Algebraic cone metric spaces and fixed point theorems of contractive mappings
© Tootkaboni and Bagheri Salec; licensee Springer. 2014
Received: 5 March 2014
Accepted: 14 June 2014
Published: 23 July 2014
In this paper we introduce algebraic cone metric spaces, prove some fixed point theorems of contractive mappings on algebraic cone metric spaces, and improve some well-known results in the non-normal case.
Keywordscone metric space Banach principle contraction fixed point
1 Introduction and preliminary
There exist a number of generalizations of metric spaces. One of them is the cone metric space initiated by Huang and Zhang . They described the convergence in cone metric spaces, introduced the notion of completeness and proved some fixed point theorems of contractive mappings on these spaces. Let us recall some notions and definitions.
is closed and nonempty and ,
if ; ; , then ,
if and , then .
Given a cone , a partial ordering ⪯ with respect to is defined by if . Furthermore, we write if and , while will stands for , where is the interior of .
The cone is called normal if there is a number , such that for every , implies . The least positive number satisfying the above condition is called the normal constant of .
is a normal cone.
For arbitrary sequences , and in E, if for each n and , then .
There exists a norm on , equivalent to the given norm , such that the cone is monotone with respect to , i.e., if , then .
for every and if and only if ;
for every ;
for every .
Then d is called a cone metric on X and is called a cone metric space.
Let be a cone metric space, be a sequence in X, and . is said to be convergent to x, if for every there is , such that for all , . Likewise, is called a Cauchy sequence in X if for every there is , such that for each , . A cone metric space X is said to be complete if every Cauchy sequence in X is convergent in X.
To replace the standard properties of a metric, the properties in the following lemma of cone metrics are often useful while dealing with cone metrics when the cone is not normal.
If for all , then .
If and , then for each there exists a natural number , such that for each .
If and , then there exists a natural number , such that for every , we have .
Proof See , p.2598. □
It follows from Lemma 1.3(c) that the sequence converges to if , and is a Cauchy sequence if as . The converse is true if is a normal cone. To see that does not necessarily imply at non-normal cones, see example (e) in Section 4.
There are several states of cone metric spaces. For example, after cone metric spaces over topological vector spaces, in  the authors introduce the concept of tvs-cone b-metric spaces over a solid cone. In the following section another state of metric spaces is presented: an algebraic cone metric space.
2 Algebraic cones and Banach cone algebras
for all , where with .
The same authors in  introduced the concept of quasicontractions on cone metric spaces over Banach algebras and proved the existence and uniqueness of fixed points of such mapping. Afterwards Xu and Radenović deleted the superfluous assumption of normality in  (see Theorems 3.1, 3.2, and 3.3 in ).
In this section, we will present the definition of the algebraic cones, Banach cone algebras, and some properties related to this concept.
Let be a cone, and X be a vector space over ℂ. Suppose the mapping satisfies:
(N1) if and only if ,
(N2) , for every ,
(N3) for every and .
Then is called a cone norm on X, and the pair is called a cone normed space (CNS).
Sometimes, to emphasize the cone , we use instead of .
the map is continuous,
the map is continuous,
the map is continuous.
When , is called a complex topological algebra and if , is called a real topological algebra.
It is obvious that every Banach algebra is a topological algebra.
Definition 2.2 Let be a Banach algebra with identity element and be a cone. is an algebraic cone if , and for each , .
Therefore, for each and if and , then .
For example if is a -algebra and is its positive elements, i.e. the set of all Hermitian element with , then is a cone in , and whenever is commutative, is an algebraic cone.
Definition 2.3 Let X be an algebra, be a real topological algebra (or as a special case be a Banach algebra), and be an algebraic cone in . Furthermore, let be a cone norm, such that is a Banach cone metric space, and for each . Then we say that is a cone Banach cone algebra.
In the following, X is an algebra with identity and is a Banach algebra with identity . Moreover, in Section 4, some examples are presented in the topological algebra case.
for every homomorphism , such that .
form a Cauchy sequence X by the normality of . Since X is complete, there exists , such that . As and , the continuity of multiplication implies that s is the inverse of .
This completes the proof. □
Note that if the condition in Theorem 2.5 is replaced by , then (a) is still true. In the next theorem, let be the group of invertible elements of X.
Theorem 2.5 If X is a Banach cone algebra on , then is open in X and the mapping is a homeomorphism.
Therefore, . Hence the mapping is continuous and homeomorphism. □
Note that for each , the spectrum of x is the set of all complex numbers λ, such that is not invertible. The complement of is the resolvent set of x. It consists of all for which there exists .
(see Theorem 10.13 in ). Since the combination with is a real-valued norm, we can present an extension of the Gelfand-Mazur theorem.
Theorem 2.6 (Gelfand-Mazur)
If X is a normal Banach cone algebra in which every nonzero element is invertible, then X is isometrically isomorphic to the complex field.
3 Fixed point theorems in algebraic cone metric spaces
In this section some fixed point theorems of generalized Lipschitz mappings with weaker conditions than the condition , are proved (see Definition 3.1 for the condition). Therefore some theorems in  and  are improved (see Theorems 3.3 and 3.9 for instance).
for each . In a similar way, we can say that F is a contraction if . In the following, we will see that this definition of contraction maps is general.
When is a complete metric space, and F is a contraction, we know that the statement ‘ if and only if ’ has an important role in the proof of Banach fixed point theorem. But in a Banach algebra , the statement ‘ if and only if ’ is not true. In fact for , does not imply that (see example (f) in Section 4). For this reason, we have to state a scholastic definition of contraction.
Definition 3.1 Let be an algebraic cone metric space, and let be a Lipschitzian map with Lipschitz coefficient L. F is a contraction map if , i.e. the series is absolutely convergent.
Let be a Lipschitz coefficient for a self-map F, then each of the properties and imply that , but example (f) of Section 4 shows that the converse cannot necessarily be established.
Lemma 3.2 Let be a Banach algebra with a unit , and . If , then is invertible and .
Proof The proof is obvious. □
Before presenting some fixed point theorems we remark that although each cone metric space over a solid cone is metrizable, as was shown by various methods, however, not all fixed point results can be reduced in this way to their standard metric counterparts. So this line of investigation is still open (see, e.g., ).
Theorem 3.3 Let be a complete algebraic cone metric space and be a contraction map with Lipschitz coefficient L. Then F has a unique fixed point .
By Lemma 1.3(b), this shows that is a Cauchy sequence and, since X is complete, there exists a with . Hence by continuity . It is obvious that u is unique. □
For each and , let and . Then the collection forms a basis for the cone metric topology of . Also it is easily seen that, for each , is an open set in cone metric space . As in the real case, the equality does not necessarily hold for all and . In the next lemma we express a necessary and sufficient condition for this equality.
For each and each , .
For each and each , there exists , such that and .
Proof (a) implies (b). If and , then . Now (a) implies that and so for each , there exists such that .
(b) implies (a). For each , if then . If , then let . So there exists such that and . Hence there exists a sequence in such that . This implies . □
Corollary 3.5 Let be an algebraic cone metric space. Then for each and each , .
Then F has a unique fixed point in .
Proof There exists a , such that , with . (Otherwise for each we have and then .) We will show that .
By Theorem 3.3, F has a unique fixed point in . Again it is easy to see that F has only one fixed point in . □
Before going to the next theorem, we note that in a Banach algebra for each with we can assume that . Indeed, if , then is invertible and . So . Now by Lemma 1.3(c), there exists a natural number N, such that .
Theorem 3.7 Let C be a nonempty, closed, and convex subset of an algebraic cone norm space X. Also let be a Lipschitzian map with Lipschitz coefficient L such that and be a subset of a compact subset of C. Then F has a fixed point.
in S. So by continuity, as in S, and therefore . □
Theorem 3.8 (See Theorem 2.1 in )
Then f and g have a unique common fixed point in X. Moreover, any fixed point of f is a fixed point of g, and vice versa.
To prove uniqueness, suppose that q is another common fixed point of f and g, then , which gives , and . □
for all , where , and are absolutely convergent. Then T has a unique fixed point in X, and for any , the iterative sequence converges to the fixed point.
Since d is continuous, so and hence .
Hence . □
Remark 3.10 In Theorem 3.9 if , then the series and show absolute convergence.
and for all , then T has a unique fixed point . Moreover, for every , .
Proof The proof is similar to Theorem 3.1 in . □
In this section, and , will denote the set of all complex-valued and the set of all continuous complex-valued functions on a topological space X, respectively. There are several natural ways of introducing a topology on and and they all follow the general approach indicated below.
- (a)Let be equipped with the pointwise topology. So in , a sequence converges to if and only if converges to for each x in X. It is obvious that under the pointwise topology is a topological algebra (see the definitions in Section 2). Define
for each and each .
for each and each .
for each .
if and only if .
is a vector-valued inner product on . Now define . It is obvious that is a cone algebraic norm on . It is obvious that . is a Banach cone algebra. For this purpose, we prove that is complete. Let be a Cauchy sequence, then for each , is a Cauchy sequence in ℂ. Therefore, there exists , such that is pointwise convergent to g. This shows that is complete.
(b) Let be a completely regular non-pseudocompact topological space. Let be equipped with pointwise convergence topology. It is obvious that is a topological subalgebra of . But is not a cone complete subalgebra of , because is not closed.
(c) Let be a completely regular non-pseudocompact topological space. Let be equipped with uniform convergence topology. A sequence converges to if and only if uniformly converges to g on X. under uniform convergence on X is not a topological algebra, because multiplication is not continuous.
for each and each .
for each and each .
for each .
if and only if .
Let with the norm and , that is, a non-normal solid cone (, p.221). We note that:
with pointwise operations is a (cone) Banach algebra.
Consider for and and let , where 0 is the constant function zero. Further, define the cone metric by . Then in the cone metric space , however, . Therefore is a Banach cone metric space with an algebraic non-normal cone.
Suppose X is as (2) and define by , and . Then is a cone metric space too and for and , since , we have in the cone metric space , but . Therefore is a Banach cone metric space with a non-continuous metric.
- (f)This example was first treated by Kaplansky for another purpose. Let be the natural orthonormal basis of . The weighted shift operators defined by
So . This implies that is convergent and .
The authors are very grateful to the anonymous referee for his or her comments and suggestions.
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