On a soft mapping and its fixed points
© Wardowski; licensee Springer 2013
Received: 15 October 2012
Accepted: 26 June 2013
Published: 11 July 2013
In the article first we introduce a new notion of soft element of a soft set and establish its natural relation with soft operations and soft objects in soft topological spaces. Next, using the notion of soft element, we define, in a different way than in the literature, a soft mapping transforming a soft set into a soft set and provide basic properties of such mappings. The new approach to soft mappings enables us to obtain the natural first fixed-point results in the soft set theory. Throughout the paper a comprehensive set of examples illustrating the discussed topics is presented.
The soft set theory, initiated by Molodtsov  in 1999, is one of the branches of mathematics, which aims to describe phenomena and concepts of an ambiguous, undefined, vague and imprecise meaning. This theory is applicable where there is no clearly defined mathematical model. Recently, many papers concerning soft sets have been published; see, e.g., [2–6]. Interesting papers merging soft set theory with fuzzy sets have also appeared; see, e.g., [7–9]. In  the authors introduced the soft set relations and many related concepts. The interesting paper is also , where the authors introduced the notion of soft topology on a soft set and proved basic properties concerning soft topological spaces. In many aspects of mathematics, fixed point theory has a tremendous application. Hence, by the present paper, we want to initiate the investigations concerning a fixed point in soft set theory for a certain class of new defined soft mappings in soft topological spaces. The author believes that the presented new results can be the inspiration for many authors.
In this article, firstly, we establish the concept of soft element of a soft set and show its natural connection with the notion of soft set and soft operations. Next, we investigate soft topological spaces and formulate the definitions and deliver some properties of a soft compact topological space. Further, on the basis of the introduced notion of soft element, we give a natural definition of a soft mapping, the image and the inverse of a soft mapping and discuss its properties. In the last, main section of this work, we define the notion of fixed point of a soft mapping and, inspired by , prove a new fixed-point result for a soft mapping in a soft compact Hausdorff topological space. The paper includes many examples illustrating presented concepts and showing the necessity of some assumptions.
Throughout the paper let ℝ be a set of all real numbers, let ℤ be a set of all integers and let ℕ be a set of all natural numbers. Denote by U an initial universe, by E a set of parameters and by the collection of all subsets of U.
where is a set-valued map such that for . is called an approximate function of . The elements of of the form will be omitted. Very often if a set of parameters A is of no importance, we will write F instead of . The collection of all soft sets on U will be denoted by .
Obviously, an element is also the element of .
Definition 2.2 An empty soft set, denoted by , is a soft set of the form .
Example 2.2 Let , . In this case the soft empty set is of the form .
Definition 2.3 A soft set is called the A-universal soft set and is denoted by if for each , i.e., .
Definition 2.4 Let . is called a soft subset of , which is denoted by , if for each .
Obviously, for each , and .
Example 2.4 Let , and let be of the form , . Then .
Definition 2.5 Let . We say that the soft sets , are equal, which is denoted by , if for all .
It is clear that if and only if and .
The basic soft operations on the soft sets are defined as follows.
The basic properties of the soft operations are described, e.g., in .
Clearly, the following equalities hold: , , .
3 Soft elements of soft sets
In this section we establish the definition of a soft element of a soft set and present its natural connection with soft sets and soft operations.
Definition 3.1 Let . We say that is a nonempty soft element of F if and . The pair , where , will be called an empty soft element of F. Nonempty soft elements of F and empty soft elements of F will be called the soft elements of F. The fact that α is a soft element of F will be denoted by .
The following proposition presents the basic properties of the soft elements.
for each nonempty soft element α.
Example 3.1 Let , . Take a soft set of the form . Then all the soft elements of F are the following: , , , .
Observe that .
4 Soft compact topological spaces
In this section we introduce the definitions and the basic properties concerning soft compact topological spaces, which will be useful in the next sections.
The foundations of the theory of soft topological spaces were given by Çağman et al. in . Let us recall the definition of a soft topological space.
If is a soft topology on F, then the pair is called a soft topological space.
The basic definitions and properties regarding soft topological spaces are also introduced in .
Using the introduced notion of soft element, the following proposition gives a natural characterization of soft open sets.
Proposition 4.1 Let be a soft topological space. A soft set is soft open if and only if for each there exists a soft set such that .
Proof Let . Then, clearly, for each we have .
Thus, . □
It is easy to verify that the family is a soft topology on G. The soft topological space is called a soft topological subspace of .
Definition 4.3 We say that a soft topological space is soft Hausdorff if for each , there exist such that and , .
Definition 4.4 Let be a soft topological space and let . The family is called a soft open cover of K if .
Definition 4.5 A soft topological space is called soft compact if for each soft open cover of K there exist , such that .
Definition 4.6 Let be a soft topological space and let . We say that the soft set K is compact in if the soft topological space is soft compact.
Proposition 4.2 Let be a soft Hausdorff topological space. Then every soft compact set in F is soft closed in F.
Proof Let K be a soft compact set in and let . For every , let be such that and , . From the soft compactness of K, there exist such that . Denote and . Then , and thus , which gives that and consequently, by Proposition 4.1, K is soft closed. □
5 Soft mappings
In this section we establish a new concept of a soft mapping, formulate related definitions and prove the properties of soft mappings.
We start with the definition of the Cartesian product and soft relation, which were firstly defined by Babitha and Sunil in . We may read them as follows.
If , then we will write .
Now, we will introduce a new definition of a soft mapping.
for each soft element , there exists only one soft element such that (which will be noted as );
for each empty soft element , is an empty soft element of G.
Remark 5.1 Observe that the above established definition of soft mapping is different from the notion of soft function introduced by Babitha and Sunil in  and also differs from the concept of soft mapping by Kharal and Ahmad in  and by Majumdar and Samanta in . Example 5.3 of a soft mapping is not a soft mapping in the sense introduced by the mentioned authors. This new approach, strictly related with the introduced notion of soft element, enables us to obtain a natural (similar to classical mappings) behavior of soft mappings. Please compare the rest of this section. Moreover, this type of soft mappings will play a fundamental role in obtaining a fixed-point theorem in the last section.
It is clear that for each soft mapping T.
Remark 5.2 Let us observe that the condition (SM2) of Definition 5.3 implies that the inverse under soft mapping T in Definition 5.5 is well defined. In particular, the inverse of the empty soft set under T is always a soft subset of F.
Now, we present the basic properties of soft mappings. The proofs are simple and hence omitted.
The formulae (5)-(8) are also true for the generalized soft operations and .
Definition 5.6 Let , be soft topological spaces and let be a soft mapping. We say that T is a soft continuous mapping (with respect to the soft topologies and ) if for each , (i.e., the inverse of a soft open set is a soft open set).
Then is a soft topological space (see Example 3 in ).
we obtain that for , and thus is not a soft continuous mapping.
then it is easy to verify that for each and hence is a soft continuous mapping.
Proposition 5.2 Let be a soft compact topological space and let be a soft continuous mapping. Then is a soft compact set in .
Thus, is a soft compact set. □
6 Fixed points of soft mappings - the main result
We start this section from the definition of a fixed point of a soft mapping.
Definition 6.1 Let be a soft set and let be a soft mapping. A soft element is called a fixed point of T if .
Example 6.1 Let be a soft mapping defined in Example 5.5. Then the soft elements , are the fixed points of .
for each ,
is soft closed for each ,
for each .
which, due to (1), is impossible. □
Proposition 6.2 Let be a soft topological space and let be a soft mapping such that for each nonempty soft element , is a nonempty soft element of F. If contains only one nonempty soft element , then α is a unique fixed point of T.
Since is a nonempty soft element, we obtain that . □
Then , but T is a fixed point-free soft mapping.
for each nonempty soft element , is a nonempty soft element of K,
for each soft closed set , if , then X contains only one nonempty soft element of K.
Then there exists a unique nonempty soft element such that .
Proof Consider a family of soft subsets of K of the form , , … , , , … . It is clear that for each . By Proposition 4.2, for each , is soft closed and, due to Proposition 6.1, a soft set D of the form is nonempty.
In order to show that , suppose that there exists such that . Denote . Let us observe that and for each . By Proposition 6.1, there exists a nonempty soft element and thus , which is a contradiction. In consequence, that, by (2) and Proposition 6.2, completes the proof. □
The assumption about the soft compactness in Theorem 6.1 is not superfluous. To see this, consider the following example.
Then T is obviously a soft continuous mapping, for any and, clearly, T does not have a fixed point.
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