Fixed-point theorems for mappings satisfying the ordered contractive condition on noncommutative spaces
© Xin and Jiang; licensee Springer. 2014
Received: 10 September 2013
Accepted: 21 January 2014
Published: 7 February 2014
In the paper, we introduce noncommutative Banach spaces which generalize the concept of Banach spaces, and the k-ordered contractive condition; we then discuss an ordered structure and several properties on noncommutative Banach spaces. Moreover, some fixed-point theorems for mappings with the k-ordered contractive condition on noncommutative Banach spaces are presented. In addition, we investigate the existence and uniqueness of fixed points for an integral equation of Fredholm type.
The well-known fixed-point theorem of Banach  is a very important tool for solving existence problems in many branches of mathematics and physics. There are a large number of generalizations of the Banach contraction principle in the literature (see [2–14] and others). The theorem has been generalized in two directions. On the one side, the usual contractive condition is replaced by weakly contractive conditions. On the other side, the action spaces are replaced by metric spaces endowed with an ordered or partially ordered structure. In particular, there is much interest in obtaining the existence and uniqueness of fixed points for self-maps by altering the action spaces. In this direction, Dhage et al.  addressed a new category of fixed-point problems for a self-map with the help of ordered Banach spaces. Further improvements in those spaces were found in . In recent years, Ran and Reurings , O’Regan and Petruşel  and others started the investigations concerning a fixed-point theory in ordered metric spaces. Later, many authors followed this concept by introducing and investigating the different types of contractive mappings, e.g., in  Caballero et al. considered contractive-like mappings in ordered metric spaces and applied their results in ordinary differential equations. Some interesting fixed-point theorems concerning partially ordered metric spaces can also be found in [3, 5].
The results obtained by Huang and Zhang  have become of interest for many scholars. They reconsidered the Banach contraction principle by initiating a new concept of cone metric spaces. Recently, also, the existence of fixed points for the given contractive type mappings in partially ordered cone metric spaces was investigated (see [4, 10]).
The purpose of this paper is to present some fixed-point theorems for mappings satisfying the ordered contractive condition in the context of noncommutative metric spaces which are noncommutative sense of those in .
The paper is organized as follows: we firstly introduce a noncommutative Banach space and the k-ordered contractive condition, and then discuss the ordered structure and several properties on noncommutative Banach spaces. Moreover, some fixed-point theorems on this space are established. Finally, we investigate the existence and uniqueness of fixed points for integral equation of Fredholm type.
Throughout this paper, the letters ℝ, , ℕ will denote the sets of all real numbers, nonnegative real numbers and natural numbers, respectively.
To begin with, we introduce some definitions and properties which will be used later.
for any , we have ;
- (2)there exists a binary continuous operation
- (3)for any , there exists a constant such that
In particular, if there exists a constant such that for , , then E is said to be uniformly bounded.
Let E be a uniformly bounded noncommutative Banach space. Taking , we conclude that , which together with the triangular inequality yields . This shows E is bounded.
Example 1.1 All Banach spaces are noncommutative Banach spaces. Let be a Banach space, then is a group with a unit θ, and there exists a metric d induced by the norm such that is a complete metric space. Firstly, the metric d satisfies for . Secondly, there exists a binary continuous mapping , satisfying the condition (2) in Definition 1.1. Finally, for any , there exists a constant such that , . According to the definition, X is a noncommutative Banach space.
for . Clearly, is a complete metric space. In order to verify that is a noncommutative Banach space. It suffices to show that for any , there exists a constant such that , for , where θ is unit in . Indeed, choose , and we get .
for . Then is a uniformly bounded noncommutative Banach space.
P is nonempty, closed, and ;
and implies ;
Then P is called a cone in E.
For , for all . This implies that .
If and , then and for all . By , we get , which implies that .
If and , then and for all , which together with the condition (2) in Definition 1.2 can infer . This shows that .
The least positive number N satisfying the above is called the normal constant of P. It is clear that .
Indeed, for any , if , then since P is a cone, thus we can get ; if , then , which means .
From the above definitions, we have the following properties.
Set , then holds for any .
If u and v are comparable, then and are comparable, and furthermore .
If u and v are comparable, then .
(compatibility) Let , and be comparable for all . If , , then and are comparable.
Proof (1) Let , we have for all . Since for any , we see , which implies that .
(2) Without loss of generality, one can suppose that , which means . Using Remark 1.1, one can see . Furthermore for all , which implies , and therefore .
As P is closed, one obtains . This says that . □
2 Fixed-point theorems on noncommutative Banach spaces
From now on, we always suppose that E is a noncommutative Banach space with a partial ordering ≲ induced by a normal cone P with the normal constant N. And some fixed-point theorems for mappings on E satisfying the ordered contractive condition will be presented. Let us begin with the following theorem.
- (1)There exists a constant such that for all , if u and v are comparable, then Au and Av are comparable and furthermore
In this case, we say A satisfies the k-ordered contractive condition.
There exists such that and are comparable.
Proof Define a sequence by the formula , . The proof can be divided into three steps.
Step I. is a Cauchy sequence.
which shows is a Cauchy sequence. The completeness of E implies that there exists such that .
Step II. is a fixed point of A.
which shows is a fixed point of A.
Step III. The uniqueness of the fixed point of A in the comparable sense.
In addition, we know . Then , which implies . On the other hand, we have . Now, from the definition of a cone, we have , and then .
For , if they are not comparable, then there exists such that u and w, v and w are comparable, respectively.
Then A has a unique fixed point. Note that condition (3) is always valid if E is a lattice.
as , which implies . This is a contradiction. Therefore, and are comparable. By Theorem 2.1, .
Corollary 2.1 Let E be a uniformly bounded noncommutative Banach space, P a normal cone with the normal constant N. For , , set . Suppose that a continuous mapping satisfies the k-ordered contractive condition and . Also, and are comparable. Then there exists a unique fixed point in in the comparable sense.
Proof It suffices to show that for any .
- (1)there exist and such that for all , if u and v are comparable, then Au and Av are comparable and furthermore
there exists such that and are comparable.
Then A admits a unique fixed point in the comparable sense.
which means is also a fixed point of . Again, since and are comparable, then and are comparable, which implies . Since the fixed point of A is also the fixed point of , the fixed point of A is unique in the comparable sense. □
- (1)there exists a constant such that for all , if u and v are comparable, then Au and Av, Au and u, Av and v are comparable, and furthermore
A is continuous.
As in the proof of Theorem 2.1, is a Cauchy sequence and there exists such that . Also, is a fixed point of A.
It remains to be shown that is a unique fixed point of A.
Since . Then , which implies . Now, applying Definition 1.2, we obtain the desired result.
Let , and . By from now on we denote the order interval . Finally, we consider the fixed-point theorem in the order interval. □
Theorem 2.3 Let be the order interval in E. If satisfies the conditions in Theorem 2.2, then A has a unique fixed point in the comparable sense.
Hence, we conclude that is a Cauchy sequence with a limit point for any .
Similarly, is a Cauchy sequence with a limit point for any .
The rest of the proof is analogue to that in Theorem 2.2. □
If one checks the proof of Theorem 2.3, then one can easily obtain the following result.
Corollary 2.3 Let be the order interval in E. If the continuous mapping satisfies the k-ordered contractive condition, then A admits a unique fixed point in the comparable sense.
We give some examples to illustrate the main result of this paper in the following.
where , . Choose such that . Clearly, Ax and Ay are comparable if x and y are comparable. Moreover, if , then . By Theorem 2.1 and Remark 2.1, we know that A has a unique fixed point.
and are continuous;
is monotonous for ;
- (3)there exist a continuous function and such that
there exists such that for any , or .
Then the integral equation has a unique solution in .
for , which implies that . Thus, A is continuous. Again, E is a lattice, by Theorem 2.1 and Remark 2.1, the integral equation has a unique solution in .
for and .
Notice that the examples given above are in linear spaces. As to the noncommutative case, it is under consideration now.
The authors would like to thank the referees for their many valuable suggestions that have greatly contributed to improve the quality of this paper. This work is supported financially by the NSFC (10971011, 11371222).
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