be a Banach space and given by , for all , which should be where , .
It is easy to verify that is a -algebra-valued metric space.
Example 2. Let and . It is obvious that A is a -algebra with the matrix norm and the involution given by , , where is the conjugate of , . Define a mapping , by:for all , , , . Then, is a -algebra-valued metric space. It is clear that it is a generalization of the complex-valued metric space given in [25], when Definition 2. Let be a -algebra-valued metric space, , and be a sequence in Ω. Then,
(i) convergent to μ whenever, for every with , there is a natural number , such thatfor all . We denote this by or as . (ii) is said to be a Cauchy sequence whenever, for every with , there is a natural number , such thatfor all . Lemmaμ
2. (i) is convergent in Ω if, for any element , there is , such that for all , .
(ii) is a Cauchy sequence in Ω if, for any there is , such that
, for all . We say that is a complete -algebra-valued metric space if every Cauchy sequence is convergent with respect to A.
Example 3. Let Ω be a compact Hausdorff space. We denote by the algebra of all complex-valued continuous functions on Ω with pointwise addition and multiplication. The algebra with the involution defined by for each and with the norm is a commutative -algebra where unit is the constant function. Let denote the positive cone of, with partial order relation if and only if . Put as . It is clear that is a complete -algebra-valued metric space.
Definition 3. [6] Let be a self map and . Then, T is called α-admissible if for all and implies . Definition 4. Let Ω be a non-empty set and be a function. We say that the self map T is -admissible if for all , , where is the unit of A.
Definition 5. Let be a -algebra-valued metric space and be a mapping. We say that T is an --contractive mapping if there exist two functions and , such thatfor all Definition 6. Suppose that A and B are -algebras. A mapping is said to be a -homomorphism if:
(a) for all and ;
(b) , ;
(c) , ; and
(d) ψ maps the unit in A to the unit in B.
Definition 7. If is a linear mapping in -algebra, it is said to be positive if . In this case, , and the restriction map increases. Every -homomorphism is contractive and hence bounded and every *-homomorphism is positive.
Definition 8. Let be the set of positive functions satisfying the following conditions:
(a) is continuous and non-decreasing, ;
(b) iff ; and
(c) , for each , where is the nth-iterate of .
3. Main Results
In this section, we give some types of Chatterjea and iri fixed-point theorems in a -algebra-valued metric space using -contraction.
Theorem 1. (Chatterjea Type) Let be a complete -algebra-valued metric space and , be a mapping satisfying: , whereand the following conditions hold: (a) T is -admissible;
(b) There exists , such that ; and
(c) T is continuous.
Then, T has a fixed point in Ω.
Proof. Let , such that , and define the sequence in , such that for all . If for some , then is a fixed point for T.
Suppose that
for all
. Because
T is
-admissible, we obtain
By induction, we have for all .
By using inequalities (
5) and (
6), we have
Because
, we obtain
Applying triangular inequality in (
7), we have
Because
is additive, we have
Putting
by induction, we have
for all
. Let
with
. We obtain
Therefore, we can prove that is a Cauchy sequence in the -algebra metric space .
Because is complete, there exists , such that as . From the continuity of T, it follows that is as .
By continuity of this limit, we have —that is, is a fixed point of T.
The proof of the uniqueness is as follows. If
is another fixed point of
T, then
This implies that
which gives a contradiction, and we can obtain
. This completes the proof. □
Corollary 1. Let be a complete -algebra-valued metric space. Suppose satisfies for all where and . Then, there exists a unique fixed point T in Ω [10]. Proof. This is an immediate consequence of Theorem 1, with , , where , . □
Theorem 2. (Banach-Chatterjea Type) Let be a complete -algebra-valued metric space and be a mapping satisfying , where the following conditions hold:
(i) T is -admissible;
(ii) there exists , such that ; and
(iii) T is continuous.
Then, T has a fixed point in Ω.
Proof. Following the first part of the proof in the Theorem 1, we obtain
By using inequalities (
8) and (
9), we have
By using triangular inequality, we obtain
Putting
, we obtain
for
. Thus, we obtain
Thus, is a Cauchy sequence in with respect to .
Because is a complete -algebra-valued metric space, we conclude that is a convergence sequence, and so as and as . Therefore, is a fixed point of T.
To prove the uniqueness, we suppose that
is another fixed point of
T. Thus,
This is a contradiction, so and . □
Corollary 2. Let be a complete real-valued metric space. Suppose satisfies for all where . Then, T has a unique fixed point in Ω. Proof. This is an immediate consequence of Theorem 2, with and and . □
Theorem 3. (Ćirić Contraction Type) Let be a complete -algebra-valued metric space and be a mapping satisfying , where the following conditions hold:
(i) T is -admissible;
(ii) there exists , such that ; and
(iii) T is continuous.
Then, T has a fixed point in Ω.
Proof. Following the first part of the proof in the Theorem 1, we obtain
By using (
10) and (
11), we have
On the other hand, we have
Because
, we obtain
By using triangular inequality, we obtain
Putting
,
; then, we obtain
Let
, such that
. We thus obtain
Thus, is a Cauchy sequence and as . Thus, we obtain as a fixed point of T.
To prove the uniqueness, we suppose that
is another fixed point of
T. Thus,
Because , this implies that , which gives a contradiction. Then, we obtain . □
Example 4. Let Ω be a Banach space and be defined as for all . is the unit of A because Ω is a Banach space. Then, is a complete -algebra-valued metric space. Define as and define as for all , where is the positive cone of A. Additionally, is defined by , where Applying , we obtain This satisfies the conditions in Theorem 2. Then, T has a fixed point of Ω.
We introduce a numerical example, assuming that the metric space is valued-non-commutative -algebra
Example 5. Let and , where is the set of all matrices entries in . It is obvious that is a -algebra with matrix norm and involution given by , where is the transpose of a, . Definefor all , . It is clear that is -algebra-valued metric space. To verify the contraction conditions in Theorem 3, we take , , . Additionally, we define by and byand , by , for , , where is the set of positive matrices of . Now, by simple calculation, we obtain Thus, we calculate the right hand side of the inequality (10) in Theorem 3 as Therefore, .
On the other hand, the left hand side of the inequality (10) in Theorem 3 is given by . Hence, it is obvious that T is -admissible and, because we can obtain Thus, all conditions of Theorem 3 are satisfied. Therefore, there exists a unique fixed point of T, and the zero matrix is the fixed point of .
We discuss a numerical example that satisfies the conditions of Theorem 3, where the metric space in this example is valued-commutative -algebra
Example 6. Let and , the set of direct sum of two copies of complex numbers. with the vector addition and pointwise multiplication defined by , and , for all , is a -algebra with the maximum norm given by , and involution given by , for all . Define a partial order ⪯ on if and only if
(a) , Im Im , and
(b) , Im Im .
Thus, iff . Additionally, if and . In addition, , Im and , Im
Let be the set of all positive element in . Suppose and be a mapping defined by for all and .
It is clear that is -algebra-valued metric space.
Now, define by and as . In addition, assume defined by ∀ a .
To verify the contraction conditions in Theorem 3, we take , . By calculation, one can obtain the following: We calculate the right-hand side of the inequality (10) in the Theorem 3 and obtain On the other hand, the left-hand side of the inequality (10) in the Theorem 3 gives It is clear that , and this satisfies all conditions of the Theorem 3.
In the following, we provide an application scenario with which to study the existence and uniqueness of the solution of a system of matrix equations. The existence and uniqueness of the solution of the linear matrix equations are very interesting and important in linear systems.
Here, we are interested in using -algebra-valued metric spaces to find a positive definite hermitian solution for a system of matrix equations with complex entries.
The proof is based on the positive cones and the linear continuous operator mapping a cone into itself.