On the fixed point theory of soft metric spaces
- Mujahid Abbas^{1, 2},
- Ghulam Murtaza^{3} and
- Salvador Romaguera^{4}Email author
https://doi.org/10.1186/s13663-016-0502-y
© Abbas et al. 2016
Received: 25 August 2015
Accepted: 14 January 2016
Published: 28 January 2016
Abstract
The aim of this paper is to show that a soft metric induces a compatible metric on the collection of all soft points of the absolute soft set, when the set of parameters is a finite set. We then show that soft metric extensions of several important fixed point theorems for metric spaces can be directly deduced from comparable existing results. We also present some examples to validate and illustrate our approach.
Keywords
soft mapping soft metric space soft contraction soft Caristi mapping1 Introduction
It is a usual practice to use mathematical tools to study the behavior of different aspects of a system and its different subsystems. So it is very natural to deal with uncertainties and imprecise data in various situations. Different kind of difficulties arise in dealing with the uncertainties and imprecision either already existing in the data or due to the mathematical tools used to solve the model featuring various situations. Fuzzy set theory, initiated by Zadeh [1], has evolved as an important tool to solve the issues of uncertainties and ambiguities. Theories such as probability theory and rough set theory have been introduced by mathematicians and computer scientists to handle the problems associated with the uncertainties and imprecision of real world models. The contribution made by probability theory, fuzzy set theory, vague sets, rough sets, and interval mathematics to deal with uncertainty is of vital importance but these theories have their own limitations. To overcome these peculiarities, in 1999, Molodtsov introduced in [2] soft sets as a mathematical tool to handle the uncertainty associated with real world data based problems. It provides enough tools to deal with uncertainty in a data and to represent it in a useful way. The problem of inadequacy of parameters has been successfully resolved by this theory. Now it has become a full-fledged research area and has attracted the attention of several mathematicians, economists, and computer scientists [3–21]. The distinguishing attribute of soft set theory is that unlike probability and fuzzy set theory, it does not uphold a precise quantity.
A vast amount of mathematical activity has been carried out to obtain many remarkable results showing the applicability of soft set theory in decision making, demand analysis, forecasting, information science, mathematics, and other disciplines (see for detailed survey [22–31]).
The notion of a soft topology on a soft set was introduced by Cagman et al. [32] and some basic properties of soft topological spaces were studied.
Das and Samanta introduced in [33] the notions of soft real set and soft real number, and discussed their properties. Based on these notions, they introduced in [34] the concept of a soft metric. They showed that each soft metric space is also a soft topological space. Abbas et al. [35] introduced the notion of soft contraction mapping based on the theory of soft elements of soft metric spaces. They studied fixed points of soft contraction mappings and obtained among others results, a soft Banach contraction principle. Almost simultaneously, Chen and Lin [36] obtained a soft metric version of the celebrated Meir-Keeler fixed point theorem. Here we show that, under some restriction, each (complete) soft metric induces a (complete) usual metric, and we deduce in a direct way soft metric versions of several important fixed point theorems for metric spaces, as the Banach contraction principle, Kannan and Meir-Keeler fixed point theorems, and Caristi-Kirk’s theorem. By means of appropriate examples we also show that the aforementioned restriction is essential.
2 Preliminaries
In the sequel, the letters U, E, and \(P(U)\) will denote the universal set, the set of parameters, and the power set of U, respectively.
According to [2] if F is a set valued mapping on \(A\subset E\) taking values in \(P(U)\), then a pair \((F,A)\) is called a soft set over U. We denote the collection of soft sets over a common universe U by \(S(U)\).
Basic notions and properties related to soft set theory may be found in [2, 34, 37, 38]. In particular, a soft set \((F,A)\) over U is said to be a soft point if there is exactly one \(\lambda \in A\) such that \(F(\lambda )=\{x\}\) for some \(x\in U\) and \(F(e)=\emptyset \), for all \(e\in A\setminus \{\lambda \}\). We shall denote such a soft point by \((F_{\lambda }^{x},A)\) or simply by \(F_{\lambda }^{x}\). A soft point \(F_{\lambda }^{x}\) is said to belong to \((F,A)\), denoted by \(F_{\lambda }^{x}\mathbin{\tilde{\in}}(F,A)\), if \(F_{\lambda }^{x}(\lambda )=\{x\}\subset F(\lambda )\).
The collection of all soft points of \((F,A)\) is denoted by \(\operatorname {SP}(F,A)\).
Now let \(\mathbb{R}\) be the set of real numbers. We denote the collection of all nonempty bounded subsets of \(\mathbb{R}\) by \(B(\mathbb{R})\).
Definition 1
[33]
A soft real set denoted by \((\widehat{f},A)\), or simply by f̂, is a mapping \(\widehat{f}:A\rightarrow B(\mathbb{R})\). If f̂ is a single valued mapping on \(A\subset E\) taking values in \(\mathbb{R}\), then the pair \((\widehat{f},A)\) or simply f̂, is called a soft element of \(\mathbb{R}\) or a soft real number. If f̂ is a single valued mapping on \(A\subset E\) taking values in the set \(\mathbb{R}^{+}\) of nonnegative real numbers, then a pair \((\widehat{f},A)\), or simply f̂, is called a nonnegative soft real number. We shall denote the set of nonnegative soft real numbers (corresponding to A) by \(\mathbb{R}(A)^{*}\). A constant soft real number c̅ is a soft real number such that for each \(e\in A\), we have \(\overline{c}(e)=c\), where c is some real number.
Definition 2
[34]
- (i)
\(\widehat{f}\mathbin{\tilde{\leq}}\widehat{g}\) if \(\widehat{f}(e)\leq \widehat{g}(e)\), for all \(e\in A\),
- (ii)
\(\widehat{f}\mathbin{\tilde{\geq}}\widehat{g}\) if \(\widehat{f}(e)\geq \widehat{g}(e)\), for all \(e\in A\),
- (iii)
\(\widehat{f}\mathbin{\tilde{<}}\widehat{g}\) if \(\widehat{f}(e)<\widehat{g}(e)\), for all \(e\in A\), and
- (iv)
\(\widehat{f}\mathbin{\tilde{>}}\widehat{g}\) if \(\widehat{f}(e)>\widehat{g}(e)\), for all \(e\in A\).
Remark 1
The notion of a soft mapping may be found in [35, 39]. Recall that if f is a soft mapping from a soft set \((F,A)\) to a soft set \((G,B)\) (denoted by \(f:(F,A)\mathbin{\tilde{\rightarrow}}(G,B)\)), then for each soft point \(F_{\lambda }^{x}\mathbin{\tilde{\in}}(F,A)\) there exists only one soft point \(G_{\mu }^{y}\mathbin{\tilde{\in}}(G,B)\) such that \(f(F_{\lambda }^{x})=G_{\mu }^{y}\).
The definition of a soft metric introduced in [34] is given below.
Definition 3
[34]
- M1.
\(d(U_{\lambda }^{x},U_{\mu }^{y})\mathbin{\tilde{\geq}}\bar{0}\).
- M2.
\(d(U_{\lambda }^{x},U_{\mu }^{y})=\bar{0}\) if and only if \(U_{\lambda }^{x}=U_{\mu }^{y}\).
- M3.
\(d(U_{\lambda }^{x},U_{\mu }^{y})=d(U_{\mu }^{x},U_{\lambda }^{y})\).
- M4.
\(d(U_{\lambda }^{x},U_{\gamma }^{z})\mathbin{\tilde{\leq}}d(U_{\lambda }^{x},U_{\mu }^{y})+d(U_{\mu }^{y},U_{\gamma }^{z})\).
The soft set Ũ endowed with a soft metric d is called a soft metric space and is denoted by \((\tilde{U},d,A)\), or simply by \((\tilde{U},d) \) if no confusion arises.
See [34] for several basic properties of the structure of soft metric spaces. In order to help the reader we recall the following notions, which will be used later on.
Given a soft metric space \((\tilde{U},d)\), a net \(\{U_{\lambda _{\alpha}}^{x_{\alpha }}\}_{\alpha \in \Lambda }\) of soft points in Ũ will be simply denoted by \(\{U_{\lambda ,\alpha }^{x}\}_{\alpha \in \Lambda }\). In particular, a sequence \(\{U_{\lambda _{n}}^{x_{n}}\}_{n\in \Bbb{N}}\) of soft points in Ũ will be denoted by \(\{U_{\lambda ,n}^{x}\}_{n}\).
Definition 4
[34]
Let \((\tilde{U},d)\) be a soft metric space. A sequence \(\{U_{\lambda ,n}^{x}\}_{n}\) of soft points in Ũ is said to be convergent in \((\tilde{U},d)\) if there is a soft point \(U_{\mu }^{y}\mathbin{\tilde{\in}}\tilde{U}\) such that \(d(U_{\lambda ,n}^{x},U_{\mu }^{y})\rightarrow \bar{0}\) as\(\ n\rightarrow \infty \). This means that, for every \(\widehat{\varepsilon }\mathbin{\tilde{>}}\bar{0}\), chosen arbitrary, there exists an \(m\in \mathbb{N}\) such that \(d(U_{\lambda ,n}^{x},U_{\mu }^{y})\mathbin{\tilde{<}}\widehat{\varepsilon }\), whenever \(n\geq m\).
Proposition 1
[34]
The limit of a sequence \(\{U_{\lambda ,n}^{x}\}_{n}\) in a soft metric space \((\tilde{U},d)\), if it exists, is unique.
Definition 5
[34]
A sequence \(\{U_{\lambda ,n}^{x}\}_{n}\) of soft points in a soft metric space \((\tilde{U},d)\) is said to be a Cauchy sequence in \((\tilde{U},d)\) if, for each \(\widehat{\varepsilon }\mathbin{\tilde{>}}\bar{0}\), there exists an \(m\in \mathbb{N}\) such that \(d(U_{\lambda ,i}^{x},U_{\lambda ,j}^{x})\mathbin{\tilde{<}}\widehat{\varepsilon }\), for all \(i,j\geq m\). That is, \(d(U_{\lambda ,i}^{x},U_{\lambda ,j}^{x})\rightarrow \bar{0}\) as \(i,j\rightarrow \infty \).
Proposition 2
[34]
Every convergent sequence \(\{U_{\lambda ,n}^{x}\}_{n}\) in a soft metric space \((\tilde{U},d)\) is a Cauchy sequence.
Definition 6
[34]
A soft metric space \((\tilde{U},d) \) is called complete if every Cauchy sequence in \((\tilde{U},d)\) converges to some soft point of Ũ. In this case, we say that the soft metric d is complete.
3 Soft metrics inducing compatible metrics
In this short section we show that if \((\tilde{U},d,A)\) is a soft metric space with A a (nonempty) finite set, then d induces in a natural way a compatible metric on \(\operatorname {SP}(\tilde{U})\). This fact will be crucial in establishing our main results in Section 4.
Theorem 1
- (1)
\(m_{d}\) is a metric on \(\operatorname {SP}(\tilde{U})\).
- (2)
For any sequence \(\{U_{\lambda ,n}^{x}\}_{n}\) of soft points and a soft point \(U_{\lambda }^{y}\), we have
- (2a)
\(\{U_{\lambda ,n}^{x}\}_{n}\) is a Cauchy sequence in \((\tilde{U},d,A)\) if and only if it is a Cauchy sequence in \((\operatorname {SP}(\tilde{U}),m_{d})\).
- (2b)
\(d(U_{\lambda }^{y},U_{\lambda ,n}^{x})\rightarrow \overline{0}\) if and only if \(m_{d}(U_{\lambda }^{y},U_{\lambda ,n}^{x})\rightarrow 0\).
- (3)
\((\tilde{U},d,A)\) is complete if and only if \((\operatorname {SP}(\tilde{U}),m_{d})\) is complete.
Proof
- (i)
\(m_{d}(U_{\lambda }^{x},U_{\mu }^{y})\geq 0\), by condition M1 of Definition 3.
- (ii)
\(m_{d}(U_{\lambda }^{x},U_{\mu }^{y})=0\Leftrightarrow U_{\lambda }^{x}=U_{\mu }^{y}\), by condition M2 of Definition 3.
- (iii)
\(m_{d}(U_{\lambda }^{x},U_{\mu }^{y})=m_{d}(U_{\mu }^{y},U_{\lambda }^{x})\), by condition M3 of Definition 3.
- (iv)\(m_{d}(U_{\lambda }^{x},U_{\mu }^{y})\leq m_{d}(U_{\lambda }^{x},U_{\upsilon }^{z})+m_{d}(U_{\mu }^{y},U_{\upsilon }^{z})\), by condition M4 of Definition 3. Indeed, we have$$\begin{aligned} m_{d}\bigl(U_{\lambda }^{x},U_{\mu }^{y} \bigr) =&\max_{\eta \in A}d\bigl(U_{\lambda }^{x},U_{\mu }^{y} \bigr) (\eta )\leq \max_{\eta \in A}d\bigl(U_{\lambda }^{x},U_{\upsilon }^{z} \bigr) (\eta )+\max_{\eta \in A}d\bigl(U_{\upsilon }^{z},U_{\mu }^{y} \bigr) (\eta ) \\ =&m_{d}\bigl(U_{\lambda }^{x},U_{\upsilon }^{z} \bigr)+m_{d}\bigl(U_{\mu }^{y},U_{\upsilon }^{z} \bigr). \end{aligned}$$
(2a) Let \(\{U_{\lambda ,n}^{x}\}_{n}\) be a Cauchy sequence in \((\tilde{U},d,A)\). Given \(\varepsilon >0\), take the constant soft real number \(\overline{\varepsilon }\mathbin{\tilde{>}}\bar{0}\). Then there exists an \(m\in \mathbb{N}\) such that \(d(U_{\lambda ,i}^{x},U_{\lambda ,j}^{x})\mathbin{\tilde{<}}\overline{\varepsilon }\) for all \(i,j\geq m\). Hence \(d(U_{\lambda ,i}^{x},U_{\lambda ,j}^{x})(\eta )<\varepsilon \) for all \(\eta \in A\) and \(i,j\geq m\). Thus \(m_{d}(U_{\lambda ,i}^{x},U_{\lambda ,j}^{x})<\varepsilon \) for all \(i,j\geq m\). We deduce that \(\{U_{\lambda ,n}^{x}\}_{n}\) is a Cauchy sequence in \((m_{d},\operatorname {SP}(\tilde{U}))\).
Conversely, let \(\{U_{\lambda ,n}^{x}\}_{n}\) be a Cauchy sequence in \((m_{d},\operatorname {SP}(\tilde{U}))\). Given \(\widehat{\varepsilon }\mathbin{\tilde{>}}\bar{0}\), there exists \(\varepsilon =\min_{\eta \in A}\widehat{\varepsilon }(\eta )>0\), because A is a finite set. Then there exists an \(m\in \mathbb{N}\) such that \(m_{d}(U_{\lambda ,i}^{x},U_{\lambda ,j}^{x})<\varepsilon \) for all \(i,j\geq m\). Hence \(d(U_{\lambda ,i}^{x},U_{\lambda ,j}^{x})(\eta )<\varepsilon \leq \widehat{\varepsilon }(\eta )\) for all \(\eta \in A\) and \(i,j\geq m\). We deduce that \(\{U_{\lambda ,n}^{x}\}_{n}\) is a Cauchy sequence in \((\tilde{U},d,A)\).
Feature (2b) follows from (2a) and (3) is a consequence of (2a) and (2b). □
4 Fixed point theorems for complete soft metric spaces
In [35] we established a Banach contraction principle for those complete soft metric spaces \((\tilde{U},d,A)\) such that A is a (nonempty) finite set, and showed that the condition that A is finite cannot be omitted. We start this section by applying Theorem 1 to deduce the soft version of Banach’s contraction principle cited above.
Theorem 2
[35]
Proof
Consider the metric \(m_{d}\) on \(\operatorname {SP}(\tilde{U})\) as constructed in Theorem 1. Since \((\tilde{U},d,A)\) is complete it follows from Theorem 1(3) that \((\operatorname {SP}(\tilde{U}),m_{d})\) is a complete metric space.
Hence f has a unique fixed point by the Banach contraction principle. □
Remark 2
Example 3.22 of [35] shows that condition ‘A is a finite set’ cannot be omitted in the above theorem. In fact, it shows that ‘A is a finite set’ cannot be replaced with ‘A is a countable set’.
Our next result provides a soft metric generalization of the celebrated Kannan fixed point theorem [40].
Theorem 3
Proof
Since \((\tilde{U},d,A)\) is complete it follows from Theorem 1(3) that \((\operatorname {SP}(\tilde{U}),m_{d})\) is a complete metric space.
Hence f has a unique fixed point by Kannan’s fixed point theorem. □
The following modification of [35], Example 3.22, shows that condition ‘A is a finite set’ cannot be omitted in the preceding theorem (compare Remark 2).
Example 1
Now we present an example where we can apply Theorem 3 but not Theorem 2.
Example 2
Let \(f:\tilde{U}\mathbin{\tilde{\rightarrow}}\tilde{U}\) such that \(f(U_{0}^{x})=f(U_{1}^{x})=U_{0}^{0}\) if \(x\in [0,2)\), and \(f(U_{0}^{x})=f(U_{1}^{x})=U_{0}^{1/2}\) if \(x\in [2,\infty )\).
Meir and Keeler proved in [41] their well-known fixed point theorem: every Meir-Keeler contractive self mapping on a complete metric space has a unique fixed point, where a self mapping T on a metric space \((X,d)\) is said to be a Meir-Keeler contractive mapping if it satisfies the following condition:
In a recent paper [36], Chen and Lin discussed the extension of the Meir and Keeler fixed point theorem to soft metric spaces. To this end, they introduced the following notion [36], Definition 15:
Let \((\tilde{U},d,A)\) be a soft metric space and let \(\varphi :A\rightarrow A\). A soft mapping \((f,\varphi ):\tilde{U}\mathbin{\tilde{\rightarrow}}\tilde{U}\) is called a soft Keir-Meeler contractive mapping if it satisfies the following condition:
Then Chen and Lin [36], Theorem 1, established that every soft Keir-Meeler contractive mapping on a complete soft metric space has a unique fixed point.
The following examples show that this result is not correct-even for the case that A is a finite set (the error in the proof seems to occur on page 4, lines 18-19: compare Definition 2 above).
Example 3
Since \(m_{d}\) is the discrete metric on \(\operatorname {SP}(\tilde{U})\) it follows from Theorem 1(3) that \((\tilde{U},d)\) is a complete soft metric space.
For \(f:U\rightarrow U\), it necessarily follows that \(f(2)=2\). Let \(\varphi :A\rightarrow A\) given by \(\varphi (0)=1\) and \(\varphi (1)=0\). Then \((f,\varphi )(U_{0}^{2})=U_{1}^{2}\) and \((f,\varphi )(U_{1}^{2})=U_{0}^{2}\). From the fact that for each \(\widehat{\varepsilon }\mathbin{\tilde{>}}\bar{0}\) we have \(d(U_{0}^{2},U_{1}^{2})(0)=0<\widehat{\varepsilon }(0)\), it follows that condition \(\widehat{\varepsilon }\mathbin{\tilde{\leq}}d(U_{\lambda }^{2},U_{\mu }^{2}) \) is not satisfied for any \(\lambda ,\mu \in A\), and thus \((f,\varphi )\) is trivially a soft Keir-Meeler contractive mapping on \((\tilde{U},d)\). However, \((f,\varphi )\) has no fixed point.
The above example suggests the following modification of [36], Definition 15.
Definition 7
Let \((\tilde{U},d,A)\) be a soft metric space and let \(\varphi :A\rightarrow A\). A soft mapping \((f,\varphi ):\tilde{U}\mathbin{\tilde{\rightarrow}}\tilde{U}\) is called a soft contraction of Meir-Keeler type if it satisfies the following condition:
Remark 3
Let \((\tilde{U},d)\) be the complete soft metric space of Example 1. Define \(f(x)=x/4\) for all \(x\in U\), and \(\varphi (\lambda )=1\) for all \(\lambda \in A\). Then \((f,\varphi )(U_{\lambda }^{x})=U_{1}^{x/4}\) for all \(x\in U\) and \(\lambda \in A\). Although \((f,\varphi )\) has no fixed point, it is easy to check that the conditions of Definition 7 hold. However, we can state the following positive result.
Theorem 4
Let \((\tilde{U},d,A)\) be a complete soft metric space with A a finite set. Then every soft contraction of Meir-Keeler type on \((\tilde{U},d,A)\) has a unique fixed point.
Proof
We first note that, by Theorem 1(3), the metric space \((\operatorname {SP}(\tilde{U}),m_{d})\) is complete.
Now let \((f,\varphi )\) be a soft contraction of Meir-Keeler type on \((\tilde{U} ,d,A)\). As in the proof of Theorem 2, the restriction of \((f,\varphi )\) to \(\operatorname {SP}(\tilde{U})\) is a self mapping on \(\operatorname {SP}(\tilde{U})\), which is also denoted by \((f,\varphi )\).
We want to show that \((f,\varphi )\) is a Meir-Keeler contractive mapping on \((\operatorname {SP}(\tilde{U}),m_{d})\). Indeed, given \(\varepsilon >0\) consider the constant soft real number ε̅. Since \(\overline{\varepsilon }\mathbin{\tilde{>}}\bar{0}\), there exists \(\widehat{\delta }\mathbin{\tilde{>}}\bar{0}\) for which the conditions of Definition 7 are satisfied. Also, \(\delta =\min_{\eta \in A}\widehat{\delta }(\eta )>0\) because A is finite.
We conclude the paper by obtaining a soft metric extension of the celebrated Caristi-Kirk’s [42, 43] theorem that a metric space \((X,d)\) is complete if and only if every Caristi mapping on \((X,d)\) has a fixed point.
Let us recall that a self mapping T on a metric space \((X,d)\) is a Caristi mapping provided that there exists a lower semicontinuous function \(\phi :X\rightarrow \mathbb{R}^{+}\) such that \(d(x,T(x))+\phi (T(x))\leq \phi (x)\) for all \(x\in X\).
Caristi proved that every Caristi mapping on a complete metric space has a fixed point, while Kirk proved that actually Caristi’s fixed point theorem characterizes metric completeness.
In Definition 9 below we propose a notion of a soft Caristi mapping. To this end, we first generalize, in a natural way, Definition 4 to the case of a net. Thus, given a soft metric space \((\tilde{U},d)\), we say that a net \(\{U_{\lambda ,\alpha }^{x}\}_{\alpha \in \Lambda }\) of soft points in Ũ is convergent in \((\tilde{U},d)\) if there is a soft point \(U_{\mu }^{y}\) such that for each \(\widehat{\varepsilon }\mathbin{\tilde{>}}\bar{0}\), there exists \(\alpha _{0}\in \Lambda \) satisfying \(d(U_{\lambda ,\alpha }^{x},U_{\mu }^{y})\mathbin{\tilde{<}}\widehat{\varepsilon }\), whenever \(\alpha \geq \alpha _{0}\).
Definition 8
Let \((\tilde{U},d,A)\) be a soft metric space. A mapping \(\phi :\operatorname {SP}(\tilde{U})\rightarrow \mathbb{R}(A)^{*}\) is called lower semicontinuous on \((\tilde{U},d,A)\) if whenever \(\{U_{\lambda ,\alpha }^{x}\}_{\alpha \in \Lambda }\) is a net of soft points in Ũ that converges in \((\tilde{U},d,A)\) to a soft point \(U_{\mu }^{y}\), the following holds: for each \(\widehat{\varepsilon }\mathbin{\tilde{>}}\bar{0}\) there exists \(\widehat{\delta }\mathbin{\tilde{>}}\bar{0}\) such that \(\phi (U_{\mu }^{y})\mathbin{\tilde{\leq}}\phi (U_{\lambda ,\alpha }^{x})+\widehat{\varepsilon }\) whenever \(d(U_{\lambda ,\alpha }^{x},U_{\mu }^{y})\mathbin{\tilde{<}}\widehat{\delta }\).
Definition 9
Let \((\tilde{U},d,A)\) be a soft metric space. A soft mapping \(f:\tilde{U}\mathbin{\tilde{\rightarrow}}\tilde{U}\) is called a soft Caristi mapping if there exists a lower semicontinuous mapping \(\phi :\operatorname {SP}(\tilde{U})\rightarrow \mathbb{R}(A)^{*}\) such that \(d(U_{\lambda }^{x},f(U_{\lambda }^{x}))+\phi (f(U_{\lambda }^{x}))\mathbin{\tilde{\leq}}\phi (U_{\lambda }^{x})\) for all \(U_{\lambda }^{x}\in \operatorname {SP}(\tilde{U})\).
Theorem 5
Let \((\tilde{U},d,A)\) be a soft metric space with A a finite set. Then \((\tilde{U},d,A)\) is complete if and only if every soft Caristi mapping on \((\tilde{U},d,A)\) has a fixed point.
Proof
Suppose that \((\tilde{U},d,A)\) is complete and let f be a soft Caristi mapping on \((\tilde{U},d,A)\). Then there exists a lower semicontinuous mapping \(\phi :\operatorname {SP}(\tilde{U})\rightarrow \mathbb{R}(A)^{*}\) such that \(d(U_{\lambda }^{x}, f(U_{\lambda }^{x}))+\phi (f(U_{\lambda }^{x}))\mathbin{\tilde{\leq}}\phi (U_{\lambda }^{x})\) for all \(U_{\lambda }^{x}\in \operatorname {SP}(\tilde{U})\).
Conversely, suppose that every soft Caristi mapping on \((\tilde{U},d,A)\) has a fixed point, and let T be a Caristi mapping on the complete metric space \((\operatorname {SP}(\tilde{U}),m_{d})\). Then there exists a lower semicontinuous function \(\phi :\operatorname {SP}(\tilde{U})\rightarrow \mathbb{R}^{+}\) such that \(m_{d}(U_{\lambda }^{x},T(U_{\lambda }^{x}))+\phi (T(U_{\lambda }^{x}))\leq \phi (U_{\lambda }^{x})\) for all \(U_{\lambda }^{x}\in \operatorname {SP}(\tilde{U})\).
5 Conclusion
In an attempt to reverse the trend of obtaining soft metric extensions of existing fixed point results in the framework of ordinary metric spaces, we showed that a soft metric space, under the restriction that a set of parameters is finite, gives rise to a compatible metric on the collection of all soft points of absolute soft set. We also studied some essential properties of the induced metric thus obtained. We then proved that some recently obtained soft fixed point results can be directly deduced from existing comparable results in ordinary metric spaces. We presented examples to show that the restriction of a finite parameter set is unavoidable. It will be an interesting problem to study the limits to which soft fixed point theory may be extended.
Declarations
Acknowledgements
Salvador Romaguera thanks the support of Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Zadeh, LA: Fuzzy sets. Inf. Control 8, 103-112 (1965) MathSciNetView ArticleGoogle Scholar
- Molodtsov, D: Soft set theory - first results. Comput. Math. Appl. 37, 19-31 (1999) MATHMathSciNetView ArticleGoogle Scholar
- Aktaş, H, Çağman, N: Soft sets and soft groups. Inf. Sci. 177, 2726-2735 (2007) MATHView ArticleGoogle Scholar
- Ali, MI, Feng, F, Liu, X, Min, WK, Shabir, M: On some new operations in soft set theory. Comput. Math. Appl. 57, 1547-1553 (2009) MATHMathSciNetView ArticleGoogle Scholar
- Feng, F, Liu, X, Leoreanu-Fotea, V, Jun, YB: Soft sets and soft rough sets. Inf. Sci. 181, 1125-1137 (2011) MATHMathSciNetView ArticleGoogle Scholar
- Jiang, Y, Tang, Y, Chen, Q, Wang, J, Tang, S: Extending soft sets with description logics. Comput. Math. Appl. 59, 2087-2096 (2009) MathSciNetView ArticleGoogle Scholar
- Jun, YB: Soft BCK/BCI-algebras. Comput. Math. Appl. 56, 1408-1413 (2008) MATHMathSciNetView ArticleGoogle Scholar
- Jun, YB, Lee, KJ, Khan, A: Soft ordered semigroups. Math. Log. Q. 56, 42-50 (2010) MATHMathSciNetView ArticleGoogle Scholar
- Jun, YB, Lee, KJ, Park, CH: Soft set theory applied to ideals in d-algebras. Comput. Math. Appl. 57, 367-378 (2009) MATHMathSciNetView ArticleGoogle Scholar
- Jun, YB, Park, CH: Applications of soft sets in ideal theory of BCK/BCI-algebras. Inf. Sci. 178, 2466-2475 (2008) MATHMathSciNetGoogle Scholar
- Kong, Z, Gao, L, Wang, L, Li, S: The normal parameter reduction of soft sets and its algorithm. Comput. Math. Appl. 56, 3029-3037 (2008) MATHMathSciNetView ArticleGoogle Scholar
- Majumdar, P, Samanta, SK: Generalized fuzzy soft sets. Comput. Math. Appl. 59, 1425-1432 (2010) MATHMathSciNetView ArticleGoogle Scholar
- Li, F: Notes on the soft operations. ARPN J. Syst. Softw. 1, 205-208 (2011) View ArticleGoogle Scholar
- Maji, PK, Roy, AR, Biswas, R: An application of soft sets in a decision making problem. Comput. Math. Appl. 44, 1077-1083 (2002) MATHMathSciNetView ArticleGoogle Scholar
- Qin, K, Hong, Z: On soft equality. J. Comput. Appl. Math. 234, 1347-1355 (2010) MATHMathSciNetView ArticleGoogle Scholar
- Xiao, Z, Gong, K, Xia, S, Zou, Y: Exclusive disjunctive soft sets. Comput. Math. Appl. 59, 2128-2137 (2009) MathSciNetView ArticleGoogle Scholar
- Xiao, Z, Gong, K, Zou, Y: A combined forecasting approach based on fuzzy soft sets. J. Comput. Appl. Math. 228, 326-333 (2009) MATHMathSciNetView ArticleGoogle Scholar
- Xu, W, Ma, J, Wang, S, Hao, G: Vague soft sets and their properties. Comput. Math. Appl. 59, 787-794 (2010) MATHMathSciNetView ArticleGoogle Scholar
- Yang, CF: A note on soft set theory. Comput. Math. Appl. 56, 1899-1900 (2008) MATHMathSciNetView ArticleGoogle Scholar
- Yang, X, Lin, TY, Yang, J, Li, Y, Yu, D: Combination of interval-valued fuzzy set and soft set. Comput. Math. Appl. 58, 521-527 (2009) MATHMathSciNetView ArticleGoogle Scholar
- Zhu, P, Wen, Q: Operations on soft sets revisited (2012). arXiv:1205.2857v1
- Feng, F, Jun, YB, Liu, XY, Li, LF: An adjustable approach to fuzzy soft set based decision making. J. Comput. Appl. Math. 234, 10-20 (2009) MathSciNetView ArticleGoogle Scholar
- Feng, F, Jun, YB, Zhao, X: Soft semirings. Comput. Math. Appl. 56, 2621-2628 (2008) MATHMathSciNetView ArticleGoogle Scholar
- Feng, F, Liu, X: Soft rough sets with applications to demand analysis. In: Int. Workshop Intell. Syst. Appl. (ISA 2009), pp. 1-4. (2009) Google Scholar
- Herawan, T, Deris, MM: On multi-soft sets construction in information systems. In: Emerging Intelligent Computing Technology and Applications with Aspects of Artificial Intelligence, pp. 101-110. Springer, Berlin (2009) View ArticleGoogle Scholar
- Herawan, T, Rose, ANM, Deris, MM: Soft set theoretic approach for dimensionality reduction. In: Database Theory and Application, pp. 171-178. Springer, Berlin (2009) View ArticleGoogle Scholar
- Kim, YK, Min, WK: Full soft sets and full soft decision systems. J. Intell. Fuzzy Syst. 26, 925-933 (2014). doi:https://doi.org/10.3233/IFS-130783 MATHMathSciNetGoogle Scholar
- Mushrif, MM, Sengupta, S, Ray, AK: Texture classification using a novel, soft-set theory based classification algorithm. Lect. Notes Comput. Sci. 3851, 246-254 (2006) View ArticleGoogle Scholar
- Roy, AR, Maji, PK: A fuzzy soft set theoretic approach to decision making problems. J. Comput. Appl. Math. 203, 412-418 (2007) MATHView ArticleGoogle Scholar
- Zhu, P, Wen, Q: Probabilistic soft sets. In: IEEE Conference on Granular Computing (GrC 2010), pp. 635-638 (2010) View ArticleGoogle Scholar
- Zou, Y, Xiao, Z: Data analysis approaches of soft sets under incomplete information. Knowl.-Based Syst. 21, 941-945 (2008) View ArticleGoogle Scholar
- Cagman, N, Karatas, S, Enginoglu, S: Soft topology. Comput. Math. Appl. 62, 351-358 (2011) MATHMathSciNetView ArticleGoogle Scholar
- Das, S, Samanta, SK: Soft real sets, soft real numbers and their properties. J. Fuzzy Math. 20, 551-576 (2012) MATHMathSciNetGoogle Scholar
- Das, S, Samanta, SK: Soft metric. Ann. Fuzzy Math. Inform. 6, 77-94 (2013) MATHMathSciNetGoogle Scholar
- Abbas, M, Murtaza, G, Romaguera, S: Soft contraction theorem. J. Nonlinear Convex Anal. 16, 423-435 (2015) MATHMathSciNetGoogle Scholar
- Chen, CM, Lin, IJ: Fixed point theory of the soft Meir-Keeler type contractive mappings on a complete soft metric space. Fixed Point Theory Appl. 2015, 184 (2015) View ArticleGoogle Scholar
- Feng, F, Li, CX, Davvaz, B, Ali, MI: Soft sets combined with fuzzy sets and rough sets: a tentative approach. Soft Comput. 14, 8999-9911 (2010) View ArticleGoogle Scholar
- Maji, PK, Biswas, R, Roy, AR: Soft set theory. Comput. Math. Appl. 45, 555-562 (2003) MATHMathSciNetView ArticleGoogle Scholar
- Wardowski, D: On a soft mapping and its fixed points. Fixed Point Theory Appl. 2013, 182 (2013) MathSciNetView ArticleGoogle Scholar
- Kannan, R: Some results on fixed points II. Am. Math. Mon. 76, 405-408 (1969) MATHView ArticleGoogle Scholar
- Meir, A, Keeler, E: A theorem on contraction mappings. J. Math. Anal. Appl. 28, 326-329 (1969) MATHMathSciNetView ArticleGoogle Scholar
- Caristi, J: Fixed point theorems for mappings satisfying inwardness conditions. Trans. Am. Math. Soc. 215, 241-251 (1976) MATHMathSciNetView ArticleGoogle Scholar
- Kirk, WA: Caristi’s fixed-point theorem and metric convexity. Colloq. Math. 36, 81-86 (1976) MATHMathSciNetGoogle Scholar