# Proving Fixed-Point Theorems Employing Fuzzy \((\sigma,\mathcal{Z})\)-Contractive-Type Mappings
## 3 Main Results
To see this, we consider the case that \(x,y\in(1,\infty)\) and take into account Remark 1; we have \(\mathcal{M}(Tx,Ty,t)=\mathcal{M}(x,y,t)\), and hence,_  
\[e^{\frac{-|x-y|}{2}} =\alpha(x,y,t)\mathcal{M}(Tx,Ty,t)\]3 \[e^{\frac{-|x-y|}{2}} =\alpha(x,y,t)\mathcal{M}(Tx,Ty,t)\]4 \[e^{\frac{-|x-y|}{2}} =\alpha(x,y,t)\mathcal{M}(Tx,Ty,t)\]5  
_which impossible; hence, \(T\) is not a fuzzy \(\mathcal{Z}\)-contractive mapping._  
Now, by an example (see also [28], Example 3.10), we show that the assumption (b) of Theorems 3 and 4 is not superfluous. **Example 7**. : _Let \(\mathcal{K}=\mathbb{N}\), and define the fuzzy metric \(\mathcal{M}\) by_  
\[e^{\frac{-|x-y|}{2}} =\alpha(x,y,t)\mathcal{M}(Tx,Ty,t)\]6_Then, \((\mathcal{K},\mathcal{M},*)\) is an \(\mathcal{M}\)-complete fuzzy metric space where \(*\) is the product \(t\)-norm. Define a mapping \(T:\mathcal{K}\to\mathcal{K}\) by \(Tx=x+1\) for all \(x\in\mathcal{K}\). Then, \(T\) is a fuzzy \((\sigma,\mathcal{Z})\)-contractive mapping with respect to the functions \(\xi:(0,1]\times(0,1]\to\mathbb{R}\) and \(\sigma:\mathcal{K}\times\mathcal{K}\times(0,\infty)\to(0,\infty)\) by_  
\[\xi(l,s)=\left\{ \begin{array}{ll}\frac{l+s}{2},&\text{ if }l>s,\\ 1&\text{ if }s\geq l,\end{array} \right.\quad\text{ and }\quad\sigma(x,y,t)=1.\]  
_From the definition of \(\sigma\), it is very clear that the conditions \((a),(c),(d),(e)\), and \((e^{\prime})\) of Theorems 3 and 4 are satisfied. Moreover, trivial calculations show that the condition \((b)\) does not hold, that is the quadruple \((\mathcal{K},\mathcal{M},T,\xi)\) does not have the property \((S)\). Notice that \(T\) does not have a fixed point._  
Next, the following example shows that the assumption \((c)\) of Theorems 3 and 4 is not superfluous. **Example 8**. : _Let \(\mathcal{K}=[1,\infty)\) and the fuzzy metric be defined by \(\mathcal{M}(x,y,t)=\frac{t}{t+d(x,y)}\), \(t>0\), where \(d(x,y)=|x-y|\), for all \(x,y\in\mathcal{K}\). Then, \((\mathcal{K},\mathcal{M},*)\) is an \(\mathcal{M}\)-complete fuzzy metric space, where \(*\) is a product \(t\)-norm. Define a mapping \(T:\mathcal{K}\to\mathcal{K}\) by_  
\[T(x)=x+1,\text{ for all }x\in\mathcal{K}.\]  
_Furthermore, we define two essential functions: \(\xi:(0,1]\times(0,1]\to\mathbb{R}\) and \(\sigma:\mathcal{K}\times\mathcal{K}\times(0,\infty)\to(0,\infty)\) by_  
\[\xi(l,s)=\frac{s}{l}\text{ \ and \ }\sigma(x,y,t)=1+\frac{|x-y|}{t}.\]  
_Then, for all \(x,y\in\mathcal{K}\) such that \(Tx\
eq Ty\) and \(t>0\), we have_  
\[1 =\sigma(x,y,t)\mathcal{M}(Tx,Ty,t)\] \[\geq\xi(\mathcal{M}(Tx,Ty,t),\mathcal{M}(x,y,t))\] \[=\frac{\mathcal{M}(Tx,Ty,t)}{\mathcal{M}(x,y,t)}=1,\]  
_which shows that \(T\) is fuzzy \((\sigma,\mathcal{Z})\)-contractive. Moreover, it is easy to show that the conditions \((a),(b),(d),(e)\), and \((e^{\prime})\) of Theorems 3 and 4 hold. Now, note that there is no \(x_{0}\) in \(\mathcal{K}\) such that \(\sigma(x_{0},Tx_{0},t)\leq 1\) for \(t>0\). Thus, the condition \((c)\) of Theorems 3 and 4 does not hold. Observe that the mapping \(T\) does not have a fixed point._  
The following theorem enables us to extend the fixed point result for the family of \(\sigma\)-\(\psi\)-fuzzy contractive mappings due to Gopal and Vetro [27] with an additional condition. **Theorem 5**. : _Let \((\mathcal{K},\mathcal{M},*)\) be an \(\mathcal{M}\)-complete FMS and \(\sigma:\mathcal{K}\times\mathcal{K}\times(0,\infty)\to(0,\infty)\).