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*

$$ d\bigl(f\bigl(U_{\lambda }^{x}\bigr),f\bigl(U_{\mu }^{y} \bigr)\bigr)\mathbin{\tilde{\leq}}\overline{c}d\bigl(U_{\lambda }^{x},U_{\mu }^{y} \bigr), $$

(1)

*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 \(U_{\lambda }^{x},U_{\mu }^{y}\in \operatorname {SP}(\tilde{U})\),

$$\begin{aligned} m_{d}\bigl(f\bigl(U_{\lambda }^{x}\bigr),f \bigl(U_{\mu }^{y}\bigr)\bigr) =&\max_{\eta \in A}d \bigl(f\bigl(U_{\lambda }^{x}\bigr),f\bigl(U_{\mu }^{y} \bigr)\bigr) (\eta )\leq \max_{\eta \in A}c \bigl( d \bigl(U_{\lambda }^{x},U_{\mu }^{y}\bigr) (\eta ) \bigr) \\ =&c\Bigl[\max_{\eta \in A}d\bigl(U_{\lambda }^{x},U_{\mu }^{y} \bigr) (\eta )\Bigr]=cm_{d}\bigl(U_{\lambda }^{x},U_{\mu }^{y} \bigr). \end{aligned}$$

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*

$$ d\bigl(f\bigl(U_{\lambda }^{x}\bigr),f\bigl(U_{\mu }^{y} \bigr)\bigr)\mathbin{\tilde{\leq}}\overline{c} \bigl\{ d\bigl(U_{\lambda }^{x},f \bigl(U_{\lambda }^{x}\bigr)\bigr)+d\bigl(U_{\mu }^{y},f \bigl(U_{\mu }^{y}\bigr)\bigr) \bigr\} , $$

(2)

*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 \(U_{\lambda }^{x},U_{\mu }^{y}\in \operatorname {SP}(\tilde{U})\):

$$\begin{aligned} m_{d}\bigl(f\bigl(U_{\lambda }^{x}\bigr),f \bigl(U_{\mu }^{y}\bigr)\bigr) =&\max_{\alpha \in A}d \bigl(f\bigl(U_{\lambda }^{x}\bigr),f\bigl(U_{\mu }^{y} \bigr)\bigr) (\alpha ) \\ \leq &\max_{\alpha \in A}c \bigl( \bigl(d\bigl(U_{\lambda }^{x},f \bigl(U_{\lambda }^{x}\bigr)\bigr)+d\bigl(U_{\mu }^{y},f \bigl(U_{\mu }^{y}\bigr)\bigr)\bigr) (\alpha ) \bigr) \\ \leq &c \bigl\{ m_{d}\bigl(U_{\lambda }^{x},f \bigl(U_{\lambda }^{x}\bigr)\bigr)+m_{d} \bigl(U_{\mu }^{y},f\bigl(U_{\mu }^{y}\bigr) \bigr) \bigr\} . \end{aligned}$$

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

$$d\bigl(U_{\lambda }^{x},U_{\mu }^{y}\bigr)= \vert \overline{x}-\overline{y}\vert +\vert \overline{\lambda }-\overline{ \mu }\vert , $$

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

$$\begin{aligned} d\bigl(f\bigl(U_{\lambda }^{x}\bigr),f\bigl(U_{\mu }^{y} \bigr)\bigr) (\eta ) =&d\bigl(U_{1}^{x/4},U_{1}^{y/4} \bigr) (\eta )=\frac{1}{4}\vert x-y\vert \leq \frac{1}{4}(x+y) \\ =&\frac{1}{3} \biggl( \biggl(x-\frac{x}{4}\biggr)+\biggl(y- \frac{y}{4}\biggr) \biggr) \\ \leq &\frac{1}{3} \bigl( d\bigl(U_{\lambda }^{x},f \bigl(U_{\lambda }^{x}\bigr)\bigr)+d\bigl(U_{\mu }^{y},f \bigl(U_{\mu }^{y}\bigr)\bigr) \bigr) (\eta ). \end{aligned}$$

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

$$d\bigl(U_{\lambda }^{x},U_{\mu }^{y}\bigr)= \vert \overline{x}-\overline{y}\vert +\vert \overline{\lambda }- \overline{\mu }\vert , $$

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

$$d\bigl(f\bigl(U_{0}^{2}\bigr),f\bigl(U_{0}^{y} \bigr)\bigr) (\eta )=d\bigl(U_{0}^{1/2},U_{0}^{0} \bigr) (\eta )=\frac{1}{2}>c(2-y)=cd\bigl(U_{0}^{2},U_{0}^{y} \bigr) (\eta ). $$

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 \(\lambda ,\mu ,\eta \in A\),

$$\begin{aligned} d\bigl(f\bigl(U_{\lambda }^{x}\bigr),f\bigl(U_{\mu }^{y} \bigr)\bigr) (\eta ) =&d\bigl(U_{0}^{0},U_{0}^{1/2} \bigr) (\eta )=\frac{1}{2}=\frac{1}{3}\biggl(2-\frac{1}{2}\biggr)\leq \frac{1}{3}\biggl(x+y-\frac{1}{2}\biggr) \\ =&\frac{1}{3} \bigl( d\bigl(U_{\lambda }^{x},U_{0}^{0} \bigr)+d\bigl(U_{\mu }^{y},U_{\mu }^{1/2}\bigr) \bigr) (\eta ). \end{aligned}$$

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 \(x,y\in X\),

$$\varepsilon \leq d(x,y)< \varepsilon +\delta\quad \Rightarrow\quad d \bigl(T(x),T(y)\bigr)< \varepsilon . $$

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 \(U_{\lambda }^{x},U_{\mu }^{y}\in \operatorname {SP}(\tilde{U})\),

$$\widehat{\varepsilon }\mathbin{\tilde{\leq}}d\bigl(U_{\lambda }^{x},U_{\mu }^{y} \bigr)\mathbin{\tilde{< }}\widehat{\varepsilon }+\widehat{\delta }\quad \Rightarrow\quad d \bigl((f,\varphi ) \bigl(U_{\lambda }^{x}\bigr),(f,\varphi ) \bigl(U_{\mu }^{y}\bigr)\bigr)\mathbin{\tilde{< }}\widehat{ \varepsilon }. $$

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:

$$\begin{aligned}& d\bigl(U_{\lambda }^{2},U_{\lambda }^{2}\bigr)= \overline{0} \quad \mbox{for all } \lambda \in A, \quad \mbox{and} \\& d\bigl(U_{0}^{2},U_{1}^{2}\bigr) (0)=d\bigl(U_{1}^{2},U_{0}^{2}\bigr) (0)=0,\qquad d\bigl(U_{0}^{2},U_{1}^{2} \bigr) (1)=d\bigl(U_{1}^{2},U_{0}^{2} \bigr) (1)=1. \end{aligned}$$

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 \(U_{\lambda }^{x},U_{\mu }^{y}\in \operatorname {SP}(\tilde{U})\),

$$\begin{aligned}& d\bigl(U_{\lambda }^{x},U_{\mu }^{y}\bigr) \mathbin{\tilde{< }}\widehat{\varepsilon }+\widehat{\delta }, \quad \mbox{and} \quad \widehat{\varepsilon }(\eta )\leq d\bigl(U_{\lambda }^{x},U_{\mu }^{y} \bigr) (\eta )\quad \mbox{for some }\eta \in A \\& \quad \Rightarrow\quad d\bigl((f,\varphi ) \bigl(U_{\lambda }^{x} \bigr),(f,\varphi ) \bigl(U_{\mu }^{y}\bigr)\bigr)\mathbin{\tilde{< }}\widehat{\varepsilon }. \end{aligned}$$

### 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

$$d\bigl(U_{\lambda }^{x},U_{\mu }^{y}\bigr) ( \eta )< \varepsilon +\delta \leq (\overline{\varepsilon }+\widehat{ \delta }) (\eta ), $$

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

$$\overline{\varepsilon }(\eta _{0})\leq d\bigl(U_{\lambda }^{x},U_{\mu }^{y} \bigr) (\eta _{0}), $$

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

$$d\bigl((f,\varphi ) \bigl(U_{\lambda }^{x}\bigr),(f,\varphi ) \bigl(U_{\mu }^{y}\bigr)\bigr)\mathbin{\tilde{< }}\overline{ \varepsilon }. $$

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

$$\Phi \bigl(U_{\lambda }^{x}\bigr)=\sum _{\eta \in A}\bigl(\phi \bigl(U_{\lambda }^{x}\bigr) ( \eta )\bigr), $$

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

$$\begin{aligned} m_{d}\bigl(U_{\lambda }^{x},f\bigl(U_{\lambda }^{x} \bigr)\bigr)+\Phi \bigl(f\bigl(U_{\lambda }^{x}\bigr)\bigr) =&d \bigl(U_{\lambda }^{x},f\bigl(U_{\lambda }^{x}\bigr) \bigr) (\eta _{x,\lambda })+\sum_{\eta \in A}\phi \bigl(f \bigl(U_{\lambda }^{x}\bigr) (\eta )\bigr) \\ \leq &\phi \bigl(U_{\lambda }^{x}\bigr) (\eta _{x,\lambda })+ \sum_{\eta \in A\setminus \{\eta _{x,\lambda }\}}\phi \bigl(f\bigl(U_{\lambda }^{x} \bigr) (\eta )\bigr) \\ =&\Phi \bigl(U_{\lambda }^{x}\bigr). \end{aligned}$$

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

$$\begin{aligned} \bigl( d\bigl(U_{\lambda }^{x},f_{T} \bigl(U_{\lambda }^{x}\bigr)\bigr)+\varphi \bigl(f_{T} \bigl(U_{\lambda }^{x}\bigr)\bigr) \bigr) (\eta ) \leq &m_{d}\bigl(U_{\lambda }^{x},T\bigl(U_{\lambda }^{x} \bigr)\bigr)+\phi \bigl(T\bigl(U_{\lambda }^{x}\bigr)\bigr) \\ \leq &\phi \bigl(U_{\lambda }^{x}\bigr)=\varphi \bigl(U_{\lambda }^{x}\bigr) (\eta ). \end{aligned}$$

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). □