On the fixed point theory of soft metric spaces

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.

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.

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]

For two soft real numbers f̂, ĝ, we say that

1. (i)

   \(\widehat{f}\mathbin{\tilde{\leq}}\widehat{g}\) if \(\widehat{f}(e)\leq \widehat{g}(e)\), for all \(e\in A\),

2. (ii)

   \(\widehat{f}\mathbin{\tilde{\geq}}\widehat{g}\) if \(\widehat{f}(e)\geq \widehat{g}(e)\), for all \(e\in A\),

3. (iii)

   \(\widehat{f}\mathbin{\tilde{<}}\widehat{g}\) if \(\widehat{f}(e)<\widehat{g}(e)\), for all \(e\in A\), and

4. (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]

Let U be a universe, A be a nonempty subset of parameters and Ũ be the absolute soft set, i.e., \(F(\lambda )=U\) for all \(\lambda \in A\), where \((F,A)=\tilde{U}\). A mapping \(d:\operatorname {SP}(\tilde{U})\times \operatorname {SP}(\tilde{U})\rightarrow \mathbb{R}(A)^{*}\) is said to be a soft metric on Ũ if for any \(U_{\lambda }^{x},U_{\mu }^{y},U_{\gamma }^{z}\in \operatorname {SP}(\tilde{U})\) (equivalently, \(U_{\lambda }^{x},U_{\mu }^{y},U_{\gamma }^{z}\mathbin{\tilde{\in}}\tilde{U}\)), the following hold:

1. M1.

   \(d(U_{\lambda }^{x},U_{\mu }^{y})\mathbin{\tilde{\geq}}\bar{0}\).

2. M2.

   \(d(U_{\lambda }^{x},U_{\mu }^{y})=\bar{0}\) if and only if \(U_{\lambda }^{x}=U_{\mu }^{y}\).

3. M3.

   \(d(U_{\lambda }^{x},U_{\mu }^{y})=d(U_{\mu }^{x},U_{\lambda }^{y})\).

4. 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.

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

Let \((\tilde{U},d,A)\) be a soft metric space with A a finite set. Define a function \(m_{d}:\operatorname {SP}(\tilde{U})\times \operatorname {SP}(\tilde{U})\rightarrow \mathbb{R}^{+}\) as

for all \(U_{\lambda }^{x},U_{\mu }^{y}\in \operatorname {SP}(\tilde{U})\). Then the following hold:

1. (1)

   \(m_{d}\) is a metric on \(\operatorname {SP}(\tilde{U})\).

2. (2)

   For any sequence \(\{U_{\lambda ,n}^{x}\}_{n}\) of soft points and a soft point \(U_{\lambda }^{y}\), we have

3. (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 \((\operatorname {SP}(\tilde{U}),m_{d})\).

4. (2b)

   \(d(U_{\lambda }^{y},U_{\lambda ,n}^{x})\rightarrow \overline{0}\) if and only if \(m_{d}(U_{\lambda }^{y},U_{\lambda ,n}^{x})\rightarrow 0\).

5. (3)

   \((\tilde{U},d,A)\) is complete if and only if \((\operatorname {SP}(\tilde{U}),m_{d})\) is complete.

Proof

(1) Let \(U_{\lambda }^{x},U_{\mu }^{y},U_{\upsilon }^{z}\in \operatorname {SP}(\tilde{U})\). Then we have:

1. (i)

   \(m_{d}(U_{\lambda }^{x},U_{\mu }^{y})\geq 0\), by condition M1 of Definition 3.

2. (ii)

   \(m_{d}(U_{\lambda }^{x},U_{\mu }^{y})=0\Leftrightarrow U_{\lambda }^{x}=U_{\mu }^{y}\), by condition M2 of Definition 3.

3. (iii)

   \(m_{d}(U_{\lambda }^{x},U_{\mu }^{y})=m_{d}(U_{\mu }^{y},U_{\lambda }^{x})\), by condition M3 of Definition 3.

4. (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). □

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]

Let \((\tilde{U},d,A)\) be a complete soft metric space with A a finite set. Suppose that the soft mapping \(f:\tilde{U}\mathbin{\tilde{\rightarrow}}\tilde{U}\) satisfies

for all \(U_{\lambda }^{x},U_{\mu }^{y}\in \operatorname {SP}(\tilde{U})\), where \(\bar{0}\mathbin{\tilde{\leq}}\overline{c}\mathbin{\tilde{<}}\bar{1}\). Then f has a unique fixed point, i.e., there is a unique soft point \(U_{\lambda }^{x}\) such that \(f(U_{\lambda }^{x})=U_{\lambda }^{x}\).

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.

Since for each \(U_{\lambda }^{x}\in \operatorname {SP}(\tilde{U})\) (or equivalently, \(U_{\lambda }^{x}\mathbin{\tilde{\in}}\tilde{U}\)) there is a unique soft point \(U_{\mu }^{y}\) such that \(f(U_{\lambda }^{x})=U_{\mu }^{y}\) (see Remark 1), the restriction of f to \(\operatorname {SP}(\tilde{U})\) is a self mapping on \(\operatorname {SP}(\tilde{U})\), also denoted by f. Note also that the real number c generating the constant soft real number c̅ satisfies \(0\leq c<1\). Finally, we obtain the following contraction condition, for each \(U_{\lambda }^{x},U_{\mu }^{y}\in \operatorname {SP}(\tilde{U})\),

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

Let \((\tilde{U},d,A)\) be a complete soft metric space with A a finite set. Suppose that the soft mapping \(f:\tilde{U}\mathbin{\tilde{\rightarrow}}\tilde{U}\) satisfies

for all \(U_{\lambda }^{x},U_{\mu }^{y}\in \operatorname {SP}(\tilde{U})\), where \(\bar{0}\mathbin{\tilde{\leq}}\overline{c}\mathbin{\tilde{<}}\overline{1/2}\). Then f has a unique fixed point.

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.

Moreover, the restriction of f to \(\operatorname {SP}(\tilde{U})\) is a self mapping on \(\operatorname {SP}(\tilde{U})\), exactly as in the proof of Theorem 2. Note also that the real number c generating the constant soft real number c̅ satisfies \(0\leq c<1/2\). Finally, we obtain the following contraction condition, for each \(U_{\lambda }^{x},U_{\mu }^{y}\in \operatorname {SP}(\tilde{U})\):

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

Let \(U=A=\{1/n:n\in \mathbb{N}\}\). According to [34], Example 4.3, the mapping \(d:\operatorname {SP}(\tilde{U})\times \operatorname {SP}(\tilde{U})\rightarrow \mathbb{R}(A)^{*}\) given by

for all \(U_{\lambda }^{x},U_{\mu }^{y}\in \operatorname {SP}(\tilde{U})\), where \(\vert \cdot \vert \) denotes the modulus of soft real numbers, is a soft metric on Ũ. Furthermore, the soft metric space \((\tilde{U},d)\) is complete [35], Example 3.21.

Let \(f:\tilde{U}\mathbin{\tilde{\rightarrow}}\tilde{U}\) such that \(f(U_{\lambda }^{x})=U_{1}^{x/4}\) for all \(x\in U\), \(\lambda \in A\). We show that f satisfies the contraction condition (2) of Theorem 3 with \(\overline{c}=\overline{1/3}\). In fact, given \(x,y\in U\) and \(\lambda ,\mu \in A\), for each \(\eta \in A\) we have

Therefore \(d(f(U_{\lambda }^{x}),f(U_{\mu }^{y}))\mathbin{\tilde{\leq}}\overline{1/3} \{ d(U_{\lambda }^{x},f(U_{\lambda }^{x}))+d(U_{\mu }^{y},f(U_{\mu }^{y})) \} \). However, f has no fixed point.

Now we present an example where we can apply Theorem 3 but not Theorem 2.

Example 2

Let \(U=\mathbb{R}^{+}\) and \(A=\{0,1\}\). Again, according to [34], Example 4.3, the mapping \(d:\operatorname {SP}(\tilde{U})\times \operatorname {SP}(\tilde{U})\rightarrow \mathbb{R}(A)^{\ast }\) given by

for all \(U_{\lambda }^{x},U_{\mu }^{y}\in \operatorname {SP}(\tilde{U})\), is a soft metric on Ũ. Since \(\mathbb{R}^{+}\) is complete for the Euclidean metric, we deduce that \((\tilde{U},d)\) is a complete soft metric space.

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 \(x\in [2,\infty )\).

Let c̅ be a constant soft real number such that \(\bar{0}\mathbin{\tilde{\leq}}\overline{c}\mathbin{\tilde{<}}\bar{1}\). Then there is a real number \(c\in [0,1)\) such that \(c=\overline{c}(\eta )\) for all \(\eta \in A\). Choose \(y\in [0,2)\) such that \(c(2-y)<1/2\). Then, for each \(\eta \in A\), we have

Therefore f does not satisfy condition (1) of Theorem 2 for any c̅ satisfying \(\bar{0}\mathbin{\tilde{\leq}}\overline{c}\mathbin{\tilde{<}}\bar{1}\).

However, taking, without loss of generality, \(x\in [0,2)\) and \(y\in [2,\infty )\), we obtain, for \(\lambda ,\mu ,\eta \in A\),

Therefore f satisfies condition (2) of Theorem 3 for \(\overline{c}=\overline{1/3}\). In fact, \(U_{0}^{0}\) is the unique fixed point of f.

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:

for each \(\varepsilon >0\) there exists \(\delta >0\) such that for each \(x,y\in X\),

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:

for each soft real number \(\widehat{\varepsilon }\mathbin{\tilde{>}}\bar{0}\) there exists \(\widehat{\delta }\mathbin{\tilde{>}}\bar{0}\) such that for each \(U_{\lambda }^{x},U_{\mu }^{y}\in \operatorname {SP}(\tilde{U})\),

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

Let \(U=\{2\}\), \(A=\{0,1\}\), and d a soft metric on Ũ defined as:

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:

for each soft real number \(\widehat{\varepsilon }\mathbin{\tilde{>}}\bar{0}\) there exists \(\widehat{\delta }\mathbin{\tilde{>}}\bar{0}\) such that for each \(U_{\lambda }^{x},U_{\mu }^{y}\in \operatorname {SP}(\tilde{U})\),

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.

Take \(U_{\lambda }^{x},U_{\mu }^{y}\in \operatorname {SP}(\tilde{U})\) satisfying \(\varepsilon \leq m_{d}(U_{\lambda }^{x},U_{\mu }^{y})<\varepsilon +\delta \). Then

for all \(\eta \in A\), so \(d(U_{\lambda }^{x},U_{\mu }^{y})\mathbin{\tilde{<}}\overline{\varepsilon }+\widehat{\delta }\). Furthermore, from \(\varepsilon \leq m_{d}(U_{\lambda }^{x},U_{\mu }^{y})\) we deduce that

where \(m_{d}(U_{\lambda }^{x},U_{\mu }^{y})=d(U_{\lambda }^{x},U_{\mu }^{y})(\eta _{0})\), \(\eta _{0}\in A\).

Since \((f,\varphi )\) is a soft mapping of Meir-Keeler type, we deduce that

From this relation it follows that \(m_{d}((f,\varphi )(U_{\lambda }^{x}),(f,\varphi )(U_{\mu }^{y}))<\varepsilon \). We deduce that \((f,\varphi )\) is a Meir-Keeler contractive mapping on \((\operatorname {SP}(\tilde{U}),m_{d})\). Hence \((f,\varphi )\) has a unique fixed point. □

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 \(U_{\lambda }^{x}\in \operatorname {SP}(\tilde{U})\).

Exactly as in Theorem 2, the restriction of f to \(\operatorname {SP}(\tilde{U})\) is a self mapping on \(\operatorname {SP}(\tilde{U})\), also denoted by f. Now define \(\Phi :\operatorname {SP}(\tilde{U}) \rightarrow \mathbb{R}^{+}\) as

for all \(U_{\lambda }^{x}\in \operatorname {SP}(\tilde{U})\). It is not difficult to check Φ is lower semicontinuous on the complete metric space \((\operatorname {SP}(\tilde{U}),m_{d})\). Furthermore, given \(U_{\lambda }^{x}\in \operatorname {SP}(\tilde{U})\) let \(\eta _{x,\lambda }\in A\) such that \(m_{d}(U_{\lambda }^{x},f(U_{\lambda }^{x}))=d(U_{\lambda }^{x},f(U_{\lambda }^{x}))(\eta _{x,\lambda })\). Then we obtain

We deduce that f is a Caristi mapping on \((\operatorname {SP}(\tilde{U}),m_{d})\), and hence it has a fixed point.

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 \(U_{\lambda }^{x}\in \operatorname {SP}(\tilde{U})\).

Let \(f_{T}:\tilde{U}\rightarrow \tilde{U}\) such that \(f_{T}(U_{\lambda }^{x})=T(U_{\lambda }^{x})\), and define \(\varphi :\operatorname {SP}(\tilde{U})\rightarrow \mathbb{R}(A)^{*}\) such that \(\varphi (U_{\lambda }^{x})=\overline{\phi (U_{\lambda }^{x})}\) for all \(U_{\lambda }^{x}\mathbin{\tilde{\in}}\tilde{U} \). Clearly φ is lower semicontinuous on \((\tilde{U},d,A)\). Then, for each \(U_{\lambda }^{x}\mathbin{\tilde{\in}}\tilde{U}\) and \(\eta \in A\) we obtain

We deduce that \(d(U_{\lambda }^{x},f(U_{\lambda }^{x}))+\varphi (f_{T}(U_{\lambda }^{x}))\mathbin{\tilde{\leq}}\varphi (U_{\lambda }^{x})\), and, consequently, \(f_{T}\) is a soft Caristi mapping on \((\tilde{U},d,A)\). Therefore \(f_{T}\), and hence T has a fixed point. So, \((\operatorname {SP}(\tilde{U}),m_{d})\) is complete by Caristi-Kirk’s theorem. Completeness of \((\tilde{U},d,A)\) is now a consequence of Theorem 1(3). □

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.

Salvador Romaguera thanks the support of Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01.

Authors and Affiliations

1. Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield, Pretoria, South Africa

   Mujahid Abbas

2. Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

   Mujahid Abbas

3. Department of Mathematics, GC University, Faisalabad, Pakistan

   Ghulam Murtaza

4. Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Valencia, 46022, Spain

   Salvador Romaguera

Corresponding author

Correspondence to Salvador Romaguera.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally in writing this article. They read and approved the final manuscript.

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