Abstract
In this paper we introduce the concept of cone metric spaces with Banach algebras, replacing Banach spaces by Banach algebras as the underlying spaces of cone metric spaces. With this modification, we shall prove some fixed point theorems of generalized Lipschitz mappings with weaker conditions on generalized Lipschitz constants. An example shows that our main results concerning the fixed point theorems in the setting of cone metric spaces with Banach algebras are more useful than the standard results in cone metric spaces presented in the literature.
MSC:54H25, 47H10.
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1 Introduction
Cone metric spaces were introduced by Huang and Zhang as a generalization of metric spaces in [1]. The distance of two elements x and y in a cone metric space X is defined to be a vector in an ordered Banach space E, and a mapping is said to be contractive if there is a constant such that
The right-hand side of inequality (1) is the vector as the result of the operation of scalar multiplication in cone metric spaces. In [1], the authors proved that there exists a unique fixed point for contractive mappings in complete cone metric spaces. Recently, scholars obtained that any cone metric space is equivalent to the usual metric space , where the real-valued metric function is defined by a nonlinear scalarization function . See, for instance, [2, 3] and [4]. In particular, for each contractive mapping T in satisfying (1), one can get
which implies that cone metric spaces are a special case of classical metric spaces. After that, some other interesting generalizations were developed. See, for instance, [5].
In this paper, we replace the Banach space E by a Banach algebra A and obtain the concept of cone metric spaces with Banach algebras. In this way, we shall prove some fixed point theorems of generalized Lipschitz mappings with weaker and natural conditions on the Lipschitz constant k. Our results generalize metric spaces and reveal the fact that the essential conditions on the contraction constant k are neither order relations nor norm relations, but spectrum radius.
Let A always be a real Banach algebra. That is, A is a real Banach space in which an operation of multiplication is defined, subject to the following properties (for all , ):
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1.
;
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2.
and ;
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3.
;
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4.
.
In this paper, we shall assume that a Banach algebra has a unit (i.e., a multiplicative identity) e such that for all . An element is said to be invertible if there is an inverse element such that . The inverse of x is denoted by . For more details, we refer to [6].
The following proposition is well known (see [6]).
Proposition 1.1 Let A be a Banach algebra with a unit e, and . If the spectral radius of x is less than 1, i.e.,
then is invertible. Actually,
A subset P of A is called a cone if
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1.
P is non-empty closed and ;
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2.
for all non-negative real numbers α, β;
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3.
;
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4.
.
For a given cone , we can define a partial ordering ⩽ with respect to P by if and only if . will stand for and . While will stand for , where intP denotes the interior of P.
The cone P is called normal if there is a number such that for all ,
The least positive number satisfying the above is called the normal constant of P [1].
In the following we always assume that P is a cone in A with and ⩽ is the partial ordering with respect to P.
Definition 1.1 (See [1])
Let X be a non-empty set. Suppose that the mapping satisfies
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1.
for all and if and only if ;
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2.
for all ;
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3.
for all .
Then d is called a cone metric on X, and is called a cone metric space (with a Banach algebra A).
We present some examples in the following.
Example 1.1 Let be the algebra of all n-square real matrices, and define the norm
Then A is a real Banach algebra with the unit e, the identity matrix.
Let . Then is a normal cone with a normal constant .
Let , and define the metric by
Then is a cone metric space with a Banach algebra A.
Example 1.2 Let A be the Banach space of all continuous real-valued functions on a compact Hausdorff topological space K, with multiplication defined pointwise. Then A is a Banach algebra, and the constant function is the unit of A.
Let . Then is a normal cone with a normal constant .
Let with the metric mapping defined by
Then is a cone metric space with a Banach algebra A.
Example 1.3 Let with convolution as multiplication:
Thus A is a Banach algebra. The unit e is .
Let , which is a normal cone in A.
And let with the metric defined by
Then is a cone metric space with A.
Definition 1.2 (See [1])
Let be a cone metric space, and a sequence in X. Then
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1.
converges to x whenever for each with , there is a natural number N such that for all . We denote this by or .
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2.
is a Cauchy sequence whenever for each with , there is a natural number N such that for all .
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3.
is a complete cone metric space if every Cauchy sequence is convergent.
Finally, we shall appeal to the following lemmas in the sequel [1].
Lemma 1.1 Let be a cone metric space, P be a normal cone with a normal constant M. Let be a sequence in X. Then converges to x if and only if ().
Lemma 1.2 Let be a cone metric space, P be a normal cone with a normal constant M. Let be a sequence in X. Then is a Cauchy sequence if and only if ().
2 Main results
In this section we shall prove some fixed point theorems of generalized Lipschitz mappings in the setting of cone metric spaces with Banach algebras.
Theorem 2.1 Let be a complete cone metric space and P be a normal cone with a normal constant M. Suppose that the mapping satisfies the generalized Lipschitz condition
where with . Then T has a unique fixed point in X. And for any , iterative sequence converges to the fixed point.
Proof Choose and set , . We have
Thus, for ,
Since P is normal with a normal constant M, and note that (), we have
Hence is a Cauchy sequence. By the completeness of X, there exists such that (). Furthermore, one has
and consequently,
Hence . This implies . So, is a fixed point of T.
Now if is another fixed point of T, then
That is,
Multiplying both sides above by
we get . Thus , which implies that , a contradiction. Hence, the fixed point is unique. □
Remark 2.1 Note that in Theorem 2.1 we only suppose that the spectral radius of k is less than 1, neither nor assumed. This is a vital improvement. In fact, the condition is weaker than that , as is illustrated by Example 2.1 in the sequel. The improvement of the condition about the generalized Lipschitz constant k shows that it is meaningful to introduce the concepts of cone metric spaces with Banach algebras and a generalized Lipschitz condition.
Theorem 2.2 Let be a complete cone metric space, P be a normal cone with a normal constant M. Suppose that the mapping satisfies the generalized Lipschitz condition
where with . Then T has a unique fixed point in X. And for any , iterative sequence converges to the fixed point.
Proof Choose , and set , . We have
That is,
We shall prove that
which implies that is a Cauchy sequence by the proof of Theorem 2.1. Note that and k commute.
Let n be large enough such that
where such that .
Denote that
where , . It is easy to see that for all .
Then
Hence
and is a Cauchy sequence.
By the completeness of X, there is such that (). To verify , we have
That is,
By the normality of P, we have
as . Hence is a fixed point of T.
Now if is another fixed point, then
Thus
for any . Since , we have
Then , which implies that , a contradiction. Hence, the fixed point is unique. □
Theorem 2.3 Let be a complete cone metric space, P be a normal cone with a normal constant M. Suppose that the mapping satisfies the generalized Lipschitz condition
where with . Then T has a unique fixed point in X. And for any , the iterative sequence converges to the fixed point.
Proof Choose , and set , . We have
That is,
As is shown in the proof of Theorem 2.2, it follows that is a Cauchy sequence, and, by the completeness of X, the limit of exists and is denoted by .
To see that is a fixed point of T, we have
Therefore,
and
as .
Now if is another fixed point of T, then
which implies that , a contradiction. Hence, the fixed point is unique. □
We conclude the paper with an example.
Example 2.1 Let . For each , . The multiplication is defined by
Then A is a Banach algebra with unit .
Let . Then P is normal with a normal constant .
Let and the metric d be defined by
Then is a complete cone metric space.
Now define the mapping by
where α can be any large positive real number.
From Lagrange mean value theorem, we have
and
Then, by Theorem 2.1, T has a unique fixed point in X.
Remark 2.2 In Example 2.1 above, we see that and (for ). Moreover, T is not a contractive mapping in the Euclidean metric on X. Hence, Example 2.1 shows that the main results in this paper are more powerful than the standard results of cone metric spaces presented in the literature.
Remark 2.3 Example 2.1 also shows that one is unable to conclude that the cone metric space with a Banach algebra A defined above is equivalent to the metric space , where the metric is defined by ; here, the nonlinear scalarization function () is defined by
See [2, 3] and [4] for more details. In fact, under this situation, we have
For and ,
and for ,
Now let the mapping be defined as in (3) with , and consider , . We have
which implies that T is not a contraction in the metric space . This shows that one is unable to prove that Theorem 2.1 above is a consequence of the corresponding results in metric spaces by means of the methods presented in the literature.
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Acknowledgements
The authors are grateful to the referees and the editors for valuable comments and suggestions, which have improved the original manuscript greatly. The research is partially supported by the PhD Start-up Fund of Hanshan Normal University, Guangdong Province, China (No. QD20110920).
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Liu, H., Xu, S. Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings. Fixed Point Theory Appl 2013, 320 (2013). https://doi.org/10.1186/1687-1812-2013-320
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DOI: https://doi.org/10.1186/1687-1812-2013-320